Entropy production in a flow through an annular packed bed

Example 4.5 Entropy production in a flow through an annular packed bed The introduction of suitable packing into a fluid flow passage considerably enhances wall-to-fluid heat transfer, and hence reduces the entropy production due to heat transfer but increases the entropy production due to fluid-flow friction. Heat transfer to a fluid flowing in an annulus has a technical importance because we can heat or cool either or both of the surfaces independently. Entropy production provides a new criterion in analyzing such processes. In terms of the velocity and temperature profiles, the local rate of entropy production per unit volume of an incompressible Newtonian fluid for a two-dimensional annular flow is 

k and /jl are the thermal conductivity and dynamic viscosity of the fluid, respectively. The terms v and T denote the velocity and temperature of the fluid. The first term on the right side of Eq. (4.38) shows the entropy production due to finite temperature differences in axial z and radial r directions, while the second term shows the entropy production due to fluid friction. We may construct the entropy production profiles using Eq. (4.38) if we know the temperature and the velocity fields. 

Assuming fully developed velocity and temperature profiles for the control volume of an annular packed bed, the energy equation is 

ae is the effective thermal diffusivity of the bed and Th the bulk fluid temperature. We assume that the plug flow conditions (v = vav) and essentially radially flat superficial velocity profiles prevail through the cross-section of the packed flow passage, and the axial thermal conduction is negligible. The uniform heat fluxes at each of the two surfaces provide the necessary boundary conditions with positive heat fluxes when the heat flows into the fluid 

Equation (4.39) can be directly integrated because the term dThldz is constant. The linearity of the energy equation allows the use of the superposition method to build solutions for asymmetric hearing by adding the two fundamental solutions (1) the outer wall heated with the inner wall insulated and (2) the inner wall heated with the outer wall insulated 

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