Abstract

Thick plates that are thermally loaded on one surface with convection on the other are often encountered in engineering practice. Given this wide utility and the limitations of most existing solutions to an adiabatic boundary condition, generalized direct thermal solutions were first derived for an arbitrary surface loading as modeled by a polynomial and its coefficients on the loaded surface with convection on the other. Once formulated, the temperature solutions were then used with elasticity relationships to determine the resulting thermal stresses. Additionally, the inverse thermal problem was solved using a least-squares type determination of the aforementioned polynomial coefficients based on the direct-solution and temperatures measured at the surface with convection. Previously published relationships for a thick-walled cylinder with internal heating/cooling and external convection are also included for comparison. Given the versatility of the polynomial solutions advocated, the method appears well suited for complicated thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties do not vary with temperature.

References

1.
Albrecht
,
W.
,
1969
, “
How Thickness and Materials Properties Influence Thermal Shock Stresses in Flat Plates and Cylinders
,”
ASME
Paper No. G9-GT-107.10.1115/G9-GT-107
2.
Chen
,
S. H.
,
1961
, “
One-Dimensional Heat Conduction With Arbitrary Heating Rate
,”
J. Aerosp. Sci.
,
28
(
4
), pp.
336
337
.10.2514/8.8974
3.
Nied
,
H. F.
,
1987
, “
Thermal Shock in an Edge-Cracked Plate Subjected to Uniform Surface Heating
,”
Eng. Fract. Mech.
,.
26
(
2
), pp.
239
246
.10.1016/0013-7944(87)90200-1
4.
Austin
,
J. B.
,
1932
, “
Temperature Distribution in Solid Bodies During Heating or Cooling
,”
J. Appl. Phys.
,
3
(
4
), pp.
179
184
.10.1063/1.1745098
5.
Vedula
,
V. R.
,
Green
,
D. J.
,
Hellmann
,
J. R.
, and
Segall
,
A. E.
,
1998
, “
Test Methodology for the Thermal Shock Characterization of Ceramics
,”
J. Mater. Sci.
,
33
(
22
), pp.
5427
5432
.10.1023/A:1004410719754
6.
Tu
,
J. J.
, and
Segall
,
A. E.
,
1997
, “
Thermomechanical Analysis of a Complex, Refractory Tundish Flow Modifier During Preheating
,”
Proceedings of the Unified International Technical Conference on Refractories, Fifth Biennial Worldwide Congress
, New Orleans, LA, Nov. 4–7.
7.
Nied
,
H. F.
, and
Erdogan
,
F.
,
1983
, “
Transient Thermal Stress Problem for a Circumferentially Cracked Hollow Cylinder
,”
J. Therm. Stresses
,
6
(
1
), pp.
1
14
.10.1080/01495738308942161
8.
Hung
,
C. I.
,
Chen
,
C. K.
, and
Lee
,
Z. Y.
,
2001
, “
Thermoelastic Transient Response of Multilayered Hollow Cylinder With Initial Interface Pressure
,”
J. Therm. Stresses
,
24
(
10
), pp.
987
1006
.10.1080/014957301753191086
9.
Lee
,
H. L.
, and
Yang
,
Y. C.
,
2001
, “
Inverse Problem of Coupled Thermoelasticity for Prediction of Heat Flux and Thermal Stresses in an Annular Cylinder
,”
Int. Comm. Heat Mass Transfer
,
28
(
5
), pp.
661
670
.10.1016/S0735-1933(01)00270-6
10.
Pisarenko
,
G. S.
,
Gogotsi
,
G. A.
, and
Grusheuskii
,
Y. L.
,
1978
, “
A Method of Investigating Refractory Nonmetallic Materials in Linear Thermal Loading
,”
Probl. Prochn.
,
4
, p.
36
.10.1007/BF01523789
11.
Segall
,
A. E.
,
2005
, “
Inverse Solutions for Determining Arbitrary Boundary-Conditions Using a Least-Squares Approach
,”
ASME J. Heat Transfer
,
127
(
12
), pp.
1403
1405
.10.1115/1.2060727
12.
Segall
,
A. E.
,
2006
, “
Thermoelastic Stresses in Thick-Walled Vessels Under an Arbitrary Thermal Transient Via the Inverse Route
,”
ASME J. Pressure Vessel Technol.
,
128
(
4
), pp.
599
604
.10.1115/1.2349573
13.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
,
1959
,
Conduction of Heat in Solids
,
Oxford University Press
,
Great Britain, UK
.
14.
Haji-Sheikh
,
A.
, and
Beck
,
J. V.
,
2000
, “
An Efficient Method of Computing Eigenvalues in Heat Conduction
,”
Numer. Heat Transfer, Part B
, (
38
), pp.
133
156
.10.1080/104077900750034643
15.
Fodor
,
G.
,
1965
,
Laplace Transforms in Engineering
,
Akademiai Kiado
,
Budapest, Hungary
.
16.
Jacquot
,
R. G.
,
Steadman
,
J. W.
, and
Rhodine
,
C. N.
,
1983
, “
The Gaver-Stehfest Algorithm for Approximate Inversion of Laplace Transforms
,”
IEEE Circuits Syst. Mag.
,
5
(
1
), pp.
4
8
.10.1109/MCAS.1983.6323897
17.
Segall
,
A. E.
,
2001
, “
Relationships for the Approximation of Direct and Inverse Problems With Asymptotic Kernels
,”
Inverse Probl. Eng.
,
9
(
2
), pp.
127
140
.10.1080/174159701088027757
18.
Segall
,
A. E.
,
2004
, “
Thermoelastic Stresses in an Axisymmetric Thick-Walled Tube Under an Arbitrary Internal Transient
,”
ASME J. Pressure Vessel Technol.
,
126
(
3
), pp.
327
332
.10.1115/1.1762461
19.
Timoshenko
,
S. P.
, and
Goodier
,
J. N.
,
1951
,
Theory of Elasticity
,
McGraw-Hill
, New York.
20.
Imber
,
M.
,
1973
, “
Temperature Extrapolation Methods for Hollow Cylinders
,”
AIAA J.
,
11
(
1
), pp.
117
118
.10.2514/3.6684
21.
Imber
,
M.
,
1979
, “
Inverse Problems for a Solid Cylinder
,”
AIAA J.
,
17
(
1
), pp.
91
94
.10.2514/3.61067
22.
Blanc
,
G.
, and
Raynaud
,
M.
,
1996
, “
Solution of the Inverse Heat Conduction Problem From Thermal Strain Measurements
,”
ASME J. Heat Transfer
,
118
(
4
), pp.
842
849
.10.1115/1.2822579
23.
Yang
,
Y. C.
,
Chen
,
U. C.
, and
Chang
,
W. J.
,
2002
, “
An Inverse Problem of Coupled Thermoelasticity in Predicting Heat Flux and Thermal Stresses by Strain Measurement
,”
J. Therm. Stresses
,
25
(
3
), pp.
265
281
.10.1080/014957302317262305
24.
Taler
,
J.
,
1997
, “
Analytical Solution of the Over Determined Inverse Heat Conduction Problem With an Application to Monitoring Thermal Stresses
,”
Heat Mass Transfer
,
33
(
3
), pp.
209
218
.10.1007/s002310050180
25.
Burggraf
,
O. R.
,
1964
, “
An Exact Solution of the Inverse Problem in Heat Conduction Theory and Applications
,”
ASME J. Heat Transfer
,
86
(
3
), pp.
373
382
.10.1115/1.3688700
26.
Segall
,
A. E.
,
Engels
,
D.
, and
Hirsh
,
A.
,
2009
, “
Transient Surface Strains and the Deconvolution of Thermoelastic States and Boundary Conditions
,”
ASME J. Pressure Vessel Technol.
,
131
(
1
), p.
011201
.10.1115/1.3006350
27.
Segall
,
A. E.
,
2001
, “
Solutions for the Correction of Temperature Measurements Based on Beaded Thermocouples
,”
Int. J. Heat Mass Transfer
,
44
(
15
), pp.
2801
2808
.10.1016/S0017-9310(00)00327-6
28.
Busby
,
H. R.
, and
Trujillo
,
D. M.
,
1985
, “
Numerical Solution to a Two-Dimensional Inverse Heat Conduction Problem
,”
Int. J. Numer. Methods Eng.
,
21
(
2
), pp.
349
359
.10.1002/nme.1620210211
30.
Segall
,
A. E.
,
Engels
,
D.
,
Drapaca
,
C.
, and
Harris
,
J.
,
2014
, “
An Inverse Method for Determining Temperature Dependent Thermophysical Properties Based on a Remotely Measured Temperature History
,”
Inverse Probl. Eng. Sci.
,
22
(
4
), pp.
672
681
.10.1080/17415977.2013.848435
You do not currently have access to this content.