**1. Introduction**

282 Mass Transfer in Chemical Engineering Processes

Yang, R. T. & Steinberg, M. (1977). A Diffusion Cell Method for Studying Heterogeneous

ISSN 0196-4313.

Kinetics in the Chemical Reaction/Diffusion Controlled Region. Kinetics of C + CO2→ 2CO at 1200-1600°C. *Ind. Eng. Chem. Fundam.*, Vol. 16, No. 2, pp. 235-242,

> Carbon combustion is a research subject, indispensable for practical utilization of coal combustion, ablative carbon heat-shields, and/or aerospace applications with carboncarbon composites (C/C-composites). Because of this practical importance, extensive research has been conducted not only experimentally but also theoretically and/or numerically, and several reviews (Batchelder, et al., 1953; Gerstein & Coffin, 1956; Walker, et al., 1959; Clark, et al., 1962; Khitrin, 1962; Mulcahy & Smith, 1969; Maahs, 1971; Rosner, 1972; Essenhigh, 1976, 1981; Annamalai & Ryan, 1993; Annamalai, et al., 1994) describe the accomplishments in this field, as mentioned in Part 1. Nevertheless, because of the complexities involved, there still remain several problems that must be clarified to understand basic nature of carbon combustion.

> In Part 1, after describing general characteristics of the carbon combustion, it was intended to represent it by use of some of the basic characteristics of the chemically reacting boundary layers (Chung, 1965; Law, 1978), under recognition that flow configurations are indispensable for proper evaluation of the heat and mass transfer, especially for the situation in which the gas-phase reaction can intimately affect overall combustion response through its coupling to the surface reactions. The flow configuration chosen was that of the stagnation-flow, which is a well-defined, one-dimensional flow, being characterized by a single parameter, called the stagnation velocity gradient, offering various advantages for mathematical analyses, experimental data acquisition, and/or physical interpretations.

> Specifically, formulation of the governing equations was first presented in Part 1, based on theories on the chemically reacting boundary layer. Chemical reactions considered were the surface C-O2 and C-CO2 reactions and the gas-phase CO-O2 reaction. Generalized speciesenthalpy coupling functions were then derived without assuming any limit or near-limit behaviors, which not only enable us to minimize the extent of numerical efforts needed for generalized treatment, but also provide useful insight into the conserved scalars in the carbon combustion. After that, it was shown that straightforward derivation of the combustion response could be allowed in the limiting situations, such as those for the Frozen, Flame-detached, and Flame-attached modes.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

appearance of CO-flame.

in the Forward Stagnation Region - Part 2 - Combustion Rate in Special Environments 285

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

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

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

(a) (b)

density; curves are results of the explicit combustion-rate expressions.

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

increases. In Fig. 1(a), the ignition surface-temperature predicted is also marked.

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 order for further comparisons with experimental results.

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 atmospheres.

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

Concluding remarks not only for Part 2 but also for Part 1 are made in Section 6, with references cited and nomenclature tables.
