Thermal Convection in Pseudocontinuum One-Phase Flow

General considerations of the heat transfer in a gas-solid flow require separate energy equations for the two phases that are coupled through a local heat transfer coefficient as well as local phase velocities. These approaches are complex and are introduced in Chapter 5. 

One of the simplified heat transfer models of two-phase flows is the pseudocontinuum one-phase flow model, in which it is assumed that (1) local thermal equilibrium between the two phases exists (2) particles are evenly distributed (3) flow is uniform and (4) heat conduction is dominant in the cross-stream direction. Therefore, the heat balance leads to a single-phase energy equation which is based on effective gas-solid properties and averaged temperatures and velocities. For an axisymmetric flow heated by a cylindrical heating surface at rw, the heat balance equation can be written as 

Using the method of separation of variables, the temperature distribution from Eqs. (4.43) and (4.44) is obtained as 

The applicability of the preceding pseudocontinuum approach to convective heat transfer of gas-solid systems without heat sources depends not only on the validity of the phase continuum approximation but also on the appropriateness of the local thermal equilibrium assumption. The local thermal equilibrium may be assumed only if the particle-heating 

A regime map of Fo versus the solid volume fraction, ap, for various gas-solid flows was presented by Hunt (1989), as shown in Fig. 4.3. Hunt (1989) suggested that except when Fo 1 and ap 0.1, use of the pseudocontinuum model is inappropriate. Thus, from Fig. 4.3, it can be seen that the pseudocontinuum model is applicable to packed beds, incipient fluidized beds, and granular flows, whereas it is not applicable to pneumatic transport flows, dilute suspensions, bubbling beds, and slugging fluidized beds [Glicksman and Decker, 1982 Hunt, 1989]. 

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