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Coarse Particles in Homogeneous Non-Newtonian Slurries: Combined Effects of Shear-Thinning Viscosity and Fluid Yield Stress on Drag and Heat Transfer from Hemispherical Particles

  • Om Prakash
  • S. A. Patel
  • A. K. Gupta
  • R. P. ChhabraEmail author
Technical Paper

Abstract

In this work, the momentum and energy equations have been solved numerically for predicting the hydrodynamic drag and heat transfer coefficient for a hemispherical particle submerged in a flow stream of yield-pseudoplastic fluids in order to elucidate the combined effects of shear-thinning viscosity and fluid yield stress. In this case, the momentum transfer aspects are influenced by the values of the Reynolds number (0.1 ≤ Re ≤ 100), Bingham number(0 ≤ Bn ≤ 100), shear-thinning index (0.2 ≤ n ≤ 1) and the orientation of the hemisphere. Similarly, the corresponding heat transfer results show additional dependence on the Prandtl number (0.7 ≤ Pr ≤ 100) and the type of thermal (isothermal or isoflux) boundary condition specified on the surface of the heated hemisphere. The numerical results are discussed in terms of the size and shape of the fluid-like yielded regions, wake lengths, hydrodynamic drag and heat transfer coefficients as functions of the preceding dimensionless parameters. Finally, the present values of the drag coefficient and Nusselt number have been fitted using simple expressions thereby enabling the interpolation of the present results for the intermediate values of the parameters and/or their prediction in a new application.

Keywords

Hemisphere Bingham number Nusselt number Herschel–Bulkley fluid Reynolds number 

List of symbols

A

Surface area of hemisphere (m2)

Bn

Bingham number, Eq. (9), dimensionless

CD

Total drag coefficient, dimensionless

CDF

Viscous (or friction) drag coefficient, dimensionless

CDP

Pressure (or form) drag coefficient, dimensionless

Cp

Specific heat of fluid (J kg−1 K−1)

d

Diameter of hemisphere (m)

\(D_{\infty }\)

Diameter of computational domain (m)

jH

Colburn-j factor, dimensionless

k

Thermal conductivity of fluid (W m−1 K−1)

K

Fluid consistency index (Pa sn)

Lr

Recirculation length measured from the center of the hemisphere, dimensionless

m

Growth rate parameter used in Papanastasiou model, Eq. (5), dimensionless

n

Shear-thinning (or fluid behaviour) index, dimensionless

Pr

Prandtl number, Eq. (10), dimensionless

\(Pr^{*}\)

Modified Prandtl number (=Pr(1 + Bn)), dimensionless

q0

Constant wall heat flux prescribed on the hemisphere (W m−2)

Re

Reynolds number, Eq. (8) dimensionless

\(Re^{*}\)

Modified Reynolds number (=Re/(1 + Bn)), dimensionless

T

Fluid temperature (K)

T0

Fluid temperature in the free stream (K)

Tw

Constant wall temperature prescribed on the hemisphere (K)

V

Velocity vector, dimensionless

V0

Free stream velocity (m s−1)

x,y

Cartesian coordinates (m)

Greek symbols

\(\dot{\gamma }\)

Rate of deformation tensor, dimensionless

\(\dot{\gamma }_{e}^{{\prime }}\)

Parameter used in Bercovier and Engelman model, Eq. (7), dimensionless

\(\mu_{{\text{HB}}}\)

Plastic viscosity of Herschel–Bulkley fluid model (=K(V 0/d) n−1) (Pa s)

\(\mu_{y}\)

Yielding viscosity used in bi-viscosity model, Eq. (6) (Pa s)

\(\rho\)

Density of fluid (kg m−3)

\(\varvec{\tau}\)

Deviatoric stress tensor, dimensionless

\(\tau_{0}\)

Fluid yield stress (Pa)

\(\phi\)

Fluid temperature (=(T − T 0)/(T w  − T 0) for constant wall temperature); (=T − T 0/(q 0 d/k) for constant wall heat flux), dimensionless

Notes

Acknowledgements

RPC gratefully acknowledges the financial support of the Department of Science and Technology (Government of India) via the award of a J C Bose fellowship to him for the period 2015–2020. Funding was provided by Department of Science and Technology, Ministry of Science and Technology (SB/S2/JCB-06/2014).

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Copyright information

© The Indian Institute of Metals - IIM 2016

Authors and Affiliations

  • Om Prakash
    • 1
  • S. A. Patel
    • 1
    • 2
  • A. K. Gupta
    • 1
  • R. P. Chhabra
    • 1
    Email author
  1. 1.Department of Chemical EngineeringIndian Institute of Technology KanpurKanpurIndia
  2. 2.Department of Chemical, Biological, and Pharmaceutical EngineeringNew Jersey Institute of TechnologyNewarkUSA

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