Temperature Distributions of Phases

For a fully developed gas-solid suspension pipe flow where the overall heat transfer rate between the suspension and the pipe wall is dominated by both thermal convection and 

The analytical solution of this problem is difficult because of the highly nonlinear radiation term in Eq. (11.97). However, under some conditions, the term can be simplified. For instance, when the temperature difference between particles and pipe wall is small (i.e., 17 — Twl 3C Tw), the radiation term in Eq. (11.97) can be linearized to 

It is noted that now the heat transfer of the system is governed by two coupled linear equations, Eqs. (11.102) and (11.104). The last boundary condition in Eq. (11.106) suggests the solution technique for Eqs. (11.102) and (11.104), i.e., separation of variables. Thus, we have 

The problem imposed by Eq. (11.112) with boundary conditions of Eq. (11.113) is noted as the Sturm-Liouville boundary-value problem [e.g., Derrick and Grossman, 1987], if 

Since U, fa, and fa are uniquely characteristic for the operating system, the validity of the preceding inequality is largely dependent on the values of X. As indicated in Eq. (11.107), only the first few successive eigenvalues starting from the smallest one are needed. Therefore, Eq. (11.114) would hold true for the first few eigenvalues. Thus, we have 

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Velocity and volume fraction distributions of phases are independent of the temperature distributions of phases. 

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