**2.1 Model definition**

The present model simulates the isobaric carbon combustion of constant surface temperature *Ts* in the stagnation flow of temperature *T*, oxygen mass-fraction *Y*O,, and carbon dioxide mass-fraction *Y*P,, in a general manner (Makino, 1990). The major reactions

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

**2.3 Boundary conditions** 

known ones, expressed as

be considered. Then, we have

 

energy are, respectively,

 

s F

 

*y*

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 255

where *B* is the frequency factor, *E* the activation energy, *Ro* the universal gas constant, the

describes the curvature of the surface such that *j* = 0 and 1 designate two-dimensional and axisymmetric flows, respectively, and the velocity components *u* and *v* of the frictionless

The boundary conditions for the continuity and the momentum equations are the well-

(9)

 (*Y*F) =0, (*Yi*) = *Yi*, (*i*=O, P, N). (10) At the carbon surface, components transported from gas to solid by diffusion, transported away from the interface by convection, and produced/consumed by surface reactions are to

<sup>F</sup> <sup>s</sup> <sup>2</sup> exp <sup>2</sup> exp *<sup>T</sup>*

 

0

In boundary layer variables, the conservation equations for momentum, species *i*, and

 

2 1

 

 

P

<sup>P</sup> <sup>s</sup> exp *<sup>T</sup>*

 

O

<sup>O</sup> <sup>s</sup> exp *<sup>T</sup>*

 

s s,O

> 

O s O

> 

P s P

N <sup>N</sup> <sup>s</sup>

*y Y*

*T Ta*

*W <sup>Y</sup> <sup>W</sup>*

*W <sup>Y</sup> <sup>W</sup>*

> 

*<sup>Y</sup> vY <sup>D</sup>* , (11)

*u ax* , *v* (*j* 1)*ay* (8)

in Eq. (1)

 

s s,P

*Ta*

 

 

P s P

*Ta*

*Ta*

s,P

 

 

*B*

s s,O

s s,P

 

*W <sup>Y</sup> <sup>W</sup>*

 

 

*vY D* . (14)

0

*<sup>j</sup>* , (15)

 

 

2

 

<sup>0</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> *<sup>L</sup> <sup>Y</sup>*<sup>F</sup> *Y*<sup>P</sup> *<sup>L</sup> <sup>Y</sup>*<sup>O</sup> *Y*<sup>P</sup> *<sup>L</sup> <sup>Y</sup>*<sup>P</sup> *<sup>T</sup> <sup>L</sup> <sup>Y</sup>*<sup>N</sup> , (16)

 

*d d f* F

s,O

*<sup>Y</sup> vY <sup>D</sup>* , (12)

s,P

*<sup>Y</sup> vY <sup>D</sup>* , (13)

*B*

s

  *B*

stoichiometric coefficient, and *W* the molecular weight. We should also note that *Rj*

flow outside the boundary layer are given by use of the velocity gradient *a* as

at *y*=0 : *u* = 0 , *v* =*vs* ,

 as *y* : *v* =*v* . For the species conservation equations, we have in the freestream as

> 

 

> 

*y*

O s O

s,O

 

O

*y*

s

 

s P

**2.4 Conservation equations with nondimensional variables and parameters** 

2 2

*d <sup>d</sup> <sup>f</sup> <sup>f</sup> <sup>d</sup> d f*

3 3

 

*B*

 

> 

> >

*W <sup>Y</sup> <sup>W</sup>*

F

considered here are the surface C-O2 and C-CO2 reactions and the gas-phase CO-O2 reaction. The surface C+O2CO2 reaction is excluded (Arthur, 1951) because our concern is the combustion at temperatures above 1000 K. Crucial assumptions introduced are conventional, constant property assumptions with unity Lewis number, constant average molecular weight, constant value of the product of density and viscosity , one-step overall irreversible gas-phase reaction, and first-order surface reactions. Surface characteristics, such as porosity and internal surface area, are grouped into the frequency factors for the surface reactions.

### **2.2 Governing equations**

The steady-state two-dimensional and/or axisymmetric boundary-layer flows with chemical reactions are governed as follows (Chung, 1965; Law, 1978):

$$\mathbf{Contimity:}$$

$$\text{Contimality:} \tag{1}$$

$$\frac{\partial \{\rho u R^j\}}{\partial x} + \frac{\partial \{\rho v R^j\}}{\partial y} = 0 \,, \tag{2}$$

Momentum:

$$
\rho \mu \frac{\partial u}{\partial \mathbf{x}} + \rho v \frac{\partial u}{\partial y} - \frac{\partial}{\partial y} \left( \mu \frac{\partial u}{\partial y} \right) = \rho\_{\alpha \flat} u\_{\alpha \flat} \left( \frac{\partial u}{\partial \mathbf{x}} \right)\_{\alpha \flat} \tag{2}
$$

$$\text{Species: } \qquad \rho u \frac{\partial Y\_i}{\partial \mathbf{x}} + \rho v \frac{\partial Y\_i}{\partial y} - \frac{\partial}{\partial y} \Big| \rho D \frac{\partial Y\_i}{\partial y} \Big| = -w\_i \quad \text{ (i = F, O)}\_{} \tag{3}$$

$$
\rho u \frac{\partial Y\_\mathbf{P}}{\partial x} + \rho v \frac{\partial Y\_\mathbf{P}}{\partial y} - \frac{\partial}{\partial y} \left( \rho D \frac{\partial Y\_\mathbf{P}}{\partial y} \right) = w\_\mathbf{P} \tag{4}
$$

$$
\rho \mu \frac{\partial \mathbf{Y\_N}}{\partial \mathbf{x}} + \rho v \frac{\partial \mathbf{Y\_N}}{\partial y} - \frac{\partial}{\partial y} \left( \rho D \frac{\partial \mathbf{Y\_N}}{\partial y} \right) = \mathbf{0} \tag{5}
$$

$$\text{Energy:}\tag{6} \qquad \text{p}\mu \frac{\partial \{\mathbf{c\_p}T\}}{\partial \mathbf{x}} + \text{p}\nu \frac{\partial \{\mathbf{c\_p}T\}}{\partial y} - \frac{\partial}{\partial y} \left(\lambda \frac{\partial T}{\partial y}\right) = qw\_{\mathbf{F}} \tag{6}$$

where *T* is the temperature, *c*p the specific heat, *q* the heat of combustion per unit mass of CO, *Y* the mass fraction, *u* the velocity in the tangential direction *x*, *v* the velocity in the normal direction *y*, and the subscripts C, F, O, P, N, g, s, and , respectively, designate carbon, carbon monoxide, oxygen, carbon dioxide, nitrogen, the gas phase, the surface, and the freestream.

In these derivations, use has been made of assumptions that the pressure and viscous heating are negligible in Eq. (6), that a single binary diffusion coefficient *D* exists for all species pairs, that *c*p is constant, and that the CO-O2 reaction can be represented by a onestep, overall, irreversible reaction with a reaction rate

$$\varepsilon\_{\rm F} = (\nu\_i \mathcal{W}\_i) \mathcal{B}\_{\rm g} \left(\frac{\rho \mathcal{Y}\_{\rm F}}{\mathcal{W}\_{\rm F}}\right)^{\nu\_{\rm F}} \left(\frac{\rho \mathcal{Y}\_{\rm O}}{\mathcal{W}\_{\rm O}}\right)^{\nu\_{\rm O}} \exp\left(-\frac{E\_{\rm g}}{R^o T}\right),\tag{7}$$

where *B* is the frequency factor, *E* the activation energy, *Ro* the universal gas constant, the stoichiometric coefficient, and *W* the molecular weight. We should also note that *Rj* in Eq. (1) describes the curvature of the surface such that *j* = 0 and 1 designate two-dimensional and axisymmetric flows, respectively, and the velocity components *u* and *v* of the frictionless flow outside the boundary layer are given by use of the velocity gradient *a* as

$$
\mu\_{\infty} = a\mathbf{x} \,, \qquad \text{or} \qquad \omega\_{\infty} = -(j+1)ay \,\tag{8}
$$

### **2.3 Boundary conditions**

254 Mass Transfer in Chemical Engineering Processes

considered here are the surface C-O2 and C-CO2 reactions and the gas-phase CO-O2 reaction. The surface C+O2CO2 reaction is excluded (Arthur, 1951) because our concern is the combustion at temperatures above 1000 K. Crucial assumptions introduced are conventional, constant property assumptions with unity Lewis number, constant average molecular weight, constant value of the product of density and viscosity , one-step overall irreversible gas-phase reaction, and first-order surface reactions. Surface characteristics, such as porosity and internal surface area, are grouped into the frequency

The steady-state two-dimensional and/or axisymmetric boundary-layer flows with

*uR<sup>j</sup> <sup>j</sup>*

 

 

*y vR*

> 

*y u*

*y <sup>Y</sup> <sup>D</sup>*

 

<sup>P</sup> <sup>P</sup> <sup>P</sup> *w*

 

 

<sup>0</sup> <sup>N</sup> <sup>N</sup> <sup>N</sup>

<sup>p</sup> <sup>p</sup> *qw*

*y y*

where *T* is the temperature, *c*p the specific heat, *q* the heat of combustion per unit mass of CO, *Y* the mass fraction, *u* the velocity in the tangential direction *x*, *v* the velocity in the normal direction *y*, and the subscripts C, F, O, P, N, g, s, and , respectively, designate carbon, carbon monoxide, oxygen, carbon dioxide, nitrogen, the gas phase, the surface, and

In these derivations, use has been made of assumptions that the pressure and viscous heating are negligible in Eq. (6), that a single binary diffusion coefficient *D* exists for all species pairs, that *c*p is constant, and that the CO-O2 reaction can be represented by a one-

*W Y*

O O

F O

 

 

 

*W Y w W B <sup>o</sup> <sup>i</sup> <sup>i</sup>*

F F <sup>F</sup> <sup>g</sup> exp

 

*y y*

*c T*

*<sup>u</sup>*

 

> 

 

*y y*

*<sup>u</sup>*

  , (1)

 

*u* , (2)

*x u*

P

F

 

, (7)

*R T E*

g

 

 

 

*u*

(*i* = F, O), (3)

 

, (4)

 

*u* , (5)

*y <sup>Y</sup> <sup>D</sup>*

> 

*y <sup>Y</sup> <sup>D</sup>*

> *y T*

 

, (6)

 

chemical reactions are governed as follows (Chung, 1965; Law, 1978):

 

*x*

*y y u v*

*<sup>i</sup> <sup>i</sup> <sup>i</sup> w*

 

> *Y v*

> > *Y v*

*v*

 

*y y Y v*

*<sup>u</sup>*

Continuity: <sup>0</sup>

 

> *x Y*

 

 

*x Y*

*x c T*

*x u*

Species: *<sup>i</sup>*

 

*x Y*

Energy:

step, overall, irreversible reaction with a reaction rate

factors for the surface reactions.

**2.2 Governing equations** 

Momentum:

the freestream.

The boundary conditions for the continuity and the momentum equations are the wellknown ones, expressed as

$$\begin{array}{ll}\text{at } y = 0: & \quad u = 0, \quad & v = v\_{s}.\\\\\text{as } y \to \sigma: & \quad v = v\_{v}.\end{array} \tag{9}$$

For the species conservation equations, we have in the freestream as

$$(\mathcal{Y}\_{\mathbb{F}})\_{\varpi} \equiv 0, \quad (\mathcal{Y}\_{i})\_{\varpi} \equiv \mathcal{Y}\_{i,\varpi} \quad (i \in \mathbb{O}, \mathcal{P}, \mathcal{N}).\tag{10}$$

At the carbon surface, components transported from gas to solid by diffusion, transported away from the interface by convection, and produced/consumed by surface reactions are to be considered. Then, we have

$$\left(\rho \boldsymbol{\nu} \boldsymbol{\gamma}\_{\rm F}\right)\_{\rm s} - \left(\rho \boldsymbol{D} \frac{\partial \boldsymbol{Y}\_{\rm F}}{\partial \boldsymbol{y}}\right)\_{\rm s} = 2\mathcal{W}\_{\rm F} \left(\frac{\rho \boldsymbol{Y}\_{\rm O}}{\boldsymbol{W}\_{\rm O}}\right)\_{\rm s} B\_{\rm s,O} \exp\left(-\frac{\boldsymbol{T}\boldsymbol{a}\_{\rm s,O}}{T\_{\rm s}}\right) + 2\mathcal{W}\_{\rm F} \left(\frac{\rho \boldsymbol{Y}\_{\rm P}}{\boldsymbol{W}\_{\rm P}}\right)\_{\rm s} B\_{\rm s,P} \exp\left(-\frac{\boldsymbol{T}\boldsymbol{a}\_{\rm s,P}}{T\_{\rm s}}\right), \tag{11}$$

$$\left(\rho v \,\mathrm{Y}\_{\mathrm{O}}\right)\_{\mathrm{s}} - \left(\rho D \frac{\partial \,\mathrm{Y}\_{\mathrm{O}}}{\partial \boldsymbol{y}}\right)\_{\mathrm{s}} = -\mathrm{W}\_{\mathrm{O}} \left(\frac{\rho \,\mathrm{Y}\_{\mathrm{O}}}{\mathrm{W}\_{\mathrm{O}}}\right)\_{\mathrm{s}} B\_{\mathrm{s},\mathrm{O}} \exp\left(-\frac{\mathrm{Ta}\_{\mathrm{s},\mathrm{O}}}{T\_{\mathrm{s}}}\right),\tag{12}$$

$$\left(\rho\boldsymbol{\sigma}\boldsymbol{\chi}\_{\rm P}\right)\_{\rm s} - \left(\rho\boldsymbol{D}\frac{\partial\boldsymbol{\chi}\_{\rm P}}{\partial\boldsymbol{\chi}}\right)\_{\rm s} = -\mathsf{W}\_{\rm P}\left(\frac{\rho\boldsymbol{\chi}\_{\rm P}}{\boldsymbol{W}\_{\rm P}}\right)\_{\rm s}B\_{\rm s,P}\exp\left(-\frac{\boldsymbol{T}\mathsf{a}\_{\rm s,P}}{T\_{\rm s}}\right),\tag{13}$$

$$\left(\rho v \,\mathrm{Y\_N}\right)\_{\mathrm{s}} - \left(\rho D \frac{\partial \mathcal{Y}\_{\mathrm{N}}}{\partial y}\right)\_{\mathrm{s}} = 0 \,\mathrm{}.\tag{14}$$

### **2.4 Conservation equations with nondimensional variables and parameters**

In boundary layer variables, the conservation equations for momentum, species *i*, and energy are, respectively,

$$\frac{d^3f}{d\eta^3} + f\frac{d^2f}{d\eta^2} + \frac{1}{2^j} \left| \frac{\rho\_{\alpha}}{\rho} - \left(\frac{df}{d\eta}\right)^2 \right| = 0 \,, \tag{15}$$

$$
\mathsf{L}\begin{pmatrix} \widetilde{\mathbf{Y}}\_{\mathcal{F}} + \widetilde{\mathbf{Y}}\_{\mathcal{P}} \end{pmatrix} = \mathsf{L}\begin{pmatrix} \widetilde{\mathbf{Y}}\_{\mathcal{O}} + \widetilde{\mathbf{Y}}\_{\mathcal{P}} \end{pmatrix} = \mathsf{L}\begin{pmatrix} \widetilde{\mathbf{Y}}\_{\mathcal{P}} - \widetilde{\mathbf{T}} \end{pmatrix} = \mathsf{L}\begin{pmatrix} \widetilde{\mathbf{Y}}\_{\mathcal{N}} \end{pmatrix} = \mathbf{0} \end{pmatrix} \tag{16}
$$

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

of composition by the chemical reactions. The boundary conditions for Eq. (15) are

whereas those for Eqs. (16) and (17) are

at =0: <sup>s</sup> <sup>s</sup>

*dY*

 

where

as : *T T Y Yi Yi*

s

*dY*

 

 

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 257

replaced by (*T*/*T*). As for the constant assumption, while enabling considerable simplification, it introduces 50%-70% errors in the transport properties of the gas in the present temperature range. However, these errors are acceptable for far greater errors in the chemical reaction rates. Furthermore, they are anticipated to be reduced due to the change

<sup>0</sup> , 0, <sup>1</sup>

(26)

*s*

 

 

<sup>~</sup> <sup>~</sup> , <sup>~</sup> <sup>~</sup> *<sup>T</sup> <sup>T</sup> Yi Yi* (*i*=F, O, P, N) ,

*<sup>d</sup> <sup>f</sup> <sup>f</sup> <sup>f</sup>*

<sup>F</sup> <sup>2</sup> <sup>~</sup> <sup>~</sup> *<sup>f</sup> <sup>Y</sup> <sup>f</sup> <sup>f</sup> <sup>d</sup>*

*<sup>f</sup> <sup>Y</sup> <sup>f</sup> <sup>d</sup>*

*dY*

 

<sup>O</sup> <sup>~</sup> <sup>~</sup>

*s*

 

*d*

<sup>~</sup> <sup>~</sup> 0, <sup>~</sup> , <sup>~</sup> <sup>~</sup> F (*i*= O, P, N) ,

which are to be supplemented by the following conservation relations at the surface:

<sup>s</sup> F,s s,O s,P

<sup>s</sup> O,s s,O

<sup>P</sup> <sup>~</sup> <sup>~</sup>

 

*<sup>f</sup> <sup>Y</sup>*

*s*

*s*

*d dY*

~ exp

 

> 

 

s

exp ~

s

*T*

  *T*

  s,O s,O

s,P s,P

*<sup>T</sup> <sup>A</sup> Da*

*<sup>T</sup> <sup>A</sup> Da*

 

 

*<sup>f</sup> <sup>Y</sup> <sup>f</sup> <sup>d</sup>*

 <sup>0</sup> <sup>~</sup> <sup>~</sup> <sup>s</sup> N,s <sup>N</sup>

<sup>s</sup> s,O s,P s,O O,s s,P P,s

s s,O

; ~

and *Da*s,O and *Da*s,P are the present surface Damköhler numbers, based only on the frequency factors for the C-O2 and C-CO2 reactions, respectively. Here, these heterogeneous

 

s s,P

*T Ta*

*T Ta*

~

~

 

*<sup>s</sup>* , (25)

 *d d f*

 

<sup>s</sup> s,P *<sup>f</sup> <sup>f</sup>* , (27)

<sup>s</sup> s,P *<sup>f</sup> <sup>f</sup>* , (28)

, (29)

<sup>~</sup> <sup>~</sup> *<sup>f</sup> <sup>f</sup> <sup>f</sup> <sup>A</sup> <sup>Y</sup> <sup>A</sup> <sup>Y</sup>* ; (31)

<sup>2</sup> <sup>~</sup> ;

*Da*

*Da*

s,P

<sup>~</sup> s,O s,O

<sup>~</sup> s,P

 

,

,

*j*

*a B*

*j*

2

*a B*

, (30)

<sup>s</sup> P,s s,P

$$
\Delta \begin{pmatrix} \widetilde{T} \\ \end{pmatrix} = -D a\_{\mathbf{g}} \alpha\_{\mathbf{g}'} \tag{17}
$$

where the convective-diffusive operator is defined as

$$
\Delta \quad = \frac{d^2}{d\eta^2} + f \frac{d}{d\eta} \,. \tag{18}
$$

The present Damköhler number for the gas-phase CO-O2 reaction is given by

$$Da\_{\rm g} = \left(\frac{B\_{\rm g}}{2^{\dot{J}}a}\right) \left(\frac{\rho\_{\rm o}}{\nu\_{\rm P} \mathcal{W}\_{\rm P}}\right)^{\nu\_{\rm F} + \nu\_{\rm O} - 1} (\mathbf{v}\_{\rm F})^{\nu\_{\rm F}} (\mathbf{v}\_{\rm O})^{\nu\_{\rm O}}\,,\tag{19}$$

with the nondimensional reaction rate

$$\alpha\_{\rm g} = \left(\frac{\widetilde{T}\_{\rm \infty}}{\widetilde{T}}\right)^{\mathbf{v}\_{\rm F} + \mathbf{v}\_{\rm O} - 1} \left(\widetilde{\mathbf{Y}}\_{\rm F}\right)^{\mathbf{v}\_{\rm F}} \left(\widetilde{\mathbf{Y}}\_{\rm O}\right)^{\mathbf{v}\_{\rm O}} \exp\left(-\frac{\widetilde{T}a\_{\rm g}}{\widetilde{T}}\right). \tag{20}$$

In the above, the conventional boundary-layer variables *s* and , related to the physical coordinates *x* and *y*, are

$$\mathbf{s} = \int\_{0}^{\mathbf{x}} \mathbf{p}\_{\alpha}(\mathbf{x}) \, \mu\_{\alpha}(\mathbf{x}) \, \mu\_{\alpha}(\mathbf{x}) \, R^{2j} \, d\mathbf{x} \,\,\,\,\,\tag{21}$$

$$
\eta = \frac{\mu\_{\infty}(\mathbf{x}) \mathbb{R}^{\cdot}}{\sqrt{2s}} \int\_{0}^{y} \rho(\mathbf{x}, y) \, dy \, \,. \tag{22}
$$

The nondimensional streamfunction *f*(*s*,) is related to the streamfunction (*x*, *y*) through

$$f(s,\eta) = \frac{\Psi(x,y)}{\sqrt{2s}}\,'{\,}$$

where (*x*, *y*) is defined by

$$
\rho u R^{\,j} = \frac{\partial \Psi}{\partial y} \prime \qquad \rho v R^{\,j} = -\frac{\partial \Psi}{\partial x} \prime \tag{24}
$$

such that the continuity equation is automatically satisfied. Variables and parameters are:

$$\begin{aligned} \widetilde{T} &= \frac{T}{q\sqrt{\left(\mathbf{c}\_{p}\alpha\_{\rm F}\right)}}, \qquad \widetilde{T}a = \frac{E\Big/\mathbf{R}^{o}}{q\sqrt{\left(\mathbf{c}\_{p}\alpha\_{\rm F}\right)}}, \qquad \alpha\_{\rm F} = \frac{\mathbf{v}\_{\rm P}\mathcal{W}\mathbf{V}\_{\rm P}}{\mathbf{v}\_{\rm F}\mathcal{W}\mathbf{V}\_{\rm F}},\\ \widetilde{Y}\_{\rm F} &= \frac{\mathbf{v}\_{\rm P}\mathcal{W}\mathbf{V}\_{\rm P}}{\mathbf{v}\_{\rm F}\mathcal{W}\mathbf{V}\_{\rm F}}Y\_{\rm F}, \qquad \widetilde{Y}\_{\rm O} = \frac{\mathbf{v}\_{\rm P}\mathcal{W}\mathbf{V}\_{\rm P}}{\mathbf{v}\_{\rm O}\mathcal{W}\mathbf{V}\_{\rm O}}Y\_{\rm O}, \qquad \widetilde{Y}\_{\rm N} = Y\_{\rm N}, \qquad \delta = \frac{\mathcal{W}\_{\rm P}}{\mathcal{W}\_{\rm C}}. \end{aligned}$$

Here, use has been made of an additional assumption that the Prandtl and Schmidt numbers are unity. Since we adopt the ideal-gas equation of state under an assumption of constant, average molecular weight across the boundary layer, the term (/) in Eq. (15) can be replaced by (*T*/*T*). As for the constant assumption, while enabling considerable simplification, it introduces 50%-70% errors in the transport properties of the gas in the present temperature range. However, these errors are acceptable for far greater errors in the chemical reaction rates. Furthermore, they are anticipated to be reduced due to the change of composition by the chemical reactions.

The boundary conditions for Eq. (15) are

$$f(0) = f\_{s\prime} \qquad \left(\frac{df}{d\eta}\right)\_s = 0, \qquad \left(\frac{df}{d\eta}\right)\_\infty = 1\,\tag{25}$$

whereas those for Eqs. (16) and (17) are

$$\begin{aligned} \text{at } \eta = 0 & \quad \widetilde{T} = \widetilde{T}\_{\text{s}}, \qquad & \widetilde{Y}\_{i} = \begin{pmatrix} \widetilde{Y}\_{i} \\ \widetilde{Y}\_{i} \end{pmatrix}\_{\text{s}} \qquad \qquad \qquad \qquad \text{(i=F, O, P, N)}, \\\\ \text{as } \eta \to \infty & \quad \widetilde{T} = \widetilde{T}\_{\text{o}, \prime} \qquad \widetilde{Y}\_{\text{F}} = 0, \qquad & \widetilde{Y}\_{i} = \begin{pmatrix} \widetilde{Y}\_{i} \\ \widetilde{Y}\_{i} \end{pmatrix}\_{\text{o}} \qquad \qquad \text{(i=O, P, N)}, \end{aligned} \tag{26}$$

which are to be supplemented by the following conservation relations at the surface:

$$-\left(\frac{d\widetilde{Y}\_{\rm F}}{d\eta}\right)\_{\rm s} + \left(-f\_{\rm s}\right)\widetilde{Y}\_{\rm F,s} = \delta\left(-f\_{\rm s,O}\right) + 2\delta\left(-f\_{\rm s,P}\right) = \delta\left(-f\_{\rm s}\right) + \delta\left(-f\_{\rm s,P}\right),\tag{27}$$

$$-\left(\frac{d\widetilde{Y}\_{\rm O}}{d\eta}\right)\_{s} + \left(-f\_{\rm s}\right)\widetilde{Y}\_{\rm O,s} = -\delta\left(-f\_{\rm s,\rm O}\right) = -\delta\left(-f\_{\rm s}\right) + \delta\left(-f\_{\rm s,\rm P}\right)\_{s}\tag{28}$$

$$-\left(\frac{d\widetilde{Y}\_{\rm P}}{d\eta}\right)\_{s} + \left(-f\_{\rm s}\right)\widetilde{Y}\_{\rm P,s} = -\delta\left(-f\_{\rm s,P}\right),\tag{29}$$

$$-\left(\frac{d\widetilde{Y}\_{\rm N}}{d\eta}\right)\_{\rm s} + \left(-f\_{\rm s}\right)\widetilde{Y}\_{\rm N,s} = 0 \,,\tag{30}$$

where

256 Mass Transfer in Chemical Engineering Processes

<sup>g</sup> <sup>g</sup>

 *<sup>d</sup> <sup>d</sup> <sup>f</sup> <sup>d</sup> d* 2 2

F O

*<sup>T</sup>* <sup>~</sup>

*Y Y*

In the above, the conventional boundary-layer variables *s* and , related to the physical

 *<sup>x</sup> <sup>j</sup> <sup>s</sup> <sup>x</sup> <sup>x</sup> <sup>u</sup> <sup>x</sup> <sup>R</sup> dx* <sup>0</sup>

*<sup>y</sup> <sup>j</sup>*

The nondimensional streamfunction *f*(*s*,) is related to the streamfunction (*x*, *y*) through

*<sup>x</sup> <sup>y</sup> <sup>f</sup> <sup>s</sup>*

*s*

2

*vR*

*s u x R*

*y uR<sup>j</sup> <sup>j</sup>*

<sup>F</sup> <sup>F</sup> *W*

O O P P

*W*

.

*<sup>W</sup> <sup>Y</sup>*

*o*

*p*

*q c <sup>E</sup> <sup>R</sup> Ta*

<sup>F</sup> , <sup>~</sup> , <sup>~</sup> , <sup>~</sup>

*<sup>p</sup>*

such that the continuity equation is automatically satisfied. Variables and parameters are:

, , ( )

Here, use has been made of an additional assumption that the Prandtl and Schmidt numbers are unity. Since we adopt the ideal-gas equation of state under an assumption of constant, average molecular weight across the boundary layer, the term (/) in Eq. (15) can be

F

O N N

exp <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>g</sup> F O

 

The present Damköhler number for the gas-phase CO-O2 reaction is given by

 

F O

 

 *<sup>a</sup> <sup>W</sup> B*

g

 

*T*

<sup>g</sup> <sup>2</sup>

 

g

with the nondimensional reaction rate

coordinates *x* and *y*, are

where (*x*, *y*) is defined by

~

*q c <sup>T</sup> <sup>T</sup>*

> F F P P

*W*

<sup>~</sup> , ( )

F O

*<sup>W</sup> <sup>Y</sup> <sup>Y</sup>*

 

P P

1

  1

*x y dy*

*x*

, , (24)

F F P P

*<sup>W</sup> <sup>Y</sup> <sup>Y</sup> <sup>Y</sup>*

*W*

F O

where the convective-diffusive operator is defined as

<sup>~</sup> *<sup>L</sup> <sup>T</sup> Da* , (17)

*L* . (18)

 

<sup>0</sup> , <sup>2</sup> . (22)

, , , (23)

C P

*W*

. (20)

<sup>2</sup> , (21)

*T Ta*

~

<sup>F</sup> <sup>O</sup>

 

F O

*Da <sup>j</sup>* , (19)

$$\left(\delta\left(-f\_{\rm s}\right)\right) = \delta\left(-f\_{\rm s,O}\right) + \delta\left(-f\_{\rm s,P}\right) = A\_{\rm s,O}\widetilde{Y}\_{\rm O,s} + A\_{\rm s,P}\widetilde{Y}\_{\rm P,s} \tag{31}$$

$$\begin{split} A\_{\rm s,O} &= Da\_{\rm s,O} \left( \frac{\widetilde{T}\_{\rm o}}{\widetilde{T}\_{\rm s}} \right) \exp \Big( -\frac{\widetilde{T}a\_{\rm s,O}}{\widetilde{T}\_{\rm s}} \right); \quad Da\_{\rm s,O} = \frac{B\_{\rm s,O}}{\sqrt{2^{\cdot^{\cdot}}a(\mu\_{\rm \infty}/\rho\_{\rm \infty})}}, \\\\ A\_{\rm s,P} &= Da\_{\rm s,P} \left( \frac{\widetilde{T}\_{\rm o}}{\widetilde{T}\_{\rm s}} \right) \exp \Big( -\frac{\widetilde{T}a\_{\rm s,P}}{\widetilde{T}\_{\rm s}} \Big); \quad Da\_{\rm s,P} = \frac{B\_{\rm s,P}}{\sqrt{2^{\cdot^{\cdot}}a(\mu\_{\rm \infty}/\rho\_{\rm \infty})}}, \end{split}$$

and *Da*s,O and *Da*s,P are the present surface Damköhler numbers, based only on the frequency factors for the C-O2 and C-CO2 reactions, respectively. Here, these heterogeneous

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

parameters in solving those are *Da*g, *Da*s,O, *Da*s,P, and (-*f*s).

**2.6 Transfer number and combustion rate** 

Eq. (17) becomes

conditions in Eq. (25) and

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 259

and a prime indicates *d*/*d*. Using the new independent variable , the energy conservation

*d d*

Therefore, the equations to be solved are Eqs. (15) and (40), subject to the boundary

*T Ts T T*

by use of (-*f*s) given by Eq. (31) and the coupling functions in Eqs. (33) to (36). Key

The influence of finite rate gas-phase kinetics is studied here. The global rate equation used has the same form as that of Howard, et al. (1973), in which the activation energy and the frequency factor are reported to be Eg=113 kJ/mol and Bg=1.3108 [(mol/m3)s]-1, respectively. The combustion response is quite similar to that of particle combustion (Makino & Law, 1986), as shown in Fig. 1(a) (Makino, 1990). The parameter , indispensable in obtaining the combustion rate, is bounded by limiting solutions to be mentioned, presenting that the gasphase CO-O2 reaction reduces the surface C-O2 reaction by consuming O2, while at the same time initiating the surface C-CO2 reaction by supplying CO2, and that with increasing surface temperature the combustion rate can first increase, then decrease, and increase again as a result of the close coupling between the three reactions. In addition, the combustion process depends critically on whether the gas-phase CO-O2 reaction is activated. If it is not, the oxygen in the ambience can readily reach the surface to participate in the C-O2 reaction. Activation of the surface C-CO2 reaction depends on whether the environment contains any CO2. However, if the gas-phase CO-O2 reaction is activated, the existence of CO-flame in the gas phase cuts off most of the oxygen supplied to the surface such that the surface C-O2 reaction is suppressed.

At the same time, the CO2 generated at the flame activates the surface C-CO2 reaction.

Fig. 1. Combustion behavior as a function of the surface temperature with the gas-phase Damköhler number taken as a parameter; *Da*s,O= *Da*s,P=108 and *Y*P,=0 (Makino, 1990). (a)

Transfer number. (b) Nondimensional combustion rate.

2 2~

 

*d*

 

 <sup>2</sup> g g

*Da*

 

*<sup>d</sup> <sup>T</sup>* . (40)

<sup>~</sup> <sup>~</sup> , <sup>~</sup> <sup>~</sup> <sup>0</sup> <sup>1</sup> , (41)

reactions are assumed to be first order, for simplicity and analytical convenience.1 As for the kinetic expressions for non-permeable solid carbon, effects of porosity and/or internal surface area are considered to be incorporated, since surface reactions are generally controlled by combinations of chemical kinetics and pore diffusions.

For self-similar flows, the normal velocity *v*s at the surface is expressible in terms of (-*f*s) by

$$(\rho v)\_{\sf s} = \left(-f\_{\sf s}\right)\sqrt{2^{j}\,a\,\mathsf{p}\_{\sf s}\mu\_{\sf s\sf s}\,}\,. \tag{32}$$

Reminding the fact that the mass burning rate of solid carbon is given by *m* = (*v*)s, which is equivalent to the definition of the combustion rate [kg/(m2s)], then (-*f*s) can be identified as the nondimensional combustion rate. Note also that the surface reactions are less sensitive to velocity gradient variations than the gas-phase reaction because *Da*s ~ *a*-1/2 while *Da*g~ *a*-1.

### **2.5 Coupling functions**

With the boundary conditions for species, cast in the specific forms of Eqs. (27) to (29), the coupling functions for the present system are given by

$$
\widetilde{Y}\_{\rm F} + \widetilde{Y}\_{\rm P} = \frac{\left(\widetilde{Y}\_{\rm P,\infty} + \mathfrak{S}\mathfrak{G}\right) + \left(\widetilde{Y}\_{\rm P,\infty} - \mathfrak{S}\right)\mathfrak{G}\widetilde{\mathfrak{G}}}{1 + \mathfrak{G}}\,,\tag{33}
$$

$$
\widetilde{Y}\_{\rm O} + \widetilde{Y}\_{\rm P} = \frac{\left(\widetilde{Y}\_{\rm O,\infty} + \widetilde{Y}\_{\rm P,\infty} - \delta \mathfrak{R}\right) + \left(\widetilde{Y}\_{\rm O,\infty} + \widetilde{Y}\_{\rm P,\infty} + \delta\right) \mathfrak{R}\widetilde{\mathfrak{s}}}{1 + \mathfrak{R}},\tag{34}
$$

$$
\widetilde{Y}\_{\rm O} + \widetilde{T} = \widetilde{Y}\_{\rm O,s} + \widetilde{T}\_{\rm s} + \left( \widetilde{Y}\_{\rm O,o} - \widetilde{Y}\_{\rm O,s} + \widetilde{T}\_{\rm o} - \widetilde{T}\_{\rm s} \right) \widetilde{\mathbf{y}}; \qquad \widetilde{Y}\_{\rm O,s} = \frac{\widetilde{Y}\_{\rm O,o} + \widetilde{T}\_{\rm o} - \widetilde{T}\_{\rm s} - \gamma}{1 + \mathfrak{B} + A\_{\rm s,O} [\mathfrak{B}/(-f\_{\rm s})]'} \tag{35}
$$

$$
\widetilde{Y}\_{\rm P} - \widetilde{T} = \widetilde{Y}\_{\rm P,s} - \widetilde{T}\_{\rm s} + \left( \widetilde{Y}\_{\rm P,\alpha} - \widetilde{Y}\_{\rm P,s} - \widetilde{T}\_{\rm \alpha} + \widetilde{T}\_{\rm s} \right) \xi; \qquad \widetilde{Y}\_{\rm P,s} = \frac{\widetilde{Y}\_{\rm P,\alpha} - \widetilde{T}\_{\rm \alpha} + \widetilde{T}\_{\rm s} + \gamma}{1 + \beta + A\_{\rm s,P} \left[ \beta \sqrt{-f\_{\rm s}} \right]}, \tag{36}
$$

$$
\widetilde{Y}\_{\rm N} = \widetilde{Y}\_{\rm N, \omega} \frac{1 + \mathfrak{P}\,\widetilde{\xi}}{1 + \mathfrak{P}} \,' \,. \tag{37}
$$

where

$$\xi = \frac{\int\_0^\eta \exp\left(-\int\_0^\eta f \, d\eta\right) d\eta}{\int\_0^\infty \exp\left(-\int\_0^\eta f \, d\eta\right) d\eta},\tag{38}$$

$$\alpha = \frac{\left(\widetilde{T}'\right)\_s}{\left(\xi'\right)\_s}, \qquad \beta = \frac{\left(-f\_s\right)}{\left(\xi'\right)\_s}, \qquad \left(\xi'\right)\_s = \frac{1}{\int\_0^\infty \exp\left(-\int\_0^\eta f \,d\eta\right)d\eta},\tag{39}$$

<sup>1</sup>The surface C-O2 reaction of half-order is also applicable (Makino, 1990).

and a prime indicates *d*/*d*. Using the new independent variable , the energy conservation Eq. (17) becomes

$$
\left(\frac{d^2\widetilde{T}}{d\xi^2}\right) = -\frac{Da\_\mathbb{g}\alpha\_\mathbb{g}}{\left(d\xi/d\eta\right)^2}\ . \tag{40}
$$

Therefore, the equations to be solved are Eqs. (15) and (40), subject to the boundary conditions in Eq. (25) and

$$\left(\widetilde{T}\right)\_{\xi=0} = \widetilde{T}\_{s\prime} \qquad \left(\widetilde{T}\right)\_{\xi=1} = \widetilde{T}\_{\infty\prime} \tag{41}$$

by use of (-*f*s) given by Eq. (31) and the coupling functions in Eqs. (33) to (36). Key parameters in solving those are *Da*g, *Da*s,O, *Da*s,P, and (-*f*s).

### **2.6 Transfer number and combustion rate**

258 Mass Transfer in Chemical Engineering Processes

reactions are assumed to be first order, for simplicity and analytical convenience.1 As for the kinetic expressions for non-permeable solid carbon, effects of porosity and/or internal surface area are considered to be incorporated, since surface reactions are generally

For self-similar flows, the normal velocity *v*s at the surface is expressible in terms of (-*f*s) by

Reminding the fact that the mass burning rate of solid carbon is given by *m* = (*v*)s, which is equivalent to the definition of the combustion rate [kg/(m2s)], then (-*f*s) can be identified as the nondimensional combustion rate. Note also that the surface reactions are less sensitive to velocity gradient variations than the gas-phase reaction because *Da*s ~ *a*-1/2 while *Da*g~ *a*-1.

With the boundary conditions for species, cast in the specific forms of Eqs. (27) to (29), the

*Y Y*

*Y Y Y Y*

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> ; <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup>

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> ; <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup>

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> P, P,

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> O, P, O, P,

<sup>O</sup> O,s <sup>s</sup> O, O,s <sup>s</sup> O,s 1

<sup>P</sup> P,s <sup>s</sup> P, P,s <sup>s</sup> P,s 1

,

1The surface C-O2 reaction of half-order is also applicable (Makino, 1990).

 *<sup>s</sup> sf* ,

*Y T Y T Y Y T T Y*

*Y T Y T Y Y T T Y*

 

1

 

1

*<sup>v</sup> <sup>f</sup> <sup>a</sup> <sup>j</sup>* <sup>2</sup> <sup>s</sup> <sup>s</sup> . (32)

*Y Y* , (33)

, (35)

, (36)

<sup>~</sup> <sup>~</sup> <sup>1</sup> *<sup>Y</sup>*<sup>N</sup> *<sup>Y</sup>*N, , (37)

O, s

P, s

*Y T T*

*Y T T*

*A f*

*A f*

, (38)

<sup>1</sup> , (39)

 

 *<sup>f</sup> <sup>d</sup> <sup>d</sup> <sup>s</sup>* 0 0 exp

*Y Y* , (34)

s,O <sup>s</sup>

s,P <sup>s</sup>

*f d d*

*f d d*

 <sup>1</sup>

 

0 0

 

exp

 

0 0 exp

controlled by combinations of chemical kinetics and pore diffusions.

coupling functions for the present system are given by

O P

 *<sup>s</sup> T <sup>s</sup>* ~

F P

**2.5 Coupling functions** 

where

The influence of finite rate gas-phase kinetics is studied here. The global rate equation used has the same form as that of Howard, et al. (1973), in which the activation energy and the frequency factor are reported to be Eg=113 kJ/mol and Bg=1.3108 [(mol/m3)s]-1, respectively. The combustion response is quite similar to that of particle combustion (Makino & Law, 1986), as shown in Fig. 1(a) (Makino, 1990). The parameter , indispensable in obtaining the combustion rate, is bounded by limiting solutions to be mentioned, presenting that the gasphase CO-O2 reaction reduces the surface C-O2 reaction by consuming O2, while at the same time initiating the surface C-CO2 reaction by supplying CO2, and that with increasing surface temperature the combustion rate can first increase, then decrease, and increase again as a result of the close coupling between the three reactions. In addition, the combustion process depends critically on whether the gas-phase CO-O2 reaction is activated. If it is not, the oxygen in the ambience can readily reach the surface to participate in the C-O2 reaction. Activation of the surface C-CO2 reaction depends on whether the environment contains any CO2. However, if the gas-phase CO-O2 reaction is activated, the existence of CO-flame in the gas phase cuts off most of the oxygen supplied to the surface such that the surface C-O2 reaction is suppressed. At the same time, the CO2 generated at the flame activates the surface C-CO2 reaction.

Fig. 1. Combustion behavior as a function of the surface temperature with the gas-phase Damköhler number taken as a parameter; *Da*s,O= *Da*s,P=108 and *Y*P,=0 (Makino, 1990). (a) Transfer number. (b) Nondimensional combustion rate.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

By using the coupling functions in Eqs. (33) to (36), it can be shown that

; <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup>

defined by

temperature distribution as

reported by Tsuji & Matsui (1976).

diffusion flame becomes a flame sheet.

*f* is the location of flame sheet, we have

 

 

*d*

   

*f f d dY*

 

 

*dY*<sup>F</sup> <sup>O</sup> ~ ~

fraction *Yi*, can be continuous across the flame.

 

*dT* <sup>F</sup> ~ ~ ~

*<sup>f</sup> <sup>f</sup> <sup>f</sup>*

 

*d dT*

the both sides of the reaction zone.

 

*d*

 

*d dY* or

3. Similarly, by evaluating the coupling function for CO and enthalpy, we have

or

 

*d*

suggesting that the amount of heat generated is equal to the heat, conducted away to

 

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 261

the other an attached flame-sheet, situated on the surface. The Flame-detached mode is

<sup>0</sup> <sup>~</sup> <sup>0</sup> <sup>~</sup>

 

s s,P

1

, <sup>~</sup> <sup>2</sup>

 

s

 1 1

O,

 

*Y <sup>f</sup> T T T T* . (50)

<sup>s</sup> O, <sup>s</sup>

Once (-*f*s) is determined from Eqs. (47) and (15), *f* can readily be evaluated, yielding the

<sup>s</sup> O, <sup>s</sup>

<sup>2</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> :

In addition, the infinitely large *Da*g yields the following important characteristics, as

1. The quantities *Y*F and *Y*O in the reaction rate g in Eq. (20) becomes zero, suggesting that fuel and oxygen do not coexist throughout the boundary layer and that the

2. In the limit of an infinitesimally thin reaction zone, by conducting an integration of the coupling function for CO and O2 across the zone, bounded between *f* - < < *f* +, where

> 

*d*

suggesting that fuel and oxidizer must flow into the flame surface in stoichiometric proportions. Here the subscript *f* + and *f* -, respectively, designate the oxygen and fuel sides of the flame. Note that in deriving Eq. (51), use has been made of an assumption that values of the individual quantities, such as the streamfunction *f* and species mass-

> 

 

 

*<sup>Y</sup>*

~ ~ O, P,

*Y Y*

*Y*<sup>O</sup> *<sup>f</sup> Y*<sup>F</sup> *<sup>f</sup>* . (46)

*f A* (47)

O, O,

*Y*

 

 

 

*<sup>f</sup> <sup>f</sup> d*

*W W*

*dY* <sup>O</sup> O O

 

*dY*

*<sup>f</sup> <sup>f</sup> <sup>f</sup> d*

 

 

*d dT*  

 

*q D*

*dT* <sup>F</sup> , (52)

*dY*

 

<sup>F</sup> <sup>F</sup> <sup>F</sup> , (51)

 

<sup>~</sup> <sup>2</sup>

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>0</sup> *<sup>f</sup>* : *<sup>T</sup> <sup>T</sup> <sup>Y</sup> <sup>T</sup> <sup>T</sup>* , (49)

*Tf T Y T T <sup>f</sup> <sup>f</sup>* (48)

It may informative to note that the parameter , defined as (-*f*s)/()s in the formulation, coincides with the conventional transfer number (Spalding, 1951), which has been shown by considering elemental carbon, (*W*C/*W*F)*Y*F+(*W*C/*W*P)*Y*P, taken as the transferred substance, and by evaluating driving force and resistance, determined by the transfer rate in the gas phase and the ejection rate at the surface, respectively (Makino, 1992; Makino, et al., 1998). That is,

$$\frac{\left(\frac{\mathcal{W}\_{\rm C}}{\mathcal{W}\_{\rm F}} + \frac{\mathcal{W}\_{\rm C}Y\_{\rm P}}{\mathcal{W}\_{\rm P}}\right)\_{\rm s} - \left(\frac{\mathcal{W}\_{\rm C}Y\_{\rm F}}{\mathcal{W}\_{\rm F}} + \frac{\mathcal{W}\_{\rm C}Y\_{\rm P}}{\mathcal{W}\_{\rm P}}\right)\_{\rm s}}{1 - \left(\frac{\mathcal{W}\_{\rm C}Y\_{\rm F}}{\mathcal{W}\_{\rm F}} + \frac{\mathcal{W}\_{\rm C}Y\_{\rm P}}{\mathcal{W}\_{\rm P}}\right)\_{\rm s}} = \frac{\left(\widetilde{\mathcal{Y}}\_{\rm F} + \widetilde{\mathcal{Y}}\_{\rm P}\right)\_{\rm s} - \left(\widetilde{\mathcal{Y}}\_{\rm F} + \widetilde{\mathcal{Y}}\_{\rm P}\right)\_{\rm s}}{\delta - \left(\widetilde{\mathcal{Y}}\_{\rm F} + \widetilde{\mathcal{Y}}\_{\rm P}\right)\_{\rm s}} = \boldsymbol{\mathfrak{B}}\,. \tag{42}$$

Figure 1(b) shows the combustion rate in the same conditions. At high surface temperatures, because of the existence of high-temperature reaction zone in the gas phase, the combustion rate is enhanced. In this context, the transfer number, less temperature-sensitive than the combustion rate, as shown in Figs. 1(a) and 1(b), is preferable for theoretical considerations.
