Hydraulic Machines D.D.

Prof Dumitru DINUHYDRAULIC MACHINES 2 CONTENTS 1. INTRODUCTORY CONCEPTS 2. HYDRAULIC PUMPS AND MOTORS 2.1 Volumetric pumps 2.1.1 Piston pump 2.1.2 Pumps with radial pistons 2.1.3 Pumps with blades 2.1.4 Pumps with axial pistons 2.1.5 Pumps with sprocket wheels 2.1.6 Others types of volumetric pumps 2.1.7 Characteristics of volumetric pumps 2.2 Hydrodynamic pumps 2.2.1 Building and classification 2.2.2 Turbo pump theory 2.2.3 Turbo pumps in network 2.2.4 Computation of centrifugal pumps 2.2.5 Parallel and series connection of centrifugal pumps 2.2.6 Suction of centrifugal pumps 2.2.7 Axial pumps 2.3 Ejectors 2.4 Volumetric hydraulic motors 2.4.1 Hydraulic cylinders 2.4.2 Motors with radial pistons 2.4.3 Motors with blades 2.4.4 Motors with axial pistons 2.4.5 Oscillating rotary motors 2.5 Turbines 2.5.1 Pelton’s turbine 2.5.2 Francis’ turbine 2.5.3 Kaplan’s turbine 3. CONTROL AND AUXILIARY APPARATUS 3.1 Control apparatus 3.1.1 Distribution apparatus 3.1.2 Flow monitoring apparatus 3.1.3 Pressure monitoring apparatus 3.2 Auxiliary apparatus 3.2.1 Conduits 3.2.2 Filters 3.2.3 Tanks 3.2.4 Accumulators 3 4. MEASURING APPARATUS 4.1 Apparatus which determine the physical properties of fluids 4.1.2 Density measurement 4.1.3 Viscosity measurement 4.2 Measuring instruments for the level of liquids 4.3 Pressure measuring instruments 4.3.1 Devices with liquids 4.3.2 Devices with elastic elements 4.3.3 Devices with transducers 4.4 Velocity measuring instruments 4.4.1 Pitot-Prandlt tube 4.4.2 Mechanical anemometers 4.4.3 Thermic anemometers 4.4.4 Optical measuring instruments 4.5 Flow measurement 4.5.1 Volumetric methods 4.5.2 Methods based on throttling the stream section of the fluids 4.5.3 Methods based on exploring the velocity field in the flow section 4.5.4 Flow – meters with variable crossing section 4.5.5 The ultrasound flow – meters 4.5.6 The electromagnetic flow – meter 4.5.7 Diluting methods BIBLIOGRAPHY 4 1. INTRODUCTORY CONCEPTS Hydropneumatic systems transmit the mechanical energy from a leading element to a led one by means of fluids. Depending on the way the energy is transmitted, hydropneumatic systems may be classified as follows: - hydropneumatic systems of hydrostatic type; - hydropneumatic systems of hydrodynamic type; - hydropneumatic systems of sonic type. For the hydropneumatic systems of hydrostatic type, potential energy is sent by means of fluids. In Fig 1. such a system is schematically shown. The hydraulic generator G H , in fact a volumetric pump, takes over the mechanical energy transmitted by the electrical engine M E , turns it into potential hydraulic energy and transmits it by means of pipes and other control, monitoring and adjusting devices to the hydraulic motor M H , which is also of volumetric type. This, in its turn, converts the hydraulic energy into mechanical energy used by the working equipment O L . Systems of hydrodynamic type use the kinetic energy of the fluid. They are also called turbo couplings or turbo transmissions. In figure 1.2 the scheme of a turbo transmission is shown. Fig.1.1. The mechanical energy received from shaft 1 is turned into kinetic energy by the hydrodynamic pump 2. In turbine 3, kinetic energy is turned into mechanical energy, which is taken over by shaft 4. 5 This transmission system has besides a coupling role, the role of variable regulator. Invented in 1904 by professor Fötinger, turbo transmission was designed to couple the shaft of a naval Diesel engine with the propeller, thus also accomplishing substantial rotation decrease. Systems of hydrodynamic type are high power systems. Fig.1.2 Systems of sonic type are based on pressure wave propagation supplied by a mono or three – phase sonic generator (a hydraulic cylinder or three hydraulic cylinders at 120 o ), to a mono three – phase receiver (motor). By the alternate movement of the piston, an area of high pressure is generated, which is sent along conduit 2 to the driving piston 3. (Fig.1.3.). So, as in the above- mentioned systems, the mechanical energy is converted into hydraulic energy (this time hydrosonic) and then back into mechanical energy. Fig.1.3. The transmission of energy is made under very high pressures 1,000 – 2,000 daN/ 2 cm . The distance between the two pistons must be a whole number multiple of wavelength ì . If we note withì the propagation speed of the pressure wave and with n the rotation in rot/s of the crank, then ì n. We must underline that sonication, i. e. energy transmission through conduits by means of pressure waves, was founded as a science by Gh. Constantinescu, a Romanian scientist. 6 A hydropneumatic system represents an assembly of elements by means of which we can produce and direct in a controllable manner the hydraulic and pneumatic energy stored in a fluid with the help of a motor that turns it again into mechanical energy. To carry out the generating functions of hydraulic energy, its reconvertion into mechanical energy, directing of the fluid agent, control and adjustment of the parameters, there are a large variety of hydraulic elements, which we shall study below. Pumps and compressors represent the generating elements of hydraulic and pneumatic energy. Hydraulic and pneumatic motors convert the energy of the fluid into mechanical energy. Within the control elements we distinguish the directing (distributing) elements, flow adjusting ones (chokes), pressure regulators (valves). Hydropneumatic systems contain auxiliary elements that in spite of their name are of vital importance for the smooth working of the assembly, achieving the fluid directing (pipes), its filtering (filters), storing (tanks), sealing, vibration and flow shock damping. We mustn’t forget the measuring equipment for the working parameters of the installation. In table 1.1. there are shown, according to STAS 7145 – 76, some of the symbols for the elements the hydropneumatic transmission systems. 7 Table 1.1 1. Pumps 1.1. One-way discharging adjustable pump 1.2. Two-way discharging adjustable pump 1.3. One-way discharging non adjustable pump 1.4. Two-way discharging non adjustable pump 2. Motors and pump-motor units 2.1. Circular irreversible hydrostatic motor with constant capacity 2.2. Reversible hydrostatic motor with constant capacity 2.3. Irreversible hydrostatic motor with adjustable capacity 2.4. Reversible hydrostatic motor with adjustable capacity 2.5. Non adjustable pump-motor with reverse direction 8 2.6. Adjustable pump-motor with reverse fluid direction 2.7. Linear motor (cylinder) with simple operating piston 2.8. Linear motor (cylinder) with double operating piston with uni and bilateral rod 2.9. Linear motor (cylinder) differential 3. Hydrostatic transmissions 3.1. Non adjustable hydrostatic transmission with one way rotation 3.2. Adjustable hydrostatic transmission pump with one way rotation 4. Hydrostatic distributors Discrete 4.1. With two channels and two positions 4.2. With two channels and three positions 4.3. With four channels and two positions 4.4. With four channels and three positions 9 Continuous (servo-distributors) 4.5. Mechanical and hydraulic distributors with one active edge 4.6. Electro hydraulic distributors 5. Pressure valves 5.1. Normal closed 5.2. Normal open 5.3. With differential control 5.4. Safety valve with external operating control 5.5. Reducing valve 6. Hydraulic resistors and flow regulators 6.1. Fixed or adjustable hydraulic resistor 6.2. Regulator for constant flow (with fixed resistor) and normal open (two-way) valve 6.3. Fixed or adjustable chok 10 6.4. Flow regulator with detour valve 6.5. Adjustable resistor with manual control 7. Auxiliary devices 7.1. Hydraulic accumulator 7.2. Filter 7.3. Cooler 7.4. Manometer 7.5. Flow -meter Compared to mechanical or electrical systems, hydropneumatic systems have a series of advantages: - a lower weight and volume, compared to their power; - reliability and silent working; - important possibilities of automation, standardization, normalization, modulation; - continuous speed adjustment; - quick at normal working parameters; - stopping within a short time; - possibility to achieve forces and important momentum, as well as high powers while control and operating can easily be done. 11 Hydropneumatic systems have also some disadvantages: - a high degree of accuracy of its components, which require complex manufacture technology; - possibilities to stop up inlets/outlets; - working under pressure with all the dangers implied; a high price, because of high quality materials required to manufacture the elements. 12 2. HYDRAULIC PUMPS AND MOTORS Pumps and hydraulic motors, i.e. hydraulic machines, are the basic elements of a hydraulic system. Hydraulic machines turn the mechanical energy into a hydraulic one and the other way round, being characterized by mechanical power N m with its components: force F, speed v or momentum M and rotation n as well as by hydraulic power N h with its components flow Q and load H. If we refer to the energetic conversion, we may group hydraulic machines by the direction of this transformation into hydraulic generators (pumps) that convert mechanical energy into hydraulic energy, and hydraulic motors, that convert hydraulic energy into mechanical energy. There is also another category of hydraulic machines, i.e. hydraulic transformers (couplings or clutches), that convert mechanical energy into mechanical energy with other parameters, by means of hydraulic energy, or hydraulic energy into hydraulic energy, by means of mechanical energy. For generating hydraulic machines (MHG), if referring to their characteristic power, the following conversion may be written: N m (M, n) ÷ ÷ ÷ ÷ MHG N h (Q, H) (2.1-1) There are generating hydraulic equipment for which the hydraulic power (secondary) is also obtained from a hydraulic power (primary). N h (Q p , H p ) ÷ ÷ ÷ ÷ MHG N h (Q s , H s ). (2.1-2) For hydraulic motors (MHM) we have the transformation: N h (Q, H) ÷ ÷ ÷ ÷ MHM N m (M, n). (2.1-3) Hydraulic transformers are in fact a combination of generating and motive hydraulic machines. By the manner in which the transformation takes place we can distinguish between hydraulic equipment in a closed circuit (2.1-4) or in an open circuit (2.1-5): N m (M p , n p ) ÷ ÷ ÷ ÷ MHG N h (Q, H) ÷ ÷ ÷ ÷ MHM N m (M s , n s ) (2.1-4) N h (Q p , H p ) ÷ ÷ ÷ ÷ MHM N m (M, n) ÷ ÷ ÷ ÷ MHG N h (Q s , H s ) (2.1-5) We must underline the fact that there is a large variety of reversible hydraulic equipment which can work both as a pump or as a motor. 13 In a hydraulic machine the conversion of position, potential or kinetic energy takes place. Referring to the type of load that is transformed we may classify hydraulic equipment into volumetric equipment and turbo equipment. Volumetric (hydrostatic) machines process potential pressure energy. Turbo machines (hydrodynamic machines) process potential pressure energy and kinetic energy. There is also another category of hydraulic machines now very rare, which convert the position potential energy, but which were widely spread in the past. They are the hydraulic elevators (MHG) and water wheels (MHM). There are also motive hydraulic motors that transform only the kinetic energy (Pelton activated turbines). Volumetric hydraulic machines can be classified into: - linear or alternative (with piston, plunger, with piston and membrane); - rotating (with radial or axial pistons, with blades, with sprocket wheels, with screws). Turbo equipment achieves the conversion of energy by hydrodynamic interactivity between the rotor with profiled blades and the fluid. From the point of view of the rotation they can be classified into pumps with a side channel, centrifugal pumps and axial pumps. When presenting the hydraulic equipment we shall take into consideration the two classifying criteria. 2.1. Volumetric pumps Volumetric pumps convert mechanical energy into hydraulic energy, which is in the form of potential pressure energy. This is achieved by means of closed spaces between the fixed and the mobile parts of the pump, this process being a discontinuous one. Volumetric pumps are, to a great extend, reversible, they can work as a pump or as a motor, according to the liquid that comes in the body of the unit with under pressure or over pressure. The pressure of the volumetric pumps is generally high-250-300 bar, and the flows extend to a very large scale 1-8,000 l/min. Their power can be up to 3,500 kW. In the case of rotating volumetric pumps, rotations range from 3,000 to 5,000 rot/min, and sometimes they can get up to 15,000 - 30,000 rot/min. 2.1.1. Piston pumps The piston pump is a volumetric hydraulic pump, which achieves the pumping effect by an alternate rectilinear movement of a piston inside a cylinder (fig.2.1.) 14 Fig.2.1. Piston pumps can be with simple or double effect (fig.2.1.) As it can be noticed from their simple working principle, for the pumps with simple effect the flow range has a strong discontinuous character (fig.2.3.), which is improved in the case of double effect pumps. (fig.2.4.) Fig.2.2. We shall calculate the mean and instantaneous flows for a piston pump. The relation gives the volume of discharged liquid for one stroke of the piston (cylinders): V = 4 2 D t h (2.1-6) where D is the diameter of the piston, and h = 2 r, its stroke. Noting with n the rotations in rot /min for the driving shaft, we can calculate the mean flow: Q med = 4 2 D t 2 r 60 n . (2.1-7) To compute the instantaneous flow, we shall first determine the speed of the piston. Starting from the value of the distance x = 1 cos o + r cos ( ) ¸ t ÷ = 1 coso - r cos ¸ (2.1-8) and noticing that 15 o sin r = ( ) ¸ t ÷ sin 1 (2.1-9) or else sin o = l r sin ¸ . (2.1-10) so cos ¸ = ¸ 2 2 2 sin 1 l r ÷ , (2.1-11) which being unfolded in this series and the first two terms retained (the error is very much decreased because 1 r is sub-unitary) we may write: cos o ¸ 2 2 2 sin 1 2 1 1 r ÷ ~ , (2.1-12) and we get: x = l – r cos ¸ - ¸ 2 2 sin 2 1 l r , (2.1-13) and v = | . | \ | ÷ = ¸ ¸ e 2 sin 21 sin r r dt dx . (2.1-14) The instantaneous flows will be: Q = | . | \ | ÷ = ¸ ¸ e t t 2 sin 21 sin 4 4 2 2 r r D v D . (2.1-15) 16 Fig. 2.3 We define the pulsation coefficient of the flow as the ratio: o % = 100 min max med Q Q Q ÷ . (2.1-16) Since max Q obtained when ¸ = 2 t , and 0 min = Q , (fig.2.3), we shall get: % 314 100 60 30 2 4 4 % 2 2 = = t e t e t o r D r D . (2.1-17) For pumps with simple effect piston the flow pulsation is high. For this reason the pumps are equipped with hover containers that are placed in the vicinity of the working cylinder. The pumps with double effect piston overflow in the returning area of the piston with a lower flow. The instantaneous flow for the area | | t t 2 , will be (fig.2.4): Q x = ( ) | . | \ | ÷ ÷ ¸ ¸ e t 2 sin 21 sin 4 2 2 r r d D . (2.1-18) Fig.2.4 17 Because curves Q and Q x intersect only on the abscissa axis, the pulsation coefficient of the flow remains approximately the same as for simple effect pumps. Their advantage, not negligible, is that they also overflow on the return stroke of the piston. The classical piston pumps are less and less frequently seen in the hydraulic installations due mainly to the high pulsation coefficient of the flow. 2.1.2. Pumps with radial pistons Pumps with radial pistons are rotary volumetric pumps with variable flow. The pulsation coefficient of the flow is very diminished, thus having beneficial effects on the extent of hydraulic oscillations introduced in the transmission system. They may be classified into pumps with external suction and with internal suction. Pumps with radial pistons and external suction (fig.2.5) mainly consist of stator 1, rotor 2, pistons 3 coupled by means of piston rods 4 to the eccentrically axle 5 (with variable eccentricity). The excentricity of the pistons axle gives the possibility that their movement be different, some being in suction, others in discharge. Fig.2.5 Pumps with radial pistons and internal suction (fig.2.6) consist of stator 1, eccentrically rotor 2, piston 3, central axle 4, which contains the suction channels 6. Due to the eccentricity e of the rotor, the pistons carry on an alternate movement of stroke 2e, being in turns in suction/discharge. The pistons are pressed to the walls of the stator 18 by the force of springs or by the centrifugal force only. By modifying the eccentricity we can adjust the flow of the pump. Fig.2.6 The cylindricality of the z cylinders of diameter d or the volume of discharged liquid for one rotation will be: e z d V 2 4 2 t = . (2.1-19) For the rotation | | min / rot n we shall have the mean flow: t e t t ze d n e z d Q med 4 60 2 4 2 2 = = . (2.1-20) Fig.2.7 19 To calculate the instantaneous flow that ranges between a minimum and the maximum value, first we establish the speed of contact point A of the piston with the stator (fig.2.7). The absolute speed v is made up of speed 1 v in relation to the center of the rotor 1 O and 2 v the movement speed of the piston inside the cylinder. We note the variable distance 1 AO with µ . Then we shall have: eµ = 1 v , (2.1-21) dt d v µ = 2 . From the triangle A O O 2 1 we get: ¢ µ µ cos 2 2 2 2 e e R ÷ + = (2.1-22) From which: ¢ ¢ ¢ ¢ µ 2 2 2 2 2 2 sin 1 cos cos cos | . | \ | ÷ ± = + ÷ ± = R e R e R e e e (2.1-23) As 1 << R e , we may leave out the second term of the radical. Then: R e + ~ ¢ µ cos . (2.1-24) The speed of the piston will be: ¢ e µ sin 2 e dt d v ÷ = = (2.1-25) For the interval | | t , o when ¢ increases; the speed 2 v decreases as the sign – from the relation (2.1-25) shows us. We shall consider speed in modulus flow of the j pistons that are in discharge, each being in the position | | 20 i ¢ : i j i i e d Q ¢ e t ¿ = = 1 2 sin 4 . (2.1-26) 20 If we note by¢ the instantaneous position angle of the first piston in discharge and by z t ¸ 2 = , the angle between two pistons, then the position angle of the piston to the given point M will be: ( )¸ ¢ ¢ 1 ÷ + = i i . (2.1-27) In the case of an even number of pistons, z = 2k, we shall have k pistons in discharge and k pistons in suction. We can rewrite the equation (2.1-26) knowing that j = k: ( ) ( ) ( ) | | { } ¸ ¢ ¸ ¢ ¸ ¢ ¢ e t 1 sin .... 2 sin sin sin 4 2 ÷ + + + + + + + = k e d Q i .(2.1-28) By transforming the sum between the braces into a product, we shall get: ( ) ( ¸ ( ¸ ÷ + = ¿ = 2 1 sin 2 sin 2 sin sin 1 ¸ ¢ ¸ ¸ ¢ k k i k i . (2.1-29) The maximum value of this sum is obviously obtained when ( ) 1 2 1 sin = ( ¸ ( ¸ ÷ + ¸ ¢ k or ( ) 2 2 1 t ¸ ¢ = ÷ + k , so ( ) 2 1 2 ¸ t ¢ ÷ ÷ = k . (2.1-30) The minimum value could be obtained for ( ) 0 2 1 sin = ( ¸ ( ¸ ÷ + ¸ ¢ k , or ( ) . 0 2 1 = ÷ + ¸ ¢ k But, because ¸ ¢ < s 0 ,(2.1-31) hence ( ) ( ) ( ) 2 1 2 1 2 1 ¸ ¸ ¢ ¸ + < ÷ + s ÷ k k k . (2.1-32) So, the minimum value of the argument of function sinus is ( ) 2 1 ¸ ÷ k or else ( ) ( ) 2 1 2 1 ¸ ¸ ¢ ÷ = + + k k . (2.1-33) 21 The minimum value of the sum in the relation (2.1-29) is obtained for 0 = ¢ . Going back to the relation (2.1-28) whose sum may be written in the form of (2.1-29) and bearing in mind the considerations on the instantaneous position angle of the first discharging piston for the maximum values of the flow, we may write: 2 sin 2 sin 4 2 max ¸ ¸ e t k e d Q = , (2.1-34) ( ) 2 1 sin 2 sin 2 sin 4 2 min ¸ ¸ ¸ e t ÷ = k k e d Q . (2.1-35) Now we are able to write the pulsation coefficient of the flow for the pumps with an even number of radial pistons: ( ) ( ) 100 4 2 100 2 1 sin 1 2 sin 1 2 100 2 1 sin 1 2 sin 2 sin 2 % k tg k k k k k k k k t t t t t ¸ ¸ ¸ t o = ( ¸ ( ¸ ÷ ÷ = = ( ¸ ( ¸ ÷ ÷ = (2.1-36) In the case of pumps with an odd number of radial pistons 2k+1, we may distinguish between two cases: either k+1 pistons are in discharge, therefore: | . | ¸ e 2 , 0 ¸ ¢ , (2.1-37) or k pistons discharge, and then: | . | ¸ e ¸ ¸ ¢ , 2 . (2.1-38) We shall compute the maximum and minimum flows for both hypotheses and we shall notice that they are identical. We shall write expressions max Q and min Q for the two cases: 1. k+1 discharging pistons 22 ( ) 2 sin 2 1 sin 4 2 max ¸ ¸ e t + = k e d Q , (2.1-39) ( ) 2 sin 2 sin 2 1 sin 4 2 min ¸ ¸ ¸ e t k k e d Q + = . (2.1-40) 2. k repressed pistons 2 sin 2 sin 4 2 max ¸ ¸ e t k e d Q = , (2.1-41) 2 sin 2 sin 2 sin 4 2 min ¸ ¸ ¸ e t k k e d Q = . (2.1-42) But ( ) ( ) t t ¸ ¸ ¸ = = + = + + 2 2 2 1 2 2 1 2 z z k k k . (2.1-43) The angles being supplemental, it results in ( ) 2 1 sin 2 sin ¸ ¸ + = k k , (2.1-43) therefore the maximum and minimum flows shall be equal for the two situations we come across with during the working of the pumps with an odd number of radial pistons. Taking into consideration the relations (2.1-41) and (2.1-42) as well as (2.1-20) we can compute the pulsation of the flow for this type of pumps: ( ) ( ) . 100 1 2 4 1 2 2 100 1 2 sin 1 1 2 sin 1 2 sin 1 2 100 2 sin 1 2 sin 2 sin 1 2 % + + = = | . | \ | + ÷ + + + = | . | \ | ÷ + = k tg k k k k k k k k k k t t t t t t ¸ ¸ ¸ t o (2.1-45) 23 In fig.2.8 the variation of the instantaneous flow for a pump with 9 radial pistons is shown. Fig.2.8 On studying table 2.1 it can be noticed that pumps with more pistons have a lower pulsation coefficient and that pumps with an odd number of pistons are from this point of view preferred to those with an even number of pistons. Table 2.1 z odd number z even number z % z % 3 14,022 2 157 5 4,973 4 32,515 7 2,527 6 14,022 9 1,526 8 7,807 11 1,020 10 4,973 12 3,444 The force required to rotate the impeller of the pump is a perpendicular force on direction 1 AO ; we shall note it by F. Force F is decomposed into two directions: 1 AO (component F - the force with which the liquid, having the pressure p, acts upon a piston of a diameter d) and 2 AO (component N which acts upon the bearing of the pump) (fig.2.9). 24 Fig.2.9 The force with which the liquid acts upon the piston is equal and has opposite direction to the force with which the pistons acts upon the liquid. p d F 4 2 t = . (2.1-46) | tg F T = . (2.1-47) We notice that: ¢ | sin sin R e = . (2.1-48) Thus: ( ) ¢ ¢ t f R e arc tg p d T = ( ¸ ( ¸ | . | \ | = sin sin 4 2 . (2.1-49) The maximum value of T is obtained for 0 90 = ¢ . The torque corresponding to a piston is: ( ) ( ¸ ( ¸ | . | \ | + = = ¢ ¢ t µ sin sin cos 4 2 R e arc tg e R p d T M r . (2.1-50) The total torque shall be: ¿ = = j i i i rt T M 1 µ . (2.1-51) where j is the number of discharging pistons. The relation shall give the power of the pump: e rt M P = . (2.1-52) 25 2.1.3. Pumps with blades Pumps with blades are volumetric pumps for which variable spaces are limited by blades, impeller, stator and front lids. They can be with external or internal suction (fig.2.10) and (fig.2.11). Fig.2.10 Fig.2.11 By the number of suction-discharge for one rotation, the pumps with blades can be with simple action (fig.2.10) and (fig.2.11) or multiple action. In fig.2.12 a double action pump with blades is shown. Fig.2.12 Pumps with blades and simple action are pumps with variable flow, their adjustment being made by modifying eccentricity e. Pumps with multiple action have a constant flow. To calculate the flow we use the scheme in fig.2.13, for which we have done the following denotations: R, r – the stator radius and the impeller radius respectively; b – the breadth of a blade; ¸ - the angle between two consecutive blades; z- the number of blades [20]. In fig.2.13 it is shown the blade coupling 1-2 in two position: at the beginning of the discharge ( ) 2 1 ,u u ÷ and at the end of discharge ( ) ' 2 ' 1 ,u u ÷ . 26 Fig. 2.13 To calculate the volume V between the blades (blades of breadth b and negligible thickness) we shall write first the elementary volume: u µ µ d d b dV = . (2.1-53) Knowing that u cos 1 e R M O + = (V.cap.2.1.2) and ¸ u u = + 2 1 , we can write ( ) | | ( ) ( ) ( ) ( ) ( ) | | . cos sin 2 2 cos 2 sin Re 4 2 2 sin 2 sin 2 1 2 sin sin 2 2 cos 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 cos 2 1 2 1 ) ` ¹ ¹ ´ ¦ ÷ + + ÷ + ÷ = = ) ` ¹ ¹ ´ ¦ ( ¸ ( ¸ + + + + + ÷ = = ÷ + = = } } } ÷ + ÷ u u ¸ ¸ u u ¸ ¸ u u ¸ u u ¸ u u µ µ u u u u u u e r R b e e R r R b d r e R b d d b V e R r (2.1-54) The maximum value of V is obtained when ( ) . 1 cos 1 2 cos 1 2 1 2 = ÷ = ÷ u u u u and (2.1-55) 27 (which means that 2 1 u u = ): ( ) ( ) ( ¸ ( ¸ + + + ÷ = ¸ ¸ ¸ ¸ sin 2 2 sin Re 4 2 2 2 2 max e r R b V . (2.1-56) At the end of discharge the relation will calculate the volume among blades: ( ) ( ) | | . cos sin 2 2 cos 2 sin Re 4 2 ' 1 ' 2 2 ' 1 ' 2 2 2 cos ' ' 2 ' 1 ) ` ¹ ¹ ´ ¦ ÷ + + ÷ ÷ ÷ = = = } } + + ÷ u u ¸ ¸ u u ¸ ¸ µ µ u u u t u t e r R b d d b V e R r (2.1-57) We calculate the extreme of the function ( ) ' 1 ' 2 ' u u ÷ V ( ) . 0 2 cos 2 cos 2 sin 2 sin * ' 1 ' 2 ' 1 ' 2 ' 1 ' 2 ' = | | . | \ | ÷ ÷ ÷ = ÷ u u ¸ u u ¸ u u e R e b d dV (2.1-58) . 0 2 sin ' 1 ' 2 = ÷u u (2.1-59) . 2 ' 1 ' 2 ¸ u u = = (2.1-60) For ( ) 1 2 ' ' 1 ' 2 , 0 2 u u u u ÷ < ÷ d dV is negative, and for ( ) ' 1 ' 2 ' ' 1 ' 2 , 0 2 u u u u ÷ > ÷ d dV is positive, so the extreme point when 2 ' 1 ' 2 ¸ u u = = represents a minimum. ( ) ( ) ( ¸ ( ¸ + + ÷ ÷ = ¸ ¸ ¸ ¸ sin 2 2 sin 4 2 2 2 2 min e e R r R b V . (2.1-61) * The term in brackets cannot be cancelled since 2 cos 2 cos ' 1 ' 2 u u ¸ ÷ >>e R 28 The relation will give the volume discharged by the couple of blades (1,2): 2 sin 4 min max 2 , 1 ¸ e R b V V V = ÷ = . (2.1-62) The z inter-blade space shall discharge for one rotation the volume: 2 sin 4 2 , 1 ¸ z e R b V z = . (2.1-63) For a rotation of n [rot/s] the mean theoretical flow of the pump is obtained with the help of the relation: z n z e R b n z e R b Q med t ¸ sin 4 2 sin 4 = = , because z t ¸ 2 = . (2.1-64) When z is big, . sin z z t t ÷ Then: ( ) . 2 4 b e n D z n z e R b Q med t t = = · (2.1-65) The formula (2.1-65) is used to calculate the flow for pumps with a finite number of blades. It obviously represents an approximation, higher or lower, according to a greater or smaller number of blades. To establish the instantaneous flow of a pump with blades, we shall first calculate the volume of fluid that exists in the interstice ¸ ¢ ¢ = ÷ + i i 1 between two blades: ( ) ( ) | | . cos sin 2 2 cos 2 sin 4 2 1 1 2 1 2 2 cos } } + ) ` ¹ ¹ ´ ¦ + + + + + ÷ = = + + + i i i i i i e R r i e e R r R b d d b V ¢ ¢ u ¢ ¢ ¸ ¸ ¢ ¢ ¸ ¸ µ µ u (2.1-66) The instantaneous flow of the couple of blades shall be: 29 . dt dV q i i = (2.1-67) By leaving out the term that contains 2 e and bearing in mind that e ¢ = dt d , we shall successively obtain: , 2 sin 2 sin 2 1 + + = i i i b e R q ¢ ¢ ¸ e (2.1-68) , 2 sin 2 sin 2 | . | \ | + = ¸ ¢ ¸ e i i b e R q (2.1-69) ( ) | |, cos cos ¸ ¢ ¢ e + ÷ = i i i b e R q (2.1-70) ( ). cos cos 1 + ÷ = i i i b e R q ¢ ¢ e (2.1-71) The total instantaneous flow of a pump with blades shall be equal to the sum of instantaneous flows of the j interstices being in discharge: ( ) ¿ = + ÷ = j i i i i b e R Q 1 1 . cos cos ¢ ¢ e (2.1-72) We shall study the pulsation of the flow first for a pump with an even number of blades: z=2k. We shall then have j = k interstices being in discharge, for any : 2 , 2 ( ¸ ( ¸ ÷ e ¸ ¸ ¢ ( ) ( ) ( ) | | . 2 sin 2 sin 2 cos cos cos cos cos cos 1 1 1 1 1 1 1 | . | \ | + = + ÷ = = ÷ = ÷ = + = + ¿ ¸ ¢ ¸ e ¸ ¢ ¢ e ¢ ¢ e ¢ ¢ e k k b e R k b e R b e R b e R Q k k i i i i (2.1-73) i Q is maximum when 2 2 , 2 2 , 1 2 sin 1 1 1 ¸ t ¢ t ¸ ¢ ¸ ¢ k k or k ÷ = = + = | . | \ | + (2.1-74) 30 and it is minimum when . 2 0 2 sin 1 1 ¸ ¢ ¸ ¢ k or k ÷ = = | . | \ | + (2.1-75) But 2 1 , 2 , 2 1 1 ¸ ¢ ¸ ¸ ¢ ± = ± = ( ¸ ( ¸ ÷ e or k so (2.1-76) Under these circumstances: 2 sin 2 max ¸ e k b e R Q = (2.1-77) and ( ) * min 2 1 sin 2 sin 2 ¸ ¸ e ± = k k b e R Q (2.1-78) The relation gives- the pulsation coefficient of the flow for a pump with an even number of blades –2k : ( ) ( ) . 100 4 2 100 2 1 sin 1 2 sin 2 100 sin 4 2 1 sin 1 2 sin 2 % k tg k k k k k z n z b e R k k b e R t t t t t t ¸ ¸ o = = ( ¸ ( ¸ ± ÷ = ( ¸ ( ¸ ± ÷ = (2.1-79) * It can be noticed that ( ) ( ) 2 1 2 1 ¸ ¸ ÷ + k and k are supplemental angles, so the value of sinus function remains the same. 31 For a pump with an odd number of blades –2k+1- we have two situations: k+1 interstices under discharge when | . | ¸ ÷ e 0 , 2 1 ¸ ¢ and k interstices under discharge when | . | ¸ e 2 , 0 1 ¸ ¢ . Computing in the same way as for the pump with the even number of interstices we shall get the relations for max Q and min Q . 1. k+1 discharged interstices ( ) 2 1 sin 2 max ¸ e + = k b e R Q . (2.1-80) ( ) . 2 sin 2 1 sin 2 min ¸ ¸ e k k b e R Q + = (2.1-81) 3. k repressed interstices . 2 sin 2 max ¸ e k b e R Q = (2.1-82) . 2 sin 2 2 min ¸ e k b e R Q = (2.1-83) The values of max Q and min Q are equal because the angles ( ) 2 1 ¸ + k and 2 ¸ k are supplemental. Bearing in mind the above shown demonstration there results that the pulsation of the flow for a pump with an odd number of blades is: 32 ( ) ( ) 100 1 2 4 1 2 2 100 1 2 sin 1 1 2 sin 1 2 sin 1 2 100 sin 4 2 1 2 sin 2 % + + = | . | \ | + ÷ + + + = = | . | \ | ÷ = k tg k k k k k k k z n z b e R k k b e R t t t t t t t ¸ ¸ e o (2.1-84) By comparing the relations (2.1-79) with (2.1-36) and (2.1-84) with (2.1-45) we can notice that the pulsation of the flow for pumps with radial pistons is identical to the one of the flow for pumps with blades (leaving out the term 2 e ), that suggests an analogy between those two types of pumps. The space between two blades behaves like a radial cylinder with piston during suction and discharge phases. By equaling the hydraulic power with the power at the shaft of the motor we can determine the necessary theoretical moment: . 2 pQ n M t = t (2.1-85) n is expressed in rotations per second. In (2.1-85) we introduce the value of the mean flow given by (2.1-64): . sin 2 2 sin 4 z e R b p z n z n z e R b p M t t t t t = = (2.1-86) Taking into account the mechanical and viscous frictions, the couple developed by the motor will be: . sin 2 z e R b p z M M t t q t q = = (2.1-87) 2.1.4. Pumps with axial pistons The pumps with axial pistons accomplish the flow of fluid by the alternate movement of a certain number of pistons inside some cylinders that are placed in an impeller, which have their axes parallel to the impeller axis of rotation. This manner of placement gives the pumps a low clearance and equilibrium due to the symmetry of the masses in rotation. The alternate movement of the piston is achieved by means of a slanted disk. Its adjustable slanting allows the change of flow of the pumps. For some pumps slanting the block of cylinders accomplishes the change of flow. 33 In fig.2.14 the working scheme of a pump with axial pistons and slanting disk is shown: 1. the block of cylinders (rotor); 2. cylinders; 3. pistons; 4. slanting disk; 5. cardan joint; 6. connecting rods with spherical joints; 7. fixed part of the suction /discharge channels (distribution element). Fig.2.14. The electrical driving motor transmits the rotation to the block of cylinders and, by means of the cardan joint 5, to the slanting disk on which the extremities of the cylinder rods are propped. The suction and discharge are accomplished by means of the fixed distribution element 7, which has channels in the area where the pistons are in suction, or in discharge. To calculate the flow of the pump with axial pistons let us consider two systems of axes (fig.2.14.) xOyz and 1 1 1 z Oy x that are rotated between them with an angle o around their common axis Oy . The coordinates of a certain M point in the system of axes that is not rotated can be written with respect to the coordinates of the same point in the rotated system of axes, (fig.2.15.) as: o o sin cos 1 1 z x x + = (2.1-88) . sin cos 1 1 1 o o x z z y y ÷ = = 34 Fig.2.15. In fig.2.16. then are shown the positions of the spherical joint A, joined with the disk and of the spherical joint B, joined to the piston, that belong to the same connecting rod, during the rotation with an o angle. [20] Fig.2.16. With respect to the systems of axes in fig. 2.14. point A has the following coordinates : - to 1 1 1 z Oy x ¢ ¢ cos sin 0 1 1 1 1 1 r z r y x A A A = = = (2.1-89) - to xOyz (see relations 2.1-88) . cos cos sin sin cos 1 1 1 o ¢ ¢ o ¢ r z r y r x A A A = = = (2.1-90) 35 Coordinates y and z of point B with respect to the system xOy are: , cos sin 2 2 ¢ ¢ r z r y B B = = (2.1-91) coordinate B x is to be determined knowing the constant length l of the connecting rod AB. We shall then write: ( ) ( ) ( ) . 1 2 2 2 2 A B A B A B z z y y x x ÷ + ÷ + ÷ = (2.1-92) Relation (2.1-92) represents an equation of 2 nd degree with the unknown . B x By solving it we get: ÷ = o ¢sin cos 1 r x B ( ). cos cos sin 2 cos cos sin 2 2 2 2 2 2 1 2 2 1 2 2 2 o ¢ ¢ o ¢ ¢ + + ÷ ÷ ÷ ÷ r r r r l (2.1-93) It can be noticed that B x is negative. This is the reason why we chose the sign “- “ before the root. The velocity of the piston can be obtained by deriving B x with respect to time: ( ) ( ) . cos cos sin 2 cos cos sin cos cos sin 2 cos sin 2 2 cos cos sin 2 cos sin 2 sin sin 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 1 2 2 1 2 1 1 . o ¢ ¢ o ¢ ¢ o ¢ ¢ e ¢ ¢ e o ¢ ¢ e ¢ ¢ e o ¢ e + + ÷ ÷ ÷ ÷ + + ÷ ÷ ÷ ÷ = = r r r r r l r r r r r x v B p (2.1-94) When the slanting angle of the disk is enough small, we may consider . 1 cos ~ o The velocity of the piston, in modulus, which becomes: , sin sin 1 o ¢ e r v p = (2.1-95) The instantaneous flow of a piston with diameter d will be: , sin sin 4 1 2 o ¢ e t r d q i = (2.1-96) and the instantaneous flow of the j pistons that are under discharge is: 36 ¿ ¿ = = = = j i i j i i i r d q Q 1 1 1 2 . sin sin 4 ¢ o e t (2.1-95) The mean flow of the pistons of d diameter and stroke o sin 2 1 r h = , inside the impeller of rotation n will be: 60 sin 2 4 1 2 n z r d Q m o t = . (2.1-96) To establish the maximum and minimum flow, we have to draw the attention that the problem is similar to that presented in chapter 2.1.2. This is also the maximum and minimum of the sums of sinuses ¿ = j l i i ¢ sin , for the j pistons that are under discharge, with an even number z = 2k or odd z = 2k + 1 of pistons. Therefore, we can write the maximum and minimum flows for the pumps with an even number of axial pistons: , 2 sin 2 sin sin 4 1 2 max ¸ ¸ o e t k r d Q = (2.1-97) ( ) . 2 1 sin 2 sin 2 sin sin 4 1 2 min ¸ ¸ ¸ o e t ÷ = k k r d Q (2.1-98) In this case the pulsation of the flow, will be: ( ) ( ) 100 4 2 100 2 1 sin 1 2 sin 1 2 100 2 1 sin 1 2 sin 2 sin 2 % k tg k k k k k k k k t t t t t ¸ ¸ ¸ t o = ( ¸ ( ¸ ÷ ÷ = = ( ¸ ( ¸ ÷ ÷ = (2.1-99) For the pump with an odd number of axial pistons we shall have: 37 ( ) 2 sin 2 1 sin sin 4 1 2 max ¸ ¸ o e t + = k r d Q . (2.1-100) ( ) 2 sin 2 sin 2 1 sin sin 4 1 2 min ¸ ¸ ¸ o e t k k r d Q + = . (2.1-101) ( ) ( ) 100 1 2 4 1 2 2 100 1 2 sin 1 1 2 sin 1 2 sin 1 2 100 2 sin 1 2 sin 2 sin 1 2 % + + = | . | \ | + ÷ + + + = = | . | \ | ÷ + = k tg k k k k k k k k k k t t t t t t ¸ ¸ ¸ t o (2.1-102) We can notice that the pulsation of the flow for the pump with axial pistons is the same with the pulsation of the flow for pumps with radial pistons and pumps with blades. To create pressure p, the piston acts upon the liquid with the force: . 4 2 p d F t = (2.1-103) Force F is decomposed into a tangent component T and a normal one N (fig.2.17). Fig.2.17 The tangent force T has the value: 38 . sin 4 sin 2 o t o p d F T = = (2.1-104) The resistant moment of a piston will be: . sin sin 4 sin sin 1 2 1 ¢ o t ¢ o µ r p d r F T M r = = = (2.1-105) z pistons will have a resistant moment: ¿ = = z i i rt r p d M 1 1 2 . sin sin 4 ¢ o t (2.1-106) The relation will give the power consumed by the pumps: , e rt M P = (2.1-107) | | | | | | | | . 310 . 97 min / 81 , 9 . . 620 . 71 min / 81 , 9 kW rot n Nm M P C rot n Nm M P rt rt = = = (2.1-108) 2.15. Pumps with sprocket wheels They are volumetric pumps that are widely spread especially due to their simple building. As the sprockets come out of gear, a variation of volume in an excessive sense is created in the suction room. The spaces between the sprocket represent active cups that carry the fluid. When the sprocket come into gear the volume decreases and a hydrostatic pressure is created (fig.2.18). Pumps with sprocket wheels are classified according to several criteria: by the type of gear (external or internal – fig.2.18 a and b), by the level of pressure (low, medium and high), by the number of rotors (with two or more, fig.2,19), by the profile of sprockets (evolventric or cycloid), by the sprockets position (straight or slanting). 39 Fig.2.18 Fig.2.19 The computation of the flow for this type of pumps can be done in a simple manner; considering the hypothesis that the cross sections of the empty spaces is equal to that of filled spaces and that the degree of coverage is equal to a unit; a hypotheses that induces a quite high error. Thus: Sg = Sp. (2.1-109) The cross section of all the cups for the two sprocket wheels that are in gear will be: ( ) 2 2 2 2 4 2 1 4 4 2 i e i e t D D D D S ÷ = | | . | \ | ÷ = t t t . (2.1-110) By considering the bottom of the sprocket equal to its head m a a = = 2 1 (the sprocket modulus) and knowing that the sprocket modulus is t p m = , we can write (fig.2.20): z m D D D D S i e i e t 2 2 2 4 2 t t = + ÷ = . (2.1-111) Let the breadth of the sprocket be m b ì = . The volume transported for one turn will be: z m V 3 2 ì t = , (2.1-112) and the flow: 40 | | min / 1 10 2 6 3 ÷ = n z m Q ì t . (2.1-113) as m is given in mm, and rotation is considered in rot/min. For a more accurate computation of the flow we can use two methods: the geometrical method (more complicated) or the method of equivalence between the energy transmitted to the liquid and the mechanical work consumed to drive the sprocket wheels. By using fig.2.20 we shall further present the second analytical method of flow computing for the pumps with sprocket wheels. [12,20]. Fig.2.20 The mechanical work consumed to rotate the sprocket wheels with angle ¢ d inculcates the energy pdV upon the liquid: ¢ Md pdV = . (2.1-114) In relation (2.1-114) M is the torque. 41 Pressure p acts upon the outline of the sprocket wheels. This intricate outline can be replaced with a simpler one 1 2 1 B CO AO . On the straight lines of this outline there act four resultant forces of pressure. This replacement has been made according to the theorem in mechanics, which states that the resultant of the projection of pressure forces on a certain surface is equal to the product between pressure and the projection of the surface on a plane, that is perpendicular on the resultant. The total torque will be: ( ) 2 2 2 1 2 2 2 ' ' 2 1 ' 1 ' ' 1 2 2 2 2 2 2 µ µ µ µ ÷ ÷ = = ÷ + ÷ = e e e r b p F r F F r F M (2.1-115) We denote the segment PC by x, and notice that r r O O 2 2 1 = , consequently we can apply the theorem of the median for the triangle C O O 2 1 : ( ) 4 4 2 2 2 2 2 1 2 r r x ÷ + = µ µ . (2.1-116) Hence: ( ) 2 2 2 2 1 2 r r x + = + µ µ . (2.1-117) Replacing in relation (2.1-115), we shall get: ( ) 2 2 2 2 x r r pb M r e ÷ ÷ = . (2.1-118) Knowing that dV = Qdt, and dt d e ¢ = and using the relations (2.1-118) and (2.1-114), we may write: ( ) 2 2 2 x r r b Q r e ÷ ÷ = e . (2.1-119) Magnitude x is variable in time: 42 ( ). 0 1 1 r b r b b tg t r tg r r P K C K x o e u o u ÷ + = = ÷ = ÷ = (2.1-120) In relation (2.1-120) we used the property of the evolvement 1 1 1 C K C K = and the fact that t e u u + = 0 (the real driving segment begins in D and ends in 0 1 1 2 u b r D K D K E = = ÷ ). Noting by P K P K l 2 1 = = the length of the half of the theoretical driving segment and by EP DP l = = 1 the length of the half of the real driving segment, we shall get: 1 0 l l r b ÷ = u . (2.1-121) So: 1 1 l t r l t r l l x b b ÷ = ÷ + ÷ = e e . (2.1-122) We can write the instantaneous flow in the form of a time function: ( ) ( ) 2 ' 1 2 2 2 2 2 2 l t l r t r r r b t Q b b r e ÷ + ÷ ÷ = e e . (2.1-123) The time in which the real driving segment is covered, is obtained by using the properties of the evolvement: , 2 1 t r l b e = (2.1-124) . 2 1 e b r l T = (2.1-125) The flow Q(t) has a periodical variation; | | T t , 0 e . To compute the pulsation of the flow first we have to establish the mean flow. The volume discharged by a pair of sprockets during a period T is: 43 ( ) ( ) | | } } ÷ ÷ ÷ = = e e b r l b r e T dt l t r r r b dt t Q V 2 0 2 1 2 2 0 . (2.1-126) By making the change of the variable , , 1 dt r dy l t r y b b e e = ÷ = (2.1-127) we get: ( ) ( ) | |. 3 3 2 2 2 2 1 2 2 2 1 1 l r r r bl dy y r r r b V r e b l l r e b ÷ ÷ = = ÷ ÷ = } ÷ (2.1-128) Knowing that the number of sprockets is z and the wheels turn with rotation n, the mean flow will be: ( ) | | 2 ' 2 2 3 3 l r r r bl z n z V Q r e b m ÷ ÷ = = e . (2.1-129) The maximum value of the relation (2.1-123) is obtained for e b r l t 1 = : ( ) 2 2 max r e r r b Q ÷ = e . (2.1-130) For t =0 or e b r l t 1 2 = the flow has the minimum value: ( ) 2 ' 2 2 min l r r b Q e ÷ ÷ = e . (2.1-131) We are now able to determine the pulsation of the flow for a pump with sprocket wheels: ( ) | | . 100 3 3 % 2 ' 2 2 1 l r r z l r r e b ÷ ÷ = t o (2.1-132) 44 The moment applied to the driving wheel is determined from the relation (2.1- 114): ( ) 2 2 2 x r r pb pQ dt pdV d pdV M r e ÷ ÷ = = = = e e ¢ . (2.1-133) The moment will be maximum for x=0: ( ) 2 2 max r e r r pb M ÷ = . (2.1-134) By making the same approximations as in relation (2.1-111) we get: ( ). 2 max l z pbm M + = . (2.1-135) The maximum force applied to the liquid will be: r r M F max = . (2.1-136) The power expressed with respect to the moment and to the angular velocity is written with the known formula: e M P = . (2.1-137) 2.1.6. Other types of volumetric pumps The pumps with diaphragm (fig.2.21) This type of pump is mostly used when the circulating fluid mustn’t come into contact with the parts of the pump or mustn’t be contaminated by the lubricating oil. 45 Fig.2.21 It consists of one or more metallic diaphragm 1 between two concave disks 2. The diaphragms move elastically under the action of the piston 8 and liquid of working 7 (oil). The volume variation in the working room, that is superior to the diaphragm, ensures suction (through valve 4) and discharge (through valve 3) of the fluid. Pump 5 carries out the compensation for the losses of oil due to the non- tightness of the piston. Valve 6 is a limiting valve for the discharge pressure. The pump with screw (fig.2.22) The number of rotors (two or more) can classify pumps with screw, by the shape of the thread (rectangular, trapezoidal, and cycloid), by the number of starts (one, two or more). In fig.2.22 it is presented the scheme of a pump with screw with two rotors (screws), of which one is driving. The driving rotor has a thread right and the other one left. Fig. 2.22 46 By the relative rotation of the two rotors the liquids get into the suction room A, and fill the clearance between the rotors in the area that is not driven. The liquid will be transported in the discharge room R, on a straight trajectory, without flow pulsations. The working of this pump is similar to that of the endless piston. The pump with cycloid gearings (fig.2.23) This type of pump consists of two cycloidal shaped rotors, of which one is driving,that rotate conversely. The hachured area represents the section of suctioned liquid due to the rotation of the cycloid gearing that is (in the next moment to that shown in the figure ) to be repressed. Fig.2.23 The pump with roll (fig.2.24) The pump with rolls is another type of volumetric rotary pump with an eccentric rotor. Suction and discharging are carried out due to the variation of volume in the space among the rotor, stator and rolls. The rolls are made of plastic with a metallic core. Due to rotation they are pushed on the walls of the stator by the centrifugal force, thus separating the variable volumes. Fig.2.24 Fig.2.24 2.17. Characteristics of volumetric pumps One of the main characteristics of the volumetric pumps is the characteristic flow-pressure. The real flow represents a slight decrease with respect to pressure, due to the increase of volumetric losses. Over a certain pressure lim p the decrease of the flow 47 is obvious (fig.2.25). Function ( ) p f N = , which represents the variation of power, is approximately linear up to value lim p , after which its increase is even more obvious. (fig.2.25). After the same value lim p , the curve of efficiency, ( ) p f = q has a strong descending carriage. In fig.2.26 there are shown the characteristics ( ) p f Q = for a pump with adjustable flow at different eccentricities (or tipping angles in the case of pumps with axial pistons). Figure 2.27 shows the mechanical characteristic moment-pressure-rotation. The slope of these curves, n M c c , shows us the litheness of the mechanical characteristic. Fig. 2.25 Fig. 2.26 Fig. 2.27 48 2.2 Hydrodynamic pumps 2.2.1 Building and classification Pumps or hydrodynamic generators process the potential energy of pressure and kinetic energy, by means of an impeller equipped with blades. The blades of the impeller are usually placed between two parallel disks; one is fixed on the shaft (the crown) and the other one that contains the inlet of the fluid (the ring). The fluid passes through the suction pipe, gets into the rotor where a kinetic energy is inculcated upon it, which afterwards is converted into potential energy in the spiral room and in the discharge pipe. Some centrifugal pumps are equipped with a stator with blades that have the role to convert the kinetic load into pressure load and to direct the fluid. In fig.2.28 it is schematically represented a centrifugal pump with the following components: Fig.2.28 1. The suction flange that makes the connection with the suction conduit. 2. Ring. 3. Network of blades. 4. The crown of the rotor. 5. The axis of the pump. 6. The tightening system of the axle. 7. The spiral room that collects the fluid from the periphery of the stator and contributes to the convention of kinetic pressure into potential pressure. 8. The stator that has the role to direct the stream and converts the kinetic energy into pressure energy. 9. The diffuser, that also contributes to the conversion of the kinetic load in pressure load and makes the connection with the discharging conduit. 49 Hydrodynamic pumps or turbo pumps may be classified by the specific rotation or dynamic rapidity, that can be considered as the rotation of a pump geometrically similar with the given one, which absorbs a power of 1 H.P. at a load of 1m: 4 / 5 H P n n HP S = (2.2-1) Specific rotation s n and rotation n measured with the tachometer, obviously cannot have the same dimension. In table 2.1 there are shown the classification of turbo pumps and the shape of the meridian suction of their rotor, with respect to specific rotation. Table 2.1 Type of pump Pump with lateral channel Centrifugal pump with rotor Axial pump slow normal rapid diagona l The shape in meridia n section of the rotor K 0,04 – 0,2 0,2 – 0,4 0,4 – 0,8 0,8 – 1,55 1,55 – 2,6 2,6 – 6,2 S n 8 – 40 40 – 80 80 – 150 150 – 300 300 – 500 500 – 1200 q n 2,2 - 11 11 – 22 22 - 41 41 – 82 82 - 135 135 - 380 In order to classify the turbo pumps we can also use their characteristic rotation or kinematic rapidity: 50 4 / 3 H Q n n q = (2.2-2) as well as the characteristic number ( ) 4 / 3 2 gH Q n K t = . (2.2-3) Between these values there are the relations: K n n q S HP 193 65 , 3 = = . (2.2-4) 2.2.2. Turbo pumps theory Inside the rotor of the turbo pump, the liquid particles carry out a complex movement. Following the outline of the blade, the particle covers a relative trajectory 1-2, but, at the same time, the rotor turns, the movement of the particle with respect to a reference system joined to the frame of the pump being ' 2 1÷ - the absolute trajectory. (fig.2-29). The basic theoretical equations of the turbo pumps applied to the case of centrifugal pumps are obtained for the following hypotheses: a) Between two consecutive blades of the rotor of the centrifugal pump, the flow of the fluid is stationary, in the shape of some streamlines that take the curvature of the blade. b) Inside the pump we don’t have hydrodynamic losses. c) The rotor consists of an infinite number of blades with negligible thickness. Thus, noting by symbol 1 the inlet in the inter blade channel, and by 2 the outlet, we shall have (fig.2.29 and fig.2.30): - the relative inlet and outlet velocities in and from the rotor 1 w and 2 w tangent in any point to the stream line that has the shape of blade; - peripheral velocities that are due to the rotation with speed e of the rotor on the circles with radii 1 R and 2 R , 1 1 R u e = and 2 2 R u e = ; - absolute velocities 1 v and 2 v that result from the making up of the relative and peripheral velocities: 51 . , 2 2 2 1 1 1 u w v u w v + = + = (2.2-5) Fig.2.29 Fig.2.30 Absolute velocity decomposes into a tangent component ,a load component: . cos , cos 2 2 1 1 2 1 o o v v v v u u = = (2.2-6) and a normal component, a flow component: . sin , sin 2 2 1 1 2 1 o o v v v v m m = = (2.2-7) The theoretical volumetric flow of liquid at inlet, equal to that at outlet, will be: , 2 2 2 1 2 2 1 1 m m v v b R v b R Q t t t = = (2.2-8) where 1 b and 2 b are the thickness of the blades at inlet and outlet, respectively. 52 The fundamental equation of turbo machines, applied in the case of centrifugal pumps can be obtained in several ways: a) by applying the theory of variation for the moment of movement quantity (impulse) We shall further consider an ideal centrifugal pump (the impeller with an infinite number of very thin blades). The movement quantities at inlet and outlet 1 and 2 are 1 u m v Q and 2 u m v Q , and their moments 1 1 R v Q u m and 2 2 R v Q u m . The variation of the moment for the movement quantity between these two points will be: ( ) ( ). 1 2 1 2 1 2 1 2 R v R v Q R v R v Q M u u v u u m t ÷ = = ÷ = A µ (2.2-9) The power, in the case of rotation with angular velocity e , will be given by the relation: ( ) ( ). 1 2 1 2 1 2 1 2 u v u v Q R v R v Q M P u u v u u v t t ÷ = ÷ = A = µ e e µ e (2.2-10) The relation expresses the power of an ideal pump with an infinite number of blades: · = T v H gQ P t µ . (2.2-11) Equaling the last two relations we get: g v u v u H u u T 1 2 1 2 ÷ = · , (2.2-12) expression that represents the fundamental equation of ideal centrifugal pumps. Euler has inferred it for hydraulic wheels long before the invention of centrifugal pumps. b) by applying Bernoulli’s equation for the relative movement between the points 1 and 2. In Bernoulli’s equation for relative movement 53 f h z p g u w z p g u w + + + ÷ = + + ÷ 2 2 2 2 2 2 1 1 2 1 2 1 2 2 ¸ ¸ (2.2-13) we consider 2 1 z z = . The pressure load created in the rotor will be: f h g u u g w w p p ÷ ÷ + ÷ = ÷ 2 2 2 1 2 2 2 2 2 1 1 2 ¸ . (2.2-14) The load · T H will be equal to the increase of the water pressure at outlet of the rotor plus the increase of kinetic energy plus the losses of load: f T h g v v p p H + ÷ + ÷ = · 2 2 1 2 2 1 2 ¸ . (2.2-15) From the relations (2.2-14) and (2.2-15) we get the expression: g v v g u u g w w H T 2 2 2 2 1 2 2 2 1 2 2 2 2 2 1 ÷ + ÷ + ÷ = · . (2.2-16) which is the fundamental equation of turbo machines applied to centrifugal pumps, in velocities. From the velocity triangle we have: . cos 2 , cos 2 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 1 2 1 o o u v u v w u v u v w ÷ + = ÷ + = (2.2-17) By replacing (2.2-17) into (2.2-16) we get the fundamental equation of turbo machines applied to centrifugal pumps, similar to equation (2.2-12): ( ) g v u v u v u v u g H u u T 1 2 1 2 1 1 1 2 2 2 cos cos 1 ÷ = ÷ = · o o . (2.2-18) The fundamental equation may also be written in the form: 1 2 1 2 u u T T v u v u gH Y ÷ = = · · (2.2-19) where · T Y is the specific energy, the energy of mass unit. 54 2.2.3. Turbo pumps in network The pump load or the pressure difference between the input and output of the liquid in a pump is independent from the network in which it works. The working parameters depend on and are defined by the network that a pump services. In fig.2.31 it is schematically shown a simple hydraulic system in which a pump P sucks liquid from the tank a R , with a pressure a p and whose level of liquid has the quote a z to the reference plane N – N and discharges it into the tank r R in which the pressure is r p and the level of liquid is at the quote r z . Vacuum gauge V measures the inlet pressure in the pump i p , and manometer M the outlet pressure from the pump e p . a h and r h are the load losses in the suction, respectively discharging conduits. The velocities of the fluid on suction and discharge are a v and r v . Applying Bernoulli’s equation to the suction route, we get: i i i i a a a a H g v p z h g v p z = + + = ÷ + + 2 2 2 2 ¸ ¸ . (2.2-20) 55 Fig.2.31 On the discharging route we shall have: e e e e r r r r H g v p z h g v p z = + + = + + + 2 2 2 2 ¸ ¸ . (2.2-21) The load of the pump will be: . 2 2 2 2 2 2 ¿ + ÷ + A + A = = + + ÷ + ÷ + ÷ = ÷ = ar a r r a a r a r a r i e h g v v p z h h g v v p p z z H H H ¸ ¸ (2.2-22) Relation (2.2-22) signifies the pump functions, namely: the liquid lifting on the height z A , the pressure rise from a p to r p , the alteration of the liquid kinetic energy by increasing its velocity, the overcome of the losses on the suction and discharging routes. The losses on the routes are local and linear: ¿ ¿ ¿ | . | \ | + = + = r a r a ar g v d l h h h , 2 2 ç ì . (2.2-23) 56 The suction and discharging routes, having conduits of diameters a d and r d are covered by the flow Q: ¿ ¿ ¿ | . | \ | + = r a r a gd Q d l h , 4 2 2 , 2 16 t ç ì . (2.2-24) and | | . | \ | ÷ = ÷ 4 4 2 2 2 2 1 1 8 2 a r a r d d g Q g v v t . (2.2-25) By replacing (2.2-24) and (2.2-25) into (2.2-22) we get: 2 , 4 4 4 2 1 1 1 1 8 Q d d d d g p z H r a a r ( ( ¸ ( ¸ | | . | \ | ÷ + | . | \ | + + A + A = ¿ ¿ ç ì t ¸ . (2.2-26) The expression: ( ( ¸ ( ¸ | | . | \ | ÷ + | . | \ | + = ¿ ¿ r a a r r d d d d g K , 4 4 4 2 1 1 1 1 8 ç ì t (2.2-27) is constant for a certain network. We denote by , ¸ p z H S A + A = (2.2-28) the static load. In this case the load expression becomes: 2 Q K H H r S + = . (2.2-29) Function (2.2-29) stands for the network characteristics and represents, as it can be noticed, a parabola. Should the flow be reversed (emptying the tank through the network), the expression would become: 2 Q K H H r S ÷ = . (2.2-30) 57 Figure 2.32 shows many more characteristics of networks at the same static load, but which have some alterations for r K (different diameters of conduits, bends, different taps, etc.). Fig.2.32 Analytically or experimentally we can determine the function ( ) Q f H = - the interior characteristics or the machine characteristic. In the case of a finite number of blades, due to the variation of velocity in the inter- blade channel, the value of the product 2 2 u v u is decreased. Consequently, the conveyed specific energy will be smaller. We may write: p H H gH gH Y Y T T T T T T ÷ = = = · · · 1 . (2.2-31) where p = 0,2 – 0,45 according to the model proposed by Pfleiderer. T H is the theoretical height for a pump with a finite number of blades for the case when we circulate a liquid without viscosity. The real height may be written in the form: ¿ ÷ = r T h H H (2.2-32) where ¿ r h is the dissipation due to viscosity, proportional to the square of the flow, 2 1 1 Q K h r = (2.2-33) 58 and the shock losses 2 r h due to the fact that for flows different from the rated flow N Q , the inlet angle of the stream of liquid 1 | differ from the inlet constructive angle of the blade. 2 2 1 | | . | \ | ÷ = N r Q Q K h r . (2.2-34) Then: 2 2 2 1 1 ¿ | | . | \ | ÷ + = N r Q Q K Q K h . (2.2-35) Returning to the fundamental equation of centrifugal pumps, we notice that · T H is bigger as 1 1 u v u is smaller and nil when the inlet in the impeller is normal ( ) 0 1 90 = o : g v u H u T 2 2 = · . (2.2-36) In fig.2.30 we notice that: 2 2 2 2 | ctg v u v m u ÷ = . (2.2-37) But the normal component of the outlet velocity has the value: 2 2 2 b D Q v m t = . (2.2-38) Taking into consideration relations (2.2-36), (2.2-37) and (22.-38) we may write: | | . | \ | ÷ = · 2 2 2 2 2 | t ctg b D Q u g u H T . (2.2-39) The theoretical load of a centrifugal pump with an infinite number of blades has a linear variation with respect to the flow. The bending of the line depends on the angle 2 | (fig.2.33). The theoretic manometer height is maximum when 0 2 90 > | , in other words when the blades of the impeller are curved forward. Pumps with 0 2 90 > | and those with 0 2 90 = | have a smaller efficiency than those with 0 2 90 < | , due to the high losses of energy at the inlet of the liquid into the collecting channel (high acceleration inculcated upon the liquid in the inter- blade 59 channel). Centrifugal pumps with 0 2 90 > | have also an instability of energy. These disadvantages make us prefer pumps with 0 2 90 < | , although their theoretical manometer height is lower. Fig.2.33 Considering the relations (2.2-31), (2.2-35) and (2.2-39) we may write the expression of the real load: ( ) 2 2 2 1 2 2 2 2 2 1 1 | | . | \ | ÷ ÷ ÷ | | . | \ | ÷ + = N Q Q K Q K ctg b D Q u p g u H | t . (2.2-40) In fig.2.34 it is shown the interior characteristic of the pump that resulted from the superposition of the linear variation of the theoretical load with the parabolic variation of the dissipation due to viscosity and shocks. The working point of a pump in a certain network is found at the crossing between the network characteristic with the interior characteristic (fig.2.35). Fig.2.34 60 Fig.2.35 The optimal running of a hydraulic system will take place when the duty point is in the area of maximum efficiency. The curve ( ) Q q is experimentally obtained, after dependence ( ) Q P has been determined. To improve the pump performances within the hydraulic system, we may change the position of the duty point, by modifying the network characteristic. This may be achieved in several ways. A simple way is to modify constant r K by varying the local strength coefficients ç and the adjusting parts. We may also change the static load of the network. Fig.2.33 shows the sliding of the duty point of the pump when the network characteristics are altered. 2.2.4. Computation of centrifugal pumps For a real pump, the thickness of blades has an influence on the velocities at inlet and outlet of the liquid to and from the rotor. In fig.2.36 we noted by s the thickness of the blade and by t the pitch of the blade. We shall analyze the state of the radial velocities in points O, little before inlet in the impeller, l, at inlet in the impeller, 2, at outlet of the impeller, and 3, immediately after outlet of the impeller. Nothing by o the circle bow corresponding to the thickness of the blade, we shall get: 1 1 1 sin | o = s . (2.2-41) From the continuity equation of the flow ( m v - the radial flow component) between the points O and 1, we get: ( ) , 1 1 1 1 1 0 1 b t v b t v m m = ÷o (2.2-42) where z D t 1 1 t = (z – the number of blades) Further we shall have: 61 l m m m v t t v v µ o 0 0 1 1 1 1 = ÷ = (2.2-43) where 1 1 1 1 t t o µ ÷ = is the decrease coefficient of the section due to the thickness of blades. To avoid shocks at the inlet section, the blades are rounded. Fig.2.36 Similarly, at the outlet from the impeller, we shall have: ( ) 3 2 2 2 2 3 2 b t v b t v m m = ÷o (2.2-44) As the construction of blades at the outlet from the rotor is edged, 0 2 = o and 3 2 m m v v = . (2.2-45) The influence of the outlet angle has been discussed in the previous chapter. The angle 2 | has values that range between 14 and 0 30 , rarely higher. When computing the impeller dimensions (fig.2.37) we start from the diameter of the driving shaft – d – computed with respect to the torque for a certain rotation of the driving motor. The power of the driving motor may be computed with respect to the load H, and flow Q of the pump, and, obviously, with respect to the efficiency of transmission. 62 The diameter of the hub is adopted ( )d d n 5 , 1 2 , 1 ÷ = . (2.2-46) The pump should be computed for a flow ' Q higher than Q, as we have to take into consideration the volumetric losses: ( )Q Q 15 , 1 ...... 03 , 1 ' = . (2.2-47) The velocity of the liquid through the conduit, S v , is adopted between 2 and 4m/s, the higher value corresponding to a load at lower suction. From the continuity equation it results that: 2 ' 4 n S S d v Q D + = t . (2.2-48) Diameter 1 D is adopted bigger than S D , so that the inlet edge should be outside the curvature area of the stream lines: ( )mm D D S 15 ..... 5 1 + = . (2.2-49) The thickness of the blade at the inlet in the rotor is computed taking into consideration the radial (flow) component of the velocity, little before the inlet in the impeller. 1 1 1 1 sin 1 0 µ o µ v v v m m = = . (2.2-50) Thus: 1 1 1 1 ' 1 sin µ o t v D Q b = . (2.2-51) Generally 1 v can be taken equal to S v . If 0 1 90 = o , we may write: 1 1 1 µ t S v D Q b = . (2.2-52) Assuming that, in a first approximation 8 , 0 1 = µ , we may determine the velocity triangle at inlet by means of formulae: 60 1 1 n D u µ = (2.2-53) 63 and 1 1 1 u v tg = | (for 0 1 90 = o ). (2.2-54) The necessary manometric load of the pump H is established beforehand depending on the necessities of the installation. For a certain hydraulic efficiency h q we may write: h T H H q = (2.2-55) and according to (2.2-31) ( ) p H H T T + = · 1 . (2.2-56) For radial pumps the computation relation of coefficient p is: 2 2 1 1 1 2 | | . | \ | ÷ = D D z p ¢ , (2.2-57) where ¢ is a coefficient experimentally established. For centrifugal pumps with a stator with blades ¢ can be established by means of the relation: ( ) 2 sin 6 , 0 65 , 0 55 , 0 | ¢ + ÷ = . (2.2-58) For a pump with bladeless stator, its values are a little higher. From the relation (2.2-39) where ' Q Q ÷ it results 2 u and then 2 D : n u D t 2 2 60 = . (2.2-59) For the case when 1 2 2D D ~ the pump is well designed, with low friction losses. When 2 D is much higher, we must choose a pump with more serial impellers, and when 2 D is lower, the flow and load characteristics require a pump with more parallel impellers. 64 A pump ensures the water directing with the greatest number of blades, but which have a detrimental effect in what regards the increase in the friction losses. When establishing the number of blades we must take into consideration these aspects. The computation for the number of blades is: 2 sin 5 , 6 2 1 1 2 1 2 | | + ÷ + = D D D D z . (2.2-60) To compute the coefficient p we need the number of blades that is established for the hypothesis that 1 2 2D D = , this is to be checked by mean of the relation (2.2-59). If the error is too high, the computation must be reconsidered, acting upon some parameters, within reasonable limits, and if not, we resort to serial or parallel of several impellers as above stated. The number of impellers i is established by the relation , H H i A = (2.2-61) Fig. 2.37 where , 2 2 2 u KD H = A (2.2-62) | | | | min / , rot m D e and ( ) 4 10 5 , 1 ..... 3 , 1 ÷ = K for a stator with blades, ( ) 4 10 4 , 1 ....... 1 ÷ = K for a stator without blades. 65 2.2.5 Parallel and series connection of centrifugal pumps To increase the flow or the load of a hydraulic system, we use parallel or series connections of pumps. a) Parallel connection (fig.2.38) In the case when two or more pumps are connected in parallel it is achieved an increase of the flow for a constant load. For two pumps we’ll have 2 1 Q Q Q c + = , (2.2-63) expression that is in fact the relation of continuity. 2 1 H H H c = = . (2.2-64) signifies self-equilibrium of the system pump-network. Fig. 2.38 When two identical pumps are parallel – connected (fig.2.39) the interior characteristic is obtained by doubling the abscissa of the points on the interior characteristic for one pump. The duty point of the system c F will be at the crossing between the interior characteristic and the network characteristic. The efficiency of parallel – connected centrifugal pumps depends on the characteristic of the network. 66 It can be noticed that in the case of network R, the increase in flow Q A as compared to a system with one pump is more important than the increase ' Q A in the case of network R’. It is noticed that in the case of parallel connected pumps there appears an increase of load, that also depends on the characteristic of the network. Fig.2.39 The efficiency of the two identical pumps is q q q = = 2 1 . The efficiency is the ratio between the useful and consumed power: 2 1 P H Q P H Q c F c F ¸ ¸ q = = . (2.2-65) In a coupled regime, each pumps works in duty point F, and c F Q Q 2 1 = . Thus: q ¸ c c H Q P P 2 1 2 1 = = . (2.2-66) The efficiency of the coupling will be: q q ¸ q ¸ ¸ q = + = + = c c c c c c c c CP H Q H Q H Q P P H Q 2 1 2 1 2 1 . (2.2-67) In the case of the parallel connected of two or more identical pumps, the general efficiency will be equal to the efficiency of each pump. When parallel connecting two pumps with different characteristics, the problem is much more complex. The characteristic of coupling is obtained in a 67 similar way, by summing up the characteristics abscissae of the two pumps at a constant load, ( ) 2 1 Q Q Q H c c + = (fig.2.40). On the diagram of the coupling the critical point cr P appears, situated at the quote of critical load cr H , corresponding to the crossing between the smaller pump characteristic and the ordinate. If the duty point of the system is below cr P , as in the case of the characteristic of the network R, then the parallel –connecting of the two pumps is justified. In the case of characteristic ' R , the duty point is above cr P , the smaller pump working on the braking characteristic. In this case the flow of the coupling is lower than the flow of one pump (the big one), thus the coupling becoming unjustified. Fig.2.40. The efficiency of a coupling of two different pumps will be given by the relation [8]: 2 2 1 1 2 2 1 1 q q q ¸ q ¸ ¸ q Q Q Q H Q H Q H Q c c c c c CP + = + = . (2.2-68) 68 b) Series – connection (fig.2.41) To increase the load we use the series connection of two or more centrifugal pumps. The flow that passes through two series connected pumps is the same: 2 1 Q Q Q c = = , (2.2-69) and the load 2 1 H H H c + = . (2.2-70) Fig.2.41 To plot the characteristic of the assembly we sum up the ordinates of the characteristic point for each pump. Fig.2.42. shows the common characteristic of two identical series – connected pumps. From fig.2.42. it can be noticed that in the case of characteristic R we get a higher increase of the load than in the case of characteristic ' R . It can be noticed that in series - connection an increase in flow is also obtained. Fig.2.42. The efficiency of the coupling is equal to the efficiency of each pump taken separately. 69 q q ¸ q ¸ ¸ ¸ q = + = + = c c c c c c c c CP H Q H Q H Q P P H Q 2 1 2 1 2 1 . (2.2-71) For different pumps series – connected, the characteristic of the coupling is also obtained by summing up the ordinates of the points on the characteristics of the two pumps (fig.2.43). Fig.2.43 Here there is also a critical point corresponding to the load abscissa O of the smaller pump. In networks whose characteristics the duty point is below cr P it is irrational to use two pumps whose total flow is lower than that of a single pump. The efficiency of the coupling when series – connecting two different pumps will be [8]: 2 2 1 1 2 2 1 1 q q q ¸ q ¸ ¸ q H H H H Q H Q H Q c c c c c CP + = + = . (2.2-72) For reason of strength of materials, the peripheral velocities of the impellers cannot exceed certain values. 70 Since the maximum theoretical load depends on the peripheral velocity of the impeller, thus being limited by it, to increase the load on a single unit, we use pumps with several series – connected impellers (fig. 2.44.) Fig.2.44. Also, obtaining higher flows is limited by rotation and by the outlet diameter from the impeller, as well as by the circulating velocity of the liquid. By using double impellers and by parallel – connecting them within a pump (fig.2.45), we achieve the increase in flow and also the self – equilibrium of the axial thrusting forces. Fig.2.45 To simultaneously obtain high loads and flows on a single pump, we can use several impellers that are axially series and parallel connected (fig.2.46) Fig.2.46 2.2.6 Suction of centrifugal pumps The suction of centrifugal pumps is due to the depression generated in the impeller; in fact it is due to the difference of pressure between the impeller and the suction tank. In the case when the pump sucks water from an atmospheric pressure (barometric) b a p p = and the depression in the impeller would attain vacuum, the theoretical maximum suction height would be: 71 m h p H b b asp t 33 , 10 = = = ¸ ¸ ¸ . (2.2-73) Fig. 2.47 shows a centrifugal pump that sucks from a pressure a p . We shall consider three reference points: a – the level of liquid, O – the highest point before inlet in the impeller, 1 – immediately after inlet in the impeller. We shall consider as reference points the level of the liquid that is to be sucked and that is under motion with velocity a v * . * If b a p p = then we have the case of a centrifugal pump that sucks from a river. 72 We apply Bernoulli’s relation between the considered points: rir ra asp ra asp a a h h H g v p h H g v p g v p + + + + = = + + + = + ¿ ¿ 2 2 2 2 1 1 2 0 0 2 ¸ ¸ ¸ . (2.2-74) where ¿ ra h are the local and linear losses on the suction itinerary, and rir h the load loss at the inlet in the channels of the impeller. This loss of load may be written under the form: , 2 2 1 g v h rir , = (2.2-75) where , is the local coefficient of loss at the inlet in the pump. If suction is being made from a tank ( ) 0 = a v , the suction load will be: ( ) ¿ ÷ + ÷ ÷ = ra a asp h g v p p H 2 1 2 1 1 , ¸ . (2.2-76) The maximum load at sucking would be when 0 1 = p , but it is known that in real practice the maximum depression in a moving liquid corresponds to the absolute saturation pressure of the liquid at the respective temperature, the moment when the cavitation phenomenon appears: v p p = 1 . Thus: ( ) ¿ ÷ + ÷ ÷ = ra v a asp h g v p p H 2 1 2 1 max , ¸ . (2.2-77) The term ( ) g v 2 1 2 1 , + depends on the design characteristics of the hydraulic machine, and it can be expressed with respect to the effective load of the pump H by means of cavitation coefficient o : Fig. 2.47 73 ( ) H g v o , = + 2 1 2 1 . (2.2-78) We rewrite expression (2.2-77): ¿ ÷ ÷ ÷ = ra v a asp h H p p H o ¸ max . (2.2-79) The cavitation coefficient is given by the experimental relation [8]: ( ) rotation specific n n a S S ÷ = 3 / 4 o . (22-80) Several values are granted for coefficient a in the literature of the subject: 4 10 29 , 2 ÷ (Thoma); 4 10 20 , 2 ÷ (Stepanoff); 4 10 16 , 2 ÷ (Escher – Wyss). It has been established that coefficient a also depends on the specific rotation. Coefficient o may also be written [8]: H C Q n 10 3 / 4 | | . | \ | = o , (2.2-81) where C is Rudnev’s cavitation coefficient and has the values: . 150 . . . .. 80 1000 . . . . . 800 80 . . . . . 50 800 . . . . . 600 = = = = S S n for C and n for C Relation (2.2-79) shows us a maximum suction height, which for different reasons doesn’t correspond to the real suction height. Thus, velocity 1 v of inlet in the impeller may have a higher value, generating cavity suction conditions. Thus, to establish the needed suction height we operate on the cavitation coefficient by considering a: ( )o o 4 , 1 ..... 2 , 1 lim = . (2.2-82) or, in a simple manner, by directly operating on the suction load, reducing it, to: max 75 , 0 asp asp H H = . (2.2-83) According to relation (2.2-81) the suction height will be: ¿ ÷ ÷ ÷ = ra v a asp h H p p H lim o ¸ . (2.2-84) 74 The suction height on the centrifugal pumps is under the influence of a series of factors. In the case of suction from an atmospheric pressure b a p p = , the suction height depends on the variation of this pressure with the weather state, latitude and especially with the height of the place. Should we denote by 0 p , the pressure at sea level, the pressure variation with height | | m z might be written under the form: ( ) z p p b 5 0 10 4 , 2 1 ÷ ÷ = . (2.2-85) The suction height depends through v p , on the nature of the vehicled fluid and on its temperature. We can go as far as that the suction height comes be negative * when: ¿ ÷ ÷ > ra v h H p p lim 0 o ¸ ¸ . (2.2-86) In this case the pressure in the tank of suction should be increased or, in the case when the tank is open, this should be mounted above the pump, at a corresponding height. 2.2.7 Axial pumps According to the classification shown in chapter 2.2.1, axial pumps are at the extremity of the specter of specific rotation for pumps ( ) 1200 ..... 500 = CP S n . For this type of pumps, the specific energy is obtained by a partial conversion of kinetic energy in the inter- blade channel, the moving of the fluid being performed axially. *In the case of circulating water at temperatures higher than C 0 60 . For water at 0 10 5÷ , m H asp 7 ..... 6 = . 75 In fig.2.48 an axial pump is schematically shown. It is mainly made up of: directing device 1, hub with blades 2, that together with axle 7 is the mobile part of the pump, rectifying device 3, carcass 4, together with elbow 5 and stuffing box 6, careening of impeller 8. Generally axial pumps have blades with a fixed pitch. For axial pumps of high powers we can use impellers with variable pitch for different load situations. The design of the blade is similar to the design of the naval propeller, namely a sequence of hydrodynamic profiles disposed under different placed angles from hub to periphery. The directing device ensures a shockless input of the fluid particles into the impeller, and the rectifying device, apart from converting a part of kinetic energy into pressure energy is designed to direct the fluid jet in an axial direction. Fig. 2.48 In fig.2.49 we have considered a cylindrical section through the pump, at a distance r, section from which we have taken only one element of the directing device, impeller and rectifying device. Unlike centrifugal pumps, the peripheral velocity at the input into the impeller 1 u is equal to the peripheral velocity at the output from the impeller 2 u : 76 e r u u u = = = 2 1 . (2.2-87) The absolute velocity at the input into impeller, 1 v , results from the composition of relative velocity, 1 w , tangent to the blade, with the peripheral velocity, 1 u . The role of the profile in the directing device is to result an absolute velocity at output, as near as possible on the direction of velocity 1 v . Also, at input in the directing device velocity a v has to have an axial direction, 0 90 = a o . At outlet from impeller, velocity 2 w , tangent to the trailing edge, composed with peripheral velocity, 2 u (equal to 1 u ), will give the absolute output from impeller velocity, 2 v . The profiles of the stator will have to direct the output velocity in point 3, as much as possible to the axial direction. The profiles of the blades influence one another. The problem is that of a network of profiles with pitch t. We can consider that the profile of the impeller in fig. 2.49 is attacked with velocity · w , a mean of velocities 1 w and 2 w . Observing the speed triangle in the same figure, we may write: Fig.2.49 . 2 , 2 2 1 2 1 2 2 u u a u u a v v u v tg v v u v w + ÷ = | | . | \ | + ÷ + = · · | (2.2-88) If we denote by l its chord and by b its span, we can write the relation of the lift and resistance forces that act upon it: 77 . 2 , 2 2 2 lb w C F b l w C F x x z z · · = = µ µ (2.2-89) We have denoted by z C and x C the coefficient of the lift force and of resistance at advancement. Resultant 2 2 z x F F F + = may decompose by an axial and a tangent direction. ( ) ( ) . sin , cos ¢ | ¢ | + = + = · · F T F A (2.2-90) For an axial pump with z blades we shall have an axial thrust: ( ), cos ¢ | + = = · F z zA F A (2.2-91) and a hydraulic power: ( ) . sin ¢ | + = = · uF z u T z P h (2.2-92) The elementary hydraulic power may be written in several forms: ( ) ( ) ( ) . sin cos 2 sin cos sin 2 ¢ | ¢ µ ¢ | ¢ ¢ | + = = + = + = · · · · dr l w C u z dF u z dF u z dP z z h (2.2-93) Through a cylindrical span section dr, at a distance r, the elementary flow a v rdr dQ t 2 = will pass. The elementary hydraulic power may be written as: a t a t t h v dr t z Y v dr r Y dQ Y dP µ t µ µ = = = 2 . (2.2-94) where t Y is the specific energy transmited by the pump, whose value we can found by equaling relations (2.2-94) and (2.2-93): ( ) ¢ | ¢ + = · · sin cos 2 2 w v t C l u Y a Z t . (2.2-95) As angle ¢ is relatively small 0 11 8÷ , we may approximate 1 cos ~ ¢ and ( ) · · = + | ¢ | sin sin . We notice as well that a v w = · · | sin . The specific energy created by the axial pump will thus be expressed by: 78 · = = w u t l C gH Y Z t t 2 . (2.2-96) It is noticed that the load of the pump or its specific energy depends in a direct proportion on rotation by peripheral velocity and on the shape of the profile by the lift coefficient Z C . Should the pitch t be decreased, thus increasing the number of blades, the load of the pump will increase. The number of blades cannot be increased due to the high hydraulic losses in the network of profiles. The cavitation phenomenon limits the increase of velocity · w . In fig. 2.50 there are plotted the characteristics of the axial pumps at different rotations as well as the efficiency curves. Considering the evolute of efficiency curves we can establish a maximum maximorum efficiency that corresponds to an optimal working regime of the axial pump [8]. The characteristics of the axial pumps are rather steep due to the blade profile that consists of a sequence of hydrodynamic profiles. It is known that for an alteration of the angle of incidence brought about by the alteration of flow in our case (more exact of a v ), the lift coefficient may abruptly decrease. Fig.2.50 2.3 Ejectors Ejectors are pumping elements of a special category, without any motioning parts. Their working principle is the following (fig.2.51). The motive fluid such as steam, air or water enters the ejector through nipple 1. In mixing chamber 2, its kinetic energy is partially converted into pressure energy that drives the pumped fluid. The mixture passes in the convergent – divergent diffuser 3, where its kinetic energy is converted into pressure energy. 79 Fig.2.51 Then the fluid can be separated from the pumped fluid, the easiest to be separated being steam (by condensation). To carry out the compensation of a water – water ejector we make use of the scheme in fig. 2.52. Fig.2.52 The flow of the driving fluid 0 Q passes with velocity 0 v through section 0 S . When entering the ejector it drives the flow 1 Q of fluid that passes with velocity 1 v through section 0 S S ÷ . The fluid has pressure 1 p in section 1. In section 2, after the mixing has been carried out and the transfer of energy and mass achieved, the flow will be Q, velocity 2 v and pressure 2 p . From the equation of continuity: ( ) 2 0 0 1 0 Sv v S v S S = + ÷ , (2.3-1) it results that: 0 0 1 0 2 1 v S S v S S v + | . | \ | ÷ = . (2.3-2) By applying the theorem of impulse: ( ) | | ( )S p p v S S v S Sv 2 1 2 1 0 2 0 0 2 2 ÷ = ÷ ÷ ÷ µ , (2.3-3) we get pressure 2 p : 80 ( ¸ ( ¸ ÷ | . | \ | ÷ + + = 2 2 2 1 0 2 0 0 1 2 1 v v S S v S S p p µ . (2.3-4) The usable power of the ejected fluid is obtained from the relation: ( ) . 2 2 2 2 2 2 2 1 2 0 2 0 0 1 2 1 2 1 0 2 2 2 2 1 2 0 0 1 2 1 1 2 2 2 2 1 2 ( ( ¸ ( ¸ | | . | \ | + ÷ | | . | \ | + ÷ ÷ + = = | | . | \ | + ÷ | | . | \ | + ÷ | | . | \ | + = = | | . | \ | + = } p v v v S S p v Sv v S S p v Sv p v Q p v Q p v Q dQ p v P u µ µ µ µ µ µ µ (2.3-5) The efficiency of ejectors is below 0.35 that is enough low; their advantage is that they have motionless parts. This makes ejectors highly reliable. Ejectors are used for compression ratios of about 5. To achieve a higher ratio several ejectors are series connected. Water ejectors use as driving fluid water under pressure. In the naval field they are used at installations of fire - fighting with lather, at draining systems, etc. Referring also to the naval field, air-air ejectors are used to ventilate small – dimensioned compartments and steam-water ejectors to supply the boilers. Ejectors are widely spread and they can be seen in many fields of activity. 81 2.4. Volumetric hydraulic motors By minimum design alterations most of volumetric hydraulic pumps may by transformed into hydraulic motors. Driving the under-pressured liquid the machine converts the hydraulic energy into mechanical energy. Volumetric motors may be classified in the same manner as the pumps they have been obtained from. Without going further into design and calculus details, we shall stress on a few elements that are characteristic for running hydraulic machines as motors. Rotary volumetric hydraulic motors may have a continuous or oscillating movement. A particular category are linear motors (hydraulic cylinders). 2.4.1 Hydraulic cylinders Hydraulic cylinders are volumetric hydraulic motors that are widely spread in hydraulic systems. In fig.2.53 a hydraulic cylinder that consists of piston 1, piston rod 2, and cylinder 3 is schematically shown. Fig.2.53 There are a large variety of hydraulic cylinders that can be classified according to several criteria: 1. By the way pressure acts upon the piston, there are: simple acting, where the piston doesn’t come back hydraulically and double acting, when pressure acts on both sides of the piston. 2. By the section of the two sides of the piston: with unilateral rod ( 2 1 S S = ) and bilateral rod ( 2 1 S S = ). Linear hydraulic motors can be mono or multi-cylinder. The stroke of the piston may be constant or variable, its adjustment being made mechanically or hydraulically. 82 From the designing point of view hydraulic cylinders can be of several kinds, depending on the working pressure, usage position and goal. Fig. 2.54 shows the main elements of a hydraulic cylinder: 1 – cylinder, 2 – lid, 3 – tightening screw, 4 – sealing gasket. In fig. 2.55 the sealing elements of the piston rod are shown (one of the design alternatives): 1 – tightening flange, 2 – screw, 3 – tightening part, 4 – sealings, 5 – piston rod. Fig.2.54 Fig.2.55 The pistons of the cylinders are also to be found in a large variety of designs. Fig.2.56 shows a piston with rings. There are pistons without sealing gaskets (pistons with circular channels). In this case surfaces are very accurate, a reciprocal smooth friction between the cylinder and the piston being accomplished. There are also pistons with circular gaskets, profiled gaskets etc. Fig.2.56 If we refer to the hydraulic cylinders where tightening is achieved by the reduced play between the surfaces of the cylinders and those of pistons, an outstanding Fig. 2.57 importance in the calculus has the determination of hydraulic losses. These losses are determined starting from the equations of flow through cylindrical centric slits. The piston of radius 1 r moves rectilinearly with a constant velocity p u ± in the cylinder of radius 2 r (fig.2.57). By integrating Navier-Stokes’ equation written in cylindrical co-ordinates, in the case of permanent linear movement, done on the direction of the axis ( 0 , 0 = = u v v r ) by leaving out massic forces and ( ) z p p = , we get: 83 2 1 2 ln 4 1 C r C r dz dp v + + = q . (2.4-1) To determine the constants 1 C and 2 C we write the limit conditions: . 0 , ; , 2 1 = = ± = = v r r u v r r p (2.4-2) In the end we get the relation of velocity: 1 2 2 2 1 2 2 1 2 2 2 2 2 ln ln ln ln 4 1 r r r r u r r r r r r r r dz dp v p ± | | | | | . | \ | ÷ + ÷ ÷ = q . (2.4-3) The flow of the lost fluid will be given by: ( ) ( ) ( ) | | . | \ | ÷ ÷ ± ± ( ( ( ( ¸ ( ¸ ÷ ÷ ÷ ÷ = = } 1 2 2 1 2 1 2 2 1 2 1 2 2 2 1 2 2 4 1 4 2 ln 2 ln ln 8 2 2 1 r r r r r r r u r r r r r r dz dp dr r r v Q p r r t q t t . (2.2-4) The fall of pressure on length b is denoted by p A . The force of viscous friction is given by: t t b r F 1 2 = , (2.4-5) where dr dv q t = . (2.4-6) Thus: 1 2 1 1 2 2 1 2 2 1 1 ln 2 ln 2 2 r r r u b r r r r r r r p r dr dv b r F p q t t q t  | | | | | . | \ | ÷ ÷ A = = . (2.4-7) The force of maximum hydraulic friction (2.4-7) is got for 1 r r = : 84 1 2 1 2 2 1 2 2 2 1 max ln 2 ln 2 r r u b r r r r r p F p q t t  | | | | | . | \ | ÷ ÷ A = . (2.4-8) An important problem is to determine the forces of friction on the cylinder for the sealing elements. This force is given by: ¿ = = z i i s p r F 1 2 2µt , (2.4-9) where: p is active pressure, µ - friction coefficient ( 008 , 0 006 , 0 ÷ = µ for leather gaskets, 01 , 0 = µ for rubber), i s - breadth of a gasket. In the case of the sealing with rings, there also appears an elastic pressure of the ring on the cylinder, c p : ( ) ¿ ÷ + = z i i c s p p r F 1 2 2µt , (2.4-10) where 15 , 0 ..... 07 , 0 = µ . To determine the work pressure that is required for the piston we determine the sum of the resistant forces, ¿ R. This sum consists of the necessary force for the working partas well as the friction forces and the inertia forces on the respective ring. Pressure for work 1 p is determined from: ¿ = ÷ ÷ 0 2 1 1 R S p S p e . (2.4-11) where 1 S and 2 S are the surfaces of the piston on the two sides, and e p is the exhausting pressure. If 2 1 S S = and 0 = e p (i.e.there is no anti-pressure at discharge valve) then the equilibrium equation becomes: ¿ = ÷ 0 1 R S p . (2.4-12) 85 2.4.2 Motors with radial pistons In the case when a hydraulic machine with radial pistons is working as a motor, the forces and moments are determined using the scheme in fig.2.58. Fig.2.58 F is the force with which the liquid of pressure p acts upon the piston of diameter | | . | \ | = 4 2 d p F d t . T is the normal force on direction AO, the one that determines the rotation of the hydraulic motor with the angular velocity e : | tg F T = . (2.4-13) Component N, having the direction 2 AO , stresses the bearings of the motor. The torque corresponding to a piston is given by: µ T M r = . (2.4-14) and the total torque given by the j pistons upon which it is acted with overpressure: ¿ = = j i i i rt T M 1 µ . (2.4-15) The values of T and µ are the same as for the pump with radial pistons (see chapter 2.1.2). 86 2.4.3 Motors with blades To compute this type of motors we start from moments 1 M and 2 M , of reverse signs, due to pressure p that act upon the blades (fig.2.59) [12]. Fig.2.59 ( ) ( ) 2 2 1 1 1 1 2 2 r pb r r r b p M ÷ = | . | \ | ÷ + ÷ = µ µ µ . (2.4-16) ( ) ( ) 2 2 2 2 2 2 2 2 r pb r r r b p M ÷ = | . | \ | ÷ + ÷ = µ µ µ . (2.4-17) The moment at the axle of motor will be: ( ) 2 2 2 1 2 1 2 µ µ ÷ = ÷ = pb M M M . (2.4-18) In the above relations ( ) , cos cos 2 1 R e and R e + + ~ + ~ ¢ u µ u µ where t ¢ = for an even number of blades k z 2 = and z t t ¢ ± = for an odd number of blades 1 2 + = k z , as 1 0 , + ÷ | . | ¸ ÷ e k z t u interstices at discharge, respectively k z , , 0 ( ¸ ( \ | e t u interstices at discharge. In the first case, when k z 2 = and t ¢ = , we’ll have: ( ) ( ) u u u cos Re 2 cos cos 2 2 2 b p R e R e pb M = + ÷ ÷ + = . (2.4-19) 87 The moment varies with respect to angle u . The definition field of the function ( ) u M is ( ¸ ( ¸ ÷ e 2 , 2 ¸ ¸ u where z t ¸ 2 = . . cos Re 2 , Re 2 min max z b p M b p M t = = (2.4-20) In the case when we have an odd number of blades 1 2 + = k z , we’ll get: ( ) ( ) | | ( ) | |. cos cos cos cos 2 cos cos 2 2 ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ + ÷ = = ) ` ¹ ¹ ´ ¦ + ÷ + + ÷ = R e b p R e R e b p M (2.4-21) For 1 + k discharging interstices, z t t ¢ + = : ( ¸ ( ¸ | . | \ | + + = z R e b p M t ¢ ¢ cos cos , (2.4-22) and for k discharging interstices, z t t ¢ ÷ = : ( ¸ ( ¸ | . | \ | ÷ + = z R e b p M t ¢ ¢ cos cos . (2.4-23) As it can be noticed the moment at the axle of motor isn’t constant. Its variation is characterized by the moment’s pulsation, whose value is equal to the flow pulsation that was computed in chapter 2.1.3 for pumps with blades. Starting from this moment we can determine the forces that act upon blades and bearings, thus being able to carry out the dimensioning of these elements. 88 2.4.4 Motors with axial pistons In the case of motors with axial pistons the force with which the pressure liquid acts upon a piston of diameter d, 4 2 d p F t = . (2.4-24) decomposes into a tangent and a normal force (fig.2.60). Fig.2.60 The tangent force o sin F T = , (2.4-25) creates the torque. For one piston we shall have: ¢ o t µ sin sin 4 1 2 r d p T M r = = . (2.4-26) The z pistons will create a total torque: ¿ = = z i r r p d M t 1 1 2 sin sin 4 ¢ o t . (2.4-27) It is noticed that the torque has a pulsatory character. The relation will give the power at the axle of motor: e t r M P = , (2.4-28) where e depends on the flow the motor is supplied with. 89 2.4.5 Oscillating rotary motors Oscillating rotary motors are a particular category of hydraulic motors that are used in actuations which do not require complete rotations of the active element; i.e. servo-controls, industrial robots, rudders of ships etc. These motors can be with one or several blades (fig.2.62), circular oscillating (fig.2.61 and fig.2.62) or linear oscillating (fig.2.63). Considering the pressure in the exhausting chamber to be nil, thus p p = A , we can write the relation of the moment at the shaft of the motor with a single blade of thickness b: ( ) ( ) 2 2 8 4 2 d D pb d D b d D p R F M ÷ = + ÷ = = . (2.4-29) Fig.2.61 Fig.2.62 Fig.2.63 90 The angular velocity will be: ( ) 2 2 8 d D b Q ÷ = e . (2.4-30) For motors with z blades the relations for the moment and angular velocity become: ( ) 2 2 8 d D b p z M ÷ = , (2.4-31) and ( ) 2 2 8 d D b z Q ÷ = e . (2.4-32) 91 2.5 Turbines Turbines are motric hydraulic machines that accomplish the conversion of energy by the hydrodynamic action of the fluid upon the impeller. Turbines are part of hydrodynamic machines (turbo-machines) category, and they can work as pumps or motors. They process the pressure potential energy and kinetic energy. In chapter 2.2 we have dealt with the general working principle of turbo- machines. An important classifying criterion of turbines is that of specific rotation (see chapter 2.2.1). 4 / 3 4 / 5 65 , 3 H Q n H P n n S = = , (2.5-1) which in fact determines the most appropriate shape of impeller. By specific rotation, there are three main types of turbines: - Pelton turbines ; 50 2÷ = S n - Francis turbines ; 550 40÷ = s n - Kaplan turbines . 1200 400÷ = S n 2.5.1 Pelton turbine Pelton turbine is an actuation turbine that processes the kinetic energy of the fluid (water). This kinetic energy comes from the conversion of a part of the potential energy of water pressure in injector 1 (fig.2.64). The jet of water from the injector hits the curved blades (buckets) 2 of impeller 3, bringing about its revolving movement with the angular velocity e . The flow can be adjusted by shifting the injector needle 4 by means of a servomechanism 5. 92 This is the simple working principle of Pelton actuated turbine. To establish the working parameters of this motive hydraulic machine, we consider its impeller being designed in such a manner as to take over the entire kinetic energy of the fluid jet (the kinetic energy of water at output from impeller should be nil): Fig.2.64 g v H n 2 2 1 = . (2.5-2) n H - net hydraulic falls available at input in the machine. The buckets of the impeller are usually ellipsoidal. Fig.2.65 shows a section through a bucket made up of two ellipsoids stuck on the big axis. The bucket is symmetrically hit by water jet with velocity 1 v . The jet is reversed by the bucket with angle 2 | , a lower value than the theoretical one of 0 180 [2]. Fig.2.65 The absolute velocity 2 v is composed of the relative velocity 2 w and peripheral velocity u : u w v + = 2 2 . (2.5-3) The jet thrusts the immobile turbine with the force: 93 ( ) 2 1 1 cos 1 | µ ÷ = Qv F , (2.5-4) and the moving turbine with the force: ( )( ) 2 1 2 cos 1 | µ ÷ ÷ = u v Q F . (2.5-5) Thus, the thrusting force will be maximum when 0 2 90 = | . The relation will give the power developed by the turbine: ( )( ) 2 1 2 cos 1 | µ ÷ ÷ = = u v Qu u F P . (2.5-6) The theoretical power being QH P t ¸ = , the hydraulic efficiency of Pelton turbine will be: ( )( ) . cos 1 1 1 2 u u v gH P P t h ÷ ÷ = = | q (2.5-7) To establish the maximum hydraulic efficiency we derive expression (2.4- 7) with respect to peripheral velocity u: ( )( ) 0 2 cos 1 1 2 = ÷ ÷ = c c gH u v u h | q . (2.5-8) Thus: 2 1 v u optim = . (2.5-9) Usually ( ) 1 485 , 0 42 , 0 v u ÷ = . 2.5.2 Francis turbine Francis turbine (fig.2.66) is a turbine with partial reaction. The force that brings about the revolving of the turbine is produced both by the action of water jet and by increasing velocity among blades (reaction). 94 The position potential energy of water in the inlet chamber 1 is partially converted into kinetic energy in diffuser 2. The impeller with curved blades, twisted in space, alters the water direction, generating a couple and achieving mechanical energy. Water comes out with an important kinetic energy, which is partly recovered by diffuser 5. Fig.2.66 The inlet chamber, that can be of three types: open tank, closed container or spiral, (as shown in the figure), connects the supply conduit with the distributor. The distributor directs the water to the impeller and converts a part of its potential energy into kinetic energy. The inlet flow in the impeller may be adjusted by altering the passing section among the blades of distributor. The flow adjustment is done by means of device 3. The impeller may have a horizontal or a vertical axle (for middle and big turbines). Its blades have the shape of a plough knife coulter and many times they are cast together with the body or the support on which the axle is fitted. The diffuser, also called suction conduit is a tube of variable section that has the role to recover a part of kinetic energy of the outlet water and to direct it to the trailing channel. We can observe the design similarity between a centrifugal pump and a Francis turbine. Both are turbo-machines whose running differs only in the sense of energetic conversion. The fundamental equation of turbo-machines (2.2-12) is the same, the indexes of velocities being obviously different since the sense of water circulation is reversed: ( ) 2 1 2 1 1 u u T v u v u g H ÷ = · . (2.5-10) 95 By applying Bernoulli’s equation between the outlet surface from impeller and water surface, we’ll get: f at a h g v p g v H p + + = + + 2 2 4 2 2 ¸ ¸ , (2.5-11) where f h is the load loss in the suction conduit. Hence: ¸ ¸ ¸ at f a at p h H g v v p p < + ÷ ÷ ÷ = 2 2 4 2 2 2 . (2.5-12) In the suction conduit a water suction is created, which increases the fall of pressure; thus a part of the outlet kinetic load, g v 2 2 2 is being recovered. However ¸ 2 p cannot be this low, being limited by the vaporization pressure, v p . We can avoid the cavitation phenomenon provided the suction height shouldn’t exceed the value: f v a a h g v v p p H + ÷ ÷ ÷ = 2 2 4 2 2 max ¸ . (2.5-13) 2.5.3 Kaplan’s turbine Kaplan turbines are the correspondents of axial pumps in the field of motive turbo-machines. Fig.2.67 schematically shows a Kaplan turbine. Fig.2.67 96 Water from the supply conduit passes in the spiral chamber 1, and from here on through distributor 2, equipped with directing blades, by means of device 3 and further on to impeller 4. The impeller is of axial type, made up of a hub on which 2 to 6 ellicoidal blades are fitted, with a loose end. The hub ends in a parabolic ogive. The blades of the impeller can be fixed or adjustable. Suction conduit 5 has an important role in Kaplan turbines since a large quantity of kinetic energy is to be recovered. Kaplan turbines may have a horizontal, vertical or slanting axle. They are used for high flows and low loads. As the Franscis turbines, they are turbines with partial reaction, the shape of the impeller being determined by their higher specific rotation. 97 3. CONTROL AND AUXILIARY APPARATUS The main aim of a hydropneumatic system is to produce hydraulic and pneumatic energy and to reconvert it into mechanical energy. The elements which produce or reconvert pneumatic or hydraulic energy, namely pumps (compressors) and motors are considered the main components of the actuating system. Yet we must underline that an outstanding importance for the smooth working of a system represent the other hydropneumatic elements, which will be studied in this chapter. These elements can be classified into control and auxiliary apparatus. The control apparatus has the role to direct and adjust the flow or the pressure of the fluid, a role that is mainly played by distributors, chokes and valves, respectively. Auxiliary apparatus leads the fluid medium (pipes), filters it (filters), stores and cools it (tanks and heat exchangers), amasses hydropneumatic energy (accumulators), and tightens (tightening system). 3.1 Control apparatus The fluid medium must be directed from the generator element to different parts of the actuating system so as to cover a route, sometimes very intricate, before it gets to the motive element and from here on, in some cases, back to the generator one. The distribution devices achieve the reverse of the sense of motor as well as the selective covering of different routes of the installation. The alteration of linear or angular velocity of hydraulic and pneumatic linear or rotary motors is accomplished by a variation of flow. This is carried out either by the generating element in the case when it is with an adjustable flow or with the help of chokes, in a resistive manner, with losses of energy. The force or couple of linear or rotary motors is modified by the variation of pressure. The control and adjustment of pressure in the installation, and other functions as well, are carried out by hydropneumatic elements that are known under the generic name of valves. 98 3.1.1 Distribution apparatus They are meant to direct the fluid from the generator element to different parts of the installation. This can also be done by means of resistive adjustment. Classification of distributors can be done taking into consideration several criteria: a) By the finite or infinite number of working positions there are discrete distributors (fig.3.1 a, b, c) or continuous distributors (fig.3.1d). b) By the shape of the distributing part: - cylindrical with translation motion - drawer (fig.3.1 a) or with rotation motion (fig.3.2 b); - plane with translation motion (fig.3.1 c) or with rotation motion . c) By the actuating manner there are: manual, electrical, hydraulic, pneumatic, electrohydraulic, pneumohydraulic and mechanical distributors; d) By the kind of the control device: direct or piloted (a smaller distributor controls a larger one – fig. 3.1 e); e) By the covering degree: (i.e. the difference between the dimension of the piston p L and that of the channel c L (fig.3.1 a)), there are: critical covering, c p L L = ; positive covering, c p L L > ; negative covering, c p L L < . The distributor working is generally a simple one. At a manual, electric, hydraulic, etc. command, the distributor part moves discretely or continuously opening or covering the passing to different routes. Piloted distributors as in fig.3.1 e work in the following manner: electromagnets 1 command the pilot plunger 2 which moves to the left or the right opening the way for the oil that actuates the main drawer 3, moving it to the right or left, respectively. This way it is achieved the main circuit of oil PA and BR, PB and AR respectively. The flow that passes through the distributor is in direct proportion with the passing section ( ) t x d A ÷ =t , where d is the diameter of the piston, x – the variable opening, and c p L L t ÷ = - the covering degree. The flow Q also depends on the fall of pressure p A : ( ) µ t µ p t x d Q A ÷ = 2 . (3.1-1) 65 , 0 6 , 0 ÷ = µ , flow coefficient. 99 Fig 3.1 We can notice the linear dependence of the flow with respect to movement. In fig.3.2 there are shown the diagrams ( ) x Q for routes AP, AR, BP and BR for distributors with critical, positive or negative covering degree. 100 Fig.3.2 a Fig.3.2 b Fig.3.2 c For a certain opening, the fall of pressure depends on the square flow. 3.1.2 Flow monitoring apparatus In this chapter we shall refer to the resistive adjustment of the flow. This adjustment is achieved with the help of some hydraulic resistances, fixed or variable, that together with other elements make up a device called choke. Chokes can be discrete or continuous according to their finite or infinite positions of work. There are also the so-called digital chokes, which represent a complex design of 3 to 5 identical chokes, which ensure, with respect to a basic flow, several flow variants that are in a geometric series. The continuous adjustment can be simultaneously done with the distributions in continuous distributors (fig.3.1 d). Fig.3.3 This basic element of a choke, the hydraulic resistance can be of several kinds: monitoring edge (fig.3.3 a, b, c), capillary (fig.3.3 d) or slit (fig.3.3 e). Chokes are mounted in low power hydraulic circuits whose generating element is a pump with constant flow. The mounting is done on the supplying circuit, on the return circuit, or parallel to the hydraulic motor (fig.3.4). 101 Fig.3.4 In fig.3.5 is shown a way choke that carries out the adjustment of the flow only if the circuit is from left to right [5]. The adjustable nut 1 moves inside the body 2 altering the choking space 3. The fluid covers the route AB 3 CD. In the reverse sense, the circuit is done unchoked on the route DEA, the reversing valve being open. Fig.3.5 Relation (3.1-1) is valid for chokes also. It can be noticed that the flow Q depends on the fall of pressure, not only on the opening x. To avoid the inevitable dependence of the flow on the pressure fall, the choke is mounted together with the regulator. The assembly choke-regulator can be of restrictive type (fig.3.6 a), where there is the regulator and then the choke, and of by-pass type (fig.3.6 b) in which the main flow passes through the choke ( in order to maintain p A constant, a larger or smaller quantity of fluid passes through the regulator in the tank). The regulator can be also mounted parallel to the choke (fig.3.7). Noting by A the section of the mobile part of the regulator and by a F the force of the spring, we can write the equilibrium of the forces: A p F A p a 0 1 = + . (3.1-2) 102 Fig.3.6 Hence, . 1 0 const A F p p p a = = ÷ = A (3.1-3) Fig. 3.7 Before stabilizing, the regulated flow has a transitory regime. The equation which describes this transitory regime is generally known: ( ) ¿ = ÷ + + ext F t x k dt dx c dt x d M 2 2 , (3.1-4) where M is the mass of the part in motion, c – the coefficient of viscous friction, K – the rigidity of the spring, t – the covering degree and ¿ ext F - the sum of external forces. The working of the choke in a stationary regime is defined by several characteristics: - - the adjustment characteristic in which the flow Q is expressed with respect to the linear stroke h, angular stroke ¢ or the number of divisions of the operating impeller (fig.3.8 a); - the operating force characteristic, where the force to actuate the choke is expressed with respect to required stroke and pressure (fig.3.8 b); - the hydraulic characteristic - ( ) Q p A (fig.3.8 c). 103 Fig.3.8 3.1.3 Pressure monitoring apparatus The mechanical work carried out in hydropneumatic actuating systems is determined, apart from flow, by pressure. The pressure, as a variable of effort, should be monitored in the system. By monitoring pressure we understand a multitude of functions: to protect the installation by limiting the maximum admitted value, to reduce pressure so as to match the particularities of different consumers, to adjust pressure, to maintain it constant irrespective of the requirement of the consumers, to provide a successive setting in motion of hydraulic motors, to create a supplementary load in installation, to forbid the circulation of fluid. The elements of the hydropneumatic actuating system, which carry out the monitoring of pressure, are valves. They are series or parallel mounted, with the hydropneumatic generator or motor, downstream or upstream. According to their function valves can be classified into: - safety valves - reduction valves - regulating valves - pressure servo-regulators - succession valves - anti pressure valves - reversing valves The adjustment of pressure can be done by discharging the overflow in the tank, by choking or by reversing the flow of the pump. Valves can be directly actuated or piloted (by altering the reference element). Essentially, directly operated valves are made up of (fig.3.9): a pressure sensor 1, reference element 2 (usually a spring), and regulating element 3 (variable opening). Fig.3.9 104 A directly operated valve works as follows: pressure acts upon the pressure sensor; at a certain value the force of the spring will be overcome and a part of the flow will leak through the outlet of the valve, bringing about the reduction of pressure in the system. In a stationary regime, the fall of pressure in the system, p A , depends on the variable section of the leaking outlet x S : 2 2 2 x d S Q C p µ = A . (3.1-5) The section can be considered linearly dependent on the opening x: x k S s x = . (3.1-6) Writing the equilibrium of forces on the direction of the axis of valve, we get: x k F F p S R H + = + A 0 0 , (3.1-7) where 0 S is the active surface, 0 F - the pre regulating force of the spring, H F - the hydrodynamic force due to the flow of fluid through the regulating section, a force which is in direct proportion with the fall of pressure. p x k F H H A = . (3.1-8) From (3.1-5), (3.1-6), (3.1-7) and (3.1-8) we get: | | . | \ | ÷ A A ÷ A = 0 0 0 2 S F p p k k S k C p Q H R S d µ . (3.1-9) As it can be noticed, the characteristic of the valve depends on a series of parameters: the constant of the spring R k , the constant, S k , the constant, h k which is influenced by the shape of the element of sesisors a.s.o. Relation (3.1-6) is valid for x relatively small openings. When the valve has the nominal opening, variation of p A with respect to the flow becomes parabolic according to relation (3.1-5), where x S is a constant. In fig.3.10 the valve characteristic is plotted. We can notice the slow increase of p A in the area of small openings, followed by the parabolic variation. Fig. 3.10 105 We shall further describe several of the multitudes of types of directly operated valves. The safety valve (fig.3.11). The aim of safety valves is to protect the installation in the case of pressure increase. The safety valve consists of valve 1, spring 2, adjustment screw of the force in the spring 3 and the valve body 4. Its working is very simple. At normal pressure the force of the spring keeps the valve pushed on the seat. The admitted value of pressure once exceeded, the force of the spring is overcome and a part of the flow of fluid is dissipated to the tank or to some other direction, thus bringing about the decrease of pressure in the installation. Fig.3.11 The reversing valve (fig.3.12). Reversing valve allows the passing of a fluid in one way, the other being blocked. They are also called return valves. There are also relieving reversing valves which allow, at a certain actuation, the fluid to pass in the other way. Fig. 3.12 The reduction valve (fig.3.13). This kind of valve enables to obtain a constant outlet pressure 2 p , even if the inlet pressure 1 p is variable, on condition that 1 p should be higher than 2 p . 106 Fig.3.13 Fig.3.13 shows a pressure regulator “Progresul” type. The spring 1 is thus tarred that its force should overcome the force created by pressure 2 p . Pressure 1 p being higher moves the piston 2, partially or totally covering orifice 3 to the balance of forces. For higher pressures and flows the actuating of the valve cannot be done directly, due to weight and clearance reasons. A smaller valve directly actuated senses the pressure and monitors the working of the actual adjusting element. This assembly makes up the valve with indirect or piloted actuation. Fig.3.14 schematically shows such a valve. When the pressure in the system exceeds the value 0 p for which the spring of the valve with direct actuation 1 S opens, the pressure in the upper part of the element 2 S decreases, thus increasing the possibility that an important part of the flow should drain through section 2 A . We can adjust the pressure in the entire system by adjusting the spring of valve 1 S . Fig.3.14 107 3.2 Auxiliary apparatus Within a hydropneumatic system there are a series of elements, that have a definite part and contribute to conversion of energy, under best circumstances. The auxiliary apparatus do not generate hydraulic or pneumatic energy, do not convert it into mechanical energy and do not actuate or adjust the working of the system. In exchange they enable the link between the different elements, store the fluid, filter and cool it, accumulate energy, ensure the tightening of the system and s.o. There is a large category of hydropneumatic auxiliary elements. In what follows we shall deal with conduits, filters, tanks and accumulators. 3.2.1 Conduits The flowing of hydraulic medium in carried out through a large variety of conduits. Conduits may be rigid or flexible (hoses). Rigid conduits of smaller diameters can be twisted in the shape of spiral, allowing a certain flexibility. The routes, which the fluid crosses, can be inside the body of elements or can be exterior. According to the pressure they are working at there are conduits of low, medium, high and ultra high pressure. There are also hoses of high pressure, that have metallic insertion. A conduit is characterized by the nominal diameter DN, which represents the diameter of the interior section. There is a relationship between the nominal diameter, the nominal pressure n p and the thickness of the conduit. This relationship is settled by standards and is given in tables. There are also relationships between the nominal pressure and the flow velocity, relationships mainly determined by the limitation of losses of pressure. Conduits are attached to the elements of hydropneumatic systems and one to the other, by means of couplings, T. bends, bends, 108 reducing elbows. These are standardized and correspond to the nominal diameter of the respective conduits. The large variety of existent links prevents us from presenting all of them, under the circumstances of our present work. There are several types of couplings. - coupling with re-edged pipe (fig.3.15 a); - coupling with re-edged pipe and jack (fig.3.15 b); - coupling with spherical jack on cone (fig.3.15 c); - coupling with elastic sleeve type ERMETO (fig.3.15 d). Fig. 3.15 When fluids flow through conduits, couplings, taps, etc due to the friction between the fluid particles and walls, losses of pressure occur which can be linear or local: 2 1 2 v D p lin µ ì = , (3.2-1) 2 2 v p loc µ , = , (3.2-2) where ì is the coefficient of linear losses of load, , - the coefficient of local losses of load, l – the length of the conduit, D – the diameter of the conduit, µ - the density of the fluid, v – the velocity of the fluid. 109 3.2.2 Filters Filtration of hydraulic or pneumatic medium has an prominent importance for the good working of a system. Filters reduce the degree of contamination of the fluid with impurities under a certain limit, therefore, apart from ensuring a smooth working of the system, they avoid a premature wear of its elements. Impurities which come form the exterior or interior (from the fluid itself or from the components of the hydropneumatic system) can be of mechanical or chemical nature. The degree of contamination is given by the value, shape and the number of impurifying particles. It is influenced by the velocity and pressure of the fluid. Fig.3.16 In fig. 3.16 is shown a filter which consists of: 1 – inlet, 2 – lid, 3 – filtering element, 4 – spring, 5 – ring 0, 6 – outlet, 7 – by – pass valve [5]. In the case when the filter is stuck, the pressure created in the system opens the by – pass valve, thus being carried out a direct circuit, by detouring the filtering element. There are filters which signal these situations, so that we may change or clean the filtering element. Some filters have sockets for manometers, whose indications allow us to establish the silting degree of the filtering element. Many times we make use of double filters, in parallel, thus we are able to change the filtering element without stopping the installation. Filtering elements can be of surface when they retain the particles on their surface or of depth when the particles are retained in the whole mass of the element. Surface filters can be made of paper, cardboard, metallic texture, that are able to regain their filtering capacity by blowing or by reversing the circulation sense of the fluid. Depth filters are made up of successive layers, each one can be made of paper impregnated with resins, of balls or other sintered metallic elements, folded fabric, etc. 110 The fineness of filtration can be rendered in absolute or relative units. As absolute unity we have the diameter of the largest spherical particle that can pass through the filtering element. From this point of view, filtration can be: - rough m µ 200 ; - medium m µ 60 50÷ ; - fine m µ 15 10 ÷ ; - micronic m µ 5 2÷ ; - sub-micronic m µ 1 . The relative fineness reports the number of particles of a certain diameter which pass through the filter to the total number of particles. Filters can be mounted on suction, discharge or return routes. If filters are mounted on suction, there is the advantage that the pump is protected and it doesn’t work under pressure. In exchange there are supplementary resistances on suction circuit, that have an effect on the decreasing the suction load of the pump and on the conditions which lead to cavitation. If filters are mounted on discharge, the so-called pressure filters, they allow to protect the elements of installation, which require a more accurate mechanical finishing (distributors, valves, etc). They have the disadvantage that they must be designed in such a manner as to resist pressures and the pump isn’t directly protected. Return filters, usually mounted on tanks, are a widely spread solution, as they work under relatively low pressure. Of course that within an installation there can be mounted several filtering elements, in different areas. Apart from protecting the installation and cleaning the fluid medium, filters can contribute to improving the pulsation regime in a hydrostatic system. 3.2.3 Tanks The storage of hydraulic medium can be done at atmospheric pressure, in tanks, at low pressure, in filling tanks and at high pressure, in accumulators. In a hydrostatic system the part of tanks is manifold: it stores the fluid medium, allows the impurities to decant, creates conditions to cool the fluid, constitutes a support for different elements of the installation (often pumps are mounted directly on tanks). 111 In fig.3.17 a hydraulic tank is schematically shown. We can notice the suction conduit 1 and the discharge conduit 2. At the end of the suction conduit there is filter 3. The two chambers are separated by a spillway wall 4. The filling is achieved by means of element 5 equipped with a filter. Taps 6 and 7 are meant to eliminate the sediments. Fig.3.17 As we have already stated, tanks allow cooling by mixing the liquid in the installation or by heat radiation. Taking into account the thermic sum, we can establish the required volume of the tank by means of relation [8]: | | 3 3 3 0 10 m T T Q V ÷ | | . | \ | ÷ = , (3.2-3) where T – the established working temperature; 0 T - the temperature of the exterior medium; ( )P Q q ÷ = 1 860 , (3.2-4) the quantity of heat emitted in the system; P – the power of the pump expressed in kW; q - the total efficiency. The volume of the tank can be also expressed by the formula: ( ) ] [ 5 ..... 3 max l Q V P = , (3.2-5) where max P Q represents the maximum flow of pump in l / min. 112 3.2.4 Accumulators The role of the accumulator in a hydraulic installation is multiple; it is used as a: - tank of liquid under pressure; - supplier of high flows in certain moments; - compensator for losses of liquid and volumetric losses due to dilation; - hydraulic generator for short periods; - recoverer of braking energy; - pulsations damper; - hydraulic chock absorber. The energy storage can be achieved hydromechanically (accumulators with weights–fig.3.18 a, with spring – fig.3.18 b or hydropneumatically. For hydropneumatic accumulators the compressed gas stores energy. In the discharge phase, the gas expands exhausting the liquid. Hydropneumatic accumulators can be with gas without separation (fig.3.18 c) or with separation of piston type (fig.3.18 d), bladder (fig.3.18 c) or membrane (fig.3.18 f). Fig. 3.18 For accumulators calculus we consider the behaviour of the gas as polytropic ( . const pV n = ). Due to the fact that the accumulator filling is done within a short time, the polytropic exponent doesn’t exceed the value 1.1, so that the hypothesis of isothermic transformation ( . const pV = ) isn’t so far from being true. In fig.3.19 three phases of the accumulator working are shown[8]. 113 Fig.3.19 a) Before the entrance of the liquid, the gas takes the entire volume V (we consider the volume of the piston as negligible) at the pressure p. b) The liquid discharged by the pump has taken the maximum volume M V at the pressure M p . The gas will be at the same pressure M p taking the entire volume M V V ÷ . c) In the discharging phase there must remain a minimum volume of liquid m V at pressure m p . Knowing V, p and M p , we determine M V from the law of isothermic transformation: M M p p V V V = ÷ . (3.2-6) Hence: | | . | \ | ÷ = M M p p V V 1 . (3.2-7) Writing the law of isothermic transformation, we shall get: m m p p V V V = ÷ , (3.2-8) or else: 114 | | . | \ | ÷ = m m p p V V 1 . (3.2-9) The relation will give the required volume of the accumulators: | | . | \ | ÷ = ÷ = M m m M u p p p p V V V V . (3.2-10) 115 4. MEASURING APPARATUS From a quantitative point of view, the assessment of a physical magnitude can be done either by calculus or by measurement. The measurement of a physical magnitude consists of its comparison with another magnitude of the same nature which is conventionally thought of as a measure unity. As a result of the measurement, we get an abstract number called numerical value, which shows us how many times the measure unity is included in the respective physical magnitude. Consequently, a symbolic product between the numerical value b and its measure unity | can always express a physical magnitude B,: | b B = . (4.1-1) Generally, the results of the measurement are affected by errors. Thus, if we denote by b the numerical value of a physical magnitude, got by measurement, i (i= 1, 2, . . . . n) and by 0 b the real numerical value of the same physical magnitude, we define: - the absolute error: ( ) 0 0 B B b b B i i i ÷ = ÷ = A | ; (4.1-2) - the relative error: 0 0 0 0 b b b B B B i i i ÷ = ÷ = c ; (4.1-3) which can also be expressed as a percentage, in the form: | | % 100 0 0 B B B i i ÷ = c . (4.1-4) The absolute error taken with the reverse sign represents the correction of the measurement i K , namely: i i i B B B K ÷ = A ÷ = 0 , (4.1-5) hence the real value of the respective physical magnitude in the form: 116 i i K B B + = 0 . (4.1-6) If the measuring proceeding has been repeated for n times, we can calculate a mean value: ¿ = = n i i B n B 1 1 , (4.1-7) and a square mean of the errors: ( ) ¿ = = n i i B n 1 2 1 o . (4.1-8) The measurement error can be considered as a sum of random errors, systematical errors and, sometimes, even mistakes (glaring errors). To identify the results that have been affected by suchlike glaring errors, we establish a tolerance field: | | ( ) ( ) | | o o Z B Z B B B D + ÷ = = , , max min , (4.1-9) where Z is a coefficient which depends on the numbers of measurements, n (thus, for n = 6, Z = 1.73 ; for n = 10, Z = 1.96 ; for n = 15, Z = 2.13). The results of the measurements i B which don’t belong to the field D, defined by the relation (4.1-9), are considered to be glaring errors and they are eliminated. 4.1 Apparatus which determine the physical properties of fluids The physical properties of fluids are characterized by a series of physical magnitudes such as: density, viscosity, temperature, coefficient of isothermic compressibility, coefficient of isobar dilatation, specific heat, superficial tension, etc. We shall further analyze the measuring proceedings of density and viscosity for fluids. 4.1.2 Density measurement For homogeneous bodies, density represents the mass in relation to the unity of volume ( V m/ = µ ). 117 Among the apparatus used to measure density there are: the areometer, the hydrostatic scales, and the U tube. a) The areometer is a floater of given weight G, which in a liquid of known density 0 µ , plunges to the level A (fig.4.1), and in the liquid of searched density, µ , plunges to the level B. By using the equation of flotability, we get: 0 0 V g V g G µ µ = = , (4.1-10) where from: h d V V V V 2 0 0 0 0 0 4 4 t µ µ µ + = = (4.1-11) where h represents the difference in level AB, d is the diameter of the cylindrical part of the areometer, and 0 V is determined from the relation: g G V 0 0 µ = . (4.1-12) Fig.4.1 b) The hydrostatic scales enable us to determine the density of a liquid by measuring the Archimedean force F that arises by submerging a solid body, of a known V volume. Under these circumstances, the relation will give the density of the liquid. V g F = µ . (4.1-13) c) The U – tube (fig.4.2) enables us to determine the density µ of a liquid by measuring the heights 0 h and h corresponding to the two columns of liquid, of which one has the known density 0 µ . By using the fundamental equation of hydrostatics, we get: 118 0 0 h g h g µ µ = , therefore: h h 0 0 µ µ = . (4.1-14) If the liquids are non-miscible we can use two communicating vessels, h and h 0 being in this case the heights of the two columns of liquid above the separation surface. Fig.4.2 4.1.3 Viscosity measurement Apparatus which are designed to measure viscosity, also called viscometers, can be classified according to following criteria: a) by the design principle we distinguish: - viscometers with band, which are based on the laminar motion between two plane surfaces; - viscometers with a capillary, which are based on the laminar motion of the fluid in a capillary; - viscometers with a plunger or in a forced motion, which are based on the laminary flow around a rigid body in a translation motion; - viscometers with a rotating body, which are based on the laminar motion around rotating bodies; - viscometers with an oscillatory body which are based on the non-permanent motion around an oscillatory body; - viscometers with flow orifices, usually designed to measure some conventional viscosity, proportional to the time of flow of the liquid through the orifices of the viscometer under certain standard conditions; b) by the kind of the determined viscosity, we distinguished: 119 - viscometers meant to determine the dynamic viscosity; - viscometers meant to determine the kinetic viscosity; - viscometers to determine the conventional viscosity. c) By their degree of accuracy viscometers can be for laboratories (with a high degree of accuracy) and technical. In the end, we should also underline that in industrial installations there are used devices which are meant both for viscosity measurement and display and for its adjustment between certain limits. Such devices are called viscosity controllers or viscositates. In what follows we shall analyze the main types of viscometers, which are used in actual practice. a) Viscometers with capillary These devices are meant to measure the dynamic viscosity q . The working principle of these viscometers consists in determining the pressure fall p A that arises at the flow of liquid, in laminary and permanent regime, through the calibrated tube 1 (fig.4.3 a). Denoting by d and l the interior diameter of the tube and its length, respectively, and according to Hagen-Poisseuille’s law, the dynamic viscosity of the liquid can be determined by means of the relation: p l Q d A = 128 4 t q , (4.1-15) where the flow of liquid Q is maintained constant, and the fall of pressure p A can be determined by measuring the difference in level h between the free surfaces of the liquid from the piezometric tubes 2: h p ¸ = A . 120 Fig.4.3 Therefore relation (4.1-15) becomes: l Q h d 128 4 t ¸ q = . (4.1-16) We should stress that the constant maintaining of the flow is essential and this is achieved either by using some flow regulators or by the use of volumetric pumps with a constant flow. A variant of this type of viscometer is Ostwald viscometer (fig.4.3 b) where the laminary flow of the liquid takes place gravitationally through a vertical capillary 1 having the dimensions d = 0.4 – 0.5 mm, and l = 10 cm. The tank 7 is filled with a liquid of unknown viscosity, by means of a calibrated dropper. Then the liquid is sucked into the tank 2, situated in the tube 3, until it reaches above the reference point 4; afterwards we can measure the time t in which the liquid flows gravitationally through the capillary, between reference points 4 and 5. The experiment is then repeated, this time using another liquid of known dynamic viscosity 0 q and which has the flow time 0 t . The unknown dynamic viscosity is determined by means of: 121 0 0 0 µ µ q q t t = , (4.1-17) where 0 µ µ and are the densities of the two liquids. b) Rotary body viscometers This type of viscometers measure the viscosity of the liquid on the basis of the moments of friction forces with act upon a solid body in rotation in the mass of the fluid. In fig.4.4 such a type of viscometer is schematically drawn; it consists of: 1 – leading shaft, 2 – leading disk, 3 – carcass, 4 – driven disk, 5 – tightening element, 6 – driven shaft. Due to friction forces, between the leading disk 2 and the driven disk 4, there arises a moment which is transmitted by the driven shaft 6. This moment is proportional to the dynamic viscosity of the fluid provided the rotation of the leading shaft 1 should be maintained constant. Fig.4.4 c) Plunger viscometers The working principle of this type of viscometers is based on measuring the resistance the fluid withstands against the advancement of a body in a translation motion. In the case of a spherical ball of radius r, which moves uniformly with velocity 0 v , the resistance at advancement is given by Stokes’ formula: 122 0 6 v r F q t = . (4.1-18) The most common viscometer with a plunger is Höppler viscometer, schematically shown in fig.4.5; it consists of:1 – tube of a special glass which is filled with the liquid to be analyzed, 2 – frame with wedging screws, 3 – exterior tube through which there flows a liquid that adjusts the temperature of the liquid under test, 4 – fixing screw, 5 – connecting pieces, 6 – thermometer, 7 – falling ball (plunger). The tube 1 is slanting with respect to the vertical at an angle of ' 0 30 10 and can revolve around the axis O. In order to determine viscosity we measure the time t, in which ball 7 falls between the two reference points marked on the glass tube 1; then we can use the relation: ( )t k b µ µ q ÷ = , (4.1-19) where k is the constant of the device, and b µ and µ are the densities of the ball and of the tested liquid, respectively. Fig.4.5 Fig.4.6 d) Conventional viscometers In actual practice the conventional viscosity of the fluid is often used; this is the magnitude determined by measuring the flow time of a certain volume of liquid through a special device, under conventionally chosen conditions. There are several conventional viscosities (Engler, Saybolt, Redwood, etc) which differ both in the measuring conditions and in the unit measures. One of the most used conventional viscometer is the Engler viscometer, schematically shown in fig.4.6; it consists of: 1 – a brass pot filled with the liquid whose density is to be measured, 2 – lid, 3 – orifice obturated with the rod 8, 4 – metallic bath 123 with an electric resistance, 5 – frame with wedge screws, 6 – stirrer, 7 – standardized balloon at 3 200cm , 9 – thermometres. Engler conventional viscosity, expressed in Engler degrees ( ) E 0 , is determined as being the ratio between the flow time of 3 200cm of the tested liquid and the flow time of 3 200cm distilled water at C 0 20 passing through the orifice 3. e) Viscosity controllers These are more complex devices, which apart from measuring the density have also the role to regulate, within certain bounds, the viscosity of the working fluid. Fig.4.7 For example, such regulators are used on board ships on the fuel oil supply – line for the ship’s main engine. The installations, which regulate the viscosity of liquids, have the classical structure of automatic systems. Their basic element is the viscosity transducer 1 TV which carries out the conversion of viscosity into a mechanical magnitude (displacement, moment, force, difference of pressure) which, in its turn, is used as a reaction signal, r x , in the comparison element E.C. (fig.4.7). The inlet magnitude 0 u is given by the prescribed magnitude of viscosity, which, before entering E.C. is converted by the transducer 2 TV into a magnitude i x of the same nature with r x . The error signal a x is amplified by the amplifying element EA, thus obtaining the control signal, which actuates the fulfillment element EE. To alter the viscosity of the fluid we heat it at a temperature corresponding to the prescribed viscosity, by using heaters (steam or electrical) denoted in the scheme by IT (technological installation). The control magnitude acts by means of the fulfillment element EE upon the technological installation, which regulates the viscosity by altering the steam, flow, the number of electrical resistances, etc. 4.2 Measuring instruments for the level of liquids The measurement of the level of liquids, which are in tanks, basins, etc, is done by means of special instruments called level indicators and they can be simple indicators or may have other functions such as recording or adjustment. In what follows the simplest indicators will be described. 124 a) The glass level, consists of a glass (or other transparent material) tube 1, mounted through two hook-ups 2, with or without a tap, in which the level of the liquid in communicating vessels is observed (fig.4.8 a). In the case when the pressure in the receptacle is between 1 – 12 bar, the tube, made of reinforced glass, is endowed with metallic setting. For higher pressures we use: - boards of special glass, directly mounted on the wall of the receptacle at different heights; - non-transparent metallic tube 1 (fig.4.8 b) ,in which there is a floater 2 that, in its turn, magnetically moves the floater 3 into the exterior transparent tube 4. Fig.4.8 b) Level indicator with floater, has as a main part a floater 1 (fig.4.9 a), situated inside the receptacle connected with the indicating element 2, situated in the exterior tube 3. For higher pressures and temperatures ( ) 0 400 25 s s t and bar p , the rod of the floater magnetically drives a second floater, which is in an exterior transparent tube. Fig.4.9 125 c) Level indicator by gas bubbling, has as basic constructive element a tube through which a gas (usually air or nitrogen) passes at constant and low velocity; its relative pressure p is indicated by a sensitive manometer (fig.4.9 b). As a consequence, on the basis of the fundamental equation of hydrostatics, we shall get the level of the liquid in the tank: ¸ p h = , (4.2-1) where g µ ¸ = is the specific weight of the liquid in the tank. d) Level indicator with capacitive transducer has as sensitive element a circular copper conductor, which is inserted in the mass of the liquid. Depending on the depth of the liquid in the receptacle,the transducer capacity varies, being measured by means of a measuring bridge. 4.3 Pressure measuring instruments The instruments designed to measure pressures may be classified according to two basic criteria, namely: a) by the kind of the measured pressure and b) by their working principle. a) According to the way in which the zero point is chosen, pressures may be absolute, when they are reported to the total vacuum (absolute zero), and relative when reported to the normal atmospheric pressure (relative zero). Between the two pressures there is always a linking relationship: rel abs p p p + = 0 , (4.3-1) where 0 p is the normal atmospheric pressure ( ) Pa k p 325 . 101 0 = . Unlike the absolute pressures which are always positive, relative pressures may be positive or negative, as the corresponding absolute pressure is higher or lower than the atmospheric pressure. Subsequently, according to the kind of measured pressure we distinguish among: - barometers that measure the absolute pressure; - manometers that measure positive relative pressures; a reason for which these pressures are also called manometric pressures; 126 - vacuumeters that measure negative relative pressures, which, in absolute value are called vacuumetric pressures ( ) 0 < ÷ = rel rel V p with p p ; - manovacuumeters that can measure both manometric and vacuummetric pressures. b) According to their working principle we distinguish among: - devices with liquid for which the measured pressure is compensated by the pressure exerted by a column of liquid; - devices with elastic element for which the measured pressure brings about a distortion within elastic limits of an element (curved tube, membrane, bladder); - devices with transducers which, in their turn, can be thermic, pneumatic and electric. 4.3.1 Devices with liquids Devices with liquid are simple devices whose working is based on the balance between the measured pressure and the static pressure of a column of liquid; they are used to measure pressures or differences in pressure of 1 bar at the most. As liquids we can use mercury, ethylic alcohol, water and carbon tetrachloride. When choosing a liquid we have to take into consideration the following basic conditions, namely: its density must be higher than that of the fluid whose pressure is measured; the two fluids must be non-miscible and mustn’t interact chemically. In what follows, we shall analyze a few constructive variants of this type of devices meant for pressure measuring. a) The device with an open tube (piezometer) which consists of a transparent tube coupled at one end to the receptacle whose pressure is being measured, the other end being open, in contact with the atmosphere (fig.4.10). The device can work both as a manometer and as a vacuumeter. In the case of the manometer, the manometric pressure of the liquid in the tank in point A ( A p ), is obtained by writing the fundamental equation of hydrostatics in a horizontal plane N-N (fig.4.10 a): H h p A 0 ¸ ¸ = + , (4.3-2) or else h H p A ¸ ¸ ÷ = 0 , (4.3-3) 127 where ¸ is the specific weight of the liquid in the tank, and 0 ¸ is the specific weight of the liquid in the tube. When it works as a vacuummeter (fig.4.10 b) the relative pressure in point A is determined by means of relation: H h p A 0 ¸ ¸ ÷ ÷ = , (4.3-4) and the vacuumetric pressure in A will be: H h p p A VA 0 ¸ ¸ + = = . (4.3-5) If in the tank there is a gas whose specific weight is negligible in relation to the specific weight of the liquid in the tube ( 0 ¸ ¸ < ), relation (4.3-3) and (4.3-4) become: H p A 0 ¸ = , (4.3-6) and H p p VA A 0 ¸ ÷ = ÷ = . (4.3-7) respectively. Fig.4.10 b) The differential manometer is used to measure the difference in pressure between two receptacles that contain liquids, which may be different. It is in fact a U-glass tube, overturned, which is connected to the two tanks (fig.4.11) Fig.4.11 by means of taps 1 R and 2 R . With the help of the tap R we adjust the pressure x p of the air situated above the columns of liquid, of height 1 h and 2 h in the two branches of 128 the tube. On the basis of the fundamental equation of hydrostatics, we can write the relation: 2 2 2 1 1 1 h p h p p x ¸ ¸ ÷ = ÷ = , (5.3-8) hence, we get: 2 2 1 1 2 1 h h p p p ¸ ¸ ÷ = ÷ = , (4.3-9) where 1 ¸ and 2 ¸ are the weights of the two liquids. c) Micromanometer with slanting tube (fig.4.12) is used to measure low pressures (up to Pa 1 . 0 ) or low differences of pressure. It consists of tank 1 and a transparent tube, slanting with respect to the horizontal at an angle o , which can be modified by means of device 3. Initially, the free level of the liquid in the tank has the position 1 N . After coupling, due to pressure p, the free level will lower in the tank with h A and, at the same time, it will go up in the transparent tube with the distance l, measured on the direction of the tube. Fig.4.12 The relation will determine pressure p: ( )¸ o sin l h p + A = (4.3-10) or else ,taking into consideration the equality of the volumes: l d h D 4 4 2 2 t t = A , we get: | | . | \ | + = o ¸ sin 2 2 D d l p (4.3-11) 129 where ¸ is the specific weight of the liquid in the device. Usually, the diameter of the tube is much smaller than the diameter of the tank ( D d s ) and, subsequently the ratio D d / id negligible, and relation (4.3-11) takes the form: o ¸ sin l p = . (4.3-12) d) Micromanometers with liquid of a special design are used for gases, to measure small differences of pressure. The variation of pressure or the difference in pressure alters the position of the nonmiscible liquid bubble 3, which is in the transparent tube 2 (fig.5.13). The number of divisions with which the micrometric screw 4 must be rotated to return the bubble to its initial position, shows, on the basis of a standard, the variation of pressure. Fig.4.13 In fig.4.13 there are shown two constructive variants of this type of device: 1 – tank with liquid, 2 – glass tube, 3 – non-miscible liquid bubble, 4 – micrometric screw, 5 – reference. 4.3.2 Devices with elastic elements Manometers with elastic elements, also called mechanical manometers, are highly reliable devices, used to measure pressures within very large fields. Their working principle consist in deforming some elastic elements 1 (bladder, membrane or Bourdon tube). This deformation, proportional to the measured pressure, i p (fig.4.14) is amplified and sent further by means of a lever system 2 to the indicator needle 3 which will indicate the pressure on the scale 4. The return to zero position is done by means of spring 5. The recommended pressure fields are: - for manometers with bladder (fig.4.14 a) between 0.5 Pa and 0.5 MPa; - for manometers with membrane (fig.4.14 b): up to 3MPa; - for manometers with Bourdon tube (fig.4.14 c) up to 300 MPa. 130 Fig.4.14 4.3.3 Devices with transducers The devices consists of a sensitive element, upon which pressure acts, and a transducer that converts the signal supplied by the sensitive element into another physical magnitude. This is sent to a measuring instrument, which has been beforehand standardized. Generally, transducers can be thermic, pneumatic and electrical. Devices with electrical transducers can be used to measure high pressures or pressures that vary rapidly under high temperature conditions. Devices with pneumatic transducers are generally used to measure pressures in explosive mediums, as they also enable to transmit the information at distance. 4.4 Velocity measuring instruments The velocity of a fluid can be determined either as a local value, or as a mean value on the flow section. To determine the local velocities we use instruments called anemometers, which have different working principles. The most used are Pitot-Prandtl tube, mechanical anemometers, thermic anemometers, and laser anemometers. To determine the mean velocity on the flow section we use the relation: } = = A m A Q dt v A v 1 , (4.4-1) 131 that requires either to know the velocity distribution in the points of the flow section – (which can be determined by measuring the local velocity in a network of points of the flow section)- or to know the flow which can be measured with the help of specific methods that are to be further described. In what follows we shall present the main types of anemometers. 4.4.1 Pitot-Prandtl tube Pitot-Prandtl tube basically converts the kinetic energy of the fluid particle into potential pressure energy, in points where its velocity cancels (cessation points). Fig. 4.15 Pitot-Prandtl tube consists of two L – shape tubes (fig.4.15). The interior tube 1 is connected with the fluid by means of orifice O whose surface in normally placed on the flow direction, and the space between the two tubes is connected with the fluid through the side slits F. At the other end, the Pitot-Prandtl tube is connected at a U- shaped differential manometer 3, in which there is a liquid of specific weight 1 ¸ . According to Bernoulli’s equation, in the point of front orifice O, the fluid velocity becomes nil and the total pressure ( ) t p will become manifest and is sent through the interior tube 2 to the right side of the differential manometer. At the side slits F, the fluid velocity being v, the static pressure ( ) s p will become manifest, which will be sent to the left side of the manometer. The difference between the two pressures represents the dynamic pressure ( ) d p : s t d p p v p ÷ = = 2 2 µ , (4.4-2) hence the fluid velocity v: 132 ( ) s t p p v ÷ = µ 2 , (4.4-3) where µ is the density of the fluid which is being measured. On the other hand, applying the fundamental equation of hydrostatics for the plane N-N (fig.4.15), we can write: 1 1 2 H h p H p s t ¸ ¸ ¸ + + = + , (4.4-4) hence, taking into account that: 1 2 H H h H + A + = , (4.4-5) we get: ( ) ( ) H h H h h p p s t A ÷ ÷ = A + ÷ = ÷ ¸ ¸ ¸ ¸ ¸ 1 1 , (4.4-6) with which the fluid velocity becomes: ( ) | | ( ) | | H h g H h v A ÷ ÷ = A ÷ ÷ = µ µ µ µ ¸ ¸ ¸ µ 1 1 2 2 . (4.4-7) For H A small enough ( ) 0 ~ AH , we may write: µ µ µ ÷ = 1 2gh v , (4.4-8) which in the case of gases 1 µ µ << , becomes: h k gh v = = µ µ 1 2 . (4.4-9) We should emphasize that in order to decrease the measuring error it is necessary that the Pitot-Prandtl tube should disturb the flow as little as possible. As a consequence, we must observe the following basic rules: - the tube is oriented with its axis parallel to the flow direction so as the surface of orifice O be perpendicular on the fluid velocity; - between the diameter of conduit D and diameter d of Pitot-Prandtl tube should exist the ratio 50 / > d D ; - the tube mounting must be done at a distance of at least 50 D away from any local load loss (the end of the conduit, bends, taps, etc). 133 4.4.2 Mechanical anemometers Their working principle is based on the dependence between the fluid velocity and the rotation velocity of an impeller placed in a fluid. This type of anemometers is calibrated and they require periodical checking, as the friction that arises in the bearing impeller may vary in time thus bringing about measurement errors. The device can be coupled with a counter, thus enabling us to determine the mean flow within a certain time interval. 4.4.3 Thermic anemometers Their working is based on the existent dependence between the velocity of the fluid stream and the intensity of heat exchange between a conductor of very small dimensions and the fluid in motion. We may distinguish among: a) anemometers with a constant electric current, where the temperature of the conductor varies with respect to velocity, thus modifying its resistance which is then measured in a bridge assembly; b) anemometers with a constant temperature, where the temperature is constant (and therefore its electric resistance as well) by varying the intensity with respect to the stream of fluid. The anemometer with a hot wire enables us to measure velocities or velocity variations. The plunger (fig.4.16 a) contains the sensitive element which is a cylindrical tungsten, platinum or platinum alloy with 10 – 20% iridium wire 1. For higher velocities it is desirable to use a m µ 4 2÷ diameter tungsten wire, which has the advantage of a higher mechanical resistance, but also the disadvantage of a very difficult welding on the frames 2. The platinum wire can be of smaller dimensions ( ) m µ 1 and is covered in a sliver coat of m µ 20 10÷ . We have denoted the body of the plunger by 3. There are various types of plungers, which differ in the way the sensitive wire is mounted on the frames. We can distinguish among: plungers with a normal wire (perpendicularly placed on the direction of mean flow), plunger with slanting wire (the slanting angle with respect to the direction of mean flow usually being 0 45 = o ), x – shaped plunger (it consists of two wires placed in perpendicular planes). 134 By means of electric connection 4, the plunger is interspersed in a Wheatstone bridge 5, where the fourth resistance is placed (fig.4.16 b). The other three resistances are constant and are made of materials that have a small temperature coefficient. The sensitive wire inserted in the stream of fluid is electrically heated, which maintains at a constant value either the intensity of the current, or the wire temperature. Fig.4.16 For the latter case, the bridge is under permanent equilibrium, by an automatic regulation of variable resistance, which is series mounted with the bridge. The intensity of the current is thus varied so as the temperature of the wire should be maintained constant; this way it becomes a measure of fluid velocity which can be determined by means of galvanometer 6, standardized in velocity units. The hot wire may have a series of disadvantages, such as: it can easily be broken by mechanical hits; it collects dust and oil particles which can alter the standard curve; it is not suitable for liquids, especially those of high electrical conductivity. These disadvantages are partially eliminated by the plunger with a hot film, which actually is a small nickel surface (of apx. 0.3/1 mm) with an extremely small thickness. In fact this film represents a layer on a glass or isolating quartz sublayer. 4.4.4 Optical measuring instruments Optical measuring instruments used to determine velocity have the great advantage that they do not disturb the flow by the presence in the mass of fluid in motion of a solid body which represents the sensitive element. But the essential problem of optical measurement systems is to obtain a reasonable resolution space and, at the same time, to supply an electrical outlet system which can easily be amplified. Among the optical procedures the most used is the laser anemometer, based on Doppler effect. Essentially, its working principle is the following: the laser beam, which crosses the liquid in motion, is spread in the solid particles, which are in discharge in the fluid (dust, smoke). The frequency of the spread light depends on the particle velocity transported in the liquid; the altering of the frequency being correlated with their velocity constitutes in fact the Doppler effect. 135 4.5 Flow measurement Flow is the most important parameter for installations that use fluids as working agents. The main determination methods for flows can be grouped into: volumetric methods; methods based on throttling the stream section (aperture stops, nozzles, Venturi nipples, etc); methods based on the velocity field exploring in the flow section, by using some measuring apparatus for the local velocity (Pitot-Prandlt tube, anemometer with hot wire and film, etc); electromagnetic methods; dilution methods; ultrasound methods, etc. Table 4.1 shows the usage field of different types of flow meters according to STAS 9280-73. Table 4.1 Type of flow meter Usage field Observations Measuring Field Informative accuracy Aperture stop D Re 10 . . . . . 10 5 7 3 1 . . . . . 5 , 0 ± u vD D = Re ; where D – the interior diameter of the conduit v – the mean velocity in the flow section u - kinematic viscosity Bearing D Re 10 ... . . 10 2 7 4 1 ± Venturi tube D Re 10 . . . . . 10 2 7 4 1 . . . . . 5 , 0 ± Pitot- Prandlt tube s m v i / 180 < 5 , 2 . . . . . 1 ± i v - local velocity Anemomet er with hot wire s m v i / 50 . . . . . 3 , 0 = 5 , 0 . . . . . 1 , 0 ± “ Rotameter h m / 30 ... . . 3 3 5 , 2 ± 136 Ultrasound flow-meter s m / 3 , 0 . . . . . 002 , 0 3 2 .. .. . 5 , 0 ± Counter with membrane h m / 600 . .... 05 , 0 3 3 ± Rotary counter h m / 3000 . . . .. 70 3 1 ± Flow- meter with ionization s m / 01 . . . . . 001 , 0 3 5 . . . . . 2 ± 4.5.1 Volumetric methods The method of the standardized tank consists in measuring the volume of liquid V that flows within a measured time interval, t, in to a standardized tank. The relation gives the flow: t V Q = . (4.5.1) Within this method, there are also used volumetric counters, which, essentially, consist of two or more chambers of a known volume, successively filled with liquid. The number of fillings of these chambers determines the total volume. 4.5.2 Methods based on throttling the stream section of the fluid Mainly, these methods are based on the fact that by throttling the flow section, there arises a difference between the pressure upstream and downstream from throttling which depends on the stream velocity and, implicitly, on flow. In fact, on the basis of this principle, any local resistance may be used for measuring the flow. To determine the flow in conduits, there are frequently used aperture stops, nozzles and Venturi tubes, and inside channels there are used spillways and Venturi channel. a) Venturi tube (venturimeter) consists of a conical converging tube 1, followed by a conical diverging tube 2 and is equipped with pressure sockets at which there are coupled the piezometric tubes 3 or another differential manometer (fig.4.17 a). The tube is interspersed on conduit 4 whose flow is to be determined. 137 By writing Bernoulli’s equation for an ideal fluid between points 1 and 2, situated in the inlet section of area 1 A and, respectively, in the minimum section, of area 2 A , we shall get: ¸ ¸ 2 2 2 1 2 1 2 2 p g v p g v + = + , (4.5-2) since the quotes of the two points are equal, and 1 v and 2 v are the velocities of the motion fluid for the considered points. Taking into account the equation of continuity: 2 2 1 1 v A v A Q = = , (5.5-3) we get: ¸ 2 1 2 1 2 2 2 1 2 p p A A g v ÷ = | | . | \ | ÷ , (4.5-4) or else: ¸ 2 1 2 1 2 2 2 2 1 1 p p g A A v ÷ ÷ = . (4.5-5) Hence the theoretical flow in the form: h g m A p p g m A v A Q 2 1 2 1 2 2 2 1 2 2 2 2 ÷ = ÷ ÷ = = ¸ , (4.5-6) where: 1 2 A A m = , (4.5-7) is called throttling coefficient, and h is the difference between the quotes of the free levels of the liquid in the piezometric tubes 3. 138 In the case of a real fluid the flow is determined by means of: h g A p p g A Q 2 2 2 2 1 2 µ ¸ µ = ÷ = , (4.5-8) where the flow coefficient µ takes into consideration both the jet contraction m and the load losses which may arise. In the specialized studies for standardized venturimeters, there are given diagrams from which we can get the values for the flow coefficient µ with respect to Reynolds number | . | \ | = u vD D Re , for different values of the contraction coefficient m [1, 10 ]. Fig. 4.17 b) The aperture stop (fig.4.17 b) is a disk with a central orifice, of a smaller diameter than that of the conduit ( ) D d < , which is coaxial, mounted on the conduit rout, between two flanges. The fall of pressure: h p p p = ÷ = A 1 2 , (4.5-9) is measured by means of two piezometric tubes or of a differential manometer. The flow is also computed by means of relation (4.5-8), the flow coefficients being determined for standardized aperture stops with the help of some diagrams with respect to the throttling coefficient m and number | | 10 , 1 Re D . c) The nipple (fig.4.17 c) replaces the aperture stop. Having a profiled inlet, the losses of load are reduced and, consequently, the flow coefficient µ of the nipple will be greater than that of the aperture stop. The flow is computed with the help of the same relation (4.5-8), and the flow coefficient µ is determined from the diagram with respect to m and D Re . 139 We should underline that, irrespective of the apparatus type (Venturi tube, nipple or aperture stop), the mounting conditions have a great influence on the accuracy of measurement, bringing about errors up to 50% in the case of a faulty centering. Also, the throttling devices should be mounted on a straight portion of conduit, bearing in mind that both upstream and downstream the apparatus there shouldn’t exist any local losses of load on a distance of at least ( D 25 1= ). d) The spillway enables us to determine the flow of the streams with a free surface. The spillways, fig.4.18 is a vertical wall with a sharp edge which blocks the stream of liquid and over which water can spill. It can extend over the entire breadth of the channel (for which case it is called without lateral contraction), or we can place a spillway only on a portion of the breadth of the channel (the lode of fluid having in this case a lateral contraction). Fig.4.18 For a rectangular spillway the flow is expressed with respect to the height h of the liquid above the crest of the spillway. This height is measured at a distance max 3h L > upstream the crest of spillway, by means of a limnograph, floater or by means of other methods. For a rectangular spillway, the flow will be computed with the help of the relation: h g h b Q 2 3 2 µ = , (4.5-10) where b is the breadth of the spillway and µ is the flow coefficient for whose computation there are a series of already established relations by different authors [ 6 ], among which the most common is that of Bazin’s: ( ( ¸ ( ¸ | | . | \ | + + | . | \ | + = 2 55 , 0 1 0045 , 0 6075 , 0 p h h h µ . (4.5-11) For of other shaped spillways (triangular, trapezoidal), the flow is obtained by using the plotting of the dependence ( ) h f Q = called flow characteristic of the spillway [ 6 ]. e)Venturi channel is a flat – bottomed channel, whose sidewalls make up a convergent – divergent nipple. It enables us determine the flows by using the relation: 140 h g h b Q 2 3 2 2 / 3 | . | \ | = µ , (4.5-12) where µ is the flow coefficient which depends on the throttling degree and on velocity. For approximate assessments we may adopt 1 = µ . 4.5.3 Methods based on exploring the velocity field in the flow section These methods are based on measuring the local velocities in the flow section, by using Pitot-Prandlt tubes, hydraulic mills, wire plungers or hot wire plungers. In circular conduits it is enough to determine the distribution of velocities by two perpendicular diameters in a flow section chosen on a straight portion of conduit and as remote as possible from the local losses of load. Carrying out these measurements we may than plot the dependence v(r), where r is the radius of the point in which there has been done the measurement of the local velocity. The flow is determined by computing the integral: ( ) dr r r v Q D t 2 2 / 0 } = . (5.5-13) For rivers or canals, where building a spillway is not possible, the method of graphical integration is used. To choose the measuring points we scale plot the bed section, the measuring points being situated on verticals whose number depend on the size and shape of the section. It is recommended that the number of measuring points, n, should satisfy the relation: S n S 25 14 < < , (4.5-14) where S is the area of the bed section, expressed in 2 m . After having carried out the measuring of the local velocities, the flow can be obtained by the graphic computation of the integral: }} = S d v Q . (4.5-15) 141 4.5.4 Flow-meters with variable crossing section The most representative is the rotameter, which consists of a transparent, vertical, truncated – cone tube 1, (fig.4.19) in which there is a free object 2. The upward stream holds the object 2 in equilibrium, its weight G, being balanced by the force of the fluid that acts upon it: S v C G 2 2 µ = , (4.5-16) where: S – the surface of the object, µ - the density of the fluid, and C the resistance coefficient. Fig.4.19 On the basis of this relation, we can get the flow velocity through the space S S ÷ 0 between the tube 1 and the floating object 2: CS G v µ 2 = , (4.5-17) and the flow is obtained by means of: ( ) ( ) S G S S CS G S S Q µ µ µ 2 2 0 0 ÷ = ÷ = . (4.5-18) Since the crossing section S S ÷ 0 varies almost linearly with the portion of the floater, the dial of the apparatus will also be straight. The shape of object 2 is thus chosen so that the viscosity of the fluid shouldn’t influence the flow coefficient. Its working simplicity, the possibility to directly read the flow, small hydraulic losses are only a few of the advantages of this instrument. Among its disadvantages we 142 should stress that it is relatively fragile, and its installing requires a vertical portion of straight conduit. On the basis of the same principle we may use spring valves as flow indicators, inter- spaced in the conduits. The valve is linked with the flow and acts upon indicating needle, which moves on a dial standardized for flow units. 4.5.5 The Ultrasound flow-meter It can be used to measure the flow of any liquid. Its working principle is based on the fact that the sonic wave given out by the emitting crystal 1, diagonal to the axis of the conduit, will be later received by the receiving crystal 2, this delay being with respect to the flow of fluid. (fig.4.20). Two sonic fluxes may be used for another designing variant; one propagates in the sense of flow, the other one in the reverse sense. The difference between the propagation times is linked to the flow. These apparatus lack inertia, they can be used in the case of non-permanent flow, for quick variations of the flow. Fig.4.20 4.5.6 The electromagnetic flow-meter Its working principle is based on the law of electromagnetic induction. It consists of a magnet placed around a conduit which produces a uniform magnetic field. Inside the conduit there are two electrodes between which an electromotive force will be generated in proportion with the flow of fluid. This method has the advantage that it can be applied to any type of flow, as it is independent of the Re number, pressure and temperature of the fluid. 4.5.7 Diluting methods These methods consist in injecting in the fluid stream a known quantity of a substance, which can act upon the density, conductivity, radioactivity, or temperature of the fluid. 143 After the mixture has become homogeneous, we can measure the propriety upon which we have acted. At first, to measure the flow in conduits, concentrated salt solutions have been injected by means of a dosing pump. Lately, there has been resorted to ionizing the fluid by means of a continuous or recurrent ionizing source (usually Plutonium 235). The flow of fluid is determined with respect to the frequency the signals appear, which varies with the quantity of ions driven into the stream of fluid. 144 BIBLIOGRAPHY 1. Anton, V. and others – “Hidraulica si masini hidraulice”, Ed. Didactica si Pedagogica, Bucuresti, 1978. 2. Benche, V. “Mecanica fluidelor si masini hidraulice”, Universitatea din Brasov, 1978. 3. Dinu, D.and Petrea, F. “Masini hidraulice si pneumatice”, Institutul de Marina Civila, Constanta, 1993. 4. Dinu D. „Mecanica fluidelor pentru navigatori”, Ed. Nautica, 2010. 5. Fatu, D. “Indrumator de exploatare si intretinere a echipamentelor hidraulice”, Ed. Tehnica, Bucuresti, 1991. 6. Florea,J.and Panaitescu,V.“Mecanica fluidelor “, Ed. 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