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13.7: Homogeneous Models

Before discussing the homogeneous models, it is worthwhile to appreciate the complexity of the flow. For the construction of fluid basic equations, it was assumed that the flow is continuous. Now, this assumption has to be broken, and the flow is continuous only in many chunks (small segments). Furthermore, these segments are not defined but results of the conditions imposed on the flow. In fact, the different flow regimes are examples of typical configuration of segments of continuous flow. Initially, it was assumed that the different flow regimes can be neglected at least for the pressure loss (not correct for the heat transfer). The single phase was studied earlier in this book and there is a considerable amount of information about it. Thus, the simplest is to used it for approximation. The average velocity (see also equation (13)) is
\[
    U_m = \dfrac{Q_L + Q_G} {A}  = U_{sL} + U_{sG} = U_m
     \label{phase:eq:Uavarge}  \tag{22}
\]  
It can be noted that the continuity equation is satisfied as

Averaged Mass Rate

\[
     \label{phase:eq:m}
    \dot{m} = \rho_m \,U_m \, A  \tag{23}
\]

Example 13.1

Under what conditions equation (23) is correct?

Solution 13.1

Under Construction

The governing momentum equation can be approximated as
\[
    \dot{m}\, \dfrac{dU_m}{dx} = - A\, \dfrac{dP}{dx}  - S\, \tau_w -  
            A\,\rho_m\,g\,\sin\theta   
     \label{phase:eq:momentum}  \tag{24}
\]  
or modifying equation (24) as

Averaged Momentum

\[
 \label{phase:eq:Pmomentum}
 -  \dfrac{dP}{dx} = - \dfrac{S}{A} \, \tau_w -  
    \dfrac{\dot{m}}{A} \, \dfrac{dU_m}{dx} + \rho_m\,g\,\sin\theta     \tag{25}
\]

The energy equation can be approximated as

Averaged Energy

\[
 \label{phase:eq:energy}
    \dfrac{dq}{dx} - \dfrac{dw}{dx} = \dot{m}\, \dfrac{d}{dx}
        \left( h_m + \dfrac{{U_m}^2}{2} + g\,x\,\sin\theta \right)          \tag{26}
\]

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.