Approximate analytical methodology for calculating friction factors in flow through polygonal cross section ducts

  • M. C. Reis
  • L. A. Sphaier
  • L. S. de B. Alves
  • R. M. Cotta
Technical Paper


An approximate analytical methodology for calculating the friction factor within ducts of irregular cross-section is herein proposed. The approximations are developed by transforming the original governing PDEs into simpler ODEs, using approximation rules provided by the coupled integral equations approach. The transformed system is directly integrated and analytical solutions for the friction factor are readily obtained. Four different approximation cases are analyzed, which yield simple closed-form expressions for calculating the friction factor in terms of geometric parameters for rectangular, triangular, and trapezoidal ducts. The results of these expressions are compared with literature data, and very reasonable agreement is seen. After performing an error analysis of the results, regions for applicability of the methodology where accuracy requirements can be maintained are highlighted. Finally, enhanced approximation formulas yielding maximum errors as low as 3% are developed by using simple weighted averages of different approximation cases.


Lumped-capacitance formulation Arbitrary geometry Friction factor Mathematical modeling 

List of symbols

a, b, c

Geometric parameters in cross-section domain


Hydraulic diameter


Fanning’s friction-factor


Dimensionless pressure gradient

\(H_{\alpha ,\beta }\)

Hermite approximation


Aspect ratio




Axial velocity


Cross-sectional averaged velocity


Dimensionless axial velocity


Axial variable


Dimensionless function for left boundary

y, z

Cross-section variables

Y, Z

Dimensionless problem variables


Function for left boundary


Reynolds number

Greek symbols

\(\alpha\), \(\beta\), \(\nu\)

Hermite approximation parameters

\(\xi\), \(\eta\)

Dimensionless left boundary function parameters


Dynamic viscosity

\(\zeta _i\)

Solution constants



The authors would like to acknowledge the financial support provided by the Brazilizan Government funding agencies CAPES, CNPq, and FAPERJ.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering – TEM/PGMECUniversidade Federal Fluminense – UFFNiteróiBrazil
  2. 2.Laboratório de Transmissáo e Tecnologia do Calor – LTTC, Departamento de Engenharia Mecânica – POLI/COPPEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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