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22.2.1 Equations of Motion for Particles



Particle Force Balance


FLUENT predicts the trajectory of a discrete phase particle (or droplet or bubble) by integrating the force balance on the particle, which is written in a Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, and can be written (for the $x$ direction in Cartesian coordinates) as


 \frac{d u_p}{dt} = F_D(u - u_p) + \frac{g_x (\rho_p - \rho)}{\rho_p} + F_x (22.2-1)

where $F_x$ is an additional acceleration (force/unit particle mass) term, $F_D (u - u_p)$ is the drag force per unit particle mass and


 F_D = \frac{1 8 \mu}{\rho_p d^{2}_{p}} \;\; \frac{C_D {\rm Re}}{24} (22.2-2)

Here, $u$ is the fluid phase velocity, $u_p$ is the particle velocity, $\mu$ is the molecular viscosity of the fluid, $\rho$ is the fluid density, $\rho_p$ is the density of the particle, and $d_p$ is the particle diameter. Re is the relative Reynolds number, which is defined as


 {\rm Re} \equiv \frac{\rho d_p \left \vert u_{p} - u \right \vert}{\mu} (22.2-3)



Inclusion of the Gravity Term


While Equation  22.2-1 includes a force of gravity on the particle, it is important to note that in FLUENT the default gravitational acceleration is zero. If you want to include the gravitational force, you must remember to define the magnitude and direction of the gravity vector in the Operating Conditions panel .



Other Forces


Equation  22.2-1 incorporates additional forces ( $F_x$) in the particle force balance that can be important under special circumstances. The first of these is the "virtual mass'' force, the force required to accelerate the fluid surrounding the particle. This force can be written as


 F_x = \frac{1}{2}\frac{\rho}{\rho_p}\frac{d}{dt} \left(u - u_{p} \right) (22.2-4)

and is important when $\rho > \rho_p$. An additional force arises due to the pressure gradient in the fluid:

 F_x = \left ( \frac{\rho}{\rho_p} \right ) {u_p}_i \frac{\partial u}{\partial x_i} (22.2-5)



Laws for Drag Coefficients


The drag coefficient, $C_D$, can be taken from either


 C_D = a_1 + \frac{a_2}{\rm Re} + \frac{a_3}{{\rm Re}^2} (22.2-6)

where $a_1$, $a_2$, and $a_3$ are constants that apply to smooth spherical particles over several ranges of Re given by Morsi and Alexander [ 253], or


 C_D = \frac{24}{\rm Re_{sph}} \left( 1 + b_1 {\rm Re_{sph}}^{b_2} \right) + \frac{b_3 {\rm Re_{sph}}}{b_4 + {\rm Re_{sph}}} (22.2-7)

where

$\displaystyle b_1$ $\textstyle =$ $\displaystyle \exp(2.3288 - 6.4581\phi + 2.4486\phi^2)$  
$\displaystyle b_2$ $\textstyle =$ $\displaystyle 0.0964 + 0.5565\phi$  
$\displaystyle b_3$ $\textstyle =$ $\displaystyle \exp(4.905 - 13.8944\phi + 18.4222\phi^2 - 10.2599\phi^3)$  
$\displaystyle b_4$ $\textstyle =$ $\displaystyle \exp(1.4681 + 12.2584\phi - 20.7322\phi^2 + 15.8855\phi^3)$ (22.2-8)

which is taken from Haider and Levenspiel [ 132]. The shape factor, $\phi$, is defined as

 \phi = \frac{s}{S} (22.2-9)

where $s$ is the surface area of a sphere having the same volume as the particle, and $S$ is the actual surface area of the particle. The Reynolds number $\rm Re_{sph}$ is computed with the diameter of a sphere having the same volume.

figure   

The shape factor cannot exceed a value of 1.

For sub-micron particles, a form of Stokes' drag law is available [ 274]. In this case, $F_D$ is defined as


 F_D = \frac{18 \mu}{{d_p}^2 \rho_p C_c} (22.2-10)

The factor $C_c$ is the Cunningham correction to Stokes' drag law , which you can compute from


 C_c = 1 + \frac{2 \lambda}{d_p} (1.257 + 0.4 e^{-(1.1 d_p/2\lambda)}) (22.2-11)

where $\lambda$ is the molecular mean free path.

A high-Mach-number drag law is also available. This drag law is similar to the spherical law (Equation  22.2-6) with corrections [ 62] to account for a particle Mach number greater than 0.4 at a particle Reynolds number greater than 20.

For unsteady models involving discrete phase droplet breakup, a dynamic drag law option is also available. See Section  22.6 for a description of this law.

Instructions for selecting the drag law are provided in Section  22.11.4.



Forces in Rotating Reference Frames


The additional force term, $F_x$, in Equation  22.2-1 also includes forces on particles that arise due to rotation of the reference frame. These forces arise when you are modeling flows in rotating frames of reference (see Section  10.2). For rotation defined about the $z$ axis, for example, the forces on the particles in the Cartesian $x$ and $y$ directions can be written as


 \left( 1 - \frac{\rho}{\rho_p} \right) \Omega^2x + 2 \Omega \left( u_{y,p} - \frac{\rho}{\rho_p} u_y \right) (22.2-12)

where $u_{y,p}$ and $u_y$ are the particle and fluid velocities in the Cartesian $y$ direction, and


 \left( 1 - \frac{\rho}{\rho_p} \right) \Omega^2 y - 2 \Omega \left( u_{x,p}- \frac{\rho}{\rho_p} u_x \right) (22.2-13)

where $u_{x,p}$ and $u_x$ are the particle and fluid velocities in the Cartesian $x$ direction.



Thermophoretic Force


Small particles suspended in a gas that has a temperature gradient experience a force in the direction opposite to that of the gradient. This phenomenon is known as thermophoresis. FLUENT can optionally include a thermophoretic effect on particles in the additional acceleration (force/unit mass) term, $F_x$, in Equation  22.2-1:


 F_x = -D_{T,p} \frac{1}{m_pT} \frac{\partial T}{\partial x} (22.2-14)

where $D_{T,p}$ is the thermophoretic coefficient. You can define the coefficient to be constant, polynomial, or a user-defined function, or you can use the form suggested by Talbot [ 367]:


 F_x = -\frac{6 \pi d_p \mu^2 C_s (K + C_t {\rm Kn})}{\rho (1... ...+ 2C_t {\rm Kn})} \frac{1}{m_pT} \frac{\partial T}{\partial x} (22.2-15)


where: ${\rm Kn}$ = Knudsen number = 2 $\lambda/d_p$
  $\lambda$ = mean free path of the fluid
  $K$ = $k/k_p$
  $k$ = fluid thermal conductivity based on translational
    $\; \; \;$ energy only = (15/4) $\mu R$
  $k_p$ = particle thermal conductivity
  $C_S$ = 1.17
  $C_t$ = 2.18
  $C_m$ = 1.14
  $m_p$ = particle mass
  $T$ = local fluid temperature
  $\mu$ = fluid viscosity

This expression assumes that the particle is a sphere and that the fluid is an ideal gas.



Brownian Force


For sub-micron particles, the effects of Brownian motion can be optionally included in the additional force term. The components of the Brownian force are modeled as a Gaussian white noise process with spectral intensity $S_{n, ij}$ given by [ 205]


 S_{n, ij} = S_0 \delta_{ij} (22.2-16)

where $\delta_{ij}$ is the Kronecker delta function, and


 S_0 = \frac{216 \nu k_B T}{\pi^2 \rho d_p^5 \left(\frac{\rho_p}{\rho}\right)^2 C_c} (22.2-17)

$T$ is the absolute temperature of the fluid, $\nu$ is the kinematic viscosity, and $k_B$ is the Boltzmann constant. Amplitudes of the Brownian force components are of the form


 F_{b_i} = \zeta_i \sqrt{\frac{\pi S_o}{\Delta t}} (22.2-18)

where $\zeta_i$ are zero-mean, unit-variance-independent Gaussian random numbers. The amplitudes of the Brownian force components are evaluated at each time step. The energy equation must be enabled in order for the Brownian force to take effect. Brownian force is intended only for nonturbulent models.



Saffman's Lift Force


The Saffman's lift force, or lift due to shear, can also be included in the additional force term as an option. The lift force used is from Li and Ahmadi [ 205] and is a generalization of the expression provided by Saffman [ 313]:


 {\vec F} = \frac{2 K \nu^{1/2} \rho d_{ij}}{\rho_p d_p (d_{lk} d_{kl})^{1/4}} \left({\vec v} - {\vec v}_p \right) (22.2-19)

where $K=2.594$ and $d_{ij}$ is the deformation tensor. This form of the lift force is intended for small particle Reynolds numbers. Also, the particle Reynolds number based on the particle-fluid velocity difference must be smaller than the square root of the particle Reynolds number based on the shear field. Since this restriction is valid for submicron particles, it is recommended to use this option only for submicron particles.


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© Fluent Inc. 2006-09-20