The boundary condition design of a three-dimensional furnace that heats an object moving along a conveyor belt of an assembly line is considered. A furnace of this type can be used by the manufacturing industry for applications such as industrial baking, curing of paint, annealing or manufacturing through chemical deposition. The object that is to be heated moves along the furnace as it is heated following a specified temperature history. The spatial temperature distribution on the object is kept isothermal through the whole process. The temperature distribution of the heaters of the furnace should be changed as the object moves so that the specified temperature history can be satisfied. The design problem is transient where a series of inverse problems are solved. The process furnace considered is in the shape of a rectangular tunnel where the heaters are located on the top and the design object moves along the bottom. The inverse design approach is used for the solution, which is advantageous over a traditional trial-and-error solution where an iterative solution is required for every position as the object moves. The inverse formulation of the design problem is ill-posed and involves a set of Fredholm equations of the first kind. The use of advanced solvers that are able to regularize the resulting system is essential. These include the conjugate gradient method, the truncated singular value decomposition or Tikhonov regularization, rather than an ordinary solver, like Gauss-Seidel or Gauss elimination.

1.
Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V., and Yagola, A. G., 1995, Numerical Methods for Solving Ill-Posed Problems, Kluwer Academic Publishers, Boston, Massachusetts.
2.
Alifanov, O. M., 1994, Inverse Heat Transfer Problems, Springer-Verlag, Berlin.
3.
Alifanov, O. M., Artyukhin, E. A., and Rumyantsev, S. V., 1995, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, New York.
4.
Beck, J. V., Blackwell, B., and St. Clair, Jr., C. R., 1985, Inverse Heat, John Wiley and Sons Inc., New York.
5.
O¨zis¸ik, M. N., and Orlande, H. R. B., 2000, Inverse Heat Transfer, Taylor and Francis, New York.
6.
Kennon
,
S. R.
, and
Dulikravich
,
G. S.
,
1985
, “
The Inverse Design of Internally Cooled Turbine Blades
,”
ASME J. Eng. Gas Turbines Power
,
107
, pp.
123
126
.
7.
Dulikravich, G. S., and Martin, T. J., 1997, “Inverse Shape and Boundary Condition Problems and Optimization in Heat Conduction,” Advances in Numerical Heat Transfer, W. J. Minkowycz, and E. M. Sparrow, eds., vol. 1, Taylor and Francis, Washington DC.
8.
Howell
,
J. R.
,
Ezekoye
,
O. A.
, and
Morales
,
J. C.
,
2000
, “
Inverse Design Model for Radiative Heat Transfer
,”
ASME J. Heat Transfer
,
122
, pp.
492
502
.
9.
Franc¸a
,
F. H. R.
,
Ezekoye
,
O. A.
, and
Howell
,
J. R.
,
2001
, “
Inverse Boundary Design Combining Radiation and Convection Heat Transfer
,”
ASME J. Heat Transfer
,
123
(
5
), pp.
884
891
.
10.
Franc¸a, F. H. R., Howell, J. R., Ezekoye, O. A., and Morales, J. C., 2002, “Inverse Design of Thermal Systems,” Advances in Heat Transfer, J. P. Hartnett and T. F. Irvine, eds., 36, pp. 1–110, Academic Press, New York.
11.
Ertu¨rk
,
H.
,
Ezekoye
,
O. A.
, and
Howell
,
J. R.
,
2002
, “
The Application of An Inverse Formulation In The Design of Boundary Conditions for Transient Radiating Enclosures
,”
ASME J. Heat Transfer
,
124
, pp.
1095
1102
.
12.
Siegel, R., and Howell, J. R., 2002, Thermal Radiation Heat Transfer, 4th Ed., Taylor and Francis, Washington DC.
13.
Federov
,
A. G.
,
Lee
,
K. H.
, and
Viskanta
,
R.
,
1998
, “
Inverse Optimal Design of the Radiant Heating in Materials Processing and Manufacturing
,”
Mater. Chem. Phys.
,
7
, pp.
719
726
.
14.
Modest, M. F., 1993, Radiative Heat Transfer, Mc Graw-Hill Book Co, Singapore.
15.
Ertu¨rk, H., Arinc¸, F., and Selc¸uk, N., 1997, “Accuracy of Monte Carlo Method Re-Examined on a Box-Shaped Furnace Problem,” Radiative Heat Transfer II: Proceedings of Second International Symposium on Radiative Heat Transfer, Mengu¨c¸, M. P., ed., Begell House, New York, pp. 85–95.
16.
Ertu¨rk
,
H.
,
Ezekoye
,
O. A.
, and
Howell
,
J. R.
,
2002
, “
Comparison of Three Regularized Solution Techniques in a Three-Dimensional Inverse Radiation Problem
,”
J. Quant. Spectrosc. Radiat. Transf.
,
73
, pp.
307
316
.
17.
Beckman, F. S., 1960, “The Solution of Linear Equations By the Conjugate Gradient Method,” Mathematical Methods For Digital Computers, A. Ralston and H. S. Wilf, eds., John Wiley and Sons, New York, pp. 62–72.
18.
Hansen, P. C., 1998, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM Publications, Philadelphia, Pennsylvania.
You do not currently have access to this content.