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22.4.5 Conservation Equations for Wall-Film Particles

Conservation equations for momentum, mass, and energy for individual parcels in the wall-film are described below. The particle-based approach for thin films was first formulated by O'Rourke [ 271] and most of the following derivation is based closely on that work.

Momentum

The equation for the momentum of a parcel on the film is

 \rho h \frac{d \vec{u}_p}{d t} + h (\nabla_s p_f)_\alpha = ... ...}_p + \dot{\vec{F}}_{n,\alpha} + \rho h (\vec{g} - \vec{a}_w) (22.4-16)

where $\alpha$ denotes the current face on which the particle resides, $h$ denotes the current film height at the particle location, $\nabla_s$ is the gradient operator restricted to the surface, and $p_f$ is the pressure on the surface of the film. On the right-hand side of Equation  22.4-16, $\tau_g$ denotes the magnitude of the shear stress of the gas flow on the surface of the film, $\vec{t_g}$ is the unit vector in the direction of the relative motion of the gas and the film surface, $\mu_l$ is the liquid viscosity, and $\tau_w$ is the magnitude of the stress that the wall exerts on the film. Similarly to the expression for $\vec{t_g}$, $\vec{t_w}$ is the unit vector in the direction of the relative motion of the film and the wall. The remaining expressions on the right-hand side of Equation  22.4-16 are $\dot{\vec{P}}_{imp,\alpha}$ which denotes the impingement pressure on the film surface, $\dot{\vec{M}}_{imp,\alpha}$ is the impingement momentum source, and $\dot{\vec{F}}_{n,\alpha}$ is the force necessary to keep the film on the surface, as determined by
 \vec{u}_p \cdot \hat{n}_\alpha = 0. (22.4-17)

Here, $\rho h (\vec{g} - \vec{a}_w)$ is the body force term. Note that the body force term can be very significant, despite the small values of film thickness due to the very high acceleration rates seen in simulations with moving boundaries. The requirement represented by Equation  22.4-17 is explicitly enforced at each time step in FLUENT for all particles representing the wall-film.

The term $h (\nabla_s p_f)_\alpha$ is the surface gradient of the pressure on the face, $p_f$. This pressure, $p_f$, is the sum of the fluid pressure and the impingement pressure from the drops on the face, given by

 p_f = P_{cell} - \dot{\vec{P}}_{imp,\alpha} \cdot \hat{n} + \dot{M}_{imp,\alpha} \vec{u}_p \cdot \hat{n}

where the impingement mass $\dot{M}_{imp,\alpha}$ is given by
 \dot{M}_{imp,\alpha} = {\int \!\! \int \!\! \int} \rho_l V_... ...v} \cdot \hat{n} f(\vec{x}_s,\vec{v},r,T_d,t) dr d\vec{v}dT_d (22.4-18)

and the impingement pressure is given by
 \dot{\vec{P}}_{imp,\alpha} = \int \!\! \int \!\! \int \rho_... ...{v}\cdot \hat{n} f(\vec{x}_s,\vec{v},r,T_d,t) dr d\vec{v}dT_d (22.4-19)

where $V_p$ is the volume of the drop. An approximation of the impingement mass in Equation  22.4-18 is given by
 \dot{M}_{imp,\alpha} = \left. \left( \sum_{n=0}^{N_s} \rho V_p \right) \right/ { A_\alpha \Delta t }, (22.4-20)

and the corresponding expression of the impingement pressure in Equation  22.4-19 is given by
 \dot{\vec{P}}_{imp,\alpha} = \left. \left( \sum_{n=0}^{N_... ...^{n+1} - \vec{u}_p^n) \right) \right/ { A_\alpha \Delta t }. (22.4-21)

The summation in Equation  22.4-20 is over all the drops which actually stick to the face $\alpha$ during the time step ( $N_s$). The summation in Equation  22.4-21 is over all the particles which impinge upon the face during the same interval ( $N_i$).

The expression for the stress that the gas exerts on the surface of the wall-film, $\tau_g$, in Equation  22.4-16 is given by

 \tau_g = C_f (\vec{u}_g - 2 \vec{u}_p)^2 = C_f V_{rel_g}^2

where $C_f$ is the skin friction coefficient and $\vec{u}_g$ is the gas velocity evaluated at the film height above the wall. The assumption made in evaluating the skin friction coefficient is that the wall shear stress from the gas is constant over the thickness of the film and the boundary layer above the film (in the normal direction from the face). The stress is tangent to the wall in the direction of the difference between the wall-film velocity and the gas velocity, so the unit vector in the direction of the velocity difference along the surface is

 \hat{t}_g = \frac{ \vec{V}_{rel_g} - (\vec{V}_{rel_g} \cdot ... ...\vec{V}_{rel_g} - (\vec{V}_{rel_g} \cdot \hat{n})\hat{n}\vert}

where $\hat{n}$ is the normal face . The expression for the stress that the wall exerts on the film, $\tau_w$, in Equation  22.4-16 is given by

 \tau_w = - \frac{ \mu_l } {h} \vert 2 \vec{u}_p - \vec{u}_w\vert = - \frac{ \mu_l } {h} \vert \vec{V}_{rel_w} \vert

where $\mu_l$ is the liquid viscosity and $\vec{u}_w$ is the velocity of the wall. Here, $\tau_w$ acts in the direction of the velocity difference between the wall and the film, as given by

 \hat{t}_w = \frac{ \vec{V}_{rel_w} - (\vec{V}_{rel_w} \cdot ... ...vec{V}_{rel_w} - (\vec{V}_{rel_w} \cdot \hat{n})\hat{n}\vert}.

Note that the tangential unit vectors, $\hat{t}_g$ and $\hat{t}_w$, are independent and can point in completely different directions.

Since FLUENT solves a particle position equation of the form

 \frac{d \vec{u}_p}{d t} = \alpha - \beta \vec{u}_p,

Equation  22.4-16 must be rearranged. The film particle acceleration is then given by
 \frac{d \vec{u}_p}{d t} = \left( \frac{ C_f \vert V_{rel_g}... ...rho h^2} + \frac{ \dot{M}_{imp} }{\rho h} \right) \vec{u}_p. (22.4-22)

The terms for $M_{imp}$ and $\vec{P}_{imp}$ are used from the previous time step and the differential equations for the particle motion are solved with the existing integration routines.

Mass Transfer from the Film

The film vaporization law is applied when the film particle is above the vaporization temperature $T_{vap}$. A wall particle has the temperature limited by the boiling temperature $T_{bp}$ and does not have a specific boiling law associated with the physics of film boiling.

The vaporization rate of the film is governed by gradient diffusion from the surface exposed to the gas phase. The gradient of vapor concentration between the film surface and the gas phase is

 \dot{N}_i = B_f \left( C_{i,s} - C_{i,\infty} \right) (22.4-23)

where $\dot{N}_i$ is the molar flux of vapor (with units of kgmol/m $^2$-s), $B_f$ is the mass transfer coefficient (in m/s), and $ C_{i,s}$ and $ C_{i,\infty}$ are the vapor concentrations on the film surface and in the bulk gas, respectively. The units of vapor concentration are kgmol/m $^3$.

The vapor concentration at the surface is evaluated using the saturated vapor pressure at the film surface temperature and the bulk gas concentration is obtained from the flow field solution. The vaporization rate is sensitive to the saturated vapor pressure, similar to droplet vaporization.

The mass transfer coefficient is obtained using a Nusselt correlation for the heat transfer coefficient and replacing the Prandtl number with the Schmidt number. The equation is

$\displaystyle Nu_x = \frac{B_f x}{k_f} = \left\{ \begin{array}{cl} 0.332 Re_x^{... ... \\ 0.0296 Re_x^{4/5} Sc^{1/3} & Re_x > 2500, 0.6 < Sc < 60 \end{array}\right.$     (22.4-24)

where the Reynolds number is based on a representative length derived from the face area. The temperature for the film surface is equal to the gas temperature, but is limited by the boiling temperature of the liquid. The particle properties are evaluated at the surface temperature when used in correlation 22.4-24.

For multicomponent vaporization, the Schmidt number based on the diffusivity of each species is used to calculate the correlation in equation 22.4-24 for each component.

The mass of the particle is decreased by

 m_p(t + \Delta t) = m_p(t) - N_i A_p M_{w,i} \Delta t (22.4-25)

where $M_{w,i}$ is the molecular weight of the gas phase species to which the vapor from the liquid is added. The diameter of the film particle is decreased to account for the mass loss in the individual parcel. This keeps the number of drops in the parcel constant and acts only as a place holder. When the parcel detaches from the boundary, the diameter is set to the height of the film and the number in the parcel is adjusted so that the overall mass of the parcel is conserved.

Energy Transfer from the Film

To obtain an equation for the temperature in the film, energy flux from the gas side as well as energy flux from the wall side must be considered. The assumed temperature profile in the liquid is bilinear, with the surface temperature $T_s$ being the maximum temperature of the gas at the film height. Furthermore, the boiling point of the liquid and the wall temperature will be the maximum of the wall face temperature $T_w$, and will be the same boiling temperature as the liquid. An energy balance on a film particle yields

 \frac{d}{d t} \left\{ m_p C_p T_p \right\} = Q_{cond} + Q_{conv} (22.4-26)

where $Q_{cond}$ is the conduction from the wall, given by

 Q_{cond} = \frac{\kappa A_p}{h} (T_w - T_p)

where $\kappa$ is the thermal conductivity of the liquid and $h$ is the film height at the location of the particle, as seen in Figure  22.4.4. The convection from the top surface, $Q_{conv}$ is given by

 Q_{conv} = h_f A_p (T_g - T_p)

where $h_f$ is the film heat transfer coefficient given by Equation  22.4-24 and $A_p$ is the area represented by a film particle, taken to be a mass weighted percentage of the face area, $A_f$. Contributions from the impingement terms are neglected in this formulation, as well as contributions from the gradients of the mean temperature on the edges of the film.

Figure 22.4.4: Assumption of a Bilinear Temperature Profile in the Film
figure

Assuming that the temperature changes slowly for each particle in the film, the equation for the change in temperature of a non vaporizing particle can be written as


 m_p C_p \frac{ d T_{p}}{ d t} = A_p \left( - \left[ h_f + ... ...appa}{h} \right] T_{p} + h T_g + \frac{\kappa}{h} T_w \right) (22.4-27)

As the particle trajectory is computed, FLUENT integrates Equation  22.4-27 to obtain the particle temperature at the next time value, yielding


 T_p (t + \Delta t) = \alpha_p + [T_p (t) - \alpha_p] e^{-\beta_p \Delta t} (22.4-28)

where $\Delta t$ is the integration time step and $\alpha_p$ and $\beta_p$ are given by

 \alpha_p = \frac{ h_f T_g + \frac{\kappa}{h} T_w }{ h_f + \frac{\kappa}{h} } (22.4-29)

and
 \beta_p = \frac{A_p (h_f + \frac{\kappa}{h}) }{m_p C_p} (22.4-30)

When the particle changes its mass during vaporization, an additional term is added to Equation  22.4-27 to account for the enthalpy of vaporization, which is given by

 m_p C_p \frac{ d T_{p}}{ d t} = A_p \left( - \left[ h_f + ... ...{p} + h T_g + \frac{\kappa}{h} T_w \right) + \dot{m}_p h_{fg} (22.4-31)

where $h_{fg}$ is the latent heat of vaporization (with units of J/kg) and the expression $\dot{m}_p$ is the rate of evaporation in kg/s. This alters the expression for $\alpha_p$ in Equation  22.4-29 so that

 \alpha_p = \frac{ h_f T_g + \frac{\kappa}{h} T_w + \dot{m}_p h_{fg} /A_p} { h_f + \frac{\kappa}{h} } (22.4-32)

When the wall-film model is active, the heat flux from the wall to the liquid film is subtracted from the heat flux from the wall to the gas phase. Additionally, enthalpy from vaporization of the liquid from the wall is subtracted from the cell to which the vapor mass goes. Since film boiling is modeled by limiting the liquid phase temperature to the boiling point of the material, energy in excess of that absorbed by the liquid will be put into the gas phase. When the thermal boundary conditions on the wall are set to a constant heat flux, the local temperature of the wall face is used as the thermal boundary condition for the wall-film particles.


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