This paper provides a solution for two-dimensional (2D) heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green's functions (GFs). Two approaches are used, which provide solutions for either semi-infinite plates or finite plates with isothermal conditions, which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The procedure involves the derivation of previously unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

References

References
1.
Roach
,
P.
,
1998
,
Verification and Validation in Computation Science and Engineering
,
Hermosa
,
Albuquerque, NM
.
2.
Ozisik
,
M.
,
1993
,
Heat Conduction
,
2nd ed.
,
Wiley
,
New York
.
3.
Arpaci
,
V.
,
1966
,
Conduction Heat Transfer
,
Addison-Wesley
,
Reading, MA
.
4.
Cole
,
K.
,
Beck
,
J.
,
Haji-Sheikh
,
A.
, and
Litkouhi
,
B.
,
2011
,
Heat Conduction Using Green's Functions
,
2nd ed.
,
Hemisphere Press
,
New York
.
5.
Yan
,
L.
,
Haji-Sheikh
,
A.
, and
Beck
,
J. V.
,
1993
, “
Thermal Characteristics of Two-Layered Bodies With Embedded Thin-Film Heat Source
,”
ASME J. Electron. Packag.
,
115
(
3
), pp.
276
283
.
6.
Aviles-Ramos
,
C.
,
Haji-Sheikh
,
A.
, and
Beck
,
J. V.
,
1998
, “
Exact Solution of Heat Conduction in Composite Materials and Application to Inverse Problems
,”
ASME J. Heat Transfer
,
120
(
3
), pp.
592
599
.
7.
McMasters
,
R. L.
,
Dowding
,
K. J.
,
Beck
,
J. V.
, and
Yen
,
D.
,
2002
, “
Methodology to Generate Accurate Solutions for Verification in Transient Three-Dimensional Heat Conduction
,”
Numer. Heat Transfer, Part B
,
41
(
6
), pp.
521
541
.
8.
Yen
,
D.
,
Beck
,
J.
,
McMasters
,
R.
, and
Amos
,
D.
,
2002
, “
Solution of an Initial-Boundary Value Problem for Heat Conduction in a Parallelepiped by Time Partitioning
,”
Int. J. Heat Mass Transfer
,
45
(
21
), pp.
4267
4279
.
9.
Beck
,
J.
, and
Cole
,
K.
,
2007
, “
Improving Convergence of Summations in Heat Conduction
,”
Int. J. Heat Mass Transfer
,
50
(
1–2
), pp.
257
268
.
10.
Yen
,
D.
, and
Beck
,
J.
,
2004
, “
Green's Functions and Three-Dimensional Steady-State Heat-Conduction Problems in a Two-Layered Composite
,”
J. Eng. Math.
,
49
(
3
), pp.
305
319
.
11.
Haji-Sheikh
,
A.
,
Beck
,
J.
, and
Agonafer
,
D.
,
2003
, “
Steady-State Heat Conduction in Multi-Layer Bodies
,”
Int. J. Heat Mass Transfer
,
46
(
13
), pp.
2363
2379
.
12.
McMasters
,
R.
, and
Beck
,
J.
,
2014
, “
A Two-Dimensional Cylindrical Transient Conduction Solution Using Green's Functions
,”
ASME J. Heat Transfer
,
136
(
10
), p.
101301
.
13.
Haji-Sheikh
,
A.
, and
Beck
,
J. V.
,
2002
, “
Temperature Solution in Multi-Dimensional Multi-Layer Bodies
,”
Int. J. Heat Mass Transfer
,
45
(
9
), pp.
1865
1877
.
14.
Beck
,
J.
,
Haji-Sheikh
,
A.
,
Amos
,
D.
, and
Yen
,
D.
,
2004
, “
Verification Solution for Partial Heating of Rectangular Solid
,”
Int. J. Heat Mass Transfer
,
47
(
19–20
), pp.
4243
4255
.
15.
Kidawa-Kukla
,
J.
,
2008
, “
Temperature Distribution in a Rectangular Plate Heated by a Moving Heat Source
,”
Int. J. Heat Mass Transfer
,
51
(
3–4
), pp.
865
872
.
16.
Beck
,
J.
,
McMasters
,
R.
,
Dowding
,
K.
, and
Amos
,
D.
,
2006
, “
Intrinsic Verification Methods in Linear Heat Conduction
,”
Int. J. Heat Mass Transfer
,
49
(
17–18
), pp.
2984
2994
.
17.
Cole
,
K.
,
Beck
,
J.
,
Woodbury
,
K.
, and
de Monte
,
F.
,
2014
, “
Intrinsic Verification and a Heat Conduction Database
,”
Int. J. Therm. Sci.
,
78
, pp.
36
47
.
18.
Schlesinger
,
S.
,
1979
, “
Terminology for Model Credibility
,”
Simulation
,
32
(
3
), pp.
103
104
.https://doi.org/10.1177/003754977903200304
19.
Oberkampf
,
W.
, and
Trucano
,
T.
,
2002
, “
Verification and Validation in Computational Fluid Dynamics
,”
Prog. Aerosp. Sci.
,
38
(
3
), pp.
209
272
.
20.
de Monte
,
F.
,
Beck
,
J.
, and
Amos
,
D.
,
2012
, “
Solving Two-Dimensional Cartesian Unsteady Heat Conduction Problems for Small Values of the Time
,”
Int. J. Therm. Sci.
,
60
, pp.
106
113
.
21.
Cole
,
K.
,
2004
, “
Computer Software for Fins and Slab Bodies With Green's Functions
,”
Comput. Appl. Eng. Educ.
,
12
(
3
), pp.
189
197
.
22.
Cole
,
K.
, and
Crittenden
,
P.
,
2001
, “
Heat Conduction in Cartesian Coordinates and a Library of Green's Functions
,”
35th National Heat Transfer Conference
, Anaheim, CA, June 10–12, pp.
1373
1382
.
23.
Cole
,
K.
, 2000, “
Green's Function Library
,” University of Nebraska, Lincoln, NE, accessed Aug. 16, 2017, http://www.greensfunction.unl.edu
24.
Polyanin
,
A.
,
2002
,
Handbook of Linear Partial Differential Equations for Engineers and Scientists
,
CRC Press
,
Boca Raton, FL
.
25.
Mansur
,
W.
,
Vasconcellos
,
C.
,
Zambrozuski
,
N.
, and
Rotunno Filho
,
O.
,
2009
, “
Numerical Solution for the Linear Transient Heat Conduction Equation Using an Explicit Green's Approach
,”
Int. J. Heat Mass Transfer
,
52
(
3–4
), pp.
694
701
.
26.
Woodbury
,
K.
,
Hamidreza
,
N.
, and
Beck
,
J.
,
2017
, “
Exact Analytical Solution for 2-D Transient Heat Conduction in a Rectangle With Partial Heating on One Edge
,”
Int. J. Therm. Sci.
,
112
, pp.
252
262
.
27.
de Monte
,
F.
,
Beck
,
J.
, and
Amos
,
D.
,
2011
, “
A Heat-Flux Based ‘Building Block’ Approach for Solving Heat Conduction Problems
,”
Int. J. Heat Mass Transfer
,
54
(
13–14
), pp.
2789
2800
.
28.
Duffy
,
D.
,
2008
,
Mixed Boundary Value Problems
, CRC Press, Boca Raton, FL.
29.
Beck
,
J.
,
2011
, “
Transient Three-Dimensional Heat Conduction Problems With Partial Heating
,”
Int. J. Heat Mass Transfer
,
54
(
11–12
), pp.
2479
2489
.
30.
Oldham
,
K.
,
Myland
,
J.
, and
Spanier
,
J.
,
2009
,
An Atlas of Functions
,
2nd ed.
,
Springer
,
New York
.
31.
McMasters
,
R.
,
de Monte
,
F.
,
Beck
,
J.
,
Nallapaneni
,
S.
, and
Amos
,
D.
,
2016
, “
Diffusion Penetration Time for Transient Heat Conduction
,”
J. Thermophys. Heat Transfer
,
30
(
3
), pp.
614
621
.
32.
Cole
,
K. D.
,
de Monte
,
F.
,
McMasters
,
R. L.
,
Woodbury
,
K. A.
,
Haji-Sheikh
,
A.
, and
Beck
,
J. V.
,
2016
, “
Steady Heat Conduction With Generalized Boundary Conditions
,”
ASME
Paper No. IMECE2016-66605.
You do not currently have access to this content.