**(IV) Flame-detached mode without H2:**

$$\mathfrak{B} \approx \left(\frac{KA\_{\mathrm{s,P}}}{1 + KA\_{\mathrm{s,P}}}\right) \left(\frac{2\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{O}}} Y\_{\mathrm{O},\alpha} + \frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{P}}} Y\_{\mathrm{P},\alpha}\right) + \left(\frac{KA\_{\mathrm{s,A}}}{1 + KA\_{\mathrm{s,P}}}\right) \left(\frac{\mathcal{W}\_{\mathrm{C}}}{\mathcal{W}\_{\mathrm{A}}} Y\_{\mathrm{A},\alpha}\right) \,. \tag{32}$$

As the correction factor *K* for the two-dimensional flow, we have Eq. (16) for the Frozen and Flame-attached modes; Eq. (18) for the Flame-detached mode, regardless of H2 ejection from the carbon surface.

### **4.3 Surface kinetic parameters and thermophysical properties**

In numerical calculations, use has been made of the kinetic parameters for the surface C-O2 and C-CO2 reactions, described in Section 5 in Part 1. For C-H2O reaction, the frequency factor *B*s,A=2107 m/s and activation energy *E*s,A=271 kJ/mol, determined after re-examining previous experimental results (Makino, et al., 1998a). As mentioned, effects of porosity and/or other surface characteristics are grouped into the kinetic parameters. Thermophysical properties are =1.10 kg/m3 and =1.9510-5 Pas for the roomtemperature airflow (*T*=320 K), while =0.276 kg/m3 and =5.1010-5 Pas for the hightemperature airflow (*T*=1280 K). As for the thermophysical properties of water vapor, =0.598 kg/m3 and =1.2210-5 Pas at *T*=370 K. Wilke's equation (Reid, et al., 1977) has been used in estimating viscosities of humid air.

### **4.4 Further consideration for experimental comparisons**

Experimental results have already been compared with theoretical results in Figs. 3 and 4, and a fair degree of agreement has been demonstrated in general, suggesting appropriateness of the analysis, including the choice of the thermophysical properties. However, Fig. 4(b) requires a further comment because theoretical result of the Flamedetached mode overestimates the combustion rate, especially at high surface temperatures *T*s. As assumed in the Flame-detached mode, CO and H2 produced at the surface reaction are to be transported to the flame and then oxidized. Generally speaking, however, H2 can easily been oxidized, compared to CO, especially at high temperatures. In addition, the velocity gradient (*a*=820 s-1) in Fig. 4(b) is not so high. In this situation, H2 produced at the surface reaction is considered to be completely consumed by the water-gas shift reaction (H2+CO2H2O+CO), so that the Flame-detached mode without H2 presented (Makino & Umehara, 2007) seems to be appropriate. A theoretical result is also shown in Fig. 4(b) by a dashed curve. We see that the agreement at high *T*s has much been improved, suggesting that this consideration is to the point.

### **5. Other results relevant to the high-temperature air combustion**

As one of the advantages for the High-Temperature Air Combustion, it has been pointed out that oxygen concentration in a furnace can be reduced without reducing combustion rate. In order to confirm this fact, an experiment has been conducted by varying O2 and CO2 concentrations in the high-temperature oxidizer-flow (Makino and Umehara, 2007). In 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 are shown.

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

298 Mass Transfer in Chemical Engineering Processes

P, P C

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

A,

 

O, O C

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

1 1

O, O C

s,P s,P

 

> 

2

*W W*

**4.3 Surface kinetic parameters and thermophysical properties** 

 

s,P s,P

*W W*

4

 

the carbon surface.

*KA*

been used in estimating viscosities of humid air.

that this consideration is to the point.

**4.4 Further consideration for experimental comparisons** 

1

**(IV) Flame-detached mode without H2:** 

*KA*

1/2

 

 

A, A C

> 

s,P s,A

A C

*Y W W*

s,A s,A

*KA KA*

> 

 

P, P C

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

As the correction factor *K* for the two-dimensional flow, we have Eq. (16) for the Frozen and Flame-attached modes; Eq. (18) for the Flame-detached mode, regardless of H2 ejection from

In numerical calculations, use has been made of the kinetic parameters for the surface C-O2 and C-CO2 reactions, described in Section 5 in Part 1. For C-H2O reaction, the frequency factor *B*s,A=2107 m/s and activation energy *E*s,A=271 kJ/mol, determined after re-examining previous experimental results (Makino, et al., 1998a). As mentioned, effects of porosity and/or other surface characteristics are grouped into the kinetic parameters. Thermophysical properties are =1.10 kg/m3 and =1.9510-5 Pas for the roomtemperature airflow (*T*=320 K), while =0.276 kg/m3 and =5.1010-5 Pas for the hightemperature airflow (*T*=1280 K). As for the thermophysical properties of water vapor, =0.598 kg/m3 and =1.2210-5 Pas at *T*=370 K. Wilke's equation (Reid, et al., 1977) has

Experimental results have already been compared with theoretical results in Figs. 3 and 4, and a fair degree of agreement has been demonstrated in general, suggesting appropriateness of the analysis, including the choice of the thermophysical properties. However, Fig. 4(b) requires a further comment because theoretical result of the Flamedetached mode overestimates the combustion rate, especially at high surface temperatures *T*s. As assumed in the Flame-detached mode, CO and H2 produced at the surface reaction are to be transported to the flame and then oxidized. Generally speaking, however, H2 can easily been oxidized, compared to CO, especially at high temperatures. In addition, the velocity gradient (*a*=820 s-1) in Fig. 4(b) is not so high. In this situation, H2 produced at the surface reaction is considered to be completely consumed by the water-gas shift reaction (H2+CO2H2O+CO), so that the Flame-detached mode without H2 presented (Makino & Umehara, 2007) seems to be appropriate. A theoretical result is also shown in Fig. 4(b) by a dashed curve. We see that the agreement at high *T*s has much been improved, suggesting

**5. Other results relevant to the high-temperature air combustion** 

As one of the advantages for the High-Temperature Air Combustion, it has been pointed out that oxygen concentration in a furnace can be reduced without reducing combustion rate. In order to confirm this fact, an experiment has been conducted by varying O2 and CO2 concentrations in the high-temperature oxidizer-flow (Makino and Umehara, 2007). In

*KA KA*

1

*KA* . (32)

*KA* , (31)

  *W*

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 the combustion rate is concerned.

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

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.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

there is no surface coating for protecting oxidation.

can also exert influences on the combustion rate.

**6. Concluding remarks** 

understanding.

gradient.

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 301

Theoretical results are also shown in Figs. 6(a) and 6(b). In obtaining these results, use has been made of kinetic parameters for the artificial graphite with higher density (C = 1.82103 kg/m3), after confirming the experimental fact that there appears no remarkable difference in the combustion rates in different graphite densities, because of the prevalence of combustion behavior in the diffusionally controlled regime in the present experimental conditions. As far as the trend and approximate magnitude are concerned, fair agreement is demonstrated, including the ignition surface-temperature. It should be noted that the combustion rate of the C/C-composite is nearly the same as that of artificial graphite when

In this monograph, combustion of solid carbon has been overviewed not only experimentally but also theoretically. As explained in Part 1, only the carbon combustion in the forward stagnation flowfield has been considered, in order to have a clear

In Part 1, by conducting the aerothermochemical analysis, based on the chemically reacting boundary layer, with considering the surface C-O2 and C-CO2 reactions and the gas-phase CO-O2 reaction, the generalized species-enthalpy coupling functions have successfully been derived, which demonstrate close coupling between the surface and gas-phase reactions that

Then, focus has been put on the ignition of CO-flame over the burning carbon in the prescribed flowfield, because establishment of the CO-flame in the gas phase can change the dominant surface reaction from the faster C-O2 reaction to the slower C-CO2 reaction, causing abrupt changes in the combustion rate. By further conducting the asymptotic expansion analysis, with using the generalized coupling functions, the explicit ignition criterion has been derived, suggesting that ignition is facilitated with increasing surface temperature and oxidizer concentration, while suppressed with decreasing velocity

Then, attempts have been made to estimate kinetic parameters for the surface and gas-phase reactions, indispensable for predicting combustion behavior, with using theoretical results obtained. A fair degree of agreement has been demonstrated between experimental and

In Part 2, a further study has been conducted in the stagnation flow with high velocity gradient, at least one order of magnitude higher than that ever used, in order to suppress the appearance of CO-flame. It is observed that the combustion rate increases monotonically and reaches the diffusion-limited value with increasing surface temperature when the velocity gradient is high, while there exists a discontinuous change in the combustion rate with increasing surface temperature, due to the establishment of CO-flame when the velocity gradient is low. In addition, an attempt has been made to obtain explicit combustion-rate expressions, presented by the transfer number in terms of the natural logarithmic term, just like that for droplet combustion. For the three limiting cases, explicit expressions have further been obtained by making an assumption of small combustion rate. It has even been found that before the establishment of CO-flame the combustion rate can fairly be represented by the expression in the Frozen mode, and that after the establishment of CO-flame the combustion rate can be represented by the expression in the Flame-attached and/or Flame-detached modes. Since the present expressions are explicit and have fair

theoretical results, through conducting experimental comparisons.

surface reaction and there is no gas phase reaction, the monotonic increase in the combustion rate is observed. The same comments as those in Fig. 3(a) can be made for the high-temperature oxidizer-flow although higher surface temperature *T*s is required in activating the surface C-CO2 reaction.

Finally, it is confirmed that as far as the combustion rates at around *T*s=2000 K are concerned, those in the high-temperature oxidizer-flows in Figs. 5(a) and 5(b) are nearly the same as that in the room-temperature airflow in Fig. 3(a) with the same mass flow rate. As pointed out (Makino, et al., 2003) that the O2 mass-fraction can be reduced down to about 0.14 in the High-Temperature Air Combustion, without reducing combustion rate, it has been confirmed that the O2 mass-fraction can further be reduced (Makino and Umehara, 2007) when there exists CO2 in the oxidizer-flow.
