Collision integral for diffusion

The collision integral for diffusion depends upon the choice of the intermolecular force law between colliding molecules and is a function of temperature. The characteristic length also depends upon the intermolecular force law selected. In comparison with the simple Eq. (6-2) for perfect gases, Eq. (6-3) takes into account the interactive forces between real molecules. But, while in the first case only two specific parameters are needed, the diffusion collision integral, Q0, is a complicated function of several parameters. 

Here Dtj incm2/s, in A, Tin K, and P in atm. The dimensionless quantity ClDij is the collision integral for diffusion, and is a function of the dimensionless temperature KrH ir The parameters (TJ and are those appearing in the Lennard-Jones potential between molecules / and j. 

Collision integral for self-diffusion in kinetic theory (—) Collision integral for diffusion in kinetic theory (—) Modified particle collision density ( 3 [ ] [ ,[)... 

The three-halves power of dimensionless temperature in the expression for eA( ) is based on the temperature dependence of gas-phase ordinary molecular diffusion coefficients when the catalytic pores are larger than 1 p.m. In this pore-size regime, Knudsen diffusional resistance is negligible. The temperature dependence of the collision integral for ordinary molecular diffusion, illustrated in Bird et al. (2002, pp. 526, 866), has not been included in ea) ). The thermal energy balance given by equation (27-28), which includes conduction and interdiffu-sional fluxes, is written in dimensionless form with the aid of one additional parameter,... 

Table 15-2. Lennard-Jones potential parameters and values of the collision integral for ideal Fickian gas diffusivity calculation with Chapman-Enskog equation (15-221 (Cussler. 2009 Hirshfelder et aL,...

Our previous study (J 6) of self diffusion in compressed supercritical water compared the experimental results to the predictions of the dilute polar gas model of Monchick and Mason (39). The model, using a Stockmayer potential for the evaluation of the collision integrals and a temperature dependent hard sphere diameters, gave a good description of the temperature and pressure dependence of the diffusion. Unfortunately, a similar detailed analysis of the self diffusion of supercritical toluene is prevented by the lack of density data at supercritical conditions. Viscosities of toluene from 320°C to 470°C at constant volumes corresponding to densities from p/pQ - 0.5 to 1.8 have been reported ( 4 ). However, without PVT data, we cannot calculate the corresponding values of the pressure. 

The parameter the diffusion collision integral, is a function of k T/e, where is the Boltzmann constant and e is a molecular energy parameter. Values of tabulated as a function of k T/e, have been published (Hirschfelder et al., 1964 Bird et al., 1960). Neufeld et al., (1972) correlated using a simple eight parameter equation that is suitable for computer calculations (see, also, Danner and Daubert, 1983 Reid et al., 1987). Values of a and e/k (which has units of kelvin) can be found in the literature—for only a few species—or estimated from critical properties (Reid et al., 1987 Danner and Daubert, 1983). The mixture a is calculated as the arithmetic average of the pure component values. The mixture e is taken to be the geometric average of the pure component values. 

In order to obtain the overall rate constants for dissociation at high pressures, equation (1.37) must be solved. This requires a complete set of collisional transition rates in phase space, i.e. k(q,p This set of course is extremely difficult to obtain. However, for the high velocity limit of collisions, the impulsive limit , k(q,p q, p ) can be deter-mined. With these values equation (1.37) can be simplified by expansion of the collision integrals, analogous to the conversion of the master equation (1.34) into a diffusion equation (1.58), at least for... 

Table 15-2 is a brief list of Lennard-Jones parameters for a few molecules and includes a brief list of values of Qq. Detailed tables are available for a variety of conpounds to calculate these parameters (e.g., Cussler. 2009 Hirshfelder et al., 1954). An enpirical fit for Qq is given by Wankat and Knaebel (2008). Because of the temperature dependence of the collision integral, the gas diffusivities are proportional to T at low temperatures and to T - at high tenperatures. Wankat and Knaebel (20081 summarize other methods to predict gas diffusivities. 

The collision integral Qg g is a ratio giving the deviation of a gas with interactions compared to a gas of rigid, elastic spheres. This value would be 1.0 for a gas with no interactions. Equation (6.2-44) predicts diffusivities with an average deviation of about... 

Senior RC, Grace JR. Integrated particle collision and turbulent diffusion model for dilute gas-solid suspensions. Powder Technol 96 48-78, 1998. 

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