Particle Heat and Mass Transfer

Particle Mass Balance

The equation of conservation of mass of a material particle is given by:

Figure 1. EQUATION_DISPLAY

\frac{{dm}_{p}}{dt} = {\dot{m}}_{p}

(3010)

where $m_{p}$ is the mass of the particle and ${\dot{m}}_{p}$ the rate of mass transfer to the particle. The latter is zero unless mass transfer occurs, such as evaporation.

If the two-way coupling is active, ${\dot{m}}_{p}$ is accumulated over all the parcels and applied in the continuous phase continuity equation. See Two-Way Coupling with the Continuous Phase.

Droplet Evaporation

Quasi-steady single-component droplet evaporation assumes droplets to be internally homogeneous, consisting of a single liquid component such as a chemical species.

The rate of change of droplet mass due to quasi-steady evaporation ${\dot{m}}_{p}$ can be written as [706]:

Figure 2. EQUATION_DISPLAY

{\dot{m}}_{p} = {- g}^{*} A_{s} \ln (1 + B)

(3011)

where $B$ is the Spalding transfer number and $g^{*}$ is the mass transfer conductance (to be precise, in the limit $B \to 0$ ).

Multi-component droplet evaporation also assumes droplets to be internally homogeneous, consisting of an ideal mixture of liquid components, such as different chemical species, some of which are transferred. This means that they can vaporize. There can also be inert components in both the droplet and the gas.

For multi-component droplet evaporation, the rate of change of mass of each transferred component due to quasi-steady evaporation ${\dot{m}}_{p i}$ can be written as:

Figure 3. EQUATION_DISPLAY

{\dot{m}}_{p i} = {- ε_{i} g}^{*} A_{s} \ln (1 + B)

(3012)

where the index $i$ refers to each component of the mixture of components and $ε_{i}$ is the fractional mass transfer rate. The fractional mass transfer rate has the property:

Figure 4. EQUATION_DISPLAY

\sum_{T}^{} ε_{i} = 1

(3013)

with the summation over the T transferred components. Effectively, the transfer number represents the driving force for evaporation, which is a function of thermodynamic conditions of the liquid and vapor. Conductance, on the other hand, represents geometrical and mechanical effects, such as the size and velocity of the droplet. Expressions for these quantities depend on the mode of evaporation, of which there are three:

Super-Critical
Heat Transfer Limited Evaporation
Vapor Diffusion Limited Evaporation

The conductance is positive by definition. The transfer number can be positive or negative, with the latter implying condensation, that is, condensation is treated as “reverse evaporation”.


Evaporation Mode	Single-Component Droplets	Multi-Component Droplets
Super-Critical This mode is active when the droplet temperature exceeds the critical temperature.	$T_{p} \geq T_{c}$ where $T_{p}$ is the particle temperature and $T_{c}$ is the critical temperature of the material.	$T_{p} \geq \max (T_{c, i})$ where $T_{c, i}$ is the critical temperature of the transferred components.
Heat Transfer Limited Evaporation This mode is active when the vapor at the droplet surface is saturated and subcritical. It enforces a balance between latent heat transfer and other modes of heat transfer so that the droplet temperature remains at the saturation temperature at the droplet surface. For cases where other modes of heat transfer are involved, such as radiation and user-supplied source, this balance is achieved by recalculating the evaporation rate required to balance other modes of heat transfer, thereby ensuring that the droplet temperature does not exceed the saturation temperature at the droplet surface. It should be noted that for cases where the saturation temperature is a function of local vapor pressure, the saturation temperature could increase over time as a result of strong evaporation.	The condition for a saturated vapor is that the surface equilibrium mole fraction of the vapor exceeds unity: $X_{v, s} = \frac{p_{s a t} (T_{p})}{p} \geq 1$ in which $p_{s a t} (T_{p})$ is the vapor saturation pressure at the droplet surface temperature. Under this condition, the transfer number is $B = \frac{c_{p} (T - T_{p})}{L}$ where $L$ is the latent heat of vaporization.	The first condition for a saturated vapor is: Figure 5. EQUATION_DISPLAY $\sum_{T}^{} X_{i s} \geq 1$ (3014) where $X_{i s}$ is the equilibrium mole fraction of transferred component i at the droplet surface. $X_{i s}$ is evaluated using Raoult’s law: $X_{i s} = \frac{γ_{i} p_{s a t}^{i} (T_{p})}{p} X_{i p}$ in which: $γ_{i}$ is the activity coefficient of component $i$ . The value of $γ_{i}$ is set to 1 for the default Raoult's law formulation. $p_{s a t}^{i} (T_{p})$ is the component saturation pressure that is evaluated at the droplet surface temperature. $X_{i p}$ is the component mole fraction in the droplet. When the condition in Eqn. (3014) is satisfied, the volatile components vaporize in proportion to their mass fractions, that is: $ε_{i} = \frac{Y_{i p}}{\sum_{T}^{} Y_{i p}}$ The second condition for a saturated vapor is: $\sum_{T}^{} Y_{i} = 1$ where $Y_{i}$ is the mass fraction of component i in the “free stream”, that is the cell containing the droplet. The transfer number is $B = \frac{c_{p} (T - T_{p})}{\sum_{T}^{} ε_{i} L_{i}}$ where $L_{i}$ is the component latent heat of vaporization.
	Finally, the conductance is $g^{*} = \frac{k {Nu}_{p}}{{c_{p} D}_{p}}$ where ${Nu}_{p}$ is the Nusselt number.
Vapor Diffusion Limited Evaporation This mode is active when the vapor at the droplet surface is subcritical and not saturated, that is, when neither of the other two modes is active. The evaporation rate then depends on the rate at which vapor can diffuse away from the droplet.	Under this condition, the transfer number is [704] Figure 6. EQUATION_DISPLAY $B = \frac{Y_{v, s} - Y_{v}}{1 - Y_{v, s}}$ (3015) where $Y_{v}$ is the vapor mass fraction in the “free stream”, that is, cell, and $Y_{v, s}$ is the surface equilibrium vapor mass fraction Figure 7. EQUATION_DISPLAY $Y_{v, s} = X_{v, s} \frac{W_{v}}{W_{s}}$ (3016) The molecular weights $W_{v}$ and $W_{s}$ are of the vapor and gas mixture at the droplet surface respectively.	From [704], the fractional mass transfer rate is: $ε_{i} = \frac{Y_{i s} (1 + B) - Y_{i}}{B}$ The transfer number is: $B = \frac{\sum_{T}^{} Y_{i s} - \sum_{T}^{} Y_{i}}{1 - \sum_{T}^{} Y_{i s}}$ where $Y_{i}$ is the mass fraction of component i in the “free stream”, that is the cell containing the droplet. $Y_{i s}$ is its surface equilibrium mass fraction: $Y_{i s} = X_{i s} \frac{W_{i}}{W_{s}}$
	Finally, the conductance is: Figure 8. EQUATION_DISPLAY $g^{*} = \frac{ρ D_{v} S h_{p}}{D_{p}}$ (3017) where $D_{v}$ is the molecular diffusivity of the vapor and $S h_{p}$ the Sherwood number.

Sherwood Number

The Sherwood number is effectively a dimensionless mass transfer conductance.

The Sherwood number is related to the actual mass transfer conductance through Eqn. (3017). As with the drag coefficient and heat transfer coefficient, the Sherwood number must be defined using a correlation.

Simcenter STAR-CCM+ provides the Ranz-Marshall correlation for defining the Sherwood number.

Ranz-Marshall Correlation

The Ranz-Marshall correlation [687] is suitable for spherical particles up to ${Re}_{p} ≃ 5000$ . It is formulated as

Figure 9. EQUATION_DISPLAY

{S h}_{p} = 2 (1 + 0.3 {Re}_{p}^{1 / 2} {S c}^{1 / 3})

(3018)

in which $Sc$ is the Schmidt number of the continuous phase. This correlation is available only when the continuous phase is viscous.

Particle Energy Balance

A material particle is assumed to be internally homogeneous which, from a thermal point of view, implies a low Biot number, for example, less than approximately 0.1.

The generic form of the equation of conservation of energy consistent with this assumption is

Figure 10. EQUATION_DISPLAY

m_{p} c_{p} \frac{d T_{p}}{d t} = Q_{t} + {Q_{r a d} + Q}_{s}

(3019)

Here, $Q_{t}$ represents the rate of convective heat transfer to the particle from the continuous phase, $Q_{r a d}$ represents the rate of radiative heat transfer, and $Q_{s}$ other heat sources.

Convective Heat Transfer

The convective heat transfer term is formulated as

Figure 11. EQUATION_DISPLAY

Q_{t} = f h A_{s} (T - T_{p})

(3020)

where $A_{s}$ is the particle surface area, and h is the heat transfer coefficient. The factor $f$ is a mass transfer correction, for which the formulation of El Wakil and others [657] is used

Figure 12. EQUATION_DISPLAY

f = \frac{z}{e^{z} - 1}

(3021)

with

Figure 13. EQUATION_DISPLAY

z = \frac{- {\dot{m}}_{p} c_{p}}{h A_{s}}

(3022)

In the limit ${\dot{m}}_{p} \to 0$ , $f \to 1$ .

The heat transfer to the particle is always active in the energy equation. It can also be written

Figure 14. EQUATION_DISPLAY

Q_{t} = \frac{{f m}_{p} c_{p} (T - T_{p})}{τ_{T}}

(3023)

where $τ_{T}$ is the thermal relaxation time-scale:

Figure 15. EQUATION_DISPLAY

τ_{T} = \frac{m_{p} c_{p}}{h A_{s}}

(3024)

If the Two-Way Coupling model is active, $Q_{t}$ is accumulated over all the parcels and applied in the continuous phase energy equation.

Heat Transfer Coefficient

As with the drag coefficient, the heat transfer coefficient h must be defined using a correlation.

The heat transfer coefficient is often given in terms of the particle Nusselt number:

Figure 16. EQUATION_DISPLAY

{Nu}_{p} \equiv \frac{h D_{p}}{k}

(3025)

where $k$ is the thermal conductivity of the continuous phase.

Simcenter STAR-CCM+ provides the Ranz-Marshall correlation for defining the heat transfer coefficient.

Ranz-Marshall Correlation

The Ranz-Marshall correlation [687] is suitable for spherical particles up to ${Re}_{p} ≃ 5000$ . It is formulated as:

Figure 17. EQUATION_DISPLAY

{Nu}_{p} = 2 (1 + 0.3 {Re}_{p}^{1 / 2} {Pr}^{1 / 3})

(3026)

in which $\Pr$ is the Prandtl number of the continuous phase. This correlation is available only when the continuous phase is viscous.

Radiative Heat Transfer

If the Particle Radiation model is active, heat transfer due to radiation is included in the energy equation. The radiative heat transfer, $Q_{r a d}$ , to the particle is defined as:

Figure 18. EQUATION_DISPLAY

Q_{r a d} = \frac{A_{s}}{4} Q_{a, p} (G - 4 σ T_{p}^{4})

(3027)

where $A_{s}$ is the particle surface area, $Q_{a, p}$ is the absorption efficiency of the particle, $G$ is the incident radiative heat flux, and $σ$ is the Stefan-Boltzmann constant.

Other Sources of Heat Transfer

The other terms in the energy equation are the user-defined energy source and the latent heat transfer:

Figure 19. EQUATION_DISPLAY

Q_{s} = V_{p} q_{u} + {\dot{m}}_{p} L_{e f f}

(3028)

where $q_{u}$ is the user energy source (per unit volume) and $L_{e f f}$ is the effective latent heat of transferred material, which depends on the mass transfer model and the composition of the transferred material. The user energy source is activated in the energy equation by the User-Defined Energy Source model.

Multiphase Mixture Evaporation

The formulation of the Multiphase Mixture Evaporation model is identical to the Multi-Component Droplet Evaporation model.

REA Spray Drying Evaporation

Spray drying is a method of producing a dried powder from a solid-containing liquid feed by rapid removal of excess moisture with a hot drying gas. The REA (Reaction Engineering Approach) model was first introduced by Chen & Xie [654]. It is a semi-empirical model developed specifically for the simulation of the drying processes of solid-containing materials such as milk droplets that were injected into a dryer. The REA model controls the moisture removal rate by correlating the instantaneous surface vapor density $ρ_{v, s}$ with its saturated value $ρ_{v, s}^{sat}$ :

Figure 20. EQUATION_DISPLAY

\frac{{d m}_{p}}{d t} = - h_{m} A_{p} (ρ_{v, s} - ρ_{v, b})

(3029)

Figure 21. EQUATION_DISPLAY

ρ_{v, s} = ψ ρ_{v, s}^{sat}

(3030)

where:

$m_{p}$ is the mass of the particle.
$h_{m}$ is the mass transfer coefficient.
$ρ_{v, b}$ is the partial density of the moisture vapor in the bulk drying gas.
$ψ$ is the rate reduction factor

During the initial stage of drying, the particle is entirely coated with liquid and, by equilibrium assumption, the vapor density is equal to its saturated value, leading to $ψ = 1$ . As a result, the formulation for evaporation of a pure liquid droplet is recovered under this condition. Progressively lower values of $ψ$ are expected as the drying continues due to increased solid fraction on the particle surface that hinders moisture removal.

The rate reduction factor is modeled in REA according to:

Figure 22. EQUATION_DISPLAY

ψ = exp (- \frac{Δ E_{v}}{R_{u} T_{s}})

(3031)

where:

$Δ E_{v}$ is the apparent activation energy.
$T_{s}$ is the particle surface temperature.
$R_{u}$ is the universal gas constant.

The surface temperature is approximated with the particle temperature under the assumption of a small Biot number.

This formulation assumes that evaporation is an activation process, having to overcome an energy barrier as the result of, for example, the formation of a porous crust on the particle surface. The apparent activation energy represents a measure of difficulty in extracting moisture, regardless of the drying stage and the causes of drying hindrance.

To further expand the applicability of the model for varying drying gas conditions, $Δ E_{v}$ is normalized with its equilibrium value of the drying gas. The equilibrium activation energy is defined as

Figure 23. EQUATION_DISPLAY

Δ E_{v, b} = - R_{u} T_{b} \ln (ρ_{v, b} / ρ_{v, b}^{sat} (T_{b}))

(3032)

where:

$T_{b}$ is the temperature of the bulk drying gas.
$ρ_{v, b}$ is the vapor density.
$ρ_{v, b}^{sat}$ is the vapor density saturation value.

The normalized activation energy $g$ is a function only of the deviation of the moisture content on the particle surface from its equilibrium value in the drying gas, $(X_{s} - X_{e q})$ :

Figure 24. EQUATION_DISPLAY

g (X_{s} - X_{e q}) = \frac{Δ E_{v}}{Δ E_{v, b}}

(3033)

The dry-basis moisture content $X$ is defined as the ratio of the mass of the evaporable liquids to the mass of the solid components (non-evaporable liquids are considered as solids in this case):

Figure 25. EQUATION_DISPLAY

X = \frac{m_{l}}{m_{p} - m_{l}}

(3034)

When a particle’s surface is initially saturated with liquids, the normalized activation energy is very low (close to zero). It gradually increases when the particle’s moisture content is reduced during drying, its value is expected to be unity at equilibrium. The normalized activation energy profile is considered as a characteristic (or fingerprint) of individual material, it is mainly influenced by the composition of the material and its initial moisture contents.

The Chen-Lin model [654] for the normalized activation energy is currently implemented in Simcenter STAR-CCM+:

Figure 26. EQUATION_DISPLAY

\frac{{Δ E}_{v}}{{Δ E}_{v, b}} = a \cdot exp (b \cdot {[X_{s} - X_{e q}]}^{c})

(3035)

The default model constants of $a$ = 0.998, $b$ = -1.405, and $c$ = 0.93 pertain to a 20 wt% skim milk solution. Correlations for differing material and initial compositions can be found from literature and implemented with user-defined field functions.

The equilibrium moisture content $X_{e q}$ is a function of the water activity $a_{w}$ of the drying gas which is constructed from experimental sorption isotherms of the material being dried:

Figure 27. EQUATION_DISPLAY

X_{e q} = f (a_{w})

(3036)

Water activity relates to water in food materials and is defined as the ratio of partial vapor pressure of water in the sample to the partial vapor pressure of pure water at the same temperature.

The GAB (Guggenheim-Anderson-de Boer) model [715] for the equilibrium moisture content is currently implemented in Simcenter STAR-CCM+:

Figure 28. EQUATION_DISPLAY

X_{e q} (a_{w}, T) = \frac{C \cdot K \cdot X_{m o} \cdot a_{w}}{(1 - K \cdot a_{w}) (1 - K \cdot a_{w} + C \cdot K \cdot a_{w})}

(3037)

Figure 29. EQUATION_DISPLAY

C = C_{0} exp (\frac{Δ H_{1}}{R T})

(3038)

Figure 30. EQUATION_DISPLAY

K = K_{0} exp (\frac{Δ H_{2}}{R T})

(3039)

The default values of the constants are:

$C_{0}$ = 1.645 x 10^-3
$K_{0}$ = 5.71
$Δ H_{1}$ = 2.4831 x 10⁷ [J/kmol]
$Δ H_{2}$ = -5.118 x 10⁴ [J/kmol]
$X_{m o}$ = 0.06156

These pertain to a 20 wt% skim milk solution. Correlations for differing material and initial compositions can be found from literature and implemented with user-defined field functions.

The mass transfer coefficient $h_{m}$ can be evaluated via appropriate Sherwood number correlations such as the Ranz-Marshall formula given in Eqn. (3018).