Fully-developed heat transfer to power-law fluids in laminar flow

4 Fully-developed heat transfer to power-law fluids in laminar flow 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z 0), the fluid temperature is uniform at Tf for z 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

At steady state, the temperature of the fluid, T, apparent viscosity will be a function of both r and z, and the rate of transfer of heat into the control volume is  

Similarly, the rate of transfer of heat out of the control voliune is given by  

Under steady state conditions, expressions 6.4 and 6.5 must be equal and slight re-arrangement yields  

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