Bulk rate-determining step pressure

From the above considerations, a plot of hydrogen flow rate as a function of the difference of pressure or the difference of pressure roots should be used to assess the presence of either surface or bulk rate-determining steps. 

Calderbank et al. (C6) studied the Fischer-Tropsch reaction in slurry reactors of 2- and 10-in. diameters, at pressures of 11 and 22 atm, and at a temperature of 265°C. It was assumed that the liquid-film diffusion of hydrogen from the gas-liquid interface is a rate-determining step, whereas the mass transfer of hydrogen from the bulk liquid to the catalyst was believed to be rapid because of the high ratio between catalyst exterior surface area and bubble surface area. The experimental data were not in complete agreement with a theoretical model based on these assumptions. 

The temperature dependence of the hydrogen permeability can be expressed by the well-known Arrhenius expression P exp( - )], formally similar to the self-diffusivity coefficient mentioned in Section 15.3. However, in this case, the activation energy for diffusion, E , is referred to the jumps of the hydrogen atoms in the bulk of the Pd-aUoy (the rate-determining step is the H diffusion according to the Richardson equation). Considering that 0 depends on the partial pressure of the carbon monoxide, in addition to the temperature of the mixture, that is 0 (T, ), the scaling factor g(T,p ) can... 

The key step in the derivation by Reuter et al. of their lattice model is the use of detailed balance to determine the sticking coefficients for each species on each type of site.31 The total adsorption rate at a particular site can be expressed as Tad = SI(p, T), where S is the local sticking coefficient and I(p,T) is the impingement rate of the species of interest from a gas phase with partial pressure p and temperature T. At steady state, the total adsorption and desorption rates must satisfy the detailed balance condition TdesjTad = exp[(Fb—/j,(T, p))/kT, where Fb is the free energy of the adsorbed species and fi(T, p) is the chemical potential of the gas phase species. The adsorption free energy is well approximated by the adsorption enthalpy, which is simply the adsorption energy calculated by a DFT calculation. This approach provides a direct link between the adsorption and desorption rates and the pressure and temperature of the bulk gas phase. 

Figure 4.3 Hydrogen flux through membranes of various thickness fabricated with Group IVB-VB elements. Data show that the rate limiting steps determining hydrogen flux vary and include diffusion through the bulk membrane material for the thicker membranes, gas phase diffusion at high pressures,...

Here, both reactions occur in the film with or without depletion of B. Let us first consider the case of no depletion of B. Because R is formed in the film, its concentration in the film would be higher than in the bulk. Also, the second step becomes second order. The situation in the film is similar to that in a plug-flow reactor where the concentration of R is determined by the relative rates of its formation and disappearance and thus exhibits a maximum at a certain position in the film. This is shown in Figure 14.7a. When the bulk concentration of R is high and the partial pressure of A is relatively low, the concentration of R at any point in the film may not be significantly different from that in the bulk (because no reaction occurs in the bulk). Also, both steps become pseudo-first-order. Such a situation would lead to concentration profiles shown in Figure 14.7b. 

In the above calculations and considerations diffusion through the nonporous layer of the membrane was assumed to be the rate-determining process and thus the only transport resistance. In every membrane process, however, additional transport steps at the feed occur, usually summarized as polarization . By the preferential transport of one component out of a mixture through the membrane the fluid layer directly adjacent to the membrane surface will be depleted of that component, and its concentration will be lower than that in the bulk of the feed mixture. This (unknown) lower concentration determines the sorption and thus the effective activity and partial vapor pressure of the component directly at the feed side of the membrane. The flux reduction caused by the additional resistance for the transport of matter by diffusion through the liquid layer adjacent to the feed side of the membrane is known as concentration polarizatioif, and effective in all membrane processes. Due to the phase change... 

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