**3.3 Combustion in humid airflow**

A further increase in the H2O mass-fraction can considerably change the combustion behavior (Makino & Umehara, 2007). The H2O mass-fraction *Y*<sup>A</sup> is now increased to be 0.10, the dew point of which is as high as 328 K (55°C). Note that this H2O mass-fraction is even higher than that ever used in the previous studies with humid airflow (Matsui, et al., 1983; 1986), by virtue of a small-sized boiler installed in the experimental apparatus. Figure 4(a) shows the combustion rate in the high-temperature airflow with *a*=3300 s-1, as a function of the surface temperature *T*s. The O2 mass-fraction is reduced, because of the increased H2O concentration. It is seen that the combustion rate increases first gradually and then rapidly with increasing surface temperature. This trend is quite different from that in Figs. 3(a) or 3(b).

In order to elucidate causes for this trend, theoretical results are obtained, with additional surface C-H2O and global gas-phase H2-O2 reactions taken into the formulation (Makino & Umehara, 2007), which will be explained later. Not only results in the Frozen and Flamedetached modes, but also that in the Flame-attached mode is shown. In the Flame-attached mode, it is assumed that combustion products of the surface reactions can immediately be

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

 *T* <sup>∞</sup> (K) *a* (s-1) ◆ 1280 3300 *Y* A=0.10, *Y* O=0.21, *Y* P=0.00 ρC=1.25×103

*T* s,ig=1380 K

Frozen

kg/m3

0

expressions.

1000 1500 2000

Surface tempareture *T* s , K

(a) (b)

0.04

0

1000 1500 2000

(c) Fig. 4. Combustion rate in humid airflow (Makino & Umehara, 2007) with the H2O massfraction *Y*A=0.10, as a function of the surface temperature *T*s; (a) in the high-temperature airflow with the velocity gradient *a*=3300 s-1; (b) in the room-temperature airflow with the same mass flow rate (*a*=820 s-1); (c) in the room-temperature airflow with the same velocity gradient. Data points are experimental and curves are results of the explicit combustion-rate

 *T* <sup>∞</sup> (K) *a* (s-1) △ 370 3300 *Y* A=0.10, *Y* O=0.21, *Y* P=0.00 ρC=1.25×103

kg/m3

Frozen Flame-detached

Flame-attached

*T* s,ig=1690 K

Surface tempareture *T* s , K

0.01

0.02

Combustion rate , kg/(m2

・s)

0.03

Flame-attached

0

1000 1500 2000

*T* s,ig=1420 K

Frozen

 *T* <sup>∞</sup> (K) *a* (s-1) 〇 370 820 *Y* A=0.10, *Y* O=0.21, *Y* P=0.00 ρC=1.25×103

kg/m3

Flame-detached without H2

Surface tempareture *T* s , K

Flame-attached

Flame-detached

0.01

0.02

Combustion rate , kg/(m2

・s)

0.03

0.04

Flame-detached

0.01

0.02

Combustion rate , kg/(m2

・s)

0.03

0.04

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 295

oxidized, so that neither CO nor H2 is ejected into the gas phase. It is seen that experimental results at temperatures lower than about 1500 K are close to the theoretical result of the Flame-attached mode, while those at temperatures higher than about 1700 K are close to the result of the Flame-detached mode. The ignition surface-temperature is predicted to be 1380 K. From these comparisons, we can deduce that because of the high H2O mass-fraction, as well as the high-temperature airflow, the CO-flame established at 1380 K adheres to the carbon surface. The combustion in the Flame-attached mode prevails until CO-ejection becomes strong enough to separate the CO-flame from the surface. As the surface temperature is increased, the CO-flame detaches, so that the combustion proceeds in the Flame-detached mode. The rapid increase in the combustion rate at high temperatures can be attributed to the participation of the C-H2O reaction.

Figure 4(b) shows the combustion rate in the room-temperature airflow with the same mass flow rate (*a*=820 s-1). The airflow temperature, being raised to *T*=370 K for preventing condensation of water vapor, cannot be called as the "room" temperature, any more, but its terminology is retained to distinguish it from the high-temperature. It is seen that the combustion rate gradually increases with increasing surface temperature. Compared to Fig. 4(a), the combustion rate around 1500 K is nearly the same as that in the high-temperature airflow. So, we can say that when the H2O concentration is high, there is no merit to use the high-temperature airflow, until the water vapor begins to participate in the surface reaction as another oxidizer at about 1700 K or higher. A difference in the combustion rates at high temperatures becomes large because no remarkable increase in the combustion rate is observed, although the water vapor is anticipated to participate in the surface reaction. A further consideration will be made later.

Theoretical results are also shown in Fig. 4(b). The ignition surface-temperature is now predicted to be 1420 K. We see that the combustion rate experimentally obtained locates in the middle of the theoretical results in the Frozen and Flame-attached modes, after the establishment of CO-flame, suggesting that the gas-phase reaction proceeds in a finite rate, because the airflow is neither fast in velocity nor high in temperature. One more thing to be noted is the combustion behavior at high temperatures, presenting that the combustion rate in the experiment cannot reach the theoretical result that the Flame-detached mode predicts, about which it will be discussed later.

Figure 4(c) shows the combustion rate in the room-temperature airflow with *a*=3300 s-1. Nearly the same trend as that in Figs. 3(a) and/or 3(b) with low velocity gradient is shown. Because the airflow temperature is low, the establishment of CO-flame is retarded until the surface temperature reaches about 1700 K, and the combustion rate up to this temperature is about double of that in the high-temperature airflow. The rapid increase at high temperatures can be attributed to the contribution of the surface C-H2O reaction.

Theoretical results are also shown in Fig. 4(c). Until the establishment of CO-flame at *T*s = 1690 K predicted, we see again that the Frozen mode can fairly represent the combustion behavior. At high temperatures at which the CO-flame has already been established, the combustion behavior is fairly represented by the Flame-detached mode.

### **4. Extended formulation for the carbon combustion**

Theoretical study (Makino & Umehara, 2007) has been conducted for the system with three surface reactions and two global gas-phase reactions, by extending the previous formulation. Although some of the assumptions introduced in Section 2 in Part 1 are not

oxidized, so that neither CO nor H2 is ejected into the gas phase. It is seen that experimental results at temperatures lower than about 1500 K are close to the theoretical result of the Flame-attached mode, while those at temperatures higher than about 1700 K are close to the result of the Flame-detached mode. The ignition surface-temperature is predicted to be 1380 K. From these comparisons, we can deduce that because of the high H2O mass-fraction, as well as the high-temperature airflow, the CO-flame established at 1380 K adheres to the carbon surface. The combustion in the Flame-attached mode prevails until CO-ejection becomes strong enough to separate the CO-flame from the surface. As the surface temperature is increased, the CO-flame detaches, so that the combustion proceeds in the Flame-detached mode. The rapid increase in the combustion rate at high temperatures can

Figure 4(b) shows the combustion rate in the room-temperature airflow with the same mass flow rate (*a*=820 s-1). The airflow temperature, being raised to *T*=370 K for preventing condensation of water vapor, cannot be called as the "room" temperature, any more, but its terminology is retained to distinguish it from the high-temperature. It is seen that the combustion rate gradually increases with increasing surface temperature. Compared to Fig. 4(a), the combustion rate around 1500 K is nearly the same as that in the high-temperature airflow. So, we can say that when the H2O concentration is high, there is no merit to use the high-temperature airflow, until the water vapor begins to participate in the surface reaction as another oxidizer at about 1700 K or higher. A difference in the combustion rates at high temperatures becomes large because no remarkable increase in the combustion rate is observed, although the water vapor is anticipated to participate in the surface reaction. A

Theoretical results are also shown in Fig. 4(b). The ignition surface-temperature is now predicted to be 1420 K. We see that the combustion rate experimentally obtained locates in the middle of the theoretical results in the Frozen and Flame-attached modes, after the establishment of CO-flame, suggesting that the gas-phase reaction proceeds in a finite rate, because the airflow is neither fast in velocity nor high in temperature. One more thing to be noted is the combustion behavior at high temperatures, presenting that the combustion rate in the experiment cannot reach the theoretical result that the Flame-detached mode predicts,

Figure 4(c) shows the combustion rate in the room-temperature airflow with *a*=3300 s-1. Nearly the same trend as that in Figs. 3(a) and/or 3(b) with low velocity gradient is shown. Because the airflow temperature is low, the establishment of CO-flame is retarded until the surface temperature reaches about 1700 K, and the combustion rate up to this temperature is about double of that in the high-temperature airflow. The rapid increase at high

Theoretical results are also shown in Fig. 4(c). Until the establishment of CO-flame at *T*s = 1690 K predicted, we see again that the Frozen mode can fairly represent the combustion behavior. At high temperatures at which the CO-flame has already been established, the

Theoretical study (Makino & Umehara, 2007) has been conducted for the system with three surface reactions and two global gas-phase reactions, by extending the previous formulation. Although some of the assumptions introduced in Section 2 in Part 1 are not

temperatures can be attributed to the contribution of the surface C-H2O reaction.

combustion behavior is fairly represented by the Flame-detached mode.

**4. Extended formulation for the carbon combustion** 

be attributed to the participation of the C-H2O reaction.

further consideration will be made later.

about which it will be discussed later.

Fig. 4. Combustion rate in humid airflow (Makino & Umehara, 2007) with the H2O massfraction *Y*A=0.10, as a function of the surface temperature *T*s; (a) in the high-temperature airflow with the velocity gradient *a*=3300 s-1; (b) in the room-temperature airflow with the same mass flow rate (*a*=820 s-1); (c) in the room-temperature airflow with the same velocity gradient. Data points are experimental and curves are results of the explicit combustion-rate expressions.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

*<sup>Y</sup>*O,s ,

P,s

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

 

s,O

O, O C

O, O C

 

O, O C

> 

 

*Y W W*

 

2

s,O s,P

 

  2

*W W*

2

*W W*

s,P s,P

s,P s,P

s,O s,O

2

*W <sup>W</sup> KA*

1

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

determined by use of other coupling functions O <sup>F</sup> <sup>H</sup>

Here, *Q*

been used, so that we have

have from Eq. (25) as

oxidized. For this mode, we have

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

as

**(I) Frozen mode:** 

 

  *KA*

1

**(II) Flame-attached mode:** 

1 2

**(III) Flame-detached mode:** 

 

2 1

 

2 1

 

*KA*

*KA KA*

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 297

phase. For evaluating , the temperature profile *T* = *T*s + (*Tf* - *T*s)(/*f*) inside the flame has

*<sup>T</sup> <sup>T</sup> <sup>Y</sup> <sup>Q</sup> <sup>Y</sup> <sup>Q</sup> <sup>Y</sup>* A, s O, A, A,s

where the coupling function in Eq. (24) is evaluated at the flame. By further using *f* and *Y*A,*<sup>f</sup>*,

*A*

A,s *<sup>Y</sup>*

s,A

 1

By use of the approximate relation in Eq. (4), analytical expressions for can be obtained

 

O, s,P

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

 

*KA KA*

*KA KA*

1

 

1

*KA*

*KA KA* , (30)

<sup>1</sup> *<sup>Y</sup>*

1 1

A,

A,

A,

<sup>1</sup> *<sup>Y</sup>*

P, P C

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

<sup>1</sup> *<sup>Y</sup>*

P, P C

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

O C

*KA KA*

s,P s,P

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

*Y Y*

The other mode that has been found (Makino & Umehara, 2007) is the **Flame-detached mode without H2**, in which there exists no H2 in the gas phase because it can easily be

) <sup>~</sup> (1 <sup>~</sup>

~ is the ratio of the heats of combustion of the H2-O2 and CO-O2 reactions in the gas

 

 

1

 

A,

*Y*

( )

*f*

s

*<sup>Y</sup>* ,

) <sup>~</sup> (1 <sup>~</sup> <sup>~</sup> <sup>~</sup> , (26)

 

<sup>~</sup> <sup>~</sup> *<sup>Y</sup> <sup>Y</sup>* , respectively, we

*f f*

 

*<sup>f</sup> Y*

*f*

<sup>~</sup> <sup>~</sup> <sup>~</sup> *<sup>Y</sup> <sup>Y</sup> Y* and H <sup>A</sup>

 

 

 

> P C

> >

  *W W*

*W W*

s,A s,A

s,A s,A

*W*

P, P C

*Y W W*

*KA* , (29)

*KA KA*

P, s,A

O, O C

O, O C

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

1 <sup>~</sup> <sup>~</sup> A, A,s

*Y* , (28)

 

s,A s,A

A C

A C

*W*

A C

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

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

*Y W W*

 

> 

 

  2

 

A, A C

 

*Y*

*Y* . (27)

2

O,

~

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

appropriate for systems with hydrogen species, use has been made of those as they are, for tractability, in order to capture fundamental aspects of the carbon combustion under prescribed situations.

### **4.1 Mass fractions of oxidizers at the carbon surface**

By extending Eq. (31) in Part 1, so as to include contribution of the C-H2O reaction, the combustion rate (–*f*s) can be expressed as

$$\delta(-f\_{\rm s}) = A\_{\rm s,O} \widetilde{Y}\_{\rm O,s} + A\_{\rm s,P} \widetilde{Y}\_{\rm P,s} + A\_{\rm s,A} \widetilde{Y}\_{\rm A,s} \,. \tag{20}$$

Again, use has been made of an assumption that all the surface reactions are the first-order. The reduced surface Damköhler number *As,i*, the surface Damköhler number *Das,i*, and the stoichiometrically weighted mass fraction, relevant to the oxidizing species *i* (=O, P, A) are also defined in the same manner as those in Section 2 in Part 1.

Although *Yi*,s must be determined through numerical calculations when the gas-phase kinetics is finite, they can be determined analytically for some limiting cases, as mentioned. One of them is the **Frozen mode,** in which we have

$$\widetilde{Y}\_{i,\sf s} = \frac{Y\_{\sf i,\sf cs}}{1 + \sf B + A\_{\sf s,i} \left[\sf B / (-f\_{\sf s})\right]} \quad \text{( $i = O$ , P, A)}.\tag{21}$$

Another is the **Flame-attached mode** in which CO and H2 produced at the surface reactions are immediately consumed, so that it looks that the CO-flame adheres to the surface. In the same manner (Makino, et al., 1998b), we have

$$
\widetilde{Y}\_{\rm O,s} = \frac{\widetilde{Y}\_{\rm O,\,\alpha} - 2\delta\mathfrak{B}}{1 + \mathfrak{B}} \; , \quad \widetilde{Y}\_{\rm P,s} = \frac{\widetilde{Y}\_{\rm P,\,\alpha} + \delta\mathfrak{B}}{1 + \mathfrak{B}} \; , \quad \widetilde{Y}\_{\rm A,s} = \frac{\widetilde{Y}\_{\rm A,\,\alpha}}{1 + \mathfrak{B}} \; . \tag{22}
$$

The third is the **Flame-detached mode** in which the gas-phase reaction is infinitely fast and the CO-flame locates in the gas phase. Although a coupling function

$$
\widetilde{Y}\_{\rm O,s} + \widetilde{Y}\_{\rm P,s} + \widetilde{Y}\_{\rm A,s} = \frac{\widetilde{Y}\_{\rm O,o} + \widetilde{Y}\_{\rm P,o} + \widetilde{Y}\_{\rm A,o} - \delta \mathfrak{B}}{1 + \mathfrak{B}} \tag{23}
$$

can easily be obtained and we can also put *Y*O,s = 0 for this combustion situation, a separation of *Y*A,s from *Y*P,s is not straightforward. For this aim, it is needed to take account of another species-enthalpy coupling function, say, (Makino & Umehara, 2007)

$$
\widetilde{T} + \widetilde{Y}\_{\rm O} + (1 - \widetilde{Q})\widetilde{Y}\_{\rm A} \,\,\,\,\,\tag{24}
$$

then we have

$$\widetilde{Y}\_{\rm A,s} = \frac{1}{1-\widetilde{Q}} \frac{\widetilde{T}\_{\rm ox} - \widetilde{T}\_{\rm s} + \widetilde{Y}\_{\rm O,\alpha} + (1-\widetilde{Q})\widetilde{Y}\_{\rm A,\alpha} - \gamma}{1 + \mathfrak{P} + A\_{\rm s,A} \left[\mathfrak{P}/(-f\_{\rm s})\right]}. \tag{25}$$

Here, *Q* ~ is the ratio of the heats of combustion of the H2-O2 and CO-O2 reactions in the gas phase. For evaluating , the temperature profile *T* = *T*s + (*Tf* - *T*s)(/*f*) inside the flame has been used, so that we have

$$\gamma = \widetilde{T}\_{\infty} - \widetilde{T}\_{\text{s}} + \widetilde{Y}\_{\text{O},\text{or}} + (1 - \widetilde{Q})\widetilde{Y}\_{\text{A},\text{or}} + (1 - \widetilde{Q}) \left( \widetilde{Y}\_{\text{A},s} \frac{1 - \widetilde{\xi}\_{f}}{\widetilde{\xi}\_{f}} - \frac{\widetilde{Y}\_{\text{A},f}}{\widetilde{\xi}\_{f}} \right) . \tag{26}$$

where the coupling function in Eq. (24) is evaluated at the flame. By further using *f* and *Y*A,*<sup>f</sup>*, determined by use of other coupling functions O <sup>F</sup> <sup>H</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> *<sup>Y</sup> <sup>Y</sup> Y* and H <sup>A</sup> <sup>~</sup> <sup>~</sup> *<sup>Y</sup> <sup>Y</sup>* , respectively, we have from Eq. (25) as

$$\widetilde{Y}\_{\rm A,s} = \frac{\widetilde{Y}\_{\rm A,\,\alpha}}{1 + \mathfrak{H} + A\_{\rm s,A} \frac{\mathfrak{P}}{(-f\_{\rm s})} \left(1 - \frac{\widetilde{Y}\_{\rm O,\,\alpha}}{2\mathfrak{S}\mathfrak{P}}\right)} \,\tag{27}$$

The other mode that has been found (Makino & Umehara, 2007) is the **Flame-detached mode without H2**, in which there exists no H2 in the gas phase because it can easily be oxidized. For this mode, we have

$$\widetilde{Y}\_{\rm O,s} = 0 \,, \quad \widetilde{Y}\_{\rm P,s} = \frac{\widetilde{Y}\_{\rm O,o} + \widetilde{Y}\_{\rm P,o} - \delta \mathfrak{R}}{1 + \mathfrak{B}} \,, \qquad \widetilde{Y}\_{\rm A,s} = \frac{\widetilde{Y}\_{\rm A,o}}{1 + \mathfrak{B}} \, , \tag{28}$$

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

By use of the approximate relation in Eq. (4), analytical expressions for can be obtained as

**(I) Frozen mode:** 

296 Mass Transfer in Chemical Engineering Processes

appropriate for systems with hydrogen species, use has been made of those as they are, for tractability, in order to capture fundamental aspects of the carbon combustion under

By extending Eq. (31) in Part 1, so as to include contribution of the C-H2O reaction, the

s s,O O,s s,P P,s s,A A,s

Again, use has been made of an assumption that all the surface reactions are the first-order. The reduced surface Damköhler number *As,i*, the surface Damköhler number *Das,i*, and the stoichiometrically weighted mass fraction, relevant to the oxidizing species *i* (=O, P, A) are

Although *Yi*,s must be determined through numerical calculations when the gas-phase kinetics is finite, they can be determined analytically for some limiting cases, as mentioned.

1 /( )

*i*

i, ,s *<sup>A</sup> <sup>f</sup> Y*

s, s

Another is the **Flame-attached mode** in which CO and H2 produced at the surface reactions are immediately consumed, so that it looks that the CO-flame adheres to the surface. In the

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

The third is the **Flame-detached mode** in which the gas-phase reaction is infinitely fast and

can easily be obtained and we can also put *Y*O,s = 0 for this combustion situation, a separation of *Y*A,s from *Y*P,s is not straightforward. For this aim, it is needed to take account

O A

) <sup>~</sup> (1 <sup>~</sup> <sup>~</sup> <sup>~</sup>

*T T Y Q Y*

A,s *A f*

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

of another species-enthalpy coupling function, say, (Makino & Umehara, 2007)

*Y*

 1

*<sup>Y</sup>* ,

 

*Y Y Y* (23)

~

. (25)

1

~

1 /( )

 

s,A s s O, A,

*Y Y Y*

<sup>~</sup> <sup>~</sup> <sup>~</sup> ( *<sup>f</sup>* ) *<sup>A</sup> <sup>Y</sup> <sup>A</sup> <sup>Y</sup> <sup>A</sup> <sup>Y</sup>* . (20)

*<sup>i</sup>* (*i* = O, P, A). (21)

1 <sup>~</sup> <sup>~</sup> A, A,s

) <sup>~</sup> (1 <sup>~</sup> <sup>~</sup> *<sup>T</sup> <sup>Y</sup> <sup>Q</sup> <sup>Y</sup>* , (24)

*Y* . (22)

*Y*

prescribed situations.

**4.1 Mass fractions of oxidizers at the carbon surface** 

also defined in the same manner as those in Section 2 in Part 1.

 1

the CO-flame locates in the gas phase. Although a coupling function

O,s P,s A,s

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

*Q*

~ 1

*Y*

*<sup>Y</sup>* ,

One of them is the **Frozen mode,** in which we have

same manner (Makino, et al., 1998b), we have

O,s

then we have

~

*Y*

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

*Y*

combustion rate (–*f*s) can be expressed as

$$\otimes \approx \left(\frac{KA\_{\rm s,O}}{1+KA\_{\rm s,O}}\right) \left(\frac{2\mathcal{W}\_{\rm C}}{\mathcal{W}\_{\rm O}} Y\_{\rm O,o}\right) + \left(\frac{KA\_{\rm s,P}}{1+KA\_{\rm s,P}}\right) \left(\frac{\mathcal{W}\_{\rm C}}{\mathcal{W}\_{\rm P}} Y\_{\rm P,o}\right) + \left(\frac{KA\_{\rm s,A}}{1+KA\_{\rm s,A}}\right) \left(\frac{\mathcal{W}\_{\rm C}}{\mathcal{W}\_{\rm A}} Y\_{\rm A,o}\right), \tag{29}$$

**(II) Flame-attached mode:** 

$$\beta \approx \left(\frac{1}{1 + 2KA\_{\mathrm{s,O}} - KA\_{\mathrm{s,P}}}\right) \left(KA\_{\mathrm{s,O}} \frac{2\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{O}}} Y\_{\mathrm{O},\mathrm{or}} + KA\_{\mathrm{s,P}} \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{P}}} Y\_{\mathrm{P},\mathrm{or}} + KA\_{\mathrm{s,A}} \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{A}}} Y\_{\mathrm{A},\mathrm{or}}\right), \tag{30}$$

**(III) Flame-detached mode:** 

$$\begin{split} \boldsymbol{\theta} & \approx \frac{1}{2} \left\{ \frac{KA\_{\mathrm{s},\mathrm{P}}}{1+KA\_{\mathrm{s},\mathrm{P}}} \left( \frac{2\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{O}}} \mathcal{Y}\_{\mathrm{O},\mathrm{o}} + \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{P}}} \mathcal{Y}\_{\mathrm{P},\mathrm{o}} \right) + \frac{KA\_{\mathrm{s},\mathrm{A}}}{1+KA\_{\mathrm{s},\mathrm{A}}} \left( \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{O}}} \mathcal{Y}\_{\mathrm{O},\mathrm{o}} + \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{A}}} \mathcal{Y}\_{\mathrm{A},\mathrm{o}} \right) \right\} \\ & + \frac{1}{2} \left[ \left\{ \frac{KA\_{\mathrm{s},\mathrm{P}}}{1+KA\_{\mathrm{s},\mathrm{P}}} \left( \frac{2\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{O}}} \mathcal{Y}\_{\mathrm{O},\mathrm{o}} + \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{P}}} \mathcal{Y}\_{\mathrm{P},\mathrm{o}} \right) - \frac{KA\_{\mathrm{s},\mathrm{A}}}{1+KA\_{\mathrm{s},\mathrm{A}}} \left( \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{O}}} \mathcal{Y}\_{\mathrm{O},\mathrm{o}} + \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{A}}} \mathcal{Y}\_{\mathrm{A},\mathrm{o}} \right) \right\}^{2} \end{split}$$

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

**5.1 Effects of O2 and CO2 in the oxidizer-flow** 

the combustion rate is concerned.

0

1000 1500 2000

 *T* <sup>∞</sup> (K) *a* (s-1) △ 320 3300 ◆ 1280 3300 〇 320 820 *Y* O=0.105, *Y* P=0.10 ρC=1.25×10<sup>3</sup>

kg/m3

Surface tempareture , K

(a) (b)

0.01

Combustion rate , kg/(m2

・s)

0.02

0.03

are shown.

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 299

addition, combustion rate of C/C-composite in the high-temperature airflow has been examined (Makino, et al., 2006) in a similar way, relevant to evaluation of protection properties from oxidation. In this Section, those results not presented in previous Sections

Experimental conditions for the O2 and/or CO2 concentrations in the high-temperature oxidizer-flow have been chosen to have the same combustion rate as that in the roomtemperature airflow, at around *T*s=2000 K, shown in Fig. 3(a). Figure 5(a) shows the combustion rate in the high-temperature oxidizer-flow, as a function of the surface temperature *T*s. The O2 and CO2 mass-fractions are set to be 0.105 and 0.10, respectively. The H2O mass-fraction *Y*A=0.001 or less. Because of the monotonic increase in the combustion rate, the combustion rate at 2000 K is nearly equal to that in the room-temperature airflow, shown in Figs. 3(a) and 3(b), experienced the abrupt decreases in the combustion rate upon the establishment of CO-flame, although it is generally suppressed, because of the reduced O2 mass-fraction. For comparisons, results in the room-temperature oxidizer-flows with the same mass flow rate and the same velocity gradient are also shown in Fig. 5(a), the general trend of which is in accordance with that in the airflow shown in Figs. 3(a) and 3(b), as far as

Figure 5(b) shows the combustion rate as a function of *T*s, with CO2 taken as the only oxidizer. The CO2 mass-fraction is set to be 0.39. Since CO2 is the only oxidizer for the

0

1000 1500 2000

 *T* <sup>∞</sup> (K) *a* (s-1) △ 320 3300 ◆ 1280 3300 〇 320 820 *Y* O=0.00, *Y* P=0.39 ρC=1.25×103

kg/m3

Surface tempareture , K

0.01

Combustion rate , kg/(m2

Fig. 5. Combustion rate in the high-temperature oxidizer-flow with the velocity gradient *a* = 3300 s-1, as a function of the surface temperature (Makino and Umehara, 2007). The H2O mass-fraction *Y*A=0.001 or less. Notation is the same as that in Fig. 3. (a) The O2 and CO2 mass-fractions are 0.105 and 0.10, respectively; (b) The CO2 mass-fraction is 0.39.

・s)

0.02

0.03

$$+4\frac{KA\_{\rm s,P}}{1+KA\_{\rm s,P}} \left(\frac{\mathcal{W}\_{\rm C}}{\mathcal{W}\_{\rm O}} Y\_{\rm O,\alpha} + \frac{\mathcal{W}\_{\rm C}}{\mathcal{W}\_{\rm P}} Y\_{\rm P,\alpha} \left| \frac{KA\_{\rm s,A}}{1+KA\_{\rm s,A}} \left(\frac{\mathcal{W}\_{\rm C}}{\mathcal{W}\_{\rm A}} Y\_{\rm A,\alpha} \right) \right|^{1/2} \right) \tag{31}$$
