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Fig. 12. Optimal powers of heaters for aspect ratio (H/L) = 1.0, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qy< 4.0 although using optimization algorithm the desired uniform heat flux distribution on the DO po = —2.0 is not achieved. Although, the heat flux along the top surface (elements: 1—25) of the design object is uniform but both with and without optimization they are different than the desired one. But, the heat flux distribution along the vertical surfaces (elements: 26—40, and 41—55) of the design object, is found non-uniform and this non-uniformity is owing to Next, we proceed with consideration of heaters along all three walls, viz., top, back and west walls. Using direct method (REM7*), and the present algorithm (REM? + MGA) the heat flux distribution along the surfaces of the design object have been estimated. In this

Figure 12 Optimal powers of heaters for aspect ratio (H/L) = 1.0, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qy< 4.0 although using optimization algorithm the desired uniform heat flux distribution on the DO po = —2.0 is not achieved. Although, the heat flux along the top surface (elements: 1—25) of the design object is uniform but both with and without optimization they are different than the desired one. But, the heat flux distribution along the vertical surfaces (elements: 26—40, and 41—55) of the design object, is found non-uniform and this non-uniformity is owing to Next, we proceed with consideration of heaters along all three walls, viz., top, back and west walls. Using direct method (REM7*), and the present algorithm (REM? + MGA) the heat flux distribution along the surfaces of the design object have been estimated. In this

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Fig. 1. Schematic of (a) the radiant enclosure with a centrally located design object and (b) the computational domain.
Fig. 1. Schematic of (a) the radiant enclosure with a centrally located design object and (b) the computational domain.
Fig. 3. Flowchart of the micro-genetic algorithm used in the present work.
Fig. 3. Flowchart of the micro-genetic algorithm used in the present work.
Fig. 4. (a) Grid independence test for direct problem using the REM’, (b) ray inde- pendence test: comparison with the analytical method. As in the REM? the radiative calculation is done through the ray tracing, choosing the correct numbers of the rays is an important criterion. After testing the direct method (REM7) for the grid
Fig. 4. (a) Grid independence test for direct problem using the REM’, (b) ray inde- pendence test: comparison with the analytical method. As in the REM? the radiative calculation is done through the ray tracing, choosing the correct numbers of the rays is an important criterion. After testing the direct method (REM7) for the grid
Fig. 5. (a) Validation of the direct method — comparison of the heat flux distribution obtained using REM? with analytical results and (b) validation of the optimization algorithm — comparison of the heat flux distribution obtained using REM? (direct method) and optimization algorithm. For the ray independency test, for the chosen model four different numbers of rays viz., 30 x 30, 60x60, 90x90 and 120 x 120 were evaluated. In the REM2, with these numbers of rays the radiative heat flux distribution along the bottom surface of the enclosure was calculated. The heat flux distribution calculated using the different numbers of rays was then compared with that of using the analytical method. The comparison is shown in Fig. 4b. From Fig. 4b, it is observed that the heat flux distribution calculated using 30 x 30, 60 x 60 rays are not matching well with that of calculated using the analytical method while the heat flux calcu- lated using 90 x 90 and 120 x 120 rays are found in good agree- ment. Therefore 90 x 90 and 120 x 120 rays are found adequate to achieve the accurate heat flux distribution using REM. Being
Fig. 5. (a) Validation of the direct method — comparison of the heat flux distribution obtained using REM? with analytical results and (b) validation of the optimization algorithm — comparison of the heat flux distribution obtained using REM? (direct method) and optimization algorithm. For the ray independency test, for the chosen model four different numbers of rays viz., 30 x 30, 60x60, 90x90 and 120 x 120 were evaluated. In the REM2, with these numbers of rays the radiative heat flux distribution along the bottom surface of the enclosure was calculated. The heat flux distribution calculated using the different numbers of rays was then compared with that of using the analytical method. The comparison is shown in Fig. 4b. From Fig. 4b, it is observed that the heat flux distribution calculated using 30 x 30, 60 x 60 rays are not matching well with that of calculated using the analytical method while the heat flux calcu- lated using 90 x 90 and 120 x 120 rays are found in good agree- ment. Therefore 90 x 90 and 120 x 120 rays are found adequate to achieve the accurate heat flux distribution using REM. Being
Fig. 6. Heat flux distribution on the surfaces of the design object with and without optimization considering 100 panel heaters along the top wall. In this study, using the considered enclosure domain, in this, by providing the heat flux distributions on the design surface obtained from the direct method, keeping the properties and constraints the same as known parameters, powers of the panel heaters are esti- mated. For this validation, thermal conditions and computational domain are the same as that considered for results given in Fig. 5a. Towards this validation, in the inverse analysis, the MGA in conjunction with REM? is allowed to run for 1000 generations. Within 500 generations the value of the objective function was
Fig. 6. Heat flux distribution on the surfaces of the design object with and without optimization considering 100 panel heaters along the top wall. In this study, using the considered enclosure domain, in this, by providing the heat flux distributions on the design surface obtained from the direct method, keeping the properties and constraints the same as known parameters, powers of the panel heaters are esti- mated. For this validation, thermal conditions and computational domain are the same as that considered for results given in Fig. 5a. Towards this validation, in the inverse analysis, the MGA in conjunction with REM? is allowed to run for 1000 generations. Within 500 generations the value of the objective function was
Fig. 7. Heat flux distribution on the surfaces of the design object with and without optimization considering 300 panel heaters along on top, back and west walls.
Fig. 7. Heat flux distribution on the surfaces of the design object with and without optimization considering 300 panel heaters along on top, back and west walls.
Fig. 8. Estimated heat flux distribution on the surfaces of the design object and corresponding deviation using different aspect ratios of the radiant enclosure For yielding the uniform thermal conditions on a 3-D design object, in the present algorithm requirement of numbers of panel heaters is checked. This study is conducted using the actual computational domain shown in Fig. 1b. Except the bottom surface of the enclosure on which the DO is located, we have considered 100 panel heaters along each of the 3 walls viz., top, back and west wall. Initially only the heaters along the top wall are considered active for the computation while rest all surfaces are treated as insulated. Fig. 6 shows the estimated heat flux distribution on the surfaces of a DO obtained using the present algorithm (REM? + MGA) and the heat flux distribution without optimization (direct method only, i.e. REM?) is also shown. In this case the heaters along the top walls are set to deliver the flux qj = 1.0. The purpose of this comparison is to show the effect of optimization algorithm and without optimization on the estimated heat flux distribution on the surfaces of the DO. In Fig. 6, heat flux distri- butions are plotted for all the surface elements ranging from 1 to 55 (shaded regions in Fig. 2a, e and f) of the three surfaces of the design object in the computational domain. It is observed from Fig. 6 that considering 100 panel heaters along the top wall only, minimized as low as 10~° and as shown in Fig. 5b, the original heat flux distribution on the heater surface was found to be recovered. The corresponding maximum error in the heat flux distribution on design surface is less than 0.1%. This much accuracy is sufficient for the practical design point of view. Having proved the accuracies of both the methods, viz., REM? and MGA used in the present analysis, in the following pages, for various cases, we provide and analyze the results of estimations of powers of the panel heaters to yield uniform thermal conditions.
Fig. 8. Estimated heat flux distribution on the surfaces of the design object and corresponding deviation using different aspect ratios of the radiant enclosure For yielding the uniform thermal conditions on a 3-D design object, in the present algorithm requirement of numbers of panel heaters is checked. This study is conducted using the actual computational domain shown in Fig. 1b. Except the bottom surface of the enclosure on which the DO is located, we have considered 100 panel heaters along each of the 3 walls viz., top, back and west wall. Initially only the heaters along the top wall are considered active for the computation while rest all surfaces are treated as insulated. Fig. 6 shows the estimated heat flux distribution on the surfaces of a DO obtained using the present algorithm (REM? + MGA) and the heat flux distribution without optimization (direct method only, i.e. REM?) is also shown. In this case the heaters along the top walls are set to deliver the flux qj = 1.0. The purpose of this comparison is to show the effect of optimization algorithm and without optimization on the estimated heat flux distribution on the surfaces of the DO. In Fig. 6, heat flux distri- butions are plotted for all the surface elements ranging from 1 to 55 (shaded regions in Fig. 2a, e and f) of the three surfaces of the design object in the computational domain. It is observed from Fig. 6 that considering 100 panel heaters along the top wall only, minimized as low as 10~° and as shown in Fig. 5b, the original heat flux distribution on the heater surface was found to be recovered. The corresponding maximum error in the heat flux distribution on design surface is less than 0.1%. This much accuracy is sufficient for the practical design point of view. Having proved the accuracies of both the methods, viz., REM? and MGA used in the present analysis, in the following pages, for various cases, we provide and analyze the results of estimations of powers of the panel heaters to yield uniform thermal conditions.
Fig. 9. Optimal powers of heaters for aspect ratio (H/L) = 0.4, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qu < 4.0
Fig. 9. Optimal powers of heaters for aspect ratio (H/L) = 0.4, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qu < 4.0
Fig. 10. Optimal powers of heaters for aspect ratio (H/L) = 0.5, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qy < 4.(
Fig. 10. Optimal powers of heaters for aspect ratio (H/L) = 0.5, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qy < 4.(
Fig. 11. Optimal powers of heaters for aspect ratio (H/L) = 0.75, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qy<4.0.
Fig. 11. Optimal powers of heaters for aspect ratio (H/L) = 0.75, (a) top wall, (b) back wall and (c) west wall; heater power range: 0.0001 < qy<4.0.
Fig. 13. Estimated heat flux distribution on the surfaces of the design object for different ranges of powers of panel heaters, (a) 0.0001 < qu < 1.0, (b) 0.0001 < qu < 2.0, (c) 0.0001 < qu < 4.0, (d) 0.0001 < qu <8.0.
Fig. 13. Estimated heat flux distribution on the surfaces of the design object for different ranges of powers of panel heaters, (a) 0.0001 < qu < 1.0, (b) 0.0001 < qu < 2.0, (c) 0.0001 < qu < 4.0, (d) 0.0001 < qu <8.0.
Fig. 14. Heat flux distribution on heaters for aspect ratio (H/L) = 0.5, (a) top wall, (b) back wall, (c) west wall; heater power range 0.0001 < qy < 1.(
Fig. 14. Heat flux distribution on heaters for aspect ratio (H/L) = 0.5, (a) top wall, (b) back wall, (c) west wall; heater power range 0.0001 < qy < 1.(
Fig. 15. (a) Convergence history for different crossover probability values, (b) convergence history for the cases using different heater power ranges. Since in the present study MGA has been used as an optimiza- tion tool, it is important to check the effect of MGA parameters. Towards this aspect, the effects of crossover probabilities (P.) are studied. Single-point crossover probabilities (P,) = 0.2, 04, 0.6, 0.8, 1.0 and uniform crossover probability of 0.5 are considered. Fig. 15a shows the convergence history of the objective function (Eq. (14)) against the numbers of generations. It is well known that in mini- mization problems like present one, as the objective function value
Fig. 15. (a) Convergence history for different crossover probability values, (b) convergence history for the cases using different heater power ranges. Since in the present study MGA has been used as an optimiza- tion tool, it is important to check the effect of MGA parameters. Towards this aspect, the effects of crossover probabilities (P.) are studied. Single-point crossover probabilities (P,) = 0.2, 04, 0.6, 0.8, 1.0 and uniform crossover probability of 0.5 are considered. Fig. 15a shows the convergence history of the objective function (Eq. (14)) against the numbers of generations. It is well known that in mini- mization problems like present one, as the objective function value
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