Heat and Mass Transfer

, Volume 48, Issue 9, pp 1647–1662 | Cite as

Entropy generation in non-Newtonian fluids due to heat and mass transfer in the entrance region of ducts

  • N. Galanis
  • M. M. Rashidi


A new solution for the Graetz problem (hydrodynamically developed forced convection in isothermal ducts) extended to power-law fluids and mass transfer with phase change at the walls is presented. The temperature and concentration spatial distributions in the corresponding entrance regions are obtained for two geometries (parallel-plates duct and circular pipe) in terms of appropriate dimensionless parameters. They are used to illustrate the effects of the fluid nature on the velocity, temperature and concentration distributions, on the axial evolution of the sensible and latent Nusselt numbers as well as on the local entropy generation rate due to velocity, temperature and concentration gradients.


Nusselt Number Entropy Generation Sherwood Number Slug Flow Circular Pipe 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols


Duct area


Brinkman number = (ζ u b n+1 )/(k (T in  − T w ) D h n−1 )


Dimensionless species concentration = (ω − ω w )/(ω in   ω w )


Specific heat of the fluid (J kg−1 K−1)


Mass diffusion coefficient (m2 s−1)


Hydraulic diameter (m)


Half distance between parallel-plates (m)


Specific enthalpy of vaporization (J kg−1)


Jakob number = (c p (T in  − T w )/h fg )


Thermal conductivity (W m−1 K−1)


Metric parameter (m = 0 for a flat duct, m = 1 for a circular pipe)


Molecular weight for species i (kg mol−1)


Flow behavior index (dimensionless)


Latent Nusselt number


Sensible Nusselt number


Pressure (Pa)


Dimensionless pressure = (p/(ρ u b 2 ))


Péclet number for mass diffusion = (u b D h )/D


Péclet number for thermal diffusion = (ρ C p D h u b )/k


Pipe radius

\( \bar{R} \)

Universal gas constant (J mol−1 K−1)


Hydraulic radius (m)


Sherwood number


Dimensionless entropy generation = (\( s_{g}^{*} D_{h}^{2} /k \))


Dimensionless entropy generation due to temperature, velocity, concentration

\( s_{g}^{*} \)

Entropy generation rate (W m−3 K−1)


Axial velocity (m s−1)


Dimensionless axial velocity = (u/u b )

x, y

Axial and transverse coordinates (m)


 = 4αX/((α + m + 1)Pe T )


 = 4αX/((α + m + 1)Pe M )

X, Y

Dimensionless axial and transverse coordinates (x/D h , y/R h )

Greek symbols


 = (n + 1)/n


Coefficient of consistence (Pa sn)


Dimensionless temperature = (T − T w )/(T in  − T w )


Dynamic viscosity (Pa s)


Density (kg m−3)


Viscous dissipation (W m−3)


Mass fraction (kg of diffusing species/kg of mixture)



Bulk or average value







This project is part of the R&D program of the NSERC Chair in Industrial Energy Efficiency established in 2006 at “Université de Sherbrooke”. The authors acknowledge the support of the Natural Sciences & Engineering Research Council of Canada, Hydro Québec, Rio Tinto Alcan and Canmet-Energy Research Center. The second author acknowledges gratefully 1 year as invited professor at Université de Sherbrooke (Canada), during 2010–2011.


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Génie mécaniqueUniversité de SherbrookeSherbrookeCanada
  2. 2.Mechanical Engineering DepartmentEngineering Faculty of Bu-Ali Sina UniversityHamedanIran

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