Multiphase flow through a converging nozzle
In this sub-section, the behavior of a two-phase mixture rather than a single phase flow through a converging nozzle will be discussed. As it is explained in the compressible single phase flow case, the first step is to calculate the choking condition. If the back pressure is lower than the critical one, then the nozzle is choked. The employed fluid is water and the liquid-vapor interaction is going to be analyzed. Details of the phase change model will be provided here, but before the following assumptions should be considered:
1. An ideal, homogenized mixture of vapor and liquid is assumed through the nozzle;
2. 1-D model is considered, so the results are averaged over the nozzle cross-sections;
3. 4-th order polynomial trends are used to introduce the physical properties of the fluid.
Refer to [1], there are two models to describe the thermodynamic behavior of the mixture: homogeneous equilibrium model in which heat transfer between the phases would occur instantaneously, and homogeneous frozen model which assumes zero heat transfer between the phases. However, in many cases the real behavior of the mixture can be modelled using a combination of these two extremes. One of the suggested approaches is to use the following combination:
\begin{equation} \epsilon_G = \frac{1-\alpha_0}{2} \ , \ \epsilon_L = \frac{\alpha_0}{2} \end{equation} where $ \epsilon_G $ and $ \epsilon_L $ rule the phase interactions according to the two above-mentioned extreme models. Subscript $ G $ and $ L $ are used for gas (or vapor) and liquid phases respectively:
The following functions, which are thermodynamic properties of the two phases, will be used to model the fluid flow:
\begin{equation} f_G = \left( \frac{\partial \ln \rho_G}{\partial \ln p} \right)_s \ , \ f_L = \left( \frac{\partial \ln \rho_L}{\partial \ln p} \right)_s \end{equation}
\begin{equation} g_G = \left( \frac{\partial \ln \rho_G}{\partial \ln p} \right)_e + \left( 1 - \frac{\rho_G}{\rho_L} \right) \left( \frac{h_L}{L} \frac{\partial \ln h_L}{\partial \ln p} + \frac{\partial \ln L}{\partial \ln p} - \frac{p}{L \rho_G}\right)_e \end{equation}
\begin{equation} g_L = \left( \frac{\partial \ln \rho_L}{\partial \ln p} \right)_e + \left( \frac{\rho_L}{\rho_G} - 1 \right) \left( \frac{h_L}{L} \frac{\partial \ln h_L}{\partial \ln p} - \frac{p}{L \rho_L}\right)_e \end{equation} where the subscripts $ s $ and $ e $ refer to isentropic and phase equilibrium derivatives respectively and $ L = h_G - h_L $ is the latent heat. Considering the following coefficients:
\begin{equation} k_G = (1 - \epsilon_G)f_G + \epsilon_G g_G \ , \ k_L = \epsilon_L g^* (p_c)^{\eta} \end{equation} it is possible to obtain the mixture velocity, $ u $ and the velocity of sound, $ c $, for the mixture:
\begin{equation} \begin{split} \frac{u^2}{2} = \frac {p_0}{\rho_L} \left( 1 - \frac{p}{p_0} + \frac{1}{1-k_G}\left[\frac{\alpha_0}{1 - \alpha_0} + \frac{k_L p_0^{-\eta}}{(k_G - \eta)}\right] \left[ 1 - \left(\frac{p}{p_0}\right)^{1-k_G} \right] - \right. \\ \left. \frac{1}{(1-\eta)} \left[\frac{k_L p_0^{-\eta}}{(k_G - \eta)}\right] \left[ 1 - \left(\frac{p}{p_0}\right)^{1-\eta} \right] \right) \end{split} \end{equation}
\begin{equation} \frac{c^2}{2} = \frac{p}{\rho_L}\frac{\left(1 + \left[\frac{\alpha_0}{1-\alpha_0} + \frac{k_L p_0^{-\eta}}{(k_G - \eta)} \right] \left(\frac{p_0}{p}\right)^{k_G} - \left[\frac{k_L p_0^{-\eta}}{(k_G - \eta)} \right] \left(\frac{p_0}{p}\right)^{\eta} \right)^2}{2\left( k_G \left[\frac{\alpha_0}{1-\alpha_0} + \frac{k_L p_0^{-\eta}}{(k_G - \eta)} \right] \left(\frac{p_0}{p}\right)^{k_G} - \eta \left[\frac{k_L p_0^{-\eta}}{(k_G - \eta)} \right] \left(\frac{p_0}{p}\right)^{\eta} \right)} \end{equation} in which $ g^* $ and $ \eta $ are obtained from a logarithmic fit on $ g_L $, subscript $ 0 $ indicates reservoir conditions, and $ p_c $ is the critical pressure of the fluid. Now in order to find the choking condition of the nozzle, we can write $ u = c $ and the pressure in sonic condition ($ p^* $) will be obtained. Then by inserting $ p^* $ in the following equation, $ \alpha^* $ (the gas volume fraction in sonic condition) will be calculated.
\begin{equation} \frac{\alpha}{1-\alpha} = \left[\frac{\alpha_0}{1-\alpha_0} + \frac{k_L p_0^{-\eta}}{(k_G - \eta)} \right] \left(\frac{p_0}{p}\right)^{k_G} - \left[\frac{k_L p_0^{-\eta}}{(k_G - \eta)} \right] \left(\frac{p_0}{p}\right)^{\eta} \end{equation} Then the values of $ p $ and $ \alpha $ will become $ p^* $ and $ \alpha^* $. Finally, by substituting $ p^* $ in Eq. (6), the mixture velocity in sonic condition, $ u^* $, can be obtained.
Applying the model in a Matlab code, it is possible to calculate the mixture thermodynamic properties and the code can be verified by comparing the $ g_L $ and $ g_G $ parameters with the reference [1] as shown in Fig. e.3.1.
Because the polynomial trends chosen for representing the fluid properties are 4-th order, it is not suitable to implement an algebraic derivation in logarithmic scales (see Eqs. (2) to (4)). To make the algorithm more robust, we used a finite difference centered scheme for calculating the derivatives and consequently a constant best fit for each of them. As one can notice from Fig. e.3.1, $ g_L $ and $ g_G $ are in good agreement with the reference [1] although such a simplification is employed.
After calculating the choking conditions, if the back pressure, $ p_b $ is higher than $ p^* $, then $ p_{out} = p_b $ and $ c_{out}, u_{out}, \alpha_{out} $ can be calculated with Eqs. (6) to (8). Otherwise, the choking conditions can be used as the outlet conditions.
Other topics on fluid flow through nozzles are listed in the main section 1-D flow model in nozzles.
The relevant Matlab codes can be downloaded using the following link:
Matlab Codes Bank
Reference
[1] C. E. Brennen, Fundamentals of multiphase flow, Cambridge University Press (2005)
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