Abstract

The asymptotic limit for perimeter averaged convection is generalized for short ducts of arbitrary cross section. A correction factor to Lévêque's original analysis is derived in terms of the state of wall shear stress under conditions of fully developed flows for walls of constant temperature (T) and constant heat flux (H1 and H2). This analysis is performed for four duct geometries: elliptic, rhombic, rectangular, and regular polygons. The magnitude of this correction is greatest for the H2 wall condition and for ducts having walls with acute corners. The results of this analysis can be incorporated into a generalized correlation for the full Graetz problem in ducts of arbitrary cross section.

References

1.
Lévêque
,
A.
,
1928
, “
Les Lois de la Transmission de Chaleur Par Convection
,”
Ann. Mines, Mem., Ser. 12
,
13
, pp. 201;
305
381
.
2.
Newman
,
J.
,
1969
, “
Extension of the Lévêque Solution
,”
ASME J. Heat Transfer
,
91
(
1
), pp.
177
178
.10.1115/1.3580091
3.
Shih
,
Y. P.
, and
Tsou
,
J. D.
,
1978
, “
Extended Lévêque Solutions for Heat Transfer to Power Law Fluids in Laminar Flow in a Pipe
,”
Chem. Eng. J.
,
15
(
1
), pp.
55
62
.10.1016/0300-9467(78)80036-7
4.
Richardson
,
S. M.
,
1979
, “
Extended Lévêque Solutions for Flows of Power Law Fluids in Pipes and Channel
,”
Int. J. Heat Mass Transfer
,
22
(
10
), pp.
1417
1423
.10.1016/0017-9310(79)90204-7
5.
Gottifredi
,
J. C.
, and
Flores
,
A. F.
,
1985
, “
Extended Lévêque Solution for Heat Transfer to Non-Newtonian Fluids in Pipes and Flat Ducts
,”
Int. J. Heat Mass Transfer
,
28
(
5
), pp.
903
908
.10.1016/0017-9310(85)90271-6
6.
Chen
,
J. D.
, and
Ju
,
Y. H.
,
1988
, “
Extended Lévêque Solution for Heat Transfer to Power‐Law Fluids in Pipes
,”
J. Chin. Inst. Eng.
,
11
(
3
), pp.
305
307
.10.1080/02533839.1988.9677072
7.
Shih
,
Y. P.
,
Huang
,
C. C.
, and
Tsay
,
S. Y.
,
1995
, “
Extended Lévêque Solution for Laminar Heat Transfer to Power-Law Fluids in Pipes With Wall Slip
,”
Int. J. Heat Mass Transfer
,
38
(
3
), pp.
403
408
.10.1016/0017-9310(94)00209-E
8.
Bennett
,
T. D.
,
2019
, “
Correlations for the Graetz Problem in Convection—Part 2: For Ducts of Arbitrary Cross-Section
,”
Int. J. Heat Mass Transfer
,
135
, pp.
1327
1334
.10.1016/j.ijheatmasstransfer.2019.02.052
9.
Lahjomri
,
J.
, and
Oubarra
,
A.
,
1999
, “
Analytical Solution of the Graetz Problem
,”
ASME J. Heat Transfer
,
121
(
4
), pp.
1078
1083
.10.1115/1.2826060
10.
Jagadeesh
,
M.
, and
Mandapati
,
K.
,
2016
, “
Effect of Axial Conduction and Viscous Dissipation on Heat Transfer for Laminar Flow Through a Circular Pipe
,”
Perspect. Sci.
,
8
, pp.
61
65
.
11.
Gulhane
,
N. P.
, and
Mahulikar
,
S. P.
,
2012
, “
Numerical Investigation on Laminar Microconvective Liquid Flow With Entrance Effect and Graetz Problem Due to Variation in Thermal Properties
,”
Heat Transfer Eng.
,
33
(
8
), pp.
748
761
.10.1080/01457632.2012.624877
12.
Nóbrega
,
J. M.
,
Pinho
,
F. T.
,
Oliveira
,
P. J.
, and
Carneiro
,
O. S.
,
2004
, “
Accounting for Temperature-Dependent Properties in Viscoelastic Duct Flows
,”
Int. J. Heat Mass Transfer
,
47
(
6–7
), pp.
1141
1158
.10.1016/j.ijheatmasstransfer.2003.10.004
13.
Barletta
,
A.
, and
Magyari
,
E.
,
2007
, “
Forced Convection With Viscous Dissipation in the Thermal Entrance Region of a Circular Duct With Prescribed Wall Heat Flux
,”
Int. J. Heat Mass Transfer
,
50
(
1–2
), pp.
26
35
.10.1016/j.ijheatmasstransfer.2006.06.036
14.
Kumar
,
M. M. J.
, and
Satyamurty
,
V. V.
,
2015
, “
Effect of Entry Temperature on Forced Convection Heat Transfer With Viscous Dissipation in Thermally Developing Region of Concentric Annuli
,”
ASME J. Heat Transfer
,
137
(
12
), p.
121001
.10.1115/1.4030908
15.
Suzzi
,
N.
, and
Lorenzini
,
M.
,
2019
, “
Viscous Heating of a Laminar Flow in the Thermal Entrance Region of a Rectangular Channel With Rounded Corners and Uniform Wall Temperature
,”
Int. J. Therm. Sci.
,
145
, p.
106032
.10.1016/j.ijthermalsci.2019.106032
16.
Coelho
,
P. M.
,
Pinho
,
F.
, and
Oliveira
,
P. J.
,
2003
, “
Thermal Entry Flow for a Viscoelastic Fluid: The Graetz Problem for the PTT Model
,”
Int. J. Heat Mass Transfer
,
46
(
20
), pp.
3865
3880
.10.1016/S0017-9310(03)00179-0
17.
Yilmaz
,
T.
, and
Cihan
,
E.
,
1993
, “
General Equation for Heat Transfer for Laminar Flow in Ducts of Arbitrary Cross-Sections
,”
Int. J. Heat Mass Transfer
,
36
(
13
), pp.
3265
3270
.10.1016/0017-9310(93)90009-U
18.
Yilmaz
,
T.
, and
Cihan
,
E.
,
1995
, “
An Equation for Laminar Flow Heat Transfer for Constant Heat Flux Boundary Condition in Ducts of Arbitrary Cross-Sectional Area
,”
ASME J. Heat Transfer
,
117
(
3
), pp.
765
766
.10.1115/1.2822644
19.
Bennett
,
T. D.
,
2020
, “
Refinement of the Generalized Graetz Problem Correlation With New Benchmark Calculations
,”
ASME J. Heat Transfer
,
142
(
5
), p.
051802
.10.1115/1.4046346
20.
Shah
,
R. K.
, and
London
,
A. L.
,
1978
,
Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data
,
Academic Press
,
New York
.
21.
Bennett
,
T. D.
,
2019
, “
A Historical Misperception on Calculating the Average Convection Coefficient in Tubes With Constant Wall Heat Flux
,”
ASME J. Heat Transfer
,
141
, p. 0
61702
.10.1115/1.4043303
22.
Muzychka
,
Y. S.
, and
Yovanovich
,
M. M.
,
2004
, “
Laminar Forced Convection Heat Transfer in the Combined Entry Region of Non-Circular Ducts
,”
ASME J. Heat Transfer
,
126
(
1
), pp.
54
61
.10.1115/1.1643752
23.
Churchill
,
S. W.
, and
Ozoe
,
H.
,
1973
, “
Correlations for Laminar Forced Convection in Flow Over an Isothermal Flat Plate and in Developing and Fully Developed Flow in an Iso-Thermal Tube
,”
ASME J. Heat Transfer
,
95
(
3
), pp.
416
419
.10.1115/1.3450078
24.
Churchill
,
S. W.
, and
Ozoe
,
H.
,
1973
, “
Correlations for Laminar Forced Convection With Uniform Heating in Flow Over a Plate and in Developing and Fully Developed Flow in a Tube
,”
ASME J. Heat Transfer
,
95
(
1
), pp.
78
84
.10.1115/1.3450009
25.
Shah
,
R. K.
,
1975
, “
Laminar Flow Friction and Forced Convection Heat Transfer in Ducts of Arbitrary Geometry
,”
Int. J. Heat Mass Transfer
,
18
(
7–8
), pp.
849
862
.10.1016/0017-9310(75)90176-3
26.
Tamayol
,
A.
, and
Bahrami
,
M.
,
2010
, “
Laminar Flow in Microchannels With Noncircular Cross Section
,”
ASME J. Fluids Eng.
,
132
(
11
), p.
111201
.10.1115/1.4001973
27.
Ramanujan
,
S.
,
1914
, “
Modular Equations and Approximations to pi
,”
Q. J. Pure Appl. Math.
,
45
, pp.
350
372
.
28.
James
,
P. A.
,
1970
, “
Forced Convection Heat Transfer in Narrow Passages
,”
Can. J. Chem. Eng.
,
48
(
3
), pp.
330
332
.10.1002/cjce.5450480320
29.
Drew
,
T. B.
,
1931
, “
Mathematical Attacks on Forced Convection Problems: A Review
,”
Trans. Am. Inst. Chem. Eng.
,
27
, p.
26
.
30.
Bennett
,
T. D.
,
2012
,
Transport by Advection and Diffusion
,
Wiley
,
New York
.
31.
Bennett
,
T. D.
,
2020
, “
Laminar Convection in Rectangular Ducts of Fully Developed Flow
,”
Int. J. Heat Mass Transfer
,
156
, p.
119846
.10.1016/j.ijheatmasstransfer.2020.119846
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