Blackwell, J. H. -- A Transient-Flow Method for Determination of Thermal Constants of Insulating Materials in Bulk Part I €”Theory

A TransientFlow Method for Determination of Thermal Constants ofInsulating Materials in Bulk Part I—Theory J. H. Blackwell Citation: J. Appl. Phys. 25, 137 (1954); doi: 10.1063/1.1721592 View online: http://dx.doi.org/10.1063/1.1721592 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v25/i2 Published by the American Institute of Physics. Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 02 Apr 2013 to 139.184.30.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions be performed under laboratory and field conditions. F. BLACKWELL Department of Physics.184. C.. 865--881 (1949). (It may be B 1 E. using methods of the operational calculus first suggested by S.30. the "probe" used must be a loose fit in the hole. Canada (Received March 2. transient probe methods for determining thermal constants may be described as follows: a body of known dimensions and thermal constants (the "probe") which contains a source of heat and a thermometer is immersed in the medium whose constants are unknown. . Number 2 A Transient-Flow Method for Determination of Thermal Constants of Insulating Materials in Bulk Part I-Theory J. F. Accordingly it was suggested that the theory be completely revised and experimental tests. Heating Ventilating Engrs. to a first approximation.Journal of Applied Physics February.l These were drawn to the attention of the author when geophysicists wished to apply the method to the determination of the thermal constants of natural rock in situ. A subsequent paper will deal with the experimental results obtained with the new theory. Reliability of this method of attack decreases as the probe radius increases and in certain applications (of which the geophysical is one) it is impossible to make the probe sufficiently small. (3) Only semi-intuitive estimates are given of the minimum probe-length to ensure radial-flow conditions and these appear to be unduly exaggerated. London. 137 Downloaded 02 Apr 2013 to 139. Hooper an F . this method has been used before but the work described here is an attempt to eliminate. physical idealizations inherent in previous applications. . van der Held and F·dG · van LeDrunen'JPhAysicaSoc15.132. is inserted in the medium and constants deduced from a record of probe temperature versus elapsed time. so that the instrument may be inserted in diamond-drill holes already in the rock. The first paper is concerned with development of a new approximate mathematical treatment. (2) Thermal contact-resistance at the boundary between "probe" and external medium is assumed zero. This is never true and in particular is a serious disadvantage in the geophysical problem. 22. 1953) This is the first of two papers concerning an improved transient-flow method for determining the thermal conductivity and diffusivity of insulating materials in bulk. noted that a cylindrical shape for the probe is necessary in the geophysical case.) The deficiencies which have existed in some or all of previous uses of the method are as follows: (1) It is assumed that. R. th d t pro a e errors m e measure constan s. long drill holes are not very straight and in consequence. GENERAL INTRODUCTION ASIC inadequacies in mathematical analysis and treatment of experimental data are apparent in previous work with the cylindrical-probe transient flow method for measuring thermal conductivity and diffusivity. Goldstein. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap. (4) Treatment of experimental data seems relatively crude and does not lead to any systematic evaluation of b bi . University of Western Ontario. m .aip. pper. in 1932. In brief. The work was first suggested by the geophysical problem of determining the thermal constants of natural rock in situ. Ontario. M.org/about/rights_and_permissions . A correction of dubious validity is then applied to allow for departure from linesource conditions. 1954 Volume 25.s. With the aid of suitable theoretical relatioJ). or evaluate the effect of. 1. H._129 (1950). the "probe" may be represented theoretically as a continuous line-source of heat. these constants are then deduced from a record of "probe" temperature versus elapsed time." containing heat-source and thermometer. using the new theory. A cylindrical "thermal probe. London~ 1947). London Math. thermal conductivity. These corrections depend on the location of the probe heater and the point of temperature measurement. q2= ptjk2• 4 H. Phys. Soc. page 16. I Reference 4. Q'=Q/21Tb and a=M1cJ'ITb.132. b =external radius of the "probe. (London) AM. Proc. Q = heat supplied/unit probe length/unit time. Conduction oj Heal in SoliiJs (Oxford University Press. C. (9) where p is the transform variable. (8) 9 2 is bounded as {J"-+oo. bounded internally by a 2 J. CarsIaw and J.. Misener. Calculations were made to determine the minimum probe length for radial-flow theory to be accurate within experimental error. This article is copyrighted as indicated in the abstract. in consequence.t. been redeveloped both theoretically and experimentally and it is felt that the objections listed have been largely overcome. as was the choice of a good or poor conductor for the material of the latter. 1132 (1951). 1 a02 1 (}O2 -+--=-8p2 p iJp h2'l at b<p<oo : t>O.2 2. with the initial and boundary conditions: (a) Zero initial temperature throughout. 082 -K2-=H(81 -82) ap i. Different cylindrical geometries and several methods of introducing heat into the probe were considered. M I . p. The validity of the rctdial-flow assumption is investi· gated in Appendix I.org/about/rights_and_permissions .p=b: 1>0." 82(P. A probe of good conductor can be treated to a first approximation as a perfect conductor and corrections to take account of its finite conductivity applied if necessary. (b) Constant heat-input per unit time to the inner cylinder. Goldstein3 as techniques of the operational calculus. dSll -Kr-=H(Sl-9 2) dp Q' ap =---9 1 P 2 (6) p=b. :Both large and small-time approximate solutions for the prohe-temperature were developed following methods suggested by S. (1) (2) p=b: t>O. H. (7) p=b. Blackwell and A. 18. (ii) Determination of the Probe-Temperature Transform We wish to solve the radial heat flow equation for an infinite region of one material. The new theoretical treatment is described below and experimental results obtained will appear in Part II of the paper. Reuse of AIP content is subject to the terms at: http://jap. K" k'J.p=b: t>O. valid for large times. mass/unit length and specific heat of the "prohe.J. H. • S. and diffusivity of external medium. D. However. Mechanical and electrical design factors indicate the suitahility of a spiral heating element wound on a groove on the outside of the probe together with temperature measurement at the inner surface of the hollow cylinder at the center of its length. Downloaded 02 Apr 2013 to 139. S. cl=temperature. I' = radial coordinate. op 2 at (4a) 82 is bounded as (J"-+OO. (3) a02 ofh . H ="outer conductivity" at the surface p=b.Kr-(21Tb) = Q. A short summary of the method has already been published. 34." A solution subject to the same boundary and initial conditions but valid for small times makes this determination relatively simple. THEORETICAL ANALYSIS (i) Introduction It bet:ame apparent early in the investigation that two approximate solutions of the heat-flow equation for the probe temperature would be required for evaluation of the thermal constants. In what follows they have been adapted to the equivalent Laplace-transformation procedure.M1'1. Modifications to the results of the basic analysis due to finite probe-conductivity are discussed in (v) helow. d28 2 1 d8'! -+ __==q dp'! p dp 2 2 82 b<p<oo. 81 =8 2 =0 1=0. BLACKWELL 138 The method has. Examination of theoretical and experimental advantages of the different arrangements led finally to a hollow cylindrical metal probe with constant heat input per unit time. Proc. Goldstein. (c) Contact-resistance at the boundary between cylinder and external medium.t)." t = time.2 = temperature. Jaeger.aip. Let 81 (t). hollow circular cylinder of another (perfectly conduct" ing) material. The magnitudes of errors introduced by certain simplifying physical assumptions have been established and other minor improvements made. Then iJ 2O'J. 51 (1932). inclusion of contact resistance at the probe boundary complicates the large-time solution and makes it desirable to determine the contact resistance "independently. Soc. since the basic theory given below assumes radial flow.' op at (4) 882 a a(it -K2-=Q'--. Previous workers had used one only.184.e. (5) Laplace transformation of differential equation and boundary conditions with respect to time results in the subsidiary equation and boundary conditions.30. (13) and (14) shows that each has a single branch point (at the origin). Inw=2K 2/bH. --- -" FIG..3 The transform 8 1 is expanded • H..""flill.184. Hence the contour of integration Re(p)='Y may in each case be replaced by the standard equivalent contour CDEFG7 (Fig. Kt(x) are modified Bessel functions of the "- ' . p. Reuse of AIP content is subject to the terms at: http://jap. 1. second kind. Jaeger. and approximate solutions for 81 are developed below. From the Inversion Theorem of the Laplace Transformation we have. we obtain +~ ln ((3P)+0(P)J. C. BT2 (19) (20) t (21) (where c is a real positive constant and t is positive) we Downloaded 02 Apr 2013 to 139. Integrating term-by~term on Br2. Operational Methods in Applied Mathematics (Oxford University Press.5722.(10) .t).p) = where 2bQ'Ko(Q2P) pQ2b[bap~+ 2K2K1 (q 2b) ] . respectively. 1947). 7 Reference 6.org/about/rights_and_permissions . q2b -<u (11) Alt.139 TRANSIENT-FLOW METHOD f>t(p) . This article is copyrighted as indicated in the abstract... Solution of Eq. (15) evaluated term-by-term. we follow the method used by Goldstein..p) satisfy the conditions of jordan's Lemma 6 and neither has poles within or on the closed contour ABCDEFGJK as the radius R of the circular arc ~OO. (10) and (12) and simplifying. (6) subject to Eqs.. deduction of thermal constants from a temperaturetime record. Furthermore. (8). 27r~ i (cp)etpdp = - [In(~)+y] j c 1 In (cp)eIPdp= --. (15) 2 w 4K2 (16) "Exact" evaluation of integrals in Eqs.aip. Le. London. These solutions are quite unsuitable however.) The result is a logarithmic term in T (T=h 22t/b2) plus an asymptotic expansion in inverse powers of T. the Probe Temperature To obtain a large-time solution. ~=~KO(q2b)+ (K2 )K1(Q2b). respectively. 1). (t -1. 76.132..... 92. for the present purpose. (18) 1In ~ f 27r~ JBT2 P (iii) Large-Time Approximate Solution for edition. Inserting the ascending-series expansions for the modified Bessel functions in Eqs.. (Many of the term integrals diverge on the original contour but all converge on Br2. 'Y is Euler's constant = 0. and (9) gives 2bQf~ 9 l (p)= p[bap~+ 2K2Kl (q 2b) ] . 82 (p.30. ore 'odius f 9 2 (p. and Inv=2'Y. (7). p.. 9 2 (p. Eh (P) and 9 2 (p. formally in ascending powers of p and the integral of Eq.. (13) (14) Examination of the integrands of Eqs. then. S. using the results.1 \ \ \ \ (12) bH \ Ko(x). second (17) where (3=vb 2/4h22.p) are the L transforms of 8I (t). Carslaw and J. Denoting this contour by Br2 we may thus write . (15) and (16) as real infinite integrals is straightforward and that for BI (t) is outlined in Appendix II. and zero and first orders. McLachlan.184. Inserting the asymptotic expansions for the modified Bessel functions in Eqs. (22) will predict temperature within a given error. We obtain bQ'[In4T-'Y+-+2K2 1 Making a term by term inverse transformation. we expand a 1 (a. h12 = the conductivity and diffusivity of the probe material. p. we obtain al~2Q'[~_ (2H)~+ (2IJ2h2)~0(~)]..K1(q1a)Il (q 1b)}. (22) reduces to that of a simpler problem already in the literature. 1946). (v) Modifications Resulting from Finite Probe Conductivity (22) Rigorous mathematical justification of the formal process described above has not yet been made. 278. 1(a. p=aj ql =pl/h1 a and other symbols as before.. the problem has been solved again as far as the temperature transform. (10) and (12) and simplifying. (22) and numerical integration of the exact real integral of Appendix II have been very satisfactory. Downloaded 02 Apr 2013 to 139. Cambridge. 8 In the limit H---4oo (perfect contact). The region of solution is now a "composite solid.9 In order to investigate the effects of finite probe conductivity. another method suggested by Goldstein in the same paper is followed. the solution Eqs. Le. of course. Reuse of AIP content is subject to the terms at: http://jap. • Reference 4. these numerical checks (using thermal constants of the order of magnitude of expected unknowns) are useful in setting the minimum value of T for which the expansion Eq.132. a. (h(t)~ . fh(a.2~2 (1l1+ A2) }+o(~) J. 283. Large Times As before. on the other hand. a p2 a aK2 p3 p7/2 p4 (23) 2Q'[t--tH+ 16IJ2h2 /5/2+0(t 3) ]• 2 a a 15 (v 1r)aK2 X {II (qla)K 1 (q 1b) . Q = heat supplied/unit probe length/unit time at the outer surface of the probe.J. Complex Variable and Operational Calculus (Cambridge University Press.t)~- 2K2 bH 2T (24) X {ln4T_'Y+1_ah22(ln4T_'Y+ 2KI) bK 2 bH Unlike the previous case rigorous justification of this type of approximation is not difficult. We now consider the approximate inverse transform (h(a. T:sMt/b2«1. Q' =Q/21rb (as before). W. p.aip. .p) = the temperature. b = the internal and external radii of the hollow probe. that in typical rock with a probe of suitable size many hours must elapse before the term of O(l/T) can be neglected without introducing appreciable error.30. In the special case considered here. p=b. a relatively small minimum T can be tolerated. An asymptotic expansion of the transform in inverse powers of "p" is made. ~ (probe walls removed). H. BLACKWELL 140 obtain and Jaeger. p.g. An intuitive general justification of the type of approximation process used is suggested by McLachlan.p) in an ascending series in "p" and integrate term-by-term on contour Brt. and this expansion integrated term by term on the original contour Re(p)='Y. the final result is a series in ascending integral and half-integral powers of "t.K1ql {Ko(q2b)+ K:2Kl (q2b)} 81(a." but in the general case it is usually a series of repeated integrals of the complementary error function. without using the perfect-conductor assumption. comparisons between the results predicted by terms to O(l/T) in Eq. It is found. 100. Then (iv) Small-Time Approximate Solution for the Probe Temperature (25) To obtain a small-time solution. e.t).t) for large and small times. for example. see Carslaw 8 N. lO In dimensionless form the general condition for validity is the converse of the previous case. (26) / 10 Reference 6. This article is copyrighted as indicated in the abstract. Nevertheless." there is a radial temperature gradient in the probe wall and the relative radial locations of heat-injection and temperature measurement become important. more complex. and temperature transform. respectively at the location of temperature measurement. if the term of O(l/T) is included. Let K 1. In addition.org/about/rights_and_permissions . The new solution is. the parameter H can be read off as the intercept on the yaxis. h22 depends intimately on the accuracy of data-fitting to the "large-time" relation. Eq. in the form. The dominant effect for small times is to reduce the tem- As discussed in 2 (iii) above.org/about/rights_and_permissions . H large (the case of very good thermal contact) or that the value of H be known. Once again Goldstein's paper suggests a suitable alternative method. Since the conductivity of the probe is high. i. however.h 22t/b2) used is high and the contribution of the 1/t term therefore Downloaded 02 Apr 2013 to 139. Reuse of AIP content is subject to the terms at: http://jap. in any event. and Al~A2~ (b-a)2/6h 12.e. the curvefitting procedure for obtaining K 2. of course.141 TRANSIENT-FLOW METHOD perature at a given time t by a constant amount which is inversely proportional to the probe conductivity. it may be necessary to include an additional term or terms on the right-hand side of (27b) and fit the solution to the data by a "selected point" method. may be expressed in terms of the constants of the problem but are not required in what follows. 1 ~OO=A~OO+B+~C~OO+m. APPLICATION OF THEORETICAL RESULTS It is seen that to O(l/T) this equation is identical with Eq. This term is usually insignificant in magnitude-it is approximately equal to -2h22(b-a)2/3h 12b2 and. that the input power to the probe and the probe parameters are known. Hence when 9 is not very small. h22 can be evaluated at once from the constant B. the small-time solution is more sensitive to the value of probe conductivity than the large-time solution. the smaller its effect on the determination of K 2. A "small-time" solution for the poorly conducting external medium is combined with a "large-time" solution for the highly conducting probe.184. h22 unless the minimum value of T(T=. Rewrite the "small-time" relation (27a) in the form U( Y=p aU1) t-A2. h22.132. This article is copyrighted as indicated in the abstract. (27) becomes (Similarly C. The final accuracy of determination of K 2. (22). (22) except for the term -2M(Al+A2)/b2 within the coefficient of l/T. qlb2. D.2Q' 16fl2h2 ti =9. ~2~ t where bQ' A=2K2 Small Times A "small-time" solution of the ordinary type will be of little use for the composite region under consideration. With H known.e. previous workers have neglected the term of O(l/t) in their temperature expansion and determined thermal constants graphically by fitting a straight line by eye to the plot of Ih against In(t). It is assumed. 1/q2b and cross products of these small quantities. h22 which is described below is independent of its value. conduction in the probe will pass over into the "large-time" approximation region a few seconds after t= 0 and it is not experimentally practicable to employ such a short time interval.. The application of Eqs. (22) and (27) (or 27a) to the determination of thermal constants is straightforward: Rewrite the "large-time" relation. Eq. The solution should be useful for calculation provided b2/h12«t«b2/h22 i. only a rough value of the parameter is required.. where 3. Term-by-term inversion is then performed. terms involving q2b in descending series. As was to be expected from physical considerations. The series are then combined discarding terms of order higher than q12a2. In order to evaluate the constant h22 from the value of B. In the particular case under consideration we obtain 9 --t2+ a 16fl2h2 15 (v'1I")aKll t6/2+O(t3) ] • (27) If b-a«b.30. (27b) Then if the quantity y is plotted against ti for small t.) It can be seen immediately that a fit of suitable experimental data to this expression will yield a value for K 2directly from the constant A. assumption of a straightline relationship can introduce significant errors in K 2. In general. In this connection it should be borne in mind that the larger the quantity H.p) involving qla and qlb are expanded in ascending series.15 (v'1I") Kll . it is necessary either that the parameter K 2/bH be negligibly small. To obtain this solution terms in 8 1 (a.aip. the time region in the immediate vicinity of t=O must be excluded. If H is not very small. Dr. we attempt to evaluate the effect on the preceding results of the thermal capacity of a finite wall thickness of good conductor. H. Professor A... and Dr.. 11 For Case (i) see (1953). D. i. Blackwell. and reliable probable errors for the thermal constants can be evaluated (a noticeable omission in previous work on the probe method). Case (ii) The semi-infinite solid bounded by the planes z=±L and a slab of "good conductor. to determine a criterion for radial flow conditions within a certain maximum error and this entails at least partial solution of the combined radial/axial flow problem. at p=b. Space does not permit of the development of the solutions herell but the results and deductions from them are summarized as follows: Case (i) Let () be the temperature at the inner surface of the cylindrical hole at the center of its length. -L<z <L. Phys.. and for the constant advice and encouragement of the writer's Ph. R. University of Western Ontario.. by the fact that a very small proportion of the heat injected into the probe is lost from the ends which." -d<x <0. such a high value of T is usually undesirable from practical considerations. are covered with insulating caps. The latter is complicated by two considerations: Firstly. The regions and boundary conditions for which the Heat-Flow Equation is solved in the two cases may be specified as follows: Case (i) The semi-infinite solid bounded by the surface p=b internally. J. and by the planes z=±L. however. 4.aip. and the Research Professor of Physics. z=O. Supervisor. ACKNOWLEDGMENTS The writer wishes to acknowledge with gratitude the financial assistance accorded to him by the Research Council of Ontario during the work described above. and Department Head. It is necessary. The planes z= ±L are maintained at temperature zero. The temperature at the center of the probe length is then compared with the corresponding radialflow temperature. in the case of the present problem. if the probe is made sufficiently long and temperature measured at the center of its length. Professor R. Secondly.D.k=O 2k+l du ' (28) where T (as before)=h22t/b2 and Q is flux of heat across the boundary p = b. E. the error caused by assuming radial flow can be made as small as we please. R. it is difficult to specify (even approximately) the boundary conditions outside the probe length. of whom the latter first suggested the problem. however. The method of "least-squares" has been used by the author to fit the experimental data to the complete Eq. Reuse of AIP content is subject to the terms at: http://jap. zero conductivity in the "axial" direction. 472-479 Downloaded 02 Apr 2013 to 139. This analysis involves the solution of two problems: (i) We assume the probe walls are of vanishingly small thickness but impose axial boundary conditions which are certainly more severe than those encountered in practice. W.org/about/rights_and_permissions .30. p>b and there is constant heat flux across the boundary p=b. The temperature is maintained at zero on z= ±L. the probe inserted in a long hole. Can.e. This article is copyrighted as indicated in the abstract. C. it is justified. is the least satisfactory part of the present axial-flow analysis and the writer has begun a new treatment using a prolate spheroidal model for the probe. in part. the objection to fitting a curve by eye is overcome.184. J. i.e. Misener.132. analytical complexity is greatly increased by the presence of a layer of good conductor on the inside of the cylindrical hole (the walls of the probe). The necessity for it. Two simplifications are introduced here: (a) For convenience. J. Nicholls... (ii) Using the same axial boundary conditions in the external medium. Uffen. There is constant heat input to the slab per unit time and all heat output from the slab is to the region x>O. and by the presence of contact resistance at the probe/ medium interface. x>O. Brannen. A much smaller value of T min can then be used. in practice. -L<z<L. the cylindrical configuration is replaced by an analogous rectangular configuration. The latter drastic assumption is made necessary by difficulties with boundary conditions. (22a). Then rT[i'" 16bQ ()= 7r 3K 2 Jo 0 exp(-uy)dy y[Jt 2(y)+N t2(y)] _ (-l)kexp { - (2k+l)2 7r2b2U 4D }] XL . (b) The probe-material is assumed to have infinite conductivity in the "radial" direction. Acknowledgments would be incomplete without recognition also of the helpful comments of the author's teaching colleagues.J. Dr. This slab has infinite conductivity in the x direction and zero conductivity in the z direction. BLACKWELL 142 negligible. 31. Sincere thanks are recorded also for the facilities of the Physics Department. It is considered that the presence of a small amount of contact-resistance has no basic influence on the criterion we are trying to determine and it is neglected throughout the analysis outlined below. H. APPENDIX I The Assumption of Radial Flow It is obvious physically that. Dearle... 1% Ingersoll.184. however. bT} (2k+ l)27r2 2 (-l)k exp { 4L2 00 (31) where M=-:E . New York. d.org/about/rights_and_permissions .e. we have M>0. 255. The results. These results are applied as follows: using values of or. and Ingersoll.e..(A2+flk2)!]2h22t} Xerfc{[A. A = Kd2Mh22. This article is copyrighted as indicated in the abstract.. the values of 8 given by Eqs. Ingersoll. must be treated with a certain amount of circumspection because of the rather artificial boundary condition employed.. say..2k+l 7r k-O 2 (A 2+flk )t +(A2+flk2)t]2Mt} (1-exp( . hence there is some justification for stating the criterion in terms of a length/diameter ratio. Downloaded 02 Apr 2013 to 139.. Zobell. i. Heat Conduction (McGrawHill. Then 4Q 00 (-I)k[erf(ftkh2vt) 8=-L.0158. (30) In using this criterion to determine a mInimUm length. and thickness. L/b> (T/0. L> (Mt/0. for example. Values of the sum of the series M are tabulated (e. M>l-(2/n).Ty2) }du (The latter result is a special case of one published previously. then we have.g.-- 4bQ L oo = -rr2K2 0 (29) 7rK2 2k+l flk X {exp{[A y3[112(y)+NI2(y)]' Xerfc{[A + (A2+flk2)!]h2vt} -exp{[A . 1948).. (28) reduces to the radial flow solution. the reasoning of Case (i).0632)t. (The assumption of zero conductivity of the slab in the z direction permits of the application at x=O of a boundary condition similar in form to the "perfect conductor" boundary condition used in 2 (ii». Should the relative axial-flow error then exceed the maximum originally imposed. Book Company. 2 Let us require the actual temperature to differ from the radial flow solution by less than lin. i.g.. Eq.0632)!. Zobell and Ingersoll)12 and for M>0. e.aip.. and Cl appropriate to the cylindrical probe being considered and a value of L given by the criterion of Case (i). p. 13 Reference 4. z=O.. Inc. we find b2T/4L2<0. Case (ii) Let (J be the temperature at the boundary between slab and semi-infinite medium at the center of its length. respectively. The corresponding linear-flow solution13 is . Reuse of AIP content is subject to the terms at: http://jap.143 TRANSIENT-FLOW METHOD As L~oo the radial/axial flow solution.(A 2+ flk2) i]h2vt} ].. roughly. a value for h22 should be used which is a safe upper limit for the type of material being measured (e. In the cases so far examined in this way this has not proved necessary.132. at x=O. It is considered that replacement of the cylindrical problem by an analogous rectangular problem is itself a valid assumption. when applied to its rectangular analogue obviously yields the same criterion. 0.99. p. It will be noted that the latter will depend both on h22 and the radial dimensions of the probe. is not appreciably different from unity M=OICl d .e.30. Cr. the value of L would be increased and the comparison process repeated. ! For a maximum axial-flow error of the order of percent. (31) and (32) are compared for the largest value of t to be used in an experiment. the problem of the effect of the wall on axial flow error is not yet completely resolved. and the remaining symbols have their usual meanings....015 cgs units for rock) and "t" should be of the order-of-magnitude of the largest time likely to be used in an experiment. d are the density. I+M (J""'--Xradial flow solution. i.99.9) If now the value of 4 k~ exp( -INMt) +---2 flk= (2k+ 1)7r/2L.. Then 01.g. 248. specific heat. of the slab. To evaluate fMt) as a real infinite integral a standard artifice is employed. e positive: Then +Lt . 1. (33) . (37) entails replacement of modified Bessel functions of imaginary argument by ordinary Bessel functions of real argument. integrating from 0 to t. (10) and (12). (35) are real and 0'. Let Inspection of the form of F(p) shows that the second integral vanishes.(-ioo Examination of F(p) shows that like 6 1 (p) it has a single branch point at the origin and that it is pos~ible in this case also to replace the contour of integration Re(p) =')' by the equivalent contour Br2.. on the portion DEF . p.132. the following substitutions are made: Put where It is not possible to obtain this result by the more direct method of making the substitutions (35) in the original integral. J (37) o Reduction of Eq. this solution reduces to a special case of one already published.i ". p=O'e.0 i since it is found that the real integrals on the component parts of the contour do not converge in this case. 14 1r F(ee i"') exp(teei"')ieei'l'drp -r _ ~oo F(O'eir)e-'ITdO'. 284. where 0'.ir on the portion CD } = Ee. = O'ei'lt on the portion FG E. Downloaded 02 Apr 2013 to 139. the probe temperature (13) where Eh (p) is the probe temperature transform given in Eqs. Thus Eq. H.184.aip. BLACKWELL 144 APPENDIX II The Probe Temperature as a Real Infinite Integral By the Inversion Theorem of the Laplace Transformation. Then by a well-known theorem of the Laplace Transformation 01(t) = It[~ f'rl211'~ o iCO F(p)eIPdP]dt. This article is copyrighted as indicated in the abstract.-.30. (39) as H-too (perfect contact) and make appropriate changes in symbolism. If we take the limit of Eq. Reuse of AIP content is subject to the terms at: http://jap. f co F(p)etPdp= -2i Br2 r Im[F(O'e i1r)]etITdO'.PdP]dt. (33) becomes 01 (t) = f t[~ Jr o 211'$ F (p )e. and simplification. (34) (39) Br2 Referring to the contour Br2 (=CDEFG) illustrated in Fig. (36) l' Reference 4. I(J Finally.org/about/rights_and_permissions . Thus.J. The substitution x= (1/h2)0'! is then made for convenience and we obtain 6 1(p)=F(p)/p.