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Transport Phenomena - Heat Transfer Problem :
Forced convection heat transfer for plug flow in plane slit
- constant wall heat flux in thermally fully developed flow region


Problem.

A very thick paste flows in a plane narrow slit formed by two parallel walls a distance 2B apart. The length L and width W of the slit are such that B << W << L, so end effects can be neglected. Since the paste flows nearly as a solid plug, the velocity profile is approximately flat over the slit cross-section and vz = V (constant) may be assumed. For z < 0, the fluid temperature is uniform at the inlet temperature T1. For z > 0, heat is added at a uniform constant flux q0 through both the slit walls. The heat conduction in the z-direction and the viscous dissipation effect may be neglected. The thermal conductivity k and thermal diffusivity a may be assumed constant.

figure : plug flow in plane narrow slit being heated by a uniform heat flux
Figure. Plug flow in plane narrow slit being heated by a uniform heat flux.

a) Show that the temperature profile T(x, z) far downstream in the thermally fully developed region (i.e., for large z) is given by the following dimensionless asymptotic solution:

equation : Theta(sigma, zeta) = zeta + 1/2 sigma^2 - 1/6

where

equation : Theta = (T-T_1)/(q_0 B/k), sigma = x/B, and zeta = alpha z / (V B^2)

b) Show that the limiting local Nusselt number far downstream for plug flow in a plane slit with constant wall heat flux is Nu = 12.


Solution.

Click here for stepwise solution


Related Problems in Transport Phenomena - Heat Transfer :

Transport Phenomena - Heat Transfer Problem : Forced convection heat transfer for plug flow in circular tube
- Similar problem for circular tube in cylindrical coordinates rather than narrow slit in Cartesian coordinates

Transport Phenomena - Heat Transfer Problem : Forced convection heat transfer for laminar Newtonian flow in plane slit
- Similar problem for parabolic velocity profile rather than flat velocity profile


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