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280 C HAP.7 M ULTIPLE H EAT F LUX E STIMATION form. Give the gain coef1icients K I j a nd K 2j ( forj= I , 2. 3) in terms o f t he influence functions cPj; w herej = I. 2 refers to the sensors a nd; = I. 2. .... M + r-I refers to time. 7 .2. W rite a c omputer p rogram for the algorithms o f P roblem 7.1. 7 .3. a. b. 7 .4. a. b. Using exact d ata from Table 1.1 a t x /L = 0 a nd I with dimensionless time steps o f 0.01, lind estimates o f t he heat fluxes at x /L=O a nd I using the c omputer p rogram for Problem 7.1. Repeat part (a) for the sensors a t x /L=0.25 a nd 0.75. Also solve p arts for i ll + = I. C HAPTER 8 H EAT TRANSFER C OEFFICIENT E STIMATION F or P roblem 7.1 d erive digital lilter algorithms to o btain e stimates o f q l(M) a nd ( /2(M). Let the tilter coefticients be .Ij;. j = I. 2; ; = - 2. - 1.0.1 . .... O btain n umeral values for the filter coefficients• .Ij;. for the flat plate o f T able 1.1. Use i ll + = 0.05 a nd s ensors a t x =O a nd L. 7 .5. Use the z eroth-order s equential regularization m ethod with r =3 to ~erive a lgorithms for estimating two heat flux histories. q l(M) a nd 2 ( Xu be a n i nput. Q 2(M). Let 7 .6. W rite a c omputer p rogram for the algorithms o f P roblem 7.5. 7.7. a. b. Using the exact d ata from Table 4.1 a tx/L=Oand I w ithill+=O.OI. find estimates o f t he h eat fluxes a t x /L=O a nd l ;use t he c omputer p rogram from Problem 7.5. Repeat p art (a) for the sensors a t x /L=0.25 a nd 0.75. 8.1 I NTRODUCTION T he e stimation o f t he heat transfer coefficient, h, from transient temperature measurements has aspects o f b oth t he inverse heat conduction problem a nd p arameter e stimation. An example o f t he t reatment a s a n I HCP is t hat o f a o ne-dimensional case with a known ambient temperature, Too(t), such a s t he transient determination o f boiling heat transfer coefficients using an initially h ot spherical c opper solid suddenly immersed in water a t its saturation temperature. F rom t ransient temperatures measured inside o r a t t he surface o f t he copper body, the methods o f t he I HCP c an be used t o e stimate the surface heat flux, qM, a nd t he surface temperature, tOM; t he definition o f t he heat transfer coefficient, h, c an be used t o o btain t he estimate o f h given by ~M qM TooM -O.5(tOM + o•M t (8.1.1) - 1) In this expression tOM is t he estimated surface temperature a t time t M; qM, TooM, a nd ~M a re usually most accurately evaluated a t t M - 1/ 2 • An example o f a case t hat c an be treated a s a p arameter e stimation problem is a flat plate over which a fluid is flowing a t a c onstant t emperature, Too; see Figure 8.1. If t he plate is s uddenly heated by some electric heaters inside the plate, the plate temperature begins to rise a nd t he heat transfer coefficient is a s trong function o f p osition from the leading edge o fthe plate. I n s ome cases the time variation is small a nd t he basic form o f h is a function o f x; t hat is, h = hex), is k nown, such as (8.1.2) 281 282 C HAP. 8 HEAT TRANSFER COEFFICIENT ESTIMATION S EC. 8 .2 a nd examples serve t o illustrate procedures which ca~ be modi~ed for ~ifferent c onditions. Section 8.2 covers some sensitivity coefficIents. SectIon 8.3 dlscus~es m ethods for lumped bodies a nd S ection 8.4 briefty covers methods for bodIes with interior temperature gradients. Fluid at T . 8 .2 FIGURE 8.1 Electrically heated flat plate. T he d etermination o f f3 utilizing various time- and space-dependent measurements o f T in the solid a nd Too in the fluid is a p arameter estimation problem. F or cases similar t o t he two j ust given, basic solution techniques are known. These examples illustrate the large diversity o f p roblems associated with determining the heat transfer coefficient. F or this reason a discussion o f v arious types o f p roblems is given next. T he h eat transfer coefficient c an be: 1. 2. 3. 4. 2 83 SENSITIVITY COEFFICIENTS A c onstant (independent o f x a nd t). A function o f t o nly; t hat is, h(t) . A function o f x o nly; t hat is, hex). A function o f x a nd t; t hat is, hex, t). I n this list, x is a c oordinate parallel t o t he heated surface; it can also be generalized t o two surface coordinates such as h = hex, z) where both x a nd z a re parallel t o t he heated surface. I n these problems the ambient temperature, Teo, is a ssumed t o be k nown b ut it can also be a function o f x a nd t. A related problem is t he determination o f t he ambient temperature, Too, with h known. T he s ame four categories o f (1) T", = C, (2) Too = Too(t), (3) Too = Too(x), a nd (4) Too = Too(x, t) c an be listed. These four estimation problems are linear if the heat conduction equation a nd b oundary conditions are linear. An example o f a p roblem wherein the ambient fluid temperature is needed is in the extrusion o f plastics. T he t emperature o f t he m olten plastic is critical a nd h eating must be controlled, but temperature sensors are placed only in the solid mold, n ot in the flowing plastics. A more complicated problem is t o simultaneously estimate the heat transfer coefficient a nd t he ambient temperature. There are many concepts a nd techniques o f t he I HCP a nd p arameter estimation t hat c an be utilized in the solution of these problems. O ne c oncept is t hat t he sensitivity coefficients c an be employed t o gain much insight into the estimation problems. Another concept is t hat t he use o f a sequential procedure can b e a dvantageous in terms o f c omputation speed a nd for insight. These concepts a re briefly explored in this chapter. D ue t o t he large variety o f cases in connection with the determination o f t he heat transfer coefficient, only a few c an be treated. T he case o f h = h(t) is t he main one covered. T he basic concepts S ENSITIVITY COEFFICIENTS I n this section the sensitivity coefficients for the heat transfer coefficie.nt, h, a~e investigated for two cases. T he first case is for a lumped . b.o~y-that I~, o ne m which the temperature is a function o f time only. SensltIvlt~ c~ffic.lents a re given for h c onstant over the complete tim~ d omain a nd for finite ~Ime mtervals. Also the sensitivity coefficient o f t he ambIent temperature, Teo, IS found, b oth for T c onstant with time a nd for Too a pproximated by a n umber o f c onstant s egm;nts. T he s econd case is for a semi-infinite bod.y th~t is suddenly exposed t o a ftuid. T he h eat t ransfer coefficient is c onstant wIth tIme. 8.2.1 L umped B ody Case T he differential e quation for a lumped body which is suddenly exposed t o a ftuid a t a t emperature Teo is dT p cV T t=hA(T",-T) (8.2.1) where V is t he volume a nd A is t he heated surface a rea o f t he lumped body. F or convenience in notation, the ratio of V divided by A is d enoted L, V A L= - (8.2.2) F or a n initial temperature o f To a nd with b oth h a nd Teo i ndependent o f time, the solution o f Eq. (8.2. 1) is T +;: T - To =1 -exp ( -hLt) Too- To pe (8.2.3) N ote t hat T+ c an b e p lotted as a function o f t he single dimensionless time t + , where + ht peL t ;:- (8 .2.4) F or a c onstant h, t he h s tep sensitivity coefficient, Zh, is given by i t Z h;: -JT =(Too-To)-ex p( - t +) iJh peL (8.2.5) 284 CHAP. 8 HEAT TRANSFER COEFFICIENT ESTIMATION A dimensionless h step sensitivity coefficient is + h aT + + Z . = - - - - = t e xp(-t ) Too-To ah (8.2.6) which is also a function o f t+. T he distinction between the pulse and step sensitivity coefficients is discussed in Section 6.2. T he Too s tep sensitivity coefficient is aT ZToo=aToo = l-exp(-t + ) (8.2.7) which is dimensionless and thus rearrangement o f terms is n ot needed. Notice that T+ =ZToo (8.2.8) a nd see the upper curve in Figure 8.2. As the temperature rise becomes large for increasing t+, the Too sensitivity coefficient, ZToo, increases in exactly the same manner. As a parameter's sensitivity coefficient becomes large, more information can be gained a bout t he parameter. Consequently Too is relatively easy t o measure for a constant Too over the complete time range. Since the Too sensitivity coefficient becomes largest as t -oo, the most information regarding Too is found by using measured temperatures a t large times; that is, t+ >3. Also shown in Figure 8.2 is t he dimensionless heat transfer sensitivity coefficient, Z;:. F or the small values o f t+ < 0.5, Z;: increases and is nearly equal to T + ; in the range o f 0.5 < t + < 1.5, Z ;: a ttains a maxim um; and for t + > 1.5, Z ;: decreases toward zero. There a re several ramifications o f these behaviors o f Z;:. First, the optimal times for measuring temperature in order to estimate h a re near t+ = 1 a nd measurements a t t he early a nd late times contain less SEC. 8 .2 285 SENSITIVITY COEFFICIENTS information regarding h. Second, the average magnitude o f Z ;: is considerably less t han T+, hence h is relatively sensitive t o measurement errors in temperature. Third, since ZToo is s o much larger o n the average t han Z;:, Too c an be estimated more accurately t han h. Finally, it is possible to simultaneously estimate both h a nd Too b ut a relatively large dimensionless time range should be covered, say t+ = 0 t o 3. When two parameters are estimated, the sensitivity coefficients must not be highly correlated if accurate estimates are to be obtained. This is also discussed in Section 1.6 a nd in Reference 1. T his case o f a l umped body is sufficiently tractable that the sensitivity coefficients for time-variable h a nd Too can be obtained in algebraic form. The case where h(t) a nd Too(t) are constant over the time segments, O <t<t l , tl < t<t2 , t 2 <t<t 3 ,··· is considered as illustrated in Figure 8.3. T he heat transfer coefficient components are hI> h2 , . ••• , a nd the corresponding components for Too a re Tool' Too2 , • ••• T he equations for T for three time intervals are for 0 <t<t 1 , -hit) T=TOOI+(To-TOOI)exp ( p eL (8.2.9a) (8.2.9b) (8.2.9c) (8.2.10a) (8.2.10b) 0.8 0.6 h (/) 1+ = h tlpeL FIGURE 8 .2 Temperatures and sensitivity coefficients for convective heat transfer from a lumped body. °0L-~~~-*--~~~~--- 'I 't2 '3 '4 FIGURE 8 .3 Heat transfer coefficient history approximated by constant segments. 286 C HAP. 8 HEAT TRANSFER COEFFICIENT ESTIMATION . T he pulse sensitivity co~~~ients can be found by differentiation as previously Illustrated. The pulse sensltlVlty coefficient for hi is aT X~,= ah. (8.2.11a) 1 a nd for TOOi' aT x T<r,,= aT . (8.2.11 b) 0 01 T o reduce the number of dimensionless grol!ps, after the differentiations are completed, the hi components are made equal, hi = h2=h3= . .. = ho and the same is done for TOOi> (8.2.12) Tool =T",,2=T003 = ' " =T"" (8 .2.13) Figure 8.4 displays the heat transfer coefficient sensitivity coefficients for 0.2r-....--.--......,..--r---....----,------, S EC. 8 .2 2 87 SENSITIVITY COEFFICIENTS X:' the two cases of dimensionless time steps of 0.25 a nd 0.5. T he values are uncorrelated as can be seen by comparing the values at t+ =h oAt/pcV =0.25, 0.5, 0.75, a nd 1 o f X ~ which are 0.19, 0.15,0.12, a nd 0.09 a nd those for x :' which are 0 ,0.15,0.12, a nd 0.09. These values are not proportional for all times and hence are uncorrelated. The first values o f X~ a nd X :. a re different but the succeeding values are identical. This suggests that a sequential procedure (which emphasizes "recent" values) o f estimating h = h(t) would be m ore effective than a whole domain estimation procedure which uses all the d ata simultaneously. Another conclusion that can be drawn from Figure 8.4 is t hat the hi components become more difficult t o estimate as i increases. This is a consequence o f the X~, values becoming smaller in magnitude as i increases. In Figure 8.5 the sensitivity coefficients for Toot a nd T002 are displayed. A significant difference between this figure and Figure 8.4b is t hat the coefficients in the former have the same magnitude for each X T oo, while the X :, curves have decreasing magnitudes with increasing i. This difference means that all components o f Too; can be estimated with equal sensitivity t o measurement errors while the h; components become much more sensitive t o errors as i increases. 8.2.2 S emi-Infinite B ody 4.t+=0.25 x +", ~ h, T_ - To xt 1 F or a semi-infinite body at an initial temperature of To a nd exposed at t =O t o a fluid a t Too, the temperature a t a position x measured from the surface and at time t is aT an: T+ 3 == ~-=-~o =erfc [2(t)1/2 ] -exp(B,,+B~t;) erfc [ 2(t)t/2 +B,..(t;)1/2] t+ ' " h oAt/peV ( a) xt 1 t+ . . ho A tipeV ( b) FIGURE 8 .4 Heat transfer sensitivity coefficients for hi constant over segments and h = h = " ·=hoandT."=T,,,= " ·=T.,. ' , t+ . . h ot/peL F IGURE 8 .5 Ambient temperature sensitivity coefficients for At+ = 0.5. (8.2.14) C HAP. 8 288 HEAT TRANSFER C OEFFICIENT ESTIMATION S EC. 8 .2 289 SENSITIVITY COEFFICIENTS where 0.3r-----~----~----~----~----~----, hx (8.2.15a, b) B "=T' 0 .25 zt .. ~ - 0 Notice that 2 + h2 a.t (8.2.15c) B "t,,=V 0 .2 which is independent o f x . The dimensionless temperature is plotted in Figure 8.6 versus B ;t; for several values o f B" . F or each B" value, the dimensionless temperature starts a t zero, and increases monotonically, and finally approaches unity. The dimensionless step sensitivity coefficients are given by 0.15 0.1 h aT 2+ 1 <t+)1/2J Z :=----=-(B"+2B,,t,,)exp(B,,+ B2t" ) e rc [ 2( +)1/2 + B" " "+ ~ Too-To ah t" + 1[712 B,,(t;)1/2 exp ( - 4!;) (8 .2.16) and are displayed in Figure 8.7. The notation Z : is used t o emphasiz~ ~hat a single h is being examined. Though the sensitivity coefficients always Initially increase, there is a maximum after which the curves decrease and approach 100 B~t! F IGURE 8 .7 body. 1000 = h 2 a tlk2 Dimensionless sensitivity coefficients for h for convectively heated semi-infinite zero. Consequently the moderate times are best for taking temperature measurements. F or example, for B,,=O , which corresponds to taking temperature measurements at the heated surface, the sensitivity coefficients are largest for B ;t; between 0.03 a nd 10. Both the very early and very large times are less effective for determining the heat transfer coefficient. This finding is consistent with that found for the lumped-parameter case depicted in Figure 8.2. Another important observation regarding Figure 8.7 is t hat the sensitivity coefficients decrease in Imagnitude as B" increases. In this regard there is no correspondence with the lumped-parameter case. Since B" is defined to be h x/k, large values of B" result from large h o r x values o r small k values. I f the ambient temperature, Too, is o f interest, the sensitivity coefficient for Too must be found. From Eq. (8.2.14) it is readily shown that O .B 0 .6 0.4 B x=O .l 0 .2 aT + ZToo=aToo = T ~L . 0-0-1---L----~L---~-----7.~---7.10~2~--~10 3 B~t~ F IGURE • •• condition. = h 2 a tlk 2 Dimensionless temperatures for semi-infinite body with a convective boundary (8.2.17) and thus Figure 8.6 also represents the Too sensitivity coefficients. Again as for the lumped-parameter case, Too can be estimated more accurately than h (if the error statistics are the same in both cases). Furthermore the optimal times to take temperature measurements to estimate Too are the largest. 2 90 8 .3 C HAP.8 H EAT TRANSFER COEFFICIENT ESTIMATION S EC.8.3 291 L UMPED B ODY ANALYSES . with T replaced by 4>, q by unity, and the solution evaluated a t t=~t. T he resulting expression is L UMPED B ODY A NALYSES In this section the determination of the heat transfer coefficient for the case of a lumped body is discussed. A thermally lumped body is one in which the temperature is uniform but varies with time. Transient calorimeters based on this approach have been used ' for measuring heat transfer coefficients and heat fluxes for a number o f applications including determination of the boiling curve.2 This case o f a lumped body involves a simple model, Eq. (8.2.1), a nd thus the details o f the estimation procedure can be readily seen. Several procedures are discussed in this section. In Section 8.3.1 exact matching of the model temperature is employed. In Section 8.3.2 some results obtained using regression analysisl are discussed. Section 8.3.3 presents a method based on the function specification procedure with the q = C assumption. Finally in Section 8.3.4, function specification with h = C is developed. (8.3.6) EXAMPLE 8 .1. F or t he c ase o f peL = 1, T... = 1, h c onstant, a nd t he m easurements equal t o t he e xact values given by YM = I -exp( - Ml1t) t he h values a re t o be e stimated using Eq. (8.3.4) for I1t = 0.02, 0.5, a nd 1. Solution. I ntroducing t he given values into Eq. (8.3.4) yields ~M - exp( - Ml1t) +exp[ - (M - 1)l1t] {I - I + O.5[exp(-Ml1t)+exp[ - (M-I)l1t]}l1t 8.3.1 E xact M atching o f t he M easured T emperatures The exact matching of the model temperatures with the measured temperatures is analogous to the Stolz method for the IHCP. This method uses Eq. (4.3.3) which is (8.3.1) T he heat transfer coefficient equation, Eq. (8.1.1), can be written for the lumped body case as e xp(l1t)-1 tanh(l1t/ 2) 0.5l1t[1 +exp(l1t)] 0.511t which is independent o f M . F or I1t = 0.02, 0.5, a nd I , t he ~M values a re 0.999967,0.979675, a nd 0.92423, respectively. Hence, smaller l1r's h ave smaller e rrors in Ii., for exact d ata; t he e rrors a re a result o f t he n umerical approximations. 0 EXAMPLE 8.2. F or t he s ame cases as in Example 8.1, calculate t he e rrors in t o a t emperature e rror o f f. = 0.001 a t M = 3. Tex>M-0.5(TM- 1 + T ) M With exact matching of the calculated and the measured temperatures, the following relations are used: t M- 1 =YM-t> t M=yM, t MI.II=O=YM- 1 (8.3.3a,b,c) The relation given by Eq. (8.3.3c) postulates uniform temperature within the body; thus it is true for lumped bodies. Then the foregoing relations yield t _ n M- ( YM - YM-tl [Tex>M-O .5(YM- 1 + YM)]4> I (8.3.4) which is the desired algorithm. An expression for 4>1 is needed. I t represents the temperature rise at time t=~t for a unit step increase in the heat flux. Specifically it is found from the solution of oT p cV-=qA ot (8.3.5) d uc Solution. T he calculated~., values are: M I 1t=0.02 111=0.5 I1t= 1 1 (8.3.2) ~M 0.999967 0.979675 2 0 .999967 0.979675 0.92423 0.92423 3 4 1.053081 0.946850 0.988114 0.971226 0.94011 0.90820 5 0.999967 0.979675 0.92423 T he e rrors a t M = 3 d ue t o £. a re a bout 0.0531, 0.00844, a nd 0.Ql588 for 111=0.02, 0.5, a nd I , respectively. T he s mallest 111 results in the largest e rrors i n ~M' T his is consistent with the sensitivity coefficients decreasing as the time steps a re r educed as shown in Figure 8.4. Consequently, s malll1t's result in smaller numerical approximation errors b ut g reater sensitivity t o r andom m easurement errors. 0 EXAMPLE 8.3. F or t he s ame c opper billet mentioned in Example 2.1 find the h eat t ransfer coefficient for the a mbient t emperature o f 81.5°F. T he first eleven measured temperatures are given in T able 8.1 a nd f urther values a re given in T able 2.2. Solution. T he e quation for ~M is Eq. (8 .3.4), TooM is 81.5°F, a nd t he y., values are t aken 2 2 from Tables 8.1 a nd 2.2. T he r eciprocal o f tPl is 31.3 Btu!hr-ft ( =98.72 W1m ). T he 292 C HAP . 8 HEAT T RANSFER COEFFICIENT ESTIMATION S EC.8.3 2 .5.---------,.------.-----,--, T ABLE 8.1 R esults f or t he C opper B illet E xample U sing t he F unction S pecification M ethod w ith q =C a nd r =2 M t",(s) Yli • OF Ii", . Btu/hr-ft1 0 1 2 3 4 5 6 7 8 9 10 96 192 288 384 480 576 672 768 864 960 279.59 264.87 251.53 239.30 228.18 217.24 207.86 199.36 191.65 184.44 177.64 - 443 .459 - 414.oI2 - 381.388 - 359.134 - 329 .576 - 296.969 - 268.527 - 245.945 - 228.649 - '212.027 h"" Btu/hr-ft1-F Function specification method (h = C, r = 2) Exact matching, EQ . (S.3.4) t"" of 2.3217 2.33498 2.31702 2.35077 2.32471 2.25382 2.18797 2.14779 2.13832 2.12261 265.422 252.195 240.010 228.536 218.006 208.518 199.939 192.082 184.777 178.003 2 .3 2.2 .... I ~ 2 .1 I ~ ~ &l 2 Function specificati on ~ [ method(Q=C,r=2) Col:!:' 1.9 ~,. ~ I .S results of the calculations are displayed in Figure 8.8 as points connected by solid lines. There are some ftuctuations in the values, particularly near time steps 5 and 12. T he method is relatively sensitive to measurement errors. as was noted in Example 8.2. 0 8 .3.2 293 L UMPED BODY ANALYSES Regression method, EQ . (8.3.S) 1.7 1.6 !:---------;1";;0------;2;';;0:-------,30 111 0 Regression M ethod Time step index, M FIGURE 8 .8 Another way o f estimating h is t o pass a regression line through the measured temperatures. Polynomials o f various degrees can be used. In Reference 1 a number o f polynomials for T such as 8.3.3 (8 .3.7) Hcat transrcr coefficient ror copper billet or Example 8.3. F unction S pecification P rocedure w ith q =Constant T he heat transfer coefficient can be estimated using the function specification method. F or the temporary assumption o f constant q. E q. (4.4.24) can be utilized. F or the lumped body case. a solution of Eq. (8.3.5) for cP (with T -+CP. a nd so on) gives were investigated and an F-test was used to determine the "best" curve. After estimating the temperature curve. 1; it can be differentiated and used in Eq. (8.2.1) which can be written in the form j At (8.3.9a) CPj=pcL' j =1.2. ... dt 1 ~=pcL--­ T",,-t dt (8 .3.8) The Acpj_1 values are constant. (8.3.9b) Results using this method are given as crosses in Figure 8.8 for Example 8.3 . These results are not as sensitive to errors as those from the exact matching method. This method can also be made a sequential procedure by considering a limited number o f measurements as was done in Example 2.1. As a consequence Eq. (3.2.29) gives AI - I tAl+j-dq.,= .. . =o=To+CPI L qi. 1= I 1 j =1.2. .. . (8.3 .9c) 294 C HAP.8 HEAT TRANSFER COEFFICIENT ESTIMATION With Eqs. (8.3.9a, b, c) used in Eq. (4.4.24), the equation for qM for the lumped bodies is ( YM+}-I-¢I qM r I~ 296 for t > t M-I' T he sensitivity coefficient, Z M+I-I' (8.3.15) q l- To)} p eL At (8.3.10) I/ j =1 T he equation for ~M is given by Eq. (8.3.2) with O.5(1' _ I + 1M) given by ' M M 0.5(1' _I + 1'M)=¢1 (~tll ql+0.5qM)+ To (8.3.11) T he algorithm for each time step is Eq. (8.3.10) followed by Eq. (8.3.2) which employs Eq. (8.3.11). F or example 8.3, calculations for r =2 a re displayed as circles in Figure 8.8. T he results are less sensitive to small irregularities in the measurements than the exact matching procedure a nd yet follow the exact matching case when the latter moves regularly, as near the time step index o f 20 in Figure 8.8. Even less sensitivity to random temperature errors can be achieved using r = 3 a nd 4. Some numerical values o f qM, ~M' a nd 1'M a re given in Table 8.1. I n o rder t o permit verification by t he reader, values are given for M = 1 t o 10. 8.3.4 L UMPED BODY ANALYSES M -I r }~I S EC.8.3 Function Specification Procedure w ith h =Constant T o estimate the heat transfer coefficient using the function specification method, a more direct procedure is t o estimate h without the intermediate calculations for q(t). Unfortunately the nonlinearity o f t he problem enters into this approach. I n this section a direct sequential estimation procedure for h is investigated for the temporary assumption o f c onstant hM • T he sum o f squares, S, r s= I ( YM+,-I-TM+,_d2 (8.3.12) 1 =1 is minimized with respect to hM where r is the number offuture times over which hM is temporarily held constant. Taking the partial derivative o f S with respect to hM' replacing hM by ~M' a nd setting the equation equal to zero gives ~ L.. ( YM + 1- 1 =1 I- t.M+I-da1'Mh+ 1_ I a M 0 (8.3.13) Expressions for the calculated temperatures 1M + 1- I c an be found analogous ' to Eq. (8.2.9). T o simplify the presentation, however, the ambient temperature, T",(t), is a pproximated by TOOM for the r future times; for this case, the temperature is given by t.M+I - I = T"'M+( t.M -I - T"'M) exp [ hM(tM+i-1 - tM-d] p eL (8.3.14) is found from Eq. (8.3.14) to be Z M+I-I = - ( t. -I- T ) exp [ M ooM h M(tM+I-I-tM-t>] t M+I-I-tM-I (8.3.16) peL peL which is valid for t > tM - I' T he 1'M - I value is the co~verged tempera~ure .for the solution for ~M _ I ' Because this coefficient is a functIon o f hM' t he ~stlmatlon p roblem is nonlinear. D ue t o this nonlinearity, an iterative solutJ~n o f Eq. (8.3.13) for hM is needed. O ne such procedure is t he Gauss metho~. I n the Gauss linearization method, it is first a~sumed t hat an es~:~ate o f hM is k nown for the (v - 1 )st iteration and then a n Improve~ value hM IS s?ught. T he sensitivity coefficient given explicitly in Eq. (8.3.13) IS e valuated WIth hM equal to h~-I) a nd is denoted Z~~l~ I ' Moreover, the c~lculated t emperature in Eq. (8.3.13) is a pproximated by the two-term Taylor sene&, 1'i;'+I-1 = 1'~~!~1 +Z~~~~I(hi;'-h~-I) (8.3.17) a nd Z is given by Eq. (8.3.16) with hM replaced by h~-I). Using these approximations in Eq. (8.3.13) a nd solving for hi;' gives r '" y; L.. ( M + i - I - hi;' =h~-I) + i - I 1'(.-1) ) Z(·-I) M+I-I M+I-I (8.3.18) r ' .. L" (Z('-~) 1 =1 M +I-I )2 T his equation is used in an iterative manner until the changes in hi;' a re less than some small amount, such as Ihi;'-h~-I)I< 1 0-4 (8.3.19) hi;' After a converged value o f hi;' is found, M is increased by one, TM is calculated, •. . and the procedure is repeated for the new ~M' T he same copper billet problem previously considered ~n thIS sectIon IS a.lso investigated using this method. Table 8.2 gives some details o f t he c~l~~latlon for the first two time steps for the case o f hM = C a nd r = 2. T he sensItIvIty coefficients are given as functions o f M a nd the iteration index, v. T he colum~ labeled N umerator is t he numerator o f t he fraction in Eq. (8.3.18) and DenomInator is the denominator o f Eq. (8.3.18). Notice that the numerator for fixed M rapidly decreases in magnitude with v, whereas the denominator (sum o f squares o f Z 's) approaches a constant. The initial guess for h~ was 2-.7 B tu/hr-ftl-F and the procedure rapidly converged to 2.3964 Btu/hr-ft -F. ThIS con~erg~ value o f ~ is used as the initial value for h2 • A few values o f ~M a re plotted m FIgure 8.8 as s~uares. T he first few h values are significantly different from the q = C, r = 2 296 C HAP. 8 H EAT T RANSFER COEFFICIENT E STIMATION D etails o f C opper B illet C alculation f or F unction S pecification M ethod w ith h",=C a nd , =2 1 1 1 2 2 2 2 II v 0 1 2 3 4 0 1 2 3 Ii: I, I, :1 _ II:,I 1 I II:, ;II' Z "+I N umerator Denominator - 5.806 - 5.863 - 5.862 - 5.862 - 10.652 - 10.862 - 10.860 - 10.860 - 45.05 0.3926 - 1.8E-4 - 1.4E-5 147.167 152.360 152.315 152.315 - 5.430 - 5.436 - 5.436 - 10.060 - 10.080 - 10.080 - 3.99 3.05E-3 - 1.56E-4 171.832 133.452 131.153 res~lts I r l T ABLE 8 .3 R esults f or t he C opper B illet E xample U sing t he F unction S pecification M ethod w ith h =C a nd , =2 V) -....J 2 97 Btu/hr-ft 2 -F 2.7 2.3939 2.3964 2.3964 2.3964 2.3964 2.3659 2.3659 2.3659 In Section 8.3 it was found that the function specification method for qM = C gave nearly the same results as for the temporary assumption of hM = C. O ne case of a lumped body does not prove that the temporary assumptions of constant q and h are equally valid for other cases. More research is needed to determine the relative merits. Until more is known, however, the q = C method is recommended because it is computationally simpler and does not involve iteration: Furthermore, existing programs for the I HCP can be employed. F or these reasons, only the function specification method with the constant heat flux temporary assumption is used in this section. 8.4.1 A nalysis f or r F uture T emperatures U sing q =C F unction S pecification M ethod A one-dimensional heat conduction model is ~~(r"aT)=~ a T r" ar ar (X (8.4.1) at where n =O is for a rectangular coordinate, n== 1 is for a radial cylindrical coordinate, and n = 2 is for a radial spherical coordinate. The boundary conditions are ar (8.4.2) a T\ -0 ar r~rl- (8.4.3) - k a T\ r~ro =h(t)[TCX)(t)- T(ro, t)] and the initial temperature is T(r, t) = To = constant M ........... Ii", which are shown as circles. This can be verified by comparing the M =.l t o Table 8.3 with those for q =C, r =2 maximum difference is 3%, which is for M = 1; for M = 4, It IS 0.3% and for M = 10 it is a bout 0.05%. F or this example, the hM = C a nd qM = C results are then very close; in general the results are closer than the variability due to small fluctuations in ~he ~eas~red temperature. The hM = C calculation is more expensive because ~teratlOn IS needed. The number o f iterations for this case is two o r three which IS not I.a~ge b ut even so there is more computation than without the iterations. In addition, the computer pro~ram for the hM = C analysis is more complicated than that for the q = C analYSIS. F or these reasons and a t least for this billet example, the q = C sequential analysis is preferred over the h = C method. e~tlma~ed hM values for given I~ !abl~ 8.1. T he I I I Z" B ODIES W ITH I NTERNAL T EMPERATURE G RADIENTS 8 .4 B ODIES W ITH I NTERNAL T EMPERATURE G RADIENTS TAB~E. 8.2. M SEC. 8 .4 t,,(s) Ii", Btu/hr-ft 2-F I terations 1 2 3 4 5 6 7 8 9 10 96 192 288 384 480 576 672 768 864 960 2.39643 2.36590 2.32878 2.35714 2.32776 2.25087 2.18316 2.14357 2.13633 2.12164 3 3 3 3 3 3 3 3 2 2 (8.4.4) F or simplicity, the thermal conductivity k and the density-specific heat product pc are assumed to be independent o f temperature so that nonlinearity does not enter the problem via the thermal properties. The heat transfer coefficient is assumed to vary only with time. O ther "inactive" boundary conditions at r = r 1 can be as readily treated as the insulation condition indicated in Eq. (8.4.3). Much more general conditions can be treated but they unduly complicate the presentation. F or a single interior sensor, a function specification algorithm for the q =C temporary assumption is Eq. (4.4.24), repeated here r qM L t ( YM + j -I - M + j - .l4"~·" ~o)cPj ~j~_I~--------r------------- L cPJ j~ 1 (8.4.5) C HAP.8 298 H EAT TRANSFER COEFFICIENT ESTIMATION where r future temperatures are used. This equation can be used to obtain a single qM after which hM is estimated. Also all o f the qM c omponents can be estimated before any o f the hM values is estimated. A heat transfer coefficient is found from hM qM (8.4.6) TooM - O.5(tO• M+ t o.M- I) The zero subscript is used t o d enote the surface a t which the fluid is located. Unless the sensor is located a t t he heated surface, the calculated temperature in Eq. (8.4.5) is n ot the same as that with zero subscripts in Eq. (8.4.6). This is a m ajor difference between this case and that o f a l umped body. Equations (8.4.5) a nd (8.4.6) a re valid for both the Duhamel integral solution and a difference method solution (e.g., F D, FE, F eV) o f Eqs. (8.4.1)-(8.4.3). In both cases, tPj in Eq. (8.4.5) represents the temperature rise a t time t j a t the location o f the sensor for a unit step increase in the surface heat flux [ r=ro, for Eq. (8.4.2)]. I f a ccurate solutions o f the heat conduction problem are obtained in both approaches, the results for hM a re nearly identical. F or simplicity of presentation, Duhamel's theorem approach is selected. The calculated temperatures in Eq. (8 .4.5) are given by Eq. (3.2.29) and can be written as S EC.8.4 BODIES WITH I NTERNAL T EMPERATURE GRADIENTS 8 .4.2 299 E xamples T he case o f a semi-infinite body suddenly exposed to a fluid a t Too is co~sidered. T he exact solution for the case of constant h is given by Eq. (8.2.14) and IS p~o~ted in Figure 8.6. Though h is actually constant, it is estimated as though It IS a ) function o f time. Some results are displayed in Figures 8.9, 8.10, a nd 8.11 (Reference 5 . 0.6 o B, = 0.05 + B, = 0.1 o B,= 1 0.4 0.2 00 2 3 4 5 6 FIGURE 8.9 Semi-infinite body example with exact matching or temperature measurements ( 41; =0.3). 8 7 M tul,.,=o=qIAtPM-I +qzAtPM-Z+ . .. +qu-tAtPt + To M -I = I (8.4.7a) qiAtPM-i+ To i =-1 M -I tu+tI,.,=,., .. -o=qIAtPM+ . .. + qM-I AtP2+ TO= I j =1 qiAtPM -i +I+To 2 (8.4.7b) ~\ A All the tPj values (recall AtPj= tPj+ I -~) a re for the location o f the sensor. The surface temperature denoted, 10.M , is found in a similar manner to the calculation o f a n internal temperature; the equation is o I qjAtPo.M-i+To (8.4.8) -1 ; =1 where the subscript 0 for AtPo.u _ I denotes the heated surface. F or the case of one future temperature, r = I, the interior calculated temperature is made to match the measured temperature. F or this case, Eq. (8.4.5) simplifies to .. q M= I! , \(/ \/ M t o.M= \\ YM-tMI,.,=o tPl N o c omparable simplifications occur in Eqs. (8.4.6) a nd (8.4.8). -2 \ t : \ , \ f I..."\! " /B' =20 \I l!, -3LO--~~2--~~4--~~6--~~8~~~110 (8.4.9) M FIGURE 8.10 Semi-infinite body example with exact matching o f temperature measurements ( 41; =0.5). " 3 00 CHAP. 8 HEAT TRANSFER COEFFICIENT ESTIMATION 6~---------------------- ______ ~ To 37.2 5 l\ 4 , 3 I , 2 I 301 P ROBLEMS time period. The needed sensitivity coefficients are denoted X a nd are for individual hM components. (The Z coefficients are always as large o r larger than the X coefficients, however.) In order to take smaller time steps, future temperatures are needed in the function specification method. Even so, for large values o f B ", such as B" ~ 10, the sensitivity coefficients are very small (see Figure 8.7) a nd .thu~ accurate estimation is still difficult. This is verified by the results shown 1 0 FIgure 8.11 which are for ~t; = 0.25, a value for which the Stolz method is unstable. Future time steps, such as r = 2 o r 3, remove the instability but inaccurate results are found, which is n ot surprising since the sensitivity coefficients are so small (Figure 8.7). A conclusion is that the sensor should .be loc~ted as near the surface as possible, at least so that hx/k < I, where x IS the dIstance from the heated surface. 8 .5 E STIMATION OF C ONTACT C ONDUCTANCE T he estimation of the contact conductance from transient measurements is quite similar to the method for determining the heat transfer coefficient. Sensitivity coefficients and optimal considerations are given in References 3 and 4. !;.B. = 10. r =2 A B.=10.r-3 DB. = 20. r =2 = 20. r -3 -1 REFERENCES -B. 2 4 6 1. Beck, J. V. a nd Arnold, K. J., Parameter Estimation in Engineering and Science, Wiley, New 8 10 M FIGURE 8.11 (I1t; =0.25). Semi-infinite body example using function specification algorithm with q =C Figures 8.9 a nd 8.10 are for exact matching o f the simulated data. Figure 8.9 is for dimensionless time steps o f ~t: = 0.3 which are relatively small (i.e., near the stability limit) for exact matching, which is equivalent to the Stolz method. F or B" = hx/k equal to 0.05 a nd OJ, the estimated values o f hM are very good for all times, whereas for B" = 1 there are some oscillations o n the order of ±20%. Evidently larger B" values cause increased difficulty for a fixed time step with exact d ata and exact matching o f measurements. More results for exact matching are shown in Figure 8.10 with the larger time step o f ~t: = 0.5. The value o f B" = 1 in this figure is more accurate than that shown in Figure 8.9. Consequently increasing the time step can aid in reducing oscillations, but information regarding changes in h can be lost as ~t becomes larger. Figure 8.10 also reaffirms the finding from Figure 8.9 t hat increasing B" makes the estimation process more difficult; this is consistent with the sensitivity coefficients becoming smaller as B" is increased, as shown in Figure 8.7. (This figure is for the sensitivity coefficient Z which is for h constant over the entire .. York,1977. Holman, J . P., Heat Transfer, 4th ed., McGraw-Hili, New York, 1976. Beck, J. V., T ransient Sensitivity Coefficients for the Thermal Contact Conductance, Int. J. Heal Mass Transfer 10,1615 - 1616 (1967). 4. Beck, J. V., D etermination o f O ptimum, T ransient Experiments for Thermal Contact C onductance, Int . J . Heat Mass Transfer 12, 621 - 633 (1969). 5. O sman, A. Personal communication, Aug., 1983. 2. 3. P ROBLEMS 8 .1. Starting with Eq. (8.2.14), derive Eq. (8.2.16). Derive also an expression in terms of a nd B" for the maximum sensitivity coefficient, Z;:, for small values o f BX" t: 8 2. A thick concrete wall initially at 30 K is suddenly exposed to a fluid at 80 K. Calculate and plot the four temperature histories at 1.2 cm from the heated surface if the heat transfer coefficient has the values of 9, 10, 90, and 100 W/m2-K. Relate the differences between the plots to the sensitivity coefficients. Use k = 1.2 W/m-K and a =7.5 x 1 0- 7 m 2/s.