**6. Concluding remarks**

300 Mass Transfer in Chemical Engineering Processes

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

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,

Figure 6(a) shows the combustion rate as a function of the surface temperature with the velocity gradient taken as a parameter. Use has been made of a test specimen of C/Ccomposite with rectangular cross section (5 mm width; 8 mm thickness). The velocity gradient used here is defined as *a* = 2*V*/, where is the width; the maximum velocity gradient is limited to be 1300 s-1, because of air-supply system. Other experimental conditions are the same as those in Figs. 1(a) and/or 3(a). An abrupt decrease in the combustion rate, as well as the general combustion response can be observed in the same manner as that of a graphite rod, reported in the previous Sections. Figure 6(b) is a similar plot with the airflow temperature

0

Fig. 6. Combustion rate of C/C-composite (Makino, et al., 2006) as a function of the surface temperature; (a) with the velocity gradient of airflow taken as a parameter; (b) with the

1000 1500 2000 Surface temperature, K

C/C-Composite C=1.4x10<sup>3</sup>

 kg/m3 *a* =1300 s-1

*T* <sup>∞</sup>=320 K

*T* <sup>∞</sup>=1280 K

0.01

Combustion rate , kg/(m2

・s)

0.02

0.03

activating the surface C-CO2 reaction.

2007) when there exists CO2 in the oxidizer-flow.

taken as a parameter, presenting the same trend as that in Fig. 3(a).

C/C-Composite C=1.4x10<sup>3</sup>

 kg/m3 *T* <sup>∞</sup>=320 K

*a* =1300 s-1

*a* =600 s-1

1980 K

**5.2 Combustion rate of C/C-composite** 

0

airflow temperature taken as a parameter.

1000 1500 2000 Surface temperature, K

*T* s,ig =1740 K

(a) (b)

0.01

Combustion rate , kg/(m2

・s)

0.02

0.03

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 understanding.

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 can also exert influences on the combustion rate.

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 gradient.

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 theoretical results, through conducting experimental comparisons.

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

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

*m* dimensional mass burning (or combustion) rate *Q* ratio of heats of combustion in the gas phase *q* heat of combustion per unit mass of CO

*s* boundary-layer variable along the surface

*L* convective-diffusive operator

*Ro* universal gas constant *R* curvature of surface or radius

*Ta* activation temperature

*V* freestream velocity

*W* molecular weight *w* reaction rate

*Y* mass fraction

**Greek Symbols** 

viscosity

 density inner variable streamfunction reaction rate

**Subscripts** 

C carbon

F carbon monoxide *f* flame sheet

profile function

*u* velocity component along *x*

*v* velocity component along *y*

*x* tangential distance along the surface

stoichiometric CO2-to-reactant mass ratio

 perturbed temperature in the outer region perturbed temperature in the inner region

A water vapor or C-H2O surface reaction a critical value at flame attachment

product(CO2)-to-carbon mass ratio or boundary-layer thickness

measure of the thermal energy in the reaction zone relative to the activation energy

boundary-layer variable normal to the surface or perturbed concentration

thermal conductivity or parameter defined in the igninition analysis

*y* normal distance from the surface

 conventional transfer number temperature gradient at the surface reduced gas-phase Damköhler number

stoichiometric coefficient

*T* temperature

*t* time

width

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 303

accuracy, they are anticipated to make various contributions not only for qualitative and quantitative studies in facilitating understanding, but also for practical utility, such as designs of furnaces, combustors, ablative carbon heat-shields, and high-temperature structures with C/C-composites in various aerospace applications.

Finally, relevant to the High-Temperature Air Combustion, carbon combustion has been studied, by varying H2O mass-fraction up to 0.10. It has been found that the high H2O massfraction is unfavorable for the enhancement of combustion rate, especially in the medium temperature range, because establishment of the CO-flame is facilitated, and hence suppresses the combustion rate. To the contrary, at high surface temperatures (>2000 K), the high H2O mass-fraction is favorable because the water vapor participates in the surface reaction as an additional oxidizer. Theoretical results, obtained by additionally introducing the surface C-H2O reaction and the global gas-phase H2-O2 reaction into the previous formulation, have also suggested the usefulness of the explicit expressions for the combustion rate. As for the combustion in the humid airflow with relatively low velocity gradient, it is found that a new mode with suppressed H2-ejection from the surface can fairly represent the experimental observation.

Although essential feature of the carbon combustion has been captured to some extents, further progresses are strongly required for its firm understanding, because wide attention has been given to carbonaceous materials in various fields.
