3.3.6. Evaluating transport coefficients

The droplet burning rate equations involve the area weighted diffusion coefficients as indicated in Eqn. (3.638) and subsequent equations. While the optimum method of determining the burning rate would involve the evaluation of these integrals as indicated in Equations (3.665) and (3.666), it is useful to estimate the effects of composition and temperature variations when such accurate evaluations are unfeasible. The kinetic theory of gases provides a starting point for such estimates, and a simplified overview of the pertinent results is provided. The single component viscosity is

(3.732) $\mu_i = \dfrac {5} {16} \dfrac {\sqrt{\pi W_k RT}} {\pi \sigma_k^2 \Omega ^{(2,2)*}}$

where $\sigma_k$ is the Lennard-Jones collision diameter and $\Omega ^{(2,2)*}$ is the collision integral. The mixture properties can be obtained using the Wilkes formula that averages based on mole fraction weighting to leading order. A square root dependence on the temperature is evident in (3.732), but the collision integral also includes a temperature dependence and it is found that the viscosities (and the other transport coefficients) are proportional to $T^{0.7}$ , an empirical fact that is referred to as Sutherland’s law. The kinetic theory of gases is only marginally successful at predicting the thermal conductivity, but the ratio of the thermal conductivity to the specific heat is closely related to the viscosity and the Prandtl number can often be approximated as constant. The binary diffusion coefficient between species $i$ and $j$ is more simply written as the product of the diffusion coefficient and the density since this removes additional pressure and temperature dependencies; this is

(3.733) $\rho D_{i,j} = \dfrac {3} {16} \dfrac {\bar{W} \sqrt{2 \pi RT / W_{i,j}}} {\pi \sigma_{i,j}^2 \Omega^{(1,1)*}}.$

Here the reduced mass and the reduced cross sections are $W_{i,j} = W_i W_j /(W_i + W_j)$ and $\sigma_{i,j}^2 = (\sigma_i + \sigma_j)^2$ .