# Heat transfer to power-law fluids in thermal entrance region with viscous dissipation for constant heat flux conditions

- Received:

DOI: 10.1007/BF03051191

- Cite this article as:
- Sundaram, K.M. & Nath, G. Proc. Indian Acad. Sci. (1976) 83: 50. doi:10.1007/BF03051191

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## Abstract

An analytical solution of the heat transfer problem with viscous dissipation for non-Newtonian fluids with power-law model in the thermal entrance region of a circular pipe and two parallel plates under constant heat flux conditions is obtained using eigenvalue approach by suitably replacing one of the boundary conditions by total energy balance equation. Analytical expressions for the wall and the bulk temperatures and the local Nusselt number are presented. The results are in close agreement with those obtained by implicit finite-difference scheme. It is found that the role of viscous dissipation on heat transfer is completely different for heating and cooling conditions at the wall. The results for the case of cooling at the wall are of interest in the design of the oil pipe line.

### Nomenclature

*A,B*constants of integration

*b*half of the distance between parallel plates

*Br*Brinkman number defined by equations (8) and (35)

*C*_{0}constant

*C*_{p}specific heat at constant pressure

*D*_{m},*D*_{p}constants defined by equations (23) and (45) respectively

*F*function in equation (10)

*G*_{m},*G*_{p}constants defined by equations (25) and (47) respectively

*h*heat transfer coefficient

*k*thermal conductivity

*K*_{m},*K*_{p}constants defined by equations (26) and (48) respectively

*L*length of the plate

*m, n*parameters in power-law model

- Nu, Nu
_{p} local Nusselt numbers defined by equations (29) and (52) respectively

*P*Pressure

*Pr*Prandtl number

*q*_{w}heat flux

*r*radial distance

*R*radius of the pipe

*Re*Reynolds number

*S*pressure gradient in

*x*-direction —*dP/dx*= constant*T*temperature

*T*_{c}constant defined by equation (8)

*v*velocity

*V*_{max}maximum velocity

*x*axial distance

*y*distance perpendicular to the plate

*Y*_{m},*Y*_{p}eigenfunctions

- β
_{m}, β_{p} eigenvalues

- η
dimensionless radial distance

- θ
dimensionless temperature

- \(\bar \theta \)
dimensionless temperature defined by equation (16)

- θ
_{b} dimensionless bulk temperature

- θ
_{w} dimensionless wall temperature

- λ
constant

- μ
viscosity

- ν
kinematic viscosity

- ρ
density

- τ
_{rx}(τ_{xy}) shear stress defined by equation (1)

- ψ
dimensionless axial distance defined by equations (8) and (35).

### Subscripts

*a*asymptotic condition

*c*critical value

*f*fully developed

*o*inlet condition

*w*wall condition

*x*axial component