Wall Impingement

Bai-Gosman Wall Impingement

The Bai-Gosman model, originally described in [648], [649], and [650], but extended for Simcenter STAR-CCM+, categorizes possible outcomes as being in one of six possible regimes, illustrated in the following diagram.

For fluid film boundaries, momentum and kinetic energy lost by the droplets are gained by the film.

The choice of regime for a given impingement event is made using four parameters:

• The incident Weber number

  $$W{e}_I=\frac{{\rho }_p{v}_{r,n}^2{D}_p}{\sigma }$$

  (3177)

  in which $v_{r,n}=\left({\mathbf{\text{v}}}_p-{\mathbf{\text{v}}}_w\right)\cdot {\mathbf{\text{n}}}_w$ is the normal component of the particle velocity relative to the wall. ${\mathbf{\text{n}}}_w$ is the unit vector normal to the boundary.

• The Laplace number

  $$\text{La}=\frac{{\rho }_p\sigma D_p}{{\mu }_p^2}$$

  (3178)
• The boundary temperature $T_w$ (provided the Energy model is active in the physics continuum).
• The wall state; a boundary can be either wet or dry. For fluid film boundaries, this is determined by the presence or absence of a fluid film. For other boundaries, this is determined by the user-selected Wall State property.

  For a dry wall state:

  • At Weber numbers below ${We}_c$ and wall temperatures below $T_{12}$ , impinging droplets spread out on the wall.

  • At higher Weber numbers or wall temperatures, droplets can spread, break up and spread, break up and rebound, or rebound.

  • Above ${We}_c$ , impinging droplets splash.

  For a wet wall state:

  • At Weber numbers below 2 and wall temperatures below $T_{12}$ , impinging droplets adhere to the wall.

  • At higher Weber numbers or wall temperatures, droplets can spread, break up and spread, break up and rebound, or rebound.

  • Above ${We}_c$ , impinging droplets splash.

Leidenfrost effect occurs when the temperature is high enough, at $T_{12}$ , reducing friction and heat transfer. It peaks at $T_{23}$ .

Regime Transition Criteria

The idealized map of impingement regimes is shown below, in which there are three temperature ranges, separated by two transition temperatures:

• $T_{12}$ , separating range 1 from range 2, is expected to be approximately the boiling temperature of the droplet.
• $T_{23}$ , separating range 2 from range 3, is expected to be approximately the Leidenfrost temperature of the droplet.

The transition criteria for wetted walls are given below for the three wall temperature ranges.

The Weber numbers 2 and 20, shown in the preceding diagram, can be adjusted in the expert properties of this model.

The transition criteria for dry walls are given below for the three wall temperature ranges.

Temperature Range 1

The first temperature range is expressed as:

$$T_w\le T_{12}$$

(3179)

It also applies when the Energy model is not active in the physics continuum. The model recognizes the following regimes in this temperature range:

• Adhere

  $$W{e}_I\le 2$$

  (3180)

  in which the impinging droplet adheres to the boundary in nearly spherical form. If the boundary is a fluid film, the droplets transfer their mass, momentum, and kinetic energy to the film.

  Simcenter STAR-CCM+ permits the specific boundary interaction mode to control the outcome of a particle satisfying the adhere condition. The default in unsteady is Stick, while in steady the default is Escape.

• Rebound

  $$W{e}_I\le 20$$

  (3181)

  in which the impinging droplet bounces off the boundary after the impact. This is functionally identical to the Rebound boundary interaction mode with restitution coefficients

  $$e_n=0.993-1.76\theta +1.56{\theta }^2-0.49{\theta }^3$$

  (3182)

  $$e_t=\frac{5}{7}$$

  (3183)

  in which $\theta$ is the droplet incidence angle that is measured from the boundary.

• Spread

  $$W{e}_I\le W{e}_c$$

  (3184)

  in which the droplet merges with an existing film on a wetted boundary. If the boundary is a fluid film, the droplets transfer their mass, momentum, and kinetic energy to the film. The limiting Weber number $W{e}_c$ is given by

  $${\text{We}}_c=A{\text{La}}^{-0.18}$$

  (3185)

  For smooth walls or wet rough walls

  $$A=A_w$$

  (3186)

  where $A_w$ is the user-defined splash onset coefficient with a default value, 1320.

  For dry, rough walls $A$ is interpolated from the table below

  Roughness $\text{μ}\text{m}$	0	0.05	0.14	0.84	3.1	12.0	$\infty$
  $\mathbf{\text{A}}/{\mathbf{\text{A}}}_{\mathbf{\text{w}}}$	3.9879	3.9879	3.4348	1.9955	1.5576	1.0015	1

  If there is a film boundary in which fluid film can form, then parcels found in the spread regime for temperature range 1 transfer their mass, momentum, and energy to the film. If there is no film boundary and so no possibility for a fluid film to be formed, then parcels in the spread regime rebound with a normal restitution coefficient of 0 ( $e_n=0$ ) and a tangential restitution coefficient of 1 ( $e_t=1$ ).

• Splash

  $$W{e}_I>W{e}_c$$

  (3187)

  in which the impinging droplet breaks up into smaller droplets, some of which are reflected from the boundary. Residue may remain stuck to the wall. Each impinging parcel, of mass $m_{\pi }$ , produces a user-specified number of secondary parcels which are randomly reflected from the wall within an ejection cone. The total mass of the secondary parcels $m_s$ is determined by the splash mass ratio:

  $$r_m=\frac{m_s}{m_{\pi }}=C_b+\left(\begin{array}{c}{C}_{rd}\\ C_{rw}\end{array}\right)X_r$$

  (3188)

  where:

  • $C_b$ is the base coefficient.
  • $C_{rd}$ is the range coefficient for a dry wall.
  • $C_{rw}$ is the range coefficient for a wet wall.
  • $X_r$ is a random number that is distributed uniformly in the range [0 — 1].

  The total number of secondary droplets that are created per splash is given by

  $$N_s=a_0\left(\frac{W{e}_I}{W{e}_m}-1\right)$$

  (3189)

  where:

  • $a_0$ is a user-specified coefficient (default 5)
  • $W{e}_m$ is a Weber number which, in this temperature range, is equal to $W{e}_c$ .

  The actual number of secondary droplets that are created by a splash event tends to $N_s$ as the number of secondary parcels tends to infinity. This is ensured by randomly selecting the diameters and number of secondary droplets per parcel from a Rosin-Rammler distribution with user-specified exponent $q$ and reference diameter

  $$D_{ref}={\left(\frac{r_m}{N_s\Gamma (1+3/q)}\right)}^{1/3}{D}_p$$

  (3190)

  in which $\Gamma \left(x\right)$ is the gamma function. The default exponent, $q=1$, produces a Chi-squared distribution.

  The velocity of the secondary droplets is calculated from two contributions; one due to the normal incident component ${\mathbf{\text{v}}}_n$ and another due to the tangential incident component ${\mathbf{\text{v}}}_t$ . They are denoted ${{\mathbf{v}}^{\prime }}_n$ and ${{\mathbf{\text{v}}}^{\prime }}_t$ respectively.

  The magnitude of ${{\mathbf{\text{v}}}^{\prime }}_n$ for each secondary parcel is calculated from the overall conservation of energy as described in [650]. The direction of ${{\mathbf{\text{v}}}^{\prime }}_n$ is chosen by randomly selecting an azimuthal angle in the range [0°, 360°] and an ejection angle from the wall in a user-specified range, default [5°, 50°].

  If there is insufficient energy to provide the proposed secondary parcels with both surface energy and kinetic energy, the splashing is abandoned. Instead, the impinging droplet simply spreads.

  The ${\mathbf{\text{v}}}_t$ component contributes directly to the secondary droplet velocity as

  $${{\mathbf{\text{v}}}^{\prime }}_t=c_f{\mathbf{\text{v}}}_t$$

  (3191)

  where $c_f$ is the wall friction coefficient which is estimated to be in the range 0.6 to 0.8; its default value is set to 0.7.

  Finally, the parent parcel sticks to the wall. Its mass is reduced by a factor $\left(1-r_m\right)$ , while the number of droplets which it represents remains unchanged.

Temperature Range 2

The second temperature range is expressed as:

$$T_{12}<T_w\le T_{23}$$

(3192)

In this range, there is no contact or deposition due to an intervening vapor film. The regimes are determined by two characteristic Weber numbers ${\text{We}}_{T1}$ and ${\text{We}}_{T2}$ as follows:

• Rebound:

  $${\text{We}}_I\le {\text{We}}_{T1}$$

  (3193)
• Break-up and Rebound:

  $${\text{We}}_{T1}<{\text{We}}_I\le {\text{We}}_{T2}$$

  (3194)

  in which the incident droplets shatter and rebound from the wall. This regime is implemented as a variant of the splash regime in the first temperature range except that no residue is left on the wall ( $r_m=1$ ) and ${\text{We}}_m={\text{We}}_{T1}$ .

• Break-up and Spread:

  $${\text{We}}_{T2}<{\text{We}}_I\le {\text{We}}_c$$

  (3195)

  in which the incident droplets shatter and spread along the wall. This regime is implemented as a variant of the splash regime in the first temperature range except that no residue is left on the wall ( $r_m=1$ ), ${\text{We}}_m={\text{We}}_{T2}$ , and the ejection angle is zero.

• Splash:

  $${\text{We}}_I>{\text{We}}_c$$

  (3196)

  which is identical to the splash regime in the first temperature range except that no residue is left on the wall ( $r_m=1$ ).

Following [648], default values of 30 and 80 for water are used for ${\text{We}}_{T1}$ and ${\text{We}}_{T2}$ respectively.

Temperature Range 3

The third temperature range is expressed as:

$$T_w>T_{23}$$

(3197)

Again, wall contact is prevented by a vapor film and the droplets are elevated from and move tangentially to the surface. The regimes are as follows

• Spread:

  $${\text{We}}_I\le {\text{We}}_{T1}$$

  (3198)
• Break-up and Spread:

  $${\text{We}}_{T1}<{\text{We}}_I\le {\text{We}}_c$$

  (3199)

  in which the incident droplets shatter and spread along the wall. This regime is implemented as a variant of the splash regime in the first temperature range except that no residue is left on the wall ( $r_m=1$ ), ${\text{We}}_m={\text{We}}_{T1}$ , and the ejection angle is zero.

• Splash:

  $${\text{We}}_I>{\text{We}}_c$$

  (3200)

  which is identical to the splash regime in the first temperature range except that no residue is left on the wall ( $r_m=1$ ).

Bai-ONERA Wall Impingement

The Bai-ONERA model is a development from the Bai-Gosman model. The model is designed to achieve a smooth transition to recover the Bai-Gosman model for situations where the wall temperature is lower than the droplet saturation temperature. It predicts the outcomes of droplets impacting on walls, with special attention to the results when the wall temperature is between the boiling point and the Leidenfrost point for the liquid.

The Bai-ONERA model categorizes possible outcomes as being in one of four possible regimes, illustrated in the following diagram.

For fluid film boundaries (wet walls), the momentum and kinetic energy that are lost by the droplets transfer to the film.

The choice of regime for a given impingement event is made using three parameters:

• A dimensionless temperature $T^{*}$ is defined as:
  $$T^{*}=\frac{T_w-T_s}{T_L-T_s}$$
  where:

  • $T_w$ is the wall temperature
  • $T_s$ is the saturation temperature
  • $T_L$ is the Leidenfrost temperature
• The droplet Weber number $We$ that is based on the normal component of relative velocities between the wall and the droplet (see Eqn. (3177)).
• The Ohnesorge number $Oh$ defined as:
  $$Oh=\frac{{\mu }_p}{\sqrt{{\rho }_p\sigma D_p}}=\frac{1}{\sqrt{La}}$$
  where $La$ is the Laplace number (see Eqn. (3178)).

The Weber and Ohnesorge numbers combine into a single dimensionless parameter $K$ defined as

Regime Transition Criteria

The maps of impingement regimes, which are shown below, are divided into three temperature ranges, which are separated by two transition temperatures:

• $T_{12}$ , separating range 1 from range 2, is expected to be approximately equal to the saturation temperature $T_s$ .
• $T_{23}$ , separating range 2 from range 3, is expected to be approximately equal to the Leidenfrost temperature $T_L$ .

The impact outcome map of the original ONERA model is shown in the following log K-T* plot. The transition between splashing and spreading is the function $K_s$ :

$$K_s\left(T^{*}\right)=\{\begin{array}{ccc}{K}_0\hfill & ;& T^{*}\le 0,T_w\le T_{12}\\ K_0+T^{*}(K_1-K_0)\hfill & ;& 0<T^{*}<1,T_{12}<T_w<T_{23}\\ K_1\hfill & ;& 1\le T^{*},T_{23}\le T_w\end{array}$$

(3202)

The transition between spreading and rebound is the function $K_r$ :

$$K_r\left(T^{*}\right)=\{\begin{array}{ccc}0\hfill & ;& T^{*}\le 0,T_w\le T_{12}\\ K_1{\cdot \left(T^{*}\right)}^{\gamma }\hfill & ;& 0<T^{*}<1,T_{12}<T_w<T_{23}\\ K_1\hfill & ;& 1\le T^{*},T_{23}\le T_w\end{array}$$

(3203)

There are three model coefficients ( $K_0$ , $K_1$ , and $\gamma$ ) in the original ONERA model. The Bai-ONERA model as implemented in Simcenter STAR-CCM+ allows a smooth transition to the Bai-Gosman model in temperature range 1, and as a result, the value of $K_0$ is determined internally by the spread-splash transition criteria of the Bai-Gosman model. Both $K_1$ and $\gamma$ are model constants with default values of 450 and 3, respectively.

Smooth Dry Wall

For a smooth dry wall:

• $K_1=450$ .
• $\gamma =3$ .

Rough Dry Wall

For a rough dry wall in temperature range 1, $K_0={We}_A{Oh}^{-0.4}$ , where ${We}_A=A{La}^{-0.18}$ . The coefficient $A$ is taken from the following table:

Wall roughness	0.05 µm	0.14 µm	0.84 µm	3.1 µm	12.0 µm
Coefficient A	5264	4534	2634	2056	1322

As a result, the transition $K_s$ lies between the two dotted lines in the following figure, the exact position depending on the wall roughness.

Wet Wall

For a wet wall in temperature range 1, $K_r\le 20.0{La}^{0.20}$ separates the rebound and the spread regimes, which are shown by the dotted line in the following figure. The actual location of the line depends on the Laplace number $La$ of the given droplet. The dotted line extends into temperature range 2 until it merges with the $K_r$ function defined in Eqn. (3203).

In addition, to maintain compatibility with the Bai-Gosman model, the Bai-ONERA model includes an adhere regime, for droplets with low impacting Weber number in temperature range 1.

The Splash Ratio

The splash ratio $r_s$ is the ratio between the splashed and incident droplet masses:

$$r_s=\frac{m_s}{m_i}$$

(3204)

where $m_s$ is the splashed mass and $m_i$ is the incident mass.

The droplet mass that is spread on the surface is $m_{spr}=m_i-m_s=(1-r_s)m_i$ .

To adjust $r_s$ to reflect the fact that there is no spreading above the Leidenfrost temperature, introduce a correction factor $f\left(T^{*}\right)$ with the following characteristics:

• $m_{spr}=(1-r_s)m_{i}f\left(T^{*}\right)$ .
• $f$ is monotonic.
• $f=1$ at $T^{*}=0(=T_{12}^{*})$ .
• $f=0$ at $T^{*}=1(=T_{23}^{*})$ .

Define $f$ as:

$$\begin{array}{c}\\ \begin{array}{c}f\left(T^{*}\right)\\ \end{array}\end{array}=\{\begin{array}{ccc}1.0& ;& T^{*}\le 0,T\le T_{12}\\ 1-{\left(T^{*}\right)}^n& ;& 0<T^{*}<1,T_{12}<T<T_{23}\\ 0.0& ;& 1\le T^{*},T_{23}\le T\end{array}$$

(3205)

where $n$ is the Smrf property of the Bai-ONERA model, controlling the shape of variation.

The amount of splashed mass is then:

$$m_s=m_i[1-(1-r_s)f\left(T^{*}\right)]$$

(3206)

Satoh Wall Impingement Model

The Satoh model aims to predict the outcomes of oil droplets impacting solid boundaries, particularly in oil-mist separators.

In the Satoh model [695], droplets impinging on boundaries are assumed to either rebound or spread depending on their diameter $D$ and velocity normal to the wall $Vp$ .

An intermediate region where a combination of rebounding and spreading occurs is defined between two lines, Initial Spread Line (ISL) and Full Spread Line (FSL). These lines both follow power law curves.

• Initial Spread Line

  $${Vp}_{\text{ISL}}=a_{\text{ISL}}{\left(\frac{D}{D_{\text{ref}}}\right)}^{n_{\text{ISL}}}$$

  (3207)
• Full Spread Line

  $${Vp}_{\text{FSL}}=a_{\text{FSL}}{\left(\frac{D}{D_{\text{ref}}}\right)}^{n_{\text{FSL}}}$$

  (3208)

where $D_{\text{ref}}$ is the reference diameter, $a$ is the pre-exponent multiplier, and $n$ is the exponent value.

According to Satoh and others [695], the particle diameter and its velocity normal to the boundary determine the outcome of a collision between a particle and a boundary. The relationship between the two is assumed to follow two power law curves as depicted below. Below the lower curve, called the Initial Spread Line, all the particles rebound. Above the upper curve, called the Full Spread Line, the particles spread over the boundary. In the area between the curves some particles rebound and the rest spread.

The droplet diameter is in the range 0 m to 8E-6 m. The droplet impingement velocity normal to the wall is in the range 0.01 m/s to 13 m/s. In this range, small droplets with low normal velocity rebound from the wall whereas large droplets with high normal velocity tend to spread over the surface to form a film.

In Simcenter STAR-CCM+, fluid film formation is dependent on both model selection and the boundary type. An order of precedence is used to model spreading behavior: if the Fluid Film model is not available, Stick mode is used. If neither mode is applicable on the incident boundary, the particle is allowed to Escape.

Rebound

Rebound mode is available only at walls, baffles, and contact interfaces. Rebounding particles remain active in the simulation; the mode is distinguished by its treatment of the particle velocity. The rebound velocity relative to the wall velocity is determined by the impingement velocity and user-defined restitution coefficients:

The superscripts R and I denote rebound and impingement respectively; the subscripts n and t denote wall-normal and tangential respectively. Since the left-hand side of Eqn. (3209) can be split into orthogonal n and t components, it can be split into two equations:

which serve to emphasize the definition of the restitution coefficients as the constants of proportionality between impingement and rebound velocities. Both coefficients may range from 0 to 1. The latter, “perfect” elastic rebound, is the default.

The tangential velocity of a wall boundary is zero unless a value is explicitly prescribed through a wall sliding option. In other words, it is non-zero only at no-slip walls.

Splash

Splash mode is available only for droplets and solid walls. In this mode, droplets impinge on a solid or wetted surface and break up into multiple rebounding droplets.

The magnitude of the velocity of splashed particles is defined by velocity ratio and its direction by the splashing angle $\alpha$ and the tangential velocity ratio $R_t$ .

If the tangential velocity ratio $R_t$ = 0 or the impingement velocity is perpendicular to the surface, the directions for the splashed particles are randomly chosen on the surface of the circular cone.

Splashing angle $\alpha$ defines the half cone angle, and the cone axis is perpendicular to the surface at the impact.

If the ratio is positive, the cone axis is inclined in the direction of the tangential component $u_{p,t}$ of the impingement velocity.

The inclination angle $\theta$ depends on tangential velocity ratio $R_t$ , the impingement angle $\phi$ and the splashing angle $\alpha$ :

Impingement Heat Transfer

The Impingement Heat Transfer model is designed to predict the transfer of heat between a wall and a stream of droplets impinging on it. In a simulation with two-way coupling, the droplets can change the temperature of the wall.

This temperature change can happen even in cases where the wall is above the Leidenfrost temperature. Heat can be transferred between droplets and wall through the gas phase, albeit more slowly than without the Leidenfrost effect. The droplets can then cool the wall below the Leidenfrost point, so that liquid films can form.

The model makes the following assumptions:

• The area through which energy flows from the droplet to the wall face is no larger than the area of the wall face.
• For a wall boundary condition where the temperature is fixed (one-way coupling), the heat transfer is driven by the difference between the particle temperature and the wall temperature. The thermal conductivity in one-way coupling is that of the droplet material.
• The liquid droplet distorts during impact and increases its effective diameter to an amount given by Eqn. (3216) from which an effective area is calculated.
• A diffuse spray is assumed.
• The droplet is absorbed into the film directly if the film thickness on the face is greater than 1E-8 meters.

At high mass loadings (seen with volume fractions larger than 0.3), these assumptions can break down, resulting in odd behavior in the gas phase. This behavior includes, but is not limited to, spurious temperature fluctuations, unphysical particle temperatures, and general difficulty converging.

The heat penetration coefficient for the solid wall, $b_w$ , in equation Eqn. (3221) is multiplied by the HPC Multiplier. This can be used to model a coating or resistance in the wall in transient simulations using constant, field function or table methods.

The energy equation for the droplet or particle is:

where:

• ${\dot{Q}}_c$ is the rate of heat transfer from convection.
• ${\dot{Q}}_r$ is the rate of heat transfer from radiation.
• ${\dot{Q}}_l$ is the rate of heat transfer from mass transfer.
• ${\dot{Q}}_s$ is the rate of heat transfer from user-defined heat sources.
• ${\dot{Q}}_{wp}$ is the rate of heat transfer from wall conduction.

Heat that is transferred from the wall to the droplet is described by the Wruck correlation [714]:

where terms with a $w$ or a $p$ refer to the wall or particle, and $\Delta t_p$ is the particle time-step.

where $D_{\text{eff}}$ is an effective diameter, time-averaged over the droplet contact time (Akao et al. [642]):

where $D_0$ is the droplet diameter and ${We}_p$ is the droplet Weber number.

Contact Time

$t_{\text{cont}}$ is the contact time between particle and wall.

$$t_{\text{cont}}=\{\begin{array}{cc}\frac{\pi }{4}\sqrt{\frac{{\rho }_p{D_0}^3}{{\sigma }_p}}& K<40\\ \sqrt{\frac{\pi }{2}}{\left(\frac{{\rho }_p{D_0}^5}{{\sigma }_p{u_r}^2}\right)}^{0.25}& K\ge 40\end{array}$$

(3217)

where:

• ${\rho }_p$ is the particle density.
• ${\sigma }_p$ is the particle surface tension.
• $u_r$ is the velocity of the particle relative to the wall.
• $K$ is a dimensionless number proportional to the impact energy. Birkhold et al. [652] define this number as:

  $K={Ca}^{5/4}{La}^{3/4}$

  where $Ca$ is the Capillary number and $La$ is the Laplace number.

The following methods are available to calculate the contact time:

Aiko

$$t_{\text{cont}}^{\text{Aiko}}=A{t}_{\text{cont}}$$

(3218)

where $A$ is a constant. This is the default.

Birkhold

This method considers the effects of boiling on the heat transfer between droplet and wall. Upon onset of boiling, contact no longer exists between the droplet and the wall. The Birkhold method introduces the boiling delay time:

$$t_s=AT+B$$

(3219)

where $A$ and $B$ are empirical constants depending on the material of the wall and the droplet [652]. $t_s$ is supposed to decrease with increasing wall temperature. (The higher the wall temperature, the sooner boiling occurs.)

The contact time is calculated as:

$$t_{\text{cont}}^{\text{Birkhold}}=\text{min}(t_{\text{cont}},t_s)$$

(3220)

Heat Penetration Coefficients

$b_w$ and $b_p$ are the heat penetration coefficients as evaluated with the materials of the wall or the particle.

$$\begin{array}{cc}{b}_w=c_h\sqrt{{\kappa }_w{\rho }_w{\left(C_p\right)}_w}\text{,}& b_p=\sqrt{{\kappa }_p{\rho }_p{\left(C_p\right)}_p}\end{array}$$

(3221)

where:

• $c_h$ is the HPC Multiplier, default value 1.0, used to represent coating or other thermal resistance.
• ${\kappa }_w$ and ${\kappa }_p$ are the thermal conductivities of the wall and the particle.
• ${\rho }_w$ and ${\rho }_p$ are the densities of the wall and the particle.
• ${\left(C_p\right)}_w$ and ${\left(C_p\right)}_p$ are the specific heats of the wall and the particle.

When there is no conducting solid, the wall temperature is used to calculate the heat flux with the Wruck correlation. Since there is no material specified for the wall, $b_w$ is set to a very large number, reducing the harmonic average to $b_p$ .

Wall and Droplet Temperatures

$T_w$ and $T_p$ are the temperatures of the wall and the droplet.

Eqn. (3214) can be written as:

$${\dot{Q}}_{wp}=A_p{h}_{wp}\left(T_w-T_p\right)$$

(3222)

where:

• $A_p$ is the cross-sectional area of the particle.
• $h_{wp}$ is:

  $$h_{wp}=\frac{2A_{\text{cont}}\sqrt{t_{\text{cont}}}\left(b_w{b}_p\right)}{A_p\sqrt{\pi }(b_w+b_p)\Delta t_p}$$

  (3223)

Film Damping

Film Damping mode allows the motion of solid particles to be damped immediately before and after wall impingement as they pass through a fluid film. This mode is only available for solid particles on fluid film boundaries.

In this situation, the equation of conservation of momentum for a material particle is assumed to reduce to:

where:

• ${\mathbf{\text{F}}}_d$ is the drag force
• ${\mathbf{\text{F}}}_{vm}$ is the virtual mass force

Drag Force

The equation for drag force can be used in its standard form. As the particles are small, spherical, and solid, the drag coefficient of the particle, $C_d$ can be calculated using the Schiller-Naumann correlation.

Virtual Mass Force

The equation for virtual mass force is simplified using two assumptions:

• The virtual mass coefficient $C_{vm}$ is equal to 0.5
• The acceleration of the fluid film is negligible compared to the acceleration at the particle:

$$\left|\frac{D\mathbf{\text{v}}}{Dt}\right|«\left|\frac{d{\mathbf{\text{v}}}_p}{dt}\right|$$

(3225)

Using these assumptions the virtual mass force becomes:

$${\mathbf{\text{F}}}_{vm}=-0.5\rho V_p\frac{d{\mathbf{\text{v}}}_p}{dt}$$

(3226)

Droplet-to-Film Transition Criteria

The Droplet Film Transition model determines whether droplets impinging on a wall form a liquid film on the wall.

The Droplet Film Transition model calculates either the coverage ratio or the equivalent film thickness, depending on which criterion is selected. The coverage ratio criterion is generally more suited to internal combustion simulations with high Weber numbers. The equivalent film thickness criterion is more generally applicable.

Coverage Ratio

The coverage ratio is the fraction of the cell wall that is covered by impinging droplets. On impact, a droplet is taken to assume a cylindrical form with a spreading diameter $D_s$:

$$D_s=0.613D_d{We}_d^{0.39}$$

(3227)

where:

• $D_d$ is the droplet diameter.
• ${We}_d$ is the Weber number for the droplet.

The area that is covered by the droplets is then $A_s$:

$$A_s=0.25N\pi D_s^2$$

(3228)

where $N$ is the droplet number density. All droplets merge into a liquid film when $A_s$ reaches the value that is given for the Coverage Ratio property.

The coverage ratio is then ${\gamma }_c=A_s/A_c$ where $A_c$ is the area of the cell wall.

Equivalent Film Thickness

The equivalent film thickness is $\delta e$:

$$\delta e=\frac{V_d}{A_c}$$

(3229)

where $V_d$ is the total volume of all the droplets on the cell wall. All droplets merge into a liquid film when $\delta e$ reaches the value that is given for the Equivalent Film Thickness property.

Heat and Mass Transfer for Wall-Adhering Droplets

Heat transfer between a stuck droplet and a wall can be accounted for in transition boiling simulations by activating the Stuck Droplet Heat Transfer model. For cases where the wall temperature is higher than droplet's saturation temperature, bulk nucleate boiling may also be induced if the Stuck Droplet Mass Transfer model is activated.

Heat Transfer

Heat transfer for droplets that are stuck to a wall is calculated using the method of Eckhause and Reitz [656]. On impacting a wall, a stuck droplet (or parts thereof) is assumed to spread into a cylindrical disc with an effective spread diameter $D_s$ given by:

$$D_s=0.61D_p{We}_p^{0.38}$$

(3230)

where:

• $D_p$ is the diameter of the impacting droplet.
• ${We}_p$ is a variant of the droplet Weber number that uses droplet density and normal-to-wall component of the relative velocity between the wall and the droplet:

  $${We}_p=\frac{{\rho }_p{|{\mathbf{v}}_w-{\mathbf{v}}_p|}_n^2{D}_p}{{\sigma }_p}$$

  (3231)

where ${\sigma }_p$ is the surface tension of the droplet.

The thickness of the spread droplet is calculated from the droplet volume and spread diameter. It is assumed that the height-to-diameter ratio of the cylindrical disc remains invariant in cases where the droplet mass changes due to evaporation/condensation.

The generic form of the energy balance equation for a stuck droplet is:

$$m_p{C}_p\frac{d{T}_p}{dt}=Q_c+Q_w+Q_s+{\dot{m}}_p{L}_{\mathrm{e}\mathrm{f}\mathrm{f}}$$

(3232)

where:

• $m_p$, $C_p$, and $T_p$ are the mass, the specific heat, and the temperature of the droplet.
• $Q_c$ is the rate of convective heat transfer from the continuum phase.
• $Q_w$ is the rate of heat transfer from a wall.
• $Q_s$ is the rate of energy addition from other models or sources.
• ${\dot{m}}_p$ is the rate of mass transfer to the particle.
• $L_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective latent heat of the transferred material.

Heat Transfer from the Continuum Phase $Q_c$

The rate of heat transfer from the continuum phase is:

$$Q_c=h_g{A}_s(T_g-T_p)$$

(3233)

$$h_g=\mathrm{N}\mathrm{u}\left(\frac{k_g}{D_s}\right)$$

$$\mathrm{N}\mathrm{u}=0.664{\mathrm{Re}}_s^{1/2}{\mathrm{Pr}}^{1/3}$$

$${\mathrm{Re}}_s=\frac{{\rho }_g|{\mathbf{v}}_g-{\mathbf{v}}_p|D_s}{{\mu }_g}$$

where:

• $h_g$ is the heat transfer coefficient.
• $A_s$ is the droplet surface area.
• $k_g$ is the thermal conductivity of the continuum phase.
• ${\mathrm{Re}}_s$, $\mathrm{Pr}$, and $\mathrm{N}\mathrm{u}$ are the Reynolds, Prandtl, and Nusselt numbers. The Nusselt number is calculated from the Pohlhausen correlation for a laminar flow over a flat plate. [651]

Heat Transfer from the Wall

Conduction is the sole heat transfer mode in absence of the Stuck Droplet Mass Transfer model (see Case 1 below). With the Stuck Droplet Mass Transfer model activated, three cases are distinguished depending on the temperature of the wall $T_w$ in relation to the saturation temperature $T_{\mathrm{s}\mathrm{a}\mathrm{t}}$ and the Leidenfrost temperature $T_L$ :

Case 1: $T_w\le T_{\mathrm{s}\mathrm{a}\mathrm{t}}$

Conductive heat transfer is modeled according to:

$$Q_w=h_w{A}_s(T_w-T_p)$$

$$h_w=k_p/\delta$$

where $\delta$ is the thickness of the spread droplet and $k_p$ is the droplet thermal conductivity.

Case 2: $T_{\mathrm{s}\mathrm{a}\mathrm{t}}{<T}_w\le T_L$

For cases where $T_w>T_{\mathrm{s}\mathrm{a}\mathrm{t}}$, a droplet experiences the same heat transfer modes as in pool boiling heat transfer, and depending on the magnitude of the excess temperature, $\delta T_w=T_w-T_{\mathrm{s}\mathrm{a}\mathrm{t}}$, there are four modes of heat transfer: natural convection, nucleate boiling, transition boiling, and film boiling. Simcenter STAR-CCM+ simulates the nucleate and the transition boiling modes. According to the Rohsenow pool boiling correlation, the heat flux passing through the wall to the droplet increases in proportion to the cubic power of $\delta T_w$:

$$q_b=\frac{L_{\mathrm{e}\mathrm{f}\mathrm{f}}{\mu }_l}{{\mathrm{Pr}}_l^n}{\left(\frac{g({\rho }_l-{\rho }_v)}{\sigma }\right)}^{\frac{1}{2}}{\left(\frac{C_{p,l}\delta T_w}{C_{sf}{L}_{\mathrm{e}\mathrm{f}\mathrm{f}}}\right)}^3$$

(3234)

where:

• Properties subscripted with $l$ and $v$ are those of the liquid droplet and the vapor phase, respectively.
• $L_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective latent heat of the transferred material.
• $\sigma$ is the surface tension of the liquid-vapor interface.
• $C_{sf}$ is a coefficient that depends on the surface-liquid combination.
• $g$ is the gravitational acceleration.
• $T_{\mathrm{s}\mathrm{a}\mathrm{t}}$ is the saturation temperature of the droplet.

The value of exponent $n$ to the Prandtl number also depends on the surface-liquid combination. For a water-brass contact, the values for $C_{sf}$ and $n$ are 0.006 and 3.0, respectively.

Note that $C_{sf}$ is raised to third power in the Rohsenow correlation and hence influences heat transfer greatly. Available data for $C_{sf}$ shows a big variation amongst different fluid-surface combinations.

The heat flux increases to a maximum value (critical heat flux) according to:

$$q_{\max }=C_{\max }{L}_{\mathrm{e}\mathrm{f}\mathrm{f}}{\rho }_v^{1/2}{\left(\sigma g[{\rho }_l-{\rho }_v]\right)}^{1/4}$$

(3235)

where $C_{\max }$ is a user-supplied coefficient, defaulted to 0.15.

The heat flux decreases linearly from $q_{\max }$ to its minimum $q_{\min }$ at the Leidenfrost temperature $T_L$, according to:

$$q_{\min }=C_{\min }{L}_{\mathrm{e}\mathrm{f}\mathrm{f}}{\rho }_v{\left(\sigma g[{\rho }_l-{\rho }_v]\right)}^{1/4}{({\rho }_l+{\rho }_v)}^{-1/2}$$

(3236)

where $C_{\min }$ is a user-supplied coefficient, defaulted to 0.09.

Equating $q_b$ and $q_{\max }$ and solve for $\delta T_w$ to obtains the excess wall temperature $\delta T_s$ at which the maximum heat flux is attained. Based on these correlations, the model calculates the rate of heat transfer from the wall as follows:

$$Q_w=A_s\left\{\begin{array}{cc}\hfill q_b,& \delta T_w<\delta T_s\hfill \\ \hfill q_{\max },& \delta T_w<C_s\delta T_s\hfill \\ \hfill q_{\min }+\frac{\delta T_L-\delta T_w}{\delta T_l-C_s\delta T_S}(q_{\max }-q_{\min }),& \delta T_w<\delta T_L\hfill \end{array}\right\}$$

(3237)

where $\delta T_L=T_L-T_{\mathrm{s}\mathrm{a}\mathrm{t}}$ and $C_s$ is a model constant with its default value set to 1.2.

Case 3: $T_w>T_L$

When the wall temperature is above the Leidenfrost temperature, the impacting droplet is assumed to be levitating on a vapor layer whose temperature is that of the carrier fluid. No direct heat transfer from the wall is accounted for under this condition: $Q_w=0$.

The total heat transfer $Q_c+Q_w+Q_s$ to the droplet is first applied to raise the droplet temperature to $T_{\mathrm{s}\mathrm{a}\mathrm{t}}$, if necessary; any excess amount is used to vaporize the droplet through latent heat transfer (${\dot{m}}_p$) if the optional Stuck Droplet Mass Transfer model is activated.

Mass Transfer

With the Stuck Droplet Mass Transfer model activated, two mass transfer sources should be considered: droplet evaporation as a result of nucleate boiling heat transfer and mass transfer from the continuum phase as described below.

Mass Transfer from the Continuum Phase

The mass transfer rate between a stuck droplet and the surrounding continuum is given by:

$${\dot{m}}_p=-\left(\frac{{\rho }_v{D}_{g}Sh}{D_s}\right)\left(\frac{\pi }{4}{D}_s^2\right)(Y_{\mathrm{s}\mathrm{a}\mathrm{t}}-Y_{\infty })$$

(3238)

$$\mathrm{S}\mathrm{h}=0.664{\mathrm{Re}}_s^{1/2}{\mathrm{S}\mathrm{c}}^{1/3}$$

where:

• ${\rho }_v$ is the droplet vapor density evaluated at the continuum phase temperature.
• $Y_{\mathrm{s}\mathrm{a}\mathrm{t}}$ the mass fraction of the particle vapor at the particle surface.
• $Y_{\infty }$ the mass fraction of the particle vapor in the surrounding carrier fluid.
• $D_g$ is the mass diffusivity of the continuum phase
• $\text{Sc}$ and $\text{Sh}$ are the Schmidt number and the Sherwood number, respectively.

The Sherwood number is calculated according to the Pohlhausen correlation [651].