https://orcid.org/0000-0002-3336-8050
Abstract
Regarding the steady-state heat conduction problem in exchanger tubes, the meshfree boundary integral equation method is employed to determine the conduction shape factor in this paper. Different from the conventional boundary element method, the present method is free of mesh generation. After using the parametric function to represent the boundary contour and adopting the Gaussian quadrature, only collocating points on the boundary is required to obtain the linear algebraic equations. By introducing the local exact solution, the singular integral in the sense of the Cauchy principal value can be novelly determined. In addition, the boundary layer effect due to the nearly singular integral in the boundary integral equation can be dealt with. Two cases of different boundary conditions are considered. One is the isothermal condition on both the inner and outer surfaces. The other is the isothermal condition on the inner surface and the convection condition on the outer surface. Besides, numerical instability is found and the nonuniqueness solution due to the degenerate scale is examined by calculating the conduction shape factor and the temperature on the outer surface.
References
1.
Langmuir
,
I.
, 1919
, “
Convection and Radiation of Heat
,” Trans. Am. Electrochem.
,
23
, pp. 299
–332
.https://books.google.com.tw/books?id=FyIZAQAAIAAJ&pg=PA299&hl=zh-TW&source=gbs_toc_r&cad=3#v=onepage&q&f=false2.
Smith
,
J. C.
,
Lind
,
J. E. J.
, and
Lermond
,
D. S.
, 1958
, “
Shape Factors for Conductive Heat Flow
,” AIChE J.
,
4
(3
), pp. 330
–331
.10.1002/aic.6900403193.
Balcerzak
,
M. J.
, and
Raynor
,
S.
, 1961
, “
Steady State Temperature Distribution and Heat Flow in Prismatic Bars With Isothermal Boundary Conditions
,” Int. J. Heat. Mass. Transfer
,
3
(2
), pp. 113
–125
.10.1016/0017-9310(61)90074-64.
Lewis
,
G. K.
, 1968
, “
Shape Factors in Conduction Heat Flow for Circular Bars and Slabs With Various Internal Geometries
,” Int. J. Heat. Mass. Transfer
,
11
(6
), pp. 985
–992
.10.1016/0017-9310(68)90004-55.
Dugan
,
J. P.
, 1972
, “
On the Shape Factor for a Hollow, Square Cylinder
,” AlChE J.
,
18
(5
), pp. 1082
–1083
.10.1002/aic.6901805376.
Griggs
,
E. I.
,
Pitts
,
D. R.
, and
Goyal
,
A. B.
, 1973
, “
Conductive Shape Factors for a Circular Cylinder Centered in a Rectangular Slab Having One and Two Adiabatic Surfaces
,” ASME J. Heat Mass Transfer-Trans. ASME
,
95
(1
), pp. 129
–130
.10.1115/1.34499847.
Naghdi
,
A. K.
, 1981
, “
Conduction Shape Factors for Certain Multi-Hole Prismatic Bars
,” ASME J. Heat Mass Transfer-Trans. ASME
,
103
(4
), pp. 821
–823
.10.1115/1.32445518.
Warrington
,
R. O.
, Jr.
,
Powe
,
R. E.
, and
Mussulman
,
R. L.
, 1982
, “
Steady Conduction in Three-Dimensional Shells
,” ASME J. Heat Mass Transfer-Trans. ASME
,
104
(2
), pp. 393
–394
.10.1115/1.32451049.
Hsieh
,
C. K.
, and
Yang
,
S. L.
, 1984
, “
A New Theory on the Critical Thickness of Insulation
,” ASME J. Heat Mass Transfer-Trans. ASME
,
106
(3
), pp. 648
–652
.10.1115/1.324673010.
Hassani
,
A. V.
, and
Hollands
,
K. G. T.
, 1989
, “
On Natural Convection Heat Transfer From Three-Dimensional Bodies of Arbitrary Shape
,” ASME J. Heat Mass Transfer-Trans. ASME
,
111
(2
), pp. 363
–371
.10.1115/1.325068611.
Hassani
,
A. V.
, and
Hollands
,
K. G. T.
, 1990
, “
Conduction Shape Factor for a Region of Uniform Thickness Surrounding a Three-Dimensional Body of Arbitrary Shape
,” ASME J. Heat Mass Transfer-Trans. ASME
,
112
(2
), pp. 492
–495
.10.1115/1.291040512.
Hassani
,
A. V.
,
Hollands
,
K. G. T.
, and
Raithby
,
G. D.
, 1993
, “
A Close Upper Bound for the Conduction Shape Factor of a Uniform Thickness, 2D Layer
,” Int. J. Heat. Mass. Transfer
,
36
(12
), pp. 3155
–3158
.10.1016/0017-9310(93)90044-713.
Nickolay
,
M.
,
Fischer
,
L.
, and
Martin
,
H.
, 1998
, “
Shape Factors for Conductive Heat Flow in Circular and Quadratic Cross-Sections
,” Int. J. Heat. Mass. Transfer
,
41
(11
), pp. 1437
–1444
.10.1016/S0017-9310(97)00351-714.
Kołodziej
,
J. A.
, and
Stręk
,
T.
, 2001
, “
Analytical Approximations of the Shape Factors for Conductive Heat Flow in Circular and Regular Polygonal Cross-Sections
,” Int. J. Heat. Mass. Transfer
,
44
(5
), pp. 999
–1012
.10.1016/S0017-9310(00)00162-915.
Rana
,
S. K.
, and
Jena
,
A.
, 2017
, “
A BEM Formulation of Two-Dimensional Steady State Heat Conduction in Exchanger Tubes of Arbitrary Cross Sections
,” Int. J. Heat. Mass. Transfer
,
106
, pp. 195
–211
.10.1016/j.ijheatmasstransfer.2016.10.05516.
Hirbodi
,
K.
, and
Jafarpur
,
K.
, 2020
, “
A Simple and Accurate Model for Conduction Shape Factor of Hollow Cylinders
,” Int. J. Therm. Sci.
,
153
, p. 106362
.10.1016/j.ijthermalsci.2020.10636217.
Chhokar
,
C.
,
Abadi
,
G. B.
, and
Bahrami
,
M.
, 2022
, “
A New Analytical Model for the Conduction Shape Factor of Annulus Sectors
,” Int. J. Heat. Mass. Transfer
,
196
, p. 123304
.10.1016/j.ijheatmasstransfer.2022.12330418.
Chen
,
J. T.
,
Kao
,
S. K.
, and
Lee
,
J. W.
, 2017
, “
Analytical Derivation and Numerical Experiment of Degenerate Scale by Using the Degenerate Kernel of the Bipolar Coordinates
,” Eng. Anal. Bound. Elem.
,
85
, pp. 70
–86
.10.1016/j.enganabound.2017.08.00619.
Chen
,
J. T.
, and
Shen
,
W. C.
, 2007
, “
Degenerate Scale for Multiply Connected Laplace Problems
,” Mech. Res. Commun.
,
34
(1
), pp. 69
–77
.10.1016/j.mechrescom.2006.06.00920.
Lee
,
J. W.
, and
Chen
,
J. T.
, 2012
, “
A Semi-Analytical Approach for a Nonconfocal Suspended Strip in an Elliptical Waveguide
,” IEEE Trans. Microw. Theory Tech.
,
60
(12
), pp. 3642
–3655
.10.1109/TMTT.2012.222113821.
Chen
,
J. T.
, and
Lee
,
J. W.
, 2013
, “
A Semi-Analytical Method for Near-Trapped Mode and Fictitious Frequencies of Multiple Scattering by an Array of Elliptical Cylinders in Water Waves
,” Phys. Fluids
,
25
(9
), p. 097103
.10.1063/1.481933222.
Lee
,
J. W.
,
Chen
,
J. T.
,
Leu
,
S. Y.
, and
Kao
,
S. K.
, 2014
, “
Null-Field BIEM for Solving a Scattering Problem From a Point Source to a Two-Layer Prolate Spheroid
,” Acta Mech.
,
225
(3
), pp. 873
–891
.10.1007/s00707-013-0988-x23.
Chen
,
J. T.
, and
Hong
,
H.-K.
, 1994
, “
Dual Boundary Integral Equations at a Corner Using Contour Approach Around Singularity
,” Adv. Eng. Softw.
,
21
(3
), pp. 169
–178
.10.1016/0965-9978(94)90019-124.
Lee
,
J. W.
,
Liu
,
L. W.
,
Hong
,
H.-K.
, and
Chen
,
J. T.
, 2016
, “
Applications of the Clifford Algebra Valued Boundary Element Method to Electromagnetic Scattering Problems
,” Eng. Anal. Bound. Elem.
,
71
, pp. 140
–150
.10.1016/j.enganabound.2016.07.00725.
Hong
,
H.-K.
,
Kao
,
Y. C.
,
Lee
,
J. W.
,
Liu
,
L. W.
, and
Chen
,
J. T.
, 2018
, “
Quaternion Boundary Element Method for Coupled Exterior and Interior Magnetostatic Fields
,” IEEE Trans. Magn.
,
54
(6
), pp. 1
–10
.10.1109/TMAG.2018.279141526.
Hwang
,
W. S.
, and
Huang
,
Y. Y.
, 1998
, “
Non-Singular Direct Formulation of Boundary Integral Equations for Potential Flows
,” Int. J. Numer. Methods Fluids
,
26
(6
), pp. 627
–635
.10.1002/(SICI)1097-0363(19980330)26:6<627::AID-FLD664>3.0.CO;2-W27.
Young
,
D. L.
,
Hwang
,
W. S.
,
Tsai
,
C. C.
, and
Lu
,
H. L.
, 2005
, “
Accuracy of Desingularized Boundary Integral Equations for Plane Exterior Potential Problems
,” Eng. Anal. Bound. Elem.
,
29
(3
), pp. 224
–231
.10.1016/j.enganabound.2004.12.00728.
Klaseboer
,
E.
,
Sun
,
Q.
, and
Chan
,
D. Y. C.
, 2012
, “
Non-Singular Boundary Integral Methods for Fluid Mechanics Applications
,” J. Fluid Mech.
,
696
, pp. 468
–478
.10.1017/jfm.2012.7129.
Moharana
,
M. K.
, and
Das
,
P. K.
, 2012
, “
Heat Conduction Through Eccentric Annuli: An Appraisal of Analytical, Semi-Analytical, and Approximate Techniques
,” ASME J. Heat Mass Transfer-Trans. ASME
,
134
(9
), p. 091301
.10.1115/1.400639130.
Leontiou
,
T.
, and
Fyrillas
,
M. M.
, 2014
, “
Shape Optimization With Isoperimetric Constraints for Isothermal Pipes Embedded in an Insulated Slab
,” ASME J. Heat Mass Transfer-Trans. ASME
,
136
(9
), p. 094502
.10.1115/1.402778031.
Sun
,
Q.
,
Klaseboer
,
E.
,
Khoo
,
B. C.
, and
Chan
,
D. Y. C.
, 2014
, “
A Robust and Non-Singular Formulation of the Boundary Integral Method for the Potential Problem
,” Eng. Anal. Bound. Elem.
,
43
, pp. 117
–123
.10.1016/j.enganabound.2014.03.01032.
Sun
,
Q.
,
Klaseboer
,
E.
,
Khoo
,
B. C.
, and
Chan
,
D. Y. C.
, 2015
, “
Boundary Regularized Integral Equation Formulation for the Helmholtz Equation in Acoustics
,” R. Soc. Open Sci.
,
2
(1
), pp. 140520
–140529
.10.1098/rsos.14052033.
Sun
,
Q.
,
Klaseboer
,
E.
,
Khoo
,
B. C.
, and
Chan
,
D. Y. C.
, 2015
, “
Boundary Regularized Integral Equation Formulation of Stokes Flow
,” Phys. Fluids
,
27
(2
), p. 023102
.10.1063/1.490727934.
Sun
,
F. L.
,
Zhang
,
Y. M.
,
Young
,
D. L.
, and
Chen
,
W.
, 2016
, “
A New Boundary Meshfree Method for Potential Problems
,” Adv. Eng. Softw.
,
100
, pp. 32
–42
.10.1016/j.advengsoft.2016.06.00935.
Zhang
,
Y. M.
,
Sun
,
F. L.
,
Young
,
D. L.
,
Chen
,
W.
, and
Gu
,
Y.
, 2016
, “
Average Source Boundary Node Method for Potential Problems
,” Eng. Anal. Bound. Elem.
,
70
, pp. 114
–125
.10.1016/j.enganabound.2016.06.00736.
Zhang
,
Y. M.
,
Sun
,
F. L.
,
Qu
,
W. Z.
,
Gu
,
Y.
, and
Young
,
D. L.
, 2018
, “
A Meshless Average Source Boundary Node Method for Steady-State Heat Conduction in General Anisotropic Media
,” Comput. Math. Appl.
,
75
(5
), pp. 1739
–1755
.10.1016/j.camwa.2017.11.03637.
Zhang
,
C.
,
Fu
,
Z.
, and
Zhang
,
Y.
, 2021
, “
A New Global and Direct Integral Formulation for 2D Potential Problems
,” Eng. Anal. Bound. Elem.
,
125
, pp. 233
–240
.10.1016/j.enganabound.2021.01.02038.
Sangani
,
A. S.
, 2019
, “
Heat Transfer From Particles Confined Between Two Parallel Walls
,” ASME J. Heat Mass Transfer-Trans. ASME
,
141
(2
), p. 022001
.10.1115/1.404180239.
Lienhard
,
J. H. V.
, 2019
, “
Exterior Shape Factors From Interior Shape Factors
,” ASME J. Heat Mass Transfer-Trans. ASME
,
141
(6
), p. 061301
.10.1115/1.404291240.
Chen
,
J. T.
,
Kao
,
S. K.
,
Kao
,
J. H.
, and
Tai
,
W. C.
, 2022
, “
An Indirect BIE Free of Degenerate Scales
,” Commun. Pure Appl. Anal.
,
21
(6
), p. 1969
.10.3934/cpaa.202111441.
Kuo
,
S. R.
,
Chen
,
J. T.
,
Lee
,
J. W.
, and
Chen
,
Y. W.
, 2013
, “
Analytical Derivation and Numerical Experiments of Degenerate Scales for Regular n-Gon Domains in Two-Dimensional Laplace Problems
,” Appl. Math. Comput.
,
219
(10
), pp. 5668
–5683
.10.1016/j.amc.2012.11.00842.
Chen
,
Y. Z.
, 2022
, “
Numerical Solution for the Degenerate Scale for Two Unequal Circles in Thermal Conduction Problem
,” J. Therm. Stresses
,
45
(5
), pp. 375
–381
.10.1080/01495739.2022.203748143.
Corfdir
,
A.
, and
Bonnet
,
G.
, 2018
, “
Degenerate Scale for 2D Laplace Equation With Mixed Boundary Condition and Comparison With Other Conditions on the Boundary
,” Eng. Anal. Bound. Elem.
,
88
, pp. 14
–25
.10.1016/j.enganabound.2017.12.004Copyright © 2023 by ASME
You do not currently have access to this content.
