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23.5.4 Interphase Exchange Coefficients

It can be seen in Equations  23.5-13 and 23.5-14 that momentum exchange between the phases is based on the value of the fluid-fluid exchange coefficient $K_{pq}$ and, for granular flows, the fluid-solid and solid-solid exchange coefficients $K_{ls}$.



Fluid-Fluid Exchange Coefficient


For fluid-fluid flows, each secondary phase is assumed to form droplets or bubbles. This has an impact on how each of the fluids is assigned to a particular phase. For example, in flows where there are unequal amounts of two fluids, the predominant fluid should be modeled as the primary fluid, since the sparser fluid is more likely to form droplets or bubbles. The exchange coefficient for these types of bubbly, liquid-liquid or gas-liquid mixtures can be written in the following general form:


 K_{pq} = \frac{\alpha_q \alpha_p \rho_p f}{\tau_p} (23.5-15)

where $f$, the drag function, is defined differently for the different exchange-coefficient models (as described below) and $\tau_p$, the "particulate relaxation time'', is defined as


 \tau_p = \frac{\rho_p d_p^2}{18 \mu_q} (23.5-16)

where $d_p$ is the diameter of the bubbles or droplets of phase $p$.

Nearly all definitions of $f$ include a drag coefficient ( $C_D$) that is based on the relative Reynolds number ( $Re$). It is this drag function that differs among the exchange-coefficient models. For all these situations, $K_{pq}$ should tend to zero whenever the primary phase is not present within the domain. To enforce this, the drag function $f$ is always multiplied by the volume fraction of the primary phase $q$, as is reflected in Equation  23.5-15.

You can specify different exchange coefficients for each pair of phases. It is also possible to use user-defined functions to define exchange coefficients for each pair of phases. If the exchange coefficient is equal to zero (i.e., if no exchange coefficient is specified), the flow fields for the fluids will be computed independently, with the only "interaction'' being their complementary volume fractions within each computational cell.



Fluid-Solid Exchange Coefficient


The fluid-solid exchange coefficient $K_{sl}$ can be written in the following general form:


 K_{sl} = \frac{\alpha_s \rho_s f}{\tau_s} (23.5-28)

where $f$ is defined differently for the different exchange-coefficient models (as described below), and $\tau_s$, the "particulate relaxation time'', is defined as


 \tau_s = \frac{\rho_s d_s^2}{18 \mu_l} (23.5-29)

where $d_s$ is the diameter of particles of phase $s$.

All definitions of $f$ include a drag function ( $C_D$) that is based on the relative Reynolds number (Re $_s$). It is this drag function that differs among the exchange-coefficient models.



Solid-Solid Exchange Coefficient


The solid-solid exchange coefficient $K_{ls}$ has the following form [ 362]:


 K_{ls} = \frac{ 3 \left( 1 + e_{ls} \right) \left( \frac{\pi... ...+ \rho_{s} d_s^3 \right) } \vert \vec v_{l} - \vec v_{s} \vert (23.5-43)


where      
  $e_{ls}$ = the coefficient of restitution
  $C_{{\rm fr},ls}$ = the coefficient of friction between the $l^{\rm th}$ and $s^{\rm th}$
      solid-phase particles $(C_{{\rm fr},ls} = 0)$
  $d_{l}$ = the diameter of the particles of solid $l$
  $g_{0,ls}$ = the radial distribution coefficient

Note that the coefficient of restitution is described in Section  23.5.5 and the radial distribution coefficient is described in Section  23.5.5.


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