**3. Combustion behavior in the limiting cases**

Here we discuss analytical solutions for some limiting cases of the gas-phase reaction, since several limiting solutions regarding the intensity of the gas-phase CO-O2 reaction can readily be identified from the coupling functions. In addition, important characteristics indispensable for fundamental understanding is obtainable.

### **3.1 Frozen mode**

When the gas-phase CO-O2 reaction is completely frozen, the solution of the energy conservation Eq. (17) readily yields

$$
\widetilde{T} = \widetilde{T}\_s + \chi \not\subset \mathfrak{z} ; \qquad \gamma = \widetilde{T}\_\alpha - \widetilde{T}\_\text{s} \ . \tag{43}
$$

Evaluating Eqs. (35) and (36) at =0 for obtaining surface concentrations of O2 and CO2, and substituting them into Eq. (31), we obtain an implicit expression for the combustion rate (-*f*s)

$$\delta(-f\_{\rm s}) = A\_{\rm s,O} \frac{\widetilde{Y}\_{\rm O,o}}{1 + \mathfrak{H} + A\_{\rm s,O} \left[\mathfrak{B}/(-f\_{\rm s})\right]} + A\_{\rm s,P} \frac{\widetilde{Y}\_{\rm P,o}}{1 + \mathfrak{H} + A\_{\rm s,P} \left[\mathfrak{B}/(-f\_{\rm s})\right]} \,\,\,\tag{44}$$

which is to be solved numerically from Eq. (15), because of the density coupling. The combustion rate in the diffusion controlled regime becomes the highest with satisfying the following condition.

$$\boldsymbol{\beta}\_{\text{max}} = \frac{\widetilde{\boldsymbol{Y}}\_{\text{O},\infty} + \widetilde{\boldsymbol{Y}}\_{\text{P},\infty}}{\delta} \tag{45}$$

### **3.2 Flame-detached mode**

When the gas-phase CO-O2 reaction occurs infinitely fast, two flame-sheet burning modes are possible. One involves a detached flame-sheet, situated away from the surface, and

It may informative to note that the parameter , defined as (-*f*s)/()s in the formulation, coincides with the conventional transfer number (Spalding, 1951), which has been shown by considering elemental carbon, (*W*C/*W*F)*Y*F+(*W*C/*W*P)*Y*P, taken as the transferred substance, and by evaluating driving force and resistance, determined by the transfer rate in the gas phase and the ejection rate at the surface, respectively (Makino, 1992; Makino, et al., 1998).

> 

P C P

Figure 1(b) shows the combustion rate in the same conditions. At high surface temperatures, because of the existence of high-temperature reaction zone in the gas phase, the combustion rate is enhanced. In this context, the transfer number, less temperature-sensitive than the combustion rate, as shown in Figs. 1(a) and 1(b), is preferable for theoretical considerations.

Here we discuss analytical solutions for some limiting cases of the gas-phase reaction, since several limiting solutions regarding the intensity of the gas-phase CO-O2 reaction can readily be identified from the coupling functions. In addition, important characteristics

When the gas-phase CO-O2 reaction is completely frozen, the solution of the energy

<sup>~</sup> <sup>~</sup> ; s

Evaluating Eqs. (35) and (36) at =0 for obtaining surface concentrations of O2 and CO2, and substituting them into Eq. (31), we obtain an implicit expression for the combustion rate (-*f*s)

s,P <sup>s</sup>

which is to be solved numerically from Eq. (15), because of the density coupling. The combustion rate in the diffusion controlled regime becomes the highest with satisfying the

> O, P,

<sup>~</sup> <sup>~</sup> *<sup>Y</sup> <sup>Y</sup>*

When the gas-phase CO-O2 reaction occurs infinitely fast, two flame-sheet burning modes are possible. One involves a detached flame-sheet, situated away from the surface, and

s,O s

*A f*

s,P

, (44)

*A*

*T Ts*

O, <sup>s</sup> s,O 1

max

~

*Y*

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

*W W Y*

 

P s C P

F C F

*W W Y*

*W W Y*

 

 

P s C P

> *W W Y*

**3. Combustion behavior in the limiting cases** 

indispensable for fundamental understanding is obtainable.

1

*f A*

F C F

 

*W W Y*

F C F

*W W Y*

F P s F P

~ ~ ~ ~ ~ ~

*Y Y Y Y*

<sup>~</sup> <sup>~</sup> *<sup>T</sup> <sup>T</sup>* . (43)

P,

*A f*

(45)

~

*Y*

. (42)

F P s

That is,

 

**3.1 Frozen mode**

following condition.

**3.2 Flame-detached mode**

conservation Eq. (17) readily yields

the other an attached flame-sheet, situated on the surface. The Flame-detached mode is defined by

$$\widetilde{Y}\_{\mathbf{O}}\left(\mathbf{0} \le \widetilde{\mathbf{y}} \le \widetilde{\mathbf{y}}\_{f}\right) = \widetilde{Y}\_{\mathbf{F}}\left(\widetilde{\mathbf{e}}\_{f} \le \widetilde{\mathbf{y}} \le \infty\right) = \mathbf{0} \,. \tag{46}$$

By using the coupling functions in Eqs. (33) to (36), it can be shown that

$$\mathfrak{S}(-f\_{\rm s}) = A\_{\rm s,P} \frac{\widetilde{Y}\_{\rm O,\infty} + \widetilde{Y}\_{\rm P,\rm o} - \mathfrak{S} \,\mathfrak{P}}{1 + \mathfrak{P}} \tag{47}$$

$$
\widetilde{T}\_f = \widetilde{T}\_\mathbf{s} + \left(\widetilde{Y}\_{\mathbf{O},\alpha} + \widetilde{T}\_\alpha - \widetilde{T}\_\mathbf{s}\right)\widetilde{\epsilon}\_f; \qquad \widetilde{\epsilon}\_f = \frac{2\delta\mathfrak{B} - \widetilde{Y}\_{\mathbf{O},\alpha}}{[2\delta + \widetilde{Y}\_{\mathbf{O},\alpha}]\mathfrak{B}'} \tag{48}
$$

Once (-*f*s) is determined from Eqs. (47) and (15), *f* can readily be evaluated, yielding the temperature distribution as

$$0 \le \xi \le \xi\_f \; : \quad \widetilde{T} = \widetilde{T}\_{\rm s} + \left( \widetilde{Y}\_{\rm O/o} + \widetilde{T}\_{\rm o} - \widetilde{T}\_{\rm s} \right) \overline{\xi} \; : \tag{49}$$

$$\widetilde{\nabla}\_{f} \le \widetilde{\xi} \le \infty: \quad \widetilde{T} = \widetilde{T}\_{\infty} - \left\{ \widetilde{T}\_{\infty} - \widetilde{T}\_{\infty} + \left( \frac{\widetilde{Y}\_{\text{O},\text{or}} - 2\delta \mathfrak{R}}{1 + \mathfrak{R}} \right) (1 - \xi) \right\}. \tag{50}$$

In addition, the infinitely large *Da*g yields the following important characteristics, as reported by Tsuji & Matsui (1976).


$$-\left(\frac{d\widetilde{Y}\_{\rm F}}{d\eta}\right)\_{f-} = \left(\frac{d\widetilde{Y}\_{\rm O}}{d\eta}\right)\_{f+} \quad \text{or} \quad -\left(\frac{dY\_{\rm F}}{d\eta}\right)\_{f-} = \frac{\mathbf{v}\_{\rm F}\mathcal{W}\_{\rm F}}{\mathbf{v}\_{\rm O}\mathcal{W}\_{\rm O}} \left(\frac{dY\_{\rm O}}{d\eta}\right)\_{f+} \tag{51}$$

suggesting that fuel and oxidizer must flow into the flame surface in stoichiometric proportions. Here the subscript *f* + and *f* -, respectively, designate the oxygen and fuel sides of the flame. Note that in deriving Eq. (51), use has been made of an assumption that values of the individual quantities, such as the streamfunction *f* and species massfraction *Yi*, can be continuous across the flame.

3. Similarly, by evaluating the coupling function for CO and enthalpy, we have

$$
\left(\frac{d\widetilde{T}}{d\eta}\right)\_{f-} - \left(\frac{d\widetilde{T}}{d\eta}\right)\_{f+} = -\left(\frac{d\widetilde{Y}\_{\rm F}}{d\eta}\right)\_{f-} \quad \text{or} \quad \lambda \left(\frac{dT}{d\eta}\right)\_{f-} - \lambda \left(\frac{dT}{d\eta}\right)\_{f+} = -q\rho D \left(\frac{dY\_{\rm F}}{d\eta}\right)\_{f-}\tag{52}
$$

suggesting that the amount of heat generated is equal to the heat, conducted away to the both sides of the reaction zone.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

1984; Matsui, et al., 1975; Matsui, et al., 1983, 1986).

results (Makino, 1990; Makino & Law, 1990; Makino, et al., 1994).

rate, after having examined its linearity on the combustion time.

2 CARS: Coherent Anti-Stokes Raman Spectroscopy

**4.1 Combustion rate and ignition surface-temperature** 

that can support it.

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 263

reactions. In spite of this theoretical accomplishment, there are very few experimental data

In the literature, in general, emphasis has been put on examination of the surface reactivities with gaseous oxidizers, such as O2, CO2, and H2O (*cf.* Essenhigh, 1981) although surface reactivities on the same solid carbon are limited (Khitrin & Golovina, 1964; Visser & Adomeit, 1984; Harris & Smith, 1990). As for the gas-phase CO-O2 reactivity, which is sensitive to the H2O concentration, main concern has been put on that of the CO-flame (Howard, et al., 1973), called the "strong" CO-oxidation, which is, however, far from the situation over the burning carbon, especially for that prior to the appearance of CO-flame, because some of the elementary reactions are too slow to sustain the "strong" CO-oxidation. Furthermore, it has been quite rare to conduct experimental studies from the viewpoint that there exist interactions between chemical reactions and flow, so that studies have mainly been confined to obtaining combustion rate (Khitrin & Golovina, 1964; Visser & Adomeit,

In order to examine such interactions, an attempt has been made to measure temperature profiles over the burning graphite rod in the forward stagnation flowfield (Makino, et al., 1996). In this measurement, N2-CARS2 thermometry (Eckbreth, 1988) is used in order to avoid undesired appearance and/or disappearance of the CO-flame. Not only the influence due to the appearance of CO-flame on the temperature profile, but also that on the combustion rate is investigated. Measured results are further compared with predicted

Here, experimental results for the combustion rate and the temperature profiles in the gas phase are first presented, which are closely related to the coupled nature of the surface and gas-phase reactions. The experimental setup is schematically shown in Fig. 2. Air used as an oxidizer is supplied by a compressor and passes through a refrigerator-type dryer and a surge tank. The dew point from which the H2O concentration is determined is measured by a hygrometer. The airflow at room temperature, after passing through a settling chamber (52.8 mm in diameter and 790 mm in length), issues into the atmosphere with a uniform velocity (up to 3 m/s), and impinges on a graphite rod to establish a two-dimensional stagnation flow. This flowfield is well-established and is specified uniquely by the velocity gradient *a* (=4*V*/*d*), where *V* is the freestream velocity and *d* the diameter of the graphite rod. The rod is Joule-heated by an alternating current (12 V; up to 1625 A). The surface temperature is measured by a two-color pyrometer. The temperature in the central part (about 10 mm in length) of the test specimen is nearly uniform. In experiment, the test specimen is set to burn in airflow at constant surface temperature during each experimental run. Since the surface temperature is kept constant with external heating, quasi-steady combustion can be accomplished. The experiment involves recording image of test specimen in the forward stagnation region by a video camera and analyzing the signal displayed on a TV monitor to obtain surface regression rate, which is used to determine the combustion

Figure 2(a) shows the combustion rate in airflow of 110 s-1 (Makino, et al., 1996), as a function of the surface temperature, when the H2O mass-fraction is 0.003. The combustion rate, obtained from the regression rate and density change of the test specimen, increases

### **3.3 Flame-attached mode**

When the surface reactivity is decreased by decreasing the surface temperature, then the detached flame sheet moves towards the surface until it is contiguous to it (*f* = 0). This critical state is given by the condition

$$\mathfrak{B}\_{a} = \frac{\widetilde{Y}\_{\mathbf{O}, \mathcal{ox}}}{\mathbf{2}\mathbf{6}} \,, \tag{53}$$

obtained from Eq. (48), and defines the transition from the detached to the attached mode of the flame. Subsequent combustion with the Flame-attached mode is characterized by *Y*F,s = 0 and *Y*O,s 0 (Libby & Blake, 1979; Makino & Law, 1986; Henriksen, et al., 1988), with the gasphase temperature profile

$$
\widetilde{T} = \widetilde{T}\_s + \left(\widetilde{T}\_\infty - \widetilde{T}\_s\right)\widetilde{\mathbf{\xi}}\_\prime \tag{54}
$$

given by the same relation as that for the frozen case, because all gas-phase reaction is now confined at the surface. By using the coupling functions in Eqs. (33) to (36) with *Y*F,s = 0, it can be shown that

$$\mathfrak{S}(-f\_{\rm s}) = A\_{\rm s,O} \frac{\widetilde{Y}\_{\rm O,\rm \alpha} - 2 \,\delta \,\mathfrak{B}}{1 + \mathfrak{B}} + A\_{\rm s,P} \frac{\widetilde{Y}\_{\rm P,\rm \alpha} + \delta \,\mathfrak{B}}{1 + \mathfrak{B}} \,\,\,\,\,\tag{55}$$

which is also to be solved numerically from Eq. (15). The maximum combustion rate of this mode occurs at the transition state in Eq. (53), which also corresponds to the minimum combustion rate of the Flame-detached mode.

### **3.4 Diffusion-limited combustion rate**

The maximum, diffusion-limited transfer number of the system can be achieved through one of the two limiting situations. The first appears when both of the surface reactions occur infinitely fast such that *Y*O,s and *Y*P,s both vanish, yielding Eq. (45). The second appears when the surface C-CO2 reaction occurs infinitely fast in the limit of the Flame-detached mode, which again yields Eq. (45). It is of interest to note that in the first situation the reactivity of the gas-phase CO-O2 reaction is irrelevant, whereas in the second the reactivity of the surface C-O2 reaction is irrelevant. While the transfer numbers are the same in both cases, the combustion rates, thereby the oxygen supply rates, are slightly different each other, as shown in Fig, 1(b), because of the different density coupling, related to the flame structures. Note that the limiting solutions identified herein provide the counterparts of those previously derived (Libby & Blake, 1979; Makino & Law, 1986) for the carbon particle, and generalize the solution of Matsui & Tsuji (1987) with including the surface C-CO2 reaction.

### **4. Combustion rate and flame structure**

A momentary reduction in the combustion rate, reported in theoretical works (Adomeit, et al., 1985; Makino & Law, 1986; Matsui & Tsuji, 1987; Henriksen, 1989; Makino, 1990; Makino & Law, 1990), can actually be exaggerated by the appearance of CO-flame in the gas phase, bringing about a change of the dominant surface reactions from the faster C-O2 reaction to the slower C-CO2 reaction, due to an intimate coupling between the surface and gas-phase

When the surface reactivity is decreased by decreasing the surface temperature, then the detached flame sheet moves towards the surface until it is contiguous to it (*f* = 0). This

> 2

obtained from Eq. (48), and defines the transition from the detached to the attached mode of the flame. Subsequent combustion with the Flame-attached mode is characterized by *Y*F,s = 0 and *Y*O,s 0 (Libby & Blake, 1979; Makino & Law, 1986; Henriksen, et al., 1988), with the gas-

*T Ts T Ts*

given by the same relation as that for the frozen case, because all gas-phase reaction is now confined at the surface. By using the coupling functions in Eqs. (33) to (36) with *Y*F,s = 0, it

 

which is also to be solved numerically from Eq. (15). The maximum combustion rate of this mode occurs at the transition state in Eq. (53), which also corresponds to the minimum

The maximum, diffusion-limited transfer number of the system can be achieved through one of the two limiting situations. The first appears when both of the surface reactions occur infinitely fast such that *Y*O,s and *Y*P,s both vanish, yielding Eq. (45). The second appears when the surface C-CO2 reaction occurs infinitely fast in the limit of the Flame-detached mode, which again yields Eq. (45). It is of interest to note that in the first situation the reactivity of the gas-phase CO-O2 reaction is irrelevant, whereas in the second the reactivity of the surface C-O2 reaction is irrelevant. While the transfer numbers are the same in both cases, the combustion rates, thereby the oxygen supply rates, are slightly different each other, as shown in Fig, 1(b), because of the different density coupling, related to the flame structures. Note that the limiting solutions identified herein provide the counterparts of those previously derived (Libby & Blake, 1979; Makino & Law, 1986) for the carbon particle, and generalize the solution of Matsui & Tsuji (1987) with including the surface C-CO2 reaction.

A momentary reduction in the combustion rate, reported in theoretical works (Adomeit, et al., 1985; Makino & Law, 1986; Matsui & Tsuji, 1987; Henriksen, 1989; Makino, 1990; Makino & Law, 1990), can actually be exaggerated by the appearance of CO-flame in the gas phase, bringing about a change of the dominant surface reactions from the faster C-O2 reaction to the slower C-CO2 reaction, due to an intimate coupling between the surface and gas-phase

1

O,

*Y*

s s,O

combustion rate of the Flame-detached mode.

**4. Combustion rate and flame structure** 

**3.4 Diffusion-limited combustion rate** 

*A*

<sup>2</sup> <sup>~</sup> P,

~ *Y*O,

*<sup>a</sup>* , (53)

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> , (54)

1

*f A* . (55)

~

*Y*

s,P

**3.3 Flame-attached mode**

phase temperature profile

can be shown that

critical state is given by the condition

reactions. In spite of this theoretical accomplishment, there are very few experimental data that can support it.

In the literature, in general, emphasis has been put on examination of the surface reactivities with gaseous oxidizers, such as O2, CO2, and H2O (*cf.* Essenhigh, 1981) although surface reactivities on the same solid carbon are limited (Khitrin & Golovina, 1964; Visser & Adomeit, 1984; Harris & Smith, 1990). As for the gas-phase CO-O2 reactivity, which is sensitive to the H2O concentration, main concern has been put on that of the CO-flame (Howard, et al., 1973), called the "strong" CO-oxidation, which is, however, far from the situation over the burning carbon, especially for that prior to the appearance of CO-flame, because some of the elementary reactions are too slow to sustain the "strong" CO-oxidation. Furthermore, it has been quite rare to conduct experimental studies from the viewpoint that there exist interactions between chemical reactions and flow, so that studies have mainly been confined to obtaining combustion rate (Khitrin & Golovina, 1964; Visser & Adomeit, 1984; Matsui, et al., 1975; Matsui, et al., 1983, 1986).

In order to examine such interactions, an attempt has been made to measure temperature profiles over the burning graphite rod in the forward stagnation flowfield (Makino, et al., 1996). In this measurement, N2-CARS2 thermometry (Eckbreth, 1988) is used in order to avoid undesired appearance and/or disappearance of the CO-flame. Not only the influence due to the appearance of CO-flame on the temperature profile, but also that on the combustion rate is investigated. Measured results are further compared with predicted results (Makino, 1990; Makino & Law, 1990; Makino, et al., 1994).

## **4.1 Combustion rate and ignition surface-temperature**

Here, experimental results for the combustion rate and the temperature profiles in the gas phase are first presented, which are closely related to the coupled nature of the surface and gas-phase reactions. The experimental setup is schematically shown in Fig. 2. Air used as an oxidizer is supplied by a compressor and passes through a refrigerator-type dryer and a surge tank. The dew point from which the H2O concentration is determined is measured by a hygrometer. The airflow at room temperature, after passing through a settling chamber (52.8 mm in diameter and 790 mm in length), issues into the atmosphere with a uniform velocity (up to 3 m/s), and impinges on a graphite rod to establish a two-dimensional stagnation flow. This flowfield is well-established and is specified uniquely by the velocity gradient *a* (=4*V*/*d*), where *V* is the freestream velocity and *d* the diameter of the graphite rod. The rod is Joule-heated by an alternating current (12 V; up to 1625 A). The surface temperature is measured by a two-color pyrometer. The temperature in the central part (about 10 mm in length) of the test specimen is nearly uniform. In experiment, the test specimen is set to burn in airflow at constant surface temperature during each experimental run. Since the surface temperature is kept constant with external heating, quasi-steady combustion can be accomplished. The experiment involves recording image of test specimen in the forward stagnation region by a video camera and analyzing the signal displayed on a TV monitor to obtain surface regression rate, which is used to determine the combustion rate, after having examined its linearity on the combustion time.

Figure 2(a) shows the combustion rate in airflow of 110 s-1 (Makino, et al., 1996), as a function of the surface temperature, when the H2O mass-fraction is 0.003. The combustion rate, obtained from the regression rate and density change of the test specimen, increases

<sup>2</sup> CARS: Coherent Anti-Stokes Raman Spectroscopy

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

establishment of CO-flame.

boundary layer of a few mm can be measured.

errors (50 K) in the present CARS thermometry.

same results as those with global gas-phase chemistry.

can be suppressed with increasing velocity gradient.

with CO-flame at the other time.

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 265

surface temperature is 1400 K, the gas-phase temperature monotonically decreases, suggesting negligible gas-phase reaction. When the surface temperature is 1500 K, at which CO-flame can be observed visually, there exists a reaction zone in the gas phase whose temperature is nearly equal to the surface temperature. Outside the reaction zone, the temperature gradually decreases to the freestream temperature. When the surface temperature is 1700 K, the gas-phase temperature first increases from the surface temperature to the maximum, and then decreases to the freestream temperature. The existence of the maximum temperature suggests that a reaction zone locates away from the surface. That is, a change of the flame structures has certainly occurred upon the

It may be informative to note the advantage of the CARS thermometry over the conventional, physical probing method with thermocouple. When the thermocouple is used for the measurement of temperature profile corresponding to the surface temperature of 1400 K (or 1500 K), it distorts the combustion field, and hence makes the CO-flame appear (or disappear). In this context, the present result suggests the importance of using thermometry without disturbing the combustion fields, especially for the measurement at the ignition/extinction of CO-flame. In addition, the present results demonstrate the high spatial resolution of the CARS thermometry, so that the temperature profile within a thin

Predicted results are also shown in Fig. 3(a). In numerical calculations, use has been made of the formulation mentioned in Section 2 and kinetic parameters (Makino, et al., 1994) to be explained in the next Section. When there exists CO-flame, the gas-phase kinetic parameters used are those for the "strong" CO-oxidation; when the CO-oxidation is too weak to establish the CO-flame, those for the "weak" CO-oxidation are used. Fair agreement between experimental and predicted results is shown, if we take account of measurement

Our choice of the global gas-phase chemistry requires a further comment, because nowadays it is common to use detailed chemistry in the gas phase. Nonetheless, because of its simplicity, it is decided to use the global gas-phase chemistry, after having examined the fact that the formulation with detailed chemistry (Chelliah, et al., 1996) offers nearly the

Figure 3(b) shows the temperature profiles for the airflow of 200 s-1. Because of the increased velocity gradient, the ignition surface-temperature is raised to be *ca.* 1550 K, and the boundarylayer thickness is contracted, compared to Fig. 3(a), while the general trend is the same.

Figure 3(c) shows the temperature profiles at the surface temperature 1700 K, with the velocity gradient of airflow taken as a parameter (Makino, et al., 1997). It is seen that the flame structure shifts from that with high temperature flame zone in the gas phase to that with gradual decrease in the temperature, suggesting that the establishment of CO-flame

Note here that in obtaining data in Figs. 3(a) to 3(c), attention has been paid to controlling the surface temperature not to exceed 20 K from a given value. In addition, the surface temperature is intentionally set to be lower (or higher) than the ignition surface-temperature by 20 K or more. If we remove these restrictions, results are somewhat confusing and gasphase temperature scatters in relatively wide range, because of the appearance of unsteady combustion (Kurylko & Essenhigh, 1973) that proceeds without CO-flame at one time, while

Fig. 2. Combustion rate of graphite rod (C=1.25103 kg/m3) as a function of the surface temperature; (a) for the velocity gradient of 110 s-1 in airflow with the H2O mass-fraction of 0.003; (b) for the velocity gradient of 200 s-1 in airflow with the H2O mass-fraction of 0.002. Data points are experimental (Makino, et al., 1996) and curves are calculated for the theory (Makino, 1990). The gas-phase Damköhler number corresponds to that for the "strong" COoxidation. The ignition surface-temperature *T*s,ig is calculated, based on the ignition analysis (Makino & Law, 1990). Schematical drawing of the experimental setup is also shown.

with increasing surface temperature, up to a certain surface temperature. The combustion in this temperature range is that with negligible CO-oxidation, and hence the combustion rate in Frozen mode can fairly predict the experimental results. A further increase in the surface temperature causes the momentary reduction in the combustion rate, because appearance of the CO-flame alters the dominant surface reaction from the C-O2 reaction to the C-CO2 reaction. The surface temperature when the CO-flame first appears is called the ignition surface-temperature (Makino & Law, 1990), above which the combustion proceeds with the "strong" CO-oxidation. The solid curve is the predicted combustion rate with the surface kinetic parameters (Makino, et al., 1994) to be explained in the next Section, and the global gas-phase kinetic parameters (Howard, et al., 1973). In numerical calculations, use has been made of the formulation, presented in Section 2.

The same trend is also observed in airflow of 200 s-1, as shown in Fig. 2(b). Because of the reduced thickness of the boundary layer with respect to the oxidizer concentration, the combustion rate at a given surface temperature is enhanced. The ignition surfacetemperature is raised because establishment of the CO-flame is suppressed, due to an increase in the velocity gradient.

### **4.2 Temperature profile in the gas phase**

Temperature profiles in the forward stagnation region are shown in Fig. 3(a) when the velocity gradient of airflow is 110 s-1 and the H2O mass-fraction is 0.002. The surface temperature is taken as a parameter, being controlled not to exceed 20 K from a given value. We see that the temperature profile below the ignition surface-temperature (*ca.* 1450 K) is completely different from that above the ignition surface-temperature. When the

Fig. 2. Combustion rate of graphite rod (C=1.25103 kg/m3) as a function of the surface temperature; (a) for the velocity gradient of 110 s-1 in airflow with the H2O mass-fraction of 0.003; (b) for the velocity gradient of 200 s-1 in airflow with the H2O mass-fraction of 0.002. Data points are experimental (Makino, et al., 1996) and curves are calculated for the theory (Makino, 1990). The gas-phase Damköhler number corresponds to that for the "strong" COoxidation. The ignition surface-temperature *T*s,ig is calculated, based on the ignition analysis (Makino & Law, 1990). Schematical drawing of the experimental setup is also shown.

with increasing surface temperature, up to a certain surface temperature. The combustion in this temperature range is that with negligible CO-oxidation, and hence the combustion rate in Frozen mode can fairly predict the experimental results. A further increase in the surface temperature causes the momentary reduction in the combustion rate, because appearance of the CO-flame alters the dominant surface reaction from the C-O2 reaction to the C-CO2 reaction. The surface temperature when the CO-flame first appears is called the ignition surface-temperature (Makino & Law, 1990), above which the combustion proceeds with the "strong" CO-oxidation. The solid curve is the predicted combustion rate with the surface kinetic parameters (Makino, et al., 1994) to be explained in the next Section, and the global gas-phase kinetic parameters (Howard, et al., 1973). In numerical calculations, use has been

The same trend is also observed in airflow of 200 s-1, as shown in Fig. 2(b). Because of the reduced thickness of the boundary layer with respect to the oxidizer concentration, the combustion rate at a given surface temperature is enhanced. The ignition surfacetemperature is raised because establishment of the CO-flame is suppressed, due to an

Temperature profiles in the forward stagnation region are shown in Fig. 3(a) when the velocity gradient of airflow is 110 s-1 and the H2O mass-fraction is 0.002. The surface temperature is taken as a parameter, being controlled not to exceed 20 K from a given value. We see that the temperature profile below the ignition surface-temperature (*ca.* 1450 K) is completely different from that above the ignition surface-temperature. When the

made of the formulation, presented in Section 2.

**4.2 Temperature profile in the gas phase** 

increase in the velocity gradient.

surface temperature is 1400 K, the gas-phase temperature monotonically decreases, suggesting negligible gas-phase reaction. When the surface temperature is 1500 K, at which CO-flame can be observed visually, there exists a reaction zone in the gas phase whose temperature is nearly equal to the surface temperature. Outside the reaction zone, the temperature gradually decreases to the freestream temperature. When the surface temperature is 1700 K, the gas-phase temperature first increases from the surface temperature to the maximum, and then decreases to the freestream temperature. The existence of the maximum temperature suggests that a reaction zone locates away from the surface. That is, a change of the flame structures has certainly occurred upon the establishment of CO-flame.

It may be informative to note the advantage of the CARS thermometry over the conventional, physical probing method with thermocouple. When the thermocouple is used for the measurement of temperature profile corresponding to the surface temperature of 1400 K (or 1500 K), it distorts the combustion field, and hence makes the CO-flame appear (or disappear). In this context, the present result suggests the importance of using thermometry without disturbing the combustion fields, especially for the measurement at the ignition/extinction of CO-flame. In addition, the present results demonstrate the high spatial resolution of the CARS thermometry, so that the temperature profile within a thin boundary layer of a few mm can be measured.

Predicted results are also shown in Fig. 3(a). In numerical calculations, use has been made of the formulation mentioned in Section 2 and kinetic parameters (Makino, et al., 1994) to be explained in the next Section. When there exists CO-flame, the gas-phase kinetic parameters used are those for the "strong" CO-oxidation; when the CO-oxidation is too weak to establish the CO-flame, those for the "weak" CO-oxidation are used. Fair agreement between experimental and predicted results is shown, if we take account of measurement errors (50 K) in the present CARS thermometry.

Our choice of the global gas-phase chemistry requires a further comment, because nowadays it is common to use detailed chemistry in the gas phase. Nonetheless, because of its simplicity, it is decided to use the global gas-phase chemistry, after having examined the fact that the formulation with detailed chemistry (Chelliah, et al., 1996) offers nearly the same results as those with global gas-phase chemistry.

Figure 3(b) shows the temperature profiles for the airflow of 200 s-1. Because of the increased velocity gradient, the ignition surface-temperature is raised to be *ca.* 1550 K, and the boundarylayer thickness is contracted, compared to Fig. 3(a), while the general trend is the same.

Figure 3(c) shows the temperature profiles at the surface temperature 1700 K, with the velocity gradient of airflow taken as a parameter (Makino, et al., 1997). It is seen that the flame structure shifts from that with high temperature flame zone in the gas phase to that with gradual decrease in the temperature, suggesting that the establishment of CO-flame can be suppressed with increasing velocity gradient.

Note here that in obtaining data in Figs. 3(a) to 3(c), attention has been paid to controlling the surface temperature not to exceed 20 K from a given value. In addition, the surface temperature is intentionally set to be lower (or higher) than the ignition surface-temperature by 20 K or more. If we remove these restrictions, results are somewhat confusing and gasphase temperature scatters in relatively wide range, because of the appearance of unsteady combustion (Kurylko & Essenhigh, 1973) that proceeds without CO-flame at one time, while with CO-flame at the other time.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

considering the coupled nature of the gas-phase and surface reactions.

while extinction in the other three regimes.

**4.3 Ignition criterion** 

and O2 concentration.

**4.3.1 Ignition analysis** 

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 267

While studies relevant to the ignition/extinction of CO-flame over the burning carbon are of obvious practical utility in evaluating protection properties from oxidation in re-entry vehicles, as well as the combustion of coal/char, they also command fundamental interests because of the simultaneous existence of the surface and gas-phase reactions with intimate coupling (Visser & Adomeit, 1984; Makino & Law, 1986; Matsui & Tsuji, 1987). As mentioned in the previous Section, at the same surface temperature, the combustion rate is expected to be momentarily reduced upon ignition because establishment of the CO-flame in the gas phase can change the dominant surface reactions from the faster C-O2 reaction to the slower C-CO2 reaction. By the same token the combustion rate is expected to momentarily increase upon extinction. These concepts are not intuitively obvious without

Fundamentally, the ignition/extinction of CO-flame in carbon combustion must necessarily be described by the seminal analysis (Liñán, 1974) of the ignition, extinction, and structure of diffusion flames, as indicated by Matalon (1980, 1981, 1982). Specifically, as the flame temperature increases from the surface temperature to the adiabatic flame temperature, there appear a nearly-frozen regime, a partial-burning regime, a premixed-flame regime, and finally a near-equilibrium regime. Ignition can be described in the nearly-frozen regime,

For carbon combustion, Matalon (1981) analytically obtained an explicit ignition criterion when the O2 mass-fraction at the surface is O(l). When this concentration is O(), the appropriate reduced governing equation and the boundary conditions were also identified (Matalon, 1982). Here, putting emphasis on the ignition of CO-flame over the burning carbon, an attempt has first been made to extend the previous theoretical studies, so as to include analytical derivations of various criteria governing the ignition, with arbitrary O2 concentration at the surface. Note that these derivations are successfully conducted, by virtue of the generalized species-enthalpy coupling functions (Makino & Law, 1986; Makino, 1990), identified in the previous Section. Furthermore, it may be noted that the ignition analysis is especially relevant for situations where the surface O2 concentration is O() because in order for gas-phase reaction to be initiated, sufficient amount of carbon monoxide should be generated. This requires a reasonably fast surface reaction and thereby low O2 concentration. The second objective is to conduct experimental comparisons relevant to the ignition of CO-flame over a carbon rod in an oxidizing stagnation flow, with variations in the surface temperature of the rod, as well as the freestream velocity gradient

Here we intend to obtain an explicit ignition criterion without restricting the order of *Y*O,s. First we note that in the limit of *Ta*<sup>g</sup> , the completely frozen solutions for Eqs. (16) and (17) are

<sup>0</sup> <sup>s</sup> <sup>s</sup>

For finite but large values of *Ta*g, weak chemical reaction occurs in a thin region next to the carbon surface when the surface temperature is moderately high and exceeds the ambient

<sup>0</sup> ,s , ,s

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> *<sup>T</sup> <sup>T</sup> <sup>T</sup> <sup>T</sup>* (56)

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> *Yi Yi Yi Yi* (*i* = F, O, P) (57)

Fig. 3. Temperature profiles over the burning graphite rod in airflow at an atmospheric pressure. The H2O mass-fraction is 0.002. Data points are experimental (Makino, et al., 1996; Makino, et al., 1997) and solid curves are theoretical (Makino, 1990); (a) for the velocity gradient 110 s-1, with the surface temperature taken as a parameter; (b) for 200 s-1; (c) for the surface temperature 1700 K, with the velocity gradient taken as a parameter.

### **4.3 Ignition criterion**

266 Mass Transfer in Chemical Engineering Processes

(a) (b)

(c)

Fig. 3. Temperature profiles over the burning graphite rod in airflow at an atmospheric pressure. The H2O mass-fraction is 0.002. Data points are experimental (Makino, et al., 1996; Makino, et al., 1997) and solid curves are theoretical (Makino, 1990); (a) for the velocity gradient 110 s-1, with the surface temperature taken as a parameter; (b) for 200 s-1; (c) for the

surface temperature 1700 K, with the velocity gradient taken as a parameter.

While studies relevant to the ignition/extinction of CO-flame over the burning carbon are of obvious practical utility in evaluating protection properties from oxidation in re-entry vehicles, as well as the combustion of coal/char, they also command fundamental interests because of the simultaneous existence of the surface and gas-phase reactions with intimate coupling (Visser & Adomeit, 1984; Makino & Law, 1986; Matsui & Tsuji, 1987). As mentioned in the previous Section, at the same surface temperature, the combustion rate is expected to be momentarily reduced upon ignition because establishment of the CO-flame in the gas phase can change the dominant surface reactions from the faster C-O2 reaction to the slower C-CO2 reaction. By the same token the combustion rate is expected to momentarily increase upon extinction. These concepts are not intuitively obvious without considering the coupled nature of the gas-phase and surface reactions.

Fundamentally, the ignition/extinction of CO-flame in carbon combustion must necessarily be described by the seminal analysis (Liñán, 1974) of the ignition, extinction, and structure of diffusion flames, as indicated by Matalon (1980, 1981, 1982). Specifically, as the flame temperature increases from the surface temperature to the adiabatic flame temperature, there appear a nearly-frozen regime, a partial-burning regime, a premixed-flame regime, and finally a near-equilibrium regime. Ignition can be described in the nearly-frozen regime, while extinction in the other three regimes.

For carbon combustion, Matalon (1981) analytically obtained an explicit ignition criterion when the O2 mass-fraction at the surface is O(l). When this concentration is O(), the appropriate reduced governing equation and the boundary conditions were also identified (Matalon, 1982). Here, putting emphasis on the ignition of CO-flame over the burning carbon, an attempt has first been made to extend the previous theoretical studies, so as to include analytical derivations of various criteria governing the ignition, with arbitrary O2 concentration at the surface. Note that these derivations are successfully conducted, by virtue of the generalized species-enthalpy coupling functions (Makino & Law, 1986; Makino, 1990), identified in the previous Section. Furthermore, it may be noted that the ignition analysis is especially relevant for situations where the surface O2 concentration is O() because in order for gas-phase reaction to be initiated, sufficient amount of carbon monoxide should be generated. This requires a reasonably fast surface reaction and thereby low O2 concentration. The second objective is to conduct experimental comparisons relevant to the ignition of CO-flame over a carbon rod in an oxidizing stagnation flow, with variations in the surface temperature of the rod, as well as the freestream velocity gradient and O2 concentration.

### **4.3.1 Ignition analysis**

Here we intend to obtain an explicit ignition criterion without restricting the order of *Y*O,s. First we note that in the limit of *Ta*<sup>g</sup> , the completely frozen solutions for Eqs. (16) and (17) are

$$\left(\widetilde{T}\right)\_0 = \widetilde{T}\_\mathbf{s} + \left(\widetilde{T}\_\infty - \widetilde{T}\_\mathbf{s}\right)\widetilde{\xi} \tag{56}$$

$$\left(\widetilde{Y}\_{i}\right)\_{0} = \widetilde{Y}\_{i,\mathbf{s}} + \left(\widetilde{Y}\_{i,\mathbf{c}\diamond} - \widetilde{Y}\_{i,\mathbf{s}}\right)\xi \qquad \left(i = \mathbf{F}\_{i}\ \mathbf{O}, \mathbf{P}\right) \tag{57}$$

For finite but large values of *Ta*g, weak chemical reaction occurs in a thin region next to the carbon surface when the surface temperature is moderately high and exceeds the ambient

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

phase, since solid phase has great thermal inertia. For the outer, non-reactive region, if we write

former allows the determination of *C*I.

  out

Eq. (65) is

matching.

we have

criterion as

I

2

at = 1:

<sup>1</sup> <sup>2</sup>

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 269

 s O,s ~ ~

Note that the situation of *Y*F,s = O() is not considered here because it corresponds to very weak carbon combustion, such as in low O2 concentration or at low surface temperature. Evaluating the inner temperature at the surface of constant *T*s, one boundary condition for

This boundary condition is a reasonable one from the viewpoint of gas-phase quasisteadiness in that its surface temperature changes at rates much slower than that of the gas

<sup>2</sup>

we see from Eq. (17) that is governed by *L* 0 with the boundary condition that () = 0. Then, the solution is () = - *C*I (1 - ), where *C*I is a constant to be determined through

By matching the inner and outer temperatures presented in Eqs. (59) and (69), respectively,

, <sup>0</sup> <sup>I</sup>

the latter of which provides the additional boundary condition to solve Eq. (65), while the

Thus the problem is reduced to solving the single governing Eq. (65), subject to the boundary conditions Eqs. (68) and (70). The key parameters are , , and O. Before solving Eq. (65) numerically, it should be noted that there exists a general expression for the ignition

1

 or <sup>0</sup> ~

 

which implies that the heat transferred from the surface to the gas phase ceases at the ignition point. Note also that Eq. (71) further yields analytical solutions for some special cases, such as

I

 

1

corresponding to the critical condition for the vanishment of solutions at

 

*<sup>O</sup> e erfc <sup>O</sup> e erfc z d <sup>O</sup>*

 1 *<sup>s</sup> <sup>d</sup> d*

   

> 

 

 *<sup>s</sup> in d*

*<sup>O</sup> e erfc <sup>O</sup> <sup>O</sup>* 

 2

*d*

*T Y*

*<sup>O</sup>* . (67)

(0)=0 (68)

*<sup>d</sup> <sup>C</sup>* . (70)

; *<sup>z</sup> erfc <sup>z</sup> <sup>t</sup> dt* <sup>2</sup> exp

*<sup>d</sup> <sup>T</sup>* , (72)

<sup>1</sup> <sup>2</sup> <sup>I</sup> , (73)

<sup>2</sup> , (71)

<sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>~</sup> *<sup>T</sup> <sup>T</sup> <sup>T</sup> <sup>T</sup> <sup>T</sup> <sup>O</sup> <sup>s</sup> <sup>s</sup> <sup>s</sup>* , (69)

temperature. Since the usual carbon combustion proceeds under this situation, corresponding to the condition (Liñán, 1974) of

$$
\widetilde{T}\_{\sf s} + \widetilde{Y}\_{\sf F, \sf s} > \widetilde{T}\_{\sf c, \sf c} \tag{58}
$$

we define the inner temperature distribution as

$$
\begin{pmatrix} \widetilde{T} \\ \widetilde{T} \end{pmatrix}\_{\text{in}} = \begin{pmatrix} \widetilde{T} \\ \end{pmatrix}\_{0} + \varepsilon \ \widetilde{T}\_{\text{s}} \lambda \,\Theta(\xi) + O(\varepsilon^{2}) \ = \widetilde{T}\_{\text{s}} \begin{bmatrix} 1 - \varepsilon(\underline{\chi} - \lambda \,\Theta) \end{bmatrix} + O(\varepsilon^{2}) \tag{59}
$$

where

$$\mathfrak{e} = \frac{\widetilde{T}\_{\mathrm{s}}}{\widetilde{T}a\_{\mathrm{g}}}, \quad \lambda = \frac{\widetilde{Y}\_{\mathrm{O},\alpha}}{\widetilde{T}\_{\mathrm{s}} - \widetilde{T}\_{\alpha}}, \quad \xi = \mathfrak{e} \left(\frac{\widetilde{T}\_{\mathrm{s}}}{\widetilde{T}\_{\mathrm{s}} - \widetilde{T}\_{\alpha}}\right) \chi \,. \tag{60}$$

In the above, is the appropriate small parameter for expansion, and and are the inner variables.

With Eq. (59) and the coupling functions of Eqs. (33) to (36), the inner species distributions are given by:

$$
\widetilde{Y}\_{\mathbf{O}} \Big|\_{\text{in}} = \widetilde{Y}\_{\mathbf{O}, \mathbf{s}} + \varepsilon \, \widetilde{T}\_{\mathbf{s}} \lambda(\mathbf{y} - \boldsymbol{\Theta}) \tag{61}
$$

$$\left(\widetilde{Y}\_{\rm F}\right)\_{\rm ln} = \left(\frac{2\delta\mathfrak{P} - \widetilde{Y}\_{\rm O,\alpha}}{1 + \mathfrak{P}}\right) + \widetilde{Y}\_{\rm O,s} + \varepsilon \left(\frac{\widetilde{T}\_{\rm s}}{\widetilde{T}\_{\rm s} - \widetilde{T}\_{\rm o}}\right) \left(\frac{\widetilde{Y}\_{\rm O,\alpha} - 2\delta\mathfrak{P}}{1 + \mathfrak{P}} \chi - \widetilde{Y}\_{\rm O,\alpha}\Theta\right). \tag{62}$$

Thus, through evaluation of the parameter , expressed as

$$\gamma = \left(\frac{d\widetilde{T}}{d\xi}\right)\_{\rm s} = \left[\frac{d\chi}{d\xi}\frac{d\{\widetilde{T}\}\_{\rm in}}{d\chi}\right]\_{\rm s} = -\left(\widetilde{T}\_{\rm s} - \widetilde{T}\_{\rm \alpha}\right) + \widetilde{Y}\_{\rm O,\alpha} \left(\frac{d\Theta}{d\chi}\right)\_{\rm s} + O(\mathfrak{e})\,\,\,\tag{63}$$

the O2 mass-fraction at the surface is obtained as

$$\widetilde{Y}\_{\rm O,s} = \frac{\widetilde{Y}\_{\rm O,s}}{1 + \mathfrak{B} + A\_{\rm s,O} \left[\mathfrak{B}/(-f\_s)\right]} \left\{ 1 - \left(\frac{d\Theta}{d\chi}\right)\_s \right\}.\tag{64}$$

Substituting , Eqs. (59), (61), and (62) into the governing Eq. (17), expanding, and neglecting the higher-order convection terms, we obtain

$$\frac{d^2\theta}{d\chi^2} = -\Delta(\chi - \theta + \eta\_{\bullet})^{1/2} \exp(\lambda\theta - \chi),\tag{65}$$

where

$$
\Delta = D a\_{\rm g} \exp\left(-\frac{\widetilde{T} a\_{\rm g}}{\widetilde{T}\_{\rm s}}\right) \left| \left(\frac{\mathfrak{J}}{(-f\_{\rm s})}\right) \left(\frac{\widetilde{T}\_{\rm s}}{\widetilde{T}\_{\rm s} - \widetilde{T}\_{\rm o}}\right) \right|^{2} \left(\frac{\widetilde{T}\_{\rm s}}{\widetilde{T} a\_{\rm g}}\right)^{3/2} \left(\frac{\widetilde{T}\_{\rm o}}{\widetilde{T}\_{\rm s}}\right)^{3/2} \frac{\widetilde{Y}\_{\rm F,s}}{\left(\widetilde{\lambda} \widetilde{T}\_{\rm s}\right)^{3/2}},\tag{66}
$$

$$
\eta\_O = \frac{\widetilde{Y}\_{O,s}}{\varepsilon \widetilde{T}\_s \lambda} \,. \tag{67}
$$

Note that the situation of *Y*F,s = O() is not considered here because it corresponds to very weak carbon combustion, such as in low O2 concentration or at low surface temperature. Evaluating the inner temperature at the surface of constant *T*s, one boundary condition for Eq. (65) is

$$0 \neq 0 = 0\tag{68}$$

This boundary condition is a reasonable one from the viewpoint of gas-phase quasisteadiness in that its surface temperature changes at rates much slower than that of the gas phase, since solid phase has great thermal inertia.

For the outer, non-reactive region, if we write

268 Mass Transfer in Chemical Engineering Processes

temperature. Since the usual carbon combustion proceeds under this situation,

*T Y T* ~ ~ ~

> 

In the above, is the appropriate small parameter for expansion, and and are the inner

With Eq. (59) and the coupling functions of Eqs. (33) to (36), the inner species distributions

O,s <sup>s</sup> <sup>O</sup> in

s <sup>s</sup> O,s

~ ~

*T T*

 

*<sup>O</sup>*

Substituting , Eqs. (59), (61), and (62) into the governing Eq. (17), expanding, and

s s

 

<sup>g</sup> ~

~ ~ ~

*T T T*

*A f*

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

*<sup>T</sup> <sup>Y</sup>*

 

s

O, O,s 1

*Y*

2 O

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

in

 *T T Y* ~ ~ ~

s

<sup>s</sup> F,s , (58)

 

<sup>~</sup> <sup>~</sup> <sup>~</sup> *<sup>Y</sup> <sup>Y</sup> <sup>T</sup>* (61)

 

> 

*d*

 

 

*<sup>d</sup>* , (65)

3 2

 

s

1 <sup>2</sup> <sup>~</sup>

O,

*Y*

<sup>s</sup> . (60)

 

, (63)

, (66)

. (64)

s F,s

*T Y*

~

1 2

 

s

~ ~

*T T*

  ~

*Y*

O,

<sup>~</sup> <sup>~</sup> <sup>~</sup> *<sup>T</sup> <sup>T</sup> <sup>T</sup> <sup>O</sup>* <sup>2</sup> <sup>s</sup> <sup>1</sup> <sup>~</sup> *<sup>T</sup> <sup>O</sup>* (59)

O, ,

 

 

*d d*

 

O s s,

exp <sup>1</sup> <sup>2</sup>

2

 

 

<sup>1</sup> <sup>2</sup>

~ ~

 

*Ta T*

g s

 

*s*

*<sup>d</sup> <sup>T</sup> <sup>T</sup> <sup>Y</sup>*

s O,

 

*Y* . (62)

 *T T T* ~ ~ ~

s

corresponding to the condition (Liñán, 1974) of

we define the inner temperature distribution as

 

 

*d dT*

the O2 mass-fraction at the surface is obtained as

F in

<sup>~</sup> <sup>~</sup> <sup>2</sup>

where

variables.

where

are given by:

<sup>2</sup>

g s ~ ~

> 

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

*d d T*

O,

Thus, through evaluation of the parameter , expressed as

  *d d*

 

*Y*

1

s

*Y*

neglecting the higher-order convection terms, we obtain

 

g

~ ~

*Ta*

exp

 

*Da*

s s

 

*T f*

 

 

2 *d*

*Ta T* ,

in 0 s

$$\left(\widetilde{T}\right)\_{\text{out}} = \widetilde{T}\_s + \left(\widetilde{T}\_\infty - \widetilde{T}\_s\right)\widetilde{\xi} + \varepsilon \,\,\widetilde{T}\_s \Theta(\xi) + O\left(\varepsilon^2\right),\tag{69}$$

we see from Eq. (17) that is governed by *L* 0 with the boundary condition that () = 0. Then, the solution is () = - *C*I (1 - ), where *C*I is a constant to be determined through matching.

By matching the inner and outer temperatures presented in Eqs. (59) and (69), respectively, we have

$$
\lambda \Theta(\infty) = -C\_{1\prime} \qquad \left(\frac{d\Theta}{d\chi}\right)\_{\phi} = 0 \cdot \tag{70}
$$

the latter of which provides the additional boundary condition to solve Eq. (65), while the former allows the determination of *C*I.

Thus the problem is reduced to solving the single governing Eq. (65), subject to the boundary conditions Eqs. (68) and (70). The key parameters are , , and O. Before solving Eq. (65) numerically, it should be noted that there exists a general expression for the ignition criterion as

$$\Delta\Delta\chi = \frac{1}{\sqrt{\eta\_{\rm{O}}} + \frac{\sqrt{\pi \eta \lambda}}{2} e^{\lambda \eta\_{\rm{O}}} \left\{ \text{erfc}(\lambda \eta\_{\rm{O}}) + (\lambda - 1) \int\_{\chi\_{\rm{I}}}^{\infty} e^{(\lambda - 1)\chi} \text{erfc}(z) d\chi \right\}} \; \text{erfc}(z) = \frac{2}{\sqrt{\pi}} \mathbf{f}\_{z}^{\text{av}} \exp(-t^{2}) dt \; \text{v} \quad (71)$$

corresponding to the critical condition for the vanishment of solutions at

$$\left(\frac{d\theta}{d\mathcal{K}}\right)\_s = \frac{1}{\lambda} \quad \text{or} \quad \left(\frac{d\left(\widetilde{T}\right)\_{\text{fin}}}{d\widetilde{\xi}}\right)\_s = 0 \,\,\,\,\tag{72}$$

which implies that the heat transferred from the surface to the gas phase ceases at the ignition point. Note also that Eq. (71) further yields analytical solutions for some special cases, such as

$$\text{at } \lambda = 1: \quad 2\Lambda\_{\text{I}} = \frac{1}{\sqrt{\eta\_{\text{O}}} + \frac{\sqrt{\pi}}{2} e^{\eta\_{\text{O}}} \text{erfc}(\lambda \eta\_{\text{O}})} \text{ . \tag{73}$$

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

C-O2 reaction.

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 271

decreases with increasing *Y*O,. In this case the CO-O2 reaction is facilitated with increasing concentrations of O2, as well as CO, because more CO is now produced through the surface

Fig. 4. Surface temperature at the establishment of CO-flame, as a function of the stagnation velocity gradient, with the O2 mass-fraction in the freestream and the surface Damköhler number for the C-O2 reaction taken as parameters. Data points are experimental (Makino, et al., 1996) with the test specimen of 10 mm in diameter and 1.25103 kg/m3 in graphite

Solid and dashed curves in Fig. 4 are predicted ignition surface-temperature for *Da*s,O=107 and 108, obtained by the ignition criterion described here and the kinetic parameters (Makino, et al., 1994) to be explained, with keeping as many parameters fixed as possible. The density of the oxidizing gas in the freestream is estimated at *T*= 323 K. The surface Damköhler numbers in the experimental conditions are from 2107 to 2108, which are obtained with *B*s,O = 4.1106 m/s. It is seen that fair agreement is demonstrated, suggesting that the present ignition criterion has captured the essential feature of the ignition of CO-

In this Section, an attempt is made to extend and integrate previous theoretical studies (Makino, 1990; Makino and Law, 1990), in order to further investigate the coupled nature of the surface and gas-phase reactions. First, by use of the combustion rate of the graphite rod in the forward stagnation region of various oxidizer-flows, it is intended to obtain kinetic parameters for the surface C-O2 and C-CO2 reactions, based on the theoretical work (Makino, 1990), presented in Section 2. Second, based on experimental facts that the ignition of CO-flame over the burning graphite is closely related to the surface temperature and the

density; curves are calculated from theory (Makino & Law, 1990).

**5. Kinetic parameters for the surface and gas-phase reactions** 

flame over the burning carbon.

$$\text{as } \eta\_{\text{O}} \to \infty; \quad 2\Lambda\_{\text{I}}\lambda = \frac{1}{\sqrt{\eta\_{\text{O}}}},\tag{74}$$

the latter of which agrees with the result of Matalon (1981).

In numerically solving Eq. (65), by plotting () vs. for a given set of and O, the lower ignition branch of the S-curve can first be obtained. The values of , corresponding to the vertical tangents to these curves, are then obtained as the reduced ignition Damköhler number I. After that, a universal curve of (2I) vs. (1/) is obtained with O taken as a parameter. Recognizing that (l/) is usually less than about 0.5 for practical systems and using Eqs. (71), (73), and (74), we can fairly represent (2I) as (Makino & Law, 1990)

$$2\Delta\Lambda\_1\lambda = \frac{1}{\sqrt{\eta\_O} + \frac{\sqrt{\pi}}{2} \left[ e^{\eta\_O} \operatorname{erfc} \{ \sqrt{\eta\_O} \} + \left\{ \frac{1}{F(\lambda)} - 1 \right\} \exp \left( - \frac{\sqrt{\eta\_O}}{2} \right) \right] \tag{75}$$

where

$$F(\lambda) = 0.56 + \frac{0.21}{\lambda} - \frac{0.12}{\lambda^2} + \frac{0.35}{\lambda^3} \tag{76}$$

Note that for large values of (l/), Eq. (75) is still moderately accurate. Thus, for a given set of and O, an ignition Damköhler number can be determined by substituting the values of I, obtained from Eq. (75), into Eq. (66).

It may be informative to note that for some weakly-burning situations, in which O2 concentrations in the reaction zone and at the carbon surface are O(1), a monotonic transition from the nearly-frozen to the partial-burning behaviors is reported (Henriksen, 1989), instead of an abrupt, turning-point behavior, with increasing gas-phase Damköhler number. However, this could be a highly-limiting behavior. That is, in order for the gasphase reaction to be sufficiently efficient, and the ignition to be a reasonably plausible event, enough CO would have to be generated at the surface, which further requires a sufficiently fast surface C-O2 reaction and hence the diminishment of the surface O2 concentration from O(l). For these situations, the turning-point behavior can be a more appropriate indication for the ignition.

### **4.3.2 Experimental comparisons for the ignition of CO flame**

Figure 4 shows the ignition surface-temperature (Makino, et al., 1996), as a function of the velocity gradient, with O2 mass-fraction taken as a parameter. The velocity gradient has been chosen for the abscissa, as originally proposed by Tsuji & Yamaoka (1967) for the present flow configuration, after confirming its appropriateness, being examined by varying both the freestream velocity and graphite rod diameter that can exert influences in determining velocity gradient. It is seen that the ignition surface-temperature increases with increasing velocity gradient and thereby decreasing residence time. The high surface temperature, as well as the high temperature in the reaction zone, causes the high ejection rate of CO through the surface C-O2 reaction. These enhancements facilitate the CO-flame, by reducing the characteristic chemical reaction time, and hence compensating a decrease in the characteristic residence time. It is also seen that the ignition surface-temperature

In numerically solving Eq. (65), by plotting () vs. for a given set of and O, the lower ignition branch of the S-curve can first be obtained. The values of , corresponding to the vertical tangents to these curves, are then obtained as the reduced ignition Damköhler number I. After that, a universal curve of (2I) vs. (1/) is obtained with O taken as a parameter. Recognizing that (l/) is usually less than about 0.5 for practical systems and

using Eqs. (71), (73), and (74), we can fairly represent (2I) as (Makino & Law, 1990)

 

*<sup>O</sup> <sup>O</sup> F e erfc <sup>O</sup>*

<sup>1</sup> <sup>2</sup> <sup>I</sup>

2

**4.3.2 Experimental comparisons for the ignition of CO flame** 

*O*

 

 

 <sup>2</sup> <sup>3</sup> 0.21 0.12 0.35 0.56

Note that for large values of (l/), Eq. (75) is still moderately accurate. Thus, for a given set of and O, an ignition Damköhler number can be determined by substituting the values of

It may be informative to note that for some weakly-burning situations, in which O2 concentrations in the reaction zone and at the carbon surface are O(1), a monotonic transition from the nearly-frozen to the partial-burning behaviors is reported (Henriksen, 1989), instead of an abrupt, turning-point behavior, with increasing gas-phase Damköhler number. However, this could be a highly-limiting behavior. That is, in order for the gasphase reaction to be sufficiently efficient, and the ignition to be a reasonably plausible event, enough CO would have to be generated at the surface, which further requires a sufficiently fast surface C-O2 reaction and hence the diminishment of the surface O2 concentration from O(l). For these situations, the turning-point behavior can be a more appropriate indication

Figure 4 shows the ignition surface-temperature (Makino, et al., 1996), as a function of the velocity gradient, with O2 mass-fraction taken as a parameter. The velocity gradient has been chosen for the abscissa, as originally proposed by Tsuji & Yamaoka (1967) for the present flow configuration, after confirming its appropriateness, being examined by varying both the freestream velocity and graphite rod diameter that can exert influences in determining velocity gradient. It is seen that the ignition surface-temperature increases with increasing velocity gradient and thereby decreasing residence time. The high surface temperature, as well as the high temperature in the reaction zone, causes the high ejection rate of CO through the surface C-O2 reaction. These enhancements facilitate the CO-flame, by reducing the characteristic chemical reaction time, and hence compensating a decrease in the characteristic residence time. It is also seen that the ignition surface-temperature

1

 

1 exp

*F* (76)

 

2

*O*

<sup>1</sup> <sup>2</sup> <sup>I</sup> , (74)

 

 

, (75)

as O:

I, obtained from Eq. (75), into Eq. (66).

where

for the ignition.

the latter of which agrees with the result of Matalon (1981).

decreases with increasing *Y*O,. In this case the CO-O2 reaction is facilitated with increasing concentrations of O2, as well as CO, because more CO is now produced through the surface C-O2 reaction.

Fig. 4. Surface temperature at the establishment of CO-flame, as a function of the stagnation velocity gradient, with the O2 mass-fraction in the freestream and the surface Damköhler number for the C-O2 reaction taken as parameters. Data points are experimental (Makino, et al., 1996) with the test specimen of 10 mm in diameter and 1.25103 kg/m3 in graphite density; curves are calculated from theory (Makino & Law, 1990).

Solid and dashed curves in Fig. 4 are predicted ignition surface-temperature for *Da*s,O=107 and 108, obtained by the ignition criterion described here and the kinetic parameters (Makino, et al., 1994) to be explained, with keeping as many parameters fixed as possible. The density of the oxidizing gas in the freestream is estimated at *T*= 323 K. The surface Damköhler numbers in the experimental conditions are from 2107 to 2108, which are obtained with *B*s,O = 4.1106 m/s. It is seen that fair agreement is demonstrated, suggesting that the present ignition criterion has captured the essential feature of the ignition of COflame over the burning carbon.

### **5. Kinetic parameters for the surface and gas-phase reactions**

In this Section, an attempt is made to extend and integrate previous theoretical studies (Makino, 1990; Makino and Law, 1990), in order to further investigate the coupled nature of the surface and gas-phase reactions. First, by use of the combustion rate of the graphite rod in the forward stagnation region of various oxidizer-flows, it is intended to obtain kinetic parameters for the surface C-O2 and C-CO2 reactions, based on the theoretical work (Makino, 1990), presented in Section 2. Second, based on experimental facts that the ignition of CO-flame over the burning graphite is closely related to the surface temperature and the

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

approximate relation (Makino, 1990)

(Makino, et al., 1994).

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 273

In order to verify this method, the reduced surface Damköhler number *Ai* is obtained numerically by use of Eq. (77) and/or Eq. (78). Figure 5 shows the Arrhenius plot of *Ai* with the gas-phase Damköhler number taken as a parameter. We see that with increasing surface temperature the combustion behavior shifts from the Frozen mode to the Flame-detached mode, depending on the gas-phase Damköhler number. Furthermore, in the present plot, the combustion behavior in the Frozen mode purely depends on the surface C-O2 reaction rate; that in the Flame-detached mode depends on the surface C-CO2 reaction rate. Since the appropriateness of the present method has been demonstrated, estimation of the surface kinetic parameters is conducted with experimental results (Makino, et al., 1994), by use of an

0.56 <sup>~</sup> 0.4

for evaluating the transfer number from the combustion rate through the relation =(-*f*s)/(s) in Eq. (39). Values of parameters used are *q* = 10.11 MJ/kg, *c*p = 1.194 kJ/(kgK), *q*/(*c*pF) = 5387 K, and *T* = 323 K. Thermophysical properties of oxidizer are also conventional ones

Fig. 6. Arrhenius plot of the surface C-O2 and C-CO2 reactions (Makino, et al., 1994), obtained from the experimental results of the combustion rate in oxidizer-flow of various velocity gradients; (a) for the test specimen of 1.82103 kg/m3 in graphite density; (b) for the

Figure 6(a) shows the Arrhenius plot of surface reactivities, being obtained by multiplying A*i* by [*a*(/)]1/2 , for the results of the test specimen with 1.82103 kg/m3 in density. For the C-O2 reaction *B*s,O =2.2106 m/s and *E*s,O = 180 kJ/mol are obtained, while for the C-CO2 reaction *B*s,P = 6.0107 m/s and *E*s,P = 269 kJ/mol. Figure 6(b) shows the results of the test specimen with 1.25103 kg/m3. It is obtained that *B*s,O = 4.1106 m/s and *E*s,O= 179 kJ/mol for the C-O2 reaction, and that *B*s,P = 1.1108 m/s and *E*s,P = 270 kJ/mol for the C-CO2 reaction. Activation energies are respectively within the ranges of the surface C-O2 and C-

test specimen of 1.25103 kg/m3 in graphite density.

*<sup>s</sup> T*<sup>s</sup> (79)

stagnation velocity gradient, it is intended to obtain kinetic parameters for the global gasphase CO-O2 reaction prior to the ignition of CO-flame, by use of the ignition criterion (Makino and Law, 1990), presented in Section 4. Finally, experimental comparisons are further to be conducted.

### **5.1 Surface kinetic parameters**

In estimating kinetic parameters for the surface reactions, their contributions to the combustion rate are to be identified, taking account of the combustion situation in the limiting cases, as well as relative reactivities of the C-O2 and C-CO2 reactions. In the kinetically controlled regime, the combustion rate reflects the surface reactivity of the ambient oxidizer. Thus, by use of Eqs. (31) and (34), the reduced surface Damköhler number is expressed as

$$A\_{i} = \frac{\delta \left(-f\_{\rm s}\right)(1+\beta)}{\widetilde{Y}\_{i,\rm \alpha} - \delta \beta} \qquad \qquad \text{( $i=O$ , P)}\tag{77}$$

when only one kind of oxidizer participates in the surface reaction. In the diffusionally controlled regime, combustion situation is that of the Flame-detached mode, thereby following expression is obtained:

$$A\_{\rm P} = \frac{\delta \left(-f\_{\rm s}\right)(1+\beta)}{\widetilde{Y}\_{\rm O,\phi} - \delta \beta} \tag{78}$$

Note that the combustion rate here reflects the C-CO2 reaction even though there only exists oxygen in the freestream.

Fig. 5. Arrhenius plot of the reduced surface Damköhler number with the gas-phase Damköhler number taken as a parameter; *Da*s,O= *Da*s,P=108; *Da*s,P/*Da*s,O=1; *Y*O,=0.233; *Y*P,=0 (Makino, et al., 1994).

stagnation velocity gradient, it is intended to obtain kinetic parameters for the global gasphase CO-O2 reaction prior to the ignition of CO-flame, by use of the ignition criterion (Makino and Law, 1990), presented in Section 4. Finally, experimental comparisons are

In estimating kinetic parameters for the surface reactions, their contributions to the combustion rate are to be identified, taking account of the combustion situation in the limiting cases, as well as relative reactivities of the C-O2 and C-CO2 reactions. In the kinetically controlled regime, the combustion rate reflects the surface reactivity of the ambient oxidizer. Thus, by use of Eqs. (31) and (34), the reduced surface Damköhler number is expressed as

> , s ~ ( ) 1 *i <sup>i</sup> <sup>Y</sup>*

In the diffusionally controlled regime, combustion situation is that of the Flame-detached

 O, <sup>s</sup> <sup>P</sup> <sup>~</sup> ( ) 1 *Y*

Note that the combustion rate here reflects the C-CO2 reaction even though there only exists

Fig. 5. Arrhenius plot of the reduced surface Damköhler number with the gas-phase

Damköhler number taken as a parameter; *Da*s,O= *Da*s,P=108; *Da*s,P/*Da*s,O=1; *Y*O,=0.233; *Y*P,=0

 

when only one kind of oxidizer participates in the surface reaction.

mode, thereby following expression is obtained:

*<sup>f</sup> <sup>A</sup>* (*i* = O, P) (77)

*<sup>f</sup> <sup>A</sup>* (78)

further to be conducted.

oxygen in the freestream.

(Makino, et al., 1994).

**5.1 Surface kinetic parameters** 

In order to verify this method, the reduced surface Damköhler number *Ai* is obtained numerically by use of Eq. (77) and/or Eq. (78). Figure 5 shows the Arrhenius plot of *Ai* with the gas-phase Damköhler number taken as a parameter. We see that with increasing surface temperature the combustion behavior shifts from the Frozen mode to the Flame-detached mode, depending on the gas-phase Damköhler number. Furthermore, in the present plot, the combustion behavior in the Frozen mode purely depends on the surface C-O2 reaction rate; that in the Flame-detached mode depends on the surface C-CO2 reaction rate. Since the appropriateness of the present method has been demonstrated, estimation of the surface kinetic parameters is conducted with experimental results (Makino, et al., 1994), by use of an approximate relation (Makino, 1990)

$$\left(\xi\_{\rm s}^{\prime}\right) = 0.4\widetilde{T}\_{\rm s} + 0.56\tag{79}$$

for evaluating the transfer number from the combustion rate through the relation =(-*f*s)/(s) in Eq. (39). Values of parameters used are *q* = 10.11 MJ/kg, *c*p = 1.194 kJ/(kgK), *q*/(*c*pF) = 5387 K, and *T* = 323 K. Thermophysical properties of oxidizer are also conventional ones (Makino, et al., 1994).

Fig. 6. Arrhenius plot of the surface C-O2 and C-CO2 reactions (Makino, et al., 1994), obtained from the experimental results of the combustion rate in oxidizer-flow of various velocity gradients; (a) for the test specimen of 1.82103 kg/m3 in graphite density; (b) for the test specimen of 1.25103 kg/m3 in graphite density.

Figure 6(a) shows the Arrhenius plot of surface reactivities, being obtained by multiplying A*i* by [*a*(/)]1/2 , for the results of the test specimen with 1.82103 kg/m3 in density. For the C-O2 reaction *B*s,O =2.2106 m/s and *E*s,O = 180 kJ/mol are obtained, while for the C-CO2 reaction *B*s,P = 6.0107 m/s and *E*s,P = 269 kJ/mol. Figure 6(b) shows the results of the test specimen with 1.25103 kg/m3. It is obtained that *B*s,O = 4.1106 m/s and *E*s,O= 179 kJ/mol for the C-O2 reaction, and that *B*s,P = 1.1108 m/s and *E*s,P = 270 kJ/mol for the C-CO2 reaction. Activation energies are respectively within the ranges of the surface C-O2 and C-

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

**5.3 Experimental comparisons for the combustion rate** 

(a) (b)

Fig. 8. Experimental comparisons (Makino, et al., 1994) for the combustion rate of test

temperature can be converted into conventional one by multiplying *q*/(*c*pF) = 5387 K.

specimen (C = 1.82103 kg/m3 in graphite density) in airflow under an atmospheric pressure with H2O mass-fraction of 0.003; (a) for 200 s-1 in stagnation velocity gradient; (b) for 820 s-1. Data points are experimental and solid curves are calculated from theory. The nondimensional

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 275

It is noted that *B*g\* obtained here is one order of magnitude lower than that of Howard, et al. (1973), which is reported to be *B*g\* =1.3108 [(mol/m3)1/2s]-1, because the present value is that prior to the appearance of CO-flame and is to be low, compared to that of the "strong" COoxidation in the literature. As for the "weak" CO-oxidation, Sobolev (1959) reports *B*g\* = 3.0106 [(mol/m3)1/2s]-1, by examining data of Chukhanov (1938a, 1938b) who studied the initiation of CO-oxidation, accompanied by the carbon combustion. We see that the value reported by Sobolev (1959) exhibits a lower bound of the experimental results shown in Fig. 7. It is also confirmed in Fig. 7 that there exists no remarkable effects of O2 and/or H2O concentrations in the oxidizer, thereby the assumption for the reaction orders is shown to be appropriate within the present experimental conditions. The choice of reaction orders, however, requires a further comment because another reaction order for O2 concentration, 0.25 in place of 0.5, is recommended in the literature. Relevant to this, an attempt (Makino, et al., 1994) has further been conducted to compare the experimental data with another ignition criterion, obtained through a similar ignition analysis with this reaction order. However, its result was unfavorable, presenting a much poorer correlation between them.

Experimental comparisons have already been conducted in Fig. 2, for test specimens with C=1.25103 kg/m3 in graphite density, and a fair degree of agreement has been demonstrated, as far as the trend and approximate magnitude are concerned. Further experimental comparisons are made for test specimens with C=1.82103 kg/m3 (Makino, et al., 1994), with kinetic parameters obtained herein. Figure 8(a) compares predicted results with experimental data in airflow of 200 s-1 at an atmospheric pressure. The gas-phase Damköhler number is evaluated to be *Da*g= 3104 from the present kinetic parameter, while *Da*g = 4105 from the value in the literature (Howard, et al., 1973). The ignition surfacetemperature is estimated to be *T*s,ig 1476 K from the ignition analysis. We see from Fig. 8(a)

CO2 reactions; *cf.* Table 19.6 in Essenhigh (1981). It is also seen in Figs. 6(a) and 6(b) that the first-order Arrhenius kinetics, assumed in the theoretical model, is appropriate for the surface C-O2 and C-CO2 reactions within the present experimental conditions.

### **5.2 Global gas-phase kinetic parameters**

Estimation of gas-phase kinetic parameters has also been made with experimental data for the ignition surface-temperature and the ignition criterion (Makino & Law, 1990) for the COflame over the burning carbon. Here, reaction orders are *a priori* assumed to be *n*F = 1 and *n*<sup>O</sup> = 0.5, which are the same as those of the global rate expression by Howard et al. (1973). It is also assumed that the frequency factor *B*g is proportional to the half order of H2O concentration: that is, *B*g = *B*g\*(*Y*A/*W*A)1/2 [(mol/m3)1/2s]-1, where the subscript A designates water vapor. The H2O mass-fraction at the surface is estimated with *Y*A,s = *Y*A,/(l+), with water vapor taken as an inert because it acts as a kind of catalyst for the gas-phase CO-O2 reaction, and hence its profile is not anticipated to be influenced. Thus, for a given set of and O, an ignition Damköhler number can be determined by substituting <sup>I</sup> in Eq. (75) into Eq. (66).

Figure 7 shows the Arrhenius plot of the global gas-phase reactivity, obtained as the results of the ignition surface-temperature. In data processing, data in a series of experiments (Makino & Law, 1990; Makino, et al., 1994) have been used, with using kinetic parameters for the surface C-O2 reaction. With iteration in terms of the activation temperature, required for determining I with respect to O, *E*g = 113 kJ/mol is obtained with *B*g\* = 9.1106 [(mol/m3)1/2s]-1. This activation energy is also within the range of the global CO-O2 reaction; *cf*. Table II in Howard, et al. (1973).

Fig. 7. Arrhenius plot of the global gas-phase reaction (Makino, et al., 1994), obtained from the experimental results of the ignition surface-temperature for the test specimens (1.82103 kg/m3 and 1.25103 kg/m3 in graphite density) in oxidizer-flow at various pressures, O2, and H2O concentrations .

CO2 reactions; *cf.* Table 19.6 in Essenhigh (1981). It is also seen in Figs. 6(a) and 6(b) that the first-order Arrhenius kinetics, assumed in the theoretical model, is appropriate for the

Estimation of gas-phase kinetic parameters has also been made with experimental data for the ignition surface-temperature and the ignition criterion (Makino & Law, 1990) for the COflame over the burning carbon. Here, reaction orders are *a priori* assumed to be *n*F = 1 and *n*<sup>O</sup> = 0.5, which are the same as those of the global rate expression by Howard et al. (1973). It is also assumed that the frequency factor *B*g is proportional to the half order of H2O concentration: that is, *B*g = *B*g\*(*Y*A/*W*A)1/2 [(mol/m3)1/2s]-1, where the subscript A designates water vapor. The H2O mass-fraction at the surface is estimated with *Y*A,s = *Y*A,/(l+), with water vapor taken as an inert because it acts as a kind of catalyst for the gas-phase CO-O2 reaction, and hence its profile is not anticipated to be influenced. Thus, for a given set of and O, an ignition Damköhler number can be determined by substituting <sup>I</sup>

Figure 7 shows the Arrhenius plot of the global gas-phase reactivity, obtained as the results of the ignition surface-temperature. In data processing, data in a series of experiments (Makino & Law, 1990; Makino, et al., 1994) have been used, with using kinetic parameters for the surface C-O2 reaction. With iteration in terms of the activation temperature, required for determining I with respect to O, *E*g = 113 kJ/mol is obtained with *B*g\* = 9.1106 [(mol/m3)1/2s]-1. This activation energy is also within the range of the global CO-O2

Fig. 7. Arrhenius plot of the global gas-phase reaction (Makino, et al., 1994), obtained from the experimental results of the ignition surface-temperature for the test specimens (1.82103 kg/m3 and 1.25103 kg/m3 in graphite density) in oxidizer-flow at various pressures, O2,

surface C-O2 and C-CO2 reactions within the present experimental conditions.

**5.2 Global gas-phase kinetic parameters** 

reaction; *cf*. Table II in Howard, et al. (1973).

in Eq. (75) into Eq. (66).

and H2O concentrations .

It is noted that *B*g\* obtained here is one order of magnitude lower than that of Howard, et al. (1973), which is reported to be *B*g\* =1.3108 [(mol/m3)1/2s]-1, because the present value is that prior to the appearance of CO-flame and is to be low, compared to that of the "strong" COoxidation in the literature. As for the "weak" CO-oxidation, Sobolev (1959) reports *B*g\* = 3.0106 [(mol/m3)1/2s]-1, by examining data of Chukhanov (1938a, 1938b) who studied the initiation of CO-oxidation, accompanied by the carbon combustion. We see that the value reported by Sobolev (1959) exhibits a lower bound of the experimental results shown in Fig. 7. It is also confirmed in Fig. 7 that there exists no remarkable effects of O2 and/or H2O concentrations in the oxidizer, thereby the assumption for the reaction orders is shown to be appropriate within the present experimental conditions. The choice of reaction orders, however, requires a further comment because another reaction order for O2 concentration, 0.25 in place of 0.5, is recommended in the literature. Relevant to this, an attempt (Makino, et al., 1994) has further been conducted to compare the experimental data with another ignition criterion, obtained through a similar ignition analysis with this reaction order. However, its result was unfavorable, presenting a much poorer correlation between them.

### **5.3 Experimental comparisons for the combustion rate**

Experimental comparisons have already been conducted in Fig. 2, for test specimens with C=1.25103 kg/m3 in graphite density, and a fair degree of agreement has been demonstrated, as far as the trend and approximate magnitude are concerned. Further experimental comparisons are made for test specimens with C=1.82103 kg/m3 (Makino, et al., 1994), with kinetic parameters obtained herein. Figure 8(a) compares predicted results with experimental data in airflow of 200 s-1 at an atmospheric pressure. The gas-phase Damköhler number is evaluated to be *Da*g= 3104 from the present kinetic parameter, while *Da*g = 4105 from the value in the literature (Howard, et al., 1973). The ignition surfacetemperature is estimated to be *T*s,ig 1476 K from the ignition analysis. We see from Fig. 8(a)

Fig. 8. Experimental comparisons (Makino, et al., 1994) for the combustion rate of test specimen (C = 1.82103 kg/m3 in graphite density) in airflow under an atmospheric pressure with H2O mass-fraction of 0.003; (a) for 200 s-1 in stagnation velocity gradient; (b) for 820 s-1. Data points are experimental and solid curves are calculated from theory. The nondimensional temperature can be converted into conventional one by multiplying *q*/(*c*pF) = 5387 K.

Mass Transfer Related to Heterogeneous Combustion of Solid Carbon

water-vapor in the oxidizing-gas are also to be taken into account.

experimental and theoretical results.

**7. Acknowledgment**

combustion of solid carbon.

*B* frequency factor

*D* diffusion coefficient *Da* Damköhler number

*E* activation energy

*k* surface reactivity

*T* temperature

*t* time

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

*Ta* activation temperature

*A* reduced surface Damköhler number

*c*p specific heat capacity of gas

*a* velocity gradient in the stagnation flowfield

*F* function defined in the ignition criterion

*q* heat of combustion per unit mass of CO

*s* boundary-layer variable along the surface

*m* dimensional mass burning (or combustion) rate

*f* nondimensional streamfunction

*L* convective-diffusive operator

**8. Nomenclature** 

*C* constant

*d* diameter

**General** 

in the Forward Stagnation Region - Part 1 - Combustion Rate and Flame Structure 277

Then, attempts have been made to estimate kinetic parameters for the surface and gas-phase reactions, indispensable for predicting combustion behavior. In estimating the kinetic parameters for the surface reactions, use has been made of the reduced surface Damköhler number, evaluated by the combustion rate measured in experiments. In estimating the kinetic parameters for the global gas-phase reaction, prior to the appearance of the COflame, use has been made of the ignition criterion theoretically obtained, by evaluating it at the ignition surface-temperature experimentally determined. Experimental comparisons have also been conducted and a fair degree of agreement has been demonstrated between

Further studies are intended to be made in Part 2 for exploring carbon combustion at high velocity gradients and/or in the High-Temperature Air Combustion, in which effects of

In conducting a series of studies on the carbon combustion, I have been assisted by many of my former graduate and undergraduate students, as well as research staffs, in Shizuoka University, being engaged in researches in the field of mechanical engineering for twenty years as a staff, from a research associate to a full professor. Here, I want to express my sincere appreciation to all of them who have participated in researches for exploring

*j j*=0 and 1 designate two-dimensional and axisymmetric flows, respectively

that up to the ignition surface-temperature the combustion proceeds under the "weak" COoxidation, that at the temperature the combustion rate abruptly changes, and that the "strong" CO-oxidation prevails above the temperature.

Figure 8(b) shows a similar plot in airflow of 820 s-1. Because of the lack of the experimental data, as well as the enhanced ignition surface-temperature (*T*s,ig 1810 K), which inevitably leads to small difference between combustion rates before and after the ignition of COflame, the abrupt change in the combustion rate does not appear clearly. However, the general behavior is similar to that in Fig. 8(a).

It may informative to note that a decrease in the combustion rate, observed at temperatures between 1500 K and 2000 K, has been so-called the "negative temperature coefficient" of the combustion rate, which has also been a research subject in the field of carbon combustion. Nagel and Strickland-Constable (1962) used the "site" theory to explain the peak rate, while Yang and Steinberg (1977) attributed the peak rate to the change of reaction depth at constant activation energy. Other entries relevant to the "negative temperature coefficient" can be found in the survey paper (Essenhigh, 1981). However, another explanation can be made, as explained (Makino, et al., 1994; Makino, et al., 1996; Makino, et al., 1998) in the previous Sections, that this phenomenon can be induced by the appearance of CO-flame, established over the burning carbon, thereby the dominant surface reaction has been altered from the C-O2 reaction to the C-CO2 reaction.

Since the appearance of CO-flame is anticipated to be suppressed at high velocity gradients, it has strongly been required to raise the velocity gradient as high as possible, in order for firm understanding of the carbon combustion, while it has been usual to do experiments under the stagnation velocity gradient less than 1000 s-1 (Matsui, et al., 1975; Visser & Adomeit, 1984; Makino, et al., 1994; Makino, et al., 1996), because of difficulties in conducting experiments. In one of the Sections in Part 2, it is intended to study carbon combustion at high velocity gradients.
