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50 C HAP.1 D ESCRIPTION OF T HE I NVERSE H EAT C ONDUCTION PROBLEM c.onduction problem is the estimation o f the initial temperature distributIOn from temperatures measured at later times. Comment on the relative difficulty o f recovering I i and T2 from measurements o f T a t x = 0 and 2a for the two cases of measurements of (1.t/a 2 equal to (a) 0.2, 0.4, 0.6, 0.8, and I ; and (b) 2, 4, 6, 8, a nd 10. r I I , I I i 1 .27. a. F or Eqs. (1.6.4), (1.6.5), and (1.6.7) a nd the convective boundary condition at x = L o f -k Ox L = h[T(L, t) oT/ T",,(t)] wher~ .h is constant, derive the differential equation and boundary b. condltl?ns for the sensitivity coefficient defined by Eq. (1.6.8). ~elate m words the results of part (a) to the problem of a unit step IDcrease o f surface heat flux. c. F or the case of x = L , t M _ I = 0, t < t M , k (x) = constant, c (x) = constant and hL/k = I, find numerical values for k X M/L for t + = 0.25, 0.5, a nd 1. C ompare the values with those for x /L in Table 1.1. CHAPTER 2 EXACT S OLUTIONS OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM 2.1 I NTRODUCTION Exact solutions of the inverse heat conduction problem are very important because ( I) they provide closed form expressions for the heat flux in terms of temperature measurements, (2) they give considerable insight into the characteristics of inverse problems, and (3) they provide standards of comparison for approximate methods. Inverse heat conduction problems can be divided into steady-state and transient problems. The steady-state inverse problem is simpler in that the only necessary thermal property is the thermal conductivity k, and a temperature history is not required. The transient inverse heat conduction problem can be divided into two categories: lumped thermal capacitance and distributed thermal capacitance. The transient case requires many discrete temperature measurements. F or lumped thermal capacitance, the important thermal property is the volumetric heat capacity p c. If the thermal capacitance is distributed, then the thermal conductivity k must be known in addition to the volumetric heat capacity. Throughout Chapter 2, the thermal properties are assumed to be independent of temperature ; this assumption is one of the weaknesses of exact solutions. Section 2.2 considers one-dimensional steady-state problems in which the temperature is known at two or more locations. Section 2.3 examines the lumped thermal capacitance case and some numerical approximations to the exact solution. Section 2.4 considers a planar semi-infinite body for which the surface temperature history is known; an approximate technique for numerically evaluating the resulting integral and an example problem are presented. Section 2.5 presents the development of an exact solution for a one-dimensional planar body with a temperature sensor at an arbitrary depth E below the heated 51 - 52 C HAP.2 S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM S EC.2.2 surface; this solution requires the existence o f all o rder derivatives o f b oth the experimental temperature d ata Yet) a nd t he heat flux q f: passing through depth E. Results for solid spheres a nd cylinders a re a lso presented. STEADY-STATE SOLUTION by t he weighting factors Wi ' I A weighted least squares criterion is defined as J L wf()j _ 1)2 S= I , II (2.2.3) j= t 2.2 F ourier's law indicates the temperature profile must be linear, STEADY-STATE SOLUTION X (2.2.4) T (x)=ax+b= - q"k+ To O ne o f t he simpler inverse solutions is for steady-state heat flow t hrough a p lanar o ne-dimensional body with constant thermal properties. F or this situation, Fourier's Law, dT q =-kdx There are two parameters (q, To) in Eq. (2.2.4) t hat a re d etermined such t hat t he weighted least squares e rror is a minimum. (To is the temperature a t x~O.) Differentiating Eq. (2.2.3) with respect t o t he two unknown parameters gives (2.2.1) as a - = - 2 ~ Wj(lj-1) - 1) = 0 L- 2 aTo A q= k ( YI - Y2 ) X 2- X I i=1 a - s =-2 is a differential equation and can be integrated directly. T he e ntering heat flux, q, is the same as a t a ny location x. S uppose t he steady-state temperature is k nown a t two depths ( XI' x 2 ) below the surface, as shown in Figure 2.1. Integrating Eq. (2.2.1) between locations X I a nd X2 a nd solving for q gives, L aq a1) 0 2 .. IwJ(yJ-T.-)-= J aq (2.2.6) Equations (2.2.5) a nd (2.2.6) involve two sensitivity coefficients which c an be evaluated from Eq. (2 .2.4), (2.2 .2) _ J=I, aT, aT, x· _ J=_...J. aTo where } j represents the experimental value o f t emperature a t d epth X i' E quation (2.2.2) demonstrates t hat a m inimum o f two experimental temperature m easurements a nd t heir corresponding locations along with the thermal conductivity must be known t o d etermine the heat flux. N ote t hat t he heat flux is l inear in the experimental temperature measurements 1'1; this linearity occurs repeatedly for constant-property I HCP's. As i ndicated previously, a minimum o f two temperature measurements is necessary for the steady-state determination o f h eat flux. S uppose t here a re J t emperature sensors located in a p lanar b ody for which the heat flow is o nedimensional. While there a re J sensors, the n umber o f d istinct sensor locations may be less t han J d ue t o m ultiple sensors a t t he same depth. T he s teady-state heat flux c an be calculated by minimizing the least squares e rror between the computed a nd e xperimental temperatures. I n o rder t o generalize the analysis, assume t hat s ome o f t he sensors a re m ore accurate t han others, as indicated aq (2.2.7) k S ubstituting Eqs. (2.2.4) a nd (2.2.7) i nto Eqs. (2.2.5) a nd (2.2.6), replacing q by its estimate q, a nd simplifying, the following two normal equations a re o btained : 1'1 I, I , I (2.2.8) q -To Lt W2jXj+ "2 L W2jXj2 = --k1 L Wj2 ljXj k k J J J j= I j= j= I Solving this system o f e quations for the unknown heat flux q= - k J-I ( J-I)( J J L wf J.=I w fxf L )J-I( - J (2.2.9) qgives, (.t wf)(.t WfXj lj) -(.t wfXj)(.t wflj) j =1 J-)1 2 . il (2.2.10) .L J-IWfXj I t c an be demonstrated t hat Eq. (2.2.10) reduces to Eq. (2.2.2) for J = 2 a nd W I' W2'" 1. Again, note t hat t he unknown heat flux is linear in the temperature measurements. Equation (2.2.10) could also be developed by (1) d etermining the two c onstants a a nd b in Eq. (2.2.4) by fitting a weighted least squares curve t o t he experimental temperature d ata a nd (2) differentiating the curve t o d etermine the heat flux (see Problem 2.1). However, the a pproach o f using sensitivity coefficients (aTjaq) is m ore c onsistent with the remainder o f t he text. Results ~-------~------~~ FIGURE 2.1 J j= ( 225) .. aTo Steady-state temperature measurements at J locations. b 4 64 C HAP.2 SEC. 2.3 S OLUTIONS OF T HE I NVERSE H EAT C ONDUCTION P ROBLEM 2.3 T RANSIENT A NALYSIS O F BODIES W ITH S MALL I NTERNAL T HERMAL R ESISTANCE E xact S olution F or bodies in which the thermal conductivity is very large a nd/or t he c haracteristic length scale L ( = volume/surface area) is small, it is possible t o ignore the internal thermal resistance. This analysis is referred t o as the lumped (thermal) capacitance analysis. T he energy balance on a body o f a rbitrary shape a nd o f surface area A, volume V, a nd uniform temperature T is V dT q (t)=pc A d t d TI (2.3.6) dt ' " Since the truncation e rror becomes smaller with increasing polynomial order, one might be inclined t o t hink t hat t he five-point equation is superior t o t he three-point (CD) equation. This would be true if the d ata were errorless; however, it is demonstrated t hat e rrors in the temperature d ata p laya crucial role in determining the accuracy o f t he computed heat flux. Instead o f r equiring a fourth-order polynomial t o pass exactly through five d ata p oint, an alternative approach is t o least-squares fit a straight line through five p oints a nd e valuate the slope a t the center point. This approximation can be determined from Eq. (2.2.10) by choosing all the weighting factors equal to unity a nd considering the five equally spaced points o f -:q... CIA', aI, 9, m;z~t; t he result is (2.3.1) N ote t hat q(t) d epends o n t he rate o f c hange o f t emperature a t time t a nd n ot o n temperature d ata for all times past a nd future. I f t he heat flux is known, the temperature response o f t he body can be calculated by integrating Eq. (2.3.1) T(t) = To + i' o A q(l) dl p cV (2.3.2) i ;<1-2 > -cf'll- d TI d t '" N ote t hat the temperature a t any time depends only o n t he p ast history a nd n ot o n the future values o f q(t). H eat flux measurement devices t hat satisfy the assumption o f negligible internal resistance are often referred t o as slug calorimeters. 2.3.2 55 T he symbol O( ) d enotes o rder o f m agnitude o f the truncation error, as determined by a Taylor series expansion. N ote t hat the C D a pproximation is a higher-order approximation a nd thus for small At h as a smaller truncation error than either B D o r F D. T he B O a nd F D results can be derived by passing a straight line through two points, a nd t he C D results can be developed by passing a parabola through three points. Higher-order results can be obtained by increasing the number o f p oints through which a polynomial is required t o pass. F or example, a fourth-order polynomial passing through five equally spaced points yields similar t o Eq. (2.2.10) c an also be developed for cylindrical a nd spherical geometries (see Problems 2.2 a nd 2.3). 2.3.1 T RANSIENT A NALYSIS O F BODIES - 2YM - 1- (.)-t,..,-t" M -f YM- 1 + YM+1 + 2YM + 1 , ~ -t~+ l .. 10At (2.3.7) N ote t hat when using any o f t he foregoing derivative approximations in c onjunction with the exact solution Eq. (2.3.1), t he heat flux is linear in the temperature measurements provided the volumetric heat capacity pc is constant. All o f these results have the appearance o f a digital filter. A pproximate S olutions 2 .3.3 Although Eq. (2.3.1) is a n exact solution for the heat flux, practical application o f this equation is generally with d ata available only a t discrete times. C onsequently, some difference approximations for d T/dt a re needed. I n t he discussion t hat follows, the temperature d ata (li) a re taken at equally spaced time increments At. T hree common two-point difference approximations t o d T/dt a re a s follows: d TI dt ' /II d TI dt '/11 dTI dt '/11 YM- y;M 1 A t - +O(At) B ackward Difference (BD) YM - YM +~t + O(At) F orward Difference (FD) YM - Y; + 1 M - I + O(At 1 ) 2At C entral Difference (CD) T emperature E rrors a nd A pproximate S olutions Any measurement o f t emperature contains errors, a nd these errors have a n i mpact o n t he computed heat flux . A convenient assumption to be used t hroughout this book is t hat t he temperature errors are additive (see Section 1.4.2). T his allows the measured temperature li t o be written as the sum o f t he true temperature T; a nd a n associated e rror 0 li (2.3.3) (2.3.8) Let us determine the e rror in the derivative approximation caused by temperature errors. F or the BD method, (2.3.4) d TI T M-TM- 1 OYM-OYM- 1 ~ +--=---~~ (2.3.9) dt ' " At At T he effect o f t he temperature errors is c ontained in the term (0 YM - 0 YM _ d/At. (2.3.5) L 56 C HAP. 2 SOLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM r Difference Approximation Backward Forward Central Five-point, fourth-order Least-squares, straight-line, five-point (2 .3.10) Equation Number Standard Deviation Derivative Approximation (2 .3.3) (2 .3.4) (2.3.5) (2 .3.6) (2 .3.7) .j~0'/61 = 1.4140'/61 . j20'/6t = 1.414O'/6t .j20'/(2/lt) = 0.7070'/61 0.950'/61 0'/ (.j1061) = 0.316a/61 TABLE 2 .2 T emperature D ata f or E xample 2 .1; f rom B eck a nd A rnold' (2 .3.11) (2.3.12) Observation Number -d t '" ;::; · L a jY",_j+j Time (s) Yj Temperature o f Dillet (O F) 9 10 11 12 13 14 15 16 17 The single error bY", affects only the calculations of q", and q", + 1 for the BD method, with all other q calculations being unaffected. For the F D method, only q", _ I a nd q", are affected by the single error bY", . At first glance, it might seem inappropriate to study only a single error because all temperature measurements are likely to have errors. However, the effects of all temperature errors can be superposed for linear problems (constant properties). I f all errors b Yj are the same, there will n ot be any heat flux error because they cancel out at each time step. This can be understood by realizing that the slope of the temperature response curve does not change if all temperature measurements are shifted by the same amount. In general, all difference approximations can be written as dTI 67 T RANSIENT A NAL VSIS OF BODIES T ABLE 2.1 S tandard D eviation o f E rror i n D erivative A pproximation f or T emperature E rrors t hat a re I ndependent, A dditive, a nd o f C onstant V ariance ( 0'2) N ote that the form of the error term is the same as the difference approximation itself. This will always occur in linear problems with additive errors. Similar results can be written down by inspection for the other difference approximations considered. I t is i mportant to understand how a single temperature error affects the heat flux computed a t the same time the error was made and how the error affects any subsequent calculations o f heat flux . Suppose the backward difference approximation is used a nd all temperature measurement errors are identically zero except bY",. F rom Eqs. (2.3.1) a nd (2.3.9), V bY",) q (t",)=q",=pc - ( T",-T"'_I +A L\t L\t SEC. 2.3 768 864 960 1056 1152 1248 1344 1440 1536 191.65 184.44 177.64 171.41 165.04 159 .89 155.19 150.78 146.68 EXAMPLE 2. I . A solid copper billet 0.0462 m (1.82 in.) long and 0.0254 m (1 in.) in diameter is heated in a furnace and then removed. Two thermocouples are attached to the billet. Some temperatures, Y" given by one of the thermocouples are listed in Table 2.2 as functions of time. See also the plot of Yj versus time in Figure 2.2. F or this test, pcV/(A61) = 200 W/m2-K (31 Dtu/hr-ft 2 -F). Compare the five methods presented i~ this section for calculating the derivative d T/dl. These data are from an actual expenment presented on p. 243 o f Deck a nd Arnold I . (2.3.13) j =1 where n is the number of points used in the difference approximation; Eq. (2.3.13) is a compact way o f writing Eqs. (2.3.3) - (2.3.7). F or all of the difference approximations considered in this section, L~ . I a j=O ; it is this characteristic that allows a uniform shift in all Yj to have no effect on the slope calculation. If certain statistical assumptions are made about the temperature errors, then it is possible to calculate the standard deviation of the error in the difference approximation. For example, if the temperature errors are independent, additive, and of constant variance (0'2), the methods of Beck and ArnoldI can be used to calculate the results in Table 2.1 (see also Section 1.4). These theoretical results favor the least squares approach because the standard deviation of the heat flux estimate is smallest. Solution. T he results of Example 2.1 a re summarized in Table 2.3 and Figure 2.3. Several conclusions can b e drawn from this example. 1 . Doth F D and DD give the same numerical results but they are displaced in time by 61 . 2 . All methods except DD use temperature measurements at times greater than the calculation time. This will be referred to as using "future (temperature) information." I b 68 CHAP. 2 SOLUTIONS OF THE INVERSE HEAT C ONDUCTION PROBLEM I , S EC.2.4 ! 410 HEAT FLUX F ROM MEASURED SURFACE TEMPERATURE HISTORY 59 8 .0 ~-"'---"--'"T"---r--...--".---r---r---r--'" 400 7.0 390 '!" ~ 380 ! E 370 ~ .. .. ! ~ . 220 .... 6.0 ! E ~ 360 5.0 180 0 0 350 0 160 BO FO CO t::.. 4 .0 2 4 6 8 10 12 14 16 Number of time steps, i 0 0 Eq . (2.3.6) V 3 40 Eq . (2.3.7) 3 2 4 5 6 7 8 9 10 Time index, M FIGURE 2.2 Temperatures for cooling billet example (Example 2.1). FIGURE 2 .3 Average temperature difference (AT) for Example 2.1. T ABLE 2.3 A verage T emperature D ifference ( AT) f or E xample 2.1. pcV - - 11M =A At AT. A T i n ' F M YJI - Y - 1 JI 9 10 - 7.21 - 6.80 -6.23 - 6.37 - 5.05 - 4.70 - 4.41 - 4.10 11 12 13 14 15 16 17 3. YJI + 1 - YJI - 7.21 - 6.80 - 6.23 -6.37 - 5 .05 - 4.70 - 4.41 - 4.10 ! (YM + 1 - YM - - 7.01 - 6.52 - 6.30 - 5.76 - 4.93 - 4.56 - 4.26 1) Five-point, f ourth o rder Five-point linear least squares 6 . T he l inear least-squares result is appealing because t he s lope continuously D decays with time in a s mooth m anner. 2.4 H EAT F LUX F ROM M EASURED S URFACE TEMPERATURE H ISTORY 2.4.1 E xact R esults f or C ontinuous S urface T emperature H istory - 6.47 - 6.35 - 5.81 - 4.85 -4.54 - 6.63 - 6.17 - 5 .64 - 5.11 - 4.58 All m ethods except F D a nd B D show a continuous decay o f s lope with time. T he physics o f t he p roblem dictates t hat t he magnitude o f t he average temperature difference s hould decrease continuously with time. 4 . By using m ore i nformation (additional temperature measurement) t o c alculate h eat flux a t a given time, some values c annot be c alculated a t b oth early a nd late times. I f the surface temperature, Y(t), o f an object is known continuously as a function of time, some relatively simple exact solutions exist for the heat flux variation with time. This surface temperature specification yields a simpler inverse problem because the known surface temperature can be treated as a boundary condition in the traditional sense. O ne approach to this problem is to determine the temperature distribution within the body and then the temperature gradient at the surface is used to determine the heat flux. I f the thermal properties are treated as constant, then Duhamel's theorem provides a convenient means of calculating the temperature field. The analysis starts with the temperature form of Duhamel's theorem 2 ,3 ,4 , d Y(,i) N -I T (x, t) = To + ' 0 u(x, t - ,i) ~ d,i + i~O u(x, t - ,ii)~ l i f. (2.4.1) where u(x, t) is the temperature response function for a body a t zero initial 60 CHAP. 2 S OLUTIONS OF T HE INVERSE H EAT C ONDUCTION PROBLEM r I , S EC.2.4 HEAT FLUX F ROM M EASURED SURFACE TEMPERATURE HISTORY 61 I t emperature a nd subjected to a unit step in surface temperature, Y(t) is the surface temperature variation with time, and To is the uniform initial (for t ~ to) temperature. T he integral in Eq. (2.4.1) allows a continuous surface t emperature in time, and the summation term allows N discrete steps in surface temperature occurring a t A.; = illt. Some understanding o f D uhamel's theorem can be gained by considering the discrete version o f Eq. (2.4.1). I f a series o f steps in the surface temperature occurs as shown in Figure 2.4, t he temperature a t p osition x a nd a t time t for 21lt < t < 31lt is given by I I I , T (x, t) = To + u(x, t)1l Yo + u(x, t - Ilt)1l Y + u(x, t - 21lt)1l Y 1 2 i (2.4.2) I L ~~Ix=o = - k au(xa~-A.)lx=o Y'(A.)dA. i x) (2",l1.t C' a - ul a -I = c= x x=O ",Xl1.t =- ~ f e - o y.'I L 2 (2.4.5) L .=0 a x x=O x 1'.=(2n+ 1) 2' n =O, 1,2, . .. O ther step function solutions for various geometries a nd/or "inactive surface" b oundary conditions are available in the literature. The analysis t hat follows is restricted t o t he semi-infinite planar solid; for this case, Eq. (2.4.3) becomes q(t) = fkPC -vn J[r.' ~dA. fkPC -vn {Jt (2.4.6) " ,t-A. = 1 1 [ ' Y(t) - Y(A.) } [ Y(t)- Y(t o)] +2 (t_A.)312 dA. J,. (2.4.7) Equation (2.4.7) comes from integration by parts o f Eq. (2.4.6). Note that b oth o f these forms have singularities a t A.=t. I f a purely numerical procedure is used, these singularities can cause some difficulty. Therefore, a n analytical integration procedure is considered. T he r eader is reminded t hat Eqs. (2.4.6) and (2.4.7) presume that the body is a t a uniform temperature Y(t o) for t~ to. l t may be possible to use a gaussian q uadrature S t o t reat this singularity; however, such an approach would not allow the experimenter freedom in choosing times a t which the surface temperature is sampled. I n the next section, a combined analytical numerical procedure will be explored. (2.4.3) where Y'(A.) = d Y /dA.. E quation (2.4.3) is exact within the restrictions o f D uhamel's theorem. I n o rder t o apply Eq. (2.4.3), the derivative o f the unit step response must be known. The simplest case o f a unit step function is for a semi-infinite planar solid. u(x, t )= I -erf au I I The magnitude o f t he actual step in surface temperature is multiplied by the response due to a unit step in surface temperature, u(x, t). T he unit step response, u, must be shifted in time to correspond t o the time when the temperature step actually occurs. Additional information on Duhamel's theorem c an be found in Carslaw a nd Jaeger 2 , Schneider 3, Myers,4 and C hapter 3. I f only the heat flux is o f interest, it is n ot necessary to calculate the entire temperature field. The heat flux a t the active surface can be determined from Fourier's law by differentiating Eq. (2.4.1) t o get [for Y(t) c ontinuous in time t], q(t) = - k " . u(x, t ) = I - 2 ~ -I e _o y. I L' sm 1'.x ~ .=01'. L I (2.4.4) 2,4.2 A pproximate R esults f or S emi-Infinite B ody w ith S urface T emperature M easured a t D iscrete T imes Another simple case is t he infinite plate o f thickness L where the step change in surface temperature occurs a t x=o a nd the "inactive surface" is perfectly insulated. Assume t hat t he surface temperature l j is measured only a t discrete times t j • Between successive times, the surface temperature is a ssumed to vary linearly with time. F or these assumptions, Eq. (2.4.6) can be integrated analytically t o give (2.4.8a) =2 fkPC -vn f ; =1 (2.4.8b) where the symbol denotes a n e stimated value. The important material property group is (kpc)tI2; values o f this parameter are presented in Table 2.4. F or a given heat flux into a semi-infinite body, a A FIGURE 2 .4 Illustration o f temperature form o f Duhamel's theorem. - 11-11-1 J tM-t;+JtM-t;-t b 62 C HAP. 2 S OLUTIONS O F T HE INVERSE HEAT C ONDUCTION PROBLEM T ABLE 2 .4 T hermal P roperty S EC.2.4 JiPC HEAT FLUX F ROM M EASURED S URFACE T EMPERATURE HISTORY 6 3 2 000r------.--,.----,---.----,.---r----.1 .5 Material Copper, pure Silver, pure Aluminum, pure Steel, low carbon Steel, 20% C r Steel, 40% Ni Glass, plate 106.8 1150 1000 703 ( q"",. = 12,000 Btu/ft2 - s) 92.9 65 .3 41.0 26.2 18.4 3.9 442 282 198 42 k pc=3.2855 (ft2~:-R) 1500 2- S 1.0 . I t: 1 -' 1000 body with a small value o f kpc has a larger surface temperature rise than bodies with large kpc. Physically, this occurs because a large-k body is able to remove the heat from the surface more rapidly. Note that the heat flux is linear in the temperature measurements, provided (kpC)I/2 is independent of temperature. E XAMPLE 2.2. I n o rder t o d emonstrate the method given by Eq. (2 .4.8), a n example problem is selected for which a n exact solution is available. Consider a heat ftux with a parabolic variation with time q - = 4 t ( 1 -t-) - q.... I.... (2.4.9) t..... Using t he analytical solution o f Carslaw and Jaegerl for q -t" a nd superposition, the surface temperature variation is given by (see also Problem 1.4). 7 ;(t)- To= 4qm•• [ r (2) ( t ~ r(5/2) tma• )3/1 - r (3) ( t rm tmax )5/1J (2.4.10) .Jr= where r (x) is the G amma function. [ r(2) = 1, r (3)=2, r (5/2)=37t lll/4 a nd r (7/2)= 157t l/l/8.] T he results o f t he foregoing analytical solution are given in Figure 2.5 for the following parameters: q. ... = 12,000 Btu/ftl-sec, Btu k pc=3.2855 ( - 1I t s-R )1 To = 5400 R -s (Copper a t elevated temperature) These conditions are intended t o b e representative o f those encountered in traversing a copper calorimeter across a high heat ftux arc jet. T he exact temperature d ata calculated from Eq. (2.4.10) for a time step o f AI =0.001 s applied t o t he inverse solution given by Eq. (2.4.8), yield t he results shown in Figure 2.6. At early times, the percent e rror in the computed heat ftux is in excess o f - 15%; after a time o f 0.01 s t he heat ftux error remains nearly constant a t approximately - 2% until approximately 0.065 s. As the final time o f 0.07 s is approached, the errors become large a nd positive. This is because the heat ftux is a pproaching zero a l 0.07 s. 0 0.01 0.02 0 .03 0.04 Time,s FIGURE 2 .6 Parabolic heat flux history and corresponding surface temperature variation. Although the results for the preceding example are good, the use of exact temperature data is not a very severe test of any inverse method. In order to simulate the effect of temperature errors on the computed heat flux history, a random error term was added to the results computed from Eq. (2.4.10) Y. = 'T.(t.. ) + (J.l1 T . where (J is a random variable o f uniform distribution with values in the range [ - 1,1] and l1 T is the maximum magnitude of the temperature error. F or the calculations that follow, it is assumed that l 1T=SR; the calculated errors are shown in Figure 2.6. In general, the results using simulated temperature errors scatter about the results using errorless temperature data. However, the realistic temperature d ata produce a much wider variation in the heat flux error. 2 .4.3 L""';O T emperature E rror P ropagation i n Eq. ( 2.4.8) I t was demonstrated in Section 2.3.3 t hat if an error {, Yj is made in a single temperature measurement Yj, the corresponding heat flux e rror becomes identically zero after n time steps where n is the number of measurement points a pp,.,imal;n, Iho dod. .I ;" d Tld,. F o, a "m;-;ofiruto b od, Ihal . ... C HAP.. 2 64 S OLUTIONS OF T HE INVERSE H EAT C ONDUCTION PROBLEM 5 I I a a • ; ;;0 G0 . .. g ° a a e a •• e a • a ee e 0 Q e e e e e e a . e a e • ..a . e ,- a ~ 1t~1 0 represents the dimensionless heat flux e rror and shows how it decays for subsequent times. The normalization is conveniently chosen so that the first value is unity. Table 2.5 presents results showing how the dimensionless heat flux error due to an initial temperature error, b Yo, decays with time. The results of Table 2.5 indicate that a positive temperature error b Yo causes a negative error in heat flux and this error damps very slowly with time. Even after 100 time steps, the dimensionless heat flux e rror is still 5% o f its initial value. I f the initial heat flux e rror was large, then this heat flux may be very inaccurate. For a given value of temperature error b Yo , the heat flux e rror is proportional to ( kpLf/l and inversely proportional to I/(~r)l/l; small time steps and large values o f kpc b oth produce large heat flux errors. Equation (2.4.11) reveals that an error in Y1 may be quite different from an error in Yo because Y appears in two places in the heat flux equation. I f a single 1 temperature error ( i YM occurs, the resulting heat flux e rror and its decay with - ~ - 10 co ~ Exact thermocouple data e 5R" Random thermocouple error e 1 0.02 J J J 1 0.03 0.04 0.05 0.06 0.07 Time,s FIGURE 2.6 Relative heat ftux error (%) using exact and inexact thermocouple data. finite internal thermal resistance, the heat flux e rror corresponding to a single temperature error takes an infinite number o f time steps to decay to zero. This can be demonstrated by considering the first few terms o f Eq. (2.4.8), Jfffr TABLE 2 .5 H eat F lux E rror f or 6 Y 0 a t I nitial T ime [ (Y1 - YoX.JM - .JM - 1)+(Y1 - Y1X.JM - I-.JM - 2)+ . . 'J, M = 1 ,2,... M (time index) (2.4.11) where the data were taken with equal increments of ~t. Note that the initial temperature measurement Yo appears in only one place in Eq. (2.4.11). Suppose there is a n error b Yo in the temperature Yo( = To + b Yo) b ut all other temperature measurements are exact. The corresponding error in heat flux can be determined from Eq. (2.4.11), q u=2 fkPC .Jruft [(Y1- -2 (2.4.13) a ~ q u=2 M = I , 2, . .. Jfffr 0 'l;j ° -201 0.0 0.01 H EAT FLUX F ROM M EASURED S URFACE T EMPERATURE H ISTORY 6 5 - I (iq'iP -:-- =.JM - .JM - I, kpc ( i Yo 2 -• . 0 >< ::J : ;: S EC.2.4 the term in Eq. (2.4.12) that contains the temperature error ( i Yo represents the corresponding heat flux error. Let (iqC;:' be the error in q u corresponding to the temperature error ( i Yo; the superscript on (iqM is important because it indicates the time at which the temperature error occurred. Then I I e 0- #. I I r ToX.JM - .JM - 1)+(Y1 - Y1X.JM - I-.JM - 2)+ . ..J .IfkPC bYo(.JM-.JM-l) n&i (2.4.12) The term inside the brackets in Eq. (2.4.12) is the heat flux for errorless data; I L I 2 3 4 5 6 7 II 9 10 100 ---- -I b,l;:' - - -- 2 ·PC bYo ~ - 7tt'11 1.000 0.414 0.31!! 0.26!! 0.236 0.213 0.196 0.1!!3 0.172 0.162 0.050 66 C HAP.2 S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM subsequent time steps are as follows: lJqW' ---=10 Pc lJY", . 21tdt 1 (2.4.14a) ~ 1 2 ~ kpc o ", i = I, 2, . . . M =I,2, . . . (2.4.14b) 1 tdt The subscript on q represents the computation time t ",=Mdt, a nd the superscript indicates the time a t which the temperature error occurred. Note that the dimensionless heat flux error in Eq. (2.4.14) is independent of the time ( M) a t which it occurred, provided M + 0. T he results of Eq. (2.4.14) are given in Table 2.6. T he heat flux e rror corresponding to lJY",(M + 0) decays considerably faster than the error corresponding to lJYo. This implies that greater care should be taken in measuring Yo t han other values o f Y. I t should be reiterated that all temperature measurements will contain errors. A complete error analysis can be accomplished by superposing the heat flux e rror calculations for a single temperature error because the problem under consideration is linear. T ABLE 2 .6 D ecay o f H eat F lux E rror R esulting F rom T emperature E rror CSY", a t T ime t,w=MIlt i (time index) I 2 3 4 5 6 7 8 9 10 100 1.00000 - 0.58579 -0.09638 - 0.04989 - 0.03188 -0.02265 -0.01716 -0.01359 -0.01110 -0.00930 - 0.00793 -0.00025 EXACT S OLUTIONS OF INVERSE HEAT C ONDUCTION PROBLEMS 67 2.5 EXACT S OLUTIONS OF INVERSE H EAT C ONDUCTION P ROBLEMS 2.5.1 lJqW! I r :;t r; r:t ~y. = ",,+ 1 -2", . +",.-1 o SEC. 2.6 L iterature R eview Few exact solutions to the inverse problem of heat conduction for which the temperature sensor is a t an arbitrary location are available in the literature. This is in contrast to the direct problem of heat conduction for which a wide range of solutions is available. Burggraf6 presented one of the earliest exact solutions. He approached the problem by assuming that both the temperature yet) and heat flux q dt) were known a t a sensor location. The temperature field .was developed in terms of an infinite series of all-order derivatives of both yet) and qE(t). I f the temperature sensor was located a t the center of a solid cylinder or sphere, then qE(t) was identically zero (for one-dimensional radial heat conduction). Langford 7 independently developed results similar to the Burggraf solution. Kover'yanov 9 deVeloped results for hollow cylinders and spheres. The heat flux a t the exposed surface was determined by differentiating the temperature field. Imber and KhanS obtained an exact solution for the temperature field using Laplace transforms when the temperature was known at two distinct interior points. Their temperature solution can be extrapolated in b oth directions toward the boundaries. The extrapolation distance is limited to the distance between the two temperature sensors. N o computational results were presented for the more difficult prob!em o f calculating heat flux a t the exposed surface. 2.5.2 D erivation o f E xact S olution f or P lanar G eometry The analysis that follows closely parallels that of Burggraf. 6 The body is divided into (I) an inverse region and (2) a direct region as indicated in Figure 1.4. The direct region has conventional boundary conditions: specified temperature yet) a t the left face a nd arbitrary boundary conditions a t the "inactive surface," L. By some means, it is necessary to solve for the temperature field in region 2. Next, the solution is differentiated a t the location of the temperature sensor in order to calculate the heat flux qE a t X I = E. Once the heat flux a t the sensor location is calculated, the inverse problem has two boundary conditions specified a t the same boundary. The Burggraf solution requires that qE(t) and all of its derivatives are known. Starting with the constant-property form of the energy equation, (2.5.1) and differentiating it with respect to time yields, 68 C HAP.2 S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM a2 T a aT ai2=ar (exV 2 T)=cxV 2 a r =cxV2(cxV2T) t; 2 aa =ex 2V4 T (2.5.2) SEC. 2.5 EXACT S OLUTIONS OF INVERSE H EAT CONDUCTION P ROBLEMS 69 E quation (2.5.10) is t he general solution for the t emperature field in the inverse region. T he r emaining p roblem is t o d etermine t he functions I (r) a nd g(r). T hese functions a re f ound by substituting Eq. (2.5.10) i nto t he differential equation, Eq. (2.5.1), G eneralizing for a n a rbitrary t ime derivative gives anT =cx"V 20 T at" (2.5.3) T he s ame t ype o f p rocedure is applied t o F o urier's law : q = - kVT (2.5.4) (2.5.5) F or t ime derivatives o f t he heat flux o f a rbitrary o rder, t he d erivative is A solution is o btained b y requiring t hat e ach t erm i nside the brackets o f Eq. (2.5.12) is identically zero, V2J,,= - 10-1 } 1 cx " =1,2• ... T he t emperature field is assumed t o b e a n infinite series involving t he t emperature g radients a t t he t emperature s ensor location r = E, T(r, t )= Jo Ho(r)VOTL£ 2I 7 (2.5. ~d o dd terms, H 2i(r)V TI.=£ + i~O H 2 i+ 1(r)V 2i +I T I.:£ (2.5.9) 1 aiTI co 1 aiql H 2i (r) i -a i + L H 2i + l (r)(- k i) -a i ex t r=f: i=O - CX t r=E ao d iy 1 ao diqE d' = L / i(r)-i - - L gi(r) - i q l. i=O dt k i=o dt / ~ Y w here H 2i } ;(r)=-.(r), ex' gi(r) H 2i + l (r) exi alTI at i r=£ ~ iY i i1' 10(E) = I , go(E) = 0, J,,(E)=go(E)=O. " =1,2, . . . (2 .5.14) a TI ao d "Y co dOq . q E=-k=-k L I~(E) - + L g~(E)--,t. o ar r=E 0=0 dt 0=0 dt co L i=O or Also t he h eat flux is given by S ubstituting Eqs. (2.5.3) a nd (2.5.5) i nto E q. (2.5.8) yields T(r, t )= The b oundary c onditions o n t he 1 a nd 9 f unctions a re d etermined from t he r equirement t hat t he s olution exactly matches the t emperature d ata Y(t), (2.5.8) m = 2 s phencal I t is convenient t o d ivide t he series in Eq. ex (2.5.7) 1 d ( d T) m =O p lanar V2 T=- - t " - m=lcylindrical t " dr dr . Jo (2.5.13) 1 V go=-go-I 2 I n t he analysis t hat follows, the geometry is restricted t o o ne d imension such t hat T(r. t) = V 2 g 0 =0 V210 = 0 (2.5.6) Jt C ~ aiql a t i r= £ = at i (2.5.10) (2.5.11) L J'~ ~ d~C go(E) = 1, lo(E) = 0, 1~(E)=g~(E)=O, " =1.2, . . . (2.5.15) The s olution t o Eq. (2.5.13) subject t o t he b oundary c onditions given by Eqs. (2.5.14) a nd (2.5.15) completely determines t he 1 a nd 9 functions. N ote t hat these functions m ust be determined in a sequential m anner s tarting with 10 a nd go. C HAP.2 70 S OLUTIONS OF T HE INVERSE HEAT C ONDUCTION PROBLEM Observations 1. F or an insulated surface a t r = E, the f-series alone determines the temperature profile. 2 . F or an isothermal surface a t r =E, the g-series alone determines the temperature profile. 3 . The functions fo a nd go represent steady-state solutions. 4 . All-order derivatives of Y(t) a nd qE(t) must exist. By direct substitution, it is shown that 1 ( E_x)2n - 1 ( E_x)2n+1 = (2n)! ]. (Xn ' ;n (2n + I)! (Xn (2.5.16) Solution f or Planar Geometry ( r=x). .r. is a solution to Eqs. (2f."1~- (2.!14f. The temperature distribution is written as co 1 ( E_x)2n d ny T (x, t )= Y (t)+ n~1 (2n)! (Xn dtn (E - x) [ co 1 (E - x)2n d nqE ] + -k- qdt) + J d2n + I)! (Xn dtn (2.5.17) The planar geometry solution is similar in appearance to a Taylor series expansion about the temperature sensor depth. The heat flux a t the active surface is o f primary interest; it is determined by using Fourier's law and Eq. (2.5.17) and evaluating at x = 0: co E2n-1 1 d ny co E 2n dnqE , (2.5.18) q (t)=qE+k n~1 ( 2n-1)!;'; de" + n~1 (2n)! dtn ;foI t is convenient to normalize the heat fluxes and introduce a dimensionless time as follows: qE (Xt qEE Q =T' Q E=T' ' E= E2 f f.+n=1 f 1 d ny _ 1_ dnQE ( 2n-l)! d'E +n=1 (2n)! d'E EXACT S OLUTIONS OF INVERSE H EAT C ONDUCTION PROBLEMS (2.5.20) This is a very important result because it is exact for continuous temperature measurements and has a simple form. The solution clearly shows the dependence of the surface heat flux on all orders oftime derivatives of the measured temperature and the related heat flux, both at x = E. The temperature level itself is not significant since only derivatives are needed. I t is surprising that the solution given by Eq. (2.5.20) shows no explicit dependence on the initial temperature distribution in the body. Burggraf6 has pointed out, however, that a polynomial fit of finite order to the experimental temperature data, Y(t), implies that the initial temperature profile must be nonuniform. 71 Since the inverse solution given by Eq. (2.5.20) is exact, one might ask if there is any need for further investigation. The answer to this is a definite yes because the exact solution has several practical limitations: 1. 2. High-order derivatives of discrete temperature data Y(t) and heat flux QE must be evaluated numerically. I t is awkward for composite bodies and not appropriate for temperature- dependent properties. 3 . I t may require a numerical procedure to solve the direct problem and determine Q,;{t). 4 . I t is not applicable to the overspecified problem of more than one interior temperature sensor. S . The method does not lend itself readily to the case of multiple heat flux determination such as the two-dimensional case. Although the above exact solution has limited applicability in a practical sense, it is extremely important because o f the insights provided. 2.5.3 Expressions f or C ylinders a nd S pheres Solid cylinders and spheres in which the heat flux is one dimensional radially and the temperature sensor is located a t the center have simple exact solutions for the IHCP. Consequently, the heat flux a t the exterior surface depends only on the temperature response and its derivatives. From Burggraf6 and Langford,' the temperature fields for solid cylinders and spheres are given by co (2.5.19) Note that Q has the units of temperature. The normalized heat flux is written as Q =Q. SEC. 2.5 L "= ( R_x)2n d ny . 22n( ,)2 n -d n (cylinder) 1 n. (X t co ( R_x)2n d Ry T (x, t )= Y (t)+ L (2 1 )'" -d" (sphere) n=1 n + .(X t T (x, t) = Y (t)+ (2.5.21) (2.5.22) Note that the coordinate x is attached to the exterior surface and that R is the external radius. The heat flux is evaluated from Fourier's law and Eqs. (2.5.21) and (2.5.22); the results are q= - k a TI co nR 2n - 1 d Ry I( ,)2 n -d " n. (X t - a x=O = k "L1 22" x = a TI co 2 nR 2n - 1 d ny q= - k =k a x x=O "= 1 (2n + 1)!(X" dt" L - (cylinder) (sphere) (2.5.23) (2.5.24) Comments on hollow cylinders and spheres can be found in Burggraf,6 Langford,' and Kover'yanov. 9 72 C HAP.2 S OLUTIONS OF T HE INVERSE H EAT C ONDUCTION PROBLEM r I 2 .5.4 E xample R esults f or P lanar G eometry .., 1 d "Y .=d2n-l)! dtE <:i 0 II ...... <l .,.. N .q ... - <Xl Q=I-- (2.5.25) 0 II ...... T he first five derivatives are approximated as follows: d lj drE ~ <l l j+l-lj_1 (2.5.26) 2ArE N >D 0 0 II . ..... dr~ ~ <l r: 2Ar~ (2.5.28) .,.. .0 ;; N 0 ::J . 0 I I) II II . ..... <l .... N 0 u (2.5.29) IC W dr~ ~ .. .. II - lj-3+ 4 lj-2- 5 lj-1 + 5lj+I-4lj+2+ lj+3 2Ar~ (2.5.30) II ...... CI CI <l ::J Some insight is gained by looking a t the coefficients o f the various order differences; the appropriate term to consider is 10 r:: .. .,.. I /) 0 GI <l II . ..... r:: (2 .5.31) ( 2n-l)!Ar1. Equation (2.5.31) is tabulated in Table 2.7 for various values o f the dimensionless time step. F or each value of ArE, the maximum coefficient is underlined. Note that as ArE becomes smaller, higher-order derivatives appear to become more important. If ArE is large ( > 10 2 ) then the first derivative is the dominant term; if ArE is much smaller than unity, then several high-order terms appear to be significant. Equation (2.5.25) can be written in an al~ernative form by using the difference approximations given by Eqs. (2.5.26)-(2.5.30) and simplifying to obtain ~ V \) Qj=D_ 3lj-3 + D_ 2lj - 2+ D_,lj_1 + Dolj+D1lj+ I + D 2lj+2 + D 3lj+3 (2.5.32) . ....... .., 8:8~~~~~ _ ""';oci""";NN""': r -r- v 0 '\ t --V'l 0 '\-0'\-.::1" :8:8 r -o..,-oo ' D'D o r-O'\r1 1")\00\o~~ 0 "'-00.,.. · 0 .... oON~vi N r-r.... - N ::<lO\N r -MI,f"')VI,f')OO >D .... r -O\oo 'OM_V)M_ \ Of"I"'IMO\O("t) 0:8:::!8gg~;g ~~~:iNOO r- ~~8~~ q~ - .ioooo \ O\OMI.I')t'C") 'DN~>D§ ~;;;; 8 oo~ .,.. I /) dSlj .., -00000 r- X X r--- X X X \ 0 f"I"\ _ N 0\ 1 oC("t)"Ct'v)V')V'I .,.. (2.5.27) d3lj - lj_2+ 2 lj_I-2lj+l + lj+2 x--- - - 0'" In order to apply the exact solution, the time derivatives must be approximated numerically. F or simplicity, suppose the temperature sensor is attached at a perfectly insulated boundary and central differences are used. Only the first five derivatives are considered; in order for the method t o be of practical utility, the series must be truncated at a reasonable limit. Applying Eq . (2.5.20), ·u I:r- .... ..,-\NO 8 .... r-CXl_ o N ........ M : 8 .... 0 N q,,<~~~§~ . .,.NOOOOO I\r- r- .,.. 00 N 0\01""'-00 q~~§~~ N OOOOO ;: ....I G 0 (J . ..... GI u <l r:: ... GI GI ..... ... C~ .. . ... , ...<l ~ N ::::.. w .... .0 .. ...... <l q~a§~ -0000 .., I '" I I I ::~~8 xx ,..... M _x x c- I oOrt"iVv) _ '-0("1"')001.1') ~~"'1~r--: O -OO-N ....I I l Ot: c (N 1-"':' D _ I'>D .... 00 .... r- .... 0 \ '" -NMVI.I')\Or- 73 75 REFERENCES where the D's a re independent o f t he time index j a nd can be written as 1 D-3=2.9!L\'t~ 4 01 10 111'\ . .. I I') ( "4 I I I I I I I 0 .... '" -S 01 ----------- 00000000000 xxxxxxxxxx O\~OOI,f')O\\OOO \ 00000 6 ~oci\CiV'i~~c_r'iNN.....;..n fO\ C"4 _ _ 1 I I I 0 0 0 _ ,... ----------xxxxxxxxxxx ~~~~~~~~~~~ ~~~~&1~C;;~~~q D Z=2 I t'')1rD ...... N \ O - M r - N \ C \ o I It'I '" N C"4 I I I I I I _ 0 -0 _ 0 N ----------xxxxxxxxxx ~~~~N~~Y"l~$~ t"'lNO("f'lt"'lt'f")O~f"'lO­ f"I"ivir-iociM"":..n . . r 'o\"": IIIIIIIII ... CD 0 '0 . .. : I I CI e( :=~ CD 1 0 ON O1· 0 ~. c~ .;; . .s;CT ... ' i ... CI ~&&I ~~ I ....! . CD CD ... > :I .- ~ 0'1 CD ... I c ,. eO .., I- > CD CD CI .. ii: 00 . . N ... &&Iii: . ...I> a l_ e( c 1 -0 74 I I 10 10 111'\ I I') , I I I I 1 N I -4 5 +--4+ 3 ·9!L\tt: 7!L\tt: 2·S!L\tE Equation (2.5.32) has the appearance o f a seven-point moving average filter, with the sum o f t he coefficients equal to zero. Again, note t hat q is linear in the temperature measurements. T he coefficients a re n ot symmetrical; t hat is, D - i'/=Dj except for certain values o f i. T able 2.8 presents the temperature coefficients defined in Eq. (2.5.33) as functions o f t he dimensionless time step L\tf:For large dimensionless time steps (-> lOz), DI a nd D _I a re the dominant terms; only D _I a nd DI c ontain IjL\tE terms. This suggests, in turn, t hat for large time steps, a two-point central difference might be adequate. T he o pposite is t rue for small time steps; all o f t he D's a ppear t o be approximately the same order o f magnitude. Thus, the seven-point formula considered in this example might not be adequate for small time steps. In summary, an exact solution has been presented for the inverse problem of heat conduction, provided the properties are constant. The solution requires that infinite-order derivatives o f t he experimental d ata Y(t) must exist. If small dimensionless time steps are used, high-order derivatives can dominate. F rom a practical point o f view, the utility o f t he Burggrafsolution is t o provide beneficial insight into inverse heat conduction problems. X t "'lOO\Q'\-.:ttt')IrDr-rt'lt--- ...... &&1'0 •- (2 .S.33) 2·9!L\~ 01 00000000000 .u .C.. !D 2 S 4 1 1 1 D I = - - - - - - - - - + - - + -2.9!L\t~ 7 !L\tt S!M~ 3 !L\tl 2L\'tf: 0'1 00000000000 : lU c .E 1 2L\tE _ I cr 1 D o= - - - - 7 ! L\tt 3 !L\t~ I o 1 1 .I")t"')-\Ot"IO\O -_V'l("f'lt"'l\O_r--~V')C"t') CI 4 1 D-l=2.9!L\'t~ -7!L\'t:+S!L\'t~+3!M~ - X r_M~~'DNN~-.,..'" o o\Or- 1 2.9!L\'t~+7!L\'tt - 2·S!L\'t~ _ 0 .... 0- ----------- 00000000000 xxxxxxxxxxX ~~~~~N~~OO~~ r_o~Or_~ _ _ : :!Nr_ REFERENCES f""')N("f'lMf'f"I:;f"l:t""'~\oM ...;-.:i-"....;~....;~"....;~"....;~"....; I. B eck.J. V. a nd A rnold. K. J .• Parameter Estimation in Ellgineeringand Science. Wiley. New York • 1977. Carslaw. H. S. a nd Jaeger. J. c.. Conduction 0/ Heat in Solids. 2nd ed .• O xford University Press• L ondon. 1959. L' ';du. i II' 2. p, J .. C "",",, . . H~' " . ...' . Add""-W~", • . ...,;'. MA, " SS, 76 C HAP.2 S OLUTIONS OF THE INVERSE H EAT C ONDUCTION PROBLEM 4. Meyers, G. E., Analytical Methods in Conduction Heat Transfer, McGraw-Hili, New York,I971. 5. Carnahan, B., Luther, H. A., and Wilkes, J . 0 ., Applied Numerical Method;, Wiley, New York, 1969. 6. Burggraf, O. R., An Exact Solution o f the Inverse Problem in Heat Conduction Theory and Applications, A SME J . Heal Transfer,86C, 373 - 382, August 1964. 7. Langford, D., New Analytical Solutions of the One-Dimensional Heat Equation for Temperature and Heat Flow Rate Both Prescribed at the Same Fixed Boundary (with applications to the phase change problem), Q. Appl. Math. 14 (4), 315 - 322 (1976). 8. Imber, M. and Khan, J., Prediction of Transient Temperature Distributions with Embedded Thermocouples, A lA A J . 10 (6),784- 789 (1972) . 9. Kover'yanov, V. A., Inverse Problem of Nonsteady-State Thermal Conductivity, TeploJizika Vysokikh Temperatur, 5 (1),141 - 143 (1967). P ROBLEMS 2 .1. Develop Eq. (2.2.10) by (a) using a weighted least-squares procedure that fits a linear equation to the experimental temperature data and (b) differentiating the curve fit to determine the average temperature gradient, and (c) use of Fourier's law to obtain the heat flux. 2 .2. F or cylindrical geometries, the temperature profile is linear in the variable lnr; T - 7; = [ - QIn(r/ri)]/(21tkL), Q= q(21trL) where Q is constant. Develop a heat flux estimating equation analogous to Eq. (2.2.10) t hat minimizes the weighted least-squares error between the computed and experimental temperatures. The heat flux is to be found at r = rio Is this result the same as the one obtained for a temperature profile linear in r ? 2 .3. F or spherical geometries, the temperature profile is linear in the variable l /r; T - 7; = (Q/41tk)(l/r-l/rj), Q = q(41tr2) = constant. Develop a heat flux equation analogous to Eq. (2.2.10) that minimizes the weighted least-squares error between the computed and experimental temperatures. Is this result the same as that obtained for a temperature profile linear in r? 2 A. Develop Eq. (2.3.7).lt is acceptable to start with Eq. (2.2.10) as suggested in the text immediately above Eq. (2.3.7). 2.5. Demonstrate that if five temperature-time points are used to determine a least-squares straight line [e.g., Eq. (2.3.7)], then a single temperature error eM will cause errors in the calculation of qM - 2, qM - I , q M, qM + I , and q M+2' W hat are the heat flux errors? 2 .6. Verify the numerical results in Table 2.3. 2 .7. Derive Eq. (2.4.7) from Eq. (2.4.6) by integration by parts. Hint: rewrite the integrand as . 77 PROBLEMS Y '().) + Y '(t)- Y'(t) . jt-). and express in terms o f two integrals. 2 .8. Verify that Eq. (2.5.16) is a solution to Eqs. (2.5.13,14,15). 2 .9. Prove that the sum of terms in Eq. (2.4.14a, b) adds to zero. What is the physical significance of this result? 2.10. F or the spherical heat flux equation given by Eq . (2.5 .24), generate a table analogous to Table 2.7. W hat conclusions can be made? 2.11. Starting with Eq. (2.4.7) a nd the assumption that the surface temperature response varies piecewise linearly with time, show that the heat flux can be expressed as (k P A A c)tI2 { YM - Yo YM - YM - t q (tM)=qM= - ;(CM - t )1I2+(CM - CM _ I)t /2 o ~ t[ + L... i =1 Y M-Yi YM - Yi - I 2 (Yi-Yi - I ) ] } 1/2 1/2 + 1/2 I2 ( tM-t i ) (CM - ti - tl ( CM-t l ) +(CM - Ci - tl I 2 .12. The result given in Problem 2.11 has been given in numerous places in the literature and is algebraically more complicated than Eq. (2.4.8). Show t hat the above expression can be simplified to Eq. (2.4.8b). Hint: Expand the first tw.o summations and show that most ofthe terms cancel.