## Abstract

In this study, the friction factor of a fully developed laminar flow in a noncircular duct, which is used in an automotive cooler system, is investigated. The Poisson equation of velocity with homogeneous boundary conditions is solved by using the method of separation of variables and analytical solutions of velocity, wall shear stress, and friction factor for arbitrary symmetric noncircular ducts are obtained. The analytical solutions are validated by comparing against analytical and experimental data available for rectangular and triangular ducts. Cross-sectional distribution of velocity, wall shear stress along the duct boundary, and friction factor are analyzed in depth. It is obtained that the friction factor for this specific duct is $λ=95Re$, with Re defined with the equivalent diameter as length scale. Moreover, the applicability of a general equation of friction factor proposed by Muzychka and Yovanovich is validated for this specific duct by comparing with the analytical solution. The result shows that the general equation is accurate enough and can be used for the evaluation of flow resistance in the design of the cooler system.

## References

1.
Muzychka
,
Y.
, and
Yovanovich
,
M.
,
2009
, “
Pressure Drop in Laminar Developing Flow in Noncircular Ducts: A Scaling and Modeling Approach
,”
ASME J. Fluids Eng.
,
131
(
11
), p.
111105
.10.1115/1.4000377
2.
Sarmiento
,
A.
,
Milanez
,
F.
, and
Mantelli
,
M.
,
2021
, “
Theoretical Models for Compact Printed Circuit Heat Exchangers With Straight Semicircular Channels
,”
Appl. Therm. Eng.
,
184
, p.
115435
.10.1016/j.applthermaleng.2020.115435
3.
He
,
S.
, and
Gotts
,
J.
,
2004
, “
Calculation of Friction Coefficients for Noncircular Channels
,”
ASME J. Fluids Eng.
,
126
(
6
), pp.
1033
1038
.10.1115/1.1845479
4.
Cornish
,
R.
,
1928
, “
Flow in a Pipe of Rectangular Cross-Section
,”
Proc. R. Soc. London, Ser. A
,
120
(
786
), pp.
691
700
.
5.
Yilmaz
,
T.
,
1990
, “
General Equations for Pressure Drop for Laminar Flow in Ducts of Arbitrary Cross Sections
,”
ASME J. Energy Resour. Technol.
,
112
(
4
), pp.
220
223
.10.1115/1.2905761
6.
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
7.
Obot
,
N. T.
,
1988
, “
Determination of Incompressible Flow Friction in Smooth Circular and Noncircular Passages: A Generalized Approach Including Validation of the Nearly Century Old Hydraulic Diameter Concept
,”
ASME J. Fluids Eng.
,
110
(
4
), pp.
431
440
.10.1115/1.3243574
8.
Duan
,
Z.
,
Yovanovich
,
M. M.
, and
Muzychka
,
Y. S.
,
2012
, “
Pressure Drop for Fully Developed Turbulent Flow in Circular and Noncircular Ducts
,”
ASME J. Fluids Eng.
,
134
(
6
), pp.
287
304
.10.1115/1.4006861
9.
Hodge
,
R. I.
,
1961
, “
Frictional Pressure Drop in Noncircular Ducts
,”
ASME J. Heat Transfer
,
83
(
3
), pp.
384
385
.10.1115/1.3682295
10.
,
P. C.
, and
Ambrose
,
C. W.
,
1980
, “
A Generalised Length Dimension for Non-Circular Ducts
,”
Lett. Heat Mass Transfer
,
7
(
5
), pp.
323
328
.10.1016/0094-4548(80)90044-2
11.
Kumar
,
R.
, and
Varun
,
A. K.
,
2016
, “
Thermal and Fluid Dynamic Characteristics of Flow Through Triangular Cross-Sectional Duct: A Review
,”
Renewable Sustainable Energy Rev.
,
61
, pp.
123
140
.10.1016/j.rser.2016.03.011
12.
Sarmiento
,
A.
,
Soares
,
V.
,
Milanez
,
F.
, and
Mantelli
,
M.
,
2020
, “
Heat Transfer Correlation for Circular and Non-Circular Ducts in the Transition Regime
,”
Int. J. Heat Mass Transfer
,
149
, p.
119165
.10.1016/j.ijheatmasstransfer.2019.119165
13.
Rehman
,
D.
,
Barattini
,
D.
,
Hong
,
C.
, and
Morini
,
G. L.
,
2021
, “
Effect of Aspect Ratio and Inlet Manifold Shape on the Laminar-to-Turbulent Transition of Gas Flow in Rectangular Microchannels
,”
Exp. Fluids
,
62
(
3
), pp.
1
19
.10.1007/s00348-021-03137-3
14.
Hildebrand
,
F. B.
,
2013
,
Advanced Calculus for Engineers
,
Martino Publishing
, Mansfield Center, CT.
15.
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
16.
Kolodziej
,
J.
,
1987
, “
Review of Application of Boundary Collocation Methods in Mechaniccs of Continuous Media
,”
SM Arch.
,
12
(
4
), pp.
187
231
.
17.
Gerhart
,
P. M.
,
Gerhart
,
A. L.
, and
Hochstein
,
J. I.
,
2016
,
Munson, Young and Okiishi's Fundamentals of Fluid Mechanics
,
Wiley
, Hoboken, NJ.
18.
Shah
,
R. K.
, and
London
,
A. L.
,
1978
,
Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data
,
, New York.
19.
Carlson
,
L. W.
, and
Irvine
,
T. F.
,
1961
, “
Fully Developed Pressure Drop in Triangular Shaped Ducts
,”
ASME J. Heat Transfer
,
83
(
4
), pp.
441
444
.10.1115/1.3683663
You do not currently have access to this content.