General Case Description

This verification problem is taken from Ambrosek and Longhurst (2008). When two species react on a surface to form a third, it is possible to predict the rate at which equilibration between the species will occur. For example, the reaction between two isotopic species, A and B, is described as

(1)

and the partial pressure of A, B, and AB in equilibrium of the reaction is defined by

(2)

where is the partial pressure of corresponding gas , is the equilibrium constant.

Assuming that the molecular species have the same mass and chemical properties such that there is no enthalpy change associated with this reaction and only configurational entropy is driving the reaction. Then

(3)

where is the Gibbs free energy, is the ideal gas constant, is the temperature.

Therefore, the partial pressure of AB in equilibrium depends on initial partial pressure of A and B:

(4)

At equilibrium, the surface concentrations of A (i.e., ) and B (i.e., ) from Sieverts' law are given by

(5)

and

(6)

where is Sieverts’ solubility. Because we are considering isotopic variants, will be the same for each homonuclear species. Under equilibrium conditions, we also expect

(7)

where is the dissociation coefficient and is the recombination coefficient. That leads to

(8)

Under ratedep condition, equilibrium is not assumed, but the relationships between the coefficients are maintained. In particular, the recombination and dissociation coefficients are assumed to be independent of the surface species concentrations and gas partial pressures, respectively. If the species molecular masses and solubilities are assumed equal, the dissociation coefficients for AB, A, and B molecules should be identical. Because two different microscopic processes can produce AB (A jumping to find B and B jumping to find A) and only one (A finding A) can form A, and similarly for B, the recombination coefficient for AB should be twice of the coefficient for homonuclear molecules. We solve the net current of AB molecules from the surface to the enclosure by

(9)

where is the time, is the surface area, is the Boltzmann’s constant, is the temperature, and is the volume in the enclosure. If diffusion is small, the almost constant numbers of A and B atoms in the gas imply that and should have an almost constant value regardless of the isotopic species composition. The production of A and B in equilibration conditions is given by

(10)

This case uses equal starting pressures of Pa of A and B and no AB. is specified to be atom/m/s/Pa. is specified to be m/atom/s, the temperature is 1000 K, the surface area for reaction is 0.05 m 0.05 m square, and the enclosure volume is 1 m.

Analytical solution

Ambrosek and Longhurst (2008) provides the analytical equation for the partial pressure of AB when the conversion rate at the surface is high as

(11)

Results

A comparison of the concentration of AB as a function of time is plotted in Figure 1. The TMAP8 calculations are found to be in good agreement with the analytical solution, with a root mean square percentage error (RMSPE) of RMSPE = 0.13 %. The concentrations of A and B as a function of time are also plotted in Figure 1.

Figure 1: Comparison of concentration of AB as a function of time calculated through TMAP8 and analytically for the solution in ratedep condition when A and B have equal pressures (Ambrosek and Longhurst, 2008).

Input files

References