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22.9.2 Particle Types in FLUENT

The laws that you activate depend upon the particle type that you select. In the Set Injection Properties panel you will specify the Particle Type, and FLUENT will use a given set of heat and mass transfer laws for the chosen type. All particle types have predefined sequences of physical laws as shown in the table below:


Particle Type Description Laws Activated
Inert inert/heating or cooling 1, 6
Droplet heating/evaporation/boiling 1, 2, 3, 6
Combusting heating;
evolution of volatiles/swelling;
heterogeneous surface reaction
1, 4, 5, 6
Multicomponent multicomponent droplets/particles 7

In addition to the above laws, you can define your own laws using a user-defined function. See the separate UDF Manual for information about user-defined functions.

You can also extend combusting particles to include an evaporating/boiling material by selecting Wet Combustion in the Set Injection Properties panel.

FLUENT's physical laws (Laws 1 through 6), which describe the heat and mass transfer conditions listed in this table, are explained in detail in the sections that follow.



Inert Heating or Cooling (Law 1/Law 6)


The inert heating or cooling laws (Laws 1 and 6) are applied when the particle temperature is less than the vaporization temperature that you define, $T_{\rm vap}$, and after the volatile fraction, $f_{v,0}$, of a particle has been consumed. These conditions may be written as

Law 1:

 T_p < T_{\rm vap} (22.9-9)

Law 6:

 m_p \leq (1 - f_{v,0}) m_{p,0} (22.9-10)

where $T_p$ is the particle temperature, $m_{p,0}$ is the initial mass of the particle, and $m_p$ is its current mass.

Law 1 is applied until the temperature of the particle/droplet reaches the vaporization temperature. At this point a noninert particle/droplet may proceed to obey one of the mass-transfer laws (2, 3, 4, and/or 5), returning to Law 6 when the volatile portion of the particle/droplet has been consumed. (Note that the vaporization temperature, $T_{\rm vap}$, is an arbitrary modeling constant used to define the onset of the vaporization/boiling/volatilization laws.)

When using Law 1 or Law 6, FLUENT uses a simple heat balance to relate the particle temperature, $T_p (t)$, to the convective heat transfer and the absorption/emission of radiation at the particle surface:


 m_p c_p \frac{dT_p}{dt} = h A_p (T_{\infty} - T_p) + \epsilon_p A_p \sigma (\theta_R^4 - T_p^4) (22.9-11)


where      
  $m_p$ = mass of the particle (kg)
  $c_p$ = heat capacity of the particle (J/kg-K)
  $A_p$ = surface area of the particle (m $^2$)
  $T_{\infty}$ = local temperature of the continuous phase (K)
  $h$ = convective heat transfer coefficient (W/m $^2$-K)
  $\epsilon_p$ = particle emissivity (dimensionless)
  $\sigma$ = Stefan-Boltzmann constant (5.67 x 10 $^{-8}$ W/m $^2$-K $^4$)
  $\theta_R$ = radiation temperature, $(\frac{G}{4\sigma})^{1/4}$

Equation  22.9-11 assumes that there is negligible internal resistance to heat transfer, i.e., the particle is at uniform temperature throughout.

$G$ is the incident radiation in W/m $^2$:


 G = \int_{\Omega=4\pi} I d\Omega (22.9-12)

where $I$ is the radiation intensity and $\Omega$ is the solid angle.

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

Equation  22.9-11 is integrated in time using an approximate, linearized form that assumes that the particle temperature changes slowly from one time value to the next:


 m_{p} c_{p} \frac{ d T_{p}}{ d t} = A_{p} \left\{ - \left[ h... ...{\infty} + \epsilon_{p} \sigma \theta_{R}^{4} \right] \right\} (22.9-13)

As the particle trajectory is computed, FLUENT integrates Equation  22.9-13 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} (22.9-14)

where $\Delta t$ is the integration time step and


 \alpha_p = \frac{h T_{\infty} + \epsilon_p \sigma \theta_R^4 }{h + \epsilon_p \sigma T_p^3(t) } (22.9-15)

and


 \beta_p = \frac{A_p(h + \epsilon_p \sigma T_p^3(t) ) }{m_p c_p} (22.9-16)

FLUENT can also solve Equation  22.9-13 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section  22.11.7 for details.

The heat transfer coefficient, $h$, is evaluated using the correlation of Ranz and Marshall [ 296, 297]:


 {\rm Nu} = \frac{h d_p}{k_{\infty}} = 2.0 + 0.6 {\rm Re}_d^{1/2} {\rm Pr}^{1/3} (22.9-17)

where


       
  $d_p$ = particle diameter (m)
  $k_{\infty}$ = thermal conductivity of the continuous phase (W/m-K)
  Re $_d$ = Reynolds number based on the particle diameter and
      the relative velocity (Equation  22.2-3)
  Pr = Prandtl number of the continuous phase ( $c_p \mu / k_{\infty}$)

Finally, the heat lost or gained by the particle as it traverses each computational cell appears as a source or sink of heat in subsequent calculations of the continuous phase energy equation. During Laws 1 and 6, particles/droplets do not exchange mass with the continuous phase and do not participate in any chemical reaction.



Droplet Vaporization (Law 2)


Law 2 is applied to predict the vaporization from a discrete phase droplet. Law 2 is initiated when the temperature of the droplet reaches the vaporization temperature, $T_{\rm vap}$, and continues until the droplet reaches the boiling point, $T_{\rm bp}$, or until the droplet's volatile fraction is completely consumed:


 T_p < T_{\rm bp} (22.9-18)


 m_p > (1-f_{v,0}) m_{p,0} (22.9-19)

The onset of the vaporization law is determined by the setting of $T_{\rm vap}$, a modeling parameter that has no physical significance. Note that once vaporization is initiated (by the droplet reaching this threshold temperature), it will continue to vaporize even if the droplet temperature falls below $T_{\rm vap}$. Vaporization will be halted only if the droplet temperature falls below the dew point. In such cases, the droplet will remain in Law 2 but no evaporation will be predicted. When the boiling point is reached, the droplet vaporization is predicted by a boiling rate, Law 3, as described in a section that follows.

Mass Transfer During Law 2

During Law 2, the rate of vaporization is governed by gradient diffusion, with the flux of droplet vapor into the gas phase related to the gradient of the vapor concentration between the droplet surface and the bulk gas:


 N_i = k_c (C_{i,s} - C_{i,\infty}) (22.9-20)

where


  $N_i$ = molar flux of vapor (kgmol/m $^2$-s)
  $k_c$ = mass transfer coefficient (m/s)
  $C_{i,s}$ = vapor concentration at the droplet surface (kgmol/m $^3$)
  $C_{i,\infty}$ = vapor concentration in the bulk gas (kgmol/m $^3$)

Note that FLUENT's vaporization law assumes that $N_i$ is positive (evaporation). If conditions exist in which $N_i$ is negative (i.e., the droplet temperature falls below the dew point and condensation conditions exist), FLUENT treats the droplet as inert ( $N_i = 0.0$).

The concentration of vapor at the droplet surface is evaluated by assuming that the partial pressure of vapor at the interface is equal to the saturated vapor pressure, $p_{\rm sat}$, at the particle droplet temperature, $T_p$:


 C_{i,s} = \frac{p_{\rm sat} (T_p)}{R T_p} (22.9-21)

where $R$ is the universal gas constant.

The concentration of vapor in the bulk gas is known from solution of the transport equation for species $i$ or from the PDF look-up table for nonpremixed or partially premixed combustion calculations:


 C_{i, \infty} = X_i \frac{p}{R T_{\infty}} (22.9-22)

where $X_i$ is the local bulk mole fraction of species $i$, $p$ is the local absolute pressure, and $T_{\infty}$ is the local bulk temperature in the gas. The mass transfer coefficient in Equation  22.9-20 is calculated from the Sherwood number correlation [ 296, 297]:


 {\rm Sh}_{AB} = \frac{k_c d_p}{D_{i,m}} = 2.0 + 0.6 {\rm Re}_d^{1/2}{\rm Sc}^{1/3} (22.9-23)


where $D_{i,m}$ = diffusion coefficient of vapor in the bulk (m $^2$/s)
  Sc = the Schmidt number, $\frac{\mu}{\rho D_{i,m}}$
  $d_p$ = particle (droplet) diameter (m)

The vapor flux given by Equation  22.9-20 becomes a source of species  $i$ in the gas phase species transport equation, (see Section  22.14) or from the PDF look-up table for nonpremixed combustion calculations.

The mass of the droplet is reduced according to


 m_p (t + \Delta t) = m_p (t) - N_i A_p M_{w,i} \Delta t (22.9-24)


where $M_{w,i}$ = molecular weight of species $i$ (kg/kgmol)
  $m_p$ = mass of the droplet (kg)
  $A_p$ = surface area of the droplet (m $^2$)

FLUENT can also solve Equation  22.9-24 in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section  22.11.7 for details.

Defining the Vapor Pressure and Diffusion Coefficient

You must define the vapor pressure as a polynomial or piecewise linear function of temperature ( $p_{\rm sat}(T)$) during the problem definition. Note that the vapor pressure definition is critical, as $p_{\rm sat}$ is used to obtain the driving force for the evaporation process (Equations  22.9-20 and  22.9-21). You should provide accurate vapor pressure values for temperatures over the entire range of possible droplet temperatures in your problem. Vapor pressure data can be obtained from a physics or engineering handbook (e.g., [ 279]).

You must also input the diffusion coefficient, $D_{i,m}$, during the setup of the discrete phase material properties. Note that the diffusion coefficient inputs that you supply for the continuous phase are not used in the discrete phase model.

Heat Transfer to the Droplet

Finally, the droplet temperature is updated according to a heat balance that relates the sensible heat change in the droplet to the convective and latent heat transfer between the droplet and the continuous phase:


 m_p c_p \frac{d T_p}{dt} = h A_p (T_{\infty} - T_p) + \frac{... ...} h_{\rm fg} + A_p \epsilon_p \sigma ({\theta_R}^4 - {T_p}^4) (22.9-25)


where $c_p$ = droplet heat capacity (J/kg-K)
  $T_p$ = droplet temperature (K)
  $h$ = convective heat transfer coefficient (W/m $^2$-K)
  $T_{\infty}$ = temperature of continuous phase (K)
  $\frac{d m_{p}}{dt}$ = rate of evaporation (kg/s)
  $h_{\rm fg}$ = latent heat (J/kg)
  $\epsilon_p$ = particle emissivity (dimensionless)
  $\sigma$ = Stefan-Boltzmann constant (5.67 x 10 $^{-8}$ W/m $^2$-K $^4$)
  $\theta_R$ = radiation temperature, $(\frac{I}{4\sigma})^{1/4}$, where $I$ is the radiation intensity

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

The heat transferred to or from the gas phase becomes a source/sink of energy during subsequent calculations of the continuous phase energy equation.



Droplet Boiling (Law 3)


Law 3 is applied to predict the convective boiling of a discrete phase droplet when the temperature of the droplet has reached the boiling temperature, $T_{\rm bp}$, and while the mass of the droplet exceeds the nonvolatile fraction, ( $1 - f_{v,0}$):


 T_p \geq T_{\rm bp} (22.9-26)

and


 m_p > (1 - f_{v,0}) m_{p,0} (22.9-27)

When the droplet temperature reaches the boiling point, a boiling rate equation is applied [ 188]:


 \frac{d (d_p)}{dt} = \frac{4 k_{\infty}}{\rho_p c_{p,\infty... ...\frac{c_{p,\infty} (T_{\infty} - T_p) }{ h_{\rm fg} } \right ] (22.9-28)


where $c_{p,\infty}$ = heat capacity of the gas (J/kg-K)
  $\rho_p$ = droplet density (kg/m $^3$)
  $k_{\infty}$ = thermal conductivity of the gas (W/m-K)

Equation  22.9-28 was derived assuming steady flow at constant pressure. Note that the model requires $T_{\infty} > T_{{\rm bp}}$ in order for boiling to occur and that the droplet remains at fixed temperature ( $T_{{\rm bp}}$) throughout the boiling law.

When radiation heat transfer is active, FLUENT uses a slight modification of Equation  22.9-28, derived by starting from Equation  22.9-25 and assuming that the droplet temperature is constant. This yields


 - \frac{d m_p}{dt} h_{\rm fg} = h A_p (T_{\infty} - T_p) + A_p \epsilon_p \sigma ({\theta_R}^4 - {T_p}^4) (22.9-29)

or


 -\frac{d (d_p)}{dt} = \frac{2}{\rho_{p} h_{\rm fg} } \left[ ... ...p}) + \epsilon_{p} \sigma (\theta_{R}^{4} - T_{p}^{4}) \right] (22.9-30)

Using Equation  22.9-17 for the Nusselt number correlation and replacing the Prandtl number term with an empirical constant, Equation  22.9-30 becomes


 -\frac{d (d_p)}{dt} = \frac{2}{\rho_{p} h_{\rm fg} } \left[ ... ...p}) + \epsilon_{p} \sigma (\theta_{R}^{4} - T_{p}^{4}) \right] (22.9-31)

In the absence of radiation, this result matches that of Equation  22.9-28 in the limit that the argument of the logarithm is close to unity. FLUENT uses Equation  22.9-31 when radiation is active in your model and Equation  22.9-28 when radiation is not active. Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

The droplet is assumed to stay at constant temperature while the boiling rate is applied. Once the boiling law is entered it is applied for the duration of the particle trajectory. The energy required for vaporization appears as a (negative) source term in the energy equation for the gas phase. The evaporated liquid enters the gas phase as species $i$, as defined by your input for the destination species (see Section  22.14).



Devolatilization (Law 4)


The devolatilization law is applied to a combusting particle when the temperature of the particle reaches the vaporization temperature, $T_{\rm vap}$, and remains in effect while the mass of the particle, $m_p$, exceeds the mass of the nonvolatiles in the particle:


 T_p \geq T_{\rm vap} \;\; {\rm and}\;\; T_p \geq T_{\rm bp} (22.9-32)

and


 m_p > (1 - f_{v,0})(1 - f_{w,0}) m_{p,0} (22.9-33)

where $f_{w,0}$ is the mass fraction of the evaporating/boiling material if Wet Combustion is selected (otherwise, $f_{w,0} = 0$). As implied by Equation  22.9-32, the boiling point, $T_{\rm bp}$, and the vaporization temperature, $T_{\rm vap}$, should be set equal to each other when Law 4 is to be used. When wet combustion is active, $T_{\rm bp}$ and $T_{\rm vap}$ refer to the boiling and evaporation temperatures for the combusting material only.

FLUENT provides a choice of four devolatilization models:

Each of these models is described, in turn, below.

Choosing the Devolatilization Model

You will choose the devolatilization model when you are setting physical properties for the combusting-particle material in the Materials panel, as described in Section  22.14.2. By default, the constant rate model (Equation  22.9-34) will be used.

The Constant Rate Devolatilization Model

The constant rate devolatilization law dictates that volatiles are released at a constant rate [ 27]:


 - \frac{1}{f_{v,0}(1-f_{w,0}) m_{p,0}} \frac{d m_p}{dt} = A_0 (22.9-34)


where $m_p$ = particle mass (kg)
  $f_{v,0}$ = fraction of volatiles initially present in the particle
  $m_{p,0}$ = initial particle mass (kg)
  $A_0$ = rate constant (s $^{-1}$)

The rate constant $A_0$ is defined as part of your modeling inputs, with a default value of 12 s $^{-1}$ derived from the work of Pillai [ 284] on coal combustion. Proper use of the constant devolatilization rate requires that the vaporization temperature, which controls the onset of devolatilization, be set appropriately. Values in the literature show this temperature to be about 600 K [ 27].

The volatile fraction of the particle enters the gas phase as the devolatilizing species $i$, defined by you (see Section  22.14). Once in the gas phase, the volatiles may react according to the inputs governing the gas phase chemistry.

The Single Kinetic Rate Model

The single kinetic rate devolatilization model assumes that the rate of devolatilization is first-order dependent on the amount of volatiles remaining in the particle [ 16]:


 - \frac{d m_p}{dt} = k [ m_p - (1 - f_{v,0})(1 - f_{w,0}) m_{p,0} ] (22.9-35)


where $m_p$ = particle mass (kg)
  $f_{v,0}$ = mass fraction of volatiles initially present in the particle
  $f_{w,0}$ = mass fraction of evaporating/boiling material (if wet combustion
      is modeled)
  $m_{p,0}$ = initial particle mass (kg)
  $k$ = kinetic rate (s $^{-1}$)

Note that $f_{v,0}$, the fraction of volatiles in the particle, should be defined using a value slightly in excess of that determined by proximate analysis. The kinetic rate, $k$, is defined by input of an Arrhenius type pre-exponential factor and an activation energy:


 k = A_1 e^{-(E/RT)} (22.9-36)

FLUENT uses default rate constants, $A_1$ and $E$, as given in [ 16].

Equation  22.9-35 has the approximate analytical solution:


 m_p (t + \Delta t) = (1 - f_{v,0})(1- f_{w,0})m_{p,0} + [ m_p (t) - (1 -f_{v,0})(1 - f_{w,0} )m_{p,0} ] e^{-k \Delta t} (22.9-37)

which is obtained by assuming that the particle temperature varies only slightly between discrete time integration steps.

FLUENT can also solve Equation  22.9-37 in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section  22.11.7 for details.

The Two Competing Rates (Kobayashi) Model

FLUENT also provides the kinetic devolatilization rate expressions of the form proposed by Kobayashi [ 185]:


 {\cal R}_1 = A_1 e^{-(E_1/RT_p)} (22.9-38)


 {\cal R}_2 = A_2 e^{-(E_2/RT_p)} (22.9-39)

where ${\cal R}_1$ and ${\cal R}_2$ are competing rates that may control the devolatilization over different temperature ranges. The two kinetic rates are weighted to yield an expression for the devolatilization as


 \frac{m_v (t)}{(1-f_{w,0})m_{p,0} - m_a} = \int_0^t \left( \... ...int_0^{t} \left( {\cal R}_1 + {\cal R}_2 \right) dt \right) dt (22.9-40)


where $m_v (t)$ = volatile yield up to time $t$
  $m_{p,0}$ = initial particle mass at injection
  $\alpha_1, \alpha_2$ = yield factors
  $m_a $ = ash content in the particle

The Kobayashi model requires input of the kinetic rate parameters, $A_1$, $E_1$, $A_2$, and $E_2$, and the yields of the two competing reactions, $\alpha_1$ and $\alpha_2$. FLUENT uses default values for the yield factors of 0.3 for the first (slow) reaction and 1.0 for the second (fast) reaction. It is recommended in the literature [ 185] that $\alpha_1$ be set to the fraction of volatiles determined by proximate analysis, since this rate represents devolatilization at low temperature. The second yield parameter, $\alpha_2$, should be set close to unity, which is the yield of volatiles at very high temperature.

By default, Equation  22.9-40 is integrated in time analytically, assuming the particle temperature to be constant over the discrete time integration step. FLUENT can also solve Equation  22.9-40 in conjunction with the equivalent heat transfer equation using a stiff coupled solver. See Section  22.11.7 for details.

The CPD Model

In contrast to the coal devolatilization models presented above, which are based on empirical rate relationships, the chemical percolation devolatilization (CPD) model characterizes the devolatilization behavior of rapidly heated coal based on the physical and chemical transformations of the coal structure [ 110, 111, 128].

General Description

During coal pyrolysis, the labile bonds between the aromatic clusters in the coal structure lattice are cleaved, resulting in two general classes of fragments. One set of fragments has a low molecular weight (and correspondingly high vapor pressure) and escapes from the coal particle as a light gas. The other set of fragments consists of tar gas precursors that have a relatively high molecular weight (and correspondingly low vapor pressure) and tend to remain in the coal for a long period of time during typical devolatilization conditions. During this time, reattachment with the coal lattice (which is referred to as crosslinking) can occur. The high molecular weight compounds plus the residual lattice are referred to as metaplast. The softening behavior of a coal particle is determined by the quantity and nature of the metaplast generated during devolatilization. The portion of the lattice structure that remains after devolatilization is comprised of char and mineral-compound-based ash.

The CPD model characterizes the chemical and physical processes by considering the coal structure as a simplified lattice or network of chemical bridges that link the aromatic clusters. Modeling the cleavage of the bridges and the generation of light gas, char, and tar precursors is then considered to be analogous to the chemical reaction scheme shown in Figure  22.9.2.

Figure 22.9.2: Coal Bridge
figure

The variable $\pounds$ represents the original population of labile bridges in the coal lattice. Upon heating, these bridges become the set of reactive bridges, $\pounds^{*}$. For the reactive bridges, two competing paths are available. In one path, the bridges react to form side chains, $\delta$. The side chains may detach from the aromatic clusters to form light gas, $g_1$. As bridges between neighboring aromatic clusters are cleaved, a certain fraction of the coal becomes detached from the coal lattice. These detached aromatic clusters are the heavy-molecular-weight tar precursors that form the metaplast. The metaplast vaporizes to form coal tar. While waiting for vaporization, the metaplast can also reattach to the coal lattice matrix (crosslinking). In the other path, the bridges react and become a char bridge, $c$, with the release of an associated light gas product, $g_2$. The total population of bridges in the coal lattice matrix can be represented by the variable $p$, where $p=\pounds + c$.

Reaction Rates

Given this set of variables that characterizes the coal lattice structure during devolatilization, the following set of reaction rate expressions can be defined for each, starting with the assumption that the reactive bridges are destroyed at the same rate at which they are created ( $\frac{\partial{\pounds^{*}}}{\partial{t}} = 0$):


$\displaystyle \frac{d \pounds}{d t}$ $\textstyle =$ $\displaystyle - k_b \pounds$ (22.9-41)
$\displaystyle \frac{d c}{d t}$ $\textstyle =$ $\displaystyle k_b \frac{\pounds}{\rho + 1}$ (22.9-42)
$\displaystyle \frac{d \delta}{d t}$ $\textstyle =$ $\displaystyle \left[ 2 \rho k_b \frac{\pounds}{\rho + 1} \right] -k_g \delta$ (22.9-43)
$\displaystyle \frac{d g_1}{d t}$ $\textstyle =$ $\displaystyle k_g \delta$ (22.9-44)
$\displaystyle \frac{d g_2}{d t}$ $\textstyle =$ $\displaystyle 2 \frac{dc}{dt}$ (22.9-45)

where the rate constants for bridge breaking and gas release steps, $k_b$ and $k_g$, are expressed in Arrhenius form with a distributed activation energy:


 k = A e^{-(E \pm E_{\sigma})/RT} (22.9-46)

where $A$, $E$, and $E_{\sigma}$ are, respectively, the pre-exponential factor, the activation energy, and the distributed variation in the activation energy, $R$ is the universal gas constant, and $T$ is the temperature. The ratio of rate constants, $\rho = k_{\delta}/k_c$, is set to 0.9 in this model based on experimental data.

Mass Conservation

The following mass conservation relationships are imposed:


$\displaystyle g$ $\textstyle =$ $\displaystyle g_1 + g_2$ (22.9-47)
$\displaystyle g_1$ $\textstyle =$ $\displaystyle 2f - \sigma$ (22.9-48)
$\displaystyle g_2$ $\textstyle =$ $\displaystyle 2(c-c_0)$ (22.9-49)

where $f$ is the fraction of broken bridges ( $f=1-p$). The initial conditions for this system are given by the following:


$\displaystyle c(0)$ $\textstyle =$ $\displaystyle c_0$ (22.9-50)
$\displaystyle \pounds(0)$ $\textstyle =$ $\displaystyle \pounds_0 = p_0-c_0$ (22.9-51)
$\displaystyle \delta(0)$ $\textstyle =$ $\displaystyle 2f_0 = 2(1- c_0- \pounds_0)$ (22.9-52)
$\displaystyle g(0)$ $\textstyle =$ $\displaystyle g_1(0) = g_2(0) = 0$ (22.9-53)

where $c_0$ is the initial fraction of char bridges, $p_0$ is the initial fraction of bridges in the coal lattice, and $\pounds_0$ is the initial fraction of labile bridges in the coal lattice.

Fractional Change in the Coal Mass

Given the set of reaction equations for the coal structure parameters, it is necessary to relate these quantities to changes in coal mass and the related release of volatile products. To accomplish this, the fractional change in the coal mass as a function of time is divided into three parts: light gas ( $f_{\rm gas}$), tar precursor fragments ( $f_{\rm frag}$), and char ( $f_{\rm char}$). This is accomplished by using the following relationships, which are obtained using percolation lattice statistics:


$\displaystyle f_{\rm gas} (t)$ $\textstyle =$ $\displaystyle \frac{r (g_1 + g_2)(\sigma+1)}{4+2r(1-c_0)(\sigma+1)}$ (22.9-54)
$\displaystyle f_{\rm frag} (t)$ $\textstyle =$ $\displaystyle \frac{2}{2+r(1-c_0)(\sigma+1)} \left[\Phi F(p) + r \Omega K(p) \right]$ (22.9-55)
$\displaystyle f_{\rm char} (t)$ $\textstyle =$ $\displaystyle 1 - f_{\rm gas} (t) - f_{\rm frag}(t)$ (22.9-56)

The variables $\Phi$, $\Omega$, $F(p)$, and $K(p)$ are the statistical relationships related to the cleaving of bridges based on the percolation lattice statistics, and are given by the following equations:


$\displaystyle \Phi$ $\textstyle =$ $\displaystyle 1+r \left[\frac{\pounds}{p} + \frac{(\sigma-1) \delta}{4(1 - p)} \right]$ (22.9-57)
$\displaystyle \Omega$ $\textstyle =$ $\displaystyle \frac{\delta}{2(1-p)} - \frac{\pounds}{p}$ (22.9-58)
$\displaystyle F(p)$ $\textstyle =$ $\displaystyle \left( \frac{p'}{p} \right) ^{\frac{\sigma+1}{\sigma-1}}$ (22.9-59)
$\displaystyle K(p)$ $\textstyle =$ $\displaystyle \left[ 1 - \left( \frac{\sigma+1}{2} \right) p' \right] \left( \frac{p'}{p} \right) ^{\frac{\sigma+1}{\sigma-1}}$ (22.9-60)

$r$ is the ratio of bridge mass to site mass, $m_b/m_a$, where


$\displaystyle m_b$ $\textstyle =$ $\displaystyle 2 M_{w, \delta}$ (22.9-61)
$\displaystyle m_a$ $\textstyle =$ $\displaystyle M_{w,1} - (\sigma +1) M_{w, \delta}$ (22.9-62)

where $M_{w, \delta}$ and $M_{w,1}$ are the side chain and cluster molecular weights respectively. $\sigma +1$ is the lattice coordination number, which is determined from solid-state nuclear magnetic esonance (NMR) measurements related to coal structure parameters, and $p'$ is the root of the following equation in $p$ (the total number of bridges in the coal lattice matrix):


 p' (1-p')^{\sigma -1} = p(1-p)^{\sigma -1} (22.9-63)

In accounting for mass in the metaplast (tar precursor fragments), the part that vaporizes is treated in a manner similar to flash vaporization, where it is assumed that the finite fragments undergo vapor/liquid phase equilibration on a time scale that is rapid with respect to the bridge reactions. As an estimate of the vapor/liquid that is present at any time, a vapor pressure correlation based on a simple form of Raoult's Law is used. The vapor pressure treatment is largely responsible for predicting pressure-dependent devolatilization yields. For the part of the metaplast that reattaches to the coal lattice, a cross-linking rate expression given by the following equation is used:


 \frac{d m_{\rm cross}}{dt} = m_{\rm frag} A_{\rm cross} e^{-(E_{\rm cross}/RT)} (22.9-64)

where $m_{\rm cross}$ is the amount of mass reattaching to the matrix, $m_{\rm frag}$ is the amount of mass in the tar precursor fragments (metaplast), and $A_{\rm cross}$ and $E_{\rm cross}$ are rate expression constants.

CPD Inputs

Given the set of equations and corresponding rate constants introduced for the CPD model, the number of constants that must be defined to use the model is a primary concern. For the relationships defined previously, it can be shown that the following parameters are coal independent [ 110]:

These constants are included in the submodel formulation and are not input or modified during problem setup.

There are an additional five parameters that are coal-specific and must be specified during the problem setup:

The first four of these are coal structure quantities that are obtained from NMR experimental data. The last quantity, representing the char bridges that either exist in the parent coal or are formed very early in the devolatilization process, is estimated based on the coal rank. These quantities are entered in the Materials panel as described in Section  22.14.2. Values for the coal-dependent parameters for a variety of coals are listed in Table  22.9.1.


Table 22.9.1: Chemical Structure Parameters for $^{13}$C NMR for 13 Coals
Coal Type $\sigma+1$ $p_0$ $M_{w,1}$ $M_{w, \delta}$ $c_0$
Zap (AR) 3.9 .63 277 40 .20
Wyodak (AR) 5.6 .55 410 42 .14
Utah (AR) 5.1 .49 359 36 0
Ill6 (AR) 5.0 .63 316 27 0
Pitt8 (AR) 4.5 .62 294 24 0
Stockton (AR) 4.8 .69 275 20 0
Freeport (AR) 5.3 .67 302 17 0
Pocahontas (AR) 4.4 .74 299 14 .20
Blue (Sandia) 5.0 .42 410 47 .15
Rose (AFR) 5.8 .57 459 48 .10
1443 (lignite, ACERC) 4.8 .59 297 36 .20
1488 (subbituminous, ACERC) 4.7 .54 310 37 .15
1468 (anthracite, ACERC) 4.7 .89 656 12 .25
AR refers to eight types of coal from the Argonne premium sample bank [ 347, 387]. Sandia refers to the coal examined at Sandia National Laboratories [ 109]. AFR refers to coal examined at Advanced Fuel Research. ACERC refers to three types of coal examined at the Advanced Combustion Engineering Research Center.

Particle Swelling During Devolatilization

The particle diameter changes during devolatilization according to the swelling coefficient, $C_{\rm sw}$, which is defined by you and applied in the following relationship:


 \frac{d_p}{d_{p,0}} = 1 + (C_{\rm sw} - 1) \frac{(1-f_{w,0})m_{p,0} - m_p}{f_{v,0} (1-f_{w,0}) m_{p,0}} (22.9-65)


where $d_{p,0}$ = particle diameter at the start of devolatilization
  $d_p$ = current particle diameter

The term $\frac{(1-f_{w,0})m_{p,0} - m_p}{f_{v,0} (1-f_{w,0}) m_{p,0}}$ is the ratio of the mass that has been devolatilized to the total volatile mass of the particle. This quantity approaches a value of 1.0 as the devolatilization law is applied. When the swelling coefficient is equal to 1.0, the particle diameter stays constant. When the swelling coefficient is equal to 2.0, the final particle diameter doubles when all of the volatile component has vaporized, and when the swelling coefficient is equal to 0.5 the final particle diameter is half of its initial diameter.

Heat Transfer to the Particle During Devolatilization

Heat transfer to the particle during the devolatilization process includes contributions from convection, radiation (if active), and the heat consumed during devolatilization:


 m_p c_p \frac{d T_p}{dt} = h A_p (T_{\infty} - T_p) + \frac{... ...t} h_{\rm fg} + A_p \epsilon_p \sigma ({\theta_R}^4 - {T_p}^4) (22.9-66)

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

By default, Equation  22.9-66 is solved analytically, by assuming that the temperature and mass of the particle do not change significantly between time steps:


 T_p (t + \Delta t) = \alpha_p + [T_p (t) - \alpha_p] e^{-\beta_p \Delta t} (22.9-67)

where


 \alpha_p = \frac{h A_p T_{\infty} + \frac{d m_p}{d t} h_{\rm... ...p \sigma {\theta_R}^4 }{h A_p + \epsilon_p A_p \sigma {T_p}^3} (22.9-68)

and


 \beta_p = \frac{A_p (h + \epsilon_p \sigma {T_p}^3) }{m_p c_p} (22.9-69)

FLUENT can also solve Equation  22.9-66 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section  22.11.7 for details.



Surface Combustion (Law 5)


After the volatile component of the particle is completely evolved, a surface reaction begins which consumes the combustible fraction, $f_{\rm comb}$, of the particle. Law 5 is thus active (for a combusting particle) after the volatiles are evolved:


 m_p < (1 - f_{v,0})(1 - f_{w,0}) m_{p,0} (22.9-70)

and until the combustible fraction is consumed:


 m_p > (1 - f_{v,0} - f_{\rm comb})(1 - f_{w,0}) m_{p,0} (22.9-71)

When the combustible fraction, $f_{\rm comb}$, has been consumed in Law 5, the combusting particle may contain residual "ash'' that reverts to the inert heating law, Law 6 (described previously).

With the exception of the multiple surface reactions model, the surface combustion law consumes the reactive content of the particle as governed by the stoichiometric requirement, $S_b$, of the surface "burnout'' reaction:


 {\rm char(s)} + S_b {\rm ox(g)} \longrightarrow {\rm products(g)} (22.9-72)

where $S_b$ is defined in terms of mass of oxidant per mass of char, and the oxidant and product species are defined in the Set Injection Properties panel.

FLUENT provides a choice of four heterogeneous surface reaction rate models for combusting particles:

Each of these models is described in detail below. You will choose the surface combustion model when you are setting physical properties for the combusting-particle material in the Materials panel, as described in Section  22.14.2. By default, the diffusion-limited rate model will be used.

The Diffusion-Limited Surface Reaction Rate Model

The diffusion-limited surface reaction rate model which is the default model in FLUENT, assumes that the surface reaction proceeds at a rate determined by the diffusion of the gaseous oxidant to the surface of the particle:


 \frac{d m_p}{dt} = -4 \pi d_{p} D_{i,m} \frac{Y_{\rm ox} T_{\infty} \rho}{S_b (T_p + T_{\infty})} (22.9-73)


where $D_{i,m}$ = diffusion coefficient for oxidant in the bulk (m $^2$/s)
  $Y_{\rm ox}$ = local mass fraction of oxidant in the gas
  $\rho$ = gas density (kg/m $^3$)
  $S_b$ = stoichiometry of Equation  22.9-72

Equation  22.9-73 is derived from the model of Baum and Street [ 27] with the kinetic contribution to the surface reaction rate ignored. The diffusion-limited rate model assumes that the diameter of the particles does not change. Since the mass of the particles is decreasing, the effective density decreases, and the char particles become more porous.

The Kinetic/Diffusion Surface Reaction Rate Model

The kinetic/diffusion-limited rate model assumes that the surface reaction rate is determined either by kinetics or by a diffusion rate. FLUENT uses the model of Baum and Street [ 27] and Field [ 106], in which a diffusion rate coefficient


 D_0 = C_1 \frac{\left [ (T_p + T_{\infty})/2 \right]^{0.75}}{d_p} (22.9-74)

and a kinetic rate


 {\cal R} = C_2 e^{-(E/RT_p)} (22.9-75)

are weighted to yield a char combustion rate of


 \frac{d m_p}{dt} = -A_p p_{\rm ox} \frac{D_0 {\cal R}}{D_0 + {\cal R}} (22.9-76)

where $A_p$ is the surface area of the droplet ( $\pi d^{2}_{p}$), $p_{\rm ox}$ is the partial pressure of oxidant species in the gas surrounding the combusting particle, and the kinetic rate, ${\cal R}$, incorporates the effects of chemical reaction on the internal surface of the char particle (intrinsic reaction) and pore diffusion. In FLUENT, Equation  22.9-76 is recast in terms of the oxidant mass fraction, $Y_{\rm ox}$, as


 \frac{d m_p}{dt} = -A_p \frac{\rho RT_{\infty} Y_{\rm ox}}{M_{w, \rm ox}} \;\; \frac{D_0 {\cal R}}{D_0 + {\cal R}} (22.9-77)

The particle size is assumed to remain constant in this model while the density is allowed to decrease.

When this model is enabled, the rate constants used in Equations  22.9-74 and  22.9-75 are entered in the Materials panel, as described in Section  22.14.

The Intrinsic Model

The intrinsic model in FLUENT is based on Smith's model [ 342], assuming the order of reaction is equal to unity. Like the kinetic/diffusion model, the intrinsic model assumes that the surface reaction rate includes the effects of both bulk diffusion and chemical reaction (see Equation  22.9-77). The intrinsic model uses Equation  22.9-74 to compute the diffusion rate coefficient, $D_0$, but the chemical rate, ${\cal R}$, is explicitly expressed in terms of the intrinsic chemical and pore diffusion rates:


 {\cal R} = \eta \frac{d_p}{6} \rho_p A_g k_i (22.9-78)

$\eta$ is the effectiveness factor, or the ratio of the actual combustion rate to the rate attainable if no pore diffusion resistance existed [ 198]:


 \eta = \frac{3}{\phi^2} (\phi \coth \phi - 1) (22.9-79)

where $\phi$ is the Thiele modulus:


 \phi = \frac{d_p}{2} \left[ \frac{S_b \rho_p A_g k_i p_{\rm ox}}{D_e \rho_{\rm ox}} \right]^{1/2} (22.9-80)

$\rho_{\rm ox}$ is the density of oxidant in the bulk gas (kg/m $^3$) and $D_e$ is the effective diffusion coefficient in the particle pores. Assuming that the pore size distribution is unimodal and the bulk and Knudsen diffusion proceed in parallel, $D_e$ is given by


 D_e = \frac{\theta}{\tau^2} \left[ \frac{1}{D_{\rm Kn}} + \frac{1}{D_0} \right]^{-1} (22.9-81)

where $D_0$ is the bulk molecular diffusion coefficient and $\theta$ is the porosity of the char particle:


 \theta = 1 - \frac{\rho_p}{\rho_t} (22.9-82)

$\rho_p$ and $\rho_t$ are, respectively, the apparent and true densities of the pyrolysis char.

$\tau$ (in Equation  22.9-81) is the tortuosity of the pores. The default value for $\tau$ in FLUENT is $\sqrt{2}$, which corresponds to an average intersecting angle between the pores and the external surface of 45 $^\circ$ [ 198].

$D_{\rm Kn}$ is the Knudsen diffusion coefficient:


 D_{\rm Kn} = 97.0 \overline{r}_p \sqrt{\frac{T_p}{M_{w, \rm ox}}} (22.9-83)

where $T_p$ is the particle temperature and $\overline{r}_p$ is the mean pore radius of the char particle, which can be measured by mercury porosimetry. Note that macropores ( $\overline{r}_p > 150$ Å) dominate in low-rank chars while micropores ( $\overline{r}_p < 10$ Å) dominate in high-rank chars [ 198].

$A_g$ (in Equations  22.9-78 and 22.9-80) is the specific internal surface area of the char particle, which is assumed in this model to remain constant during char combustion. Internal surface area data for various pyrolysis chars can be found in [ 341]. The mean value of the internal surface area during char combustion is higher than that of the pyrolysis char [ 198]. For example, an estimated mean value for bituminous chars is 300 m $^2$/g [ 53].

$k_i$ (in Equations  22.9-78 and 22.9-80) is the intrinsic reactivity, which is of Arrhenius form:


 k_i = A_i e^{-(E_i/RT_p)} (22.9-84)

where the pre-exponential factor $A_i$ and the activation energy $E_i$ can be measured for each char. In the absence of such measurements, the default values provided by FLUENT (which are taken from a least squares fit of data of a wide range of porous carbons, including chars [ 341]) can be used.

To allow a more adequate description of the char particle size (and hence density) variation during combustion, you can specify the burning mode $\alpha$, relating the char particle diameter to the fractional degree of burnout $U$ (where $U = 1 - m_p/m_{p,0}$) by [ 340]


 \frac{d_p}{d_{p,0}} = (1-U)^{\alpha} (22.9-85)

where $m_p$ is the char particle mass and the subscript zero refers to initial conditions (i.e., at the start of char combustion). Note that $0 \leq \alpha \leq 1/3$ where the limiting values 0 and $1/3$ correspond, respectively, to a constant size with decreasing density (zone 1) and a decreasing size with constant density (zone 3) during burnout. In zone 2, an intermediate value of $\alpha=0.25$, corresponding to a decrease of both size and density, has been found to work well for a variety of chars [ 340].

When this model is enabled, the rate constants used in Equations  22.9-74, 22.9-78, 22.9-80, 22.9-81, 22.9-83, 22.9-84, and 22.9-85 are entered in the Materials panel, as described in Section  22.14.

The Multiple Surface Reactions Model

Modeling multiple particle surface reactions follows a pattern similar to the wall surface reaction models, where the surface species is now a "particle surface species''. For the mixture material defined in the Species Model panel, the particle surface species can be depleted or produced by the stoichiometry of the particle surface reaction (defined in the Reactions panel). The particle surface species constitutes the reactive char mass of the particle, hence, if a particle surface species is depleted, the reactive "char'' content of the particle is consumed, and in turn, when a surface species is produced, it is added to the particle "char'' mass. Any number of particle surface species and any number of particle surface reactions can be defined for any given combusting particle.

Multiple injections can be accommodated, and combusting particles reacting according to the multiple surface reactions model can coexist in the calculation, with combusting particles following other char combustion laws. The model is based on oxidation studies of char particles, but it is also applicable to gas-solid reactions in general, not only to char oxidation reactions.

See Section  14.3 for information about particle surface reactions.

Limitations

Note the following limitations of the multiple surface reactions model:

Heat and Mass Transfer During Char Combustion

The surface reaction consumes the oxidant species in the gas phase; i.e., it supplies a (negative) source term during the computation of the transport equation for this species. Similarly, the surface reaction is a source of species in the gas phase: the product of the heterogeneous surface reaction appears in the gas phase as a user-selected chemical species. The surface reaction also consumes or produces energy, in an amount determined by the heat of reaction defined by you.

The particle heat balance during surface reaction is


 m_p c_p \frac{d T_p}{dt} = h A_p (T_{\infty} - T_p) - f_h\fr... ..._{\rm reac} + A_p \epsilon_p \sigma ({\theta_R}^4 - {T_p}^4 ) (22.9-86)

where $H_{\rm reac}$ is the heat released by the surface reaction. Note that only a portion ( $1 - f_h$) of the energy produced by the surface reaction appears as a heat source in the gas-phase energy equation: the particle absorbs a fraction $f_h$ of this heat directly. For coal combustion, it is recommended that $f_h$ be set at 1.0 if the char burnout product is CO and 0.3 if the char burnout product is CO $_2$ [ 39].

Radiation heat transfer to the particle is included only if you have enabled the P-1 or discrete ordinates radiation model and you have activated radiation heat transfer to particles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

By default, Equation  22.9-86 is solved analytically, by assuming that the temperature and mass of the particle do not change significantly between time steps. FLUENT can also solve Equation  22.9-86 in conjunction with the equivalent mass transfer equation using a stiff coupled solver. See Section  22.11.7 for details.



Multicomponent Particle Definition (Law 7)


Multicomponent particles are described in FLUENT as a mixture of species within droplets/particles. The particle mass $m$ is the sum of the masses of the components


 m=\sum_{i}m_{i} (22.9-87)

The density of the particle $\rho_{p}$ can be either constant, or volume-averaged:


 \rho_{p}=\left(\sum_{i}\frac{m_{i}}{m\rho_{i}}\right)^{-1} (22.9-88)

For particles containing more than one component it is difficult to assign the whole particle to one process like boiling or heating. Therefore it can be only modeled by a law integrating all processes of relevance in one equation. The source terms for temperature and component mass are the sum of the sources from the partial processes:


 m_{p}c_{p}\left(\frac{dT_{p}}{dt}\right)=A_{p}\epsilon_{p}\s... ...}(T_{\infty}-T_{p})+\sum_{i}\frac{dm_{i}}{dt}(h_{i,p}-h_{i,g}) (22.9-89)


 \left(\frac{dm_{i}}{dt}\right)=M_{w,i}k_{c,i}(C_{i,s}-C_{i,\infty}) (22.9-90)

The equation for the particle temperature $T$ consists of terms for radiation, convective heating (Equation  22.9-11) and vaporization. Radiation heat transfer to the particle is included only if you have enabled P-1 or Discrete-Ordinates (DO) radiation and you have activated radiation heat transfer to the particles using the Particle Radiation Interaction option in the Discrete Phase Model panel.

The mass of the particle components $m_{i}$ is only influenced by the vaporization (Equation  22.9-20), where $M_{w,i}$ is the molecular weight of species $i$. The mass transfer coefficient $k_{c,i}$ of component $i$ is calculated from the Sherwood correlation (Equation  22.9-23). The concentration of vapor at the particle surface $C_{i,s}$ depends on the saturation pressure of the component.

Raoult's Law

The correlation between the vapor concentration of a species $C_{i,s}$ over the surface and its mole fraction in the condensed phase $X_{i}$ (under the assumption of an ideal gas law) is described by Raoult's law:


 C_{i,s}=\frac{p_{i}}{RT}=\frac{X_{i}p}{RT} (22.9-91)

You can define your own law other than Raoult's Law for vapor concentration at the particle surface using a user-defined function.

Click here to go to the FLUENT UDF Manual for details.


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