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23.5.3 Conservation Equations

The general conservation equations from which the equations solved by FLUENT are derived are presented in this section, followed by the solved equations themselves.



Equations in General Form


Conservation of Mass

The continuity equation for phase $q$ is


 \frac{\partial}{\partial t} (\alpha_q \rho_q) + \nabla \cdot... ...= \sum_{p=1}^n ( {\dot{m}_{\rm pq} - \dot{m}_{\rm qp}} ) + S_q (23.5-4)

where ${\vec v}_q$ is the velocity of phase $q$ and $\dot{m}_{pq}$ characterizes the mass transfer from the $p^{\rm th}$ to $q^{\rm th}$ phase, and $\dot{m}_{\rm qp}$ characterizes the mass transfer from phase $q$ to phase $p$, and you are able to specify these mechanisms separately.

By default, the source term $S_q$ on the right-hand side of Equation  23.5-4 is zero, but you can specify a constant or user-defined mass source for each phase. A similar term appears in the momentum and enthalpy equations. See Section  23.7 for more information on the modeling of mass transfer in FLUENT's general multiphase models.

Conservation of Momentum

The momentum balance for phase $q$ yields


 \frac{\partial}{\partial t} ( \alpha_q \rho_q \vec v_q ) + \... ...cdot \overline{\overline{\tau}}_q + \alpha_q \rho_q {\vec g} +


 \sum_{p=1}^n ( \vec R_{pq} + \dot{m}_{pq} \vec v_{pq} - \dot... ...p}) + (\vec F_q + \vec F_{{\rm lift},q} + \vec F_{{\rm vm},q}) (23.5-5)

where $\overline{\overline{\tau}}_q$ is the $q^{\rm th}$ phase stress-strain tensor


 \overline{\overline{\tau}}_q = \alpha_q \mu_q(\nabla \vec v_... ...frac{2}{3}\mu_q) \nabla \cdot \vec v_q \overline{\overline{I}} (23.5-6)

Here $\mu_q$ and $\lambda_q$ are the shear and bulk viscosity of phase $q$, $\vec F_q$ is an external body force, $\vec F_{{\rm lift},q}$ is a lift force, $\vec F_{{\rm vm},q}$ is a virtual mass force, $\vec R_{pq}$ is an interaction force between phases, and $p$ is the pressure shared by all phases.

$\vec v_{pq}$ is the interphase velocity, defined as follows. If $\dot{m}_{pq} > 0$ (i.e., phase $p$ mass is being transferred to phase $q$), $\vec v_{pq}=\vec v_{p}$; if $\dot{m}_{pq} < 0$ (i.e., phase $q$ mass is being transferred to phase $p$), $\vec v_{pq}=\vec v_{q}$. Likewise, if $\dot{m}_{qp} > 0$ then $v_{qp} = v_q$, if $\dot{m}_{qp} < 0$ then $v_{qp} = v_p$.

Equation  23.5-5 must be closed with appropriate expressions for the interphase force $\vec R_{pq}$. This force depends on the friction, pressure, cohesion, and other effects, and is subject to the conditions that $\vec R_{pq} = - \vec R_{qp}$ and $\vec R_{qq} = 0$.

FLUENT uses a simple interaction term of the following form:


 \sum_{p=1}^n \vec R_{pq} = \sum_{p=1}^n {K_{pq} ( \vec v_p - \vec v_q)} (23.5-7)

where $K_{pq}$ ( $=K_{qp}$) is the interphase momentum exchange coefficient (described in Section  23.5.4).

Lift Forces

For multiphase flows, FLUENT can include the effect of lift forces on the secondary phase particles (or droplets or bubbles). These lift forces act on a particle mainly due to velocity gradients in the primary-phase flow field. The lift force will be more significant for larger particles, but the FLUENT model assumes that the particle diameter is much smaller than the interparticle spacing. Thus, the inclusion of lift forces is not appropriate for closely packed particles or for very small particles.

The lift force acting on a secondary phase $p$ in a primary phase $q$ is computed from [ 90]


 {\vec F}_{\rm lift} = - 0.5 \rho_q \alpha_p (\vec v_q - \vec v_p) \times (\nabla \times \vec v_q) (23.5-8)

The lift force ${\vec F}_{\rm lift}$ will be added to the right-hand side of the momentum equation for both phases ( ${\vec F}_{{\rm lift},q}=-{\vec F}_{{\rm lift},p}$).

In most cases, the lift force is insignificant compared to the drag force, so there is no reason to include this extra term. If the lift force is significant (e.g., if the phases separate quickly), it may be appropriate to include this term. By default, ${\vec F}_{\rm lift}$ is not included. The lift force and lift coefficient can be specified for each pair of phases, if desired.

figure   

It is important that if you include the lift force in your calculation, you need not include it everywhere in the computational domain since it is computationally expensive to converge. For example, in the wall boundary layer for turbulent bubbly flows in channels, the lift force is significant when the slip velocity is large in the vicinity of high strain rates for the primary phase.

Virtual Mass Force

For multiphase flows, FLUENT includes the "virtual mass effect'' that occurs when a secondary phase $p$ accelerates relative to the primary phase $q$. The inertia of the primary-phase mass encountered by the accelerating particles (or droplets or bubbles) exerts a "virtual mass force'' on the particles [ 90]:


 {\vec F}_{\rm vm} = 0.5 \alpha_p \rho_q \left( \frac{d_q {\vec v}_q}{d t} - \frac{d_p {\vec v}_p}{d t} \right) (23.5-9)

The term $\frac{d_q}{dt}$ denotes the phase material time derivative of the form


 \frac{d_q (\phi)}{dt} = \frac{\partial (\phi) }{\partial t} + (\vec{v}_q \cdot \nabla ) \phi (23.5-10)

The virtual mass force ${\vec F}_{\rm vm}$ will be added to the right-hand side of the momentum equation for both phases ( ${\vec F}_{{\rm vm},q}=-{\vec F}_{{\rm vm},p}$).

The virtual mass effect is significant when the secondary phase density is much smaller than the primary phase density (e.g., for a transient bubble column). By default, ${\vec F}_{\rm vm}$ is not included.

Conservation of Energy

To describe the conservation of energy in Eulerian multiphase applications, a separate enthalpy equation can be written for each phase:


 \frac{\partial}{\partial t} ( \alpha_q \rho_q h_q ) + \nabla... ...m_{p=1}^n (Q_{pq} + \dot{m}_{pq} h_{pq} - \dot{m}_{qp} h_{qp}) (23.5-11)

where $h_q$ is the specific enthalpy of the $q^{\rm th}$ phase, $\vec q_q$ is the heat flux, $S_q$ is a source term that includes sources of enthalpy (e.g., due to chemical reaction or radiation), $Q_{pq}$ is the intensity of heat exchange between the $p^{\rm th}$ and $q^{\rm th}$ phases, and $h_{pq}$ is the interphase enthalpy (e.g., the enthalpy of the vapor at the temperature of the droplets, in the case of evaporation). The heat exchange between phases must comply with the local balance conditions $Q_{pq} = - Q_{qp}$ and $Q_{qq} = 0$.



Equations Solved by FLUENT


The equations for fluid-fluid and granular multiphase flows, as solved by FLUENT, are presented here for the general case of an $n$-phase flow.

Continuity Equation

The volume fraction of each phase is calculated from a continuity equation:


 \frac{1}{\rho_{rq}} \left ( \frac{\partial}{\partial t} ( \a... ... ) = \sum_{p=1}^n { ( \dot{m}_{pq} - \dot{m}_{qp} ) } \right ) (23.5-12)

where $\rho_{rq}$ is the phase reference density, or the volume averaged density of the $q^{\rm th}$ phase in the solution domain.

The solution of this equation for each secondary phase, along with the condition that the volume fractions sum to one (given by Equation  23.5-2), allows for the calculation of the primary-phase volume fraction. This treatment is common to fluid-fluid and granular flows.

Fluid-Fluid Momentum Equations

The conservation of momentum for a fluid phase $q$ is


$\displaystyle \frac{\partial}{\partial t} ( \alpha_q \rho_q \vec v_q ) + \nabla \cdot ( \alpha_q \rho_q \vec v_q \vec v_q )$ $\textstyle =$ $\displaystyle -\alpha_q \nabla p + \nabla \cdot \overline{\overline{\tau}}_q + \alpha_q \rho_q \vec g +$  
    $\displaystyle \sum_{p=1}^n (K_{pq}(\vec v_p - \vec v_q) + \dot{m}_{pq} \vec v_{pq} - \dot{m}_{qp} \vec v_{qp} ) +$  
    $\displaystyle (\vec F_q + \vec F_{{\rm lift},q} + \vec F_{{\rm vm},q})$ (23.5-13)

Here $\vec g $ is the acceleration due to gravity and $\overline{\overline{\tau}}_q$, $\vec F_q$, $\vec F_{{\rm lift},q}$, and $\vec F_{{\rm vm},q}$ are as defined for Equation  23.5-5.

Fluid-Solid Momentum Equations

Following the work of [ 7, 52, 81, 121, 199, 222, 268, 364], FLUENT uses a multi-fluid granular model to describe the flow behavior of a fluid-solid mixture. The solid-phase stresses are derived by making an analogy between the random particle motion arising from particle-particle collisions and the thermal motion of molecules in a gas, taking into account the inelasticity of the granular phase. As is the case for a gas, the intensity of the particle velocity fluctuations determines the stresses, viscosity, and pressure of the solid phase. The kinetic energy associated with the particle velocity fluctuations is represented by a "pseudothermal'' or granular temperature which is proportional to the mean square of the random motion of particles.

The conservation of momentum for the fluid phases is similar to Equation  23.5-13, and that for the $s^{\rm th}$ solid phase is


$\displaystyle \frac{\partial}{\partial t} ( \alpha_s \rho_s \vec v_s ) + \nabla \cdot ( \alpha_s \rho_s \vec v_s \vec v_s )$ $\textstyle =$ $\displaystyle -\alpha_s \nabla p - \nabla p_s + \nabla \cdot \overline{\overline{\tau}}_s + \alpha_s \rho_s \vec g +$  
    $\displaystyle \sum_{l=1}^{N} (K_{ls}(\vec v_l - \vec v_s) + \dot{m}_{ls} \vec v_{ls} - \dot{m}_{sl} \vec v_{sl}) +$  
    $\displaystyle (\vec F_s + \vec F_{{\rm lift},s} + \vec F_{{\rm vm},s})$ (23.5-14)

where $p_{s}$ is the $s^{\rm th}$ solids pressure, $K_{ls} = K_{sl}$ is the momentum exchange coefficient between fluid or solid phase $l$ and solid phase $s$, $N$ is the total number of phases, and $\vec F_q$, $\vec F_{{\rm lift},q}$, and $\vec F_{{\rm vm},q}$ are as defined for Equation  23.5-5.

Conservation of Energy

The equation solved by FLUENT for the conservation of energy is Equation  23.5-11.


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Up: 23.5 Eulerian Model Theory
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© Fluent Inc. 2006-09-20