**2. Combustion response in high stagnation flowfields**

It has been recognized that phenomena of carbon combustion become complicated upon the appearance of CO-flame, as pointed out in Part 1. Then, another simpler combustion response, being anticipated to be observed by suppressing its appearance, by use of the high velocity gradients, would provide useful insight into the carbon combustion, as well as facilitate deeper understanding for it. In addition, under simplified situations, there is a possibility that we could find out an explicit combustion-rate expression that can further contribute much to the foundation of theoretical understanding of the carbon combustion, offering mathematical simplifications, just like that in droplet combustion. Various contributions to practical applications, such as designs of furnaces, combustors, ablative carbon heat-shields, and high-temperature structures with C/C-composites in oxidizing atmosphere, are also anticipated.

### **2.1 Experimental results for the combustion rate**

Figure 1(a) shows the combustion rate (Makino, et al., 1998b) as a function of the surface temperature, with the velocity gradient taken as a parameter. The H2O mass-fraction in

Next, after presenting profiles of gas-phase temperature, measured over the burning carbon, a further analytical study was conducted about the ignition phenomenon, related to finite-rate kinetics in the gas phase, by use of the asymptotic expansion method to obtain a critical condition for the appearance of the CO-flame. Appropriateness of this criterion was further examined by comparing temperature distributions in the gas phase and/or surface temperatures at which the CO-flame could appear. After having constructed these theories, evaluations of kinetic parameters for the surface and gas-phase reactions were then conducted,

In this Part 2, it is intended to make use of the information obtained in Part 1, for exploring carbon combustion, further. First, in order to decouple the close coupling between surface and gas-phase reactions, an attempt is conducted to raise the velocity gradient as high as possible, in Section 2. It is also endeavored to obtain explicit combustion-rate expressions, even though they might be approximate, because they are anticipated to contribute much to the foundation of theoretical understanding of carbon combustion, offering mathematical simplifications, just like that in droplet combustion, and to the practical applications, such as designs of ablative carbon heat shields and/or structures with C/C-composites in oxidizing

After having examined appropriateness of the explicit expressions, carbon combustion in the high-temperature airflow is then examined in Section 3, relevant to the High-Temperature Air Combustion, which is anticipated to have various advantages, such as energy saving, utilization of low-calorific fuels, reduction of nitric oxide emission, etc. The carbon combustion in the high-temperature, humid airflow is also examined theoretically in Section 4, by extending formulations for the system with three surface reactions and two global gas-phase reactions. Existence of a new burning mode with suppressed H2 ejection from the surface can be confirmed for the carbon combustion at high temperatures when the velocity gradient of the humid airflow is relatively low. Some other results relevant to the

Concluding remarks not only for Part 2 but also for Part 1 are made in Section 6, with

It has been recognized that phenomena of carbon combustion become complicated upon the appearance of CO-flame, as pointed out in Part 1. Then, another simpler combustion response, being anticipated to be observed by suppressing its appearance, by use of the high velocity gradients, would provide useful insight into the carbon combustion, as well as facilitate deeper understanding for it. In addition, under simplified situations, there is a possibility that we could find out an explicit combustion-rate expression that can further contribute much to the foundation of theoretical understanding of the carbon combustion, offering mathematical simplifications, just like that in droplet combustion. Various contributions to practical applications, such as designs of furnaces, combustors, ablative carbon heat-shields, and high-temperature structures with C/C-composites in oxidizing

Figure 1(a) shows the combustion rate (Makino, et al., 1998b) as a function of the surface temperature, with the velocity gradient taken as a parameter. The H2O mass-fraction in

in order for further comparisons with experimental results.

High-Temperature Air Combustion are further shown in Section 5.

**2. Combustion response in high stagnation flowfields** 

references cited and nomenclature tables.

atmosphere, are also anticipated.

**2.1 Experimental results for the combustion rate** 

atmospheres.

airflow is set to be 0.003. Data points are experimental and solid curves are results of combustion-rate expressions to be mentioned. When the velocity gradient is 200 s-1, the same trend as those in Figs. 2 and 8 in Part 1 is observed. That is, with increasing surface temperature, the combustion rate first increases, then decreases abruptly, and again increases. In Fig. 1(a), the ignition surface-temperature predicted is also marked.

As the velocity gradient is increased up to 640 s-1, the combustion rate becomes high, due to an enhanced oxidizer supply, but the trend is still the same. A further increase in the velocity gradient, however, changes the trend. When the velocity gradient is 1300 s-1, which is even higher than that ever used in the previous experimental studies (Matsui, et al., 1975; 1983; 1986), the combustion rate first increases, then reaches a plateau, and again increases, as surface temperature increases. Since the ignition surface-temperature is as high as 1970 K, at which the combustion rate without CO-flame is nearly the same as that with CO-flame, no significant decrease occurs in the combustion rate. On the contrary, a careful observation suggests that there is a slight, discontinuous increase in the combustion rate just after the appearance of CO-flame.

Since the ignition surface-temperature strongly depends on the velocity gradient (Visser & Adomeit, 1984; Makino & Law, 1990), as explained in Section 4 in Part 1, the discontinuous change in the combustion rate, caused by the appearance of CO-flame, ceases to exist with

Fig. 1. Combustion rate as a function of the surface temperature, with the velocity gradient taken as a parameter (Makino, et al., 1998b), (a) when there appears CO-flame within the experimental conditions; (b) when the velocity gradient is at least one order of magnitude higher than that ever used in the previous studies. Oxidizer is air and its H2O mass-fraction is 0.003. Data points are experimental with the test specimen of 1.25103 kg/m3 in graphite density; curves are results of the explicit combustion-rate expressions.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

(a) (b)

curves are those of the explicit expressions.

one order of magnitude smaller than the third term (/2)1/2.

of the following relation (Makino, 1992; Makino, et al., 1998b).

approximate expressions for the transfer number.

 

*KA KA*

1

**Frozen mode:** 

Fig. 2.(a) A profile of the streamfunction *f* for the two-dimensional stagnation flow, as a function of the boundary-layer variable , when the surface temperature *Ts* 1450 K, the ambient temperature *T* 320 K, and the combustion rate (-*fs*) = 0.10. The solid curve is the result obtained by a numerical calculation, and the dashed curve is the simplified profile used to find out the approximate expression (Makino, et al., 1998b). (b) Combustion rates for the three limiting modes in the stagnation airflow as a function of the surface temperature when the surface Damköhler number for the C-O2 reaction, *DasO*, and that for the C-CO2 reaction, *DasP*, are 108. The solid curves are results of the implicit expressions and dashed

Equation (2) shows that the combustion rate (-*fs*) can be expressed by the transfer number in terms of the logarithmic term, ln(1+). Note that the first and second terms in Eq. (3) are

In order to obtain the specific form of the transfer number , a two term expansion of the exponential function is expected to be sufficient because (-*fs*)<<1, so that use has been made

<sup>1</sup> exp . <sup>1</sup> *<sup>s</sup> <sup>s</sup> <sup>K</sup> <sup>f</sup> <sup>K</sup> <sup>f</sup>*

By virtue of this relation, Eqs. (44), (47), and (55) in Part 1 can yield the following

 

O, s,P

 

*Y KA*

P,

 

s,O s,O

 

~

*KA*

1

(4)

 

s,P

 

~

*Y*

 

(5)

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 287

increasing velocity gradient, as shown in Fig. 1(b). Here, use has been made of a graphite rod with a small diameter (down to 5 mm), as well as airflow with high velocity (up to 50 m/s). We see that the combustion rate increases monotonically with increasing surface temperature. Note that the velocity gradient used here is at least one order of magnitude higher than those in previous works.

As for the "negative temperature coefficient" of the combustion rate, examined in the literature (*cf*. Essenhigh, 1981), a further comment is required because it completely disappears at high velocity gradients. This experimental fact suggests that it has nothing to do with chemical events, related to the surface reactions, hitherto examined. Although it is described in the literature that some (Nagel and Strickland-Constable, 1962) attributed it to the sites of surface reactions and others (Yang and Steinberg, 1977) did it to the reaction depth, Figs. 1(a) and 1(b) certainly suggest that this phenomenon is closely related to the gas-phase reaction, which can even be blown off when the velocity gradients are high.

### **2.2 Approximate, explicit expressions for the combustion rate**

In order to calculate the combustion rate, temperature profiles in the gas phase must be obtained by numerically solving the energy conservation equation for finite gas-phase reaction kinetics. However, if we note that carbon combustion proceeds with nearly frozen gas-phase chemistry until the establishment of the CO-flame (Makino, et al., 1994; Makino, et al., 1996) and that the combustion is expected to proceed under nearly infinite gas-phase kinetics once the CO-flame is established, analytically-obtained combustion rates (Makino, 1990; Makino, 1992), presented in Section 3 in Part 1, are still useful for practical utility.

However, it should also be noted that the combustion-rate expressions thus obtained are implicit, so that further numerical calculations are required by taking account of the relation, (-*fs*)/()*s*, which is a function of the streamfunction *f*. Since this procedure is slightly complicated and cannot be used easily in practical situations, explicit expressions are anxiously required, in order to make these results more useful.

In order to elucidate the relation between the nondimensional combustion rate (-*fs*) and the transfer number (Spalding, 1951), dependence of ()*s* on the profile of the streamfunction *f* is first to be examined, by introducing a simplified profile of *f* as

$$f = \begin{cases} f\_s & \text{( $0 \le \eta \le \eta\_\*$ )} \\ b\eta + c & \text{( $\eta \le \eta \le \eta\_{\*\*}$ )}, \\ \eta + d & \text{( $\eta \ge \eta\_{\*\*}$ )} \end{cases} \tag{1}$$

as shown in Fig. 2(a), and then conducting an integration. Here, *b*, *c*, and *d* are constants, *f*(\*) = *fs*, and *f*(\*\*) = *fo*.

Recalling the definitions of and ()*s*, and making use of a relation, (-*fs*)<<1, as is the case for most solid combustion, we have the following approximate relation:

$$\mathbf{1} + \emptyset \approx \exp\left[\mathbf{K}(-f\_s)\right] \text{ or } \left(-f\_s\right) \approx \frac{\ln(1+\emptyset)}{K},\tag{2}$$

where

$$K = \eta\_{\sf{v}} + \sqrt{\frac{\pi}{2}} \left[ \left\{ \exp\left(\frac{(b-1)f\_o}{2b}\right) - 1 \right\} \text{erfc}\left(\frac{f\_o}{\sqrt{2}}\right) - \left(1 - \frac{1}{\sqrt{b}}\right) \text{erf}\left(\frac{f\_o}{\sqrt{2}}\right) \right] + \sqrt{\frac{\pi}{2}}.\tag{3}$$

Fig. 2.(a) A profile of the streamfunction *f* for the two-dimensional stagnation flow, as a function of the boundary-layer variable , when the surface temperature *Ts* 1450 K, the ambient temperature *T* 320 K, and the combustion rate (-*fs*) = 0.10. The solid curve is the result obtained by a numerical calculation, and the dashed curve is the simplified profile used to find out the approximate expression (Makino, et al., 1998b). (b) Combustion rates for the three limiting modes in the stagnation airflow as a function of the surface temperature when the surface Damköhler number for the C-O2 reaction, *DasO*, and that for the C-CO2 reaction, *DasP*, are 108. The solid curves are results of the implicit expressions and dashed curves are those of the explicit expressions.

Equation (2) shows that the combustion rate (-*fs*) can be expressed by the transfer number in terms of the logarithmic term, ln(1+). Note that the first and second terms in Eq. (3) are one order of magnitude smaller than the third term (/2)1/2.

In order to obtain the specific form of the transfer number , a two term expansion of the exponential function is expected to be sufficient because (-*fs*)<<1, so that use has been made of the following relation (Makino, 1992; Makino, et al., 1998b).

$$\frac{\mathfrak{B}}{1+\mathfrak{B}} = 1 - \exp[-K(-f\_s)] \approx K(-f\_s) \tag{4}$$

By virtue of this relation, Eqs. (44), (47), and (55) in Part 1 can yield the following approximate expressions for the transfer number.

#### **Frozen mode:**

286 Mass Transfer in Chemical Engineering Processes

increasing velocity gradient, as shown in Fig. 1(b). Here, use has been made of a graphite rod with a small diameter (down to 5 mm), as well as airflow with high velocity (up to 50 m/s). We see that the combustion rate increases monotonically with increasing surface temperature. Note that the velocity gradient used here is at least one order of magnitude

As for the "negative temperature coefficient" of the combustion rate, examined in the literature (*cf*. Essenhigh, 1981), a further comment is required because it completely disappears at high velocity gradients. This experimental fact suggests that it has nothing to do with chemical events, related to the surface reactions, hitherto examined. Although it is described in the literature that some (Nagel and Strickland-Constable, 1962) attributed it to the sites of surface reactions and others (Yang and Steinberg, 1977) did it to the reaction depth, Figs. 1(a) and 1(b) certainly suggest that this phenomenon is closely related to the gas-phase reaction, which can even be blown off when the velocity gradients are high.

In order to calculate the combustion rate, temperature profiles in the gas phase must be obtained by numerically solving the energy conservation equation for finite gas-phase reaction kinetics. However, if we note that carbon combustion proceeds with nearly frozen gas-phase chemistry until the establishment of the CO-flame (Makino, et al., 1994; Makino, et al., 1996) and that the combustion is expected to proceed under nearly infinite gas-phase kinetics once the CO-flame is established, analytically-obtained combustion rates (Makino, 1990; Makino, 1992), presented in Section 3 in Part 1, are still useful for practical utility. However, it should also be noted that the combustion-rate expressions thus obtained are implicit, so that further numerical calculations are required by taking account of the relation, (-*fs*)/()*s*, which is a function of the streamfunction *f*. Since this procedure is slightly complicated and cannot be used easily in practical situations, explicit expressions are

In order to elucidate the relation between the nondimensional combustion rate (-*fs*) and the transfer number (Spalding, 1951), dependence of ()*s* on the profile of the streamfunction *f*

as shown in Fig. 2(a), and then conducting an integration. Here, *b*, *c*, and *d* are constants,

Recalling the definitions of and ()*s*, and making use of a relation, (-*fs*)<<1, as is the case

*<sup>s</sup>* <sup>1</sup> exp *<sup>K</sup> <sup>f</sup>* or , ln <sup>1</sup>

1

 *<sup>o</sup> <sup>o</sup> of erf <sup>b</sup> <sup>f</sup> erfc <sup>b</sup>*

 

 

*d b c f*

 

\*\* \* \*\* \* 0

*<sup>K</sup> fs*

 . <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>1</sup>

 

2

*<sup>b</sup> <sup>f</sup> <sup>K</sup>* (3)

  , (1)

(2)

 

 

 

**2.2 Approximate, explicit expressions for the combustion rate** 

anxiously required, in order to make these results more useful.

is first to be examined, by introducing a simplified profile of *f* as

for most solid combustion, we have the following approximate relation:

2 1

  2

 

exp <sup>2</sup>

 

 

\*

 

*s*

*f*

*f*(\*) = *fs*, and *f*(\*\*) = *fo*.

where

higher than those in previous works.

$$\beta \approx \left(\frac{KA\_{\rm s,O}}{1 + KA\_{\rm s,O}}\right) \left(\frac{\widetilde{Y}\_{\rm O,o}}{\delta}\right) + \left(\frac{KA\_{\rm s,P}}{1 + KA\_{\rm s,P}}\right) \left(\frac{\widetilde{Y}\_{\rm P,o}}{\delta}\right) \tag{5}$$

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

the coupling function in Eq. (33) in Part 1, we have

**2.4 Approximate expression for the correction factor** *K*

*D <sup>h</sup> <sup>x</sup> Nu*

*<sup>R</sup> <sup>x</sup>*

two-dimensional stagnation flow with

   

 *a T Sc <sup>T</sup> <sup>h</sup> <sup>R</sup>* <sup>2</sup> 0.540 <sup>~</sup> ~

based on the analogy between heat and mass transfers, where

 

*<sup>T</sup> <sup>h</sup> <sup>R</sup>* D 0.6 0.570 <sup>~</sup> ~

 

 *a T Sc*

temperature *TR* in the boundary layer.

as (Katto, 1982; White, 1988). Two-dimensional Stagnation Flow:

Axisymmetric Stagnation Flow:

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 289

Furthermore, by evaluating mass fluxes at the surface and in the gas phase, with the elemental carbon, (*W*C/*W*F)*Y*F+(*W*C/*W*P)*Y*P, taken as the transferred substance, and by use of

suggesting that the correction factor *K* depends on both ()*s* and the representative

In obtaining an approximate expression for the factor *K*, it seems that we can use the accomplishment in the field of heat and mass transfer. The mass-transfer coefficient is given

<sup>D</sup> 0.6 from 0.4 0.763 *Sc*

*R*

However, this kind of expression is far from satisfactory because Eqs. (13) and (14) are originally obtained for heat-transfer problems without mass transfer. In addition, this relation is obtained under an assumption that there is no density change, even though there

Because of the simultaneous existence of temperature and concentration distributions in the carbon combustion, we are required to obtain an approximate expression for the factor *K* in another way. In this attempt, (/2)1/2 in the factor *K* in Eq. (3) is kept as it is because it is 1/()*s* for inviscid stagnation flow without mass ejection from the surface. The remaining part of the factor *K* is then determined by use of numerical results (Makino, 1990; Makino, et al., 1994; Makino, et al., 1996). In this determination, use has been made of a curve-fitting method, with (*T*/*Ts*) taken as a variable, to have a simple function. It has turned out that we can fairly represent the combustion rate for the Frozen and/or Flame-attached modes in

*<sup>R</sup> <sup>x</sup>*

, , , . <sup>D</sup>

*<sup>U</sup> ax Sc Ux Re*

exist temperature and/or concentration distributions in the gas phase.

*T <sup>T</sup> <sup>h</sup> <sup>j</sup> <sup>s</sup> <sup>R</sup>* <sup>2</sup> <sup>~</sup> ~

   

*a*

<sup>D</sup> , (12)

 from 0.4 0.570 *Sc Re Nu*

*<sup>x</sup>* , (13)

*<sup>x</sup>* , (14)

*D*

*x*

*Re Nu*

 (15)

,

*<sup>T</sup> <sup>K</sup>* (16)

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

 

2

*T T*

s s

1 ~ ~

 

*T*

 

~

*x*

**Flame-detached mode:** 

$$\beta \approx \left(\frac{KA\_{\rm s,P}}{1 + KA\_{\rm s,P}}\right) \left(\frac{\widetilde{Y}\_{\rm O,\alpha} + \widetilde{Y}\_{\rm P,\infty}}{\delta}\right) \tag{6}$$

### **Flame-attached mode:**

$$\beta \approx \left(\frac{KA\_{\rm s,O}}{1 + 2KA\_{\rm s,O} - KA\_{\rm s,P}}\right) \left(\frac{\widetilde{Y}\_{\rm O,o}}{\delta}\right) + \left(\frac{KA\_{\rm s,P}}{1 + 2KA\_{\rm s,O} - KA\_{\rm s,P}}\right) \left(\frac{\widetilde{Y}\_{\rm P,o}}{\delta}\right) \tag{7}$$

Although these are approximate, the transfer number can be expressed explicitly, in terms of the reduced surface Damköhler numbers, *A*s,O and *A*s,P, and O2 and CO2 concentrations in the freestream.

In addition, we have

$$KA\_{\mathbf{s},i} = k\_{\mathbf{s},i} \frac{K}{\sqrt{2^{j} \, a \left(\mu\_{\infty}/\rho\_{\infty}\right)}} ; \; k\_{\mathbf{s},i} \equiv B\_{\mathbf{s},i} \Big(\frac{\widetilde{T}\_{\infty}}{\widetilde{T}\_{\mathbf{s}}}\Big) \exp\left(-\frac{\widetilde{T}a\_{\mathbf{s},i}}{\widetilde{T}\_{\mathbf{s}}}\right) \tag{8}$$

where *k*s,*<sup>i</sup>* is the specific reaction rate constant for the surface reaction. Note that the factor, *K*/[2*j a* (/)]1/2, in Eq. (8) also appears in the combustion rate defined in Eq. (32) in Part 1, by use of the relation in Eq. (2), as

$$\dot{m} \approx \rho\_{\infty} \frac{\sqrt{2^{\frac{1}{d}} a \left(\mu\_{\infty}/\rho\_{\infty}\right)}}{K} \ln(1+\emptyset). \tag{9}$$

### **2.3 Correction factor** *K* **and mass-transfer coefficient**

In order to elucidate the physical meaning of the factor, *K*/[2*j a* (/)]1/2, let us consider a situation that << 1, with the Frozen mode combustion taken as an example. Then, Eq. (9) leads to the following result:

$$\dot{m} \approx \frac{1}{\frac{1}{k\_{\rm s,O}} + \frac{K}{\sqrt{2^{\frac{1}{d}}a\left(\mu\_{\infty}/\rho\_{\infty}\right)}}} \rho\_{\rm o} \left(\frac{\tilde{\mathbf{Y}}\_{\rm O,o}}{\delta}\right) + \frac{1}{\frac{1}{k\_{\rm s,P}} + \frac{K}{\sqrt{2^{\frac{1}{d}}a\left(\mu\_{\infty}/\rho\_{\infty}\right)}}} \rho\_{\rm o} \left(\frac{\tilde{\mathbf{Y}}\_{\rm P,o}}{\delta}\right). \tag{10}$$

We see that this expression is similar to the well-known expression for the solid combustion rate,

$$\dot{m} = \frac{1}{\frac{1}{k\_s} + \frac{1}{h\_D}} (\rho Y\_O)\_{\infty} \,\prime \tag{11}$$

for the first-order kinetics (Fischbeck, 1933; Fischbeck, et al., 1934; Tu, et al., 1934; Frank-Kamenetskii, 1969). Here, *hD* is the overall convective mass-transfer coefficient. It is seen that the factor, [2*j a* (/)]1/2/*K*, corresponds to the mass-transfer coefficient *h*D, suggesting that the specific form of *h*D is of use in determining a form of the correction factor *K*.

Furthermore, by evaluating mass fluxes at the surface and in the gas phase, with the elemental carbon, (*W*C/*W*F)*Y*F+(*W*C/*W*P)*Y*P, taken as the transferred substance, and by use of the coupling function in Eq. (33) in Part 1, we have

$$\mathcal{H}\_{\rm D} = \left(\frac{\widetilde{T}\_{\rm R}}{\widetilde{T}\_{\infty}}\right) (\xi')\_{s} \sqrt{2^{\frac{1}{3}} a \left(\mu\_{\infty}/\rho\_{\infty}\right)}\,,\tag{12}$$

suggesting that the correction factor *K* depends on both ()*s* and the representative temperature *TR* in the boundary layer.

### **2.4 Approximate expression for the correction factor** *K*

In obtaining an approximate expression for the factor *K*, it seems that we can use the accomplishment in the field of heat and mass transfer. The mass-transfer coefficient is given as (Katto, 1982; White, 1988).

Two-dimensional Stagnation Flow:

288 Mass Transfer in Chemical Engineering Processes

 

> 

 

 ln <sup>1</sup> . <sup>2</sup> 

where *k*s,*<sup>i</sup>* is the specific reaction rate constant for the surface reaction. Note that the factor, *K*/[2*j a* (/)]1/2, in Eq. (8) also appears in the combustion rate defined in Eq. (32) in Part 1,

In order to elucidate the physical meaning of the factor, *K*/[2*j a* (/)]1/2, let us consider a situation that << 1, with the Frozen mode combustion taken as an example. Then, Eq. (9)

s,

*Y*

*k*

1 <sup>1</sup> <sup>~</sup>

. (10)

1 *Y*

 

We see that this expression is similar to the well-known expression for the solid combustion

 *<sup>O</sup>*

*k h <sup>m</sup>* <sup>1</sup> <sup>1</sup>

the specific form of *h*D is of use in determining a form of the correction factor *K*.

*s D*

for the first-order kinetics (Fischbeck, 1933; Fischbeck, et al., 1934; Tu, et al., 1934; Frank-Kamenetskii, 1969). Here, *hD* is the overall convective mass-transfer coefficient. It is seen that the factor, [2*j a* (/)]1/2/*K*, corresponds to the mass-transfer coefficient *h*D, suggesting that

2

*a*

s s, s, ~

*T <sup>T</sup> <sup>k</sup> <sup>B</sup> <sup>i</sup>*

exp ~ ~

*Y KA*

 

Although these are approximate, the transfer number can be expressed explicitly, in terms of the reduced surface Damköhler numbers, *A*s,O and *A*s,P, and O2 and CO2 concentrations in

P,

 

 O, P, s,P s,P

 

s,O s,P s,O

*m*

**2.3 Correction factor** *K* **and mass-transfer coefficient** 

*K*

*a*

*KA KA KA*

*KA KA*

> 

 

~

<sup>2</sup> <sup>s</sup> , s, ;

 *K a*

> 

*j P j O*

*Y*

O,

 

*j*

1

 

s,O s,P

 

*<sup>i</sup> <sup>i</sup>* (*i* = O, P) , (8)

s s,

*T Ta*

(9)

<sup>1</sup> , (11)

*K*

~

*KA KA*

 

 

(6)

 

> 

 

 

P,

~

(7)

 

 

~

*Y*

 

O, s,P

1 2

~ ~

*Y Y*

**Flame-detached mode:** 

**Flame-attached mode:** 

the freestream. In addition, we have

 

by use of the relation in Eq. (2), as

leads to the following result:

*m*

rate,

2

*a*

s,

*k*

1

*<sup>K</sup> KA <sup>k</sup> <sup>j</sup> <sup>i</sup> <sup>i</sup>*

1 2

$$h\_{\rm D} = \left(\frac{\widetilde{T}\_R}{\widetilde{T}\_\infty}\right) \frac{0.570}{Sc^{0.6}} \sqrt{a \left(\mu\_\alpha/\rho\_\alpha\right)} \quad \text{from} \quad \frac{Nu\_\chi}{\sqrt{Re\_\chi}} = 0.570 \text{ Sc}^{0.4} \tag{13}$$

Axisymmetric Stagnation Flow:

 *a T Sc <sup>T</sup> <sup>h</sup> <sup>R</sup>* <sup>2</sup> 0.540 <sup>~</sup> ~ <sup>D</sup> 0.6 from 0.4 0.763 *Sc Re Nu x <sup>x</sup>* , (14)

based on the analogy between heat and mass transfers, where

$$\text{Nu}\_{\text{x}} = \frac{\text{h}\_{\text{D}} \text{x}}{\text{D}\_{R}}, \qquad \text{Re}\_{\text{x}} = \frac{\rho\_{R} \text{IJx}}{\mu\_{R}}, \qquad \text{UI} = \text{ax}\_{\text{A}} \qquad \text{Sc} = \frac{\mu/\rho}{\text{D}}. \tag{15}$$

However, this kind of expression is far from satisfactory because Eqs. (13) and (14) are originally obtained for heat-transfer problems without mass transfer. In addition, this relation is obtained under an assumption that there is no density change, even though there exist temperature and/or concentration distributions in the gas phase.

Because of the simultaneous existence of temperature and concentration distributions in the carbon combustion, we are required to obtain an approximate expression for the factor *K* in another way. In this attempt, (/2)1/2 in the factor *K* in Eq. (3) is kept as it is because it is 1/()*s* for inviscid stagnation flow without mass ejection from the surface. The remaining part of the factor *K* is then determined by use of numerical results (Makino, 1990; Makino, et al., 1994; Makino, et al., 1996). In this determination, use has been made of a curve-fitting method, with (*T*/*Ts*) taken as a variable, to have a simple function. It has turned out that we can fairly represent the combustion rate for the Frozen and/or Flame-attached modes in two-dimensional stagnation flow with

$$K = \left(\frac{\widetilde{T}\_{\infty}}{\widetilde{T}\_{\text{s}}}\right) \left(1 - \frac{\widetilde{T}\_{\infty}}{2\widetilde{T}\_{\text{s}}}\right) + \sqrt{\frac{\pi}{2}}\,'\,\tag{16}$$

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

combustion situation is that just after the establishment of CO-flame.

about 2500 K in the surface temperature.

**3. High-temperature air combustion** 

**3.1 Combustion in relatively dry airflow** 

High-Temperature Air Combustion.

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 291

When the surface temperature is higher than the ignition surface-temperature, Eq. (2) with for the Flame-detached mode in Eq. (6) and *K* in Eq. (18) can fairly represent the experimental results, except for the temperatures near the ignition surface-temperature, especially, in airflow with low velocity-gradient, say, 200 s-1. In this temperature range, we can use Eq. (2) with for the Flame-attached mode in Eq. (7) and *K* in Eq. (16) although accuracy of this prediction is not so high, compared to the other cases. This is attributed to the fact that we cannot assume the gas-phase reaction rate infinitely fast because the

When the velocity gradient is high, as shown in Fig. 1(b), the expression in Eq. (2) with for the Frozen mode in Eq. (5) and *K* in Eq. (16) fairly represents the experimental results, up to

Here, carbon combustion has been examined, relevant to the High-Temperature Air Combustion, characterized by use of hot air (~1280 K) and attracted as one of the new technology concepts for pursuing energy saving and/or utilization of low-calorific fuels. Although it has been confirmed to reduce NO*x* emission through reduction of O2 concentration in furnaces, without reducing combustion rate of gaseous and/or liquid fuels (Katsuki & Hasegawa, 1998; Tsuji, et al., 2003), its appropriateness for solid-fuel combustion has not been examined fully. Since solid fuels are commonly used as one of the important energy sources in industries, it is strongly required to examine its appropriateness from the fundamental viewpoint. Here, focus is put on examinations for the promoting and suppressing effects that the temperature and water vapor in the airflow have. From the practical point of view, the carbon combustion in airflow at high temperatures, especially, in high velocity gradients, is related to evaluation of ablative carbon heat-shield for atmospheric re-entry. As for that in airflow at high H2O concentrations, it is related to evaluation of protection properties of rocket nozzles, made of carbonaceous materials, from erosive attacks of water vapor, contained in working fluid for propulsion, as well as the coal/char combustion in such environments with an appreciable amount of water vapor.

Figure 3(a) shows the combustion rate as a function of the surface temperature *T*s, with the airflow temperature *T* taken as a parameter. The H2O mass-fraction *Y*A=0.003 in the airflow, considered to be dry, practically. The combustion rate in the high-temperature airflow (*T*=1280 K), shown by a solid diamond, increases monotonically and reaches the diffusion-limited value with increasing *T*s. Monotonic change in the combustion rate is attributed to the high velocity gradient (*a*=3300 s-1), which is too high for the CO-flame to be established (Makino, et al., 2003), so that the combustion here is considered to proceed solely with the surface C-O2 reaction. Note that this velocity gradient has been chosen, so as to suppress the abrupt changes in the combustion rates, in order to clarify effects of the

Results in the room-temperature airflow (*T*=320 K) with the same mass flow rate (*a*=820 s-1) are also shown. The combustion rate first increases, then decreases abruptly, and again increases, with increasing *T*s, as explained in the previous Section. The ignition surfacetemperature observed is about 1800 K, in accordance with the abrupt decrease in the

within 3% error when the O2 mass-fraction *Y*O, is 0.233 (cf. Fig. 2(b); Makino, et al., 1998b); for *Y*O,=0.533, error is within 5%; for *Y*O,=1, error is within 8%. Examinations have been made in the range of the surface Damköhler numbers *Da*s,O and *Da*s,P from 106 to 1010, that of the surface temperature *T*s from 1077 K to 2424 K, and that of the freestream temperature *T* from 323 K to 1239 K. The Frozen and Flame-attached modes can fairly be correlated by the single Eq. (16) because the gas-phase temperature profiles are the same. Note that the combustion rate in high O2 concentrations violates the assumption that (-*fs*)<<1. Nonetheless, the expressions appear to provide a fair representation because these expressions vary as the natural logarithm of the transfer number.

For axisymmetric stagnation flow, it turns out that the combustion rate in the Frozen and/or Flame-attached modes can fairly be represented with

$$K = \sqrt{\frac{2}{3} \left( \frac{\widetilde{T}\_{\alpha}}{\widetilde{T}\_{s}} \right) \left( 1 - \frac{\widetilde{T}\_{\alpha}}{2 \widetilde{T}\_{s}} \right)} + \sqrt{\frac{\pi}{2}} \,\tag{17}$$

within 3% error for *Y*O,=0.233 (Makino, et al., 1998b); within 5% error for *Y*O,=0.7. Difference in the forms between Eq. (16) and Eq. (17) can be attributed to the difference in the flow configuration.

For the combustion rate in the Flame-detached mode, not only the surface and freestream temperatures but also the oxidizer concentration must be taken into account. It has turned out that

$$K = \left(\frac{\widetilde{T}\_{\infty}}{\widetilde{T}\_{s}}\right)\left(1 - \frac{\widetilde{T}\_{\infty}}{2\widetilde{T}\_{s}}\right) - 0.05\left(1 + 2\widetilde{Y}\_{\text{O,\,\,\,\text{O}}}\right) + \sqrt{\frac{\pi}{2}}\tag{18}$$

can fairly represent the combustion rate in two-dimensional stagnation flow, within 4% error when the O2 mass-fraction *Y*<sup>O</sup> is 0.233 and 0.533, although the error becomes 6% near the transition state for the flame attaches. In an oxygen flow, the error is within 6% except for the transition state, while it increases up to 15% around the state.

For axisymmetric stagnation flow, the combustion rate in the Flame-detached mode can be represented with

$$K = \sqrt{\frac{2}{3} \left( \frac{\widetilde{T}\_{\alpha}}{\widetilde{T}\_{s}} \right) \left( 1 - \frac{\widetilde{T}\_{\alpha}}{2 \widetilde{T}\_{s}} \right)} - 0.05 \left( 1 + 2 \widetilde{Y}\_{\text{O},\alpha} \right) + \sqrt{\frac{\pi}{2}}.\tag{19}$$

The error is nearly the same as that for the two-dimension case.

### **2.5 Experimental comparisons at high velocity gradients**

In order to verify the validity of the explicit combustion-rate expressions, comparisons have been made with their values and the experimental results (Makino, et al., 1998b). Kinetic parameters are those evaluated in Section 5 in Part 1. The values of thermophysical properties are those at *T*=320 K, which yields =2.1210-5 kg2/(m4・s) and /=1.7810-5 m2/s. Results for the explicit combustion-rate expressions are shown in Figs. 1(a) and 1(b) by solid curves. As shown in Fig. 1(a), up to the ignition surfacetemperature, a reasonable prediction can be made by Eq. (2), with the transfer number for the Frozen mode in Eq. (5) and the correction factor *K* in Eq. (16), for two-dimensional case. When the surface temperature is higher than the ignition surface-temperature, Eq. (2) with for the Flame-detached mode in Eq. (6) and *K* in Eq. (18) can fairly represent the experimental results, except for the temperatures near the ignition surface-temperature, especially, in airflow with low velocity-gradient, say, 200 s-1. In this temperature range, we can use Eq. (2) with for the Flame-attached mode in Eq. (7) and *K* in Eq. (16) although accuracy of this prediction is not so high, compared to the other cases. This is attributed to the fact that we cannot assume the gas-phase reaction rate infinitely fast because the combustion situation is that just after the establishment of CO-flame.

When the velocity gradient is high, as shown in Fig. 1(b), the expression in Eq. (2) with for the Frozen mode in Eq. (5) and *K* in Eq. (16) fairly represents the experimental results, up to about 2500 K in the surface temperature.
