9.3. Diffusional Kinetics

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9.3. Diffusional Kinetics

In the case of heterogeneous surface burning of a particle, consideration must be given to the question of whether diffusion rates or surface kinetic reaction rates are controlling the overall burning rate of the material. In many cases, it cannot be assumed that the surface oxidation kinetic rate is fast compared to the rate of diffusion of oxygen to the surface. The surface temperature determines the rate of oxidation, and this temperature is not always known a priori. Thus, in surface combustion the assumption that chemical kinetic rates are much faster than diffusion rates cannot be made.

Consider, for example, a carbon surface burning in a concentration of oxygen in the free stream specified by ρm_o∞. The burning is at a steady mass rate. Then the concentration of oxygen at the surface is some value m_os. If the surface oxidation rate follows first-order kinetics, as Frank-Kamenetskii [21] assumed,

$G_{ox} = \frac{G_{f}}{i} = k_{s} ρ m_{os}$ $G_{ox} = \frac{G_{f}}{i} = k_{s} ρ m_{os}$

(9.9)

where G is the flux in grams per second per square centimeter; k_s the heterogeneous specific reaction rate constant for surface oxidation in units reflecting a volume to surface area ratio, that is, centimeters per second; and i the mass stoichiometric index. The problem is that m_os is unknown. But one knows that the consumption rate of oxygen must be equal to the rate of diffusion of oxygen to the surface. Thus, if h_Dρ is designated as the overall convective mass transfer coefficient (conductance), one can write

$G_{ox} = k_{s} ρ m_{os} = h_{D} ρ (m_{o \infty} - m_{os})$ $G_{ox} = k_{s} ρ m_{os} = h_{D} ρ (m_{o \infty} - m_{os})$

(9.10)

What is sought is the mass burning rate in terms of m_o∞. It follows that

$h_{D} m_{os} = h_{D} m_{o \infty} - k_{s} m_{os}$ $h_{D} m_{os} = h_{D} m_{o \infty} - k_{s} m_{os}$

(9.11)

$k_{s} m_{os} + h_{D} m_{os} = h_{D} m_{o \infty}$ $k_{s} m_{os} + h_{D} m_{os} = h_{D} m_{o \infty}$

(9.12)

$m_{os} = (\frac{h_{D}}{k_{s} + h_{D}}) m_{o \infty}$ $m_{os} = (\frac{h_{D}}{k_{s} + h_{D}}) m_{o \infty}$

(9.13)

$G_{ox} = (\frac{ρ k_{s} h_{D}}{k_{s} + h_{D}}) m_{o \infty} = ρ K m_{o \infty}$ $G_{ox} = (\frac{ρ k_{s} h_{D}}{k_{s} + h_{D}}) m_{o \infty} = ρ K m_{o \infty}$

(9.14)

$K \equiv \frac{k_{s} h_{D}}{k_{s} + h_{D}}$ $K \equiv \frac{k_{s} h_{D}}{k_{s} + h_{D}}$

(9.15)

$\frac{1}{K} = \frac{k_{s} + h_{D}}{k_{s} h_{D}} = \frac{1}{h_{D}} + \frac{1}{k_{s}}$ $\frac{1}{K} = \frac{k_{s} + h_{D}}{k_{s} h_{D}} = \frac{1}{h_{D}} + \frac{1}{k_{s}}$

(9.16)

When the kinetic rates are large compared to the diffusion rates, K = h_D; when the diffusion rates are large compared to the kinetic rates, K = k_s. When k << h_D, m_os ≅ m_o∞ from Eqn (9.13); thus

$G_{ox} = k_{s} ρ m_{o \infty}$ $G_{ox} = k_{s} ρ m_{o \infty}$

(9.17)

When k_s >> h_D, Eqn (9.13) gives

$m_{os} = (\frac{h_{D}}{k_{s}}) m_{o \infty}$ $m_{os} = (\frac{h_{D}}{k_{s}}) m_{o \infty}$

(9.18)

But since k_s >> h_D, it follows from Eqn (9.18) that

$m_{os} < < m_{o \infty}$ $m_{os} < < m_{o \infty}$

(9.19)

This result permits one to write Eqn (9.10) as

$G_{ox} = h_{D} ρ (m_{o \infty} - m_{os}) ≅ h_{D} ρ m_{o \infty}$ $G_{ox} = h_{D} ρ (m_{o \infty} - m_{os}) ≅ h_{D} ρ m_{o \infty}$

(9.20)

Consider the case of rapid kinetics, k_s >> h_D, further. In terms of Eqn (9.14), or examining Eqn (9.20) in light of K,

$K = h_{D}$ $K = h_{D}$

(9.21)

Of course, Eqn (9.20) also gives one the mass burning rate of the fuel

$\frac{G_{f}}{i} = G_{ox} = h_{D} ρ m_{o \infty}$ $\frac{G_{f}}{i} = G_{ox} = h_{D} ρ m_{o \infty}$

(9.22)

where h_D is the convective mass transfer coefficient for an unspecified geometry. For a given geometry, h_D would contain the appropriate boundary layer thickness, or it would have to be determined by independent measurements giving correlations that permit h_D to be found from other parameters of the system. More interestingly, Eqn (9.22) should be compared to Eqn (6.179) in Chapter 6, which can be written as

$\begin{matrix} G_{ox} ≅ \frac{λ}{c_{p} δ} m_{o \infty} \frac{H}{L_{v}} = \frac{λ}{c_{p} ρ} \frac{ρ}{δ} m_{o \infty} \frac{H}{L_{v}} \\ = \frac{α}{δ} \frac{H}{L_{v}} ρ m_{o \infty} = \frac{D}{δ} \frac{H}{L_{v}} ρ m_{o \infty} \end{matrix}$ $\begin{matrix} G_{ox} ≅ \frac{λ}{c_{p} δ} m_{o \infty} \frac{H}{L_{v}} = \frac{λ}{c_{p} ρ} \frac{ρ}{δ} m_{o \infty} \frac{H}{L_{v}} \\ = \frac{α}{δ} \frac{H}{L_{v}} ρ m_{o \infty} = \frac{D}{δ} \frac{H}{L_{v}} ρ m_{o \infty} \end{matrix}$

(9.23)

Thus one notes that the development of Eqn (9.22) is for a small B number, in which case

$h_{D} = \frac{D}{δ} \frac{H}{L_{v}}$ $h_{D} = \frac{D}{δ} \frac{H}{L_{v}}$

(9.24)

where the symbols are as defined in Chapter 6. (H/L_v) is a simplified form of the B number. Nevertheless, the approach leading to Eqn (9.22) gives simple physical insight into surface oxidation phenomena where the kinetic and diffusion rates are competitive.

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Table of Contents for 9.3. Diffusional Kinetics

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9.3. Diffusional Kinetics

Table of Contents for
9.3. Diffusional Kinetics