212 C HAP. 6 INVERSE CONVOLUTION FOR A SINGLE SURFACE H EAT FLUX P ROBLEMS 2 13 EXAMPLE 5 .6. Repeat Example 5.4 for the whole d omain regularization method. Find a n o ptimal value o f a +. 10. Beck, J. V., C riteria for Comparison o f M ethods o f S olution o f the Inverse H eat C onduction P roblem, Nuc/. Eng. Des. 53, 11 - 22 (1979). Using t he R value o f 0 .0283 in Eq. (5 .6.15) gives a+ = 0 .0008 which is quite close t o t he a+ =0.001 values used in the examples for the triangular heat flux test P ROBLEMS Solurion. 0 ~ 5.1. a. C alculate the 4>(r, t) function for a solid cylinder for r = 0. Give the EXAMPLE 5.7. Repeat Example 5.5 for the whole domain regularization method. values for a steel cylinder 0.664 cm in diameter. The dimensionless . Solution. T he A t+ value is 0.028 a nd thus Eq. (5 .6.15) gives a +=R 1 Since the standard statistical assumptions are n ot valid, a value o f R =O.OI is used a nd then Eqs. (S.6.14) a nd (S.6.1S) give U a = 0.0000 k l =0.0001 + 2 . GIve 4> values I n SI units for 6 time steps a nd give the times in seconds. Use values given in Table 1.3 a nd the properties given in Table 5.4. b . Repeat the problem for a cylinder 4 cm in diameter. c. Repeat p art (a) for brick. 5 2. a. A solid steel sph.ere is 4 c m in diameter. Calculate the center temper- e·~!S4y = U S x 10- 12 m4_KI/Wl N ote t hat t he temperatures are in K and t hat t he heat flux has units o f W1m2. . tl~e s tep IS A t•. = aAt/a = 0.05 where a = radIUS o f the cylinder. atures for ten tIme steps o f A t = 1.818 s for the heat flux given by t<O, q =O O <t<4At, 0 T he time step did enter the above selection o f a. S uppose that t he R =0.01 value is appropriate for A t=0.9 s. I t is interesting t o see t he 'effect o f choosing the time step o f 0.3 s. W ith the time step decreased by a factor o f 3, the ()qr value is also decreased by this factor. T hen Eq. (5.6.16) shows t hat ex is increased by a factor o f9. q=I00,OOOW/m 2 O <t<8At, q=200,OOO W /m 2 t >8At, q =OW/m 2 T he initial temperature is 50°C. b. Add to the temperatures some random errors that have zero mean have a s tandard deviation o f2 C, are uncorrelated, and are normal.' 5.3. Show that the expressions M REFERENCES ~ 0) 1. Keltner, N. R. a nd Beck, J. V., U nsteady Surface Element Method, J . Heat Transfer 103, 759- 764 (1981). 2. Beck, J. V. a nd Keltner, N. R., T ransient Thermal Contact o f T wo Semi-Infinite Bodies O ver a C ircular Area, Spacecraft Radiative Transfer and Temperature Control, T . E. Horton, Ed. Vol. 83 o f P rogress in Astronautics a nd Aeronautics, AIAA, New York, 1982, p p. 61 - 82. 3. Litkouhi, B. a nd Beck, J . V~ Intrinsic Thermocouple Analysis Using M ultinode Unsteady Surface Element Method, AIAA P apcr N o. 83-1437, 1983. 4. Stolz, G., J r ., Numerical Solutions t o a n Inverse P roblem o f H eat C onduction for Simple Shapes, J . H eat Transfer 81, 2 0- 26 (1960). 5. Beyer, W. H ~ Ed., Handbook o f Tables for Probability and Statistics, 2 nd ed., T he Chemical Rubber Co., Cleveland, O H, 1968, p. 484. 6. Beck, J . V. a nd Arnold, K. J ., Parameler Estimation in Engineering and Science, Wiley,. New York,1977. 7. Farnia, K., Computer-Assisted Expcrimental a nd Analytical Study o f T ime{fempcrature· D epcndent Thermal Propcrties o fthe Aluminum Alloy 2024-T351, P h.D . Dissertation, Dept. o f Mechanical Engineering, Micbigan State University, 1976. 8. Twomey, S., O n t he N umerical Solution o f F redholm Integral Equations o f t he F irst Kind by the Inversion o f t he Linear System P roduced by Quadrature, J . Assoc. Compo Mach. 10, 9 7-101 (1963). 9. Twomey, S., T he Application o f N umerical Filtering t o t he Solution o f Integral Equations Encountered in Indirect Sensing Measurements, J . Franklin Inst. 179, 95- 109 (1965). T M=To+ I Vqn4>M-n+1> 11=1 T M=To+ I qnA4>M-ft n =1 a nd V qn=qn-qft-t M a re equivalent where M is a n arbitrary positive integer. 5 .4. A plate ~f thickne~s 2 L initially a t the constant temperature o f To is d ropped Into a flUId a t Teo. T he boundary conditions for one-half the plate can be given by oT ox - =0 a t x =O ' -L~x~L, - -k aTI ox L = h[T(L, t )- Teo ] where h is the heat transfer coefficient. a. Using the solution o f t he problem for a step change in the surface heat flux a t x = L a nd insulated a t x = 0, d enoted cJ)(x, t), derive T (L, t) = To + f~ h (Teo - T(L, A)] OcJ)(L~: - A) dA where h is assumed constant. (How is </>(x, t) related to cJ)(x, t)?) (a) 214 C HAP . 5 b. INVERSE CONVOLUTION FOR A SINGLE SURFACE HEAT FLUX E quation (a) can be considered an integral equation for the unknown function T IL, t). By using exact matching, derive the following: 215 PROBLEMS show that q =qo+oq where qo is t he solution with OY,.=O a nd hq is the solution in Problem 5.7. U nder what conditions does a bounded value o f h Y,. result in a n u nbounded response in qj? Show t hat your answer is equivalent to tPz > 3tP, c. where TM = T (L, tM)' T he foregoing analysis is not restricted to a flat plate. Describe some other geometries for which this analysis applies. This problem is related to the surface element method.' - J 5 .5. Modify the analysis o f Problem 5.4 to treat the case of T... = T... (t). 5 .6. S tarting with Eq. (a) of Problem 5.4, derive a sequential function specification algorithm for estimating Tx ( t) given the temperature history T (L, t). Use r = 2 and assume that Tx (t) is temporarily constant. 5 .7. T he Stolz method can be investigated by considering a single temperature error, bY,., a nd a special sensitivity matrix. The Stolz method can be represented by the set of n algebraic equations a b b b b a b q. 5.10. Repeat Problem 5.9 for the center point in a sphere. Use Table 1.4. Answer part (a), At+ ::::0.19. 5.11. Write a computer program for Eq. (5.3.4). Let r be variable from 1 to 5 Use the program developed in Problem 5.11 to verify the results o f Figure 5.7. b. Use the program developed in Problem 5.11 to verify the results o f Figure 5.10. c. Use the program developed in Problem 5.11 to verify the results o f Figure 5.12. 0 where by" is the r th entry in the right vector. Show t hat 5.13. a. b Y,., b ' y. u q.= - , o q.+,=- 2 0. a a < b qj=(I- ~)6.j-" j =r+2, r +3, . .. 5.15. Solve Problem 5.13 for a solid sphere with the sensor at the center. i =r+l.r+2, . . . 5 .8. This problem uses the results of Problem 5.7. F or the right side of the matrix equation equal to Y where Y =[Y, Y2 . .. Y,.+c'iY. b. c. Use the program developed in Problem 5.11 t o investigate the stability o f t he function specification method for the flat plate with the sensor at the insulated surface. Let r = 2. Repeat p art (a) for r = 3. Repeat part (a) for r =4. 5.14. Solve Problem 5.13 for a solid cylinder with the sensor at the center. Using these expressions, show that b ( b)j -,-'' b qj=- a2bY• 1 - u 1'1 a nd ,pI values are 5.12. a. bY,. ... Using the criterion for stability for the Stolz method. ,pz > 3,p" given in Problem 5.8, find the stability limit for a solid cylinder with the sensor a t the cl!nter. Use Table 1.3. Answer: At+ ::::0.23. b. F ind the stability limit for the Stolz m ethod for the same geometry as in part (a) by using the temperature impulse test case. c. Give conclusions regarding your work for parts (a) a nd (b). 5.9. a. a nd make the maximum value of M be 40. T he inputs. 0 0 hq, bq: a ba bba bbb when tPl+' -,pI ::::constant for i = 1.,; • .... Use this criterion to find the stability limit for the Stolz procedure for the finite plate of Table 1.1 a nd the semi-infinite body of Table 1.2. y.+, y .] 5.16. Write a computer program for Eq. (5.3.4a). The K j values are inputs. Write the program t o t reat r =3 a nd solve for the flat plate geometry with the sensor at x =L a nd heated at x = 0. T he t imestepsareAr+ =0.05. Investigate the effect of arbitrary K j values by modifying those that are obtained for the function specification method which are K , =0.292, 216 C HAP.5 INVERSE CONVOLUTION FOR A SINGLE SURFACE HEAT FLUX PROBLEMS time Consider the heat impulse test to check for conservation of energy. W hat conclusions can you draw? 5.17. Repeat Problem 5.16 for the temperature impulse test case. 5 .18. Write a computer program for the function specification method for the I HCP with q temporarily constant for r = 3 and two interior temperature histories. 5 .19. a . Calculate the dimensionless components of XTX for n = 10 and at + = 0.05 for the flat plate case with the sensor a t x = L and the plate heated a t x =O. Also display the components for XTX+cxHfH. a nd for cx=O.OOI. P lot several rows for XTX a nd XTX+cxHfH., including the last row, versus the column numbers. What conclusions can you draw from your results? b. Repeat part (a) for the case o f a solid sphere with the sensor at the center. Use at; =0.05. 5 .20. a. Write a computer program for the whole domain first-order regularization method for the I HCP . b. Use the program to reproduce the results shown in Figure 5.15 . c. Use the program to reproduce the results shown in Figure 5.16. d. Use the program to reproduce Figure 5.17. 5 21. Modify the program developed in Problem 5.20 t o solve the zerothorder regularization method and repeat Problem 5.20. 5 .22. Write a computer program for the zeroth-order sequential regularization method for r =5 a nd compare the gain coefficients with those obtained using the function specification method for at + = 0.05 for a flat plate with the sensor at x =L a nd heated a t x =O. (Use Table 4.1.) Vary cx over a large range of values. 5 .23. An experiment for a flat plate heated at x = 0 and insulated at x = 1 in. is discussed in Reference 6, p . 402. The plate is Armco iron with k =43 .3 Btu/hr-ft-F and p c= 55 .6 Btu/ft 3 -F. The initial temperature is 81.36"F. Calculate the surface heat flux using the function specification method and the following actual data. - .j time 0 1.8 3.6 F Temp., O time Temp., O F 81.36 81.52 81.36 16.2 18.0 19.8 85 .85 86.91 88.05 a. b. c. Temp., O F time Temp., OF 5.4 7.2 9.0 10.8 12.6 14.4 K2 = 8.56, a nd K3 = 31.8 . F or example, try a. K l =0.292, K 2=8.56, and K 3= 15.9 b. K l =0.292, K 2=8.56, a nd K 3=47.4 c. K l = 0, K2=O, and K 3=31.8 - 2 17 81.36 81.85 82.42 83.48 84.48 85 .28 21.6 23.4 25.2 27.0 28.8 88.79 89.69 89.77 89 .93 90.09 Use r =3. Use r =4. Use r =5. 5 .24. Repeat Problem 5.23 using the whole domain first-order regularization method. Select several values of cx. 5 .25. Repeat Problem 5.23 using the sequential first-order regularization method with r = 5 and several cx values. 5.26. a. F ind the digital filter coefficients for Problem 5.23 for the function specification method for r =4 . b. Find the digital filter coefficients for Problem 5.23 for the whole domain regularization method. Select an optimal oc value. T he· midrange o f times is to be used to obtain the filter coefficients. 5.27. Using a computer program for the sequential regularization method investigate the effect of the regularization order. Consider the zeroth, first and second orders. Let cx vary over a wide range. Investigate both the deterministic and random errors. 5 .28. F or the whole domain regularization method generate a figure analogous to Figure 5.22. Consider values o f cx= 1 0- 6 ,0.001, and 1. 5 .29. F or the whole domain regularization method generate a figure analogous to Figure 5.23. Consider values of cx= 1 0- 6 ,0.001, and 1.