OVERDUE FINES: . 25¢ per day per item (mm *f W= ‘M ‘\’:.;’5 . m' 1 Place in book return to remove "‘ ‘4‘,” "4"- charge from circulation records ABSTRACT THE OPTIMUM ANALYTICAL DESIGN OF TRANSIENT EXPERIMENTS FOR SIMULTANEOUS DETERMINATIONS OF THERMAL CONDUCTIVITY AND SPECIFIC HEAT by James Vere Beck Accurate values of thermal properties of new and conventional materials are indispensable for the design of space vehicles. In order to (a) determine the thermal conductivity and specific heat for some modern materials which degrade when heated and (b) provide a more rapid method of simultaneously determining both these prOper- ties, a transient experiment and a method of analysis are needed. A method of calculating the properties from transient tempe ra- ture and heat flux measurements has been developed and includes (a) the calculation of temperatures and (b) the iteration procedure called nonlinear estimation. The temperatures can be accurately calculated using a modified Crank-Nicloson finite-difference approximation applied to the heat conduction equation. The nonlinear estimation pro- cedure is an extension of linear regression analysis. Properties can be readily calculated with less than 0. l % error due to approximations 1n the numerical procedure. To determine the optimum experiment a criterion 18 developed and is applied to find the optimum boundary conditions and locations of thermocouples. It is proved that both thermal prOperties can not be . My .. y . ..~-1' '- I .,...‘ A“... “D .. up ..- nu»... independently calculated from temperature measurements in a homo- geneous body unless the heat flux is measured. The optimum heat flux boundary condition is one which causes a step rise in the surface temperature. The optimum locations for the thermocouples for a finite body are at the boundaries. Constant and temperature-variable thermal conductivity and specific heat are calculated from experimental data for nickel and copper. THE OPTIMUM ANALYTICAL DESIGN OF TRANSIENT EXPERIMENTS FOR SIMULTANEOUS DETERMINATIONS OF THERMAL CONDUCTIVITY AND SPECIFIC HEAT BY James Vere Beck A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Mechanical Engineering 1964 3/ ./'1/7 \ “‘ 's '\_z\ ACKNOWLEDGEMENT The author is very grateful for guidance and encouragement during the period of research and during the preparation of this thesis by the Chairman of his Guidance Committee, Professor Amritlal M. Dhanak and the members of the Committee, Professors Maria Z. v. Krzywoblocki, Charles P. Wells and Richard J. Reid. The original Committee also included Professors Gerald P. Weeg and Harold G. Elrod, Jr. who were not at the University during the past academic year. The author wishes to particularly acknowledge many helpful discussions with Professor Elrod. The support of the Division of Engineering Research is greatly appreciated. Much of the research was made possible by the use of the excellent computing facilities and the aid of the cooperative per- sonnel at the Computer Center of Michigan State University. To his wife Barbara, the author dedicates this dissertation for her understanding and cooperation during graduate study and research. ii ‘nu \, ‘1 N- '. A... qv‘ a... I‘— V- n.— '1 TABLE OF CONTENTS Chapte r I. II. III. IV. VI. VII . VIII. IX. DESCRIPTION OF THE PROBLEM . . GENERAL RELATIONS BETWEEN TEMPERA- TURE RISE AND PROPERTY DERIVATIVES . BASIC CRITERIA . SEMI-INFINITE BODIES . FINITE BODIES . CALCULATION OF CONSTANT THERMAL PROPERTIES . TEMPERATURE-VARIABLE THERMAL PROPERTIES . CONCLUSIONS RECOMMENDED FUTURE WORK . iii Page .ll .21 .39 .53 .69 . 85 100 102 ".Il Figure LIST OF FIGURES F-contour for B : 0°. F-contour for B : 1' 90° F—contour for B = 45°. F—contour for B = -45° Heat flux to produce constant surface temperature Surface temperature history . Criterion E versus T Derivatives kT and cT at x = O for T -T. : kl c1 m 1 constant . Derivatives Tk and TC for semi-infinite body with q=C Errors Akj and A5. for semi-infinite body with q=Cfor Tm=1.5 . Derivatives Tk and TC for semi-infinite body with A q = a(9n) 2- Errors AR. and AEJ. for semi-infinite body with La NIH q=a(9n) for Tm=1.25. Temperatures in a finite body with q : C at x : O andq=0atx:L. Derivative Tk for q=C at x:0 and q:0 at x=L. Derivative for q=C atx : 0 andqzo atsz. iv Page 26 26 26 26 26 26 26 26 43 43 50 50 55 55 56 . - . —IU 49:. .t l 0 " I- A vut.‘ ‘JIE .1.- 7.3.1 7.3.2 7.3.3 7.3.4 Errors Akj and ch for q : C at x : 0 and q : 0 atx=L. T =0.64. m Temperatures in a finite body with q at x : O to causeTmzCandq:Oatx:L Derivatives T for a finite body with q at x O to k causeTm=Candq:0atx:L Oto Derivatives TC for a finite body with q at x cause TmzCandqz-Oatsz Errors Akj and AEJ. for a finite body with q at x : O tocause Tm :C andq: Oatsz. Tm: 0.76. Derivatives Tk and TC for a finite body with q at x:0tocauseTrh:CandqforT :Oatx:L . Errors Akj and ch for q at x = 0 to cause Tm : C andqforT =0atx=L Spatial nodes for interface, equation 6. 1. 3 Spatial nodes for surface node with given heat flux, (6. 1.10). Differences between calculated and measured tem- peratures for Hsu #1 Differences between calculated and measured tem- peratures for Hsu #2 Derivatives Tk and Tk for a finite body with a q I 2 atx=0tocauseTmzCandq:Oatsz. Derivatives TC and Tc for a finite body with a q l 2 atx=0to cause Tm=C andquatsz. Derivatives T and T for a finite body with a q at k1 kz x=0 to cause Tm=C anda qatx=L to cause T=Ti:0. Derivatives TC and TC for a finite body with a q at 1 Z x:0 to cause Tm=C anda qatx=L to cause TzTizo. V .56 6O 6O 61 61 64 64 7O 7O 82 82 90 9O 91 91 , v I- 3..., H 'L .h . ]} u.“~ ’fl I") ’r! H'1 nv Table 4.1.1 4.1.2 LIST OF TABLES Partial differential equations and boundary conditions for T-T., - cT and - kT 1 c k Cases for which k and c can not be independently determined. Quantities for q = C with thermocouple at x Maximum values of A for semi-infinite bodies . 1 Quantities for q : a(0n)-zwith thermocouple at x . Maximum 5 values for different experiments with finite bodies . Percent error at x = 0 for q = C at x = 0 and q = 0 at x = L. Percent error in k and c for q = C at x = 0 and q = 0 at x = L with measurements at x = 0 and L and A7 = 0.02 for data . Percent error in k and c for q to make T = T at x = 0, q = 0 at x = L with measurements at x = 0 and L and AT : 0.02 for data. Hsu temperature data [12] . Thermal properties calculated from Hsu data . Comparison of a. values calculated by present method with others . vi Page 15 20 41 45 48 57 70 75 75 77 77 77 Parameters for temperature-variable properties Possible experiments to determine k and c as functions of temperature Lindholm and Kirkpatrick temperature data for copper [36]. Thermal properties for copper . Calculated thermal properties for copper from Lindholm and Kirkpatrick data . 94 94 97 97 97 ’ LIST OF APPENDICES APPENDIX Page A. Derivation of Equations (1.3.8) and (1.3.9) . . . . 108 viii I. DESCRIPTION OF THE PROBLEM I. 1 Introduction In the space program the values of thermal properties of new and conventional materials are indispensable for the design of compo- nents such as reentry vehicle heat shields, rocket nozzles and others. With the development of a host of new materials some of which degrade when heated, a rapid transient method of determining thermal proper- ties is needed. The method should be rapid, be amenable to automation and be competitive in cost with present methods of measuring thermal properties. With the advent of the large scale digital computer it is not necessary that the computations associated with the method be simple enough to be readily performed by hand calculations. The method given herein is intended to help meet this need. The method is not restricted, however, to the determination of only thermal properties but can be utilized for finding a wide category of physical prope rties.’ The basic objective is to investigate the design of optimum experiments for the simultaneous determination of thermal conductivity and specific heat of solids from transient temperature measurements. The thermal properties can be functions of temperature. An "opti- mum” experiment permits the properties to be determined more accurately than any other similar experiment with the same tempera- ture range and duration of the experiment. The large-scale digital computer is to be used whenever needed and thus the optimum experiment -1- -2- need not be one possessing a solution convenient for hand calculations. On the other hand, the optimum experiment should have boundary conditions that can be simulated readily in the laboratory. Literally hundreds of papers have been written on the subjects of determining thermal conductivity k, specific heat c and density p [1-3]. Recently a number of papers have been written about the determination of thermal diffusivity, o. : k/p cp [4—10]. Thermal conductivity can be found using solely steady-state measurements while cp and (1 require transient measurements. To determine k and cp the heat flux and temperatures must be measured; while for a tem- peratures alone are required. Within these limitations the methods proposed for determining thermal properties are quite varied. There are a number of respects in which the present work differs substant- ially from the preceding, however. Some of these differences are listed below. a) Each previous method uses a relatively simple exact solution for the temperature-distribution to facilitate the calculation of the properties (except [10]). b) Because of a), certain boundary conditions must be main- tained and heat losses must be reduced at the heated surface. These heat losses for certain transient cases can be difficult to eliminate. c) All the solutions employ the assumption that the thermal properties do not vary with temperature. (1) Only a limited number of temperature measurements are used, usually only two or three. This is frequently true even for transient techniques for which temperature histories are obtained with one, two or more thermocouples. Not all the data is used; or if it is, ,. ,4 ‘. .-l . . M ’. .vL ,. . . Coy ’ ’ .... ~— 0. o ... - A .,_ ,. .— u. ...,. IE— o.,~. ,. .-~ ..i-. . ....... ‘ — P ..--,. .~ ‘I n: , . . r‘ ' s A..‘.. -- -c.. ' n H. C. ‘. -A .. -‘ u... ”‘- g "V" D) “"H tv ~. “6, o,_" "-. "'C. '-,. ~u,, ' 5.. . s; _ r. ‘: -~.: .2». uh”. 5. n“., x...» G L‘- ‘ .. “'3 ‘Lv be‘ - e, ., I- .,. ‘. u- , . -N~, "\ 'i. n .. . u ‘. cc ~4‘. . a fi “Ans ; ‘_~ it . “"o u it is not used efficiently. e) Questions relating to the optimum heat flux, heating time or best location of thermocouples for the transient cases are not investi- gated theoretically. f) Except for a few papers [6, ll, 12] each experiment is designed to determine only a single property. The method to be described utilizes the finite-difference solution of the heat-conduction equation and thus is flexible enough to treat any given time-variable heat flux for constant or tempe rature-dependent properties. Boundary conditions are suggested for which heat losses are negligible. flthe transient temperatures measured are utilized in a straight forward manner using a method which is an extension of least-squares analysis. It is called "nonlinear estimation. " A cri- terion for determining the optimum conditions is developed and employed. Particular emphasis is given to the determination of both thermal conductivity k and specific heat cp from a single experi- ment. Actually we shall find k and p cp from the experiment. For convenience let c: c 1.1.1 9 p ( ) The density p is relatively easy to measure accurately from gravi- metric and linear measurements. Moreover, the density is much less sensitive to temperature than either k or cp. For this reason the effect of elongation of the body during heating upon the tempera- ture distribution is usually negligible and is not considered in this analysis. Parker and co-workers [6] utilized a pulse of high-intensity short-duration light impinging on the blackened surface of the specimen. " w . ,‘ H...- - .- .n- .C .. ..,. O I .Iul LI." ‘ . .c-oid .... -. V ‘ ....-..; k. . »..,.. .3..--u. ‘o. -,., ' § >- ‘-- vit'u . V--. “— -.‘ “"0‘. A .... ... .a F ' a.-. ‘ h ”.- "\ .., -4- The other surface, which was insulated, was instrumented with a thermocouple from which the temperature data was obtained. The relatively simple analytical solution was derived for the temperature at the insulated surface for a body exposed to heat pulse of known magnitude. This solution was then utilized to calculate the thermal diffusivity o. : k/c, thermal conductivity k and heat capacity c. The accuracy obtained was about I 5%. A Russian, Smekalin [11], considered two semi-infinite speci- mens of the same material separated by an electrical heater producing a known constant heat flux. A thermocouple was located inside one of the specimens 7-10 mm. from the interface and the temperature was measured at times 2, 3, 4, 5 and 6 minutes. From temperatures at two different times the thermal diffusivity and thermal conductivity were determined. He reported accuracies of f 3% for o. and ‘1‘ 5% for k. Hsu [12] performed an experiment with two semi-infinite solids initially at two different uniform temperatures. If one of the solids (the standard) has known thermal properties, then properties can be determined for the other (the specimen). The experiment was begun by suddenly pressing the two solids together. Several thermocouples were carefully placed in the specimen at different positions and the temperatures were measured at a number of instants. When the two semi-infinite bodies were brought into contact, the temperature at the interface suddenly increased to some intermediate temperature between the initial temperature of the two solids. If the properties were independent of temperature and the contact resistance was negli- gible, the interface temperature remained constant as long as the bodies were in contact. He performed a very careful experiment with p » . ,-4 ---»’ ’- \— , - .-..-.uo . ’; .‘A .-.. It“ I ‘f'z'l .-.. --.| —— ._ A -- . ‘v .- u. E _ i i U: FT " .~ 5. -...‘ . ’v- ."‘ -~.. ‘;*~«.. My. ...~ ~ Kr . 3 ~- ‘ ~I m.“ EX: ."I ‘1 ’3‘. .., - ‘ '.“"t “M. ~ 1 ‘n- A‘I.“: a )- " 't I‘ 1 .Lec “ c . 9. 'x I‘ .J‘ ' n, 0‘- u“ f- ‘\ ~;"'¥. *Q -5- two semi-infinite specimens of nickel and gave the temperature data in his paper. His data is analyzed in section 6. 3. l. 2 Problem Definition In this dissertation a number of problem areas relative to the efficient determination of the thermal properties are investigated. They are suggested in this section and analyzed later. Boundary conditions: For simplicity the bodies considered will have a plane, one-dimensional heat flow. The boundary conditions neces- sary to separately calculate k and c are determined and those that are insufficient are also indicated. Typical boundary conditions are: given temperatures, prescribed insulated surface, prescribed heat flux and convective heating. Location of thermocouples: The optimum location and number of thermocouples are determined. The optimum thickness of a finite specimen for a given heating time is also found. Development o_fa general criterion: In order to determine an "Opti— mum" experiment some measure is needed of the effectiveness of an experiment to determine thermal properties. This measure would provide a criterion to be used to determine if one experiment is superior to another. A criterion is developed and applied for a number of different boundary conditions. Effect itemperature errors: The calculation of the properties can be affected by errors from the following sources: inaccurate tempera- ture measurement, errors in the finite-difference calculations and an imperfect model. The effect of errors due to the first two cases are examined. For most materials the heat-conduction equation (or model) . u" ~- A. _ v . ' 6- ”H" .- ' . . ’_‘>.'. . . . _ _ ' Q ‘,.J V . \. ‘ ‘v .‘-* — o .. . . - ,..q_ .. - ,...¢ .- - ~\ 1 ... ‘5‘.- . - ‘— .-_¢. ., .t ‘ ’ 9- 04, ..‘ .h OI 6‘ P .\"A._ .- '\ ‘4 r ““(t. -6— describes the heat flow very accurately and thus errors from this source are not considered. An improved model is needed for decom- posing materials such as are used in heat shields of reentry vehicles. That is beyond the present scope of this work, however. 1 . 3 Nonlinear Estimation Procedure The problem of calculating parameters appearing in a differen— tial equation which describes a physical process is called “nonlinear estimation. " Supplied with experimental data obtained from the pro- cess and a method of solving the differential equation it is possible using the nonlinear estimation procedure to calculate the parameters. The method has been developed from a statistical viewpoint in the past decade by G. E. P. Box and co-workers [13-16]. The first known reference to the method was written by Gauss in 1809 [17]. Due to the large number of calculations involved it has only recently become practical with the advent of the large-scale digital computer. Box‘s work is mainly related to the statistical development of the method rather than the application to any specific case. He has, however, given some examples involving first-order ordinary dif- fer-ential equations arising in chemical engineering. He has not given an application utilizing a partial differential equation. The governing equation describing one-dimensional heat con- duction is 8 3T _ 8T 5'; (k 3—K) -— Cs—é- (1.301) where T is temperature, x is position and 6 is time. The thermal properties, thermal conductivity k and heat capacity c, are to be determined. For continuous transient temperature measurements > I 1. .¢ -. "~ .— I . .._. . u . .1 ~ ‘ ,_-..‘a *‘a w " A .0.» 4. ‘ '-¢"\. . U. -.l~..- 3‘: . t“P- .. - I ‘ w y . 5 c I" . ‘\ - r;~ n.»- - ‘¢.‘P «b, s“ - i \. \ o. ‘ I .'~ - -7- the sum of squares function F for n thermocouples, M: F(k,c) : j]. J 82.1 em A. S (TJ.(9)-T .(e))2de (1.3.2) 0 is to be minimized with respect to k and c. The temperature TJ.(0) is found using a finite-difference approximation for (1. 3. l) at posi— tion Xj and time 8; it is a function of k and c. Temperature Te, .(8) is the experimental temperature at x3. and 9 . The quantity Aj is a weighting factor for each position and is frequently equal to unity. The sum of squares function F is a parametric function with the pro- perties k and c being the parameters. The properties can, in general, be prescribed functions of temperature. The method of minimizing F can be illustrated by assuming k and c as constants. A number of procedures have been suggested for minimizing F; they include the method of steepest descent and modifications to the Taylor series approach [13, 15, 17-19]. For a well—designed experiment and with an estimate in error less than i” 30% of the correct properties, the Taylor series approach permits very rapid convergence. The Taylor series approach in its simplest form has proven to be adequate in our work; it is out- lined below. The calculated temperature is a nonlinear function of k and c. The Taylor series method is an iterative procedure, however, which assumes at each step that the temperature is a linear function of k and c, or T(k,c) z T ko'C0+Tk Ak + TC Ac (1.3.3) For convenience the subscriptj and functional dependence upon 9 have —8- been omitted. The derivatives T and Tc are defined by k 3T(k,c) Tk : 3-11—— (1.3.4:) c ,k 0 o 8T(k,c) T : - (1.3.5) C 8c c ,k 0 o and Ak and Ac are given by Ak = k-kO (1.3.6) Ac : c - c0 (1.3.7) The properties kO and c0 are the zeroth estimate of k and c. At the point (k* ,c*) at which F is a minimum the first derivatives of F with respect to k and to c are equal to zero. These two equations can then be solved simultaneously for the first correction to the proper- ties k and c. These corrections are given by (see Appendix A) em 1 n Ak = Z [(23 AJ. [Tc’jmj-TeJme )(Tk,TC) Fl 0 e n m - . T .T.-T .d NT ; 0 1. .8 (1:31:43) k’J(J e,J)6)( CHAaf: (3) O 9m 1 n Ac: X [.(2A 3‘ Tk (T.-Te’.)de)(Tk,TC) J:1 J :J J 0 e n m — . T .T.-T .d NT ; o 1.. (3,221.35 m” e...) an k>la=1= < 3 9) 0 where p .L... ‘« -9- 2 A .. (NTk) (NTC) - (Tk,TC) (1.3.10) 0 m n 2 NTk :2 A, f (de) d0 (1.3.11) F1 0 e m n 2 NT =ZA. f (T .) d0 (1.3.12) C ._ J C,J 1—1 O 0 m n (Tk,TC) 3:21.41). 3‘ (Tk,jTC’J.)d9 (1.3.13) 0 The integrals NT and NTC are norms of TR and TC and (Tk’Tc) is an k inner product. In the simplest iterative procedure improved values of k and c are given by k 1 k0 + Ak (1.3.14) c cO+Ac (1.3.15) 1 after which another smaller pair of values Ak and Ac are obtained. The iteration procedure is continued until, say, A—kk. < 0.0001 (1.3.16) éci < 0.0001 (1.3.17) For a well-designed experiment and with the initial estimate of k and c about 10 % in error, only three iterations are usually necessary to satisfy (1. 3. 16) and 1. 3. 17). On the next iteration after satisfying these relations Ak/k and Ac/c are much smaller -- about 0. 00001. In any new experiment F- values should be calculated surrounding the point which is thought to be minimum to verify that it is actually the local minimum. Cases for which a local minimum is not the true “4 ,.- I .. o .. .-- “ r_.. .4 ‘.-l. ' 40". v- " c ‘_ . .4- ' , u . .u‘t‘ .- o.‘ -.-‘— I- I .1 o . .0... . ’~-~ ’."u.-.. -10- minimum have not been found for well-designed experiments, however. If temperature measurements Te, . are not made continuously with time but at discrete times (which is the most common case), then the integrals in (1. 3. 2) and subsequent equations would be replaced by a summation over time. For example, (1. 3.11) would become n m 2 NT 2 Z A. Z T . (1.3.18) k i=1 3 1:1 k'3 where the I superscript is for time [A0 for I : 1, 2, . . . m. A tendency of some investigators who measure thermal pro— perties is to use only part of the data [6, 11]. The nonlinear estima- tion method by using all the transient temperature data reduces the errors in the calculated property if the errors in the data are random which is a common case. (If the errors are not random, then corrections for the biased errors can be made.) Others, like Hsu, use all the data by repeatedly applying a simple equation to calculate a number of values for the properties which are then averaged. (See section 6. 3) It can be proved, however, that the ”best" estimates of the parameters are given by the least-squares procedure incorporated in nonlinear estimation [20]. »»-. ..v.. -»~~- u» . I- '- "wuuuu .~ ._ .-. ‘- 3:3.1 p . 0‘ v», p G 5.. x II. GENERAL RELATIONS BETWEEN TEMPERATURE RISE AND PROPERTY DERIVATIVES 2. 1 Relations for Properties k and c Much helpful information can be obtained from general rela- tions between the temperature, T , and the derivatives Tk=%—EC;TC=g—Zk (2.1.1) This information is useful to aid in determining the optimum boundary conditions, to help check accuracy of numerical calculations and to provide certain insights into the property-determination problem. We will restrict our attention to a one—dimens ional, Cartesian system and to constantthermal properties k and c. (Temperature—variable thermal properties are considered in chapter VII.) Let us first consider the transient heat-conduction equation 8 T _ 8T k—-2 _ C8_0 (2.1.2) 3x with the temperature boundary conditions, T(o,0) = TO(9) (2.1.3) T(L,6) = TL(6) (2.1.4) and the initial condition, T(x, 0) = Ti(x) (2.1.5) The temperatures TO(0), TL(0) and Ti(x) are known. -11.. uq. -12- In order to simplify the notation, the operator D is defined as 2 _ 8 8 D : k 7 " Cé-e- (2.1. 6) 8x and then (2. 1. 2) can be written DT : 0 (2.1.7) Equations for the calculation of TC are obtained by taking the partial derivative of eqs. (2.1. 2) to (2. 1. 5) with respect to c holding k constant; we obtain DT = -- (2.1.8) TC(0,0) : TC(L,0) : TC(x,O) : 0 (2.1.9) In a similar manner taking the partial derivative with respect to k yields BZT 1 0T DTk Z --——-2- Z -a- 56 (2.1.10) 8x Tk(0,0) = Tk(L,0) : Tk(x,0) : 0 (2.1.11) By comparing eqs. (2. 1. 8) and (2.1. 9) with (2.1.10) and (2.1.11) after multiplying the latter two by -a, we observe that TC : -0. TR 01‘ cTC + ka = (1 (2.1.1m If the insulation boundary condition, viz., 8T '83? 0 were to replace one or both of the temperature boundary conditions (2.1. 3) or (2.1. 4), relation (2.1. 12) would still be obtained. Further- -13- more, this relation can be shown to be valid also for these boundary conditions if the body is semi-infinite. Other very important boundary conditions are in terms of prescribed heat fluxes at the boundaries, viz. , _k____g:(0»9) _ qo(e) (2.1.13) «W = qL(0) (2.1.14) The initial temperature distribution considered here is simply T(x, 0) : Ti = constant (2.1.15) rather than the more complicated boundary condition (2. 1. 5). Taking the partial derivative with respect to c of (2. 1. 2), (2. 1.13), (2.1.14) and (2.1.15) gives 8T D(-cTC) : -ca—e— (2.1.16) 8(-cTC(0.9)) a(-cTC(L,e)) 3x = 8x = -cTC(x,0) : 0 (2.1.17) In a similar manner taking the partial derivative with respect to k yields _ 8T D(-ka) — C-a—é' (2.1.18) 8Tk(0,0) 8T(0,0) -k————— - ____= 0 8x 8x or using (2. l. 13), k 8Tk(0.9) C10(9) 8x — k and multiplying by -k gives 8(-ka(0,0 )) -kax =q0(6) (2.1.19) . 9..- .'k. ‘i. | o . c“ .5 ‘1‘ - -14- and similarly -k :q (0) (2.1.20) 8k L The initial condition can be written -ka(x,0) = 0 (2.1.21) We note that the. partial differential equations (2. 1. 16) and (2. 1. l 8) contain the same (except for sign) non-homogeneous term which can be evaluated from the solution of (2.1. 2), (2.1.13), (2.1. 14) and (2. 1. 15). In a standard method of solution of linear, non-homo- genous partial differential equations [21] the general problem is split into three distinct problems and then the results added to obtain the desired solution. Since these equations are linear, superposition is valid. Then the temperature rise T-Ti with heat flux boundary condi- tions can be shown to be related to cTC and ka by (2.1.22) T-T. : -cT -kT 1 c k This result can be obtained readily from Table 2. 1. 1 since the sum of the non-homogeneous terms for -cTC and -ka gives the partial differential equation and boundary conditions for T - Ti' If the body is insulated at one boundary only or if the body is semi-infinite (L-' co), (2. 1. 22) still applies. It is not valid for the heat flux given on one boundary and a prescribed temperature history on the other, however. (Incidentally, (2. 1. 22) applies if T varies with the radius in the cylin- drical or spherical coordinate systems.) It is proved in section 2. 3 that the relation derived for the temperature boundary condition, (2. l. 12), indicates that the proper- ties k and c can not be separately determined for this boundary condition. Table 2. 1. 1 Partial differential equations and boundary conditions for T-Ti, -cTC and —kT k Function P.D.E. Boundary Conditions 8(T (0,0) -Ti) T - Ti D(T-Ti) : 0 —k 8x : qO (0) (Ti = constant) 8(T (L,0) -Ti) -k = q (9) 8x L T(X,O) -T1 : O 8(-cT (0,9) ) 8T c -cTC D(-cTC)_-c5§ 8x _ 0 a(—CT (L29) ) C — 0 8x '— -cTC(x,O) : 0 8(-kT (0,0) 8T k -ka D(-ka) — C 8—0- 'k a X — q0(0) a<-ka(L,e) 1 'k a X : (114(9) -ka(x,O) : 0 Note: 2 _ 8 8 D — k 2 - C 8-? 8x -15- . . .. .fi - —. ‘ 1 1- .1... 0 ok . )— ’ ‘ :-. ...___ ‘i 5Q h 1., c,‘.-‘ "7..., n a 1...“, a: r .- 1.. “1». -\ :--. ‘ 9;. . ~. .,, N. . ._‘ . I’P _ I a" 3.,4 ‘. .5- -16- For the heat flux boundary the important relation (2. 1. 22) applies; this relation has a number of uses as will be suggested later. 2. 2 Surface Temperature of a Semi-Infinite Body with Heat Flux Boundary Condition The surface temperature rise of a semi-infinite body which is heated with a constant heat flux is (see section 4.1) NIH T «Ti : 2qo(0 /kc1r) (2. 2. 1) where Ti is the uniform initial temperature and qo is the constant heat flux beginning at time 0 : 0 (for 0< 0, qO : 0). For this case RT and cTC are equal and are given by k 1 _. _ 2 ka — q0(0 /kc1r) (2. 2. 2) 1 _ _ 2 cTC — qo(8 /kc11) (2.2.3) and thus ka : cTC (for x : 0 only) (2. 2.4) or ka - cTC : 0 From (2.2.1), (2. 2. 2) and (2. 2. 3) we find — : - : - i :2 . . l T Ti 2ka 2cTC (for x 0) (2 2 5, which is consistent with the relation T-Ti = -ka-cTC (2.2.6) which was derived for a homogeneous, constant property, finite or semi-finite body with heat flux boundary conditions. Eqs. (2. 2. 4) and (2. 2. 5) are given specifically for the special case of the surface temperature of a semi-infinite body with q = C. -17- We now demonstrate the validity of (2. 2. 4) for an arbitrary heat flux. The superposition integral [22] for a given heat flux at a boundary and uniform initial temperature can be given by 0 T(x, 0) :Mq BMX 9 M81). (2.2.7) 0 where q(0) is a known, but arbitrary time-variable heat flux and A is the temperature response for a unit constant heat flux. Let us consider only the heated surface, x : 0, for which A can be calculated from (2. 2.1), A = (1/kcen)% (2.2.8) Then (2.2.7) becomes 8 T(0,0)-Ti :()1rkc35q().)k)(0-).)zd>. (2.2.9) o By taking the derivative of (2. 2. 9) with respect to k and c it can be shown that (2. 2. 4) is valid at the surface of a semi-infinite body initially at a uniform temperature and exposed to an arbitrary time- variable heat flux. Since (2. 2. 6) is true for still more general condi- tions, (2. 2.5) also is true. Another position for which (2. 2. 4) applies is for the interface between a finite body of known thermal properties and a semi-infinite body of unknown thermal properties. At the free face of the finite body the surface temperature can be given or the heat flux can be given. The initial temperature distribution is uniform. This case is equivalent to prescribing the heat flux at the interface between the materials. -18- The problem of calculating the heat flux at a surface or inter- face when the transient temperature history near the heated surface is prescribed has been considered in some detail by the author [23]. 2. 3 Cases for Which k and c Can Not Be Independently Determined The thermal conductivity k and specific heat c can not be inde- pendently determined when A given by (1. 3.10) is equal to zero. In these cases it may be possible to determine only one property or per- haps just the ratio or product of the properties. It is well-known that A is equal to zero if and only if the functions T and TC are linearly k dependent [24]. Two functions are said to be linearly dependent if and only if the relation aT +bT : 0 (2.3.1) is satisfied and one of the constants (a or b) does not equal zero. Several boundary conditions have been found for a homogeneous body with temperature-independent properties for which (2. 3. 1) applies. These cases are listed in Table 2. 3.1. One would anticipate diffi- culty in finding k and c for any boundary conditions which approach those listed. For example, the given surface temperature case (case 1) is approached by the convective boundary condition when hL/k is large compared to unity. (The quantity h is the heat transfer coefficient and L is the thickness of the specimen.) At any boundary at which the temperature is prescribed, we have T = T = 0 (2.3.2) for a finite or semi-infinite body. This relation also applies if the -19- body is homogeneous or composite. Hence any temperature measure- ment near a temperature boundary condition yields very little informa- tion about the thermal properties. Table 2. 3.1 determined. Cases for which k and c can not be independently Case 1 Location of Measured Temp. Histories Description of Body and Boundary Conditions Initial T emp . Distribution m 1 Any no. of measurements and at any loca- tion. Any no. of measurements and at any loca- tion. Heated surface only Heated surface only Interface between bodies Finite or semi-infinite l-D body with a) given, arbi- trary time-variable T at the boundaries or b) insu- lated boundaries . See (2.1.12). Arbitrary Finite 1-D body with given, [ Arbitrary arbitrary time-variable T at one boundary and insu- lated at the other boundary. See (2.1.12). Semi-infinite l-D body with given, arbitrary time- variable heat flux at surface. See (2. 2. 4), (This also applies approxi— mately for finite body for T < 0.2. At the unheated end, the boundary condi- tion can be either BT/ax = 0 or T : Semi-finite l-D body with the convective boundary condition, q : h(Tee-T(0, Q) where h and T0,, are constants. See (2. 2. 4), Composite of finite length of material A and semi- infinite length of material B. Material A has the one free surface at which a time- variable T (or q) is given. Material A has known k and c while those of Material B are to be determined. See (2. 2. 4). const.) Uniform Uniform Uniform -20- "‘-v. - I ....51 gV-A’ "r. ...5.. r L. n, ‘ -. ' Q‘w‘,‘ u\_‘.. -. u. ‘3. ‘- ‘ "‘- R- “C . ‘ "war-l, xr. Iv. ;.. ‘ s ‘~“;FF-I~ V‘.‘u->‘ ~. I. ~) I . u . Il~ \" P. u III. BASIC CRITERIA 3. 1 Criterion for Determining Optimum Experiments In order to efficiently determine the Optimum experiment for determining the thermal properties a single criterion is needed. This criterion, if it exists, would enable us to select the boundary conditions for which a random distribution of errors in the temperature measure- ments would cause the least inaccuracies in k and c. Box [14] found a single criterion for a closely related problem. He considered, for his first example, the case of a transient, chemical reaction governed by two ordinary, first-order differential equations. Hence he had only an initial condition to consider for each variable. Instead of seeking to determine the optimum initial condition, however, he sought to find the Optimum two times at which to make measurements for calculating his two parameters. Since the problem at hand is not identical to Box's, a criterion is derived in this section. The following analysis is subject to these conditions: 1) The values of the thermal conductivity k and density- specific heat product c at the minimum of F(k, c) are known and are designated k* and c* . 2) The minimum value of F(k, c) need not be zero; however, the errors in the temperature measurements are assumed to be small. 3) While applying the criterion we fix the (a) number of tempera- ture histories, (b) maximum temperature rise and (c) maximum duration -21- ax. .b‘. -,-u a- _. -22- of the experiment. 4) The sum of squares function F(k, c) is examined in the local region near F(k* , c*). We do not explicitly require that the criterion select an experi- ment for which k and c are of equal accuracy; we shall find, however, that for a particular optimum experiment the properties can be deter- mined to the same accuracy. The sum of squares function is 0 m n F(k,c) = z A. §(T.(8)—T .(9))Zd9 (3.1.1) jzl J O J 62.] and near the minimum a Taylor series expansion gives 2 2 8F* 8F* 1 8 E* 2 18 F* 2 * _ * _ __ — — F(k +Ak,c*+Ac)~F +8] Ak+ac Ac-l-a——2-—a (Ak) +2—T(AC) >1. +——CAk AC (3.1.2) where the starred terms are evaluated at (k* ,c*). At the minimum of F , 8F* 8F* a—k— 2 8C 2 0 (3.1.3) The higher derivatives of F are found from (3.1.1) to be 2 2 3k 81;“ = 22 A. S(T*. - T )3—g—de + 22A. [@13sz 3k 1 1 e.1 8k J (3.1.4) 32F* 82T* Z -_- 2 EA. (T*. - T .)_—Z—de + 22A. (T*) 88 8c2 J J 6"] 8c J C (3.1.5) -23- 2 5 3 I : * - __. a): :1: 22A). (T j Te,j)al 8C d9 '1' ZZAJ 5T 1T Cde (3.1.6) 82F* 8k8c For the properties k* and c* when T.* : Te j (zero error in the T- 3 measurements) the first term on the right hand side in each of the preceeding three equations is equal to zero. The second terms in (3.1.4) and (3. 1. 5) are not equal to zero unless T or TC is equal k to zero (which does not occur in any case in which k and c can be found). For sufficiently small values of Tj - Te . (condition 2) the first terms on the right hand side of (3. 1. 4), (3.1. 5) and (3.1. 6) are negligible and hence the sum of the squares function is approximated by F(k*+Ak *+A )—F*+(NT*)(Ak)2+(NT*)(AC)2+ 2(T* T*)(-’33‘—)(£) ’C C_ k k—* c —c_’l_‘ k ’ c k* c* (3.1mm where 9 m n 2 NT* 2 E.A.S‘W*T*)de (31.& k . j k i=1 0 am n 2 NT*C = z A. Sung) d0 (3.1.9) i=1 J O 0 m n * * -— * * * * (Tk ,TC ) _ 3A]. 5 c Tcdk Tk,j)d8 (3.1.10) 0 We now desire to find a new coordinate system for the F- contour. This new coordinate system should be one in which the length and angle are preserved. This is readily done using the standard pro- cedures for treating quadratic forms. In the procedure we solve for -g4- the values of A in the determinantal equation * - 3k * WTpx ukxc) (T a11;,TC) (NT *C)->. to obtain for the two roots T 2 2 : * * * - * 3k * Al [NTk +NTC + x/(NTk NTC ) +4(Tk ,TC ) ]/2 (3.1.12) 2 . 2 : * _ 2k _ >1: * 12 [NTk + NTC* .flNTk NTC ) + 4(Tk*,TC ) ]/2 (3.1.13) Clearly ). 1 and A 2 are real and thus the F-contour in the region of k* andc’l‘ is an ellipse and is given by 2 2 F(pl,pZ)-F* =k1p1+k2p2 (3.1.14) where (NT *4. )(Ak/k*) - (T *,'I‘ *)(Ac/c*) pl: C l k C (3.1.15 2. Z J -(T *,T *)(Ak/k*) + (NT *—>. )(Ac/c*) p2 = k C k 2 (3.1.16) 2 2 NT *4 T‘* T * J( k 2) +( k ’ c ) The angle B between the p1 and k axes is 0 = tan-1 [(NTC*-).l) / (T *,TC*)] (3.1.17) k The angle B for four interesting cases is shown by Figures 3.1.1 to 3.1.4. If B is near zero degrees, Figure 3.1.1, then c can be determined more accurately on the average than k. For B = If 90° (Figure 3.1. 2), k is determined more accurately. For B = 45° and -25- -45°, the values k/c : c1 and kc are respectively determined more accurately. The length of the major axis of the ellipse is [(F-F*) /). 21% and of the minor axis is [(F -F*)/). 1]; Thus the ratio R of the major axis divided by the minor axis is 1 _ 3 R—(kl/kz) (3.1.18) which is always greater than unity. The area of the ellipse Acr is given by 1 : — * -E A..- "(F F 10,121 (3.1.191 If the errors in the measured temperatures Te,’ are stochastically independent, then the area Acr is that of the confidence region when F-F* is calculated using standard statistical procedures. Unfor— tunately, the errors are probably not independent in most cases. See references 25 and 26. Let us, however, consider two different experi- ments with the same (a) material, (b) number of temperature measure- ments, (c) maximum temperature rise, (d) duration of the experiment and (e) distribution of errors in the temperature measurements. For these two experiments the area Acr is smaller for fixed F — F* ( from (e) ) for the experiment having the largest value of 1. 1. = (NTk*)(NTC*) - (T *,TC*)2 = A(k*,c*) 2 1 . k (3. 1. 20) We have noted in section 2. 3 that the properties k and c can not be determined separately if A : 0. The criterion A given by (3. 1. 20) must be modified to include the constraints of maximum temperature rise Tm-Ti and maximum duration of the experiment, 0 m' The opti- mum dimensionless time is given by -..'.' . C ~shy an. on. .. l b 5". sss~ 1A \.~ -Z6- Dete rmine s c more ac cu- rately “C-C* /\ K/ Fig. 3.1.1 B=0° ¢ k_k* F -contour for Determines k :c-c* more accu- rately \ (3 = :1 90° B=i90° \ Fig. 3.1. 2 F-contour for trk_k* Determines a. c-c* Dete rmines kc c-c* more accu- more accu- ( 1 rately l) 0/7 91:3; Fig. 3. 1. 3 F-contour for B = 45° rately W \V. = Fig. 3. 1. 4 F-contour for [3 = -45° gt >z' Fig. 3. 4.1 Heat flux to pro- duce constant surface temp. T1) £31- 7.. (“I , A'z—u) L- l I 1 7‘ 17. :2 1 Fig. 3. 4. 2 Surface tempera- ture history. _11 A >‘E Fig. 3.4. 3 Criterion A versus T O (1 an; 1 or $117], or SCT _, c ”-1: . I 1 t 1 A't—u: 1 I (I. A V v Fig. 3.4.4 Derivatives ka 8: cTcl at x: Ofor 'Rn-Ti = cons ant. -37- T : 0* 0 /L2; 0* Z ka/IC* (3~ 1'21) m m for fixed values of 0m and (1* -, thus determining the optimum value of T m is equivalent to finding the optimum specimen thickness L. For any prescribed boundary conditions and given specimen (for T— 95 independent thermal properties) the derivatives TR and T * can c both be shown to be directly proportional to the temperature rise T-Ti for the same time and position. Due to this linear behavior of Tk* , TC* , and T-Ti, the criterion A can be made independent of the ' __ ' >:< >’,< >5: _ magnitude of T Ti by replacmg T and TC by Tk /(Tm Ti) and k T * /(T -T.). The maximum time 0 can be introduced simply in c m 1 m the following manner. The first term on the right hand side of (3. 1. 20) can be written 0 T m m S 2 L2 ,, *2 * ._ 4 4 _ _ NTk _ ZAJ. (kaj)d9—a~gg-ZAJ‘S(kaj)dT O 0 (3.1.22) Dividing (3.1.22) now by the fixed maximum real time 0 m causes the right side of (3. 1. 22) to be a function of the maximum dimensionless time T m (for a given heating condition). The other integrals are treated in the same manner. Hence the optimum experiment is one for which 1: e (NTk) (NTC) - (Tkx‘rc)Z (3.1.23) is a maximum for fixed Tm-Ti, 0m, number of temperature measure- ments and thermal properties. The terms on the right hand side of (3. l. 23) are defined by o \ o - 5.— ‘mv -~-v c4... ., . v.-. “---'.c. In‘ (P O T 2 _ n m(k*Tk .*) dT N k = z A. S ”J, (3.1.24) 5:1 3 (T -T.)" r O m 1 m Trn 2 n (c’l‘TC .*) dT NTC e Z A. 5 'J 2 T (3.1.25) 3'21 3 (T -T.) m 0 m 1 T _ _ n .. m(k*T {1‘1vaC J.*) d. (Tk.TC) = Z A. - ’ 2 ' (3.1.26) j=1 3' (T - T.) T O m 1 m The important criterion A is made a maximum by varying both the boundary conditions and dimensionless duration of the experiment; it is examined for both finite and semi-infinite bodies for a variety of boundary conditions in chapters IV and V. Another quantity of interest for a given experiment is the correlation coefficient p NIH p = (Tk.TC)/1NTkNTC) (3. 1. 27) The correlation coefficient p can have values equal to and between -1 and +1. For p equal to either -1 or +1, A is equal to zero and the properties k and c can not both be determined. It does not follow that A is a maximum when p : 0, however. 3. 2 Error Analysis It is important to investigate the effect of errors upon the cal- culated thermal properties. There are three major sources of errors. First, there are experimental inaccuracies in measuring the tempera- tures. Next, in calculating the temperatures in the body an approxi— mate finite-difference method is utilized. This second source of error -29- as shown in section 6. 2 can be made as small as desired (0. 1% is readily obtained). Finally, the model used to describe the tempera— tures in the body can be imperfect. For example, the thermal pro- perties actually vary with temperature but the thermal properties may be found as temperature-independent thermal properties. This error can be made, in general, as small as desired by reducing the temperature range. The main source of error is consequently the experimental error in the temperature measurements. Let us then investigate the errors in k and c due to some conti- nuous distribution of small errors OJ. in the measurements, 0j(0) :2 Te,j(e) - Te""j(9) (3.2.1) where Te .(0) is an experimentally measured temperature distribu- ’ tion which contains errors and Te .* is the "true" temperature. Using J (3. 2.1) in the sum of squares function F(k, c) given by (3.1.1) yields 9 n F(k,c) : Z A. 5 (T.(9) - T .*(9) - 0(8)) d9 (3.2. 2) :1 O The temperature Tj(0) is the calculated temperature for the assumed model which need not be perfect. If 01(0) 2 0, then the correct pro— perties k* and c* are obtained by differentiating F(k,c) with respect to k and c and setting both expressions equal to zero, 8T.* aF(k.c1_ ) ,_ ,, J _ ——ak .. 22 Aj. (Tj Te’j ) 5k d9 - 0 (3.2.3) 8T3i< 8F(k,c) _ Y _ J _ ————————8C _ 22 AJ. . (Tj* T651) ac d8 _ 0 (3.2.4) where Tj* is the calculated temperature for (SJ. : 0. -30- The quantities Ake and Ace are the errors introduced due to the 0's, .1 Ak e AC e k-k* : C—c* (3.2.5) (3. 2.6) Using the standard approximation of linearity of Tj for small 6. ives J8 T.z .1 where typically Tk j* : T.*+T .*Ak +T .*AC J k9.) e C, 8 8T. .1 8T. k*,C* (3. 2.7) Introducing (3. 2. 7) in (3. 2. 2) and differentiating with respect to k and c, setting the resulting expressions equal to zero and utilizing (3.2. 3) and (3.2. 4) yields 2:, Algal“... Ake + TC J 2 AjS(Tk,j* Ake + TC .*AC ,1 e 1' mi (3.2.8) (3.2.9) Since the 5j are small, iteration for Ake and Ace is not necessary. Solving for Ake and Ace and rearranging slightly gives Ake _ (NTc) Ik - (Tk’Tc) IC k* 5 Ace _ (NTk) IC - (Tk’Tc) 1k c* A T n mk*Tk * 5 d. where 1k: 2) A g ’ _ (3.2.10) (3.2.11) (3.2.12) -31- I T n mc*TC .* 0. dT z A. 5 "3 23 (3.2.13) :1 J (T -T O m 1 C —j .) T and the other terms are defined by (3.1.24) to (3.1. 26). Though it is not obvious from (3. 2.10) and (3.2.11) that Ak and Ac are reduced if X is maximized, this fact is demonstrated in chapters IV and V. We also note that the errors Ake and Ace are directly proportional (for small errors) to the error distribution, 5).. Generally the derivatives Tk and TC are continuous, slowly-varying functions and thus the inte- grals in the above equations involving 63‘ are larger for biased one- sided errors than for random (or even sinusoidal) errors which are both positive and negative. The biased, consistent errors, however, can be usually computed and then the measured temperatures corrected. The author has analyzed some problems with biased errors for thermo- couples embedded in solids [26, 27]. Equations for Ake and Ace for discrete measurements in time can be obtained from (3. 2.10) and (3.2.11) simply by replacing the time integral by a summation over time and the dT by AT . From the linearity of (3. 2.10) and (3. 2.11) with respect to 53., the errors in k and c for two different error distributions can be added to obtain the error due to both sources. More generally we can ex— amine the errors Ake and Ace due to a §i_n_g_l§ error (SJ. at a time T . In order to compare the effect of a single error for different experi- ments, Ak and Ac are normalized by multiplying by (T )Tm/(éjAT) -T_ m 1 to obtain _ lek *(T) _ _ c*TC’15(T) (NT 1—4-—- ‘(T . ) ’J Ak(T,x.)/T -T. T c T -T. k c T -T - _ e j/ m 1 m _ m 1 m Ak.— _ _ J k* \ (SJ. AT A (3.2.14) c’i‘T .*(-r) k*T .*(T) (NT) C” —(T .T1 1‘” Ac (T,x.) T -T T \ k T - . k c T -T. e j m 1 m _ m 1 m 1 AC.: _ _ J C* 0]. AT/ A (3.1.15) The errors All.j and ch are caused by a single error in the tempera- ture, 6j’ at time T. By plotting Akj and ch versus time T and positions xj, the information is given for determining the error in k and c for any distribution of small errors for the experiment con- sidered. For a number of different boundary conditions, ARj and ch are given usually for the Tm - value of the maximum A for each case and for two values of Xj’ usually 0 and L. 3. 3 Errors in Properties for Certain Cases A biased error instructive to consider is an error in each temperature rise which is proportional to the temperature rise, or 61' = €(Tj-Ti) (3.3.1) where E is a constant much less than unity. Now for a finite or a semi-infinite body with specified heat flux boundary conditions we have T.—T. e -cT .-kT . (3.3.2) where T1 is the uniform initial temperature. The terms. involving 53. in (3. 2.10) and (3. 2. 11) can then be written 1k = -E(NTk + (Tk,TC)) (3.3.3) _33- and IC = - 13(NTC + (Tk,TC)) (3.3.4) then with these latter expressions (3. 2.10) and (3. 2.11) readily yield Ake/k* : -6 (3.3.5) Ace/c* -6 (3.3.6) Thus if all the temperature rises were measured a constant one per- cent too large, then both k and c would be calculated one percent too small; the thermal diffusivity c1 : k/c would not be in error, however. In the analysis of section 3. 2 the error was assumed to be only in Te,j and not in Tj. If the temperature rise Tj-Ti were calculated a constant fraction 6 too large due to the same fraction 6 error in the prescribed heat flux, then the error in the properties would be the same as given by a negative constant error in Te, .. Thus for a constant fraction error in q the errors in k and c are 6 (3.3.7) Ak /k* e Ace/c* (3.3.8) ll m or a constant one percent error in the heat flux curve causes a positive one percent error in k and c. Another error which introduces a bias is the result of in- correct measurement of the location of a thermocouple. Suppose a thermocouple junction is thought to be located at the surface heated with a known q but the junction is in reality located a small distance Ax inside. The temperature at x : Ax can be approximately given by 8T(0 ,0) T(0,x) mT(8,0) + M Ax (3.3.9) Hence, 6 for this thermocouple is given by “Ti‘ 0 = T -T *ng9,O)+g——:—‘—9—ig—)Ax - T(0,0) e e ox : WAX : —-(—l-(LE—’—g—)—Ax (3.3.10) Then the error in the surfacetemperature is proportional to q and Ax. At an insulated surface q : 0 and thus 5 : 0 at such a boundary. Evidently the thermocouples should be located much more precisely at heated surfaces than at insulated ones. 3.4 Determination of a Criterion for the Optimum Heat Flux Input The determination of the optimum heat flux input for finite or semi-infinite bodies is pertinent to the efficient calculation of thermal properties. Anticipating results given in chapters IV and V, the case of two thermocouples is examined; one is located at the heated surface and the other is in the interior or at the other surface. The heat flux which maximizes A produces a step rise in the heated surface tem- perature. For conciseness this heat flux alone is discussed. Though the problem of maximizing A is not directly amenable to solution using the classical methods of the calculus of variations, the method . of determining a necessary condition for A: to be a maximum is simi- lar in some respects to the proof of the necessary conditions of Weier- strass and Legendre given by Bliss [29]. For two temperature histories A is given by Z = [(NTkl1+(NTkZ1][NTC11+(NT 1]-[(Tk.'r1 + (Tkrr (3.4.1) whe re typically Tm ka dT - 1 1 m 1 m 0 and. T - - Tkl cTCl dT (Tk’Tc)1:§ T -T. T -T. T (3'4'3) O m 1 m 1 m / The subscripts l and 2 refer respectively to the measurements at x = 0 and x = L. The heat flux q, surface temperature Tmo’ cri- terion A and the derivatives kT and cTC are shown on page 26 by k Figures 3.4.1 to 3.4.4. The vallue of A flor the constant surface temperature is designated 50' The effect upon A for a small step increase in the surface temperature, 6T, 1S now examined. It has a short duration AT and is shown by Figure 3. 4. 2; (ST is caused by an appropriate éq. The cases that we shall consider are those for which at x = 0 (T - T.) (3.4.4) Due to the linearity of kT cTC and T, superposition is valid k, (see Table 2.1.1) and thus we have 0(ka) = 0(CTC) :: — l l 0Tl (3.4.5) The terms with the "1" subscript in (3. 4.1) can be written for small values of OT as, (T -T.1‘Z ] m0 1 1 1 k1 (T -T.)2 m 1 (NT )2 [(NTk )O+2E k (3.4.6) ' ' (Tmo-Ti)2 (NT ) ,4, [(NT ) + 2E ] (3.4.7) C1 C10 C1 (T -T.)2 m 1 - - [ - - ](Tmo-Ti)2 T ,T A ,T + E 3.4.8 ( k c)1 ( k c)1’0 k,cl (T -T )2 ( ) where T m (Mk )0 65(ka ) dT 1 1 Ek : 2 : 1 (T -T) T T mo 1 m 0 Tm[__1_(T -T)][—30T]d.— 2 mo 1 2 _i 5T A T (T -T.)2 T 4 (TmO-T) T m To 0 1 m (3.4.9) The value of (NTk) With 6T : 0 and Tm : Tmo is de51gnated (NTkl)0. In deriving (3.4. 9), equations (3. 4. 4) and (3. 4. 5) are used and it is noted that CT has a non-zero value only between T o and T o + A T . Similar expressions can be found for Ec and Ek c ; these are l ’ 1 related to E by k1 E = E (3.4.10) Ek’C : 21:k (3.4.11) If instead of a single increase in the surface temperature, 6T, a finite number of increases (or decreases) in the surface tempera- ture is considered, Ek is readily modified. Here we have a dis- 1 tribution of n values 6Ti each of which represents an increase in the Surface temperature for the time duration AT i' The only res- triction is that GT. is small compared to T -T.. Then E becomes 1 mo 1 k1 (3.4.12) and (3.4. 10) and (3. 4.11) also apply. The effect of 5T upon the integrals in (3. 4. 1) with the “2" subscript is not as readily given and depends upon the boundary condi— tions at the location of thermocouple 2. The values Ek , and Ec , 2 2 and Ek c are evaluated for the particular cases examined. ’ 2 Let us return to the consideration of a single (ST. The tem- perature rise for negative 5T remains TmO-Ti; but for a positive 6T we can use the approximation (Tm-T)Z A (T -T.)‘2 [1_T_Z_QI__T-] (3.4.13) 1 A - A0 A, 4[Bl(ro) - O T -T (3.4.14) mo 1 and for a negative OT (and éq) A - AO z 4Bl(1-O) (3.4.15) where = - \ - — - - . . 8,601 [Ek(NTC. + EC Ek.C( k.TC11/2 (3 4 161 and typically Ek : Ek +Ek (To) (3.4.17) 1 2 Ek,c = Ek,c +Ek,c (To) (3.4.18) 1 2 NTk = (NTkl)0 + (NTkZ)O (3.4.19) The integrals (NTk), (NTC) and (T k,TC) are constants for a given experiment duration Tm. The value Ek is independent of T0 as 1 -38- shown by (3. 4. 9); however, E , E and E are functions of k2 cZ k, c2 To. It is shown by (3. 4. 9) that Ek is directly proportional to 5T 1 and AT and this proportionality can be also demonstrated for (To)° Then B (T ) can be written E (T),E (T)andE \ kZ 0 c2 0 k,c2 l 0 0T AT Bl(TO) : T__—-T_ (T ) B(TO) (3.4. 2.0) mo 1 m and hence (3.4.14) is given by - — 0T AT — A — AO % 4 'T—-"T.‘ T B(.O) —- AO (3.4.21) mo 1 m which is valid only for positive 6T. Note that B(TO) is independent of 6T and AT . The criterion B can aid in the determination of the optimum heat flux. We shall show in two cases of interest that B(TO) is posi- tive for any value of To. If 6T is negative, then from (3. 4. 20) and (3. 4.15) a small 5T at any time during the experiment makes A less than AD. For positive 5T, (3.4.21) is used. From the Schwarz inequality Z0 must be equal to or greater than zero [24]; for an opti— mum experiment it can not be equal to zero. Usually B(TO) is about as large as 50; regardless of its value, however, for a sufficiently small AT the right hand side of (3. 2. 21) is negative. Thus if B(TO) can be shown to be positive, a value of 5T either positive or nega- tive makes A less than A0. Hence the optimum heat flux boundary condition appears to be one which produces a step rise in the tem- perature of the heated surface ( provided B(TO) is positive ). IV. SEMI-INFINITE BODIES 4. 1 Semi-Infinite Body with Constant Heat Flux The temperature at a point x in a semi-infinite body which is subjected to a constant heat flux qO is [30] _1- _l T-Ti : 2 q()(e/kc)a ierfc(O. 5TX 2) (4.1.1) where TX = C10/x2 : kG/cx2 (4.1.2) Taking the partial derivative of (4.1.1) with respect to k and c gives L 3 1 _ 8T _ qox T x 1 ka _ k ("61‘1'7‘ - .1. exp 64% > + erfc l 2 2 n ZTX (4.1. 3) 1 2 __ 3T __ qOX Tx 1 cTC — C(8c) — — k 1 exp — 7T?— (4.1.4) k "E X Using the identity [30] . -1. 2 ierfc(z) : 17 2exp(-z ) - z erfc(z) (4.1.5) it can be verified that T-Ti = -ka -cTC as derived in chapter II for general q boundary conditions. It is instructive to evaluate (4.1.1), (4.1. 3) and (4. 1. 4) at the heated surface, -39- -40- l T(0,0) -Ti : 2qO(E)/kca)7)2 (4.1.6) 1 ka(0,0) : -q0(0 /cha (4.1.7) 1 cTC(0,6) : -q0(0 /cha (4.1.8) and we note as derived in chapter 11 ka(0,0) = cTC(0,0) : -(T(0,0) - Ti)‘/2 (4.1.9) We readily find at x : O T m NT 2 u/T ) RT HT -T))2dT ::l/8 (41 MD k m k m i x ' ' o NTk = 1/8 (4.1.1n T m - - 2 (Tk,TC) = (lflnn) S (k k oTc/(Tnnfifp ) dTX _ 1/8 0 (4.1.12) Then the criterion A for x : 0, A = (NT )(NT ) - (T T )C = (1/8)(l/8\ - (1/8)‘2 = 0 c k k’ c ’ ' (4.1.13) is equal to zero. With A equal to zero, it is impossible to determine both pro- perties k and c. The surface temperature history can be used, however, to calculate the product kc. Results for the various quantities of interest have been cal- culated and are tabulated for one interior measurement at x : x in Table 4.1.1. The dimensionless derivatives _ ka _ cTC : ' 2 4 T1. W 3 Tc W (~ “41 Table 4.1.1 1. X ~J|\I.&HHNNNMHHPP \fl 0 18.00 20.00 3u,nn 40.00 51,00 80.00 70,00 90,00 9'1.00 11)II,THI T(0) .2523 .3568 .4370 .5046 .5642 .5180 .6676 .7135 .7569 .7979 ,9740 .9441 1.0003 1.0705 1.1284 1.2616 1.1820 1.4927 1.595“ 1.6926 1.7841 1.8712 1.9544 2.1110 2.?569 2.3937 2.5231 2.6463 2.7640 2.8765 2.9854 1.0902 1.1915 3.2898 3.3651 3.4779 1.5692 3.9068 4.2220 4.5135 4.7873 5.€463 4.1404 7.1365 7.9788 8.7404 9.4407 10.0925 10.7047 11.2838 Quantities for q = C With thermocouple at x TC -.0009 -.0145 -.0413 -.0723 -.1038 -.1343 -.1634 -.1910 -.2171 -.2420 -.2861 -.1303 -.3692 -.4054 -.4394 -.5164 -.5649 -.6470 -,7041 -.7573 -.8072 o.4543 -.6991 -.9827 -1.0600 -1.1321 -1.2000 -1.2643 -1.3256 -1.3841 -1.4403 -1,4944 -1.5467 -1.5972 -1,6462 -1.6938 -1.7401 -1.9141 02.0736 -2.2218 -203606 -2.4916 -3,0645 -3.5490 -z,0495 —1.5520 .4,7035 ~5,010% -5.3375 ‘5.5278 Tk .0007 .0107 .0266 .0416 .0535 .0624 .0686 .0726 .0747 .0753 .0732 .0678 .0506 .0401 -.0212 -.0540 -.0871 ‘6110" -.15?5 -.1645 -.2160 '02773 -.3363 -.191J -,4492 -.5”13 -.55°7 -,AU7h -.6511 -,A¢n2 -,7441 -.7848 -,0395 -,R?R2 -.9170 T(x) ,0001 .0130 .0147 .0307 .0501 .0710 ,0044 .1184 .1425 .1666 .2149 .2625 .3099 ,354n .3993 .5058 .6061 .7010 .7912 .877? .9594 1.0388 1.1151 1.2600 1.3964 1.5254 1.6482 1.7657 1.8781 1,9868 2.0914 2.1924 2,2900 2,3860 2,4707 2.5690 2.6571 -1,07RU .1,9>15 o1.3671 '1049311 °l.517‘ -201673 -?.6$%P -3.0402 -3.4346 -3.7709 ~4."935 9.1?41 '4.3969 9.754% -4.684219,3120 2.9900 3,2979 3,5559 3.8534 4,1099 5.2310 6.1811 7.0187 7,7764 8,4744 NTc (Tk'Tc, .0000 -.0000 .010? -.000? .1015 -.8010 .0041 -.00?5 .0075 -.0043 .0115 ‘.0060 .0156 '.0876 .9106 '.0989 .0236 -.0099 .9974 '.0108 .0344 -.0118 .0407 -.01?? .0462 -.0121 .0511 -.“110 .0654 -.0109 .0643 -.0085 .971? -.0056 .0767 -.00?6 .081? .0003 .0849 .003? .0880 .0059 .0906 .0085 .0929 .0110 .0967 .0156 .9997 .0196 .1071 .0733 .1041 .0?66 .1057 .0296 ,1071 .0323 .1084 .0348 .1094 .0171 .1104 .039? .1112 .041? .1119 .0431 .1126 .0948 .1112 .0464 .1137 .0479 .‘155 .0532 .1188 .0575 .1177 .0611 .1185 .0641 .1191 .0668 .1219 .076? ,4970 .0820 .1296 .0862 .1739 .9893 .1233 .0918 .1235 .0038 .1216 .0955 .1238 .0969 .-41- .0000 .0001 .0007 .0016 .0094 .0032 .0037 .0041 .0043 .0044 ,0042 .9019 .0035 .0031 .0096 .0018 .0012 .0010 .0010 .0012 .0016 .0090 .0096 .0039 .0053 .0048 .0043 .0098 .0112 ,0197 oqt‘i .0154 .0168 .0101 .0103 .0205 .9216 .0249 .0297 .0310 .0340 .0646 .0449 .9540 .0614 .0665 .0609 .0718 .0742 .0763 DI .000000-l,0010 ?09.?1 ,nnnnnn ,nnnnnn .nwnnnn .000000 .000000 .000001 ,030001 .000002 .000004 ,000007 .000011 .000016 .000092 ,000028 .000041 .000057 .000071 .0000I3 .000004 .000104 .000112 .000170 .000112 .000142 .000149 ,000184 .000189 .000142 ,000144 .000145 ,000146 .000166 .000147 .000166 .000166 ,000145 .00014? .000187 .000142 .000147 .00014? .000190 ,000104 .000062 ,nnnn-z ,000075 .000048 .0"0063 .000059 -,9997 -.9991 -.9992 .,9060 -,995? -,993? -,9907 -,9874 -.9841 -,9755 -,9530 -,9483 -,9279 -,901? -,7915 .,5040 -.3003 .0351 .3177 .5034 .6276 .700? ,4040 .8548 .8855 .9054 ,9200 ,9305 ,9304 ,9440 .9501 .9541 ,9579 .9609 ,9635 ,9868 .9726 ,9771 .9801 .9896 .9845 ,9898 .9974 .9936 .9949 .9956 .9961 .9966 .9969 81.36 50.05 36.74 29.43 24.43 21.70 19.45 17.74 16.42 14,50 13.19 12.23 11.52 10.07 10.04 9.48 9.13 8.90 8.76 8.67 8.63 8.61 “0‘3 4.70 4,00 4.93 4,06 9,21 9,15 9.51 9.66 9,42 4,97 10.19 10,98 10,43 11."3 11.60 12.15 12.68 13.18 15.45 17.41 19.16 20,75 22.29 21.59 24.89 24,10 40,30 34,97 34,17 31,7: 99,69 97,71 24,94 24,34 22,88 71,46 1n_94 16,74 14,71 17,90 11.21 7,64 4,54 2.00 -,21 -?,14 -3,91 -5.44 -6,BK -o,24 '11,?! -13 01 -14,51 -15,89 -14,90 ~1n,01 -1n,94 ~19,7o -20,56 '21,26 -?1,91 -22,51 -93,07 -?4,9% -?A,41 -27,6‘ o98.69 -2o.44 -37.}. -34,05 -35.22 ~36.0o -34,76 -37.30 -37,74 -38.1? -42- are plotted in Fig. 4.1.1 for both x : O and x : x. (For x : O, the value of x used in (4.1. 2) and (4.1.14) in Fig. 4.1.1 is for the interior point x.) It is necessary for separate determination of k and c that Tk and TC are not proportional and neither of them is equal to zero. Note that the results for the interior location satisfy both conditions. The derivative TC is always equal to or larger than Tk in magnitude. Both Tk and TC for x = O are negative for TX >1. 4 though it is significant that Tk is initially positive and then later decreases and becomes negative. Physically, this means for a given heat flux boundary condition the interior temperature initially rises more rapidly when the conductivity is increased slightly but later the increased conductivity reduces the temperature. An increase in k or c always decreases the surface temperature. The integrals NT NTC and (T TC) for the interior point x k’ k’ are given in Table 4.1. 1. Observe that NTC is always larger in absolute value than NTk and (Tk,TC). One would expect that the specific heat-density product c could be obtained much more accur- ately than the thermal conductivity k for this experiment. This is proved by the ratio R = 9. 97 and angle B : -21. 3° at the maximum value of X which occurs at 7%8. 5 for a single interior measure- ment. See Fig. 3.1.1. Hence, for a given time duration of the experiment am the Optimum location of the thermocouple is 1 _ 5 x1 _ (kOm/8.5c) (4.1.15) for a single thermocouple. Greatly improved accuracy can be obtained using two thermo- couples, one at the heated surface and the other in the interior. A h—. y— _ .— ’2} Fig. 4.1.1 Derivatives Tk and TC for semi-infinite body with q = C. j l l l I 6 J _ and 2F- " AC5. O 2 _ Mr] T __ All-Ofor Soot x=0 _ _4_ _ - 6 L. A52 _. l l l l l o 0.5 0.5 0.75 /.o 1.25 /.5 7. Fig. 4. 1. 2 Errors Alzj and A52} for semi-infinite body with q : C for-r = 1.5. m -44- summary of the results for this case at the time of maximum K i 5 given as case 3 in Table 4.1.2. In order to compare the value of Z for two thermocouple measurements with the value of E for one measurement it should be divided by 4.0 since two interior thermocouples at x increase 5 by a factor of four. The equivalent value of K for the measurements at x : 0 and x is 0.0026 compared to the value 0.00017 for a single interior measurement. Clearly the two measurements are superior. The optimum position for the interior thermocouple located atx=x2 isfoundfromT 20.9 /x22 : 1.50 or m m x2 = (k em/1.5c)%5 (4.1.16) The criterion X (T13 for measurements at positions x : 0, yl, yZ, yj... yn at time TL 2 a. B/L2 can be readily calculated from Table 4.1.1. The length L can be any significant length dimension. The terms in Z are typically given by 1'1 NT =0.125+z AXNT ( k) T i=1 Jt k)T 1-( L L y 2 (4.1.17) 2 where (NTk)T is found from Table 4.1.1 at time T L (——§l:—) _ J For three thermocouples A is maximized by placing the third thermocouple near either x z 0 or x2. For four thermocouples, the fourth should be located near the other position not occupied by the third. If the third and fourth thermocouples are placed some- where between the surface and the interior thermocouple, the K values are not greatly reduced from the values for the optimum -45- Jada: Hooo o Sou oudmmoe maxm Momma: o 0 ms 0 .v w w . . m H .4. .33.. m .usouaoonh mo moxm posit ow gowns: mo 03mm ** X N E . E u c. 835388 m mg N £0??? ad GETH. ... 0 d 3:. mam- :.~ 34 $.35 x 63. 27:3me 0 :55 m .3. mm .m o .3 ~28 .o x n x 37?on u u m o I 8 ....III N N .F .m H U o H 3 o o ~32 8 4. Ed m .3. is m 4 mos .o x .o n x o n a m wmo .o m .3. 3.9 m .m $88 .o x n x o n a N o .4 3. a II. o o u x o u 5 H L a meadou do“ no 333.3000 ... ... ... ... * ... xg M noguonfi twuhmumsom ommU coflmfimuuoo mo acidood v mofluon OHMGCGTMEOm .HOw 4 mo mod~d> ESEMXNE N J .«a «3nt -45- placement of the thermocouples. The dimensionless errors in the thermal conductivity k and specific heat-density product c due to a single error at time T for the optimum experiment with two thermocouples (x : 0 and x2) are shown by Fig. 4.1.2. At a given time TX the values of Ako and AcO for x: 0 are the result of a single error at x = 0 and time TX. For example, the errors introduced due to a single error in the tem- perature measurement at the surface at time TX : 0.8 are Ako : -3. 69 and Aco = -0. 327. This means that. the fractional error in k due to this single error is Ak/k : -3. 69(AT /1' m) (6/(Tm-Ti)) (4.1.18) where 6 is the actual temperature error at x = 0 andAT /'r m is the reciprocal of the actual number of discrete temperature measure- ments at x = 0 utilized in the sum of squares function F. The actual error in k is the sum of all the errors due to each discrete tempera- ture measurement at x = 0 and the optimum interior location x2. NIH 4. 2 Semi-Infinite Body with q : a(0 17)- Another semi-infinite case of practical interest is for the heat flux q = a(6 17) i (4. 2.1) where a is a known constant with appropriate units. For this case the temperature is uniform until time 6 = 0 when the surface tem- perature takes a step jump to Tm-Ti and then remains constant with time. If the temperature boundary condition were prescribed rather than the heat flux, the thermal properties could not be deter- mined independently. k c T—T. l i - _ 1 (kc) l _ 3E / c 2 T— T -T.-(T'T1) a rfc 1 — erfc 2 k8) m 1 2 E T x (4.2.2) 1 .1. : ka _k8T> (kc)‘2:__1.[emfC l _ k T -Ti Bkc a 2 1 2T 2 x -1 1 2 (177x) exp(— 4T )] (4.2.3) x CTc 8T (kc)% - _ _ _ .- _i TC — T -T — C<8c)k a _ 2 [erfc + m 1 3 21' x -l 1 a (17 TX) exp < - 4- > 1 (4. 2.4) x For x = 0 the following results are obtained, ka(0,9) cT (0,0) 'T—-T. ‘ ‘T"—-T ' ’ a (4'2'5) m 1 m i " _ ‘ _ ' ' _ _1- NTk - NTC _ (Tk’Tc) —. 4 (4.2.6) Results for the various quantities of interest are tabulated for one interior measurement in Table 4. 2. l. The dimensionless deriva- tives Tk and Tc are plotted in Fig. 4.2.1 for both x = 0 and x. The maximum Z for the single interior measurement is 0. 0023 and occurs at the dimensionless time Tm a, 10.0 and thus the opti- mum position for a single thermocouple is N)»- x1 = (kBm/loc) (4.2.7) Table 4. 2.1 T X 005 .10 .15 .20 .25 .30 .35 0‘0 .45 .50 .60 .70 .90 .90 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 .7000 7.50 8.00 8.50 9.00 9.50 10.00 12.00 14.00 16.00 18.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 1(0) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 T c -.0093 “.0859 '61715 '02376 “.2862 -.3222 -.3494 -03705 '03872 -..oo. -.4207 -.4349 -..453 -.4533 -.4594 “.4701 “.4768 '64813 -0‘8‘6 ’64870 -.4888 -.4902 “.4914 -.4931 -6‘943 ’o4952 ‘64959 -6‘965 -64969 -6497? “.4975 v.4978 '64980 -.4981 -.4983 -.4984 'o4985 '04989 -.4991 -.4993 ..4994 0.4995 c.4997 -64998 -.4999 -.4999 -64999 -.4999 '6‘999 -.5000 Tk .0077 .0606 .1036 .1238 .1289 .1255 .1174 .1070 .0954 .0833 .0594 .0369 .0161 -60028 “.0201 -.0570 ’.0869 -.1116 ‘.1325 -01504 -01659 -61796 -61917 -62123 -02293 -.2‘36 -.2559 -02666 -62760 -028‘3 -02918 -02985 -030‘6 .6310? -03154 -63201 .032‘5 .03394 .63510 ‘63609 .63682 -03749 -63976 '94112 .6‘205 -6‘274 .0‘327 -04371 -.4406 ”.9937 Quantities for q : a(90)- “TM .0016 .0253 .0679 .1138 .1573 .1967 .2320 .2636 .2918 .3173 .3613 .3980 .4292 .4561 .4795 .5271 .5637 .5930 .6171 .6374 .6547 .6698 .6831 .7055 .7237 .7389 .7518 .7630 .7728 .7815 .7893 .7963 .8026 .8084 .8137 .8185 .8231 .8383 .8501 .8597 .8676 .8744 .8973 .9110 .9203 .9273 .9326 .9370 .9406 .9436 1/2 NT C .0000 .0012 .0067 .0157 .0265 .0376 .0484 .0586 .0680 .0768 .0922 .1052 .1163 .1258 .1341 .1506 .1629 .1724 .1800 .1862 .1914 .1958 .1996 .2057 .2104 .2143 .2174 .2200 .2222 .2242 .2258 .2273 .2286 .2297 .2307 .2317 .2325 .2352 .2372 .2387 .2399 .2408 .2438 .2453 .2462 .2468 .2472 .2476 .2478 .2480 with thermocouple at x (RFC) ..0000 -.0009 -000‘3 -.0092 -60141 -00182 -.0215 -.0239 -60255 -.0264 -00269 -00260 -.0242 -.0218 -.0191 -60116 .600‘0 .0034 .0104 .0169 .0229 .0285 .0337 .0432 .0514 .0587 .0652 .0711 .0764 .0812 .0857 .0897 .0935 .0970 .1003 .1033 .1062 .1161 .1242 .1308 .1365 .1414 .1588 .1697 .1773 .1831 .1877 .1914 .1945 .1972 -48- NTk .0000 .0007 .0029 .0055 .0076 .0091 .0099 .0102 .0102 .0100 .0092 .0082 .0073 .0065 .0058 .0050 .0051 .0058 .0060 .0084 .0101 .0119 .0137 .0176 .0215 .0254 .0291 .0327 .0361 .0393 .0425 .0454 .0483 .0510 .0536 .0561 .0585 .0671 .0746 .0811 .0868 .0920 .1113 .1245 .1342 .1418 .1480 .1531 .1575 .1613 A .000000 .000000 .000000 .000001 .000003 .000008 .000017 .000031 .000048 .000070 .000124 .000189 .000261 .000339 .000419 .000619 .000808 .000982 .001138 .001277 .001400 .001509 .001605 .001766 .001892 .001992 .002070 .002132 .002181 .002220 .002249 .002272 .002289 .002301 .002309 .002314 .002317 .002306 .002276 .002236 .002191 .002145 .001923 .001739 .001590 .001469 .001367 .001281 .001207 .001143 c.9999 168.64 -.9994 -69981 c.9958 0.9924 -69878 -.9818 -.9742 -.9649 -Ogssb -.9242 c.8892 c.8315 -.7646 -.6829 .0‘241 -.1388 .1081 .2938 .4268 .5221 .5918 .6441 .7164 .7634 .7961 .8201 .8386 0.532 .8650 .8749 .8832 0.903 .8964 .9018 .9066 .9108 .9241 .9335 .9405 .9459 .9503 .9639 .9711 .9756 .9788 .9811 .9830 .9844 .9856 59.98 35.15 24.90 19.47 16.16 1309‘ 12.37 11.20 10.30 9.01 8.13 7.51 7.05 6.69 6.09 5.73 5.50 5.35 5.26 5.19 5.15 5.13 5.12 5.14 5.18 5.23 5.28 5.34 5.41 5.47 5.54 5661 5.68 5.74 5.81 5.88 6.13 6.38 6.61 6.83 7.04 7.97 8.75 90‘: 10.04 10.59 11.10 11.58 12.03 40.13 36.34 33.20 30.49 28.12 25.99 24.07 22.31 20.89 19.19 16.48 14.10 11.97 10.05 8.30 4.55 1.45 -1017 -3.42 -5.37 -7.09 '8.61 -9.98 ~12.33 “1‘628 915.93 “17.35 ~18.89 -19069 '20.66 -2105: “22.31 .23.“; .23068 024.28 524.83 .25634 ‘27606 “28.39 “29.47 030.37 '31.12 “33.68 .35621 ”35.24 “37.00 .37059 '38.07 ‘38.47 ~38.80 -49- As noted for the constant heat flux case a much better experi— ment can be designed using two thermocouples, one very near the heated surface and the other in the interior as indicated by Table 4. l. 2, case 6. The maximum 3 occurs at time Tm : l. 25 and its value is 0. 0452. which compares with the much smaller value of Z = O. 0105 for the constant heat flux boundary condition, case 3. The optimum location for the interior thermocouple is x2 = (kem/l. 25c)% (4.2.8) Additional thermocouples should be placed at x : 0 and x2 or else between these two positions as previously discussed in section 4.1. The criterion X for more than two thermocouples is calculated as described in section 4.1. A typical term is given by (4. 1.17) except the constant 0. 125 is replaced with 0.25. The dimensionless errors AR and [3.5: due to a single error at time TX for the optimum experiment with two thermocouples (x : 0 and x2) are shown by Fig. 4. Z. 2. The interpretation of Fig. 4. 2. 2 is discussed in section 4.1. We note that the values of A12 and A5 .1. for q : a(0 1r) 2 are generally smaller than for the q = constant case shown by Fig. 4.1. 2.. This is a natural consequence of having a 1 larger value of E for the q = a(0 ")‘2 case than the q = C case. Examination of Criterion for Optimum Heat Flux: In section 3. 4 the parameter B(TO) is derived and it is shown that a necessary condition for Z to be a maximum is that B(TO) be positive. The boundary condi- tion considered was for the heat flux which causes a step rise in the surface temperature. There were two temperature measurements considered, one at x = 0 and the other in the interior of the body. These conditions apply to the case considered in this section. 7: for X=x 7; For x=x - 0.3 — -i - Q4— 7: and 7; for x=0 - _ 0.5 l 1 l l O I 2. 3 4 5 K Fig. 4. 2.1 Derivatives Tk and TC for semi-infinite body with -l 2 q = a(90) / . Al}; 2 f AZZ for 82 at x2 and _ F— AC} _ L060 "’ F FA“ for SO at X=O “Z- \ j—AEZ ‘ I l l l o 0.25 0.5 0.75 1.0 x. 25 ’2} Fig. 4. 2. 2 Errors ARJ. and AEJ. for semi-infinite body with _ -l/2. _ q-a(9fi) for Tm— 1.25. -51- The terms E , E and E needed in (3.4.16) are given kl cl k,cl by (3. 4. 9), (3. 4.10) and (3.4.11). The corresponding terms for the interior thermocouple (those with "2" subscript) must be evaluated for this particular case. For the optimum time of Tm = 1.25, ka and GkT are both on the average much smaller than CTc or 0CTC k Then it follows that E as 0 (4-2-9) E m 0 (4.2.10) The integral Ec is not negligible, however; it is defined by 2 Tm (CT )0 0(CT ) dT c2 CZ E (T) : Z — (4.2.11) c o 2 (T -T.) T TO m 1 m It can be proved using the linearity of TC and T that the magnitude of 5cTC is directly prOportional to the pulse 6T. Also for small 2 . AT it can be shown that BCT C2 c 8T 2 Hence Ec (To) is directly proportional to the product of GT and AT , 2 or _ I ’ EC2(TO) — AT 0T ECZ(TO) (4. 2.13) where 13": (To) is independent of AT and GT. Then B(TO) as defined 2 by (3.4. 20) is B(TO) = (1/8) [1\ITk +NTC - 2(T )] + (E; NTk]/2 kT 2 (4. 2.14) -52- Now E; is positive since from (4. 2.11) EC (To) is positive for 2 positive 5T because then both cTC and 0CTC are negative. From Z 2 Table 4. 2. 1 and (4.2.6) at time Tm = 1. 25, we have NTk = 0. 255, NTC = 0.4006 and (Tk,TC) = 0. 2384. Thus B(TO) is at least 0.0224. The positive contribution by E232 to B(TO) is usually less than 0. 0224. At Tm : 1.25,Table 4. 2. 2 gives A0 : 0.0452. Using these values in (3.4. 21) we note that even if AT/Tm approaches unity, A is less than 30. According to the analysis it is sufficient to demonstrate that B(TO) is positive, however. This has been proved for this case; )— hence based on the B(‘TO) criterion, q : a(9 n)—§is the optimum heat flux for the case of a semi-infinite body with one thermocouple at the surface and another located at x2. V. FINITE BODIES 5. 1 One Surface Insulated A number of different heat flux conditions at x = O is considered in this section. In each case the surface at x 2 L is insulated. The optimum thickness of the specimen can be calculated for each case in this chapter for a prescribed experiment duration 0m using L = (kem/Tm c)%where Tm is the optimum dimensionless dura- tion of the experiment. 5.1.1 Constant Heat Flux For the case of constant heat flux qO at x : 0 and insulated at x = L the temperature distribution and the associated property deri- vative s a re -_ 1 — Tzq—Of7E—T+X ———2— 2 (‘1) e'r1 7’ T cosnn(1— 1) (5.1.1) 2 , L 2 n=1 n 0 RT T — k —X k qOL7k on 2 1 +22 (~1)nen 7' T [cosng(1—i{-)][ +T] (5.1.2) L 2 2 n=1 1" n -54- CT 00 .. = C _ - _ n -n 11’ _1{_ TC _m_ T[1+2f_1(1) e eosan L)] (5. 1. 3) where T :0. 0/L2 andX : +[(x/L)2 /2] - (x/L) + (1/3). These expressions are plotted versus dimensionless time T for several positions x/L in Figures 5.1.1, 5.1. 2 and 5.1. 3. The derivative Tk can be either positive or negative and approaches the quasi- steady state distribution -X for times T greater than 0. 5. Conse— quently Tk approaches zero at x/L : 0. 422. Thus unlike the semi- infinite body it appears that an internal thermocouple located near x/L : 0. 5 in ineffective in aiding the determination of k. On the other hand for T > 0. 3, the derivative TC approaches -T . Various parameters of interest for the maximum value of A are given in Table 5.1.1 for case 1 with x = O and case 2 with x : L. The case of using two thermocouples is important; it is found that the optimum locations to maximize A are at the boundaries: x = 0 and x = L. The results for this case are summarized in Table 5.1.1 as case 3. The ratio of the major axis of the F-contour to the minor axis, R, is 2. 05 which is much smaller than for the single tempera- ture histories at x : 0 and x : L (which have values of 7. 76 and 14. 5 respectively). If three thermocouples are used, A is maximized by placing the third near x = O; for four thermocouples A is maximized by locating two near x = 0 and two near x : L. This is fortunate since in many cases the specimen itself need not be instrumented with thermocouples but the thermocouples can be embedded near the sur- faces of the heating and insulating elements. 1 00 0., 0.2 _ «a 0.3 0.4- 0.5 Fig. 5.1.1 Temperatures in a finite body with q = C at x = 0 and q=0atx=L. 0.2 T I l 1 f: /.0 0.! - / L 0.75 ‘ 0 AA \ HF \— 0.5 “'5‘8- 00’ "' O. 25 —‘ H‘s: -a2~ — A = 0 L - 0.3- — g J I u ’ O 0.! 0.2. T 0.3 614- 0.5 Fig. 5. 1. 2 Derivative Tk for q : C at x = 0 and q = 0 at x = L. -56- —05- -Q4- 0.5' ‘ 0-3” 0.25 I‘lx II C H'Sx- 0.2— ” M" II lb TOJ- /.O O J l I o 0.: 0.2 2.3g 0.3 0.4 La Fig. 5.1.3 Derivative forq:Catx:0andq:0atx=L. 4 I I If I T 3,_— _ AI} 2 ... Aklforxd. _ Md ... / _ _ AC! 0 —/ P .fla _ _“3 L O 0.] 0.2. 0.3 Q4- 05' 0.6 Fig. 5.1.4 Errors ARj and AEJ. forq:Catx=Oandq=O atsz. T 20.64 m Table '5. 1.! 54110110111111 2 values for'different experiments with finite bodies. B ‘ d3 Time at Ratio of major Angle of major 0W! rY Location of _ which ‘A' i. to minor axes axis measured Corr ‘1”. Case Conditions A max m""imum, of F-contour, from c coordi- coeffeicieftn Thermocouplei T _ “ m R nate, 0°. ' ' m’ '17— p 1 I qsc 81:0 38:0 0.00098 1.2 _ 7.76 -22.5 0.94 .z ; q=C q.0 3521.. 0.00019 1.3 14.5 10.5 -0.94 3 1 q=C q . 0 = 0, 1. 0.0235 0. 65 2.05 -16.5 0.39 4 0<15 0.5? q -.- 0 x = 0, 1. 0.0356 0. 75 2.18 -9.0 0.25 q=C I a I T>0.5: ' i : anliJ 5 q=c0' q :0 x : 0, L 0. 0981 0.48 1.855 -18.2 0. 36 «col/Z q =0 3: = 0, 1. 0.00989 0. 84 2.20 -16.0 0.42 7 q=COn q=0 x=0snd 0.0 --- --- --- -..- ”(.00 5 X 8 o, L 8 h-b.c. q = 0 x -.- 0 0.00083 1.16 7.41 : -23.7 0.94 Bi=0.l I 9 h-b.c. q = 0 x = 0 0.00016 0.86 7.96 -34.0 0.963 3181 10 h-b.c. q = 0 x .-. 0 0.00004 3.8 ' 2.73 -51.3 0.76 " 31:2 ‘ 11 5-15.15. q s 0 x = 0. 1. 0.0216 0.64 1.96 -15.8 0.36 Bis-0.1 _ 12 h-b.c. q = 0 x = 0.1. 0.0109 0. 56 1.44 -9.0 » 0.11 I 31:1 13- I h-b.c. q = 0 x .. 0, 1.. 0.00559 0.48 1.12 -1.8 0.01 1‘ 81:2 ‘ 14 ‘qfor’l‘ufl'm q=0 ' ...-0 0.0291 1.8 4.21 410.3 0.57 15 quor Tam q . 0 x . 0.1. 0.1432 0.76 2.38 -8.2 0.265 16 l qu '1' =0 :30 0.0111 2.96 6.42 -84.3 0.528 17 100:3 2.2: 'r . 0 x - 0 0. 0196 2. 9 3.7 -86. 5 0.20 q - C 1’ >Z.Z <1 = 0 . 10 qforTfl' q for x I 0, 1.. 0.250 a . 1.0 -45.0 0.0 m T I: 0 -51. .75," ‘v ‘ .4- H" d 5 'IQO‘E nus». . —58- The errors in k and c for a single error at time T for the optimum experiment using two thermocouples (case 3) with constant q are shown in Figure 5.1. 4. We note that k is more sensitive to errors in the temperature measurements than c. The effect of the weighting factor Aj upon Z and R was found to be insignificant in this constant-q case. In general, K was reduced slightly and R was increased for Aj's other than unity. For this reason and to reduce the scope of the problem the value of A). for all subsequent cases is unity for each temperature history. The case of constant q until time T and q = O thereafter is of interest and is case 4 of Table 5.1. 1. The optimum heating time is T = O. 5 and the soaking time is AT : O. 25. This heating curve increases K over that for q : C. 5.1.2 Heat Flux with q = c 0r1 Results for the heat flux of the form q = c e“ (5. 1. 4) with n = -0. 5, 0. 5 and n < -O. 5 results are tabulated as cases 5, 6 and 7 in Table 5. 1.1. In the first case the surface temperature suddenly increases to a value and remains constant until T g 0.4 when it begins to increase slowly. For n = 0. 5 the surface tempera- ture increases very gradually at first and then increases more rapidly at later times. For n < -0. 5 the surface temperature sud— denly increases at T : 0+ to infinity and thus Tm equals infinity and K equals zero. 5. 1. 3 Convective Heat Transfer at the Surface The convective heat transfer boundary condition is q=h(Tm-T(0.9)) (5.1.5) -59- Results for a thermocouple at x : 0 are given in Table 5.1.1 by cases 8, 9 and 10 and for thermocouples at x : 0 and L by cases 11, 12 and 13. Results are given for the Biot number Bi : h L/k equal to 0.1, l and 2. 0. For Bi : 0.1 the surface temperature rises very slowly and q is almost constant for a relatively long dimensionless time. For large values of Bi the surface rapidly approaches Too and the boundary condition of a given surface temperature rather than a given surface heat flux. When the surface temperature is given, we know that the properties can not be separately determined and E : 0. Hence the decreasing value of K as Bi increases. As Bi becomes larger the sum (-ka -cTC) becomes much smaller than T - Ti while we know the equality holds for a given heat flux boundary condition. 5. 2 Prescribed q at x : 0 to Produce Constant Surface Temperature and q = 0 at x = L The heat flux at x : O to make 5 a maximum for thermocouples at x = 0 and x = L (the insulated surface) is the one which makes the surface temperature at time T = 0 increase from T.1 to Tm and then remain constant. Results for a single thermocouple at x : 0 and for two thermocouples are shown by cases 14 and 15 of Table 5.1.1. The experiment is more accurate for determining c than k since [3 is near 13 = 0° and R is 2. 38 for two thermocouples. The temperature T', derivatives ka and cTC and the errors AR and A5 are shown in Figures 5. 2.1 to 5. 2.4. This is the optimum experiment for a finite body if the surface at x = L must be insulated. This is perhaps the best boundary condi- tion for high conductivity mate rials since these are relatively easy to O l l l l O 0.2 0.4- 0.6 0.8 /.0 ’2’ Fig. 5.2.1 Temperatures in a finite body with q at x : O to causeTm:Candq=Oatx=L. O 0.2 0.4- " , 0.6 0. 8 [.0 Fig. 5. 2.2 Derivatives Tk for a finite body with q at x = O to causeTm=Candq20atx:L. .. ['0 "O-a __ T. TQ6r ..Q4 I— —0.2 - O ‘ L 1 l I o 0.?- 0.4 0.6 0.8 T Fig. 5. 2. 3 Derivatives TC for a finite body with q at x = 0 to causeTm=Candquatx:L. Fig. 5. 2. 4 Errors AkJ. and ch for a finite body with q at x:0tocause Tm=Candq=0atx=L. T =0.76. m 60 -62- insulate. Unfortunately the B(TO) criterion for the optimum heating condition is not easy to apply in this case since T and TC are not of k simple form. Computer calculations verify, however, that B(TO) is positive and hence this is the optimum experiment if it is required thatq=0atx:L. 5.3 Constant Heat Flux at x = O and Given T = O at x = L An experiment for which k can generally be more accurately determined than c is the case of a prescribed constant heat flux at x : 0 and a constant temperature at x : L. In the standard method of determining k the temperature distribution is allowed to reach steady-state and then k alone is determined. The optimum value of K for a thermocouple at x : 0 occurs at time Tm : 2. 96 (case 16, Table 5.1.1). The angle 6 for this Optimum is -84. 3° and R = 6. 42 indicating that k can be calculated with considerably greater accuracy than c. Since the temperature at x = L is prescribed, no informa- tion about the properties is obtained at this point. To maximize K using two thermocouples the second thermocouple should also be located near x = 0. If the surface at x : O is insulated at time T = 2. 2 and then the body is allowed to thermally soak, the value of Z is increased and R is decreased measurably to 3. 7 (case 17). It is worthwhile to compare the E, R and B values of case 16 with the case of constant q at x : O and insulated at x = L, case 1 of Table 5.1.1. The former is best for determining k and the latter for c. If only a single thermocouple can be used and we require C, the T = 0 boundary condition gives a larger K than for q = O at q X L. Hence, in this comparison for a single thermocouple the T = 0 -63- boundary condition at x : L is superior to the insulation condition at L. For two or more thermocouples, however, the q = O boundary condi- tion at x = L is better because it causes a larger 5 than does the T : 0 boundary condition. 5. 4 Heat Flux Prescribed at Both x : 0 and L The largest possible value of E that can be obtained for a homo- geneous body with heat flux boundary conditions and with two or more thermocouples is found by determining the optimum heat fluxes at both x = 0 and L. The heat fluxes that will maximize E are the q at x : 0 to cause a step rise in the surface temperature and the q at x = L to cause the temperature at L to remain at its initial value, T = 0. Note if the temperatures rather than the heat flux boundary conditions were imposed, k and c could not be obtained. This case is unlike the other finite cases in-several respects. The derivatives Tk and TC at x = 0 are related by (see Fig. 5. 4.1) ka/T :- cT /T : -O.5 (5.4.1) m c m and at x = L by Mk 2 -cTC (5.4.2) and thus if either temperature history at x : O or L is used alone to find k and c the properties can not be separately determined. In the case of thermocouples at x = O and L the integrals NT NTc and k, (Tk,TC) approach as T -* on respectively 0. 5, 0.5 and 0. 0. Then the correlation coefficient goes to zero and R goes to unity indicating that both prOperties are found to equal accuracy. Unfortunately, K reaches its maximum value of 0. 25 only as T goes to infinity. This means that ... 0,5 2 4 l I O Fig. 5. 4.1 Derivatives Tk and TC for a finite body with q at x: O to cause Tmzc andq for T:O at sz. 12 I r . . a...) 0.4- — I —. _ 0.8 _ /—AK and A30 _ I Z I \l\ . O 0.2 0.4 II Q6 0.8 /.0 Fig. 5.4.2 Errors Akj and ch for q at x = 0 to cause Tm :C andqforT:0atx:L. —65- the specimen thickness must tend to zero for fixed finite duration of the experiment 9m Fig. 5.4. 2 gives the errors [ski and ch due to a single error at different times at x : 0 and x = L in the measured temperatures for Tm = 2. 0 for which E : 0.222. Note that this experiment has the smallest average errors at x = 0, A50 and Ako, and at x = L, A51 and Akl, of any experiment considered in this thesis, this is a result of the larger Z for this case than any other. The difference is most outstanding when the errors for q = C for a semi-infinite body, Fig. 4.1.2, are compared with those shown in Fig. 5.4. 2°, the same error in a measured temperature for the semi-infinite case as for this case can cause six times as large an error as for this case. To experimentally introduce a heat flux at x = O to maintain essentially a constant surface temperature is not difficult. It can be done by utilizing another finite body of accurately known thermal pro- perties at either a higher or lower temperature than the specimen. The two bodies are suddenly brought together at time 9 = 0. The heat flux into the specimen can be calculated accurately from the tempera- ture history of the surface of the body with known properties. To accurately measure the q at x : L without the temperature increasing can be very difficult if the specimen is heated rather than cooled. If the specimen is cooled at x = 0 then an electric heater at x = L auto- matically adjusted could maintain the temperature constant at x = L. This is an interesting possibility since the heat capacity of the heater would not introduce any error into the calculation for the properties. Examination of Criterion for Optimum Heat Flux: For the heat flux at x = O to be an Optimum boundary condition it is necessary that B(TO) -66- be positive (see section 3. 4). By an analogous development it can be shown that another B(TO) for the surface at x : L also must be positive for the optimum heat flux at x = L. Both values of B(TO) are positive; for conciseness B(TO) for x : O is solely considered. Due to the unique condition (5. 4. 2) the following relations apply, E z E (5.4.3) E = -213 (5.4.4) and (3.4.16) becomes 231(70) = Ekl[(NTk)+(NTc)-(Tk,TC)]+ECZ[(NTk) + (NTC) + 2(ik, TC)] (5. 4. 5) where E is given by (3.4. 9) and BC by (4.2.13). The quantities k l . 2 ER and EC are positive for positive GT and for large T's, (NT ) : .1 2 - - 1‘ (NTC) = 0. 5 and (Tk,TC) = 0; hence B(TO) given by (3. 4. 20) must be positive. Consequently the necessary condition that B(TO) be positive for the optimum heat flux at x : O is satisfied. 5. 5 Summary of Observations for Heat Flux Boundary Conditions A number of general observations can be drawn from the cases considered for the finite and semi-infinite bodies. The Optimum heat- ing boundary condition at x = O for at least two thermocouples appears to be the heat flux which makes the surface temperature take a step rise. This is true for both finite and semi-infinite bodies; it is indi- cated by the value of B(TO) being positive for a positive GT in each case. See section 3.4. For all the finite cases, except for case 16, two thermocouples with one at x : 0 and the other at x = L provide —67- a much more efficient experiment than a single thermocouple at x = O. In adding additional thermocouples the best locations to maxi- mize K are at the boundaries, x = 0 and L (except for case 16). In many cases since the Optimum locations for thermocouples are at x = O and L the specimen itself need not be instrumented with thermo- couples but the standard materials (which do not change from experi- ment to experiment) can be instrumented. This can mean a considerable saving in time and expense in performing experiments. In general, the optimum finite experiments provide larger values of E and smaller values of R than the Optimum semi-infinite experiment. Hence the finite-body experiments are preferred. For high thermal conductivity materials the Optimum experiment corres- ponds to case 15, Table 5. l. 1. For large k's (greater than 30 Btu/hr -ft-° F) the body can readily be insulated at x = L. On the other hand, for low conductivity materials (k less than 10 Btu/hr-ft-° F) it is difficult to insulate at x : L but relatively easy to maintain a con- stant temperature at x : L since in this case the heat fluxes are much smaller in magnitude than for the high k case. Hence, case 18 with Tm about 2 is recommended for low values of k. For intermediate values of k case 18 should be approximated as closely as possible. The step change in surface temperatures for cases 15 and 18 can be experimentally obtained by utilizing a standard of accurately known thermal properties and at an initial temperature different from that of the specimen. When the standard and the specimen are brought into intimate contact a common interface temperature To is suddenly attained (provided the contact resistance is negligible). Assuming temperature-independent thermal prOperties one can show that To is I- .0,- ~|:.UA .PC‘W 011).. .22 .. -.. fl «C336: . . .. ..CP '39- \y‘l A. S‘.::.Cie :0 1:. (Elise -68- given by NIH i,st O O i,sp kspcsp/kst Cst) (5. 5.1) where Ti is initial temperature and the subscripts sp and st refer respectively to the specimen and standard materials. In order for the interface to remain constant with time it is necessary for the standard to have the same boundary condition at its other face as the specimen has: q : 0 for case 15 and T : constant for case 18. It is sufficient to also require for both cases that 1 _ E Lst/Lsp - (QSt/Qsp) (5.5.2) Before performing the experiment one does not know accurately the thermal diffusivity of the specimen asp; thus the ratio of the thick- nesses of the standard to specimen given by (5. 5. 2) can not be satis- fied precisely. One usually does have an estimate Of asp which is sufficiently accurate to find LSt and approximately satisfy (5. 5. 2), however. VI. CALCULATION OF CONSTANT THERMAL PROPERTIES 6.1 Finite —Difference Equations for the Heat-Conduction Equation In order to be able to conveniently analyze transient tempera- ture data a general digital computer program is needed that can treat a variety of boundary and initial conditions. The procedure used in the program should have the potential of being extended to consider temperature-variable thermal properties. The method of finite- difference satisfies these requirements. The only doubtful point relates to the accuracy that can be Obtained with this method with a moderate expenditure of computer time. These requirements can be also satisfied since the properties can be frequently calculated to less than 0. 1 percent error due to the finite difference approximations while using less than 30 seconds of CDC 3600 computer time. In order to present concisely the finite-difference equations utilized, the interface equation between two dissimilar materials is developed. The transient heat conduction equation for constant k and c can be written 82 k : 8T C __ 8X2 88 H (6.1.1) An energy balance written for the node n shown in Fig. 6.1. 1 is -69- -70- k V— Interface h-I k ' n nH Cn-I ,n 0,, 0 Le >I< .4. Axn-I Axn Fig 6.1.1 Spatial nodes for interface, eq (6.1. 3) kl) 6I I I % ——>—61 .0 I 02 I I T Ax, r] Fig 6. 1. 2 Spatial nodes for surface node with given heat flux, eq. (6.1.10). Table 6.1.1 Percenterroratx:0forq:Catx:0andq:0at at x = L. 1) = 0.5 and 1.1: 0.75 except cases 6 and 7. Percent of error at x=0 at timesi 1 0.005 10 2 2 0.01 10 1 14.5 -O.l70 0.0347 0.0176 3 0.02 10 0.5 20.4 -l.64 -l.64 -0.0281 4 0.005 20 0. 20.4 -1.64 0.0075 0.0052 -., 5 0.01 20 0.25 23.3 -3.51 -0.897 -0.0267 6* 0.005 10 2 -27.6 -6. 88 -l.59 -O.580 7* 0.02 10 0.5 -23.3 -3.95 ~3.95 -l.39 * n=1.0andx1=l.0 -71- l l Tm+l _Tm+l m+- _Tm+ Trn -T m Trn _Tm n-l n n+1 n n-l n n+1 n ,,[1. ]T + [2 k ->. ————-]T 2 II 1 1 A9 1 7? 1 2 A6 2 c Ax) c (Ax )2 42wk4) —J—4L—]TH‘+[ka-x l l ]Tnh26x(6qunq“”5 l l 1 l 2 Z 1 A0 A0 (6. 1.10) Experience shows that 1. 1 : O. 75 gives more accurate results than the more common approximation x1 :- 1. Eq. (6.1. 10) is also used at an insulated surface by letting q : 0. To provide a comparison of the accuracy for selected time steps and number of nodes, Table 6.1. 1 is given. The percent error in the temperature rise at x : 0 is given for a finite body ini— tially at temperature Ti and with q : C at x = O and q = 0 at x = L. The temperatures for this case are shown in Fig. 5.1.1. The quan- tities AT , N and M are defined as AT = aAe/L2 (61.1n N = L/Ax (6.1.12) M = (AX)Z/O.A9 (6.1.13) -73- where A0 is the time step and Ax is the spatial node spacing. We can draw several conclusions from the data of Table 6.1.1. In each case the calculation Of the temperature at time AT is rather inaccurate and in some cases even at time 4AT the error is still large. Since the large errors do not persist with time, however, the initial few temperatures calculated need not be used to calculate the properties. For economy in performing the experiments the time step for reading the experimental transient temperature data can be considerably larger than that required for the computer to accurately calculate the temperatures. For these reasons the program developed to cal- culate prOperties permits the calculational time step to be any even divisor of the time step used to read the experiment data. Another conclusion is that the quantity M significantly affects the accuracy. The temperatures are not necessarily calculated with greater pre- cision for a fixed time step if N is doubled as can be observed by com- paring cases 1 and 4. Omitting the first few times steps for N : 10, a value of M of either 1 or 2 yields temperatures which are well within 0. 1% accuracy. For N = 20, M = 0. 5 is satisfactory to provide 0. 01% accuracy after several times steps. Cases 6 and 7 are for 77 : X l = 1. 0 while the previous cases are for 77 : O. 5 and x1: 0. 75; a com- parison of cases 1 and 6 (or 3 and 7) demonstrates that the latter values of 71 and x yield much better accuracy. Temperatures at 1 interior locations have comparable accuracy with the given values. In the iteration procedure for determining k and c it is neces- sary to evaluate the derivatives Tk and TC. The derivative Tk can be found by using the same program that is used to calculate the tempera- tures since -74- BT T(k-tAk,c) - T(k,c) 513 “ Ak C Tk: (6.1.14) The CDC 3600 computer uses 11 or 12 significant figures in its calcu- lation so Ak can be made very small compared to k. The value of Ak : 0. 0001k gives excellent accuracy. 6. 2 Accuracy of Properties Calculated from Exact Data In evaluating the accuracy of the mathematical procedure given in this paper it is necessary to use exact temperatures rather than experimentally determined values. When the procedure is used to find values of k and c from experimental data, the error introduced into the properties due to the finite-difference approximations should be an order of magnitude less than those caused by the data itself. With reasonable care in the choice Of M, N, n and X1 this can always be accomplished. For the optimum heating time Of T : O. 64 for case 3 of Table 5. l. 1 (q = C at x : 0 and q = O at x : L), the percent errors in k and c are shown in Table 6. 2. l. The exact temperatures at x : O and L at times T = 0. 02, 0. 04, 0. O6, . . . , O. 64 are used to find the pro— perties k and c. Cases 1 through 5 which are for 77 : O. 5 and x1 = 0. 75 give about 0.1 percent accuracy or better. The r) 2: x l : 1. 0 results, cases 6 and 7, have much poorer accuracy which is consistent with Table 6.1. 1. Another boundary condition of interest is the heat flux to make the surface temperature constant at T]m and q = 0 at x : L (case 15, Table 5. l. 1). The errors in k and c are shown in Table 6. 2. 2 for again AT = 0. 02 for the temperature data at x = 0 and L. For the -75- Table 6.2.1 Percent error in k and c for q : C at x : 0 and q = 0 at x = L with measurements at x : 0 and Land AT :0.02 for data. Tm=0.64. k .Percent Percent 1 errorin. error in k c 1 2.0 0.5 0.75 -0.0446 0.0727 2 0.01 10 1.0 0.5 0.75 —0.0543 0.0714 3 0.02 10 0.5 0.5 0.75 0.1305 0.1092 4 0.005 20 0.5 0.5 0.75 -0.0221 0.0209 5 0.01 20 0.25 0.5 0.75 -0.1475 0.0072 6 0.005 10 2.0 1.0 1.0 -O.4878 -O.1554 7 0.02 10 O. 1.0 1.0 -1.410 -0.0716 Table 6. 2. 2 Percent error in k and c for q to make T : Tm at x = 0, q = 0 at x = L with measurements at x = O andLand AT 20.02for data. Tm=0.8. 17:0.5 and X 1 = O. 75. Percent Percent Case AT IQ N4 error hi errorin k c 1 0.02 5 2.0 2.85 0.725 2 0.005 10 2.0 0.251 0.0559 3 0.01 10 1.0 —O.803 0.0728 4 0.02 10 0.5 4.28 0.960 5 0.00125 20 2.0 0.0299 0.0139 6 0.0025 20 1.0 0.0608 0.0164 7 0.005 20 0.5 0.659 0.0074 8 0.00125 40 0.5 0.0173 0.0051 -76- same values of AT and N in Table 6. 2.2 as in Table 6.2.1 the q : C case has greater accuracy. This better accuracy is due to the errors in the finite-difference approximation which are greatest at the earliest times at which time the constant temperature case is more sensitive to errors. (See Figures 5. l. 7 and 5. 2. 4). For a random distribution of experimental errors the constant temperature case is superior as indicated by the larger A; the errors in the calculation for the properties can be made as small as desired simply by reduc- ing AT and holding M fixed. Whenever a new experiment is being analyzed to determine k and c the accuracy of the calculation can be easily checked by running the same data on the computer with two or three values of AT for a fixed M value about unity. From Table 6. 2. 2 it appears that the error is reduced by a factor of about 10 each time AT is halved for fixed M. TO run cases 1, 2 and 3 of Table 6.2.1 it took a total of 65 seconds on the CDC 3600 computer. 6. 3 Analysis of Hsu Transient Temperature Data Hsu performed in 1956 [12] an experiment using two semi- infinite sections of nickel at different initial temperatures. The nic- kel sections were suddenly pressed together with a pressure of about 500 psi. His data is given in. Table 6. 3. 1. From each temperature (except at 0 : 0) he calculated an independent value of thermal dif- fusivity 0. using (4. 2. 2). This equation requires the interface tem- perature Trn which was not measured. Assuming the contact resis- tance between the nickel sections to be negligible, Hsu by eye drew a smooth curve through his temperature data plotted versus position. He found that the interface temperature Tm was about 46. 7° C. After Table 6.3.1 Hsu temperature data [12] Temperature, 0C Thermocouple Time Seconds positions, cm 0 5 10 15 20 25 1.495 69.9 64.6 60.85 58.65 57.3 56.15 0.892 69.9 59.0 55.65 54.2 53.2 52.5 Hsu“ 0.501 69.9 53.9 51.85 50.95 50.45 50.0 0.209 69.9 49.85 48.9 48.45 48.25 48.1 -0.269 23.7 42.75 43.8 44.45 44.7 44.95 Hsu#2 -0.996 23.7 33.7 37.1 38.65 39.7 40.4 Table 6. 3.2 Thermal properties calculated from Hsu data. Results for Hsu #1 are given by rows 1-3 and for Hsu#2 by rows 4-6. a k c p ' cp Row Analyst mZ/hr Kcal/m-hr-C Kcal/m5-C Kg/m3 Kcal/Kg-C ft2/hr Btu/ft-hr-F Btu/ft3-F lbm/ft3 Btu/lbm-F , 1 Hsu 0. 0558 53. 9 967 8870 0.109 1 2 Beck 0.05510 53.55 971.8 8892 0.1093 3* Beck 0.5931 35.98 60.70 555.1 0.1093 4 Hsu 0. 0568 54. 9 967 8870 0. 109 5 Beck 0.05925 56.30 950.3 8900 0.1068 6* Beck 0.6378 37.83 59.34 555.6 0.1068 ’1‘ -English Units Comparison of 0. values calculated by present method Table 6.3. 3 with others To k* 73* 0* 0* BSCk Has“ _ - _ ‘ 3 2 F Btu/hr ft F Btu lbm F lbm/ft ft /hr ftZ/hr ftZ/hr 75 38.0 0. 1055 555.9 0.649 95 37.5 0.1062 555.6 0.636 0.6378 0.612 116 36.7 0.1072 555.4 0.618 137 36.0 0.1082 555.1 0.600 0.5931 0.601 35.5 0.1090 554.8 0.588 * From Handbook of Thermophysical Properties of Solid Materials [34] -77- -78~ obtaining the twenty values of a. in the high temperature specimen (Hsu #1), the averaged was calculated. Utilizing the following values, p = 8870 Kg/m3 and cp : 0. 109 Kcal/KgC, the thermal conductivity k was calculated from k = p (6.3.1) Cp Qavg He repeated the same procedure with the same value of p cp to obtain a k-value from the average 0. Obtained from the low temperature specimen (Hsu #2). Hsu in his analysis of the data incorporated a number of assump- tions or conditions that can be eliminated using non-linear estimation. These are as follows: 1) The contact resistance was assumed to be negligible and thus the temperature at the interface was assumed to be unique. In our calculation the surface temperature Of each specimen-was assumed to be constant with time after 9 : 0, then the surface temperature of each specimen was calculated separately and no further assumption was made relative to the contact resistance. 2) Hsu found the interface temperature to be 46. 7° C by a manual curve fit. 3) Hsu used the same value of c : p cp for both specimens. In our analysis it was necessary to give the q at the surface of only 2133 specimen or equivalently, to give the c value of _O_n_e_ specimen. Then the value of k (and c if q is given) Of that specimen and the values Of k and c for the other specimen were determined. If Hsu had given the temperature history of a point near the surface of only one specimen with much finer time steps than 5 second -79- intervals, the heat flux at the interface could have been calculated for given values of k and c. Since this information was not available, we assumed the heat flux to be q = (Tm - Ti) (kc/0 17)::- (6. 3. 2) which is the heat flux for the surface temperature suddenly increasing from Ti to Tm and then remaining constant at Tm' (See section 4. 2.) Utilizing the program described in sections 6. l and 6. 2 the high tem- perature data (Hsu #1) was analyzed first using the q calculated from (6. 3. 2) with the values of Tm - Ti and kc that Hsu found from the low temperature data (Hsu #2). Next, the low temperature data was analyzed with the Tm - Ti and kc values determined from the Hsu #1 data. The property values calculated for Hsu #1 and #2 are shown in Table 6. 3. 2 along with values obtained by Hsu. English units are used in rows 3 and 6 and metric units in the other rows. The proper- ty calculated most independently is the thermal diffusivity 0. since neither the heat flux nor a c-value need be given to obtain 0.. The values of 0. calculated by Hsu and Beck for Hsu #1 are 0. 0558 and 0.05510 and for Hsu #2, 0. 0568 and 0. 05925 mZ/hr. For the Hsu #1 data our values are about 1. 3% lower than Hsu's and for the Hsu #2 data, about 4. 6% higher than Hsu's values. The discrepancy is quite significant particularly for the Hsu #2 case. The discrepancy is due in a large measure to the interface temperature used by Hsu compared to the calculated values using nonlinear estimation. The surface tem- perature calculated for Hsu #1 and Hsu #2 are respectively 46. 671 and 46. 605° C while Hsu from his manual curve fit Obtained the single value 46. 7° C. Though the difference between these values appears negli- gible, in reality Hsu's procedure for calculating a is very sensitive ~80- to the interface temperature particularly for the Hsu #2 data. The average (1.-value calculated from the Hsu #2 data and using Hsu's method with Trn = 46. 60°C yields about 0. 05975 which is slightly higher than the "Beck" value given in Table 6. 3. 2 which is 0. 05925 mZ/hr. If average values of o. are calculated separately for the two thermocouple temperature responses, (x : -0. 269 and -0. 996 cm), using as the surface temperature the Hsu value Of 46. 7°C, one obtains 0. 0553 and 0. 0583 mZ/hr; while using our value Of 46. 60°C for the surface tem- perature one Obtains 0. 0598 and 0. 0597 mZ/hr. Since the latter two values are much more consistent, it is probable that surface tempera- tures are more accurately calculated by nonlinear estimation than the manual curve fit. Furthermore, the properties Obtained by the non- linear estimation procedure should be more accurate than those calcu- lated by Hsu. To Obtain the k and c results given in Row 2 Of Table 6. 3. 2 the q given by (6. 3. 2) was used as described above. The specific heat cp was found from c by utilizing the accepted value of the density of nickel [34]. The density p is relatively easy to measure and is much more insensitive to temperature changes than either k or cp as can be noted by comparing values given for Hsu #1 and #2 (rows 2 and 5). The values of CI) and p were fixed by Hsu for both specimens. In our calculation only p was needed and the values given in reference [34] for the average temperature were used to Obtain the values in Table 6. 3. 2. This reference is result of a thorough literature search and analysis of published data of thermal properties for a large number of materials. The recommended values given in this reference are the generally accepted values at the present time. Before comparing the —81— prOperty values given in Table 6. 3. 2 with the recommended values the consistency of the Hsu data is examined. Hsu does not state the accuracy of his measurements; however, from the number of significant figures used to write his measured temperatures (Table 6. 3. l) evidently he thought the temperatures were measured no more accurately than about J: 0. 05° C. The values of the rms differences between the temperatures that we calculated and those that Hsu measured are for Hsu #1 and #2 respectively 0. 048 and 0. 050°C. The temperatures differences are given by Figures 6. 3.1 and 6. 3. 2. For the most part, the differences appear to be random and no one temperature or thermocouple contributes much more than the average to the rms value. This indicates that the temperatures were rather carefully measured and possess the accu- racy implied by Hsu. It is interesting to note that the temperature differences for the thermocouples nearest the interface indicate the presence of a contact resistance at the interface. The resistance would have the greatest effect at the earliest times when the heat flux is greatest. For Hsu #1 the calculated temperature is less than the measured tem- perature for the first thermocouple for times 5 and 10 seconds. For the Hsu #2 data the reverse is true. The effect Of a contact resistance would coincide with these Observations. Since these temperature dif- ferences are small, however, the assumption of constant surface tem- perature of each specimen is permissible in this case. The Handbook o_f Thermophysical Properties _O_f Solid Materials [34] gives the values for the thermal properties of nickel shown in Table 6. 3. 3. The values are given in English units and are to be ‘ 0.05 — .- 81‘ 3‘}. 0.05 — \ —- ‘ O V1 Ix‘“ ~0.05 — — 0:)“ 0.05 —’ /\ / '7 o / \/ N” -o.05 — -— ‘1' 0.05 _ / I F6 -0.05 — W '— ~ I I I l 0 5 IO /5 20 25 TIME (5150 Fig. 6. 3. 1 Differences between calculated and measured tempera- tures for Hsu #1. l T I I 0" 0.05 — //\ - k I O I¢ -o.05 — W ‘3'], 0.05 — 4 1 V \ 1““ -0005 - I l l l 0 5 /0 I5 20 25 TIME (51:0) Fig. 6. 3. 2 Differences between calculated and measured tempera- tures for Hsu #2. -83— compared with those values given in rows 3 and 6 of Table 6. 3. 2. Since the thermal conductivity varies 7% in magnitude for the tem- perature range 75 - 158°F (or 23. 7 - 69. 9°C) the property values are also given in Table 6. 3. 3 for Hsu #1 and #2 at the average tem- peratures which are 137 and 95° F. In each case the values of a calculated by us from Hsu's data are between the values for the initial temperature and the average temperature. The a-values calculated ' by Hsu are closer together and do not agree as well with the recom- mended values. A comparison Of the k and cp values calculated using the nonlinear estimation procedure (rows 3 and 6 of Table 6. 3. 2) with the published values shows excellent agreement. The values of the properties obtained for the Hsu #1 data using nonlinear estimation are based on the kc-value calculated by Hsu from his #2 data; the kc- product calculated by us is only 1% greater than the Hsu product. Hence, the property values in Table 6. 3.2 need be corrected only slightly. If the specific heat-density ratio were known, for say Hsu #1, then k for Hsu #1 and k and c for Hsu #2 can be calculated. The values of cp = 0. 1086 and p1 = 554. 9 which are obtained from [34] and are 1 consistent with a : 0. 5931 ftZ/hr were used to calculate kl : 35. 74, k = 37. 57 and c : 0.1061. (The c value is obtained from c with 2 p2 p2 2 p2 = 555. 6 lbm/ft?) These values are very close to the recommended values; the difference is less than 0. 5%. Hsu's experiment was not designed using the concepts of Opti- mum design developed in this research. It is interesting to note how close it compares with an optimum experimental design for a semi- infinite body. It does have the Optimum heating condition represented by a step rise in the surface temperature. The surface temperature -84— was not measured in either specimen but thermocouples were located near the surface (about 0.1 inch). (It is possible to place the thermo- couples as close to the surface as 0.01 inch as demonstrated by Lind- holm and Kirkpatrick; their experiment is discussed in section 7. 4.) The maximum dimensionless times based on the maximum distances from the heated surfaces to the furthest thermocouples are l. 72 and 4.15 respectively for Hsu #1 and #2. The optimum value of T is about 1. 25‘, this means that the furthest thermocouple in each case should have been located still deeper in each specimen. VII. TEMPERATURE-VARIABLE THERMAL PROPERTIES 7.1 Choice of Properties As far as the author can determine, there is no reference in the open literature to a transient method for calculating temperature- dependent thermal properties from a single experimental run. This is probably due to the great difficulty Of obtaining a convenient solu- tion by exact methods of the heat-conduction equation with tempera- ture-variable k and c. Temperatures can be calculated for this case using finite-difference methods almost as readily as for constant thermal properties, however. There are a number Of reasons why the variable properties case is important in this analysis. One is that thermal properties do vary significantly even over small tempera- ture ranges as noted in section 6. 3 for nickel. Next, the nonlinear estimation method has the potential of being much more accurate than any method used heretofore for calculating k and c or 0.. This is due to the negligible effect of random errors on the temperature and the ability to eliminate the problem of heat losses at the heated surface. Hence, the temperature-variation case is more important than for less precise methods. This case is also important to make the experi- ments more efficient; it is desirable to perform discrete experiments covering as large temperature ranges as practical. That is, instead of possibly performing ten experiments each covering 50° F intervals, it might be possible using temperature-variable properties to perform -85- -86- one experiment covering a 500°F range. The transient heat-conduction equation for temperature-variable kand c is a_ an" - 8_T 8320379480 (7.1.1) We shall assume that k and c are linear functions Of temperature; the temperature range considered can always be made small enough to permit this assumption. The properties k and c can then be expressed as leZ -k2Tl (kZ-kl)T k: T - T '1' —T-—:-,I.——' (7.1.2) 2 1 2 1 ‘ and C _C1T2 - cZTl + (CZ-cl)T (713) Tz'Tl T2'T1 where k:klandczclatT:Tl kzkzandczczathT2 The temperatures T1 and T2 can be chosen to be any convenient values such as the minimum and maximum temperatures obtained in the experi- ment. Instead of the expressions (7.1. 2) and (7. 1. 3) we could write . T-Ta kz'ka+kT -T (7.1.4) a 1 . T-Ta czca+cT -T (7.1.5) a 1 where -87- k=kandc=c atT:T a a a In the first two equations for k and c we need these thermal properties at two different temperatures. In the next two equations we find the properties at the temperature Ta (usually the mean temperature) and the changes of the properties with temperature. In theory it should make no difference whether we choose (7. 1. 2) and (7. 1. 3) or (7.1.4) and (7. 1. 5) since the same properties for k and c as functions of tem- perature must be found. But the convergence of the iteration proce- dure is affected in nonlinear estimation by the particular choice of properties functions [16]. For the special case of klzkzzka,cl:czzcaandTl:0 certain relations between the property derivatives can be derived in the manner discussed in section (2. 1). We find that T H |-1 k k +Tk (7.1.6) TC=T +T (7.1.7) but from (2. 1. 22) for Ti : O, T = -ka -cTC : -ka(Tk +Tk ) -ca(TC +TC ) 1 2 1 2 (7.1.8) where Tk and TC are the derivatives for constant properties. Generally Tk and Tk have the same sign and thus both usually are smaller in l 2 absolute value than Tk' The same statement can be made for (7.,1. 7). For the case of k = c = 0, T2 = 2 Ta and Ti : 0, one can derive T = T (7.1.9) Tk = 2Tk urk = Tk -Tk (7.1.10) 2 2 1 T = T (7.1.11) C C a T.C = 2T -TC : TC -TC (7.1.12) C2 2 1 If Tk and Tk are of approximately the same magnitude and sign 2 1 (which frequently occurs), then lTk 1>>1Ti<1 (7.1.13) m and similarly for T and T , C1 C2 ITc |>>|Té| (7.1.14) m Then if all the four properties are calculated simultaneously, the F-contour for the properties kl, k2, c1 and C2 can be shown to be much less attenuated than for properties km, k, cm and 6. Hence, fewer iterations are needed for the former. Even if only two prOper- ties are calculated with the other two fixed, the properties k and c as described by (7.1.2) and (7. 1. 3) are usually the better choice. 7. 2 Finite-Difference Equations For temperature-variable properties (7. l. 1) is nonlinear. The finite-difference equations (6.1. 3) and (6.1. 10) can be readily modified for this case, however. The thermal conductivity k appear- ing on the left hand side of (6. 1. 3) should be evaluated for the tem- peratures at time (m+l)A0; k‘s on the right hand side should be evaluated at time mAe . But the temperatures at time (m+1)A0 are unknown. Rather than iterating for each time step to make both krinfl and Tum-*-l consistent, sufficient accuracy can be obtained by -89- evaluating k and c in (6.1. 3) and (6.1110) at time mAe when the tem- peratures are known. This method accounts for the large variation of the properties with position and the smaller variation with time (prO- vided the time steps are made small). Another way to treat the non- linear case is to use a predictor-corrector method [35]. The thermal conductivity k: is evaluated at the temperature 1 and then T is used in (7. 1. 2) to calculate k. For a homogeneous body with a uniform Ax, Bn given by (6.1. 5) can be written as Bn=(Ak)ch/A 0, where cn is evaluated at temperature Tum. The quantity cl used in m 1 At the interface between two materials (6. 1.10) is evaluated at T or regions of different Ax, crfl in (6.1. 3) is evaluated at the tempera- ture T, T = 0.25Tm +0.75Tm n- n l 7. 3 Certain Cases In analyzing the efficacy of an experiment to determine thermal properties, it is necessary to examine the derivatives of the tempera- ture with respect to the properties. Two basic cases are considered in this section. The first case is for a heat flux producing a step rise in temperature at x = 0 and insulated at x = L. The derivatives for this case at x = 0 and L are shown in Figures 7. 3.1 and 7. 3. 2. This case has the same boundary conditions as case 15 of Table 5.1.1. The next case has the same boundary conditions as case 18 in the same table; these boundary conditions are a heat flux at x = 0 producing a x311 _. OJ ... -QZ _ -0.3 -04 I 1 I I 0 0,2 0.4- 0.6 0.8 /.0 2’ Fig. 7. 3.1 Derivatives Tk and Tk for a finite body with a q 1 2 atx:0to cause TmzC andq: 0atx=L. I _l_ I l 70,0"2310 - 0.5 — “ c’forfd) _a4 .- —03— — 72. Qatari-=0 -02~ . — -001 "’ _X__ ‘7 O I I I 1 I O 0.2 0.4- 0.6 0.8 /.0 Fig. 7. 3. 2 Derivatives TC and TC for a finite body with a q 1' 2 atx:0tocauseTm=Candq:0atx=L. 0.4 I I I ‘ I 0.3 " 0.2 _. 71: 0.1 -. O -0.I -‘ -0.2 T TO.3 _- _0.4. I I I I O 0.2 0.4- 0.6 0.8 #0 Z Fig. 7. 3. 3 Derivatives Tk and Tk for a finite body with a q _ 1 2 x = 0 to cause TmzC and a q at x = L to cause T = T. = 0. 1 I 1 73 - O 1 l O 0.2. 0.4 0.6 0.3 /.0 Fig. 7. 3. 4 Derivatives TC and TC for a finite body with a q at 1 2 x=0 to cause TmzC and a q at x=L to cause T=Ti:0. -92- step rise in temperature at that surface and a heat flux at x = L to maintain the temperature at T 2 Ti : 0 at that boundary. The deriva- tives for this case are shown in Figures 7. 3. 3 and 7. 3. 4. For both cases T1 and T of (7.1. 2) and (7.1.3) are 2 T1 2 0 (7.3.1) T2 = Tm (7.3-2) and for simplicity the special case of k = k : k (7.3.3) c : c = c (7.3.4) is treated. Then the derivatives calculated numerically can be checked with the relations (7. 1. 6), (7. 1. 7) and (7.1.8). For any two properties p1 and p2 to which the derivatives are related for a particular experiment by aplTpl : 6152sz (7.3.5) for all the positions at which temperatures are measured, we know that p1 and p2 can not be independently determined as discussed in section 2. 3. The quantities a and b are constants and one of which is not equal to zero. If (7. 3. 5) is not identically true but is approxi- mated in a particular experiment, the properties p1 and p2 can be independently determined; the accuracy of the calculated properties in such a case of correlated properties is usually not satisfactory, however. For both cases the derivatives Tk and Tk are highly corre- 1 2 lated and the same also is true for T and Tc . These high corre- 1 2 lations indicate that the property pairs (kl,k2) and (Cl’CZ) are difficult -93_ to accurately determine from data generated by experiments similar to those mentioned. Even greater inaccuracy would result if all four properties were determined simultaneously. This is not to say that the properties k1, k2, c1 and c2 can not be determined simultaneously; greater accuracy would be expected, however, if the two properties k and c were obtained from a series of experiments as discussed in a later paragraph. The property pairs (kl,cl) and (k 2) are much less correlated 2'C than (kl,k2) or (cl,c2). The derivatives with respect to c1 and c2 are always negative. The derivatives with respect to R1 and k2 are both negative at x : 0 and both positive at x = L. Because the derivatives with respect to k change sign from x = 0 to L but those for c do not, (or k the properties k1 and c and c2) are .less correlated than the l 2 pairs (k1,k2) and (c The A-values for the pairs (k1, cl) and 1’C21' (k2, c2) are given in Table 7. 3. 1 along with other quantities of interest. Since the case with T : 0 at x = L has the peculiarity that Z is maxi- mum at T equal to infinity the values for this case are given at the times indicated which do not correspond to maximum A for cases 3 and 4. For the two different boundary conditions considered, the low- temperature properties k1 and c1 have a larger A - value than for the high-temperature properties (though the difference is more marked for the last two cases than the. first two). Perhaps a more precise way to refer to properties kl and Cl is to call them initial-temperature proper- ties rather than "low-temperature" properties. Hence, in general based on these cases it is better to determine the initial—temperature properties rather than the final temperature properties. -94- com 000 moguomoum 30306 on So??? um ensue.“ mag-OH. coo oo-v 0.25.2698 on 833255. .mo 508:0on mo unseen magma HNSMCH oow coo com com ho .Uumpcmum mo ohdumuomeg amid: “Goeflhomxm 0.5330983 mo mcofiocsw mm o cam x madcap 30p 0“ mucosa-Comes ofiflmmom N .m .w OHQMH. E N N o ... e Hue 36 53. EN o.N Neood e . a hoe-.6 8:. a. E a H o u 2. Huh NN.o- 6.: 3.4 o; 6mNo.o e . a some .810 m E N N Hue Ne oé- ONN 34 good 0 . x o n a hoe-.6 N E a a Bus 26 m. - meN 0.0 88.0 e . .- ome See A 3 a O a 1H H x O 11- Uh a 16.800 o 80: .m HSOHGOOth mo 4 8 x 8 m m “cowowwmooo venommoe mad-ZN moxm Hogan on .QIHWIM ... m 4 2» uwmmu mMofizvcoo ommU Gown-20.300 uoflmcu mo 03964 Honda: mo 03m. Pd I u 0 ma umpcsom mofluomoum ofinmwumN’IousumuomEm-u no.“ mugoamumm a .m .N. 3an -95- For both boundary conditions the initial temperature could be either the highest or lowest temperature in a particular experiment. To experimentally Obtain the boundary condition at x = 0 most efficiently a standard material of accurately known thermal prOperties at initially a uniform temperature can be suddenly brought into contact with the specimen at another uniform temperature. After the first experiment is completed, the standard and specimen can be both heated (or in the case of the insulation condition at x = L, just the specimen need be heated) to higher temperatures as illustrated in Table 7. 3. 2. For experiment 1 the specimen is first heated to 400°F and the standard is cooled to 0 ° F; if the thermal properties are about equal for the standard and specimen the interface temperature during the experiment is about 200° F. If the properties k and c are known at 200°F in the specimen the properties kl and c at the initial temperature, 400°F, can be determined. After 1 this first experiment the temperatures can be adjusted to perform the second experiment suggested in Table 7. 3. 2 and properties found at 600°F. By performing a series of such experiments the properties k and c can be found as a function of temperature. The temperature range of each experiment is governed by the linearity of k and c with tempe ra- ture. For many materials a 500 - 1000°F range could be covered in a single experiment instead of the 200° F used in Table 7. 3. 2. Since these are transient experiments which simultaneously determine both k and c, the experimental time can be a small fraction of that required for conventional measurement of these properties. If the properties k and c are not known at 200° F to start the illustrated problem, then the temperature range for solely the first experiment is reduced so that the assumption of temperature -independent properties is valid. -96- 7. 4 Analysis of Lindholm and Kirkpatrick Transient Temperature Data Some accurate data providing transient temperature measurements over a large temperature range was obtained by Lindholm and Kirkpatrick in 1963 [36]. In their experiment with an instrumented copper rod a very large heat flux (about 1000 cal/cm2 - sec) was applied to the specimen by an arc-imaging furnace. The net heat flux to the specimen was mea— sured using a unique scanning-type radiation sampler. The heat flux was so large that the surface temperature reached the melting tempera- ture of about 1100°C in about one second. The experiment was designed to have an approximately uniform heat flux applied to a semi-infinite body. (See section 4.1.) The heat flux and temperature data obtained by Lindholm and Kirkpatrick is given in Table 7. 4. 1. Even though the experiment was very carefully performed the accuracy of the data is not as good as is desired for determining thermal properties. Lindholm [37] estimated the accuracy of the higher temperatures to be 1" 1% but the lower tem- peratures to be much less accurate and the heat flux to be within 'f 6%. The primary purpose of this experiment, however, was not the deter- mination of. thermal properties but theinvestigation Of the effect of very large heat fluxes upon the temperature gradients. An experiment designed using the guidelines given in this paper would be much better for determining k and c accurately. Lindholm and Kirkpatrick calculated the temperature distribu- tion of the copper and compared the temperatures with those measured. The properties at T1 = 20° and T : 1020°C that they used in their cal- 2 culations are given in Table 7.4. 2. Their values came from a handbook; more widely accepted values are given in reference 34. In Table 7. 4. 2 Table 7. 4. 1 Lindholm and Kirkpatrick temperature data for copper [36]. Time Heat flux, Temperature rise, T-Ti, C)C where Ti : 200C seconds cal/enn-secz 0.0254 0.280 0.660 1.296 1.930 2.532 cnn cni cnn cni cnn cum 0 910 0 0 0 0 0 0 0.1 918 372 174 65 0 0 0 0.2 940 541 305 128 25 4 .2 0.3 967 660 414 211 54 11 3 0.4 940 757 529 282 84 23 5 0.5 922 822 595 349 118 38 9 0.6 895 868 649 399 152 55 14 0. 7 842 909 698 444 186 72 20 0.8 900 972 749 489 222 83 28 Table 7. 4. 2 Thermal properties for copper. Temperature KL 0 RTP 0 CL 3 o LT133 0 °C Btu/hr-ft- F Btu/hr-ft- F Btu/ft - F Btu/ft - F 20 222 234 50.4 51.3 1020 _ 192 185 68.0 62.8 Table 7. 4. 3 Calculated thermal properties for copper from Lindholm and Kirkpatrick data Case Given properties Calculated properties ATrms’ 0C 1. k:230 C:52.6 16.7 2 Lindholm values CI: 47.0 k1 = 234 13.9 1< and.c 2 2 3 .TP values for c :48. 5 k :255 16.4 1 l lc and c 2 2 4 Lindholm values cl 239.5 C2 =92.7 8.7 for k.l andk2 -97- -98- the property values used by Lindholm and coworker have an L subs- cript and the recommended values with subscript TP. In reference 34 the k-curve for copper was Obtained from the results of fourteen experimenters one of which stated a '1‘ 10% accuracy, three gave 1 5% and the others did not say. Thus. the property values in Table 7.4. 2 are reasonably close considering the scatter in the data. Since the criterion A is larger for constant properties than for temperature-variable properties we expect the properties calculated for the temperature-variable case to beimore sensitive to errors than. for constant k and c. This is illustrated by the results for four different cases summarized in Table 7.4. 3. Case 1, which is for constant pro- perties, gives remarkably accurate values for k and c considering the accuracy of the given data. For both k and c the values are between the recommended values at 20 and 1020°C designated TP in Table 7. 4. 2. Further, the values are nearer the initial temperature values than the high temperature. This would be expected since many more low temperature measurements are made than high temperature mea- surements (see Table 7.4. 1). The rms temperature difference be- tween the calculated and measured temperatures is 16. 7° C or about 1. 7% of the maximum temperature rise. In section 7. 3 we noted that the initial-temperature properties could be found more accurately than the high-temperature properties for two heat flux boundary conditions. For this reason the low- temperatures properties were calculated using the high-temperature values of k2 and c2 in Table 7.4.2. The results are cases 2 and 3 of Table 7.4. 3. The error in the properties calculated in case 2 com- pared with the Lindholm values of the previous table are for k and c, -99- + 9. 5% and -6. 7% respectively; the error in k and c in case 3 with respect to kTP and CTP at 20° C are + 9. 0% and -5. 5%. Note that k has the largest error; this is consistent with case 1 of Table 7. 3.1 for which 8 = 3. 5°. These errors are not large considering the accu- racy of the given data. (We know that a constant six percent error in q will cause about a six percent error in a calculation for constant properties.) Also note that in calculating a kl or cl value we are not calculating a 2285 property value but are really calculating a property curve. Though the value given at 20°C may be 9. 0% in error, the value Obtained using (7. 1. 2) for 500° C say, usually would be much less. Using the Lindholm values for_‘k1 and k2, the calculated values c1 and c2 are given in case 4, Table 7.4.3. Since c1 and c2 tend to be highly correlated (as noted in the previous section), these c-values are not as accurate as found for the other cases. The values of c1 and c2 do exhibit the correct temperature behavior, however; that is, cl is less than c2. Except for case 4 the accuracy of the calculated pro- perties is better than might be expected from the given accuracy of the data. It is a characteristic, however, of nonlinear estimation to minimize the error introduced by random errors in the data. VIII. C ONC LUSIONS A general criterion is developed for dete rmining the Optimum experiments for calculating simultaneously the thermal conductivity and specific heat from transient temperature and heat flux data for solids. The criterion (designated A) for fixed maximum temperature rise, duration of the experiment and number Of thermocouples should be made a maximum for the optimum experiments. A number of boundary conditions for finite and semi-infinite bodies are considered. The two optimum experiments found by utilizing the A criterion are for finite bodies of thickness L. Each has a heat flux q at x = 0 to cause a step rise in the surface temperature; at x = L one Optimum experiment has q = 0 and the other has a q to cause the temperature at L to remain constant. The former is best for high and the latter for low thermal conductivity materials. The Optimum locations for thermocouples for these two experiments are at x : 0 and L. The effect of errors in the temperature measurements is derived and general results for a number of heat flux boundary conditions are given. Both thermal properties are calculated simultaneously from experimental data for (a) constant and (b) temperature-variable proper— ties. The errors due to the finite-difference procedure utilized can be readily made less than 0.1%. For one set of experimental data analyzed, -100- -101- the error in the calculated values Of properties is estimated to be less than 0 . 5% . IX. RECOMMENDED FUTURE WORK 9. 1 Design of Experimental Equipment There are two basic experiments that are best for finding thermal properties, cases 15 and 18 of Table 5.1.1. Both have a q condition at x = 0 which causes a step rise in temperature. This condition is easily obtained experimentally as discussed in section 5. 5. For the first case the finite specimen is insulated at x : L and the second has a q to make the temperature at x = L constant at the initial temperature. The first experiment is best for high conduc- tivity materials and the second for low k materials. The insulation condition is relatively easy to obtain. For case 18 if the specimen is the high temperature body, the initial temperature at x = L can be maintained by an electric heater. On the standard material side (for case 18) the surface at x = L can be maintained near its initial tem- perature by simply having this surface in good contact with a high conductivity material such as copper. The heater is needed on the specimen side to obtain the heat flux boundary condition at x = L. For transient experiments to be performed and analyzed in the most efficient manner an automatic data-reducing device is needed to read the analog thermocouple signals and reduce this data to punched cards. These cards are then used as input into the nonlinear estima- tion program. Before using the data, the program would direct the computer to transform the thermocouple readings to temperatures, -102- -103- correct the temperatures for a bias if necessary and then calculate the thermal properties. This automated method of measuring thermal properties can determine thermal conductivity and specific heat much more rapidly,‘ more accurately and probably less expensively than the conventional methods. It is well-adapted to meet the needs for a rapid, high- volume device for determining k and c. 9. 2 Other Applications of Nonlinear Estimation The nonlinear estimation method is very flexible and is by no means restricted to determining solely k and c. In the ablation prob- lem, for example, transient chemical reactions and transpiration occur in certain impregnated plastic materials. Also at high tempera- tures some solid materials simultaneously transport energy by both radiation and conduction. Other materials, such as iron, have second- order phase transformations which have not been thoroughly examined using a transient method to investigate the effects upon k and c. The total emissivity of solids as a function of temperature under transient conditions is of interest. In each Of these problem areas (which are related to heat—conduction) properties can be determined using non- linear estimation. In addition to determining properties, improved models to describe the physical phenomena can be developed in a rational manner by an extension of the nonlinear estimation method [38]. We need not even limit the application of the method to problems connected with heat-conduction. There are a number of problems in momentum and mass transfer that Obey similar equations. For example, the viscosity Of a fluid could be calculated from -104- transient velocity measurements in a fluid near a wall which is impul- sively moved. The equation describing this flow is identical in form to the transient heat-conduction equation. One might also use the method for calculating eddy viscosities and diffusivities in turbulent flow. The nonlinear estimation method is particularly useful for determining parameters from experimental data for problems in which there are two dimensions or one dimension and a time-variation to be considered and for which two or more parameters are to be calculated. 10. 11. LIST OF REFERENCES The Institution of Mechanical Engineers and ASME, Proceedings o_f the General Discussion 2 Heat Transfer (Atlantic City, New Jersey, 1951), pp. 290 - 295. W. D. Kingery, Property Measurements_al High Temperatures (New York: John Wiley and Sons, Inc. , 1959), pp. 125 - 129, 146 - 148, 273, 274. ASME, Progressfl International Researchfli ThermodynaLmic and Transport Properties (New York: Academic Press, 1962) pp. 395 - 518. S. B. Martin, "Simple Radiant Heating Method for Determining the Thermal Diffusivity of Cellulosic Mate rials, " Journal 3f Applied Physics, XXXI (June, 1960), 1101. B. Abeles, G. D. Cody, and D. S. Beers, "Apparatus for the Measurement of the Thermal Diffusivity of Solids at High Tem- peratures, " Journal o_f Applied Physics, XXXI (September, 1960). 1 W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, "Flash Method of Determining Thermal Diffusivity, Heat Capacity, and Thermal Conductivity, " Journal Of Applied Physics, XXXI (September, 1961), 1679. — E. L. Woisard, "Pulse Method for the Measurement of Thermal Diffusivity of Metals, " Journal 9f Applied Physics, XXXII (January, 1960), 40. M. Cutler and G. T. Cheney, "Heat-Wave Methods for the Measure- ment of Thermal Diffusivity, " Journal p_f Applied Physics, XXXIV (July, 1963), 1902. R. D. Cowan, "Pulse Method of Measuring Thermal Diffusivity at High Temperatures, " Journal 2f Applied Physics, XXXIV (April, 1963). 926. J. V. Beck, "Calculation of Thermal Diffusivity from Temperature Measurements, " Journal 2f Heat Transfer, LXXXV (May, 1963), 181. V. I. Smekalin, "A Method for the Rapid Determination of Coef- ficients of Temperature and Heat Conduction of Noncurrent- Conduct‘ing Materials, " (In Russian), Inzhen. - Fiz. Zh. , V (January, 1962), 99. ~105- 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. -lO6- S. T. Hsu, “Determination of Thermal Conductivities of Metals by Measuring Transient Temperatures in Semi-Infinite Solids, " Transactions o_ftthSME, LXXIX (July, 1957), 1197. . E. P. Box and G. A. Coutie, "Application of Digital Computers to the Exploration of Functional Relationships, " The Institution p_f Electrical Engineers, C111 (1956), 100. — . E. P. Box and H. L. Lucas, "Design of Experiments in Non- Linear Situations," Biometrika, XLVI (1959), 77. . E. P. Box, "Use of Statistical Methods in the Elucidation of Basic Mechanisms," Bull. Inter. Statist. Inst. , XXXVI (1957), 215. ’ . E. P. Box, "Fitting Empirical Data, " Annals N _Y;_. Academy o_fSciences, LXXXVI (1960), 792. . F. Gauss, ”Theoria Motus Corporum Coelestium, " Werke, v11 (1809), 240. . O. Hartley, "The Modified Gauss-Newton Method for the Fit- ting of Non-Linear Regression Functions by Least Squares," Technometrics, III (May, 1961). . W. Marquardt, "An Algorithm for Least-Squares Estimation of Nonlinear Parameters, ” Journal o_f the Society for Industrial and Applied Mathematics, XI (June, 1963). . Hald, Statistical Theory with Engineering Applications, (New York: John Wiley and Sons, Inc. , 1952) p. 527. . W. Dettman, Mathematical Methods in Physics and Engineering (New York: McGraw-Hill Book Co. , 1962‘) p. 186. . B. Hildebrand, Advanced Calculus for Engineers (New Jersey: Prentice-Hall, Inc. , 1948) p. 44.5. . W. Beck and H. Wolf, “Digital Program to Calculate Surface Heat Fluxes from Internal Temperatures in Heat-Conducting Bodies, " Avco Corp. , Research and Advanced Development Division, Wilmington, Massachusetts, RAD-TR-62-27 (August, 1962). . Courant and D. Hilbert, Methods of Mathematical Physics (New York: Interscience Publi‘sh'e'r's, 155‘. , 1953) p. 49. . Hald, Statistical Theory with Engineering Applications (New York: John Wiley and Sons, Inc. , 1952) p. 661. . M. L. Beale, "Confidence Regions in Non-Linear Estimation," Royal Statistical Society Journal, XXIIB (1960), 41. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. -lO7- J. V. Beck and H. Hurwicz, "Effect of Thermocouple Cavity on Heat Sink Temperature, " Journal o_f Heat. Transfer, LXXXII (February, 1960), 27. . V. Beck, "Thermocouple Temperature Disturbances in Low Conductivity Materials, " Journal if. Heat Transfer, LXXXIV (May, 1962), 124. . A. Bliss, Calculus of Variations (LaSalle, Illinois: The Mathematical Association of America, 1925) p. 138. . S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Second Edition (London: Oxford University—Press,—l959). . D. Richtmyer, Difference Methods for Initial-Value Problems (New York: Interscience Publishers, Inc. , 1957). . H. Crandall, Engineering Analysis (New York: McGraw-Hill Book Co. , 1956). . E. Forsythe and W.R. Wasow, Finite-Difference Methods for Partial Differential Equations (New York: John Wiley and Sons, 1960), ' . Goldsmith, T. E. Waterman and H. J. Hirschhorn, Handbook o_f Thermophysical Properties_q_f_Solid Mate rials, Volume I (New York: Pergamon Press, 1961). ' . Douglas and B. F. Jones, "On Predictor-Corrector Methods for Nonlinear Parabolic Differential Equations, "_J_. if the Society for Industrial ang Applied Mathematics, XI (March, 1963), 195. . S. Lindholm and R. C. Kirkpatrick, "Transient Heat Conduc- tion at High Thermal Flux, " ASME Paper Number 63-WA-347 (Presented at the Winter Annual Meeting, Philadelphis, Pa. , November, 1963). U. S. Lindholm, personal letter, December 18, 1963. . E. P. Box and W. Hunter, "A Useful Method for Model- Building, " University of Wisconsin, Department of Statistics, Technical Report NO. 2 (August, 1961). APPENDIX A Derivation p_f Equations (1.3. 8) and (1.3.9) Introducing (1.3. 3) into (1. 3. 2) and taking the derivative of F with respect to k gives 8 m 8F n 812' = 2; AJ. 5 (Tj(k0,co) +Tk,j(k0,cO)Ak+ Tc,j(ko,cO)Ac i=1 0 — Te,j)Tk,j(kO,cO)de (A-l) or 8F ‘1 — = A. T - _ .- ak 2[NTkAk+ (Tk,TC)Ac +3213 k,j(Tj Te’J)d0] (A 2) where (1.3.11) and (1. 3.13) are used in (A-2). At the minimum v value of F it is necessary that ar-a_0 Elk—8c- Then also evaluating BF/Bc, the corrections Ak and Ac are found by solving, _ — _ — A _ S . .- . NTk (Tk,TC) k ZAJdemJ Te,J)d0 . .(T. -T .)de 6.1 J 6.1 d — — l— —J (A-3) C L(Tk, TC) NT Ac EAJST The expressions for Ak and Ac are (1. 3. 8) and (l. 3. 9). -108- 14.0033. USE 011121., "IIIIIIIIIIiIiIIt