2. A very thick slurry flows in a circular tube of radius R. Since it flows nearly as a solid plug, the velocity profile is approximately flat over the pipe cross- section and v_{x} = v (constant) may be assumed. The governing equation for the forced convection heat transfer under plug flow conditions is given by rho dot C p V partial T partial z = k/r dot partial partial r (r partial T partial r ) For the constant heat flux conditions, the temperature gradient in the axial direction is constant and expressed in the form partial T partial z = pi Dq w rho V (pi * R ^ 2) hat C p

Combine the above two equations and using the following dimensionless variables, theta = (T - T_{b})/(T_{w} - T_{b}) xi = r/mu and determine the Nusselt number for a thermally developed plug flow in a circular tube. Note that theta = theta(E) alone, q_{w} is a constant wall heat flux. T_{b} is a bulk fluid temperature, and T_{w} is a wall temperature.