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23.6.2 Wet Steam Flow Equations

The wet steam is a mixture of two-phases. The primary phase is the gaseous-phase consisting of water-vapor (denoted by the subscript v) while the secondary phase is the liquid-phase consisting of condensed-water droplets (denoted by the subscript l).

The following assumptions are made in this model:

From the preceding assumptions, it follows that the mixture density ( $\rho$) can be related to the vapor density ( $\rho_v$) by the following equation:


 \rho = \frac{\rho_v}{(1-\beta)} (23.6-1)

In addition, the temperature and the pressure of the mixture will be equivalent to the temperature and pressure of the vapor-phase.

The mixture flow is governed by the compressible Navier-Stokes equations given in vector form by Equation  25.5-4:


 \frac{\partial {\mbox{\boldmath$W$}}}{\partial {\mbox{\boldm... ...t d{{\mbox{\boldmath$A$}}} = \int_V {\mbox{\boldmath$H$}}\, dV (23.6-2)

where ${\mbox{\boldmath$Q$}}$=(P,u,v,w,T) are mixture quantities. The flow equations are solved using the same density-based solver algorithms employed for general compressible flows.

To model wet steam, two additional transport equations are needed [ 153]. The first transport equation governs the mass fraction of the condensed liquid phase ( $\beta$):


 \frac{\partial \rho \beta}{\partial t} + \nabla \cdot \left (\rho \overrightarrow{v} \beta \right ) = \Gamma (23.6-3)

where $\Gamma$ is the mass generation rate due to condensation and evaporation (kg per unit volume per second). The second transport equation models the evolution of the number density of the droplets per unit volume:


 \frac{\partial \rho \eta}{\partial t} + \nabla \cdot \left (\rho \overrightarrow{v} \eta \right ) = \rho I (23.6-4)

where $I$ is the nucleation rate (number of new droplets per unit volume per second).

To determine the number of droplets per unit volume, Equation  23.6-1 and the average droplet volume $V_d$ are combined in the following expression:


 \eta = \frac{\beta}{(1 - \beta) V_d ({\rho_l}/{\rho_v})} (23.6-5)

where $\rho_l$ is the liquid density and the average droplet volume is defined as


 V_d = \frac{4}{3} \pi \overline{r}_d^3 (23.6-6)

where $r_d$ is the droplet radius.

Together, Equation  23.6-2, Equation  23.6-3, and Equation  23.6-4 form a closed system of equations which, along with Equation  23.6-1, permit the calculation of the wet steam flow field.


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