RIJKSUNIVERSITEIT GRONINGENAbsorption of formaldehyde in water Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op vrijdag 6 juni 2003 om 14.15 uur door Jozef Gerhardus Maria Winkelman geboren op 2 juli 1961 te Warnsveld Promotores: Prof. dr. ir. L.P.B.M. Janssen Prof. dr. ir. H.J. Heeres Prof. dr. ir. A.A.C.M. Beenackers † Beoordelingscommissie: Prof. dr. A.A. Broekhuis Prof. dr. P.D. Iedema Prof. dr. ir. G.F. Versteeg Aan mijn ouders Acknowledgements The research reported on in this thesis was supported financially by Dynea B.V., Delfzijl, The Netherlands, a formaldehyde producer, and by the Technology Foundation (STW) in the Netherlands. Contents Contents 1. Introduction 1 2. The kinetics of the dehydration of methylene glycol. Abstract 5 Introduction 5 Experimental 7 Results 8 Discussion 14 Physical properties 16 Conclusions 17 3. Simultaneous absorption and/or desorption with reversible first-order chemical reaction. Abstract 19 Introduction 19 Absorption with first-order reversible reaction and desorption 20 Special and limiting cases 23 Applications 26 Conclusions 31 4. Kinetics and chemical equilibrium of the hydration of formaldehyde Abstract 33 Introduction 33 Experimental 35 Results 37 Discussion 45 Physical properties 46 Conclusions 46 Addendum 47 5. Modeling and simulation of industrial formaldehyde absorbers. Abstract 49 Introduction 49 Reactions in aqueous formaldehyde solutions 50 Development of absorber model 52 Mass transfer rates 53 Energy transfer rates 55 Physical and chemical parameters 56 Vapour liquid equilibria 56 Contents Computational method 59 Results 61 Conclusions 66 6. Simulation of industrial formaldehyde absorbers: the behaviour of methanol and non- equilibrium stage modelling. Abstract 69 Introduction 69 Reactions in aqueous methanolic formaldehyde solutions 70 Vapour liquid equilibria in formaldehyde-water-methanol mixtures 73 Model development 74 Mass transfer rates 76 Energy transfer rates 78 Method of solution 79 Results 83 Conclusions 92 7. Epilogue 93 Symbols 95 References 99 Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 103 Appendix B: The equilibrium composition of aqueous methanolic formaldehyde solutions. 117 Appendix C: The reaction order of formaldehyde in its hydration reaction. 123 List of publications 129 Samenvatting in het nederlands (Summary in Dutch) 131 Dankwoord 137 Chapter 1: Introduction 1 Chapter 1 Introduction This thesis describes theoretical and experimental work on the absorption of formaldehyde in water and the development of chemical engineering models for the description and optimization of industrial formaldehyde absorbers. This introduction gives a short description of the industrial formaldehyde production process, and of the formaldehyde absorption step therein. The introduction ends with an outline of the other chapters and appendices. Formaldehyde is an important base chemical in the process industry with a world production rate of approximately 10 million metric tons annually (Weirauch, 1999). Historically, one of the first important applications was in the production of artificial indigo. Nowadays, its main applications are in the production of engineering plastics and resins, especially urea, phenol and melamine resins, of which large quantities are used in the plywood and particle board manufacturing industry, and also in the manufacturing of rubber, paper, fertilisers, explosives, preservatives, etc. (Walker, 1964; Cancho et al., 1989). Formaldehyde is industrially produced from methanol. The production is perfomed at approximately atmospheric pressure. Three major steps can be identified, see Fig 1. In a first step, the liquid methanol is vaporized into an air stream, and steam is added to the resulting gaseous mixture. In a second step, the gaseous mixture is lead over a catalyst bed, where the methanol is converted to formaldehyde via partial dehydrogenation and partial oxidation. The temperature of the gaseous product increases to approximately 870 K because of the highly exothermic character of the conversion of methanol to formaldehyde. steam air methanol tail gas water 55 wt% formalin VAPORIZER REACTOR ABSORBER Fig. 1. Simple scheme of the formaldehyde production process. Chapter 1: Introduction 2 To prevent thermal decomposition of formaldehyde, the gas stream is cooled directly after passing over the catalyst. In a third step, the formaldehyde is absorbed in water in an absorption column, because formaldehyde in its pure, gaseous form is highly unstable, and also because the reactor product stream contains the formaldehyde diluted in other gases, mainly nitrogen. From the absorber, the commercial product is obtained: an approximately 55% by weight solution of formaldehyde in water, or formalin. The design, operation and optimization of formaldehyde absorbers is complicated by a number of factors, of which two important ones are the reactions in the liquid phase and the exothermicity of the processes in the absorber. Formaldehyde absorbers operate less efficient than could have been expected based on the good apparent solubility of formaldehyde in water. The reason is that, in aqueous solutions, formaldehyde reacts with water to methylene glycol and higher poly(oxymethylene) glycols via a series of reversible reactions 2 2 2 2 ) OH ( CH O H O CH + (1) O H H O) HO(CH H O) HO(CH (OH) CH 2 n 2 1 n 2 2 2 + + − . (2) The good apparent solubility of formaldehyde in water is actually the good solubility of methylene glycol, and the capacity of the solution to accommodate poly(oxymethylene) glycols. Formaldehyde itself, like most gases, is only sparingly soluble in water. The rate of the hydration reaction (1) is relatively fast, causing chemical enhancement of the gas-to-liquid transfer of formaldehyde. The formation rate of the higher poly(oxymethylene) glycols is low, with reaction times in the order of tens of minutes to hours, depending on the temperature. The absorption of formaldehyde, and its consequent hydration, as well as the condensation of steam are exothermic processes. Therefore, the temperature of the liquid increases as it flows down the absorber. Because of factors such as the ones mentioned above, formaldehyde absorbers generally are divided into different absorption sections. Each of the absorption sections, or beds, is provided with a relatively large, externally cooled liquid recirculation stream. A typical example of a formaldehyde absorber is shown in Fig. 2. This thesis The major aim of this work is the development of reliable models that are capable, first, of accurately describing the performance of current formaldehyde absorbers, second, of predicting the influence of changing various operating parameters, and third, of optimising the performance of the absorber columns towards formaldehyde absorption efficiency and capacity. To achieve this goal, knowledge of the kinetics of the principal reaction (1) and the consecutive polymerisation reactions (2) is of major importance. The kinetics of the latter, the formation of the poly(oxymethylene) glycols, have been investigated extensively by other research groups (p.e. Dankelman et al., 1988; Hasse & Maurer, 1991; Hahnenstein et al., 1994, 1995). Chapter 1: Introduction 3 feed product tail gas water Fig. 2. Scheme of a typical formaldehyde absorber. The investigations into the kinetics of the principal reaction are treated in the next three chapters. Following these are two chapters on the development of chemical reaction engineering models for formaldehyde absorbers, a concluding chapter, and some additional material. Chapter 2 describes the experimental work on the determination of the dehydration rate of methylene glycol, the reverse of reaction (1). Using a traditional wet chemistry methodology, the dehydration rates where obtained from the measured formation rates of hydroxymethane sulphonate from the reaction of formaldehyde with - 2 3 SO , at various temperatures. The results could be correlated to an Arrhenius type expression. Chapter 1: Introduction 4 In Chapter 3, a theoretical treatment is presented of the problem of simultaneous absorption and/or desorption of two components, accompanied by a first-order reversible liquid phase reaction among the two. The analytical solutions developed here for the concentration profiles in the mass transfer film and for the enhancement factors are used in the next chapter. Chapter 4 describes the experimental determination of the kinetics of the hydration of formaldehyde in water. The measurements are based on the chemically enhanced absorption of formaldehyde gas into water in a stirred cell and mathematical modelling of the transfer process. The temperature dependency of the hydration rate constant correlates well to an Arrhenius type expression. From the results of this chapter, combined with those of chapter 2, the equilibrium constant of the hydration of formaldehyde is obtained. In Chapter 5 a model is developed for formaldehyde absorbers, based on differential equations for the mass and energy balances in each phase. The resulting set of coupled boundary value problems is solved by a semi-transient method. In Chapter 6 the absorber model is extended with a description of the behaviour of unconverted methanol that enters the absorber. Also modelled now are vaporisation and reabsorption of methylene gylcol and hemiformal, the principal reaction products of formaldehyde with water and methanol. The modelling is based here on a non-equilibrium stage model. Concluding remarks can be found in Chapter 7. In the appendices, some additional material can be found: Appendix A presents the results of experimental work on the determination of the viscosity of aqueous formaldehyde solutions and correlations for the density and viscosity of aqueous formaldehyde solutions as a function of the temperature and the strength of the solution; Appendix B elucidates some calculation methods to obtain the equilibrium composition of solutions containing formaldehyde. Some additional material to chapter 4, on the determination of the reaction order of formaldehyde in the hydration reaction, is included in Appendix C. Chapter 2: The kinetics of the dehydration of methylene glycol. 5 Chapter 2 The kinetics of the dehydration of methylene glycol. Abstract The kinetics of the dehydration of methylene glycol were measured under conditions relevant in industrial formaldehyde absorbers (293 K ≤ T ≤ 333 K; 6.0 ≤ pH ≤ 7.8). The rapid reaction between formaldehyde and SO 3 2− to hydroxymethane sulphonate (HMS − ) was used as a formaldehyde scavenger: O H O CH (OH) CH 2 2 2 2 + → d k , − − − → + 3 2 fast 2 3 2 )SO (O CH SO O CH , − + − − → + 3 2 fast 3 2 (OH)SO CH H )SO (O CH . At the experimental conditions, the rate determining step in the formation of HMS − appeared to be the dehydration of methylene glycol. The observed reaction rate constant for the dehydration of methylene glycol could be correlated as 1 / 6705 7 s 10 96 . 4 − − ⋅ = T d e k . The dehydration rate is shown to be independent of the concentrations of both sulphite and hydroxide ions at the conditions applied here. Introduction For a detailed model of the absorption process of formaldehyde, as well as in the further applications of formaldehyde, the kinetics of the dehydration of methylene glycol (see eq. (1)), are important. The literature data on the dehydration rate of methylene glycol are limited to ambient temperatures. Using various chemical scavengers, LeHénaff (1963) and Bell et al. (1966) obtained k d = 4.5×10 -3 s -1 at 293 K and 5.1×10 -3 s -1 at 298 K, respectively. From the formaldehyde production and subsequent reaction in the radiolysis of methanol in a flow measurement system, Sutton and Downes (1972) calculated k d = 4.4×10 -3 s -1 at 295 K. Los et al. (1977) found k d = 5.7×10 -3 s -1 at 298 K with pulse polarography. Funderburk et al. (1978) obtained k d = 4.2×10 -3 s -1 at 298 K using carbazides and hydrazine as trapping agents. In this contribution we report on the reaction rate of the dehydration of methylene glycol, characterized by the reaction rate constant k d , at the conditions prevailing in industrial formaldehyde absorbers; at the wider temperature range of 293-333 K and at a pH between 6 and 7. The experimental method applied uses the reaction of sodium sulphite with formaldehyde. In aqueous solution sulphite ions react specifically with the carbonyl group of aldehydes or ketones to form α-hydroxy sulphonates (Blackadder and Hinshelwood, 1958). The sulphite ions do not react with methylene glycol. The reaction of sulphite with formaldehyde is fast Chapter 2: The kinetics of the dehydration of methylene glycol. 6 (LeHénaff, 1963; Boyce & Hoffmann, 1984), and under suitable conditions the formation rate of the product, hydroxymethane sulphonate, CH 2 (OH)SO 3 − (HMS − ), is completely determined by the dehydration rate of methylene glycol. Sulphite then is a trapping agent, or chemical scavenger, of any formaldehyde produced by the dehydration of methylene glycol. In the mechanism of the HMS − formation, the following reactions are relevant here O H O CH ) OH ( CH 2 2 2 2 + h d k k , (1) + − − + H SO HSO 2 3 2 3 a K , (2) − − − − + 3 2 2 2 2 3 2 SO ) O ( CH SO O CH k k , (3) − − − − − + + OH SO ) OH ( CH O H SO ) O ( CH 3 2 1 ) HMS ( 2 2 3 2 a w K K . (4) Sørensen and Andersen (1970) studied the kinetics of the HMS − formation from sodium sulphite and formaldehyde at 298 K in strongly alkaline aqueous solutions (pH 9-12). At these high pH values they found the rate determining step of the overall reaction to be the dehydration of methylene glycol catalysed by hydroxide ions, and obtained k d = 1.7×10 3 [OH − ] s -1 at 298 K. The initial concentrations of formaldehyde, 20-38 mol m -3 , and sulphite, 29-37 mol m -3 , were of comparable magnitudes. Boyce and Hoffmann (1984) studied the rate of formation of HMS − from formaldehyde in acidic solutions (pH = 0-3.5) at temperatures from 288 K to 313 K. They concluded that the rate determining step in the HMS − formation involves the nucleophilic addition of bisulphite and/or sulphite ions to the carbonyl group of formaldehyde. They also found bisulphite to be the principal reactant at pH < 2, whereas at pH > 4 the reaction is carried by sulphite. The initial concentrations of formaldehyde, 10-100 mol m -3 , were considerably higher than the initial sulphite + bisulphite concentrations, 0.25-1.25 mol m -3 . From the overall reaction equation, which results from adding eqs (1), (3) and (4), it is seen that the formation of HMS − is accompanied by the release of hydroxide ions − − − + → + OH (OH)SO CH SO (OH) CH 3 2 2 3 2 2 . (5) By using sodium bisulphite in combination with sodium sulphite, a buffering capacity is introduced and the conditions can be chosen in such a way that the pH rises only modestly for a considerable length of time as compared to the time needed to get close to the maximum conversion of sulphite. Chapter 2: The kinetics of the dehydration of methylene glycol. 7 Experimental A stock solution containing approximately 1% by weight of formaldehyde was prepared by dissolving paraformaldehyde (Janssen Chimica) in water and allowing to equilibrate several days. The overall formaldehyde concentration in the stock was determined accurately with the sulphite method (Walker, 1964). The low overall formaldehyde content in the stock solution ensures that any polymeric forms of formaldehyde can be neglected (see Appendix B). Reagent grade sodium bisulphite and sodium sulphite (Janssen Chimica) were used to prepare 100 ml aliquots of buffer solutions of these components just prior to the experiments. Because carbon dioxide can interfere with the pH measurements, and oxygen can induce oxidation of bisulphite or sulphite, the water used in the experiments and in the preparation of the solutions was double distilled, and boiled out and stored with nitrogen purging. A Metrohm type E561/1 pH meter was used. The output signal from the pH meter was send to a computer with a frequency of exactly 1 Hz for later analysis. The experiments were carried out in a closed, double walled vessel, kept at the desired temperature by circulating water. The vessel was equipped with a nitrogen inlet, for purging prior to the experiment, a pH electrode and a septum sealed inlet for adding reagent solutions via syringes. Before an experiment was started, the desired amount of buffer solution was added to 100 ml water in the stirred reaction vessel, and allowed to attain thermal equilibrium. Meanwhile, a sample of the formaldehyde stock solution was brought to the same temperature in a separate vessel also under nitrogen. The injection of a desired amount of this formaldehyde stock solution into the reaction vessel marked the start of the experiment. Experiments were performed at 5 temperatures ranging from 293 to 333 K. At each temperature 8-10 experiments were performed with various sulphite/bisulphite ratios of the buffer solution, and with various amounts of buffer solution and formaldehyde stock solution added to the reaction vessel. This way, the following ranges of variation in initial concentrations were realised: pH 0 = 6-7, [NaHSO 3 ] 0 = 1.5-8.0 mol m -3 , [Na 2 SO 3 ] 0 = 0.3-5.2 mol m -3 , and [CH 2 (OH) 2 ] 0 = 4-25 mol m -3 . The concentrations were chosen in such a way that [CH 2 (OH) 2 ] 0 > S tot in all experiments, where S tot denotes the total concentration of sulphur: 0 3 2 0 3 ] SO Na [ ] NaHSO [ + = tot S . (6) It should be noted here that [NaHSO 3 ] 0 and [Na 2 SO 3 ] 0 are the initial molar concentrations of reagentia used to make up the solution. They are not a priori identical with the true initial concentrations of bisulphite and sulphite ions, [HSO 3 − ] 0 and [SO 3 2− ] 0 , since these are determined by the dissociation constants governing the system. The build-up of pyrosulphate according to O H O S HSO 2 2 2 5 2 3 + − − is proportional to the square of the HSO 3 − concentration (Hayon et al., 1972). However, the equilibrium constant of the reaction is very small, and the S 2 O 5 2− concentration becomes Chapter 2: The kinetics of the dehydration of methylene glycol. 8 negligible in solutions containing less than 50 mol m -3 of sulphur (Golding, 1960; Deister et al., 1986). Since the total sulphur concentrations employed here were always less than 15 mol m -3 , we can safely neglect any pyrosulphate formation. Results An example of the pH readings recorded during an experiment is shown in Fig. 1. The profile is typical for all experimental results: after a gradual increase there is a sharp upturn of the pH corresponding to a decrease of the concentration of hydrogen ions. In this section we will first explain qualitatively the origin of the typical profiles of the measured pH curves, and secondly show how quantitative kinetic information was extracted from the experimental data. 12 10 8 6 0 20 40 60 80 100 t (s) pH Fig. 1. pH as a function of time for a typical experiment (no. 20.1): T = 293 K, [NaHSO 3 ] 0 = 4.311 mol m -3 , [Na 2 SO 3 ] 0 = 1.146 mol m -3 , and [CH 2 (OH) 2 ] 0 = 22.84 mol m -3 . For each of the data points of an experiment, [H + ] was used to calculate the concentrations of the other relevant species, [H 2 SO 3 ], [HSO 3 − ], [SO 3 2− ], [HMS − ], [OH − ], [Na + ] and [CH 2 (OH) 2 ], from the sulphur balance ] HMS [ ] SO [ ] HSO [ ] SO H [ 2 3 3 3 2 − − − + + + = tot S , (7) the sodium balance ] Na [ ] SO Na [ 2 ] NaHSO [ 0 3 2 0 3 + = + , (8) Chapter 2: The kinetics of the dehydration of methylene glycol. 9 the carbon balance ] [HMS ] (OH) [CH ] (OH) [CH 2 2 0 2 2 − + = , (9) the charge balance ] [OH ] [HMS ] 2[SO ] [HSO ] [Na ] [H 2 3 3 − − − − + + + + + = + , (10) the water ionisation equilibrium W K = − + ] OH ][ H [ , (11) and the two sulphite dissociation equilibria 1 3 2 3 ] SO H [ ] H ][ HSO [ a K = + − , (12) 2 3 2 3 ] HSO [ ] H ][ SO [ a K = − + − . (13) In writing the carbon balance, the contribution of CH 2 O was neglected because the equilibrium of eq. (1) is far to the left and the concentration of free formaldehyde is very low (Zavitsas et al., 1970). The calculations showed that [H 2 SO 3 ] was always negligible. Therefore this species was neglected in the further considerations. For the remaining sulphur containing components the concentrations vs. pH are shown in Fig. 2 for experiment no 20.1. It may be noted that before the large pH jump, [SO 3 2− ] remains constant. This was found to be the case for all experiments, and can be explained by subtracting the sulphur balance, eq. (7), from the charge balance, eq. (10), substituting eqs (6) and (8) for S tot and [Na + ], and rearranging, giving ] H [ ] H [ ] SO Na [ ] SO [ 0 3 2 2 3 + + − − + = W K . (14) Chapter 2: The kinetics of the dehydration of methylene glycol. 10 6 7 8 9 pH 5 4 3 2 1 0 HSO 3 - SO - 3 2 HMS - 10 11 c o n c e n t r a t i o n ( m o l / m ) 3 Fig. 2. Concentrations of sulphurous components vs. pH of experiment no 20.1, see Fig. 1. Symbols: calculated points. Lines: for illustrative purposes only. It can easily be shown that, for the range of [Na 2 SO 3 ] 0 and pH 0 values employed here, in order to observe a decrease of the sulphite concentration from [SO 3 2− ] 0 with say 1%, an increase of the pH is needed varying from pH > 9 at 293 K to pH > 8 at 333 K. The highest pH measured before the large pH jump occurred was 7.8. It can therefore safely be assumed that [SO 3 2− ] is constant for pH < 7.8. Now, the typical pH profile of Fig. 1 can be explained qualitatively. The SO 3 2− ions that react with formaldehyde to HMS − are replenished with new ones obtained from the dissociation of HSO 3 − . The gradual decrease of [HSO 3 − ] results in a decrease of the buffering capacity of the mixture and a gradual increase of the pH. At the point where all HSO 3 − initially present is consumed, the buffering capacity breaks down completely, resulting in the large pH jump. Any data points beyond the pH jump, i.e. those with pH>8, are not used in the current analysis. In writing the mechanism of the HMS − formation, eqs (1)-(4), it is assumed that the ion equilibria, eqs (2) and (4), are established rapidly. The equilibrium of eq (4) is far to the right at pH < 8, and the concentration of HMS 2− is negligible. Because SO 3 2− is the better nucleophile by a factor of 10 5 compared to HSO 3 − (Burnett, 1982), and based on the results of Boyce and Hoffmann (1984), see introduction, the direct addition of bisulphite ions to formaldehyde was neglected in the scheme of the reaction mechanism. Since the equilibrium of the overall HMS − formation reaction, eq (5), is far to the right (Dasgupta et al., 1980; Deister et al., 1986) the decomposition of HMS − , via the reverse of reactions (4) and (3), can be neglected for the first part of the measured pH profiles, i.e. before the pH jump occurs. The rate of HMS − formation then is Chapter 2: The kinetics of the dehydration of methylene glycol. 11 ] SO ][ O CH [ d ] HMS d[ 2 3 2 2 − − = k t . (15) Applying the steady state principle to [CH 2 O] in reactions (1) and (3) gives ] SO [ ] ) OH ( CH [ ] O CH [ 2 3 2 2 2 2 − + = k k k h d , (16) where k h is the pseudo-first-order rate constant for the hydration of formaldehyde, see eq (1). Substitution of eq (16) into eq (15) gives ] SO [ ] ) OH ( CH ][ SO [ d ] ) OH ( CH d[ d ] HMS d[ 2 3 2 2 2 2 3 2 2 2 − − − + = − = k k k k t t h d . (17) Before the pH jump, i.e. pH < 7.8, the sulphite concentration remains constant. This allows the integration of eq (17) t k obs − = − 0 2 2 2 2 ] ) OH ( CH ln[ ] ) OH ( CH ln[ , (18) where ] SO [ ] SO [ 2 3 2 2 3 2 − − + = k k k k k h d obs . (19) The experimental data appear to be very well described by eq (18). An example is shown in Fig. 3 for the data also used in Figs 1 and 2. Rewriting eq (19) as d d h obs k k k k k 1 ] SO [ 1 / 1 2 3 2 + = − (20) shows that the values of (k h /k 2 ) and k d at each temperature should follow from the k obs data obtained whilst varying [SO 3 2− ]. However, at each temperature applied, a regression analysis of the data according to eq (20) showed that the variation of 1/k obs with 1/[SO 3 2− ] was not significant (95% confidence level). A plot of the data obtained at 323 and 333 K according to eq (20), shown in Fig. 4, illustrates this nicely. Apparently we have k 2 [SO 3 2− ] >> k h , and eq (19) reduces to k obs = k d . Chapter 2: The kinetics of the dehydration of methylene glycol. 12 3.2 3.1 3.0 2.9 2.8 0 10 20 30 40 50 t (s) l n ( ) m o l / m 3 [ C H ( O H ) ] 2 2 Fig. 3. Data of experiment no 20.1, see Fig. 1, plotted according to eq (18). 1/[SO ] (m /mol) 3 2- 3 0 1 2 3 4 5 1 / k ( s ) o b s 10 2 10 1 1 333K 323K Fig. 4. Data plotted according to eq (20). Symbols: 1/k obs obtained from eq (18) for a single experiment. Lines: mean value of 1/k obs at a given temperature. The reaction rate, eq (17), then reduces to ] (OH) CH [ d ] (OH) CH d[ d ] HMS d[ 2 2 2 2 d k t t = − = − . (21) This way, it is demonstrated that, under the experimental conditions applied, the rate of formation of HMS − is completely controlled by the dehydration rate of methylene glycol. Chapter 2: The kinetics of the dehydration of methylene glycol. 13 Regression of the reaction rate constants with the reciprocal temperature gave − × = T k d 6705 exp 10 96 . 4 7 s -1 . (22) Equation (22) describes the data obtained from the individual experiments with a mean absolute relative deviation of 13%, see Fig. 5. From the temperature coefficient the activation energy was found as E a = (55.8 ± 2.7) kJ mol -1 . Regression of the reaction rate constants according to the transition-state theory resulted in ∆H ≠ = (53.2 ± 2.7) kJ mol -1 and ∆S ≠ = (-106.3 ± 8.7) J mol -1 K -1 for the dehydration reaction. 2.7 2.9 3.1 3.3 3.5 3.7 10 10 10 -1 -2 -3 1 1/ T x 10 (K ) 3 1 - k ( s ) d - 1 Fig. 5. Arrhenius plot of the methylene glycol dehydration rate constant k d . Symbols: mean value of at least 8 experiments. Horizontal bars: standard deviation. Line: eq (22). Chapter 2: The kinetics of the dehydration of methylene glycol. 14 Discussion The reaction rate constant for the dehydration of methylene glycol obtained from eq (22) at 293 K, k d = 5.7×10 -3 s -1 compares well with the literature data mentioned in the introduction. However, the mechanism obtained here for the reaction of sulphite with formaldehyde in aqueous solution, i.e. the rate limiting dehydration of methylene glycol, differs from both the mechanism obtained by Sørensen and Andersen (1970) and by Boyce and Hoffmann (1984), see the introduction of this chapter. Since the variation of k d with [SO 3 2− ] appeared not to be significant, see Fig. 4, the addition of SO 3 2− to formaldehyde can not be rate controlling. Probably, Boyce and Hoffmann (1984) found otherwise because of the large excess of methylene glycol and the high acidities in their experiments. Hydrogen ions are known to catalyse the dehydration of methylene glycol under acidic conditions (Funderburk et al., 1978). In order to check for the possible catalysis of the dehydration of methylene glycol by hydroxide ions, which Sørensen and Andersen (1970) found rate determining at their conditions, eq (21) can be extended to ( ) ] (OH) CH [ ] OH [ d ] (OH) CH d[ 2 2 OH , 2 2 − + = − d d k k t . (23) Because this equation cannot be integrated analytically, the method of Himmelblau et al. (1967) was used to fit the experimental data. Integration of eq (23) gives ∫ − − − = − i t d i d i t k t k 0 OH , 0 2 2 2 2 d ] OH [ ] (OH) CH ln[ ] (OH) CH ln[ , (24) where the subscript i denotes the ith data point from an experiment. The integral on the right hand side of eq (24) was evaluated numerically for each data point. The trapezoidal rule suffices for this purpose because the values of [OH − ] vary smoothly, and data points were obtained every second. The constants k d and k d,OH were calculated using multiple regression according to eq (24) applied to all the data points from all the experiments at a specific temperature simultaneously. The k d,OH values thus obtained appeared to decrease with increasing temperature, and k d,OH [OH − ]<<k d for all experiments. From this it follows that catalysis of the dehydration of methylene glycol by hydroxide ions is not significant at the conditions used here. This conclusion is consistent with the measurements of Funderburk et al. (1978) who showed that the influence of OH − is significant only for pH > 8 at 298 K. The same conclusion can be drawn from inspecting the relative residuals of the k d values, by comparing the values obtained from eq (22) with those obtained from the individual experiments, as a function of the measured initial pH, see Fig. 6. The residuals were calculated as Chapter 2: The kinetics of the dehydration of methylene glycol. 15 % 100 ) ( ) ( ) ( residual experiment experiment eq(22) × − = d d d k k k . (25) If the dehydration rate would be susceptible to catalysis by hydroxide ions at the conditions employed here, a systematic deviation would be expected in Fig. 6 from more positive residuals at lower pH, i.e. low [OH − ], to more negative ones at higher pH, with higher [OH − ]. The absence of such a trend indicates the absence of catalysis by hydroxide ions in this pH region. Similarly, any catalysis of the dehydration of methylene glycol by hydrogen ions, as observed under acidic conditions (Funderburk et al. ,1978), can be excluded in the p|H region considered her, based on Fig. 6. The observation of Sørensen and Andersen (1970) that the HMS − formation rate is completely determined by k d,OH [OH − ] can be attributed to the much higher alkalinity of their solutions, where the pH ranged from 9 to 12. 6 7 pH 6.5 100 50 0 -50 -100 r e s i d u a l o f k ( % ) d +13% -13% Fig. 6. Relative residuals of the k d values from individual experiments compared to the value obtained from eq (22). Physical properties All the calculations mentioned in this chapter were corrected for non-ideal behaviour. This section explains how and why these corrections were performed. Because of the electrical interactions between ionic species, even quite dilute electrolyte solutions exhibit non-ideal behaviour. Thermodynamic treatments then require the use of activities in stead of concentrations for the charged species. Here, activity coefficients were used Chapter 2: The kinetics of the dehydration of methylene glycol. 16 to correct for the deviations from ideal behaviour, allowing the thermodynamic dissociation equilibrium constants to be expressed as a function of the temperature only. The temperature dependence of the thermodynamic equilibrium constant for the dissociation of H 2 SO 3 , K a1 th , was obtained by regression of the data measured by Devèze and Rumpf (1964) − = = + − + − 08 . 4 2009 exp ] SO H [ ] H ][ HSO [ ) H ( ) 3 HSO ( 3 2 3 1 T K th a γ γ . (26) Similarly, K a2 th , for the dissociation of HSO 3 − , was obtained from the data measured by Hayon et al. (1972) − = = − + − − + − 39 . 14 1400 exp ] HSO [ ] H ][ SO [ ) 3 HSO ( ) H ( ) 2 3 SO ( 3 2 3 2 T K th a γ γ γ . (27) The thermodynamic dissociation constant of water was taken from Stumm and Morgan (1970, p.76) T T K th w × − − = = − + − + 01706 . 0 99 . 4470 0875 . 12 ) ] OH ][ H log([ log ) OH ( ) H ( 10 10 γ γ . (28) The concentration based equilibrium constants K a1 , K a2 and K w of eqs (11)-(13) were obtained by correcting the corresponding thermodynamic equilibrium constants for the influence of the ionic strength according to the Güntelberg modification of the Debye-Hückel limiting law (Stumm & Morgan, 1970, p.83) 5 . 0 5 . 0 5 . 1 2 6 1 ) ( 10 20 . 4 ln µ µ ε γ + × − = T z i i , (29) where ε is the dielectric constant of water, varying from ε = 80.14 at 293 K down to ε = 66.78 at 333 K (Handbook of Chemistry and Physics), z i denotes the charge number of ion i, and the ionic strength, µ, is given by ∑ − = i i i z C ) 10 ( 2 1 2 3 µ , (30) where the summation runs over all ionic species. Because of the non-linearity of the equations, the concentrations of the components and the concentration based equilibrium constants had to be calculated using an iterative method. A Chapter 2: The kinetics of the dehydration of methylene glycol. 17 successive approximation method, initialised by taking all activity coefficients equal to 1.0, proved to be efficient for this purpose. Conclusions For the nearly neutral conditions employed here, pH-range 6.0-7.8, the rate determining step in the reaction of sulphite with formaldehyde in aqueous sulphite bisulphite buffer solutions is the dehydration of methylene glycol. The dehydration rate appeared to be independent of concentrations of both sulphite, [SO 3 2− ], and hydroxide, [OH − ], ions as opposed to the results obtained by Boyce and Hoffmann (1984) at very acidic conditions, pH-range 0-3.5, and by Sørensen and Andersen (1970) at more alkaline conditions, pH-range 9-12. The methylene glycol dehydration rate constant obtained in this work at 293 K agrees well with the values obtained by other authors at 293-298 K. The experimental k d values at temperatures ranging from 293 to 333 K could be described with an Arrhenius-type equation, from which an activation energy of E a = (55.8 ± 2.7) kJ mol -1 was obtained. Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 19 Chapter 3 Simultaneous absorption and/or desorption with reversible first-order chemical reaction. Abstract The problem of gas absorption accompanied by a first order reversible reaction, producing a volatile reaction product, is considered. A general analytical solution is developed for the film model, resulting in explicit relations for the concentration profiles in the film and for the mass transfer enhancement factors. The solution is not restricted to equal diffusivities, and is also applicable to the simultaneous absorption and reversible reaction of two gases. Several limiting cases are derived. It is shown that enhancement factors are possible with values larger than one, but also smaller than one, and even negative. The latter occurs in the industrially important absorption of formaldehyde in aqueous solution. Introduction In the description of interphase mass transfer accompanied by chemical reactions, it is common practice to incorporate the effects of reactions on the mass transfer rates in enhancement factors. In the extensive literature on mass transfer enhancement by chemical reaction, many contributions can be found on analytical and approximate analytical expressions for the enhancement factors for a number of reaction stoichiometries, with single or multiple reactions and with irreversible or reversible reactions [e.g. Westerterp et al. (1984, Chap. 7 and 8) and the references therein]. The greater part of this literature is devoted to the absorption of a reacting species from a gaseous phase to a liquid phase, where it reacts to non-volatile products. For this type of operation, the enhancement factor for the transferred component, defined as Eq (12) has a lower limit of one. Recently, Landau (1992) published analytical solutions for the case of first order reactions, irreversible and reversible, involving a non-volatile reactant present in the liquid phase only, and desorption of the volatile reaction product. Here also, the enhancement factor for the transferring species has a minimum of one. In fact, by linking the description of the processes in the interfacial region to those in the bulk, Landau showed that the enhancement of product desorption is negligible if the reactant originates completely from the liquid. A different situation arises if two gases absorb in a liquid, accompanied by reversible reactions. Cornelisse et al. (1980) numerically simulated the absorption of H 2 S and CO 2 in an amine solution accompanied by complex reversible reactions, and showed that, under certain conditions, negative enhancement factors are possible for such systems. As far as we know, Cornelisse et al. were the first to point out the possibility of negative enhancement factors. Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 20 In our work on the simulation of the absorption of formaldehyde in water we came across a very simple system displaying negative enhancement factors: the absorption of a gas, accompanied by a first order reversible reaction and partial desorption of the reaction product. The purpose of this contribution is to analyse this type of absorption kinetics. Absorption with first-order reversible reaction and desorption Gas absorption accompanied by a first order reversible reaction and partial desorption of the reaction product ) ( ) ( A A l → g (1) ) ( ) ( P A l l ↔ (2) ) ( ) ( P P g → l (3) with the reaction rate equation ) ( 1 K C C k R P A A − = (4) can be described according to the film model by the following equations for diffusion with reaction: A A A R x C D = 2 2 , d d l (5) A P P R x C D − = 2 2 , d d l (6) and boundary conditions I P P I A A C C C C x = = = , : 0 (7) P P A A C C C C x = = = , : δ . (8) The concentrations in the bulk liquid are not necessarily at equilibrium. Equations (5) and (6) can be solved analytically for the concentrations of A and P in the film: Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 22 Substitution of eq (13) into (15) gives R R R P I P P A P I P A I A R R P v K C C C C K C C C C K v E φ φ φ φ φ ] tanh[ ) ( ] cosh[ 1 1 1 ) ( ] tanh[ 1 1 + ( ¸ ( ¸ − ( ( ¸ ( ¸ − − + ( ( ¸ ( ¸ − − − ( ¸ ( ¸ − − = . (16) From eqs (12) and (14) it is seen that the sign of E depends on the direction of mass transfer, i.e. the sign of 0 ) /d (d = x x C , and on the sign of the concentration difference over the film, ) ( C C I − . If the combined effects of mass transfer and reaction cause the concentration of a component at the interface to be equal to the concentration in the liquid bulk, an asymptote is found for the enhancement factor of that component. Small deviations from this situation result in either positive or negative enhancement factors. We will return to this remarkable behaviour below. For l l , , P A D D = (i.e. v = 1) a solution of eqs (5)-(8) was presented earlier by Shah and Sharma (1976). Their eq (269) gives a relation for the desorption rate of the product P through the gas-liquid interface, ) ( , I P P P P C C E k − l , which contains E P implicitly. However, their equation seems to contain some misprints because calculations using it gave erroneous results. In engineering calculations, the bulk phase concentrations are usually available from macroscopic balances. A combination of eqs (13) and (15) with the rate balances ) ( ) / ( , , , A I A A A A I A g A A g C C E k m C C k − = − l (17) ) ( ) / ( , , , P I P P P P I P g P P g C C E k m C C k − = − l (18) allows the calculation of the unknown interface concentrations and enhancement factors from the known bulk phase concentrations. If the gas phase resistance to mass transfer is negligible so that mass transfer is completely liquid-film-controlled, eqs (17) and (18) can be replaced by g A A I A C m C , = and g P P I P C m C , = . From the general scheme outlined above, several interesting special cases can be derived. Here we will look at some of these in detail. Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 21 v K C C v x v K C vC x v K C C K x v K C KC x C P A I P I A P A R R I P I A R R A + + + + + − + + − + + − ( ¸ ( ¸ − = ) ( ) 1 ( ] sinh[ ] sinh[ ] sinh[ ) 1 ( sinh δ δ φ δ φ φ δ φ (9) and )] ( ) ( [ ) ( P I P A I A A I A I P P C C C C v x C C v C C − + − − − + = δ . (10) The parameters φ R and v are defined as l l l , , , 1 and ) ( P A A R D D v K D v K k = + = δ φ . (11) The enhancement factor for component A for a non-instantaneous reaction defined as ) ( d d , 0 , A I A A x A A A C C k x C D E − | | . | \ | − ≡ = l l (12) is found from Eq (9) as R R R A I A P A A I A P I P R R A v K C C C C K C C C C K E φ φ φ φ φ ] tanh[ ) ( ] cosh[ 1 1 ] tanh[ 1 1 + ( ¸ ( ¸ − ( ( ¸ ( ¸ − − + ( ( ¸ ( ¸ − − − ( ¸ ( ¸ − + = . (13) Similarly, the enhancement factor of component P for a non-instantaneous reaction, defined as ) ( d d , 0 , P I P P x P P P C C k x C D E − | | . | \ | − ≡ = l l (14) is found from eqs (10), (12) and (14) as P I P A I A A P C C C C E v E − − − + = ) 1 ( 1 (15) Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 23 Special and limiting cases Instantaneous reversible reaction, volatile reactant and product If the reaction is instantaneous, the rates of the forward and backward reactions of eq (2) have infinite values, and chemical equilibrium will prevail everywhere in the liquid. For this situation, the right-hand sides of eqs (5) and (6) become undefined. Elimination of the reaction term A R from (5) and (6) the gives 0 2 2 2 2 = + dx C d D dx C d D P P A A . With the equilibrium condition A P KC C = and the boundary condition (7), this equation can easily be solved for A C in the liquid film: ) ( A I A I A A C C x C C − − = δ . (19) A balance at the interface states that the amounts of A and P arriving at the interface from the gas phase equal the amounts of A and P that diffuse from the interface into the liquid film: 0 0 , , , , ) ( ) ( ) / ( ) / ( = = − − = − + − x P P x A A P I P g P P g A I A g A A g dx dC D dx dC D m C C k m C C k .(20) Substitution of eqs (17)-(19) and the equilibrium condition A P KC C = into eq. (20) gives the relation between the enhancement factors A E and P E : K E v E A P 1 ) 1 ( 1 − + = . (21) Now, with a known set of bulk phase concentrations, the interface concentrations and enhancement factors can be calculated from eqs (17), (18) and (21) together with the equilibrium condition I A I P KC C = . Volatile reactant, nonvolatile reaction product For the case where the reaction product P is nonvolatile, the mass transfer enhancement factor for P is not defined anymore. For this case, from eqs (10) and (12) and the condition 0 ) /d (d 0 = = x P x C , an expression is found for I P C ) )( 1 ( A I A A P I P C C E v C C − − + = . (22) Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 24 Substitution of eq (22) into (13) results in an equation for E A which is only valid if 0 ) /d (d 0 = = x P x C : R R R A I A P A A v K C C v C C K v K E φ φ φ ] tanh[ 1 ] cosh[ 1 1 ) ( 1 + ( ¸ ( ¸ − ( ( ¸ ( ¸ − − + + = . (23) This expression is similar to the one obtained by Onda et al. (1970). If also the reaction is instantaneous, taking the limit for ∞ → R φ of eq (23) gives v K E A + =1 (24) which is contained in the solution given for this case by Astarita and Savage (1980). Nonvolatile reactant, volatile reaction product Another special case is the situation where the product P is volatile, but where the reactant A originates completely from the liquid phase and is nonvolatile, 0 ) /d (d 0 = = x A x C . Then we have from eq (10) )] ( ) ( [ 1 d d 0 P I P A I A x P C C C C v x C − + − − = | . | \ | = δ (25) Substitution of eq (25) into (14) gives P I P A I A P C C C C v E − − + =1 (26) Furthermore, from eq (9) together with 0 ) /d (d 0 = = x A x C an expression for I A C is found, which is substituted into eq (26) to find R R R P I P P A P v K C C C C K v K E φ φ φ ] tanh[ ] cosh[ 1 1 1 + ( ¸ ( ¸ − ( ( ¸ ( ¸ − − − + = . (27) Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 25 If the reaction is instantaneous, there is chemical equilibrium in the liquid: I A I P KC C = and A P C K C = . For this case eq (27) reduces to: K v E P + =1 . (28) This equation was derived by Westerterp et al. (1984, p.423). It is also implicitly present in the expression obtained by Shah and Sharma (1976) for the desorption flux of P, but their eq (61) unfortunately contains a misprint. Irreversible reaction The first order irreversible reaction can be regarded as a limiting case of the reversible reaction, with K going to infinity. While eq (15) is still valid, E A can be found for this case by taking the limit of eq (13). In this way we indeed get the classical result: A I A R A I A R R A K A C C C C E E − − = = ∞ ∞ ∞ ∞ → ∞ ] cosh[ / ] tanh[ ) ( lim , , , , φ φ φ (29) with l , 1 , ) ( lim A R K R D k δ φ φ = = ∞ → ∞ . (30) Simultaneous absorption and reaction of two gases Recently, van Swaaij and Versteeg (1992) presented a review on gas-liquid mass transfer with reversible reactions. For the simultaneous absorption and reaction of two gases they noted that reversible reactions have not been studied so far. For this case, if the reaction between the two absorbed gases is first-order-reversible according to the scheme ) ( ) ( A A l → g (31) ) ( ) ( P P l → g (32) ) ( ) ( P A l l ↔ (33) it may be noted that the problem is also stated by eqs (5)-(8), and that these equations are independent of the direction of mass transfer. Also the solution in terms of the concentration profiles (9) and (10) and the enhancement factors (13) and (16) are independent of the direction of mass transfer. Thus, for the case of absorption of two gases, A and P, accompanied by a first order reversible reaction in the liquid, the mass transfer rates and enhancement factors can be obtained from eqs (13) and (16) combined with eqs (17) and (18). Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 27 φ R =0.1 2 4.47 10 50 ∞ x/δ 0 0.2 0.4 0.6 0.8 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 C C A A - m C A A,g Fig. 2. Dimensionless concentration of A in the liquid film. Parameter values as mention in example 1. φ R =∞ 2 4.47 10 50 x/δ 0 0.2 0.4 0.6 0.8 1 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 v m C A A,g C C P P - 0.1 Fig. 3. Dimensionless concentration of P in the liquid film. Parameter values as mention in example 1. These are shown in Figs 2 and 3 as dimensionless concentration differences vs. the depth in the liquid film, for various R φ values. At the critical value of 47 . 4 = R φ , the value of I P C equals that of P C . This results in the asymptotic behaviour of P E because ) ( , P i P C C − is in the denominator of eq. (14). However, note that although I P C equals P C , there is still desorption of P as indicated by the positive gradient P C at the interface. At larger values of R φ the reaction is fast enough to make I P C larger than P C , and E P becomes negative. At smaller values of R φ , the Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 28 production of P by reaction cannot keep up with the vaporisation of P: I P C becomes smaller than P C and P E is positive. In Figs 2 and 3, the lines indicated with ∞ = R φ denote the limiting instantaneous case where the concentration profiles are given by eq. (19) and the equilibrium condition A P KC C = . The influence of the volatility of the reaction product P on A E is shown in Fig. 4, using the same set of parameters except for the value of ) /( , , P l P P g k m k . The line marked with ) /( , , P l P P g k m k = 0 denotes the limiting case of volatile reactant with a nonvolatile reaction product. The dimensionless the concentration profiles of P vs. the depth in the liquid film are shown in Fig. 5 for a relatively fast reaction ( 10 = R φ ). It may be noted that the positive gradient of P at the interface, and therefore the desorption rate of P, decreases with a decreasing volatility of P. Also, in the limiting case of a nonvolatile reaction product P, we indeed have 0 ) / ( = x P dx dC = 0. 1 10 10 10 10 2 3 4 φ R E A 10 2 10 1 2 k g,P m k P l,P =0 10 20 Fig. 4. The influence of the volatility of P on E A . Parameter values as mention in example 1. Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 26 Applications In this section we will illustrate the remarkable behaviour of gas absorption accompanied by a reversible first order reaction, producing a volatile reaction product. The first two examples show the asymptotic behaviour of the enhancement factor for product desorption and for reactant absorption with selected sets of parameters. The third example describes an industrially important case of negative enhancement. Example 1 The following values are chosen for the parameters: 0 , = g P C mol m -3 , ) /( , , A l A A g k m k = ) /( , , P l P P g k m k = 10 , ) /( , g A A A C m C = 04 . 0 , ) /( , g A A P C vm C = 2 . 0 , and 0 . 5 / = v K . Figure 1 show the variation of A E and P E with R φ calculated from eqs (13), (15), (17) and (18). A E increases monotonously with increasing R φ until the limiting instantaneous value of ∞ → R A E φ ) ( = 37 . 73 is reached for very large R φ . P E displays an asymptote at 47 . 4 = R φ . If 47 . 4 < R φ then we have 1 ≥ P E , but surprisingly, if 47 . 4 > R φ then P E is always negative, with a limiting value of ∞ → R p E φ ) ( = 47 . 13 − for very large R φ . This remarkable behaviour can be understood by inspection of the concentration profiles in the liquid film. 100 50 0 -50 -100 0.1 1 10 10 10 10 2 3 4 φ R E , E A P P A P Fig. 1. Variation of E A and E P with φ R . Parameter values as mention in example 1. Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 29 x/δ 0 0.2 0.4 0.6 0.8 1 2.5 2.0 1.5 1.0 0.5 0.0 v m C A A,g C C P P - 2 k g,P m k P l,P =0 10 20 Fig. 5. The influence of the volatility of P on C P in the liquid film. Parameter values as mention in example 1. 0.1 1 10 10 10 10 2 3 4 φ R E A 10 10 10 4 3 2 10 1 1 10 10 2 10 3 ∞ K/ = v Fig. 6. The influence of the reversibility of the reaction on E A . Parameter values as mention in example 1. The influence of the reversibility of the reaction, as indicated by the value of v K / , on A E is shown in Fig. 6. Again, 0 , = g P C mol m -3 , ) /( , , A l A A g k m k = ) /( , , P l P P g k m k = 10 and ) /( , g A A P C vm C = 2 . 0 were used. The liquid bulk phase concentration of A was taken in chemical equilibrium with that of P. A E increases with v K / until the limiting case of a first order irreversible reaction is reached for v K / ∞ → , where A E is given by eq. (29). Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 30 Example 2 The same set of parameters as in example 1 is used here, i.e. 0 , = g P C mol m -3 , ) /( , , A l A A g k m k = ) /( , , P l P P g k m k = 10 and 0 . 5 / = v K . The only difference being that now the concentration of A in the gas phase is lowered in such a way that we now have ) /( , g A A A C m C = 5 . 0 and ) /( , g A A P C vm C = 5 . 2 . This change in the conditions drastically alters the variation of A E and P E with R φ , as illustrated in Fig. 7. Now it is the enhancement factor of the absorbing gas, A E , that displays an asymptote. This occurs at 5 . 17 = R φ . The dimensionless concentration profiles in the liquid film associated with 5 . 17 = R φ are shown in Fig. 8. The value of I A C equals that of A C , but there is clearly absorption of A at the interface as is revealed by the initially large negative gradient of A C in the film. At large values of R φ , the absorption of A cannot keep up with the disappearance of A due to reaction in the film; the reaction product acts as a sink for component A, and I A C becomes smaller than A C which causes negative enhancement factors for A. On the other hand, if 5 . 17 < R φ , then the mass transfer of A is relatively faster, and A E is positive because A I A C C < . The limiting instantaneous reaction enhancement factors for this situation are 5 . 26 ) ( − = ∞ → R A E φ and 5 . 6 ) ( = ∞ → R P E φ . The linear profiles of A C and P C in the liquid film for ∞ → R φ , obtained from eq. (19) and the equilibrium condition A P KC C = , are also shown in Fig. 8. 50 25 0 -25 -50 0.1 1 10 10 10 10 2 3 4 φ R E , E A P A P A Fig. 7. Variation of E A and E P with φ R . Parameter values as mention in example 2. Chapter 3: Simultaneous absorption and/or desorption with reversible first-order chemical reaction. 31 φ R =17.5 ∞ x/δ 0 0.2 0.4 0.6 0.8 1 C C A A - m C A A,g 0.0 -0.1 -0.2 -0.3 -0.4 0.0 -0.5 -1.0 -1.5 -2.0 φ R =17.5 ∞ v m C A A,g C C P P - Fig. 8. Dimensionless concentration of A and P in the liquid film. Parameter values as mention in example 2. Practical example Formaldehyde is commercially produced by gas phase oxidation of methanol with air. The gaseous reactor product stream contains formaldehyde and is fed to an absorber to dissolve the formaldehyde in water. In water, formaldehyde (A) reacts reversible to methylene glycol (P). Because of the large excess of water, this reaction can be considered here as pseudo first order. At the bottom part of the absorber, the entering gas does not yet contain any methylene glycol ( 0 , = g P C mol m -3 ). For this region of the absorber, the following parameter values are typical for an industrial formaldehyde absorber:, ) /( , , A l A A g k m k 95 . 0 = , ) /( , , P l P P g k m k 3 10 38 . 3 − × = , v K / 2 . 21 = , 4 . 1 = φ , ) /( , g A A A C m C 24 . 0 = , ) /( , g A A P C vm C 14 . 5 = . From eqs (13), (15), (17) and (18) the enhancement factors are calculated as 55 . 1 = A E and 13 . 0 − = P E , indicating that negative enhancement factors may occur in the absorption of formaldehyde. Conclusion A new analytical solution is presented for the concentration profiles and mass transfer enhancement in the general case of gas absorption accompanied by a first order reversible reaction, producing a partly volatile product. The solution is not restricted to equal diffusivities. From this general solution, several known limiting cases are readily derived. The solution is also applicable for the case of the simultaneous absorption and first order reversible reaction of two gases. It is shown that enhancement factors are possible with values not only larger than one but also smaller than one, and even negative enhancement factors are possible. The latter occurs in the industrially important absorption of formaldehyde in aqueous solution. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 33 Chapter 4 Kinetics and chemical equilibrium of the hydration of formaldehyde Abstract The reaction rate of the hydration of formaldehyde is obtained from the measured, chemically enhanced absorption rate of formaldehyde gas into water in a stirred cell with a plane gas liquid interface, and mathematically modelling of the transfer processes. Experiments were performed at the conditions prevailing in industrial formaldehyde absorbers, i.e. at temperatures of 293-333 K and at pH values between 5 and 7. The observed rate constants could be correlated as k h = 2.04×10 5 ×e -2936/T s -1 . Using the results, and the dehydration reaction rate constant, obtained previously at similar conditions, the chemical equilibrium constant for the hydration is obtained as K h = e 3769/T-5.494 . Introduction Formaldehyde is an important industrial intermediate in the manufacturing of resins, plastics, adhesives and many other products. Because formaldehyde is unstable in its pure, gaseous state it is usually produced as an aqueous solution. In such a solution, formaldehyde is almost completely hydrated to methylene glycol, 2 2 2 2 ) (OH CH k k O H O CH d h + . (1) Methylene glycol, in turn, depending on the strength of the solution, may polymerize to form a series of polyoxymethylene glycols, O H H O CH HO H O CH HO OH CH n n 2 2 1 2 2 2 ) ( ) ( ) ( + + − . (2) In the design of formaldehyde absorption and distillation processes, as well as in downstream processing, the kinetics and chemical equilibria of both reactions are important. The research group of Maurer recently studied the kinetics and chemical equilibria of the polyoxymethylene glycol formation (Hasse & Maurer, 1991; Hahnenstein, Hasse, Kreiter & Maurer, 1994; Hahnenstein, Albert, Hasse, Kreiter & Maurer, 1995) In the open literature, only two entries with experimental data on the reaction rate constant of the hydration of aqueous formaldehyde, k h , were encountered. Schecker and Schulz (1969) obtained formaldehyde hydration rates from temperature jump experiments and measurement of the approach of the formaldehyde concentration to the new equilibrium value with UV- absorption. From experiments at 298-333 K they obtained k h = 7800×e -1913/T s -1 at pH = 4-7. Sutton and Downes (1972) obtained k h = 9.8 s -1 at 295 K from radiolysis of aqueous solutions of methanol containing oxygen and semicarbazide hydrochloride in a flow system. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 34 Most of the literature data on the chemical equilibrium constant of the hydration of formaldehyde, K h , were obtained from the carbonyl specific UV-absorption at approximately 290 nm (Bieber & Trümpler, 1947; Iliceto, 1954; Landqvist, 1955; Gruen & McTigue, 1963; Siling & Akselrod, 1968; Schecker & Schulz, 1969; Zavitsas, Coffiner, Wiseman & Zavitsas, 1970). At this wavelength an electron from a non-bonding oxygen n-orbital is promoted to an anti-bonding π*-orbital of the carbonyl double bond (e.g. Calvert & Pitts, 1966). From their measurements mentioned above, Sutton and Downes (1972) could calculate also K h at 295 K. Valenta (1960) used oscillographic polarography with pulses of short duration. Then, unhydrated formaldehyde is the only reducible species and its concentration could be obtained. Finally, Bryant and Thompson (1971) derived an expression for K h from a consistent set of, partly experimental, thermochemical data. The values of K h obtained by the various authors show a considerable spreading; differences of more than a factor 3 are found. The reaction enthalpy for the hydration obtained from the sources mentioned varies from -21.4 to -39.4 kJ mol -1 . In previous work, h K was usually determined from the concentrations of free formaldehyde and methylene glycol in aqueous solutions, i.e. from the equilibrium value of F MG C C / . We think that much of the spreading of the reported h K data in the literature can be explained from the, occasionally unrecognized, complications in establishing these concentrations. In aqueous solutions, the concentration of unhydrated formaldehyde is very low because the equilibrium (1) is far to the right. In addition, the molar extinction coefficient of formaldehyde in UV-absorption is small and its value for aqueous solutions is not known. In more concentrated solutions (say above 1 M) the amount of unhydrated formaldehyde is higher, but the then the polymerization reactions (2) become significant and the methylene glycol concentration is not known anymore. Also, with more concentrated solutions, UV-absorption measurements are hampered by substantial nonspecific absorbance. Finally, commercial formaldehyde solutions often contain considerable amounts of methanol for stabilization; a fact that was not always recognized in older literature. In this contribution, we report on the reaction rate of the hydration of formaldehyde, characterized by the rate constant k h , at conditions prevailing in industrial formaldehyde absorbers, i.e. at temperatures of 293-333 K and at pH values between 5 and 7. Using these results, and the dehydration reaction rate constant, k d , obtained previously at similar conditions (Winkelman, Ottens and Beenackers, 2000), the chemical equilibrium constant for the hydration is obtained as d h h k k K / = . The h K data obtained this way, as the ratio of measured reaction rates rather than as the ratio of concentrations, are believed to be more reliable for the reasons mentioned in the section above. The experimental technique for obtaining the hydration rate is based on the measurement of the chemically enhanced absorption rate of formaldehyde gas into water. In general, chemical enhancement occurs if the absorption of a gaseous component into a liquid is accompanied by a chemical reaction of that component in the liquid. If the reaction is fast enough, then the concentration of the component is reduced in the liquid already near the gas-liquid interface. This results in a larger gradient, and thus a larger flux, of the component, when compared to the gradient and flux without any reaction. The phenomenon is characterized by the so-called enhancement factor, i E , which is defined as Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 35 reaction without ) ( reaction with ) ( 0 0 = = ≡ x i x i i J J E , (3) where both fluxes at the gas-liquid interface, 0 ) ( = x i J , are based on the same concentration difference over the liquid film ) ( , , , l l i IF i C C − (see e.g. chapter VII in Westerterp, Van Swaaij & Beenackers, 1984). Formaldehyde absorption experiments are carried out in a stirred cell reactor. The water in the reactor is stirred with a constant, but limited intensity, in such a way that the liquid surface always remains flat, and the value of the gas-liquid interfacial area remains accurately known. This way, the formaldehyde gas to liquid molar flux can be obtained from observed mass transfer rate, which, in turn, allows the calculation of the liquid phase chemical enhancement factor for formaldehyde mass transfer and the reaction rate of formaldehyde hydration, respectively, from mathematical modeling of the transfer processes. The calculation of the desired hydration reaction rate from the observed mass transfer rate requires accurate knowledge of both the gas and liquid phase mass transfer coefficients. Therefore, prior to the kinetic measurements, the mass transfer coefficients prevailing in the stirred cell reactor were accurately measured. Experimental The experimental set-up for the kinetic measurements was build around a stirred cell reactor, see Fig. 1. The double walled glass reactor, 0.08 m diameter, 0.1 m height, was equipped with 4 baffles. Stirring was provided by a 0.046 m 8-bladed turbine stirrer in the gas phase, and a 0.050 m flat blade stirrer in the liquid phase on the same shaft, driven via a magnetic coupling. The liquid surface remained perfectly flat for stirring rates of up to 22 Hz. Beyond this limit, some rippling and heightening of water against the baffles was observed. Nitrogen was passed through a saturation column, approximately 1 m in height, filled with a 32 wt.% aqueous formaldehyde solution, and flowed either via the head space of the reactor, or directly via a by- pass, to the analysis unit. The signals of the temperature control units, pressure measurement, flow measurement and stirring rate were stored into a computer. The composition of the solution in the saturation column was accurately determined using the sodium sulphite method (Walker, 1964). The formaldehyde concentration of the gas stream entering and leaving the reactor were measured with UV-spectrophotometry using the carbonyl specific absorption around 295 nm. For this purpose, a special measuring unit was constructed by Macam Photometrics Ltd.. UV-light was obtained from a low pressure deuterium discharge lamp with a stabilised power supply and reference intensity measurement for stable radiation output. Further, the unit was provided with collimating and alignment optics, a 2 nm band width monochromator, a special long-path measuring cuvet with temperature control, and a photomultiplier tube. The measuring cuvet, with a volume of only 40×10 -6 m 3 , was equipped with two flat mirrors and a concave mirror, in such a way that a total optical path length of 1 m was achieved by 12 passes of the light through the cuvet, see Fig. 2. This way, the low extinction coefficient of the absorption band of gaseous formaldehyde was compensated for, and accurate measurements could be made. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 36 1 2 3 4 Fig. 1. Experimental set-up, 1: gas supply, 2: saturation column, 3: stirred cell, 4: to analysis unit. In a typical kinetic experiment, the reactor was filled with the desired amount of distilled water, and the system, including the saturation column, the UV absorption measuring cuvet, and all connections, was allowed to equilibrate to the desired temperature. Next, the nitrogen flow to the saturation column was set to the desired rate, and passed directly to the analysis unit, via the by-pass, to measure the reactor inlet formaldehyde concentration. The stirrer was switched to the desired rate, and the flow was led through the reactor until the achievement of a pseudo steady state, as indicated by a constant UV-absorption reading from the spectrophotometer, marked the end of the experiment after a few minutes. This way, with an experiment, the hydration was measured at a single pseudo steady state point. to photo- mutiplier from mono- chromator Fig. 2. Measuring cuvet for UV-absorption and optical path of the light. For the liquid phase mass transfer measurements, the gas supply was switched to CO 2 , and a vacuum pump was connected to the gas phase reactor outlet. Before an experiment, the water was degassed for 15 minutes by lowering the pressure to just above the water vapour pressure at the temperature employed. Next the pressure was lowered even further, and the water allowed to boil Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 37 for a short time, to drive out any gasses remaining. An experiment was started by pressurising the reactor with CO 2 within a few seconds to approximately 0.12 MPa, closing the inlet valve, and recording the pressure decrease at a rate of exactly 1 Hz until the pressure decrease had diminished. Gas phase mass transfer coefficients were obtained by measuring the evaporation rates of pure liquids into an inert gas stream. For this purpose, a continuous N 2 , CO 2 or He gas supply was used, while the gas phase reactor outlet was now connected to 3 cold traps in series, cooled with liquid nitrogen, or, when CO 2 gas was used, a mixture of water, ice and CaCl 2 at approximately 258 K. A fourth trap, filled with CaCl 2 , prevented any moisture from the environment to enter. The amount of evaporated liquid was obtained from the weight increase of the cold traps. The following combinations of gases and pure liquids were used: N 2 with water, octane and ethanol; CO 2 with water; He with butyl ether, acetone, anhydrous proprionic acid and butanol. Results Liquid phase mass transfer coefficients The liquid phase mass transfer coefficients were determined by monitoring the pressure drop during the absorption of CO 2 in water, whilst operating both gas and liquid phases in batch mode. The variation of the CO2 partial pressure with time, obtained from molar balances for CO 2 over both phases, is given by l l l l l V t V V m S k V m V P P V m V g CO CO g CO CO CO g ) 1 ( ) 1 ( ln 2 2 2 2 2 0 + − = − + . (4) where the distribution coefficient, 2 CO m , was taken from Versteeg and Van Swaaij (1988). Pressure readings were taken every second, for a total monitoring time varying from about half an hour to one hour. The individual experiments could be described by equation (4) very well: the mean absolute relative residual (MARR) of the calculated and observed pressures never exceeded 0.1%. The experimental l k -values were correlated using dimensionless groups, resulting in % . Sc Re . Sh . . 5 4 178 0 36 0 58 0 ± = l l l , (5) where a 4-fold variation of Sc and a 22-fold variation of Re were achieved by measuring at different temperatures and stirring rates, see Fig. 3. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 38 10 4 10 5 Re L 10 100 S h L / S c L 0 . 3 6 Fig. 3. Liquid phase mass transfer coefficients. Symbols: experimental data. Line: Eq. (5). Gas phase mass transfer coefficients The gas phase mass transfer coefficients were obtained from the rate of evaporation of pure liquids into carrier gases. This way, any liquid side resistance against mass transfer was eliminated. The g k values were calculated by solving simultaneously the balance equations for the vapour content of the carrier gas stream, tot i i in g v i sat i i g P P P p p S k / 1 ) ( , , − = − φ (6) and the weight of condensed vapour accumulated in the cold traps, RT P P P t M W tot i i in g v i ) / 1 ( , − = φ (7) for i g k , and i P . No influence of the gas flow rate on g k was observed for any of the components. The g k data were also correlated via dimensionless groups: % Sc Re . Sh . g . g g 11 163 0 50 0 70 0 ± = . (8) By using eight combinations of carrier gases and liquids, and various stirring rates, a 23-fold variation of Sc and a 50-fold variation of Re were achieved, see Fig. 4. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 39 10 2 10 3 10 4 100 10 1 S h / S c g g 0 . 5 0 Re g N N N CO He He He He 2 2 2 2 - water - octane - ethanol - water - butyl ether - acetone - prop. anhydr. - butanol Fig. 4. Gas phase mass transfer coefficients. Symbols: experimental data. Line: Eq. (8). UV-analysis of formaldehyde The formaldehyde concentration in the gas stream was obtained spectrophotometrically from the intensity of the characteristic absorption band of the carbonyl group at a wavelength of 295 nm. Simple aldehydes, p.e. acetaldehyde, propionaldehyde and butyraldehydes, show nearly continuous absorption spectra in this wavelength region, and obey Beer's law (Calvert & Pitts, 1966; Müller & Schurath, 1983) C I I l ε = ) / ln( 0 . (9) However, because of its simple structure, the absorption band of formaldehyde shows considerably more vibrational and rotational structure, and has a discrete line structure. Because the spectral line widths are narrower than the resolution of 2 nm used in the measurements, the absorbance varies nonlinear with the formaldehyde concentration. This has been reported previously by Moortgat, Seiler and Warneck (1983), Müller and Schurath (1983) and Rogers (1990). Müller and Schurath (1983) measured the absorption of gaseous formaldehyde in a 2.48 m cell in the concentration range of 0-0.12 mol m -3 . They correlated their results according to ) 702 . 2 186 . 2 ( ) / ln( 0 C C I I − = l . This correlation is illustrated in Fig. 5, together with our measured data, showing a good agreement. The gas phase formaldehyde concentrations of Fig. 5 are the ones in the gas stream entering the reactor, and are calculated from the composition of the solution in the saturation column, using the vapour-liquid equilibrium model of developed by the research group of Maurer (Maurer, 1986; Albert, García, Kuhnert, Peschla, & Maurer, 2000), and assuming that the gas leaving the saturation column has reached physical equilibrium with the liquid. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 40 0.0 0.05 0.10 0.15 0.20 0.4 0.3 0.2 0.1 0.0 C F,g [mol/m ] 3 l n ( I / I ) 0 Fig. 5. Absorbance vs. gaseous formaldehyde concentration. Symbols: this work, horizontal bars: standard deviation. Line: Measured by Müller and Schurath (1983) at 12 . 0 0 , − = g F C mol/m 3 and extrapolated to 0.2 mol/m 3 . The absorbances vs. the formaldehyde gas concentrations for the entire concentration range employed here are shown in Fig. 6. They could be correlated with an expression similar to the one obtained by Müller and Schurath, and which is also shown in Fig. 6: ) 488 . 0 652 . 1 ( ) / ln( 0 C C I I − = l (10) 0.0 0.2 0.4 0.6 0.8 C F,g [mol/m ] 3 l n ( I / I ) 0 1.2 0.8 0.4 0.0 Fig. 6. Absorbance vs. gaseous formaldehyde concentration. Symbols: this work, horizontal bars: standard deviation. Line: Eq. (10). Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 41 Kinetic measurements The gaseous phase above an aqueous formaldehyde solution contains formaldehyde gas, water vapour, and also methylene glycol vapour (Maurer, 1986). Therefore, the gas stream entering the stirred cell from the saturation column will contain these three components, and the absorption of formaldehyde in the reactor is accompanied by absorption of methylene glycol. Since the vapour pressure of water above an aqueous formaldehyde solution is lower than above pure water at the same temperature, some evaporation of water will also occur ) ( 2 ) ( 2 l O CH O CH g → , (11) ) ( 2 2 ) ( 2 2 ) ( ) ( l OH CH OH CH g → , (12) ) ( 2 ) ( 2 g O H O H → l . (13) Once absorbed, formaldehyde will be hydrated according to the reversible reaction (1). Since water is present in large excess, the hydration reaction can be characterised by a first-order rate constant, k h (Bell, 1966): ) ( , , h MG F h F K C C k R l l − = (14) The reaction rates of the polyoxymethylene glycol formation reactions, Eq. (2), are very low. Furthermore, if the overall concentration of formaldehyde is low (say below 1 wt.%), the equilibrium constants of these reactions do not favour the formation of polyoxymethylene glycols and the solution will contain formaldehyde and methylene glycol. Therefore, any polyoxymethylene glycol formation is neglected in this work. The experiments are evaluated using the two-film model. This model for mass transfer is based on the assumption that near the interface in the liquid phase a stagnant film exists, of thickness l l l k D / = δ , where any transport of the components is by diffusion only. Similarly, in the gas phase a film is present at the interface of a thickness g g g k D / = δ . Using the film model for mass transfer, the following equations describe the processes in the stirred cell ) , , , ( ) ( 2 , , , , 0 N W MG F i C C S J g i g v in g i in g v x i = − = = φ φ , (15) ) , , ( ) / ( ) ( , , , , 0 W MG F i m C C k J i IF i g i i g x i = − = = l , (16) ) , ( ) ( ) ( , , , , 0 MG F i C C E k J i IF i i i x i = − = = l l l . (17) For the general case of absorption and/or desorption of two gases accompanied by a first-order reversible reaction in the liquid, Winkelman and Beenackers (1993) derived analytical solutions for the enhancement factors of the components involved. Translated to the notation used here, their equation for the enhancement factor of methylene glycol reads Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 42 l l l l , , , , , , ) 1 ( 1 MG IF MG F IF F F MG C C C C E v E − − − + = . (18) With the measured g F C , , and with 0 2 = N J and l l , , , W IF W C C = , the set of Eqs (15)-(18) can be solved for the other gasphase concentrations, gas flow rate leaving the cell, interface concentrations, and mass transfer enhancement factors. In the calculations, it was assumed that the formaldehyde and methylene glycol concentrations in the bulk of the liquid are negligible compared to the interface concentrations, see the addendum at the end of this chapter. The film thickness was obtained from Eq (5) using l l l k D / = δ , and varied in the experiments from approximately 50 to 130 µm. The observed reactor outlet gas phase formaldehyde concentrations of the individual experiments increased with increasing gas flow rates, and decreased with increasing stirring rates and temperatures. In Fig. 7 the ratio of the reactor outlet and inlet formaldehyde concentrations is plotted versus the quantity ) /( ) ( 3 , T N in g v φ , which was chosen intuitively for illustrative reasons only, to reduce the scattering of the data in the plot. 0.8 0.6 0.4 0.2 0.0 ( ) /(NT) φ v,g in 3 10 10 10 -19 -18 -17 C / C F , g F , g i n Fig. 7. Ratio of reactor outlet and inlet gas phase formaldehyde concentrations obtained experimentally. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 43 The enhancement factor of formaldehyde calculated from the experimental data, as described in the previous paragraph, can be equated to the one obtained from the differential equations for diffusion with parallel reaction in the liquid according to the film model ) 0 ( ) ( 2 2 , l l δ ≤ ≤ − = x K C C k dx C d D h MG F h F F , (19) ) 0 ( ) ( 2 2 , l l δ ≤ ≤ − − = x K C C k dx C d D h MG F h MG MG . (20) An analytical solution of (19)-(20) for the E F , with the assumption of negligible liquid bulk phase concentrations, reads (Winkelman & Beenackers, 1993) v K C C K E h IF F IF MG h R R F + − − + = ) )( 1 ] tanh[ ( 1 , , , , l l φ φ , (21) where the reaction factor, R φ , is defined by h F h h R K D v K k ) ( + = l δ φ . (22) With the values of E F obtained from Eqs (15)-(18), the reaction rate constants, k h , were calculated numerically from (21)-(22), where the equilibrium constant K h was written as K h = k h /k d , and k d was taken from Winkelman, Ottens and Beenackers (2000). The values of R φ obtained numerically ranged from 3.7 to 11.4. Finally, regression of the reaction rate constants with the reciprocal temperature gave 1 5 ) 2936 exp( 10 04 . 2 − − × × = s T k h . (23) The mean absolute relative deviation between the data from the individual experiments and from Eq. (23) is 9.6%, see Fig. 8. From the temperature coefficient the activation energy was found as E a = (24.4 ± 2.7) kJ mol -1 . Regression of the results according to the transition-state theory resulted in ∆H ≠ =(21.8 ± 2.7) kJ mol -1 and ∆S ≠ =(-152.0 ± 9.5) J mol -1 K -1 . The enhancement factors obtained from Eqs (21)-(23) are plotted against the ones obtained from the experimental data and Eqs (15)-(18) in Fig. 9. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 44 2.8 3.0 3.2 3.4 3.6 10 30 50 5 3 1000/T k h [ 1 / s ] Fig. 8. Arrhenius plot of the formaldehyde hydration rate constant k h . Symbols: ●: this work, mean of 8 to 12 experiments, horizontal bars: standard deviation; □: Sutton & Downes (1972). Solid Line: this work, Eq. (23). Dashed line: Schecker & Schulz (1969). 0 5 10 15 (E ) F observed ( E ) F c a l c u l a t e d 15 10 5 0 Fig. 9. Parity plot of the formaldehyde enhancement factors. Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 45 With the hydration rate according to Eq. (23), and the dehydration rate obtained previously (Winkelman, Ottens and Beenackers 2000) the equilibrium constant for the hydration of formaldehyde can be obtained as ) 494 . 5 3769 exp( − = = T k k K d h h . (24) Equation (24) is illustrated in Fig. 10. From the temperature coefficient in Eq. (24) the reaction enthalpy of the hydration was obtained as ∆H = -31.4 kJ mol -1 . The experimental data obtained here can also be used to actually establish the reaction order of formaldehyde in the hydration reaction, see Appendix C. It turns out that the reaction is indeed first order in formaldehyde under the experimental conditions applied. 2.9 3.0 3.1 3.2 3.3 3.4 3.5 1000/T K h 1000 500 100 300 3000 1 2 3 4 5 Fig. 10. Van 't Hoff plot of the formaldehyde hydration chemical equilibrium constant. Solid line: this work, Eq. (24). Symbols: ∇: Landqvist (1955); ∆: Bieber & Trümpler (1947); □: Valenta (1960); Ο: Iliceto (1954); ◊: Sutton & Downes (1972). Dashed lines: 1: Schecker & Schulz (1969); 2: Zavitsas et al. (1970); 3: Gruen &McTigue (1963); 4: Bryant & Thompson (1971); 5: Siling & Akselrod (1968). Discussion The relation for the liquid phase mass transfer coefficients, Eq. (5) is of a form often encountered in the literature. Usually, the exponent of l Sc is taken as 1/3 based on theoretical reasons (Westerterp, Van Swaaij & Beenackers, 1984). The exponent of 0.36, obtained from fitting the data, is in good agreement with this theoretical value. The value of the exponent of l Re obtained here, 0.58, is at the lower end of the wide range encountered in the literature for Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 46 this type of reactor: from 0.5 to 1.0. This may be attributed to the specific design of our liquid phase stirrer. Data on gas-side mass transfer coefficients in stirred cell reactors are more scarce. Tamir, Merchuk and Virkar (1979) and Yadav and Sharma (1979) investigated the influence of the diffusivity on g k in stirred cell reactors. For n g i g D k , ∝ they obtained exponents n = 0.632 and n = 0.487-0.518, respectively. Our exponent of 0.5 of g Sc is in good agreement with these data. The reaction rate constants of this work are in reasonable agreement with the results of Schecker and Schulz (1969), see Fig. 8. However, the activation energy of the hydration, obtained here as 24.4 kJ mol -1 , is substantially higher compared to the value of 15.9 kJ mol -1 obtained by Schecker and Schultz. The agreement of Eq. (22) with the value established by Sutton and Downes (1972) is excellent. The chemical equilibrium constant for the hydration of formaldehyde, obtained here as Eq. (24), appears to be within the range of data and relations found in the literature, see Fig. 10. A thorough comparison however is not possible due to considerable spreading of the literature data as mentioned before. Also, the heat of reaction, ∆H = -31.4 kJ mol -1 , appears to be in the wide range data, from -21.4 to -39.4 kJ mol -1 , encountered in the literature. Physical properties Pure component properties were taken from Daubert and Danner (1985), or predicted using the methods given by Reid, Prausnitz and Poling (1988). The distribution coefficient and diffusivity of CO 2 in water were taken from Versteeg and Van Swaaij (1988). Mixture properties were calculated using the mixing rules given by Reid et al. (1988). The density and viscosity of aqueous formaldehyde solutions were taken from Winkelman and Beenackers (2000). Diffusion coefficients in water were calculated with the Wilke-Chang method (Reid et al., 1988), where the volume of formaldehyde at its normal boiling point was taken form Daubert and Danner (1985) and that of methylene glycol was obtained with the Le Bas method (Reid, et al., 1988). The distribution coefficients m F , m MG and m W were calculated with the model of Maurer for vapour- liquid equilibria of aqueous formaldehyde solutions (Maurer, 1986; Albert, García, Kuhnert, Peschla, and Maurer, 2000). Conclusions The reaction rate of the hydration of formaldehyde is obtained from measuring the chemically enhanced absorption of formaldehyde gas into water in a stirred cell with a plane gas liquid interface, and mathematically modelling of the transfer processes. At the conditions prevailing in industrial formaldehyde absorbers, i.e. at temperatures of 293-333 K and at pH values between 5 and 7, the rate is found as k h = 2.04×10 5 ×e -2936/T s -1 . Using these results, and the dehydration reaction rate constant, k d , obtained previously at similar conditions, the chemical equilibrium constant for the hydration is obtained as K h = e 3769/T-5.494 . Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 47 Addendum In the calculation of the hydration rate constants, the bulk liquid concentrations of formaldehyde and methylene glycol were neglected. In this addendum, we take a closer look at the influence of this assumption by modelling the system without neglecting the bulk liquid concentrations, and comparing the results to the ones previously obtained. Model equations During a kinetic experiment, gas flow continuously through the headspace of the stirred cell, while the liquid is in batch mode. The transient component balances over the headspace and the liquid bulk read ) , , ( ) ( 0 , , , , , W MG F i J S C C dt C d V x i g i g v in g i in g v g i g = − − = = φ φ (A-1) ) , ( ) ( , MG F i J S R V n dt C d V x i F i i = + = =δ l l l (A-2) with the initial conditions . , , , , , , ; 0 : 0 sat g W g W MG F g MG g F C C C C C C t = = = = = = l l . (A-3) In Eqs (A-1) and (A-2), x denotes the distance into the liquid, thus 0 ) ( = x J denotes the flux of a component at the gas-liquid interface and δ = x J) ( denotes the flux into the liquid bulk at the interface of the liquid film and bulk, F R is the rate of the hydration reaction in the liquid bulk, see Eq (14), and the stoichiometric coefficients, i n , are given by 1 − = F n and 1 = MG n . The rate constant h k was determined for each experiment by adjusting it until g F C , , obtained by integration of Eqs (A-1)-(A3) over the experimental run time, was equal to its measured value. The Fourier times for diffusion in the gas and liquid film (typically of the order of 0.1 s) are much smaller than the time scale at which variation of the bulk concentrations take place (the gas phase residence time was of the order of 10 to 20 s). Therefore, during the integration, the fluxes of formaldehyde and methylene glycol at the interface are calculated by solving simultaneously Eqs (16)-(18) and the expression for the enhancement factor of formaldehyde, which reads in this case (Winkelman and Beenackers, 1993): R R h R F IF F MG F h F IF F MG IF MG h R R F v K C C C C K C C C C K E φ φ φ φ φ ] tanh[ ) ( ] cosh[ 1 1 ] tanh[ 1 1 , , , , , , , , , , , + − − − + − − − − + = l l l l l l l l (A-4) Chapter 4: Kinetics and chemical equilibrium of the hydration of formaldehyde 48 where R φ is given by Eq (22). The fluxes from the liquid film into the liquid bulk are obtained from the analytical expressions for the concentrations in the film (Winkelman and Beenackers, 1993) by differentiation according to = =δ x i J ) ( δ = − x i i dx dC D ) / ( ,l , giving ) ] cosh[ 1 1 ( ] tanh[ ) ( ) ( ) ( ) ( ) ( , , , , , , , 0 R R R h MG IF MG F IF F h F x F x F v K C C C C K k J J φ φ φ δ − + + − + − = = = l l l l l (A-5) and ] [ ) ( ) ( , , , , , , , v C C C C k J J MG IF MG F IF F F x F x MG l l l l l − + − + − = = = δ δ (A-6) Results The transient model described above was solved numerically for h k for each experiment. When compared to the h k data obtained Eqs (15)-(18) and (21), the differences were small. On average, the rates obtained here were 0.48 % larger, with a maximum of 1.02 %. Since this is well within the estimated experimental uncertainty, we conclude that the influence of the bulk concentration of formaldehyde and methylene glycol can safely be neglected in the evaluation of the experiments. Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 49 Chapter 5 Modelling and simulation of industrial formaldehyde absorbers Abstract The industrially important process of formaldehyde absorption in water constitutes a case of multicomponent mass transfer with multiple reactions and considerable heat effects. A stable solution algorithm is developed to simulate the performance of industrial absorbers for this process using a differential model. Good agreement with practice was achieved. Using the model, the conditions of one of the absorbers of Dynea B.V. were optimized, leading to considerable methanol savings. Introduction Formaldehyde, CH 2 O, is produced on a large scale as a raw material for a great variety of end products. Its industrial production starts from methanol. Air and vaporized methanol, combined with steam and recycled gas, are passed over hot silver grains, at ambient pressure. Here the methanol is converted to formaldehyde by partial oxidation and by reduction at about 870 K. To prevent thermal decomposition of formaldehyde, the gases are cooled immediately after the catalyst bed. The reactor product gas stream has a temperature of 420 K and consists typically of N 2 (50%), H 2 (15%), water vapor (20%) and formaldehyde (15%). Minor amounts of by-products and unreacted methanol are neglected in this study. This stream is passed through a partial condenser, where the temperature is reduced to 328 K and part of the water vapor and formaldehyde are condensed. The resulting mixed gas-liquid stream is subsequently fed to the absorber. Because it is impossible to handle in its pure, gaseous form, formaldehyde is almost exclusively produced and processed as an aqueous solution: formalin. The latter is obtained commercially by absorbing the gases leaving the reactor in water. The goals in optimizing the absorber performance seem conflicting. On the one hand the formaldehyde content of the tail gas should be minimized. On the other hand however, the formaldehyde concentration in the product liquid leaving the absorber should be as high as possible, thereby reducing the absorbing ability of the liquid in the column. A scheme of the absorber studied is shown in Fig. 1. The gaseous part of the two-phase stream entering at the bottom of the column passes upwards through two beds, randomly filled with modern high performance Pall-ring like packing. The tail gas is partly recirculated to the reactor. Make up water enters at the top of the column and flows downward, meanwhile taking up heat and absorbing formaldehyde and water from the gas stream. Each of the absorption beds has an external liquid recirculation with heat exchangers. Just below the top bed, the absorber is equipped with a partial draw off tray to provide a buffer for the upper liquid recirculation pump. At the bottom of the column a liquid layer is kept as a buffer for the lower liquid reciculation pump. Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 50 feed product tail gas water Fig. 1. Scheme of a formaldehyde absorber. Reactions in aqueous formaldehyde solutions Besides direct heat transfer between gas and liquid, and the mass transfer of water and formaldehyde, a number of reactions have to be considered in modeling the performance of formaldehyde absorbers. In formalin the dissolved formaldehyde is present principally in the form of methylene glycol, CH 2 (OH) 2 , and a series of low molecular weight polyoxymethylene glycols, HO(CH 2 O) n H (e.g. Walker, 1964). As an example the equilibrium composition of an aqueous 50 wt.% formaldehyde solution is shown in Fig. 2. Although the concentration of the unhydrated monomeric formaldehyde is well under 0.1% even in concentrated solutions, all dissolved aldehyde remains available for chemical reaction in downstream processing because of the reversibility of the reactions. The following reactions take place in the absorber: hydration 2 2 2 2 ) OH ( CH O H O CH + (1) with kinetics Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 51 ) / ( 1 WF W F 1 h h K C C C k r − = (2) were eq W F WF h C C C K = 1 (3) polymerization reactions O H H O) HO(CH H O) HO(CH (OH) CH 2 n 2 1 n 2 2 2 + + − ) 2 ( max n n ≤ ≤ (4) with kinetics ) / ( 1 1 n W WF WF WF n n K C C C C k r n n − = − (5) were eq WF WF W WF n n n C C C C K = −1 1 . (6) Here, n max denotes the largest polymer considered. The concentration of the polyoxymethylene glycols decreases rapidly with increasing molecular weight, even for concentrated solutions (Fig. 2). Therefore, the largest polymer considered here is WF 10 (n max = 10). This way, there are ten reactions in the liquid between twelve components. The reaction rate of a species can be represented as ∑ = = max 1 , n j j j i i r R ν . (7) From eq (7) and the stoichiometry of reactions (1) and (4) it follows 1 r R F − = (8) max ... 3 2 1 n W r r r r R + + + + − = (9) max 1 ... 2 4 3 2 1 n WF r r r r r R − − − − − = (10) ) 2 ( max 1 n n r r R n n WF n < ≤ − = + (11) ) ( max n n r R n WF n = = . (12) Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 52 1 10 10 10 10 -1 -2 -3 -4 F W 1 2 3 4 5 6 7 8 9 10 WF n x Fig. 2. Equilibrium molar fractions of F, W, and WF n (n=1..10) in 50 wt% formalin at 343 K. Development of absorber model The methods found in the literature for the modeling of packed columns generally belong to either of two types. In the first type the packed height is divided into a number of segments. Within each segment the conditions are supposed to be uniform in both phases. This type of models can be subdivided into equilibrium stage models (p.e. King, 1980), where the streams leaving each stage are assumed to be in equilibrium with each other and departures from this assumption are accounted for by one of several types of stage efficiencies, and nonequilibrium stage models (p.e. Krishnamurty & Taylor, 1985), where material and energy balance relations for each phase are solved simultaneously with mass and energy transfer rate equations. In the second type of models, differential mass and energy balances for each phase are written for a small section of packing, and the differential equations are numerically integrated over the total packed height (p.e. Hitch et al., 1986). Our model for the formaldehyde absorber belongs to the second type. In a subsequent paper a stage model will be introduced for this type of column. Since our first goal is the simulation of existing industrial formaldehyde absorbers, the height of the packing in the absorption beds is fixed. Major assumptions of the model are: (1) the absorption beds operate adiabatic; (2) the packing is fully wetted, therefore the interfacial area is the same for heat and mass transfer; (3) heat and mass transfer relations are based on the resistances in series model; (4) the counter current gas and liquid streams in the absorption beds are in plug flow; (5) gas to liquid mass transfer of N 2 and H 2 is negligible because of their low solubility in formalin; (6) the liquid in the partial draw off tray and at the bottom of the column (the buffers for the recirculation pumps) is ideally mixed and heat and mass transfer to this liquid is negligible. With these assumptions, the component balances for the gas and liquid phases read Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 53 ) , ( , W F i aS J dz dv g i i = − = (13) ) .. 1 ; , , ( max , n n WF W F i S R aS J dz dl n i i i = = − − = l l ε . (14) The energy balances for the gas and liquid phases are aS q dz dT Cp v g g i g i i − = ∑ ) ( , (15) ∑ ∑ ∆ − − − = j j R j i i i H r S aS q dz dT Cp l ) ( ) ( , l l l l ε . (16) The component and energy balances for the liquid on the partial draw off tray and at the bottom of the column read i B in i out i R V l l + = (17) ∑ ∑ ∆ − = − j j R j B i i in i in out H r V Cp l T T ) ( ) ( ,l . (18) Mass transfer rates The gas phase mass fluxes are calculated from ) , ( ) ( , , , , W F i C C k J I g i g i i g g i = − = (19) were the interfacial concentrations are coupled by ) , ( , , W F i C m C I g i i I i = = l . (20) The fluxes on both sides of the interface are equal: I i g i J J l , , = . (21) In the liquid, the diffusional transport is accompanied by chemical reactions, which causes mass transfer enhancement. Therefore the enhancement factor, E i , is incorporated in the fluxes at the liquid side of the interface ) ( , , , , l l l l i I i i i I i C C E k J − = . (22) Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 54 Also, the fluxes into the liquid bulk, l , i J , become different from the fluxes at the interface, I i J l , . In a preliminary study this situation was assessed by solving the differential equations for mass transfer with reactions in the film ) .. 1 ; , , ( max 2 2 , n n WF W F i R dx C d D n i i i = = − = l . (23) Assuming the polyoxymethylene glycols are nonvolatile, the boundary conditions of eq (23) are 0 , , : 0 , , , , = − = − = = dx dC D J dx dC D J dx dC x n WF W I W W F I F F l l l l (24) l l , : i i C C x = = δ . (25) The eqs (19)-(25) were solved numerically for a variety of conditions expected to prevail in formaldehyde absorbers, using a fourth order Runge-Kutta method in combination with a shooting method. The results showed that the polymerization reactions (4) are by far too slow to have any influence on the diffusion fluxes, and the gradients of the concentrations of the higher polyoxymethylene glycols in the film are negligible, ) .. 2 ( max , ) 0 ( n n C C n n WF x WF = = ≤ ≤ l l δ . (26) On the other hand, the hydration reaction (1) causes significant mass transfer enhancement. In column simulations it is not convenient to calculate the fluxes from the numerical integration procedure described above. Therefore, an approximate method was developed, similar to the method of Onda et al. (1970). The differential equations (23) for the formaldehyde and methylene glycol concentrations in the film are linearized by taking the water concentration in the film equal to that in the bulk. The resulting set of equations can be solved analytically to give expressions for E F and l , F J R R W h F WF R F I F WF F W h F WF W F WF F C K D D C C C C C K D D C K D D E φ φ φ ] tanh[ 1 ) ] cosh[ 1 1 ( 1 , , , , , , , , , , , 1 , , 1 1 1 1 l l l l l l l l l l l l l + − − − + + = (27) [ ] I F F I F F WF I F W h WF WF F W h WF R R W h F WF R I F F J C C k C C C K k C C C K k C K D D J J l l l l l l l l l l l l l l l l l , , , , , , , , , , , , , , , , , ) ( ) ( ) ( ] tanh[ ] cosh[ 1 1 1 1 1 1 1 − − + − + − − − = φ φ φ (28) Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 55 with ) ( , , , , 1 h WF F W F h R K D D C D k l l l l l + = δ φ . (29) Given the bulk concentrations in both phases, all the fluxes can now be calculated iteratively from eqs (19)-(22), (27)-(29) and the balances ) ( , , , , l l l l F I F I W W J J J J − − = (30) ) ( , , , 1 l l l F I F WF J J J − = . (31) The mass transfer rates calculated this way proved almost identical to the ones obtained by numerically solving eqs (19)-(25). Energy transfer rates The energy transfer rates contain a conductive and a convective part ) ( ) ( , , l or g j H J q E i j T j i j i j j = + = ∑ . (32) The film model of simultaneous mass and energy transfer leads to (p.e. Krishna and Taylor, 1986) ) ( ) ( l or g j T T A h q I j f j j = − = (33) where A f is the Ackermann heat transfer correction factor for non-zero mass transfer fluxes in phase j 1 − = f C f f e C A (34) ) ( , , l or g j h Cp J C j i j i j i f = = ∑ . (35) The interfacial temperature, I T , follows from a balance around the interfacial region, l E E g = : ∑ ∑ + = + i T i i i g T g i g i g H J q H J q ) ( , , ) ( , , l l l l . (36) Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 56 This balance can be rewritten as h R F I F i I i i i i vap I g g i g i g H J J T T Cp J H T T Cp J q q ) )( ( ) ( ] ) ( [ , , , , , , , ∆ − − + − + ∆ + − + = ∑ ∑ l l l l l l . (37) Physical and chemical parameters The mass transfer coefficients, g i k , and l , i k , and the specific area, a, were calculated from Onda et al. (1968). The partial liquid hold up, l ε , was taken from Otake and Kunigita (1958). The heat transfer coefficients, g h and l h , were evaluated from the mass transfer coefficients using the Chilton-Colburn analogy. To predict the heat and mass transfer coefficients, pure component and mixture physical properties are needed. The pure component properties were taken from Daubert and Danner (1985), or if not available, calculated using the predictive methods recommended by Reid et al. (1988). Densities and viscosities of the liquid were taken from Appendix A. Other mixture properties were calculated from the mixing rules recommended by Reid et al. (1988). The reaction rate and equilibrium constant of the hydration reaction, h k and h K , were taken from the results reported in Chapters 2 and 4 of this thesis. The reaction rate and the equilibrium constants of the polymerization reactions, k j and K j (j=2..n max ), were measured by Dankelman et al. (1988) over the temperature range of 293-353 K. Vapor liquid equilibria Generally, at low pressure, vapor phase non-ideality can be neglected, and vapor-liquid equilibria can be described with s i i i i P x P y γ = (38) where the liquid phase non-ideality is accounted for by the activity coefficient, i γ , which usually is temperature and composition dependent. The interpretation of published vapor-liquid equilibrium data for the formaldehyde-water system is complicated by the fact that the analytical methods, used by various authors, do not distinguish between different forms of formaldehyde present in the system, and the experimental results are presented in terms of overall compositions. To find vapor liquid equilibrium relationships for formaldehyde and water, experimental data were used of Kogan et al. (1977), Maurer (1986) and Hasse (1990). First the true molar fractions were calculated from the reported overall composition and the equilibrium relations (3) and (6). After this, for each experimental point, the quantity s i i P γ was calculated. Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 57 25 20 15 10 8 6 4 2 0 0 20 40 60 80 W [wt%] F γ F F P S [MPa] 293 K 313 K 323 K 343 K 353 K 363 K Fig. 3. Values of γ F P F S . Lines: calculated from eq (49). Symbols: experimental data, ( , ): Kogan (1977), ( , ): Maurer (1986), ( , ): Hasse (1990). Finally, s F F P γ and s W W P γ were smoothed as a function of T and liquid phase composition ] 10 9198 . 7 10 0723 . 3 10 3681 . 7 86 . 3146 4822 . 25 exp[ 2 5 2 5 F F F s F F W W T W T P − − − × − × + × − − = γ (39) ] 65 . 74 75 . 3140 0428 . 22 exp[ − − = T P s W W γ . (40) Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 58 100 5 0 0 20 40 60 80 W [wt%] F γ W W P S [kPa] 293 K 313 K 323 K 343 K 353 K 363 K 10 80 60 40 Fig. 5. Values of γ W P W S . Lines: calculated from eq (49). Symbols: experimental data, ( , ): Kogan (1977), ( , ): Maurer (1986), ( , ): Hasse (1990). Equations (39) and (40) are illustrated in Figs 3 and 4. The deviation of the calculated lines from the experimental points is generally within the experimental accuracy limits, the mean deviations are 3.4% for formaldehyde and 1.5% for water. The equilibrium ratio m i (eq 20) can now easily be calculated from s i i tot i P RT C m γ l , = . (41) Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 59 Computational method The eqs (13)-(16) form a set of 16 coupled non-linear differential equations which describe an absorption bed. If the entering gas and liquid streams are specified, a two point boundary value problem results. Several strategies for solving this type of problems for absorbers have been put forward. Treybal (1969) used a shooting method to model single solute systems. This method starts by assigning trial values to the outlet gas stream conditions, and calculating the outlet liquid conditions from overall balances. The numerical integration from bottom to top has to be repeated until convergency is achieved on the trial values. Feintuch and Treybal (1978) extended this model to multicomponent systems, but reported convergence problems due to equations becoming mathematical indeterminate in one of several nested iteration loops. Kelly et al. (1984) used the shooting method to model the physical absorption of acid gases in methanol. Another way to solve the set of equations is by dynamic simulation. Von Stockar and Wilke (1977), using a relaxation technique developed by Stilchmair (1972) and Bourne at al. (1974) for plate columns, could avoid convergence problems by simulating the packed column start-up and integrating the model equations with respect to time up to steady state. Hitch et al. (1987) also developed an unsteady-state solution algorithm. Besides the advantage of providing information on transient behaviour, they state that the relaxation method can handle complex systems without the convergence problems encountered with other methods. This type of method is highly stable because it follows the actual behaviour of start-up procedures. Srivastava and Joseph (1984) solved the two-point boundary value problem associated with packed separation columns by using orthogonal collocation for the spatial discretisation. Though the method worked well for the steady state calculation of a binary system, with multicomponent systems no steady state solution could be obtained directly. A dynamic simulation had to be carried out to find the steady state as the asymptote of the transient response. All methods mentioned above have been applied for single absorption beds, with specified entering gas and liquid streams. The formaldehyde absorber considered here, however, consists of two packed beds (Fig. 1). A further complicating factor, from the problem solving point of view, is the presence of the liquid recirculations around each packed bed. Figure 5 shows the streams that have to be considered in modeling the formaldehyde absorber. Specified are the input streams 1A, 7 and 10, and the temperature and total mass flow of the liquid recycle streams 5A and 5B. The actual entering streams of the packed beds are a priori unknown. Because of its superior convergency and stability characteristics, a transient approach was used to solve the equations that describe the absorber. Initially, the composition of the liquid in both packed beds, and of the liquid streams shown in Fig. 5, are all set equal to that of the make- up stream 1A (pure water). The liquid temperatures in and around each bed are initially set equal to the corresponding liquid recycle stream temperatures, which are input parameters. Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 60 tail gas 9 make up top bed 1A 2A 5A 3A 4A 8 7 10 2B 1B 3B 4B 5B 6 bottom bed gas feed liquid feed formalin gas stream liquid stream Fig. 5. Flow diagram of the absorber of Fig. 1. After initialization, a two-step cycle procedure is started. The first step starts with the specified conditions of the gas feed stream 7, to integrate the eqs (13) and (15) up the bottom bed. The resulting molar flows and temperature of stream 8 are used as the starting values to integrate eqs (13) and (15) up the top bed. During the integrations, the interphase fluxes are calculated using the stored liquid phase conditions. The calculated values of g T , v F and v W in the absorption beds are stored. In the second step, new liquid phase conditions are calculated. Starting with the specified conditions of stream 1A and the old values of stream 5A, the molar flows and temperature of stream 2A are calculated, and used as starting values for the integration of eqs (14) and (16) down the top bed. During this integration, the calculated liquid phase conditions are used in conjunction with the stored gas phase conditions to obtain the mass and heat fluxes. Next, eqs (17) and (18) are used to calculate stream 4A. The molar flows of stream 5A follow from the molar fractions, and the specified total mass flow rate of stream 5A. The temperature of 1B is set equal to that of 4A, and the molar flow rates follow from the differences between 4A and 5A. Now, the same sequence is repeated for the lower section of the formaldehyde absorber, to arrive at stream 6. The sequential update of gas and liquid phase conditions is repeated until the differences between successively calculated molar flow rates and temperatures have fallen below preset criteria, and steady state is reached, see Fig. 6. In practice 40 to 60 iterations were required, depending somewhat on the operating conditions. In contrast to the unsteady-state algorithms found in the literature, this algorithm does not need a separate routine for the calculation of time derivatives and advancement in the time direction. Furthermore, our algorithm contains only one level of nested iteration (which is the calculation of the interfacial conditions necessary to obtain the fluxes in the packed beds, and the calculation of the streams 4A and 4B from the non-linear algebraic eqs (17) and (18)). Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 61 start input: feed streams, mass flow rates initialize liquid phase conditions store gas phase conditions integrate gas phase equations up the packed beds using Runge-Kutta and temperature of liquid recycles integrate liquid phase equations down the packed beds and calculate liquid streams store liquid phase conditions convergence achieved? stop Fig. 6. Logic flow diagram for computing the absorber. Results A typical set of calculated temperature and vapor phase molar flow profiles is shown in Fig. 7. The presence of two packed absorption beds is clearly revealed in this figure. The occurence of temperature maxima, found here in the bottom bed, is often encountered in exothermic absorption with solvent evaporation (e.g. experimental observations of Raal and Khurana (1973) and Bourne et al. (1974), and calculations of Stockar and Wilke (1977) and Krishnamurty and Taylor (1986)). The liquid flowing down from the upper part of the bottom bed increases in temperature due to the heats of absorption and reaction. But lower in the bottom bed, the liquid meets an unsaturated gas flow, and the heat effect of solvent evaporation causes a drop in l T . Because l T T in g < the gas temperature rises while flowing upwards, until a maximum is reached. Above the maximum g T exceeds l T , with the former decreasing because the energy transfer is now directed towards the liquid. The features of the temperature profiles are reflected in the profile of the molar flow rate of water in the gas phase, see Fig. 7. The mass transfer of formaldehyde , on the other hand, is always directed towards the liquid and is not influenced much by the changes in temperature. Therefore, the gas phase molar flow rate of formaldehyde is monotonously decreasing. The profiles of the liquid phase molar flows are not very exciting due to the large liquid recycle ratio's used in formaldehyde absorbers. The influence of several operating parameters on the performance of the absorber was investigated. Important output parameters are the temperature rise of the liquid in each of the packed beds, l T ∆ , the relative amount of formaldehyde that leaves the absorber with Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 62 vapor liquid water formaldehyde v v in T [C] 70 60 50 40 2 1 0 0.0 0.5 1.0 (bottom) (top) relative height Fig. 7. Typical calculated profiles. the tail gas, in F out F v v / , and the composition of the liquid streams leaving each of the packed beds, calculated here as the overall weight percentage formaldehyde, W F . The influence of the recycle ratio's is shown in Figs 8 and 9. R is defined here as the ratio of the liquid mass flow rates of 5A and 1B for the top bed, and 5B and 6 for the bottom bed (see Fig. 5). Increasing the recycle ratio increases the absorber efficiency, i.e. reduces in F out F v v / . This is caused by a combination of more favourable hydrodynamics and lower mean temperatures. For example, increasing R b has virtually no influence on b T , l ∆ , but decreases the internal maximum of the temperature profile. Figures 10 and 11 show that increasing the temperature of a liquid recycle has a negative effect on the absorber efficiency. Increase of T t Rec causes a decrease of the amount of formaldehyde absorbed in the top bed, but an even larger decrease of the amount of water absorbed in this bed. This results in a reduction of the temperature rise of the liquid in the top bed, and a larger weight percentage formaldehyde in the liquid leaving both beds, see Fig. 10. The overall temperature rise of the liquid in the bottom bed with increasing T b Rec becomes even negative, see Fig. 11, due to the increasing water evaporation. This is of course a hypothetical situation, because it would mean that the heat exchanger in the recycle would have to heat up instead of cooling the liquid stream. Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 63 top-bed top-bed bottom-bed bottom-bed ∆T L W F [Wt%] v F out v F in W F [Wt%] 57 56 55 54 R t 40 60 80 100 43 41 39 37 16 12 8 4 0 0.08 0.06 0.04 0.02 0 Fig. 8. Influence of t R on l T ∆ , in F out F v v / and F W . top-bed top-bed bottom-bed bottom-bed ∆T L W F [Wt%] v F out v F in W F [Wt%] 57 56 55 54 R b 40 20 30 43 41 39 37 16 12 8 4 0 0.08 0.06 0.04 0.02 0 Fig. 9. Influence of b R on l T ∆ , in F out F v v / and F W . Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 64 top-bed top-bed bottom-bed bottom-bed ∆T L W F [Wt%] v F out v F in W F [Wt%] 57 56 55 54 T t Rec 40 25 30 43 41 39 37 10 8 6 4 2 0 0.08 0.06 0.04 0.02 0 35 45 Fig. 10. Influence of c t T Re on l T ∆ , in F out F v v / and F W . top-bed top-bed bottom-bed bottom-bed ∆T L W F [Wt%] v F out v F in W F [Wt%] 57 56 55 54 T b Rec 70 55 65 43 41 39 37 15 10 5 0 0.08 0.06 0.04 0.02 0 60 -5 Fig. 11. Influence of c b T Re on l T ∆ , in F out F v v / and F W . Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 65 top-bed top-bed bottom-bed bottom-bed ∆T L W F [Wt%] v F out v F in 60 50 40 30 20 10 3 0 2 10 8 6 4 2 0 0.08 0.06 0.04 0.02 0 1 make-up [m /hr] 3 Fig. 12. Influence of the amount of make-up water on l T ∆ , in F out F v v / and F W . Increasing the amount of make-up water has a positive influence on the absorber efficiency, see Fig. 12. This is caused by a reduction of the formaldehyde content of the liquid, which reduces the backpressure from the liquid phase. However, this reduction in W F is often undesirable because of extra storage and transport costs. A higher pressure increases the driving forces for absorption, thus reducing in F out F v v / , see Fig. 13. However, the extra water absorption is even higher, so that W F still will be reduced. In practice, this can be compensated for by reducing the amount of make-up water. Finally, the influence of varying the total feed rate (streams 7 and 10 in Fig. 5) is illustrated in Fig. 14 for both a constant and a proportionally increased make-up water supply. With increasing column loads, ∆T t L increases while the absorption efficiency ) / 1 ( in F out F v v − goes down. With a constant make-up, not only a decrease but surprisingly also an increase in the relative feed rates results in a diluted liquid product. The latter effect is caused by a stronger reduction of formaldehyde absorption efficiency relative to water: a maximum of W F results. The model was tested by simulating industrial absorbers of two plants of Dynea B.V., The Netherlands: one absorber with a configuration as shown in Fig. 1, and another absorber with three packed beds and a slightly different configuration. In both cases, using the actual operating parameters, the model could very well predict the performance of the columns. Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 66 top-bed top-bed bottom-bed bottom-bed ∆T L W F [Wt%] v F out v F in W F [Wt%] 57 56 55 54 1.0 1.8 43 41 39 37 10 8 6 4 2 0 0.08 0.06 0.04 0.02 0 1.4 P [bar] Fig. 13. Influence of the total pressure on l T ∆ , in F out F v v / and F W . Next, the model was used to optimize the performance of the first absorber, by simultaneously varying the parameters shown in Figs 8-12 and 14. This resulted in a savings of 1% in the costs of methanol (the basic material for making formaldehyde) for this particular formaldehyde plant. Conclusions The absorption of formaldehyde is an important step in the industrial formalin production. In the absorber column, the process of formaldehyde and water absorption is accompanied by a number of reactions in the liquid phase and considerable heat effects, necessitating separate liquid recycles with external heat exchangers. For the simulation of this type of column we developed a model based on the appropriate differential equations, without using HETP or HTU concepts. The model is completely predictive. The convergence problems often encountered with this type of complex modeling could be avoided by using a stable, semi-transient solution-algorithm. Chapter 5: Modeling and simulation of industrial formaldehyde absorbers 67 3 2 1 0 0.08 0.04 0 56 55 54 53 0.5 1.0 1.5 15 10 5 0 45 35 25 bottom bed bottom bed top bed top bed constant make-up make-up pro rata ∆T L ∆T L v F out v F in W F (wt%) W F (wt%) Fig. 14. Influence of the relative feed rate on l T ∆ , in F out F v v / and F W . Our results from simulations with varying process parameters suggest favourable absorber performance with high liquid recycle ratio's (Figs 8 and 9), low temperatures of the recirculated liquid (Figs 10 and 11) and a large amount of make-up water (Fig. 12), although the latter may be limited by a minimum desired formaldehyde content of the product stream. The suggestions mentioned above have been tested in practice and have led to a more efficient absorption column. Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 69 Chapter 6 Simulation of industrial formaldehyde absorbers: the behaviour of methanol and non-equilibrium stage modelling. Abstract A model is presented for the commercially important formaldehyde absorption in the presence of methanol. Incorporated in the model are a large number of liquid phase reactions, gas liquid heat transfer, and mass transfer with reaction of water, formaldehyde, methylene glycol, methanol and hemiformal. The evaporation of water, methylene glycol and hemiformal in the lower part of the column creates an internal circulation of these components. In this part of the column negative enhancement factors are obtained for mass transfer with reaction of methylene glycol and hemiformal. This indicates that approximate methods for calculating mass transfer enhancement factors due to reaction might fail for absorption with chemical reaction and simultaneous product desorption. The performance of an industrially applied packed column with liquid recirculations is simulated using a non-equilibrium stage model. A solution algorithm is developed and carefully described. The influence of a number of process parameters on the behaviour of methanolic species in the absorber is investigated. Introduction Formaldehyde is industrially produced by partial oxidation and dehydrogenation of vaporised methanol in air over a solid catalyst at approximately atmospheric pressure. The reactor product gas stream consists of water vapour, formaldehyde and some unreacted methanol in an inert matrix of nitrogen, hydrogen and carbon dioxide (minor amounts of by-products are neglected in this study). This gas mixture is passed through a partial condenser, where the temperature is reduced to 328 K and part of the water vapour and formaldehyde are condensed. The resulting stream is subsequently fed to the absorber to extract the formaldehyde from the gas, and to obtain the commercial product: a concentrated aqueous formaldehyde solution containing some not-converted methanol. A scheme of the absorber studied is shown in Fig. 1. The gas stream entering at the bottom of the column passes upwards trough two packed beds, randomly filled with a modern high performance Pall-ring like packing. Make-up water enters the top of the column and flows downward, meanwhile exchanging heat and mass with the gas stream. Each of the absorption beds is equipped with an external liquid recirculation with heat exchangers. Just below each bed an amount of liquid is kept to provide buffers for the liquid recirculation pumps. In a previous paper (Winkelman et al., 1992) a model for the formaldehyde absorber was developed, based on differential equations for the mass and energy balances in each phase. The resulting set of coupled boundary value problems was solved by a semi-transient method. The presence of methanol in the absorber was neglected, and only formaldehyde and water were assumed to be absorbed or desorbed. In this contribution the behaviour of methanol in the absorption process is fully incorporated in the model. Also included in the model are vaporisation and re-absorption of methylene glycol and hemiformal, which are the primary reaction products Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 70 feed product tail gas water Fig. 1. Scheme of the formaldehyde absorber. of formaldehyde with water and methanol, and which are formed in the liquid. The absorber is modelled using a non-equilibrium stage model. Reactions in aqueous methanolic formaldehyde solutions Besides heat and mass transfer between gas and liquid, a number of reactions have to be considered in modelling the performance of formaldehyde absorbers because formaldehyde reacts with both water and methanol. In aqueous solutions the dissolved formaldehyde (F) reacts fast with water (W) to form methylene glycol, CH 1 (OH) 1 , denoted by WF 1 2 2 2 2 (OH) CH O H O CH + . (1) The reaction rate is given by ) / ( 1 1 h WF W F h K C C C k r − = , (2) where K h is the chemical equilibrium constant for this hydration, defined as Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 71 eq W F WF h C C C K = 1 . (3) The equilibrium of reaction (1) is far to the right, which means that in an aqueous solution the concentration of free formaldehyde is very low. Methylene glycol, formed by reaction (1), slowly polymerises to form a series of low molecular weight poly(oxymethylene) glycols, HO(CHO) n H, denoted here by WF n O H H O) HO(CH (OH) CH H O) HO(CH 2 n 2 2 2 1 n 2 + + − ) .. 2 ( max n n = , (4) with reaction rates ) / ( 1 1 n W WF WF WF n n K C C C C k r n n − = − ) .. 2 ( max n n = , (5) and the equilibrium constants eq WF WF W WF n C C C C K N n = − 1 1 ) .. 2 ( max n n = . (6) If methanol (M) is present in the solution, formaldehyde reacts with it in a similar manner as with water, producing hemiformal, CH 3 OCH 2 OH (MF 1 ) OH OCH CH OH CH O CH 2 3 3 2 + , (7) with the reaction rate ( ) 1 1 1 / 1 KM C C C km rm MF M F − = , (8) where the equilibrium constant is defined as eq M F MF C C C KM = 1 1 . (9) The formed hemiformal also polymerises slowly to a series of polymers, higher hemiformals in this case, CH 3 O(CH 2 O) n H, denoted by MF n OH CH H O) O(CH CH OH OCH CH H O) O(CH CH 3 n 2 3 2 3 1 - n 2 3 + + ) .. 2 ( max n n = (10) with reaction rates Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 72 ( ) n MF M MF MF n n KM C C C C km rm n n / 1 1 − = − ) .. 2 ( max n n = , (11) and equilibrium constants eq MF MF M MF n C C C C KM n n = − 1 1 ) .. 2 ( max n n = . (12) Here, n max denotes the largest polymer. In Fig. 2 the equilibrium molar fractions are shown of an aqueous solution containing 55 wt % formaldehyde and 1 wt % methanol. The concentrations of the higher poly(oxymethylene) glycols and hemiformals decrease rapidly with increasing molecular weight. Therefore the largest polymers considered here are WF 10 and MF 10 , with n max = 10 for both types of polymers. This way, there are 20 chemical reactions in the liquid phase, and the total number of liquid phase components amounts to 23. The production rates of the individual species are found from ∑ = + = max 1 , , ) ( n k k k i k k i i rm m r R ν ν , (13) where k i, ν denotes the stoichiometric coefficient of component i in the reaction forming the k-th poly(oxymethylene) glycol (negative for reactants and positive for reaction products). Likewise, k i m , ν denotes the stoichiometric coefficient in the formation of the k-th hemiformal. From eq (13) and the stoichiometry of the reactions (1), (4), (7) and (10) it follows 1 10 10 10 -1 -2 -3 x i F W M i=1 i=1 2 3 4 5 6 2 3 4 5 6 7 MF i WF i Fig. 2. Equilibrium molar fractions in an aqueous solution containing 50% by weight formaldehyde and 1% by weight methanol, at 333 K. Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 73 1 1 rm r R F − − = , (14) max 2 1 n W r r r R + + + − = K , (15) max 1 3 2 1 2 n WF r r r r R − − − − = K , (16) ) 2 ( max 1 n n r r R n n WF n < ≤ − = + , (17) ) ( max n n r R n WF n = = , (18) max 2 1 n M rm rm rm R + + + − = K , (19) max 1 3 2 1 2 n MF rm rm rm rm R − − − − = K , (20) ) 2 ( max 1 n n rm rm R n n MF n < ≤ − = + , (21) ) ( max n n rm R n MF n = = . (22) The concentration of the unhydrated monomeric formaldehyde is very low due to the reactions mentioned above: well under 1%, even in concentrated solutions. However, the total amount of dissolved aldehyde remains available for chemical reactions in downstream processing because of the reversibility of the reactions. Vapour-liquid equilibria in formaldehyde-water-methanol mixtures Several models have been put forward to describe the vapour-liquid equilibria in pseudo binary formaldehyde-water systems (e.g.: Kogan et al., 1977; Brandani et al., 1980). A method to model the vapour-liquid equilibria in pseudo ternary formaldehyde-water-methanol mixtures has been presented by the group of Maurer (Maurer 1986; Albert et al., 2000). This method is used here to calculate the vapour liquid equilibria at the gas-liquid interface in the absorber. Several substances in a formaldehyde-water-methanol mixture can vaporise. These are not only the monomeric formaldehyde, water and methanol, but also the first reaction products of formaldehyde with water and methanol which are methylene glycol and hemiformal, respectively. The higher polymers always remain in the liquid phase because of their high boiling points and negligible vapour pressure (Maurer, 1986). With the method of Maurer, the thermodynamic equilibrium of the vapour-liquid system is calculated from the overall composition of the liquid (e.g. weight percentages formaldehyde and methanol) using chemical equilibrium conditions and overall composition balances in the liquid phase, combined with the physical equilibria for the components that can vaporise, ) , , , , ( 1 1 MF M WF W F i P x P y s i i i tot i = = γ , (23) where the activity coefficients, γ i , are calculated with the UNIFAC method (Gmehling et al., 1982). In the column simulations, however, the liquid phase in a stage is not at chemical equilibrium. To calculate the gas phase concentrations at the interface, in equilibrium with the liquid phase concentrations at the interface, we proceed in the following way. The equilibrium molar fractions are calculated from the overall composition of the liquid according to the method of Maurer. Once the equilibrium molar fractions are known, the activity coefficients and partial pressures can be calculated. These partial pressures are then corrected for the deviations of the Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 74 actual liquid phase molar fractions at the interface from those calculated from the chemical equilibrium conditions. Here it is assumed that the activity coefficients do not vary substantially with the change in composition from chemical equilibrium to the actual composition at the interface. Model development Simulation of continuous absorption processes is often based on stage models. The column is assumed to consist of a sequence of stages, each representing a section of the packing. Within each stage the temperature and composition of the gas and liquid phases are assumed to be constant. In this context, equilibrium stage models are widely used: the streams leaving a stage are assumed to be in equilibrium with each other and departures from this assumption are accounted for by a stage efficiency. Krishnamurthy and Taylor (1985a,b) pointed out several drawbacks of the equilibrium stage approach for separation processes. They developed a non-equilibrium stage model, where material and energy balances for each phase are solved simultaneously with the mass and energy transfer rate equations. In modelling the formaldehyde absorber, we followed the same approach. A schematic representation of a non-equilibrium stage is shown in Fig. 3. The packed beds shown in Fig. 1 consist each of a number of such stages, see Fig. 4. Vapour and liquid streams from adjacent stages are brought into contact on the stage and are allowed to exchange mass and energy across their common interface. The model of a stage consists of material and energy balances for each phase and rate equations for inter-phase mass and energy transfer. ∆Z j H v T g i g j j j H v T g i g j +1 j +1 j +1 H T l l j -1 j -1 j -1 l i H T l l j j j l i H T l l j ,F j ,F j ,F l i H T l l j ,D j ,D j ,D l i J i ,l j q l j q g j J i ,g j A j R i j gas liquid Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 75 Fig. 3. Representation of a non-equilibrium stage. v i N+1 v i 1 l i 1,F l i 1,F l i N,D l i N,D l i N,F l i N l i 0 T T 1 2 T T N-1 N . . . . B B 1 2 B B N-1 N . . . . Fig. 4. Absorber layout for non-equilibrium stage model. The balance equations are denoted by Ψ. The component balance for component i on stage j reads for the gas phase 0 , 1 , = + − ≡ Ψ + j j g i j i j i j g i A j v v , (24) and for the liquid phase 0 , , , 1 , = + − ∆ − − − ≡ Ψ − j D i j F i j j i j j j i j i j i j i l l Z SR A J l l l l l ε . (25) The energy balance on stage j for the gas phase is given by 0 , , 1 , 1 , , = + + − ≡ Ψ ∑ ∑ ∑ + + i j g i j g i j j g j i j g i j i i j g i j i j g E H J A q A H v H v , (26) and for the liquid phase Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 76 0 , , , , , , , , 1 , 1 , , = + − − − − ≡ Ψ ∑ ∑ ∑ ∑ ∑ − − i j D i j D i i j F i j F i i j i j i j j j i j i j i i j i j i j E H l H l H J A A q H l H l l l l l l l l l . (27) With the assumption of constant heat capacities of the components over the temperature range of adjacent stages, the energy balance eqs (26) and (27) are rewritten in terms of temperatures and molar flows. For the gas phase this gives 0 ) ( , 1 1 , = + − ≡ Ψ ∑ + + j g j i g i j i j g j g j g E q A Cp v T T . (28) Similarly, for the liquid phase it follows ( ) 0 ) ( ) ( ) ( ) ( , , , , 1 1 , = ∆ − + ∆ − ∆ − − + − − ≡ Ψ ∑ ∑ ∑ − − k k R k k R k j j i i j F i j F j j j i i j i j j j E Hm rm H r Z S Cp l T T A q Cp l T T ε l l l l l l l l . (29) The lowest stage in each of the packed sections represents the buffer for the liquid recirculation pumps (see Figs 1 and 4). The interfacial area is very small here, and in the model it is assumed to be zero. Mass transfer rates For clarity, the subscript j, indicating the stage number, has been dropped from all symbols throughout this section. The calculation of the fluxes is based on the two film concept, with the positive direction defined from the gas to the liquid phase. The gas phase mass transfer rates are given by ) , , , , ( ) ( 1 1 , , , , MF M WF W F i C C k J I g i g i i g g i = − = , (30) where the gas phase concentrations at the interface, I g i C , , are coupled to those in the liquid phase by ) , , , , ( 1 1 , , MF M WF W F i C m C I g i i I i = = l . (31) For the calculation of the equilibrium ratios, m i , see the section on vapour-liquid equilibria. The fluxes on either side of the interface are equal: ) , , , , ( 1 1 , , MF M WF W F i J J I i g i = = l . (32) Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 77 At the liquid side of the interface, the diffusional transport of the transferred components is accompanied by chemical reactions. This causes enhancement of the mass transfer rates. Also, the fluxes into the liquid bulk, l , i J , differ from those at the interface, I i J l , . In a previous paper, we showed the polymerisation reactions to be too slow to have any influence on the diffusion fluxes in the film, and the gradients of the concentrations of the higher polymers in the liquid film to be negligible (Winkelman et al., 1992). So, the hydration of formaldehyde, eq. (1), and the hemiformal formation, eq. (7), are the only reactions affecting the fluxes of the transferred components. To account for these parallel reactions in the liquid phase, the film model is applied ) , , , , , 0 ( 1 1 1 1 , 1 1 , 2 2 MF M WF W F i x rm m r dx C d D i i i i = ≤ ≤ − − = δ ν ν , (33) with the boundary conditions I i i i J dx dC D x l , : 0 = − = , (34) l , : i i C C x = = δ . (35) The set of eqs (30)-(35) can not be solved analytically because of the non-linearity of the reaction rates in eqs (33). Therefore, an iterative shooting method was used to calculate the interfacial concentrations and the mass fluxes, given the bulk phase concentrations. From an initial guess of the interfacial concentrations, the gradients at the interface were calculated with eqs (30)-(32) and (34), and the differential eqs (33) were numerically integrated from 0 = x to δ = x using a fourth order Runge-Kutta method. The interfacial concentrations were repeatedly updated, using a multi-dimensional Newton-Raphson method, until the obtained concentrations at δ = x match the liquid phase bulk concentrations. From the numerically calculated gradients at δ = x , the flux into the liquid bulk is obtained δ = − = x i i i dx dC D J l , . (36) The numerical effort of the above procedure is greatly reduced by noting the mutual dependency of several interfacial concentrations and fluxes. This can be understood from considering methanol and hemiformal as an example. Addition of eqs (33) for these two components results in 0 2 2 2 2 1 1 = + dx C d D dx C d D MF MF M M . (37) Integrating twice while applying boundary conditions (34) and (35) gives an explicit relation for the interfacial concentration of methanol as a function of the interfacial concentration of hemiformal: Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 78 ] / /[ )] ( ) / ( [ , , , , , , , , , , , , , 1 1 1 1 1 1 1 M M g M MF I MF MF MF I MF g MF MF g M M g M M g I M m k k C C k m C C k C k C k C + − − − + + = l l l l l l l l . (38) Similar equations are obtained for I W C l , as a function of I WF C l , 1 , and for I F C l , as a function of I MF C l , 1 and I WF C l , 1 . Thus, if the values for I WF C l , 1 and I MF C l , 1 are chosen or updated, the values of I F C l , , I W C l , and I M C l , can be calculated from eq (38) and its analogs. This way, the problem of mass transfer of five components, with two parallel reactions can be solved by iteration on two interfacial concentrations only. Energy transfer rates The film model gives the following expressions for the conductive heat fluxes from the gas phase, g q , and into the liquid phase, l q , (Krishna and Taylor, 1986) ) ( , I g g f g g T T A h q − = , (39) ) ( l l l T T h q I − = . (40) The Ackermann factor, A f , corrects the conductive energy transfer rate in the gas phase for non- zero mass transfer rates. ∑ = − = i g i g i g Cf f Cp J h Cf e Cf A , , 1 where , 1 . (41) For the liquid phase, this correction is negligible. If heat and mass transfer occur simultaneously, the total energy transfer rate contains a conductive and a convective contribution on either side of the interface. From a balance around the interface it follows that the total energy fluxes out of the gas phase and into the liquid phase must be equal ∑ ∑ + = + i i i i g i g i g H J q H J q l l l , , , , , (42) where the summations are over all transferred species. Expressing the enthalpies in terms of heat capacities and temperature differences, and introducing heats of vaporisation and reaction gives ) )( ( ) )( ( ) ( ) ( 1 , , , 1 , , , , , , , , , 1 1 1 1 R g MF MF R g WF WF i i i I i i vap g i i g i g i I g g Hm J J H J J Cp J T T H J Cp J T T q q ∆ − − + ∆ − − + − + ∆ + − + = ∑ ∑ ∑ l l l l l l . (43) Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 79 In deriving eq (43), it is assumed that the variation of the heat capacities is negligible if the temperature changes from T g to T I and from T I to l T and that the heats of reaction of both the hydration of formaldehyde and the hemiformal formation in the film are liberated at the interface. Further, the reaction products of these two reactions are taken as key components, thus allowing the reaction rates in the film to be expressed as flux differences of these key components. From eqs (39)-(43), both the interfacial temperature and the values of the heat transfer rates, g q and l q , were calculated. Method of solution. Newton's method for simultaneous correction was used to solve the model because it is more effective than tearing algorithms (Krishnamurthy and Taylor, 1985a). The total number of unknown bulk phase variables on a stage j is 30: 5 gas phase molar flow rates, the gas phase temperature, 23 liquid phase molar flow rates and the liquid phase temperature. These are stored in a vector ( j X ) j MF MF M WF WF W F g MF M WF W F T j T l l l l l l l T v v v v v X ) , , , , , , , , , ( ) ( 10 1 10 1 1 1 , , l K K = . (44) Other quantities, such as the mass and energy transfer rates and the temperature and concentrations at the gas-liquid interface, are functions of the bulk phase variables, and are therefore not considered as independent variables in the solution process. The unknown variables in ( j X ) must be found by solving the component balance eqs (24) and (25), and the energy balance eqs (28) and (29). These equations are ordered in a vector ( j Ψ ) j E MF MF M WF WF W F g E g MF g M g WF g W g F T j ) , , , , , , , , , , , ( ) ( , , , , , , , , , , , , , , 10 1 10 1 1 1 l l l l l l l l K K Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ = Ψ . (45) Specified quantities are the molar flow rates and the temperature of the feed to the last stage, the make-up water molar flow rate and its temperature, the mass flow rates of the two liquid recirculation streams and the temperatures of the recirculation streams entering the first stage of each of the two packed beds. From initial estimates, the variables are repeatedly updated with Newton's method, using the equation current current next ) ( ] ) ( ) ][( [ Ψ − = − X X J , (46) where (X) current and (X) next denote the current and next estimate for the vector (X) which contains all bulk phase variables T n T X X X )) ( ),.., (( ) ( 1 = , (47) Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 80 (Ψ) denotes the vector of current discrepancies of the set of equations to be solved and whose solution is given by 0 ) ( = Ψ ) ) ( ) (( ) ( 1 T n T T Ψ Ψ = Ψ K , (48) and [J] denotes the Jacobian matrix with elements j i j i X J ∂ Ψ ∂ = , . (49) The variables and equations are grouped in such a way that the Jacobian matrix has the well known block tridiagonal structure, shown in Fig. 5. The top bed of the absorber contributes to the Jacobian the submatrices [S] j , [T] j and [U] j , which contain the partial derivatives of the functions pertaining to stage j with respect to the variables of the stages j-1, j, and j+1, respectively. In a similar way, the submatrices [A] j , [B] j and [C] j originate from the bottom bed in the absorber. Formally, the submatrices have dimensions (30×30), but [S] j and [A] j are very sparse. The solution algorithm is adapted to use this sparseness, in order to save on computer storage requirements and calculation time. Because of the presence of the liquid recirculations, the composition of the liquid on the first stage of each absorption bed depends partly on the composition of the last stage. This is reflected in the presence of the two off-diagonal submatrices R T and R B in Fig. 5. Most of the partial derivatives in the submatrices [T] j and [B] j of the Jacobian are too complicated to calculate analytically, and are therefore obtained from finite difference approximations. The entries of all the other submatrices are obtained from analytical expressions. T 1 S 2 U 1 T 2 U 2 R T S N-1 T N-1 U N-1 S N T N U N R B A 2 B 2 C 2 A 1 B 1 C 1 A N-1 B N-1 C N-1 A N B N Fig. 5. Structure of the Jacobian matrix for the formaldehyde absorber. Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 81 Despite the presence of the two off-diagonal submatrices, the matrix generalisation of the Thomas algorithm can be used to solve eq. (46) for (X) next . With this algorithm, the Jacobian matrix is first converted to the so-called upper diagonal form by a recursive elimination. Most of the sparsety of the Jacobian is preserved during the elimination process. Additional entries appear only in the columns between the submatrices [R T ] and [T N ], and between [R B ] and [B N ]. From the upper triangular form, the solution for (X) next is found by repeated back substitution. Unfortunately, the solution method described above showed a poor convergence (e.g. large computation time) or did not result in a solution at all. The latter particularly if the initial estimates of the variables were not very close to the final solution. The reason for this unsatisfactory behaviour is not completely clear, but possibly part of the problem is caused by the great difference in the orders of magnitude of the terms in the energy balances as compared to those in the mass balances. Therefore, the solution method was changed in such a way that the energy balance equations and the temperatures where skipped from eq. (46). The remaining modified eq. (46) now contains only the component balance equations and the molar flow rates, and is solved in an inner loop j MF MF M WF WF W F MF M WF W F T j l l l l l l l v v v v v X ) , , , , , , , ( ) ( 10 1 10 1 1 1 , , loop inner K K = (50) j MF MF M WF WF W F g MF g M g WF g W g F T j ) , , , , , , , , , ( ) ( , , , , , , , , , , , , loop inner 10 1 10 1 1 1 l l l l l l l K K Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ = Ψ . (51) The inner loop iterations continued until the sum of square residuals of the mass balances, SSR M , satisfied the condition ( ) 2 6 2 , , 2 , , ] / [ 10 ) ( ) ( s mol SSR j i g j i g j i M − < Ψ + Ψ = ∑∑ . (52) Subsequently, the temperatures were found by solving the energy balances in an outer loop using eq. (46) with (X j ) and (Ψ j ) now defined as j g T j T T X ) , ( ) ( loop outer l = , (53) j E g E T j ) , ( ) ( , , loop outer l Ψ Ψ = Ψ . (54) Convergency of the energy balances was supposed to be obtained once the sum of square residuals of the energy balance equations, SSR E , satisfied the condition ( ) 2 3 2 , 2 , ] / [ 10 ) ( ) ( s J SSR j g E g E E < Ψ + Ψ = ∑ . (55) Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 82 The solution algorithm is summarised in Fig. 6. The calculations are initialised by assigning guessed values to the molar flow rates and to the temperatures at the first stage, and by simply taking the conditions at the other stages identical to the first stage. Although the initialisation is not very sophisticated, the algorithm converges readily to a solution defined by eqs (52) and (55). Once calculated solutions were available for some situations, these were used as starting values, to initialise the calculations for other sets of operating parameters. This leads to a considerable reduction of the calculation time. The transfer coefficients and other physical and most of the chemical properties needed in the calculations were obtained as described by Winkelman et al. (1992). The chemical equilibrium constants for reactions involving formaldehyde and methanol where taken from Maurer (1986). Because no open literature is available on the rates of reactions involving methanol these rates were taken equal to the rates of the corresponding reactions involving water, i.e.: km i = k i (i = 1..n max ). START input parameters initialize variables calculate ( ) and [ ] Ψ J for inner loop update flowrates SSR <10 M -6 calculate ( ) and [J] Ψ for outer loop update temperatures SSR <1 E 0 3 END No o u t e r l o o p i n n e r l o o p No Fig. 6. Algorithm for computing absorber performance. Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 83 Results Fig. 7 shows a typical example of the convergence behaviour of the double-loop solution algorithm. In this figure, the values of SSR M calculated with inner loop iterations are plotted against the values of SSR E from the outer loop. The calculations were initialised by using the same guessed molar flow rates and temperatures for all the stages. To cut calculation time, the value of SSR E was not calculated during the inner loop calculations. In constructing Fig. 7, the value of SSR E during the inner loop iterations was therefore assumed to equal the one calculated from the next outer loop iteration. It is seen that the number of inner loop iterations per outer loop iteration reduces steadily from four to just one as the conditions approach the final solution. Fig. 7 also illustrates that the improvement per individual iteration is greater the closer we are to the final solution (up to several decades in terms of SSR-values). This is typical for Newton's method. The concentrations necessary for the calculations of the mass transfer fluxes from eqs (30)-(38) can be calculated from the molar flow rates in several ways (Krishnamurthy and Taylor, 1985a,b). The simplest option is to assume constant bulk compositions, and to calculate the concentrations from the molar flow rates leaving the stage. If the bulk compositions are assumed to vary linearly, then the concentrations have to be calculated from the average of the molar flow rates entering and leaving the stage. For the simulation of packed columns, Krishnamurthy and Taylor (1985c) obtained the best results using the average composition for the bulk vapour j g tot i j i j i i j i j i j g i C v v v v C , 1 1 , 5 . 0 + = ∑ ∑ + + , (56) 11 0.4 0.02 4x10 -4 2x10 -5 END START 10 2 10 6 10 10 10 14 10 10 10 10 10 10 10 5 0 -5 -10 -15 SSR /(W) E 2 SSR E (mol/s) 2 Fig. 7. Typical convergence path. Numbers indicate the largest temperature correction Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 84 in either phase with an outer loop iteration step. Dashed lines: inner loop iterations. Solid lines: outer loop iterations. 20 40 60 80 100 0 0.24 0.23 0.22 0.21 0.20 total number of stages v M out v M in Fig. 8. Relative amount of vapour phase methanol leaving the bottom-bed vs. the total number of stages applied; ( ): fluxes based on average vapour composition; ( ): fluxes based on exit vapour composition. and using the exit composition for the bulk liquid j tot i j i j i j i C l l C l l , , ∑ = . (57) We, therefore, applied the same method. The influence of the method of calculating the vapour bulk composition is illustrated in Fig. 8, where, as an example, the relative methanol vapour phase molar flow rate leaving the bottom-bed is shown as function of the total number of stages applied. The bulk liquid phase composition was always calculated from the exit liquid molar flows. Using the stage exit molar flows to calculate the fluxes leads to lower driving forces for mass transfer, and therefore less absorption of methanol. The effect of the total number of stages on the solution was studied by varying this number while always allocating 50% to the top- and the bottom-bed, respectively. The many calculated output variables (molar flow rates and temperatures) all asymptotically approached a constant value with increasing the number of stages. Again, Fig. 8 may serve as a typical example. Generally, the variation in the results appeared to become negligible above 60 stages. Therefore all remaining calculations were performed with 60 stages (30 in each absorption bed). A typical set of calculated bulk vapour and liquid temperature profiles is shown in Fig. 9. The cooler gas stream entering the bottom-bed is quickly heated up to the liquid temperature because of a large heat transfer capability and a much lower heat capacity of the gas stream relative to the liquid stream. Similarly, the warm gas stream entering the top-bed from the Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 85 bottom-bed is rapidly cooled down to the liquid temperature level. As the liquid flows 0 1 0.25 0.5 0.75 relative height 65 64 63 62 61 60 44 42 40 38 36 34 T [C] T [C] Fig. 9. Typical temperature profiles in the absorber; ( ):liquid; ( ):gas. downwards, its temperature rises, predominantly due to the heat effects connected with condensation and reaction. In the lower part of the bottom-bed, solvent evaporation results in a temperature maximum. Interestingly, the temperature profiles obtained here using a non- equilibrium stage model are quantitatively similar to those obtained previously with a differential model (Winkelman et al., 1992). After inspection of the results, it turned out that the Ackermann correction factor for the vapour phase, A f,g , deviates very little from unity. This can be easily understood. For mass transfer to have a significant influence on the vapour phase conductive energy flux, say more than 5% (A f,g ≤ 0.95), it follows from eq. (41) that Cf > 0.1. With typical values for the vapour phase heat capacity Cp i,g ≈ 40 J/(mol K) and the heat transfer coefficient h g ≈ 40 J/(m 2 K s) this results in a condition for the fluxes: 1 . 0 , ≥ ∑ v i J mol/(m 2 s), which is larger than can be expected in absorption columns for the solubilities typical in formaldehyde absorption. Further, the temperature differences in the liquid film, j j I T T l − , , are generally well below 0.1 K because of the low resistance against heat transfer in the liquid phase. According to these observations, the following assumptions are justified: 1 = f A and j j I T T l = , . This allows for considerable simplifications in the calculation of the energy transfer rates because apparently there is no need to calculate the interfacial temperature. This way, eqs (39)-(43) reduce to ) ( l T T h q g g g − = , (58) ) )( ( ) )( ( ) ( 1 , , , 1 , , , , , , , 1 1 1 1 R g WF WF R g WF WF i i vap g i i g i g i g g H J J H J J H J Cp J T T q q ∆ − − + ∆ − − + ∆ + − + = ∑ ∑ l l l l . (59) Fig. 10 shows typical profiles of relative vapour phase molar flow rates in the formaldehyde absorption column. Because of the large liquid/gas ratio, the liquid phase composition does not Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 86 change much with the stage number of the packed section and is therefore not shown in Fig. 10. v i v i in 0 1 0.25 0.5 0.75 relative height 2 1 0 i=W i=F i=M Fig. 10. Typical profiles of relative vapour phase molar flow rates. Similar to the temperature profiles, also these profiles clearly reveal the division of de packed height in two absorption beds. The gas feed to the absorber is undersaturated with water. In the lower part of the bottom bed this leads to desorption of water. In the upper part of the bottom- bed, but even more in the top-bed, the water vapour condenses. This way an internal circulation of water is created. The mass transfer of both formaldehyde and methanol is always directed towards the liquid. Again, the profiles obtained here for formaldehyde and water using a non- equilibrium stage model are very similar to those obtained previously using a differential model. The relative vapour molar flow rates of the reaction products methylene glycol and hemiformal are sown in Fig. 11. These components are not present in the gas feed, and similar to water, they are desorbed in the lower part of the bottom-bed and re-absorbed higher in the column. v i v F 0 1 0.25 0.5 0.75 relative height 0.03 0.02 0.01 0.00 i=WF 1 i=MF 1 Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 87 Fig. 11. Typical profiles of relative molar flow rates of the vaporised reaction products. v v WF MF 1 1 + v F 0 1 0.25 0.5 0.75 relative height 0.10 0.05 0.00 v MF 1 v M 1.5 1.0 0.5 0.0 Fig. 12. Fraction of converted formaldehyde and methanol in the gas phase relative to the uncombined, free components. Fig. 12 shows that the amount of formaldehyde that is transported in the gas phase in associated form is relatively low compared to the amount of free formaldehyde: it always accounts for less than 10% of the total amount of formaldehyde in the gas phase. On the other hand, the amount of hemiformal is substantial compared to the amount of free methanol in the gas phase (in the bottom-bed it becomes even the larger one of the two). But both the amounts of methanol and hemiformal are small relative to formaldehyde and water. The numerical evaluation of the fluxes, as described in the section on mass transfer rates, allows for the calculation of the mass transfer enhancement factors, E i ) ( , , , 0 l l l i I i i x i i i C C k dx dC D E − − = = . (60) Remarkable results are obtained for the reaction products methylene glycol and hemiformal, see Fig. 13. The enhancement factor of hemiformal is smaller than one in many of the stages. Methylene glycol has even negative enhancement factors in some of the stages. This phenomenon is not some numerical peculiarity, but results from the combined action of mass transfer and reaction. This becomes clear by taking a closer look at the concentration profiles of methylene glycol in the stagnant liquid film adjacent to the gas-liquid interface. These profiles are shown in Fig. 14 for the stages 50 to 54 (the stage numbers are also indicated in Fig. 13). The profiles were obtained from the numerical method described in the section on mass transfer rates. In these stages, the positive gradient of the concentration near x = 0 indicates that the flux of methylene glycol at the interface is directed towards the gas phase. Therefore the numerator of eq (60) is always negative here. However, the concentration difference of methylene glycol over the Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 88 film in the denominator of eq (60), l l , , 1 1 WF I WF C C − , changes sign in going from stage 52 to 53. E WF 1 0 1 0.25 0.5 0.75 relative height E MF 1 5 0 -5 -10 -15 -20 2.0 1.5 1.0 0.5 0.0 54 50 Fig. 13. Mass transfer enhancement factors of the reaction products; ( ): methylene glycol; ( ): hemiformal. C C WF 1 , WF 1 , l l − mol/m 3 1 0 -1 0 1 0.25 0.5 0.75 x/δ stage 50 51 52 53 54 Fig. 14. Concentration profiles of methylene glycol in the film at the interface for the stages 50-54. For the stage numbers 53 and 54 the denominator is negative, resulting in positive enhancement factors, whereas for the stage numbers 50-52 the denominator is positive, resulting in negative enhancement factors. Recently, the possible occurrence of negative enhancement factors was shown also by analytically solving the case of mass transfer with a single first-order reversible reaction (Winkelman and Beenackers, 1993). In a previous contribution, we already discussed the influence of several operating parameters on the temperature rise of the liquid in the packed beds, on the formaldehyde absorption efficiency and on the overall weight percentage of formaldehyde in the liquid leaving Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 89 the packed beds (Winkelman et al., 1992). New aspects here are the role of methanol and hemiformal. Below, we focus on these aspects only. Fig. 15 illustrates the influence of the temperature of the liquid recycle stream of the top- bed, Rec t T . Increasing Rec t T results in higher partial pressures of methanol and hemiformal, and therefore in increasing relative molar flows of these components in the gas stream leaving the top-bed and thus in less absorbed methanol in the liquid leaving the bottom-bed. With increasing Rec b T the vapour pressures of methanol and hemiformal in the bottom-bed increase. This causes increased levels of these components in the top-bed, and higher amounts leaving the absorber with the gas stream, see Fig. 16. Although the higher amounts of absorbed methanolic compounds enter the bottom-bed with the liquid flowing down from the top-bed, this cannot completely compensate for the decreased absorption of methanol and hemiformal in the bottom- bed and the overall amount of methanol leaving the bottom-bed with the liquid stream decreases. 0.14 0.12 0.10 0.08 0.06 0.04 v i out v M in 25 30 35 40 45 T [C] t Rec W M [wt%] 2.0 1.9 1.8 1.7 i=M i=MF 1 bottom Fig. 15. Influence of Rec t T on the relative amounts of methanolic compounds in the exit gas and in the exit liquid stream. 0.12 0.10 0.08 0.06 v i out v M in 55 60 65 70 75 T [C] b Rec W M [wt%] 2.0 1.9 1.8 1.7 i=M i=MF 1 bottom Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 90 Fig. 16. Influence of Rec b T on the relative amounts of methanolic compounds in the exit gas and in the exit liquid stream. 0.12 0.10 0.08 0.06 v i out v M in 30 40 50 60 70 80 90 R t W M [wt%] 1.95 1.90 1.85 1.80 i=M i=MF 1 bottom Fig. 17. Influence of R t on the relative amounts of methanolic compounds in the exit gas and in the exit liquid stream. v i out v M in 20 25 30 35 40 R b W M [wt%] 1.95 1.90 1.85 1.80 i=M i=MF 1 bottom 0.11 0.10 0.09 0.08 0.07 0.06 Fig. 18. Influence of R b on the relative amounts of methanolic compounds in the exit gas and in the exit liquid stream. The influence of t R , the amount of liquid recycled around the top-bed, is illustrated in Fig. 17. The amounts of methanol and hemiformal leaving the absorber with the gas stream decrease with increasing t R , while at the same time the weight percentage of methanol in the liquid leaving the absorber at the bottom is almost constant. This might seem conflicting, but is caused mainly by a decrease of the temperature rise of the liquid in the top-bed with increasing t R . The Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 91 result is even more formaldehyde and water absorption, and therefore an increase of the total amount of liquid entering the bottom-bed from the top-bed which tends to dilute the methanolic species. Similarly, the temperature rise of the liquid in the bottom-bed decreases with increasing b R , leading to more formaldehyde and water absorption in the bottom-bed, dilution of the methanol and lower weight percentages of methanol in the liquid leaving the absorber, see Fig. 18. On the other hand, the increased absorption of formaldehyde and methanol in the bottom-bed results in lower amounts of these components absorbed in the top-bed. This reduces the liquid flow in the top-bed, resulting in a higher temperature rise, increased vapour pressures of methanol and hemiformal, and higher amounts of these components leaving the top-bed with the gas stream. Interesting phenomena were observed if the amount of make-up water is varied. In both absorption beds, the amount of methanol in the gas stream increases with increasing amounts of make-up water, whereas the amount of hemiformal decreases. This is caused by a shift in the relative amounts of reaction products in the liquid phase. Due to the lower overall amounts of formaldehyde present in the liquid, less methanol combines with formaldehyde to form hemiformal, and more methanol is present in the uncombined form. This results in a higher partial pressure of methanol and a lower partial pressure of hemiformal. The amounts of these components leaving the absorber with the gas stream react correspondingly to increasing amounts of make-up water, see Fig. 19. v i out v M in 0 2 4 6 W M [wt%] 2.5 2.0 1.5 1.0 i=M i=MF 1 bottom 0.12 0.10 0.08 0.06 0.04 relative amount of make up water Fig. 19. Influence of the relative amount of make-up water on the relative amounts of methanolic compounds in the exit gas and in the exit liquid stream. Chapter 6: Simulation of … absorbers: the behaviour of methanol and non-equilibrium stage modeling. 92 Conclusions. The industrially important process of formaldehyde absorption in the presence of methanol is simulated using a non-equilibrium stage model. The model takes into account the mass transfer of formaldehyde, water and methanol, and their primary reaction products methylene glycol and hemiformal. In the liquid a large number of reactions take place, giving rise to numerous liquid phase components. The double loop solution algorithm, solving the component balances in an inner loop and the energy balances in an outer loop, proved to be stable and converged readily despite a simple initialisation and a large number of equations. The non-zero mass flux correction of the conductive energy fluxes proved to be negligible. Similarly, it was found that the resistance against energy transfer in the liquid phase is negligible under practical conditions. In this type of modelling it is useful to check whether these simplifications are possible, because they reduce the calculation effort significantly. The evaporation of water, methylene glycol and hemiformal in the lower part of the column creates an internal circulation of these components. In this part of the column negative enhancement factors are obtained for mass transfer with reaction of methylene glycol and hemiformal. This indicates that approximate methods for calculating mass transfer enhancement factors due to reaction might fail for absorption with chemical reaction and simultaneous product desorption. Chapter 7: Epilogue. 93 Chapter 7 Epilogue The research reported in this thesis was instigated upon a request from Dynea B.V. concerning the absorbers in their formaldehyde plants. Since formaldehyde is a bulk chemical, its cost price is critical, and the production facilities need constant adjustments to increase the efficiency and to reduce the costs. But despite the need for a remunerative production process, the absorption columns didn't operate optimally for many years. They were bottlenecking production levels, hampering efficiency increases, and preventing reduction of the consumption of methanol, which is the raw material in the production of formaldehyde. Internally, tests were conducted with the absorbers aimed at getting insight into the influence of various process parameters on the performance, without much success though. With the lapse of time, some tests were even repeated, without the results getting any better. It turned out that the number of possible variations and process parameters was too large, and their influence on the absorber performance too complicated to be able to achieve an understanding, let alone optimisation, this way. The strategy opted for to tackle the absorber problems was the development of reaction engineering models from first principles, to describe the performance of the columns and the effects of variations in the operating parameters on the efficiencies. From an initial survey of the literature, concerning the availability of physical en chemical data on the reactions in, and properties of aqueous formaldehyde solutions, it was concluded that experimental work was needed on the kinetics of the main reaction, the hydration of formaldehyde, and the density and viscosity of the solutions. Consequently, the kinetics of the dehydration of methylene glycol were measured, the influence, in general, of a reversible reaction on gas liquid mass transfer was studied theoretically, the kinetics and equilibrium of the hydration were measured, and the density and viscosity of aqueous formaldehyde solutions were determined, see Chapters 2, 3 and 4 and Appendix A of this thesis, respectively. Data on the rate and equilibrium of the relatively slow formation of the series of poly oxymethylene glycols, and on the vapour liquid equilibria pertaining to formaldehyde containing systems, are readily available from the literature, especially from the publications by the research group of Maurer (e.g. Maurer, 1986; Hasse, 1990; Hahnenstein et.al, 1994, 1995; Albert et.al, 2000). Various reaction engineering models were developed, see Chapters 5 and 6 of this thesis, and translated to computer code. They were adapted to, p.e., the different absorption column configurations that were operational. In a first instance, the models were successfully tested for their ability to describe the current performance of the absorbers. Next, the effect of variations of the operational parameters on the column performances was investigated. This resulted in a new optimised set of operational conditions and an increased efficiency of the absorber performance. Chapter 7: Epilogue. 94 The advantages of the absorber models are threefold. The model predicts exactly the consequences of any alterations in the tuning, and the process-engineers now have complete control over the column performance. Because of the higher efficiency of the absorbers, the same formaldehyde production level can now be achieved with less methanol consumption and considerable savings. An advantage of a more psychological nature is the ending of internal discussions for years on absorber operation among the engineers, and the clarity and univocal nature of the instructions that can now be given to the operators in the plant. The results obtained here are not just confined to usage in the modelling and optimisation of formaldehyde absorbers. They were also, for example, successfully applied in the modelling of industrial formaldehyde-methanol-water distillation, where methanol is separated from formaldehyde water mixtures. In general, with the availability now of a complete description of aqueous formaldehyde systems, the results are of use in the design of many separation processes involving formaldehyde. Symbols 95 Symbols a interfacial area, m -1 f A Ackermann heat transfer correction factor j A interfacial area in stage j, m 2 C concentration, mol m -3 Cp i molar heat capacity of i, J mol -1 K -1 D diffusivity, m 2 s -1 d stirrer diameter, m E energy flux, W m -2 E chemical enhancement factor for mass transfer, - H i molar enthalpy of i, J mol -1 j R H ) (∆ reaction enthalpy of reaction j, J mol -1 i vap H , ∆ molar heat of vaporization of i, J mol -1 h heat transfer coefficient, W m -2 K -1 HMS − hydroxymethane sulphonate, CH 2 (OH)SO 3 − J molar flux, mol m -2 s -1 K chemical equilibrium constant, - K a1 , K a2 acid dissociation constants of H 2 SO 3 and HSO 3 − , respectively, in terms of concentrations, mol m -3 K h chemical equilibrium constant of the formaldehyde hydration, - K w dissociation constant of water in terms of concentrations, (mol m -3 ) 2 k reaction rate constant, m 3 mol -1 s -1 1 k first order reaction rate constant, s -1 k 2 rate constant for the reaction of formaldehyde with − 2 3 SO , m 3 mol -1 s -1 h d k k , rate constants for the dehydration of methylene glycol, and the pseudo-first-order hydration of formaldehyde, respectively, s -1 g k gas phase mass transfer coefficient, m s -1 l k liquid phase mass transfer coefficient, m s -1 l optical path length, m l i liquid phase component molar flow rate, mol s -1 M molar weight, kg mol -1 m distribution coefficient ( g C C / l , at equilibrium), - N stirrer rate, s -1 max n number of CHO segments in the largest polymers considered P pressure, Pa s i P saturated vapor pressure of i, Pa q conductive heat flux, W m -2 R ideal gas constant, J mol -1 K -1 R i production rate by reaction of i, mol m -3 s -1 Symbols 96 R t , R b liquid recycle ratio for the top and bottom bed, - Re Reynolds number, η ρ / 2 N d , - j r rate of reaction j, mol m -3 s -1 S Chapter 4: gas liquid interfacial area, m 2 S cross sectional area of the column, m 2 S tot total concentration of sulphur, mol m -3 Sc Schmidt number, D ρ η/ , - Sh Sherwood number, g g g D d k / or l l l D d k / , -. E SSR sum of square residuals of the energy balances, W M SSR sum of square residuals of the mass balances, mol s -1 T temperature, K ∆T l temperature rise of the liquid, K t time, s V volume, m 3 V B volume of liquid on the partial draw off tray at the bottom of the column, m 3 v diffusivity ratio (= l l , , / P A D D in Chapter 3; = l l , , / MG F D D in Chapter 4), - v i gas phase molar flow rate of i, mol s -1 W weight W F overall weight percentage of formaldehyde in the liquid, wt.% W M overall weight percentage of methanol in the liquid, wt.% x liquid phase molar fraction, -; distance in the film, m x ~ overall liquid phase molar fraction, - y gas phase molar fraction, - Z height of packing, m z charge number of ionic species, - j Z ∆ height of stage j, m Greek letters γ activity coefficient δ film thickness, m ε extinction coefficient, m 2 mol -1 ; dielectric constant of water l ε partial liquid holdup, - η viscosity, Pa s µ ionic strength ν i,j stoichiometric coefficient of component i in reaction j, - ρ density, kg m 3 R φ reaction factor, - g , v φ gas phase volumetric flow rate, m 3 s -1 Ψ vector of discrepancies of the set of balance equations Subscripts 0 initial 2,3,.. polymerization reactions Symbols 97 A reactant b bottom bed F formaldehyde g gas phase h hydration reaction i component i j reaction j l liquid phase M methanol n MF hemiformal, CH 2 O(CH 2 O) n H, max 1 n n ≤ ≤ MG methylene glycol max maximum n number of CH 2 O segments in a polyoxymethylene glycol P reaction product t top bed tot total W water WF n polyoxymethylene glycol with n CH 2 O segments ∞ irreversible reaction, limit for ∞ → K Superscripts — (overbar) bulk conditions D liquid draw off F liquid feed I interface in incoming or feed conditions j stage j out outgoing stream Rec liquid recirculation stream sat saturated th thermodynamic equilibrium constant based on activities References 99 References Albert, M., García, B.C., Kuhnert, C., Peschla, R., & Maurer, G. 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Chem., 74, 2746-2750. Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 103 Appendix A Correlations for the density and viscosity of aqueous formaldehyde solutions Abstract Empirical correlations are presented for the density and viscosity of aqueous formaldehyde solutions as a function of temperature (T) and overall weight percentage of formaldehyde (W F ). Experimental density data from the literature, at T = 288-338 K and with W F = 1.6-50 wt %, are described with an average absolute residual (AAR) of 0.14%. Experimental viscosity data, both new and from the literature, at T = 288-333 K and with W F = 1.6-50 wt %, are described with an AAR of 1.8%. The residuals of the correlations are free of trending effects as a function of T and W F . It is shown that both properties can be described using liquid mixture correlation methods from the literature with almost the same accuracy relative to the empirical correlations. Introduction Formaldehyde is an important industrial base chemical. One of the key steps in its production is the absorption of gaseous formaldehyde in water, usually in a packed absorber. The performance of the absorbers depends on the process operation variables, such as the temperature, the pressure, and the flow rates, and on the hydrodynamic properties of the packing, such as the mass-transfer coefficients, the specific interfacial area, and the liquid phase hold-up in the packing. Therefore, in modelling, design, and optimisation calculations of the formaldehyde absorbers, the hydrodynamic properties have to be evaluated. In the literature these parameters are usually correlated to, among other things, the liquid-phase physical properties, especially the density and viscosity. Also, in the specification sheets of packing manufacturers, the performance of the packing types is often given as a function of these liquid-phase properties, along with various flow-rate parameters. Two literature sources were found giving correlations for the density of aqueous formaldehyde solutions, ρ m . Walker (1964) gives a correlation for ρ m as a function of the strength of the solution, W F , which is valid at 291 K only, ) 291 ( 3 10 00 . 1 ) ( 3 291 K T W F K m = + × = ρ , (1) and the temperature coefficients in the range of T = 288-303 K for W F = 15 and 45 wt %, from which, using eq (1), the following correlations can be obtained: ) 303 288 %, 15 ( ) 291 ( 2 . 0 1045 ) ( % 15 K T wt W T F wt m − = = − + = ρ , (2) ) 303 288 %, 45 ( ) 291 ( 4 . 0 1135 ) ( % 45 K T wt W T F wt m − = = − + = ρ . (3) The Kirk-Othmer Encyclopedia of Chemical Technology (1994) presents a correlation for ρ m , which reads (slightly modified to yield consistent units) Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 104 )] 328 ( 10 55 . 0 0 . 1 )][ 45 ( 3 1119 [ 3 T W F m − × + − + = − ρ . (4) No information is given on the accuracy of eq (4), nor on the temperature and concentration range for which it is valid. The same source also presents a correlation of the viscosity of aqueous formaldehyde solutions, η m , which slightly modified reads )] 15 . 273 ( 024 . 0 039 . 0 28 . 1 [ 10 3 − − + = − T W F m η . (5) Equation (5) is valid for rather concentrated solutions, W F = 30-50 wt %, and for T = 298-313 K. No information is given on the accuracy of eq (5). In this contribution the results of a study to correlate the available literature data on ρ m and η m as a function of T and W F are reported. Also, the results of a series of viscosity measurements of aqueous formaldehyde solutions are reported. Density Three literature sources were found reporting data on ρ m ; see Table 1. Lileev et al. (1982) specified the strength of the solutions in terms of the overall formaldehyde molar fraction, F x ~ , from which we calculated W F , because these units were used by the other authors mentioned in Table 1: % 100 ) ~ 1 ( ~ ~ × − + = W F F F F F F M x M x M x W . (6) The experimental results show that ρ m varies with T and W F and that always W m ρ ρ > (for 0 > F W ). Fig. 1 shows the density difference ) ( W m ρ ρ − as a function of W F , with ρ W from Perry et al. (1984) It shows that a considerable fraction of the observed variation of ρ m can be accounted for by introducing a linear dependency of ) ( W m ρ ρ − on W F . Least-squares regression of the data accordingly, followed by an analysis of the residuals, ∆ i , defined as % 100 ) ( ) ( ) ( exp exp × − = ∆ i m m calc m i ρ ρ ρ , (7) showed that the residuals have a clear trend as a function of T, indicating an inadequacy in the relation which makes extrapolation unreliable outside the applied experimental conditions. The residuals tend to increase monotonically with increasing T, justifying the introduction of an additional temperature-dependent parameter. Using multiple regression the following equation was thus obtained: F W m W T) 10 8166 . 6 0950 . 5 ( 3 − × − + = ρ ρ . (8) Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 105 Table 1. Literature data on the density of aqueous formaldehyde solutions. Ref. T (K) W F (wt %) ρ m (kg m -3 ) data points 1 291-338 2-50 1005.4-1570.0 27 3 288, 298 6-43 1018.4-1135.4 16 4 288, 298, 308 1.6-17 997.4-1045.4 15 all data 288-338 1.6-50 997.4-1570.0 58 0 20 40 60 160 120 80 40 0 ρ ρ m W − ( k g / m ) 3 W (wt%) F Walker Skelding & Ashbolt Lileev et al. Fig. 1. Density difference between aqueous formaldehyde solutions and water, both at the same temperature. ρ m : from the literature sources indicated. ρ W : Perry et al. (1984). When eq (8) was applied to a 15 wt % solution, at 288 ≤ T ≤ 303 K, the difference with eq (2) (Walker, 1964) was always less than 0.1%. Similarly, with a 45 wt % solution, at the same temperatures, the difference between eqs (8) and (3) (Walker, 1964) was no more than 0.3%. For eq (8) an average absolute residual (AAR) of 0.14% was found, with a maximum absolute residual (MAX) of 0.69%. The AAR is calculated from ∑ = ∆ = n i i n 1 1 AAR . (9) More importantly, however, the residuals obtained with eq (8) do not show any systematic variation with T or W F ; see Figs 2 and 3. Therefore, eq (8) is a reliable empirical equation for ρ m . With eq (4) (Kirk-Othmer Encyclopedia of Chemical Technology, 1994) an AAR of 0.42% (MAX of 1.8%) was observed, which is 3 times as high as the value of eq (8). Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 106 280 300 320 340 T (K) r e l a t i v e r e s i d u a l s ( % ) 4 2 0 -2 -4 Fig. 2. Relative residuals of the empirical density correlation (8) as a function of T. Symbols: see Fig. 1. 0 20 40 60 W (wt%) F r e l a t i v e r e s i d u a l s ( % ) 4 2 0 -2 -4 Fig. 3. Relative residuals of the empirical density correlation (8) as a function of W F . Symbols: see Fig. 1. Many literature methods for the calculation of liquid mixture densities use the critical properties and acentric factors, i.e., vapour pressure vs. temperature correlations, of the individual components (p.e. Reid et al., 1988) and are therefore not suitable here because the required properties of the higher poly(oxymethylene) glycols (POMs) are unknown. Although Amagat’s law originally holds strictly only for mixtures of ideal gases, it is also recommended for the calculation of liquid densities of mixtures of similar components (Perry et al., 1984), ∑ = i i i m V x V . (10) Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 107 To apply eq (10), the composition of the liquid has to be considered. In aqueous solutions, formaldehyde is hydrated to methylene glycol and a series of POMs: 2 2 2 2 (OH) CH O H O CH + , (11) ) .. 1 ( O H O) HO(CH (OH) CH O)H HO(CH 2 1 i 2 2 2 2 ∞ = + + + i . (12) Methylene glycol and the POMs only exist in formaldehyde solutions. They cannot be isolated in a pure form, and their pure-component properties cannot be measured directly. The equilibrium of eq (11) is far to the right and the concentration of free formaldehyde in aqueous solutions is negligible compared to those of methylene glycol and the higher POMs. Then, for the formaldehyde-water system, Amagat’s law can be written as ∑ ∞ = + + = 1 ) ( i WFi F W W W m x iV V x V V , (13) where it is assumed that the molar volumes of the POMs can be written as the sum of the volumes of the constituent groups. The subscript WFi denotes HO(CH 2 O) i H, i.e., the component consisting stoichiometrically of water and i formaldehyde units. With the molar balance 1 1 = + ∑ ∞ = i WFi w x x , (14) the overall formaldehyde balance ∑ ∑ ∞ = ∞ = + + = 1 1 ) 1 ( ~ i WFi W i WFi F x i x ix x , (15) and the substitution ρ / M V = , eq (13) can be rewritten as F F F W W m m V x x M M ~ 1 ~ − + = ρ ρ . (16) Because every molecule in the solution is either a free water molecule or a water molecule chemically bonded to one or more formaldehyde units, the true total concentration in the solution is equal to the overall water concentration. Therefore, the true mean molar weight of the solution, M m , can be obtained as ) 100 / ( 1 F W m W M M − = , (17) Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 108 and the model equation for the density of aqueous formaldehyde solutions, from inserting eqs (6) and (17) in eq (16) and rewriting, becomes F F W F F F W m V W M W M ρ ρ ρ + − = ) 100 ( 100 . (18) The only parameter in eq (18) to be determined from the experimental data is the molar volume of the CH 2 O groups in the POM molecules, V F . By taking V F constant, an AAR of 0.45% was obtained (MAX of 1.7%). Not surprisingly, however, taking V F constant resulted in a clear trend of the residuals of eq (18) as a function of T, varying in the expected direction, i.e., from negative values at the lower temperatures to positive values at the higher temperatures. Because of the clear trend of the residuals, a second parameter to account for the influence of T on V F seems justified. Least-squares analysis of the experimental data according to eq (18) resulted in the following optimum parameters for V F : T V F 6 3 10 59 . 30 10 709 . 12 − − × + × = . (19) Figs 4 and 5 illustrate the relative residuals of ρ m calculated with eqs (18) and (19) as a function of W F and T. No trend in the residuals was found. Here, an AAR of 0.22% was found (MAX of 0.69%). At first glance, eqs (8) and (18) might seem paradoxical: eq (8) correlates ρ m linearly with W F , while eq (18) correlates 1/ρ m similarly. This is not a true inconsistency because the coefficient of W F is positive in eq (8), resulting in an increase of ρ m with an increase of W F , whereas the overall coefficient of W F in the denominator of eq (18) is negative, giving the same direction of variation of ρ m with W F . 280 300 320 340 T (K) r e l a t i v e r e s i d u a l s ( % ) 4 2 0 -2 -4 Fig. 4. Relative residuals of the density correlation obtained from Anmagat’s law (eq 18) as a function of the temperature. Symbols: see Fig. 1. Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 109 0 20 40 60 W (wt%) F r e l a t i v e r e s i d u a l s ( % ) 4 2 0 -2 -4 Fig. 5. Relative residuals of the density correlation obtained from Anmagat’s law (eq 18) as a function of W F . Symbols: see Fig. 1. Viscosity Table 2 summarizes literature data on the viscosity of aqueous formaldehyde solutions. Because this data set is rather limited, we performed additional viscosity measurements with a Schott automated viscosity meter (described in more detail by Soliman & Marschall, 1990) The solutions were prepared by dissolving a desired amount of paraformaldehyde (Janssen Chimica) in distilled water. By keeping high efflux times (120-360 s), the error due to kinetic energy was assumed negligible. Although the vapour pressure of pure formaldehyde at the highest temperature of the measurements, 325 K, is more than 1.1 MPa (Reid et al., 1988), its concentration is so low because of the reactions (11) and (12) that the formaldehyde vapour pressure over a 33 wt % solution is only approximately 1 kPa (Maurer, 1986). Thus, the influence of possible evaporation of formaldehyde on the measurements is neglected. The viscometer was calibrated at each temperature using pure water. The absolute viscosity was determined from the measured kinematic viscosity using the density obtained from eq (8). The results are shown in Table 3, where each data point is the mean of three measurements whose flow times were within 0.15 s. The total uncertainty of the viscosity data was estimated to be ±1.5%. Table 2. Literature data on the viscosity of aqueous formaldehyde solutions. T (K) W F (wt %) 3 10 × m η (Pa s) data points 1 298, 333 5-50 0.54-1.87 16 4 288, 298, 308 1.6-17 0.7487-1.6086 15 all data 288-333 1.6-50 0.54-1.87 31 Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 110 Table 3. New experimental data on the viscosity of aqueous formaldehyde solutions. 3 10 × m η (Pa s) W F (wt %) T = 297.85 K T = 307.15 K T = 318.05 K T = 325.25 K 5 1.0295 0.8417 0.6830 0.6011 15 1.2853 1.0537 0.8470 0.7465 25 1.6377 1.3365 1.0619 0.9279 33 2.0456 1.6483 1.3129 1.1277 Over a wide temperature range, the logarithm of the kinematic viscosity, η/ρ, uses to correlate linearly with 1/T for pure liquids (Reid et al., 1988). This appears also to hold for formaldehyde solutions for a constant W F . The influence of the composition could be accounted for by correlating ) / ln( m m m M ρ η linearly both to 1/T and W F . Finally, from an analysis of the residuals it was found that an additional term, linearly with T, was needed to obtain a correlation free of trending effects of the residuals. The empirical correlation developed this way is T W T M F m m m 0404 . 0 10 36 . 9 5644 90 . 47 ln 3 + × + + − = − ρ η , (20) with an AAR of 1.8% (MAX of 7.5%) for 288 ≤ T ≤ 333 K. ρ m and M m are obtained from eqs (8) and (17), respectively. Equation (20) is illustrated in Fig. 6. The residuals of eq (20) did not show any clear trend as a function of W F or T; see Figs 7 and 8. 0 20 40 60 W (wt%) F 100 80 60 40 20 Walker Lileev c.s. Perry c.s. this work η ρ M m m m x 1 0 9 k m o l . m k g . s 2 ( ) Fig. 6. η m /ρ m M m as a function of W F for various T. Symbols: η m from the sources indicated, ρ m and M m from eqs (8) and 17, respectively. Lines: η m /ρ m M m calculated with the empirical viscosity correlation (20). Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 111 0 20 40 60 W (wt%) F 20 10 0 -10 -20 r e s i d u a l ( % ) Fig. 7. Relative residuals of the empirical viscosity correlation (20) as a function of W F . Symbols: see Fig. 6. 20 10 0 -10 -20 r e l a t i v e r e s i d u a l ( % ) 280 300 320 340 T (K) Fig. 8. Relative residuals of the empirical viscosity correlation (20) as a function of T. Symbols: see Fig. 6. In addition to eq (20), we also tested an Antoine-type of temperature dependency, augmented with a linear term in W F , i.e., ln(η m /ρ m M m ) = p 1 +p 2 /(T+p 3 )+p 4 W F . After optimization of the parameters using nonlinear regression, the same AAR (1.8%) was observed; however, MAX was somewhat larger (8.6%) as compared to eq (20). Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 112 The methods found in the literature for obtaining the viscosity of liquid mixtures are often based on the mole fraction average of the logarithms of the pure-component viscosities, extended with various types of correction factors (Perry et al., 1984; Reid et al., 1988). Applying mole fraction averaging to the formaldehyde-water system gives ∑ ∞ = + = 1 ln ln i WFi WFi W w m x x η η η . (21) In the literature it is shown that for various homologous series the logarithm of the pure- component viscosities varies linearly with the molecular size (p.e. Chase, 1984; Allan & Teja, 1991; Nhaesi & Asfour, 1998). This concept cannot be tested directly for methylene glycol and the POMs, because they cannot be obtained in pure form. However, experimental viscosity data are available for the closely related series of ethylene glycol and the poly(ethylene glycols) HO(CH 2 CH 2 O) i H or PEG i . Here, we will use these data just to illustrate the concept before returning attention to the aqueous formaldehyde solutions. We found that for 294 ≤ T ≤ 333 K the viscosities of PEG i can be described by ib a i PEG + = η ln , (22) with T a / 3406 60 . 15 + − = and T b / 8 . 132 1925 . 0 + − = . Fig. 9 shows experimental data of the viscosities of PEG i (i=1..6) and the straight lines calculated with eq 22. Although the viscosity of the monomer, ethylene glycol, deviates somewhat, the overall agreement is satisfactory considering the simplicity of the correlation. 10 10 10 -1 -2 -3 1 2 3 4 5 6 molecular size i η ( P a . s ) Lee & Teja Bohne c.s. Fig. 9. Viscosity of poly(ethylene glycols) as a function of the molecular size i at various T. Symbols: experimental data from the sources indicated. Lines: calculated with eq (22). Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 113 Assuming this concept also applies to the series of methylene glycol and the higher POMs gives iB A WFi + = η ln . (23) With eqs (14), (15) and (23), eq (21) can be rewritten as B x x A x x F F W W W m ~ 1 ~ ) 1 ( ln ln − + − + = η η . (24) Least-squares regression of the experimental data to eq (24) resulted in the following parameter values: T A / 7174 97 . 17 − = , T B / 5048 72 . 14 + − = . (25) The true molar fraction of water, x W , in the solutions was calculated by solving the equilibrium equations for the reactions (12) 2 2 1 2 K x x x WF W WF = , (26) ) 3 ( 3 1 1 ≥ = − i K x x x x WF WFi W WFi , (27) simultaneously with the balances (14) and (15), where F x ~ was obtained from eq 6. The equilibrium constants K 2 and K 3 for the formaldehyde-water system were taken from Hahnenstein et al. (1994) 0 20 40 60 W (wt%) F η m x 1 0 ( P a . s ) 3 2 1 0.5 Fig. 10. Viscosity of aqueous formaldehyde solutions. Symbols: experimental data, see Fig. 6. Lines: calculated with eq (24). Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 114 The accuracy of eq (24) is comparable to that of eq (20) (AAR of 2.0% and MAX of 7.6%). The correlation is illustrated in Fig. 10. The residuals do not show any trend as a function of T or W F as shown in Figs (11) and (12). When applied to all of the data, the errors of eq (5) (Walker, 1964) for the viscosity of aqueous formaldehyde solutions were large (AAR of 15.6% and MAX of 93%). When only the data within the ranges of W F = 30-50 wt % and T = 298-313 K were considered, an AAR of 3.5% (MAX of 6.2%) was obtained, thereby demonstrating the more limited applicability of eq (5). 0 20 40 60 W (wt%) F 20 10 0 -10 -20 r e s i d u a l ( % ) Fig. 11. Relative residuals of the viscosity correlation (24) as a function of W F . Symbols: see Fig. 6. 20 10 0 -10 -20 r e l a t i v e r e s i d u a l ( % ) 280 300 320 340 T (K) Fig. 12. Relative residuals of the viscosity correlation (24) as a function of T. Symbols: see Fig. 6. Appendix A: Correlations for the density and viscosity of aqueous formaldehyde solutions. 115 Conclusions The density and viscosity of aqueous formaldehyde solutions can be accurately and reliably obtained as a function of the temperature and the strength of the solution with the simple empirical correlations obtained here. The empirical density correlation (eq 8) employs two adjustable parameters that were optimised using three literature sources of density data. The empirical viscosity correlation (eq 20) has four coefficients that were optimised using two literature data sources together with a series of new additional measurements. The residuals of the correlations presented are free of trending effects as a function of both the temperature and the weight percentage of formaldehyde. Therefore, we conclude that the correlations can be used reliably in engineering calculations with a small extrapolation to cover the entire range of conditions prevailing in formaldehyde absorbers, i.e., 280 ≤ T ≤ 340 K and 0 ≤ W F ≤ 60 wt %. A mixture density correlation method from the literature, where the molar volume of the mixture is obtained as the molar fraction average of the pure-component molar volumes, appeared to represent the data with almost the same accuracy. In this case two coefficients were fitted to the data, to correlate the molar volume of the CH 2 O groups linearly to the temperature. Similarly, a literature method for liquid mixture viscosities, where the logarithm of the pure-component viscosities are molar fraction averaged, resulted in almost the same accuracy relative to the empirical correlation. In this case, it was assumed that the logarithm of the viscosities of the homologous series of methylene glycol and the higher POMs varies linearly with the molecular size of the components. This way, the molar fraction average method contains two temperature-dependent parameters, i.e., four adjustable coefficients. At first glance it seems surprising that the empirical relations for the density (eq 8) and viscosity (eq 20) both result in somewhat lower AAR values as compared to the relations that were arrived at starting from methods found in the literature (eqs 18, 19 and 24, respectively), even though in both cases the same number of coefficients were adjusted to the experimental data. This may reflect, however, the difficulties still encountered at present in the development of theory applicable to estimating liquid mixture properties. Appendix B: Equilibrium molar fractions in aqueous methanolic formaldehyde solutions 117 Appendix B Equilibrium molar fractions in aqueous methanolic formaldehyde solutions In aqueous mixtures of formaldehyde and methanol, the equilibrium composition is determined by a series of reactions: 2 2 2 2 (OH) CH O H O CH + , (1) O H H O) HO(CH (OH) 2CH 2 2 2 2 2 + , (2) ) 3 (i O H H O) HO(CH (OH) CH H O) HO(CH 2 i 2 2 2 1 - i 2 ∞ = + + L , (3) OH OCH CH OH CH O CH 2 3 3 2 + , (4) ) 2 (i OH CH H O) O(CH CH OH OCH CH H O) O(CH CH 3 i 2 3 2 3 1 - i 2 3 ∞ = + + L . (5) The equilibrium conditions, in terms of molar fractions, for the reactions read W F WF x x x K 1 1 = , (6) 2 2 1 2 WF W WF x x x K = , (7) ) 3 ( 1 1 3 ∞ = = − L i x x x x K WF WF W WF i i , (8) M F MF x x x KM 1 1 = , (9) ) 2 ( 1 1 2 ∞ = = − L i x x x x KM MF MF M MF i i . (10) In addition, three overall balances combine the overall molar fractions, F x ~ , W x ~ and M x ~ , with the true molar fractions in the mixture: ∑ ∑ ∑ ∑ ∞ = ∞ = ∞ = ∞ = + + + + = 1 1 1 1 1 ~ i MF i WF i MF i WF F F i i i i ix ix ix ix x x , (11) Appendix B: Equilibrium molar fractions in aqueous methanolic formaldehyde solutions 118 ∑ ∑ ∑ ∞ = ∞ = ∞ = + + + = 1 1 1 1 ~ i MF i WF i WF W W i i i ix ix x x x , (12) ∑ ∑ ∑ ∞ = ∞ = ∞ = + + + = 1 1 1 1 ~ i MF i WF i MF M M i i i ix ix x x x . (13) The set of eqs (6)-(13), in principle, provides enough information to calculate all molar fractions. However, we have an infinite number of eqs (8) and (10), and the summations in (11)- (13) have no upper limit. These problems can be overcome by using some simple properties from the theory of power series. Thus, by substitution of the molar fractions of the reaction products obtained from (7)-(10), the summations in (11)-(13) can be written as 1 1 1 WF i WF x S x i = ∑ ∞ = , v K v K S 3 2 1 1 1 − + = , (14) 1 2 1 WF i WF x S ix i = ∑ ∞ = , 2 3 2 3 2 2 ) 1 ( 1 1 v K v K v K v K S − + − + = , (15) M i MF x Sm x i 1 1 = ∑ ∞ = , v KM KM K v KM Sm 2 1 1 1 1 − = , (16) M i MF x Sm ix i 2 1 = ∑ ∞ = , 2 2 1 1 1 1 2 ) ( v KM KM K v K KM Sm − = . (17) where the quantities S 1 , S 2 , Sm 1 and Sm 2 are introduced for ease of notation. With (14)-(17) the overall balances, eqs (11)-(13) can be rewritten as ) 1 ( ) ~ 1 ( ) ( ~ ) 1 )( ~ ( 1 2 1 1 2 1 1 1 Sm S x K K v Sm x Sm v x K x F M F WF + − − + + − = , (18) 2 1 2 ~ 1 ) 1 ( ~ 1 Sm x Sm x S x x M WF M M − + + = , (19) M W WF W W W x Sm x x S S x x x 2 1 2 ~ ) ~ ( ~ 1 + − + = , (20) where v is defined as W WF x x v 1 = . (21) Appendix B: Equilibrium molar fractions in aqueous methanolic formaldehyde solutions 119 The set of eqs (18)-(21) can easily be solved for 1 WF x , M x and W x by iteration on v, where v is limited to 1 0 < ≤ v . The other molar fractions, F x , ) 2 ( ≥ i x i WF and ) 1 ( ≥ i x i MF , can be obtained from (6)-(10) straight forward. Figure 1 illustrates the smooth variation of the ratio v with the overall formaldehyde molar fraction. Here, 1 K was taken from Winkelman et. al (2002), 2 K , 3 K and 2 KM were taken from Hahnenstein et. al (1995), and 1 KM was obtained by multiplying WM K (Hahnenstein et. al, 1995) and 1 K , where WM K the equilibrium constant is of the reaction O H OH OCH CH OH CH (OH) CH 2 2 3 3 2 2 + = + . 0.0 0.1 0.2 0.3 0.4 0.5 0.20 0.16 0.12 0.08 0.04 0.00 x ~ F v T Fig. 1. Variation of the ratio v with F x ~ at temperatures of 300, 320 and 340 K. Solid lines: 0 ~ = M x ; dotted lines: 05 . 0 ~ = M x . Simplifications 1. Free formaldehyde is not important If the very small molar fraction of free formaldehyde is not important, then eq (18) reduces to ) 1 )( ~ 1 ( ~ ) 1 ( ~ 1 2 2 1 1 Sm x S x Sm Sm x x F M F WF + − − + = . (22) The system now consists of eqs (19)-(22), and can be solved in the same way as before, by iteration on v. Appendix B: Equilibrium molar fractions in aqueous methanolic formaldehyde solutions 120 2. No methanol present If the mixture does not contain any methanol then of course M x ~ , M x and ) 1 ( ≥ i x i MF all are zero, and eqs (18) and (20) for obtaining 1 WF x and W x reduce to 2 1 1 ) ~ 1 ( ) ~ ( 1 S x K v x K x F F WF − − = , (23) 1 ) ~ ( ~ 1 2 WF W W W x S S x x x − + = . (24) For this case, Fig. 2 illustrates the relative amount of methylene glycol with increasing overall formaldehyde content in aqueous solutions. The figure shows that at low concentrations, say below 1 mmol/l, virtually all the formaldehyde is present as methylene glycol, and the amount of poly oxymethylene glycols is negligible. 0.0 0.1 0.2 0.3 0.4 0.5 1.0 0.8 0.6 0.4 0.2 0.0 x ~ F T x ~ F x WF 1 Fig. 2. The relative amount of methylene glycol at temperatures of 300, 320 and 340 K. If also the very small molar fraction of free formaldehyde is not important, p.e. in the calculation of the viscosity (see Appendix A), then the ratio v can be obtained from the cubic equation 0 4 3 2 2 3 1 = + + + a v a v a v a , (25) with the coefficients ) ~ 2 1 )( ( 2 3 3 1 F x K K K a − − = , (26) ) )( 2 ~ 3 ( ) 1 ( ~ 2 3 3 3 2 K K x K x K a F F − − + − = , (27) ) 1 ( ~ 2 1 3 3 K x a F − − = , (28) F x a ~ 4 − = . (29) Appendix B: Equilibrium molar fractions in aqueous methanolic formaldehyde solutions 121 The physically significant root of eq (24) can easily be identified: either the cubic has only one real root, or the cubic has only one root in the correct region, i.e., 1 0 < ≤ v . The true molar fractions of water and methylene glycol are now obtained from ) 2 ( ) 1 ( ) 1 ( ) 1 ( ~ 1 ~ 1 3 2 2 3 3 2 2 3 v K v K v K v K v K v K x x x F F W − + − − + − − − = , (30) v x x W WF = 1 . (31) A further remark: if the overall molar fractions of formaldehyde and water are exactly equal, i.e. 5 . 0 ~ ~ = = W F x x , then the cubic equation (25) degenerates to a quadratic one in v, and true molar fractions of water and methylene glycol can be obtained directly from 3 2 2 1 1 1 K K x WF + + = , (32) ) 1 ( 1 2 1 K x x WF W + − = . (33) Appendix C: The reaction order of formaldehyde in the hydration. 123 Appendix C The reaction order of formaldehyde in its hydration reaction. Method The measurements described in Chapter 4 can be used to obtain the reaction order of formaldehyde in the hydration, as well as the reaction rate constant. Note that the experimental conditions and measured data allowed for the calculation of the interface concentrations and the enhancement factors, i.e. the gradients at the interface, of formaldehyde and methylene glycol without any information on the kinetics of the hydration reaction. These quantities are indexed here as observed. To establish the reaction order of formaldehyde in the hydration, the equations for diffusion with parallel reaction in the liquid film are used F F F R dx C d D = 2 2 ) 0 ( δ ≤ ≤ x , (1) F MG MG R dx C d D − = 2 2 ) 0 ( δ ≤ ≤ x , (2) with the boundary conditions observed IF MG x MG observed IF F x F C C C C ) ( ) ( ; ) ( ) ( , 0 , 0 = = = = , (3) MG x MG F x F C C C C = = = = δ δ ) ( ; ) ( . (4) Here, the rate of the reaction is written as MG d n F h F C k C k R F − = ) ( , (5) where F n denotes the reaction order of formaldehyde. The additional condition observed x F x F dx dC dx dC , 0 0 ) ( ) ( = = = , (6) allows the determination of the reaction rate constant, h k . The gradient of methylene glycol at the interface is not independent, but is determined by the one of formaldehyde and the interface concentrations. This can easily be seen by adding eqs (1) and (2), integrating twice, and applying boundary conditions (3) and (4), giving Appendix C: The reaction order of formaldehyde in the hydration. 124 ] ) ( [ ) ( , , , , MG IF MG F IF F F IF F IF MG MG C C C C v x C C v C C − + − − − + = δ , (7) and ] ) ( [ 1 ) ( ) ( , , 0 0 MG IF MG F IF F x F x MG C C C C v dx dC v dx dC − + − − − = = = δ . (8) Thus, for individual experiments we have no further information available to determine F n . Therefore, the following strategy was adopted. For a given value of F n , the reaction rate constants, ) ( F n h k , were calculated for all experiments by solving eqs (1)-(7) (see below). Next, the individual rate constants were fitted to an Arrhenius type expression RT E F n h a e k k / ) ( ˆ − ∞ = , (9) and the mean absolute relative residual, marr, of the reaction rates was calculated i all eriments i F n h F n h F n h k k k marr ∑ = − = exp 1 ) ( ) ( ) ( ˆ . (10) This procedure was repeated for F n values ranging from 0.0 to 2.0. Analytical and approximate analytical solutions In general, the equations (1)-(7) can be solved numerically only. However, an approximated analytical solution for the enhancement factor can be obtained by linearization of F n F C ) ( according to ) 1 /( ) ( 2 1 , + − F F n IF F n C C F . Argument for this linearization is found in the solution for irreversible nth order kinetics, which, to a good approximation, equals the solution for first order kinetics, provided that the reaction rate constant, k, is replaced by ) 1 /( 2 1 + − n kC n IF (Westerterp, Van Swaaij & Beenackers, 1984). This way, the enhancement factor is very similar to the analytical solution obtained by Winkelman & Beenackers (1993) for first order reversible reactions, ' ] ' tanh[ ) ' ( ) ] ' cosh[ 1 1 ( ' ) ' )( ' ] ' tanh[ 1 ( 1 , , , φ φ φ φ φ v K C C C C K C C C C K E F IF F MG F F IF F MG IF MG F + − − − + − − − − + = , (11) Appendix C: The reaction order of formaldehyde in the hydration. 125 however, here with ' ) ' ( ' ' K D v K k F + = δ φ , (12) 1 , ) ( 1 2 ' − + = F n IF F F h C n k k , (13) 1 , ) ( ) 1 ( 2 ' ' − + = = F n IF F d F h d C k n k k k K . (14) For a given value of F n , the reaction rate constants were obtained from eqs (11)-(14) iteratively, using the experimental data and the observed values of the formaldehyde enhance- ment factors. The simple secant iteration method proved adequate for this purpose. Note that for a first order reaction in formaldehyde, i.e. 1 = F n , eqs (11)-(14) represent the exact analytical solution. The only other case that allows for an exact solution is the zero order reaction, 0 = F n , where the enhancement factor is given by 0 0 , , 0 0 , 0 0 ] tanh[ ) ] tanh[ 1 ( / ) ] cosh[ 1 1 ( 1 ) ( = = = = = = − − − − − − − + = F F F F F F n n F IF F MG IF MG n n F IF F MG d h n n F v C C C C C C C k k E φ φ φ φ φ (15) where MG d n D k F δ φ ≡ =0 . (16) Numerical solution Equation (1), with F R given by (5), MG C by (7), and the boundary conditions by (3), (4) and (6), was solved for the formaldehyde concentration profile and the reaction rate constant simultaneously by replacing the differential equation by finite difference equations on a grid of mesh points on the interval ) 0 ( δ ≤ ≤ x . Here, j F C ) ( denotes F C at mesh point j, i.e. at ) ( x j x ∆ = , where N j L 1 = and N x / ) ( δ = ∆ . A finite difference approximation of eq (1) with second order accuracy reads j F j F j F j F F R x C C C D ) ( ) ( ) ( ) ( 2 ) ( 2 1 1 = ∆ + − + − ) 1 1 ( − = N j L . (17) Appendix C: The reaction order of formaldehyde in the hydration. 126 The boundary conditions (3) and (4) give two more equations IF F F C C , 0 ) ( = , (18) F N F C C = ) ( . (19) An additional equation is obtained from a second order Taylor series approximation of 1 ) ( F C : 0 2 2 2 0 0 1 ) ( 2 ) ( ) )( ( ) ( ) ( dx C d x dx dC x C C F F F F ∆ + ∆ + = . (20) The second derivative in (20) is equal to F F D R / ) ( 0 , see eq (1), while the first derivative is set equal to the observed gradient at the interface. The 2 + N equations (17)-(20) can be solved for the unknowns h k and ) 0 ( ) ( N j C j F L = . Because of the nonlinearity in the reaction rates j F R ) ( Newton-Raphson iteration was used. For this purpose, the equations, labeled by j F , are written as 0 ) ( 2 ) ( ) )( ( ) ( ) ( 0 2 , 0 0 1 1 = ∆ − ∆ − − = = − F F observed x F F F D R x dx dC x C C F , (21) 0 ) ( , 0 0 = − = IF F F C C F , (22) 0 ) ( ) ( ) ( ) ( 2 ) ( 2 1 1 = ∆ − + − = + − j F F j F j F j F j R D x C C C F ) 1 1 ( − = N j L , (23) 0 ) ( = − = F N F N C C F . (24) The vector of unknowns T N F F h C C k ] ) ( , , ) ( , [ 0 L = y is updated with a correction y ∆ , i.e. y y y ∆ + = current new , until convergence is achieved, where the vector of corrections y ∆ is obtained from the matrix equation ∑ − = − = − = ∆ ∂ ∂ N k j k k j N j F y y F 1 ) 1 ( L . (25) Results The calculated marr data from eq (10), obtained with the approximate analytical solution and with the numerical method, are shown in Fig. 1 below as a function of the reaction order of formaldehyde, F n . The data show a clear minimum around 1 = F n , allowing the conclusion that the hydration is indeed of the first order in formaldehyde. Appendix C: The reaction order of formaldehyde in the hydration. 127 A second conclusion is that the results obtained with the approximate analytical solution method are virtually identical to those obtained from the numerical method. Therefore, at least at the circumstances considered here, the approximate analytical method is suitable for calculating mass transfer enhancement factors. Fig. 1. Marr of the reaction rate constants, see eq (10), vs. the order of formaldehyde in the hydration. Line: numerical solution; symbols: approximate analytical solution. 0.0 0.4 0.8 1.2 1.6 2.0 40 30 20 10 0 reaction order of formaldehyde m a r r [ % ] List of publications 129 List of publications The following publications originated from this work: J.G.M. Winkelman, H. Sijbring, A.A.C.M. Beenackers & E.T. De Vries (1992). Modeling and simulation of industrial formaldehyde absorbers. Chemical Engineering Science, 47, 3785. (included as Chapter 5) J.G.M. Winkelman, S.J. Brodsky, & A.A.C.M. Beenackers (1992). Effects of unequal diffusivities on enhancement factors for reversible reactions: numerical solutions and comparison with DeCoursey’s method. Chemical Engineering Science, 48, 2951-2955. J.G.M. Winkelman & A.A.C.M. Beenackers (1993). Simultaneous absorption and desorption with reversible first-order chemical reaction: analytical solution and negative enhancement factors. Chemical Engineering Science, 48, 2951-2955. (included as Chapter 3) J.G.M. Winkelman, M. Ottens & A.A.C.M. Beenackers (2000). The kinetics of the dehydration of methylene glycol. Chemical Engineering Science, 55, 2065-2071. (included as Chapter 2) J.G.M. Winkelman & A.A.C.M. Beenackers (2000). Correlations for the density and viscosity of aqueous formaldehyde solutions. Industrial & Engineering Chememistry Research, 39, 557-562. (included as Appendix A) J.G.M. Winkelman, O. Voorwinde, M. Ottens, A.A.C.M. Beenackers & L.P.B.M. Janssen (2002). The kinetics and chemical equilibrium of the hydration of formaldehyde. Chemical Engineering Science, 57, 4067-4076. (included as Chapter 4) Samenvatting in het Nederlands 131 Samenvatting in het Nederlands Deze dissertatie beschrijft theoretisch en experimenteel werk aan de absorptie van formaldehyde in water. Met resultaten hiervan zijn chemisch-technische modellen ontwikkeld voor de beschrijving en optimalisatie van industriële formaldehydeabsorbeurs. Deze samenvatting geeft eerst algemene informatie over formaldehyde, en de commerciële productie ervan. Daarna wordt ingegaan op het doel van dit werk, en vervolgens het uitgevoerde onderzoek en de resultaten. Formaldehyde Formaldehyde is een belangrijke grondstof in de chemische industrie. Het is met name een hoofdbestanddeel van veel soorten kunststoffen en -harsen. Daarnaast worden kleinere hoeveelheden formaldehyde gebruikt als ontsmettings- en conserveringsmiddel (‘sterk water’), en bij de productie van rubbers, speciale betonsoorten, explosieven, meststoffen, genees- middelen, papier, etc. In 2000 werd wereldwijd ongeveer 10 miljoen ton formaldehyde geproduceerd. In zuivere vorm is formaldehyde een gas. Het kan echter niet in zuivere vorm worden opgeslagen of vervoerd omdat formaldehydegas niet stabiel is. Het zuivere gas reageert snel, bijvoorbeeld met de wand van een container waarin het is opgeslagen, tot een onbruikbare vaste stof. Daarom wordt formaldehyde vrijwel uitsluitend geproduceerd, verhandeld en vervoerd als een oplossing van het gas in water. Industriële productie van formaldehyde Bij de industriële productie van formaldehyde is methanol, ofwel methylalcohol, de grondstof. De methanol wordt verdampt en gemengd met lucht. Dit gasmengsel gaat naar een reactor. Methanoldamp reageert in de reactor met zuurstof uit de lucht tot formaldehyde. Het gas dat uit de reactor komt, bestaat voornamelijk uit stikstof en formaldehyde. Soms bevat het gas ook nog een hoeveelheid niet-omgezette methanol. Dit gasmengsel wordt naar een absorbeur geleid waarin het formaldehydegas oplost in water. Hierbij onstaat het commerciële product: een gecon- centreerde oplossing van formaldehyde in water, ofwel formaline. In de praktijk wordt formaline vaak verhandeld met een sterkte van 37 of 55 gewichtsprocent formaldehyde. methanol- verdamper reactor absorbeur formaline afgas lucht methanol water Fig. 1. Belangrijke stappen in de industriële productie van formaline. Samenvatting in het Nederlands 132 Absorbeur Een absorbeur wordt gebruikt om een gasvormige stof op te lossen in een vloeistof. Het gas wordt onderaan het kolomvormige apparaat naar binnen geleid, en stroomt naar boven. De vloeistof wordt bovenaan geïntroduceerd, en stroomt neerwaarts. Om zoveel mogelijk gas op te lossen is het voordelig om de neerstromende vloeistof en het omhoog stromende gas intensief met elkaar in contact te brengen. Dit kan onder meer worden bereikt door de absorbeur te vullen met een pakking. In de praktijk wordt vaak een gestorte pakking gebruikt: de kolom wordt gevuld met een willekeurige stapeling van bijvoorbeeld bolletjes, ringen, of één van de vele andere vormen die commercieel beschikbaar zijn. Door de vloeistof over de pakking te versprei- den ontstaat een groot contactoppervlak tussen de vloeistof en het gas. In het onderste deel van een absorbeur wordt de oplossing verzameld, zodat deze kan worden afgevoerd voor verdere verwerking of opslag. Bij formaldehydeabsorbeurs wordt een deel van de oplossing die beneden aankomt weer teruggevoerd naar de bovenzijde van de absorbeur. De vloeistof loopt dan nogmaals door de kolom, en absorbeert meer gas. Dit leidt tot een meer geconcentreerde oplossing. In figuur 2 is een voorbeeld te zien van een absorbeur met een vloeistofterugvoer. De pakking is in de figuur schematisch aangeduid met bolletjes. afgas water gas oplossing koel- water Fig. 2. Een absorbeur met één absorptiebed, en extern gekoelde vloeistofterugvoer. Samenvatting in het Nederlands 133 Over het algemeen werken absorbeurs efficiënter als het gas goed oplost in de vloeistof, en minder efficiënt als de oplosbaarheid van het gas klein is. Formaldehyde is schijnbaar goed oplosbaar in water, gezien het feit dat oplossingen met wel 55 gewichtsprocent formaldehyde gangbaar zijn. Maar de formaldehydeabsorbeurs opereren minder efficiënt dan op grond van deze goede schijnbare oplosbaarheid verwacht mag worden. De belangrijkste reden hiervoor is dat in de oplossing het formaldehyde met water reageert tot methyleenglycol. Deze hydratatiereactie is een evenwichtsreactie: methyleenglycol kan ook weer terugreageren tot formaldehyde en water. Op zijn beurt kan methyleenglycol weer verder reageren tot een serie polymere vormen van formaldehyde, de zogenaamde poly-oxy-methyleenglycolen. Deze polymerisatiereacties zijn langzaam, en het zijn ook weer evenwichtsreacties, zie figuur 3. methyleen- formaldehyde h ydratatie d e h ydrata tie water glycol methyleen- glycol p o ly merisa tie d e p o lymerisa t ie poly-oxy- Fig. 3. Reacties van formaldehyde opgelost in water. De goede schijnbare oplosbaarheid van formaldehyde in water is dus eigenlijk de goede oplosbaarheid van methyleenglycol en de capaciteit van de oplossing om poly-oxy-methyleen- glycolen op te nemen. Formaldehyde zelf is, zoals de meeste gassen, slecht oplosbaar in water. De hydratatiereactie is relatief snel, en bij evenwicht is de hoeveelheid vrij formaldehyde veel kleiner dan de hoeveelheid methyleenglycol: het hydratatie-evenwicht ligt sterk aan de zijde van methyleenglycol. Dit zorgt ervoor dat de overdracht van formaldehyde vanuit het gas naar de vloeistof chemisch versneld is. Formaldehydeabsorbeurs werken dus minder efficiënt dan op basis van de goede ‘schijnbare’ oplosbaarheid verwacht mag worden, maar meer efficiënt dan wanneer het formaldehyde niet zou reageren in de oplossing. Een verdere complicerende factor is dat bij de absorptie van formaldehyde en de daaropvolgende hydratatiereactie veel warmte vrij komt. Ook bevat het gas dat de absorbeur binnenkomt een hoeveelheid stoom die, vooral bij condensatie, veel warmte afgeeft . De temperatuur van de neer- stromende vloeistof neemt hierdoor toe. De warmte die vrijkomt wordt afgevoerd door de vloeistofterugvoer te koelen met koelwater (zie fig. 2). Onder meer vanwege dit type complicaties zijn formaldehydeabsorbeurs vaak onderverdeeld in meerdere secties of absorptiebedden. Elke sectie heeft dan een vloeistofverdeler, -opvang en - terugvoer met externe koeling. De vloeistof uit de bodem van de ene sectie wordt voor een deel teruggevoerd naar de top van dezelfde sectie. De rest wordt naar de top van de volgende sectie geleid. Iets dergelijks geldt voor de gasstroom: het gas dat uit de top van een sectie treedt, wordt Samenvatting in het Nederlands 134 aan de onderzijde van een vorige sectie ingevoerd. Er ontstaat een patroon zoals geïllustreerd is voor een tweetal secties in figuur. 4. In de praktijk zijn de secties boven elkaar geplaatst, in één enkele kolom, zodat de installatie compact blijft. afgas water gas koel- water oplossing koel- water Fig. 4. Een schakeling van twee absorptiebedden, elk met een extern gekoelde vloeistofterugvoer. Dit proefschrift Het belangrijkste streven van het onderzoek is de ontwikkeling van betrouwbare chemisch- technische modellen voor industriële formaldehydeabsorbeurs. Het doel hiervan is drieledig. Ten eerste het nauwkeurig beschrijven van het gedrag van bestaande formaldehydeabsorbeurs. Vervolgens het voorspellen van de invloed die veranderingen in de manier van bedrijven hebben op de prestatie van de absorbeurs. En ten slotte het optimaliseren van de absorptie-efficiëntie en - capaciteit van formaldehyde in de absorbeurs. Om deze doelstellingen te bereiken is het van belang om de snelheid te kennen van de reacties in de oplossing (zie fig. 3) bij verschillende concentraties en temperaturen. Anders gezegd: de kinetiek van de reacties moet bekend zijn. De kinetiek van de polymerisatie- en depolymerisatiereacties is uitgebreid onderzocht en in de literatuur beschreven door andere researchgroepen. Over de reactiesnelheid van de hydratatie van formaldehyde en de dehydratatie van methyleenglycol is slechts fragmentarisch gepubliceerd. Er zijn een handjevol data bekend, over het algemeen slechts bij kamertemperatuur. De gegevens die in de literatuur zijn te vinden over de chemische evenwichtsconstante van de hydratatie Samenvatting in het Nederlands 135 vertonen een grote spreiding: als we verschillende publicaties met elkaar vergelijken dan zit daar soms meer dan een factor 3 verschil tussen. Daarom zijn de kinetiek en het chemisch evenwicht van de hydratatie/ dehydratatiereactie nader onderzocht. Dit onderzoek en de resultaten ervan worden, na een inleidend hoofdstuk, beschreven in de hoofdstukken 2, 3 en 4 van dit proefschrift. In de hoofdstukken 5 en 6 worden modellen ontwikkeld voor de simulatie en optimalisatie van industriële formaldehydeabsorbeurs. In een drietal appendices wordt onderzoek naar de fysische eigenschappen van formaline en enig aanvullend materiaal op de eerdere hoofdstukken gepresenteerd. In hoofdstuk 2 is het onderzoek aan de orde naar de reactiesnelheid van de dehydratatie van methyleenglycol. Hierbij is gebruik gemaakt van de snelle reactie van sulfiet met formaldehyde waarbij hydroxide-ionen vrijkomen. De concentratie van de hydroxide-ionen kan eenvoudig worden gemeten. In een oplossing van methyleenglycol en sulfiet wordt formaldehyde gevormd via de dehydratatiereactie. Dit formaldehyde reageert zeer snel met sulfiet. De toename van de concentratie van hydroxide-ionen is dan een maat voor de snelheid van de dehydratatie van methyleenglycol. De reactiesnelheid van de dehydratatie is op deze wijze gemeten bij temperaturen die relevant zijn voor formaldehydeabsorbeurs (circa van 20 tot 60 o C). Hoofdstuk 3 geeft een theoretische verhandeling over de absorptie en/of desorptie van twee componenten samen met een evenwichtsreactie in de vloeistof tussen die twee. De analytische oplossingen die hier worden afgeleid geven de absorptie- en/of desorptiesnelheid als functie van chemische en hydrodynamische eigenschappen van het systeem en de concentraties van de componenten. De theoretische resultaten die hier zijn bereikt worden toegepast in het volgende hoofdstuk. De bepaling van de kinetiek van de hydratatie van formaldehyde wordt beschreven in hoofdstuk 4. De metingen zijn gebaseerd op de chemisch versnelde absorptie van formaldehyde in water en het mathematische model van hoofdstuk 3 voor dit proces. Als een gasstroom met daarin form- aldehydegas over water wordt geleid, dan zal het formaldehyde in het water gaan oplossen. Als het opgeloste formaldehyde niet zou reageren, dan zou een bepaalde mate van verzadiging optreden aan het oppervlak van de vloeistof. Deze mate van verzadiging is mede bepalend voor de oplossnelheid. Nu echter het opgeloste formaldehyde wegreageert via de hydratatiereactie, zal de mate van verzadiging aan het vloeistofoppervlak veel kleiner zijn. Dit resulteert in een hogere oplossnelheid dan wanneer de reactie niet zou optreden: de absorptie is chemisch versneld. De mate waarin de absorptie chemisch versneld wordt, is afhankelijk van de reactiesnelheid. Door de absorptiesnelheid te meten kan via een wiskundig model de reactiesnelheid worden berekend. Op deze manier is de hydratatiesnelheid van formaldehyde gemeten over het traject van 20 tot 60 o C. Met deze resultaten, en die van hoofdstuk 2, is ook de chemische evenwichtsconstante bepaald over hetzelfde temperatuurtraject. De industriële absorptie van formaldehyde in water wordt gekenschetst door simultane stof- overdracht van meerdere componenten, gepaard met meerdere simultane reacties in de vloeistof en aanzienlijke warmte-effecten. In hoofdstuk 5 wordt een model gepresenteerd waarmee dit proces wordt gesimuleerd. Het model is gebaseerd op differentiaalvergelijkingen voor de stof- en energiebalansen in de gasfase en in de vloeistoffase. Voor de resulterende set van gekoppelde grenswaardeproblemen is een stabiele oplosmethode ontwikkeld via een semi-tijdsafhankelijke Samenvatting in het Nederlands 136 benadering. Het model bleek zeer goed in staat om het gedrag van bestaande formaldehyde- absorbeurs te simuleren. Met het model zijn vervolgens de procescondities van een absorbeur bij Dynea B.V. geoptimaliseerd waardoor een hogere capaciteit en een aanzienlijke besparing van de grondstof methanol zijn bereikt. In hoofdstuk 6 wordt het absorbeurmodel uitgebreid met de beschrijving van het gedrag van niet- omgezet methanol dat via de gasstroom de absorbeur binnenkomt. In het model zijn een groot aantal reacties in de vloeistoffase opgenomen, alsmede warmte-uitwisseling tussen de gas- en vloeistofstroom, en chemisch versnelde stofoverdracht van formaldehyde, water, methyleen- glycol, methanol en hemiformal. Het gedrag van industriële formaldehydeabsorbeurs met extern gekoelde vloeistofrecirculatie wordt gesimuleerd met het model van niet-evenwichts-trappen. Dit model is toegepast om de invloed te onderzoeken die de verschillende procesomstandigheden hebben op het gedrag van methanol in de absorbeurs. De appendices geven enig aanvullend materiaal. Zo worden in appendix A de resultaten gepresenteerd van de experimentele bepaling van de viscositeit van formaline bij verschillende temperaturen en verschillende formaldehydegehaltes. Ook worden hier correlaties gegeven om de viscositeit en dichtheid van formaline nauwkeurig te berekenen als functie van de temperatuur en de concentratie. In appendix B worden enkele methoden toegelicht voor de berekening van de evenwichtsamenstelling (zie fig. 3) van oplossingen van formaldehyde in water, en van formaldehyde en methanol in water. Appendix C ten slotte bevat additioneel materiaal bij hoofdstuk 4. Hier wordt aangetoond dat de hydratatiereactie eerste orde in formaldehyde is, ofwel dat de reactiesnelheid van de hydratatie recht evenredig met de concentratie van form- aldehyde is. 137 Dankwoord Iedereen die aan dit proefschrift heeft meegewerkt wil ik van harte bedanken. Enkele mensen wil ik hier speciaal noemen. Allereerst Prof. dr. ir. A.A.C.M. (Ton) Beenackers, die mijn leermeester is geweest. Zijn enthousiasme en vakkennis waren van onschatbare waarde, niet alleen voor het promotieonderzoek, maar ook tijdens de vele projecten waar we nadien samen bij betrokken waren. Zijn plotselinge heengaan blijft ontstellend. Prof. dr. ir. L.P.B.M. Janssen en Prof. dr. ir. H.J. Heeres dank ik voor hun bereidheid de promotie af te ronden en voor hun suggesties ten aanzien van de uiteindelijke tekst. De leden van de leescommissie, Prof. dr. A.A. Broekhuis, Prof. dr. P.D. Iedema en Prof. dr. ir. G.F. Versteeg, dank ik voor hun aandacht en snelle beoordeling. Dynea B.V. heeft dit werk niet alleen financieel ondersteund; zeer nuttig waren ook de uren die ik doorbracht met dhr S. Doorn en dhr E.T. de Vries. Veel werk is gedaan door de afstudeerders Hermen Steven Giesbert, Henk Sijbring, Marcel Ottens en Olaf Voorwinde. Hun bijdragen waren onmisbaar voor de totstandkoming van dit proefschrift. Evenals die van Luuk Balt voor zijn prachtige ontwerpen van de meetopstellingen; Oetse Staal voor zijn inbreng in de automatisering; Karel van der West en Jaap Struik voor het betere sleutel- en schroefwerk; Rob Cornelissen voor de ware kunstwerken in glas; en Jan Henk Marsman voor wie elk analyseprobleem weer een uitdaging bleek. Ten slotte dank ik mijn familie en vrienden voor hun directe en indirecte bijdragen bij de totstandkoming van dit proefschrift, in het bijzonder Piety Groeneveld, Wiveca Jongeneel en Anthony Runia.