**1. Introduction**

250 Mass Transfer in Chemical Engineering Processes

Wen, Q., Wu, Y., Cao, D., Zhao, L., Sun, Q.(2009). *Electricity generation and modeling of* 

Yi, H., Nevin, K.P., Kim, B.C., Franks, A.E., Klimes, A., Tender, L.M., Lovley, D.R.(2009).

*production in microbial fuel cells*. Biosensors and Bioelectronics 24, 3498-3503. Zhang, T., Zeng, Y., Chen, S., Ai, X., Yang, H.(2007). *Improved performances of E. coli-catalyzed* 

100, 4171-4175.

Communications 9, 349-353.

*microbial fuel cell from continuous beer brewery wastewater*. Bioresource Technology

*Selection of a variant of Geobacter sulfurreducens with enhanced capacity for current* 

*microbial fuel cells with composite graphite/PTFE anodes*. Electrochemistry

Carbon combustion has been a research subject, relevant to pulverized coal combustion. However, it is not limited to basic research on coal/char combustion, but can benefit various aerospace applications, such as propulsion due to its high energy density and evaluation of protection properties of carbon-carbon composites (C/C-composites) used as hightemperature, structural materials for atmospheric re-entry, gas-turbine blades, scram jet combustors, etc., including ablative carbon heat-shields. Because of practical importance, extensive research has been conducted experimentally, theoretically, and/or numerically, as summarized in several comprehensive 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). Nonetheless, because of complexities involved, further studies are required to understand basic nature of the combustion. Some of them also command fundamental interest, because of simultaneous existence of surface and gas-phase reactions, interacting each other.

Generally speaking, processes governing the carbon combustion are as follows: (i) diffusion of oxidizing species to the solid surface, (ii) adsorption of molecules onto active sites on the surface, (iii) formation of products from adsorbed molecules on the surface, (iv) desorption of solid oxides into the gas phase, and (v) migration of gaseous products through the boundary layer into the freestream. Since these steps occur in series, the slowest of them determines the combustion rate and it is usual that steps (ii) and (iv) are extremely fast.

When the surface temperature is low, step (iii) is known to be much slower than steps (i) or (v). The combustion rate, which is also called as the mass burning rate, defined as mass transferred in unit area and time, is then determined by chemical kinetics and therefore the process is kinetically controlled. In this kinetically controlled regime, the combustion rate only depends on the surface temperature, exponentially. Since the process of diffusion, being conducted through the boundary layer, is irrelevant in this regime, the combustion rate is independent of its thickness. Concentrations of oxidizing species at the reacting surface are not too different from those in the freestream. In addition, since solid carbon is more or less porous, in general, combustion proceeds throughout the sample specimen.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

comparisons, further.

**2. Formulation** 

system parameters involved.

**2.1 Model definition** 

strongly convective burning in the forward stagnation region of a particle.

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 253

interpretations have been facilitated by its introduction. Therefore, we will confine ourselves to studying carbon combustion in the axisymmetric stagnation flow over a flat plate and/or that in the two-dimensional stagnation flow over a cylinder. From the practical point of view, we can say that it simulates the situations of ablative carbon heat-shields and/or

In this Part 1, formulation of the governing equations is first presented in Section 2, based on theories on the chemically reacting boundary layer in the forward stagnation field. Chemical reactions considered include the surface C-O2 and C-CO2 reactions and the gas-phase CO-O2 reaction, for a while. Generalized species-enthalpy coupling functions are 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. In Section 3, it is shown that straightforward derivation of the combustion response can be allowed in the limiting

situations, such as those for the Frozen, Flame-detached, and Flame-attached modes.

In Section 4, after presenting profiles of gas-phase temperature, measured over the burning carbon, a further analytical study is made about the ignition phenomenon, related to finiterate kinetics, by use of the asymptotic expansion method to obtain a critical condition. Appropriateness of this criterion is further examined by comparing temperature distributions in the gas phase and/or surface temperatures at which the CO-flame can appear. After having constructed these theories, evaluations of kinetic parameters for the surface and gas-phase reactions are conducted in Section 5, in order to make experimental

Concluding remarks for Part 1 are made in Section 6, with references cited and nomenclature tables. Note that the useful information obtained is further to be used in Part 2, to explore carbon combustion at high velocity gradients and/or in the High-Temperature

Among previous studies (Tsuji & Matsui, 1976; Adomeit, et al., 1976; Adomeit, et al., 1985; Henriksen, et al., 1988; Matsui & Tsuji, 1987), it may be noted that Adomeit's group has made a great contribution by clarifying water-catalyzed CO-O2 reaction (Adomeit, et al., 1976), conducting experimental comparisons (Adomeit, et al., 1985), and investigating ignition/extinction behavior (Henriksen, et al., 1988). Here, an extension of the worthwhile contributions is made along the following directions. First, simultaneous presence of the surface C-O2 and C-CO2 reactions and the gas-phase CO-O2 reaction are included, so as to allow studies of surface reactions over an extended range of its temperatures, as well as examining their coupling with the gas-phase reaction. Second, a set of generalized coupling functions (Makino & Law, 1986) are conformed to the present flow configuration, in order to facilitate mathematical development and/or physical interpretation of the results. Third, an attempt is made to identify effects of thermophysical properties, as well as other kinetic and

The present model simulates the isobaric carbon combustion of constant surface temperature *Ts* in the stagnation flow of temperature *T*, oxygen mass-fraction *Y*O,, and carbon dioxide mass-fraction *Y*P,, in a general manner (Makino, 1990). The major reactions

Air Combustion, with taking account of effects of water-vapor in the oxidizing-gas.

On the other hand, when the surface temperature is high, step (iii) is known to be much faster than steps (i) and (v). The combustion rate is then controlled by the diffusion rate of oxidizing species (say, oxygen) to the solid surface, at which their concentrations are negligibly small. In this diffusionally controlled regime, therefore, the combustion rate strongly depends on the boundary layer thickness and weakly on the surface temperature (*T*0.5~1.0), with exhibiting surface regression in the course of combustion.

Since oxygen-transfer to the carbon surface can occur via O2, CO2, and H2O, the major surface reactions can be

$$\text{C} + \text{O}\_2 \rightarrow \text{CO}\_2 \tag{R1}$$

$$\text{?}\,\text{?}\,\text{+}\,\text{O}\_{2}\rightarrow\text{2CO}\,,\tag{\text{R}}\tag{\text{R}}$$

$$\text{C} + \text{CO}\_2 \rightarrow 2\text{CO}\_2\tag{\text{R}}$$

$$\text{C} + \text{H}\_2\text{O} \rightarrow \text{CO} + \text{H}\_2\text{.} \tag{\text{R4}}$$

At higher temperatures, say, higher than 1000 K, CO formation is the preferred route and the relative contribution from (R1) can be considered to be negligible (Arthur, 1951). Thus, reaction (R2) will be referred to as the C-O2 reaction.

Comparing (R2) and (R3), as alternate routes of CO production, the C-O2 reaction is the preferred route for CO production at low temperatures, in simultaneous presence of O2 and CO2. It can be initiated around 600 K and saturated around 1600 K, proceeding infinitely fast, eventually, relative to diffusion. The C-CO2 reaction of (R3) is the high temperature route, initiated around 1600 K and saturated around 2500 K. It is of particular significance because CO2 in (R3) can be the product of the gas-phase, water-catalyzed, CO-oxidation,

$$\text{2CO} + \text{O}\_2 \rightarrow \text{2CO}\_2,\tag{\text{R5}}$$

referred to as the CO-O2 reaction. Thus, the C-CO2 and CO-O2 reactions can form a loop. Similarly, the C-H2O reaction (R4), generating CO and H2, is also important when the combustion environment consists of an appreciable amount of water. This reaction is also of significance because H2O is the product of the H2-oxidation,

$$\rm 2H\_2 + O\_2 \to 2H\_2O \, , \tag{R6}$$

referred to as the H2-O2 reaction, constituting a loop of the C-H2O and H2-O2 reactions.

The present monograph, consisting of two parts, is intended to shed more light on the carbon combustion, with putting a focus on its heat and mass transfer from the surface. It is, therefore, not intended as a collection of engineering data or an exhaustive review of all the pertinent published work. Rather, it has an intention to represent the carbon combustion by use of some of the basic characteristics of the chemically reacting boundary layers, 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.

Among various flow configurations, it has been reported that the stagnation-flow configuration has various advantages, because it provides a well-defined, one-dimensional flow, characterized by a single parameter, called the stagnation velocity gradient. It has even been said that mathematical analyses, experimental data acquisition, and physical interpretations have been facilitated by its introduction. Therefore, we will confine ourselves to studying carbon combustion in the axisymmetric stagnation flow over a flat plate and/or that in the two-dimensional stagnation flow over a cylinder. From the practical point of view, we can say that it simulates the situations of ablative carbon heat-shields and/or strongly convective burning in the forward stagnation region of a particle.

In this Part 1, formulation of the governing equations is first presented in Section 2, based on theories on the chemically reacting boundary layer in the forward stagnation field. Chemical reactions considered include the surface C-O2 and C-CO2 reactions and the gas-phase CO-O2 reaction, for a while. Generalized species-enthalpy coupling functions are 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. In Section 3, it is shown that straightforward derivation of the combustion response can be allowed in the limiting situations, such as those for the Frozen, Flame-detached, and Flame-attached modes.

In Section 4, after presenting profiles of gas-phase temperature, measured over the burning carbon, a further analytical study is made about the ignition phenomenon, related to finiterate kinetics, by use of the asymptotic expansion method to obtain a critical condition. Appropriateness of this criterion is further examined by comparing temperature distributions in the gas phase and/or surface temperatures at which the CO-flame can appear. After having constructed these theories, evaluations of kinetic parameters for the surface and gas-phase reactions are conducted in Section 5, in order to make experimental comparisons, further.

Concluding remarks for Part 1 are made in Section 6, with references cited and nomenclature tables. Note that the useful information obtained is further to be used in Part 2, to explore carbon combustion at high velocity gradients and/or in the High-Temperature Air Combustion, with taking account of effects of water-vapor in the oxidizing-gas.
