Diffusion in ideal gas

Equations 8.3.52-8.3.57 were presented as an exact solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures by Burghardt (1984). Equations 8.3.52-8.3.57 are somewhat less useful than Eqs. 8.3.15-8.3.24 because we need to know the composition profiles in order to evaluate the matrizant. Even if the profiles are known, the computation of the fluxes from either of Eqs. 8.3.62 or 8.3.63 is not straightforward and not recommended. It is with the development in Section 8.4 in mind that we have included these results here. 

In 1964 Toor and Stewart and Prober independently put forward a general approach to the solution of multicomponent diffusion problems. Their method, which was discussed in detail in Chapter 5, relies on the assumption of constancy of the Fick matrix [D] along the diffusion path. The so-called Tinearized theory of Toor, Stewart, and Prober is not limited to describing steady-state, one-dimensional diffusion in ideal gas mixtures (as we have already demonstrated in Chapter 5) however, for this particular situation Eq. 5.3.5, with [P] given by Eq. 4.2.2, simplifies to... 

In Section 8.3 we presented a derivation of an exact matrix solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures. Although the final expression for the composition profiles (Eq. 8.3.12), is valid whatever relationship exists between the fluxes (i.e., bootstrap condition), the derivation given in Section... 

The concentration cch4 0 is obtained via the ideal gas law to f - For the diffusion of gases, the two most important cases are molecular diffusion (diffusion in the gas phase) and Knudsen diffusion (diffusion through pores, while the number of collisions between the pore wall is larger than the number of collisions between the gas... 

Causes for deviations from ideal plug flow are molecular diffusion in the gas and dispersion caused by flow in the interstitial channels of the bed, and uneveness of flow over the cross section of the bed. 

Diffusion of the probe in the gas and polymer phases, and adsorption on the support and on the polymer surface (both types of adsorption have nonlinear isotherms), simultaneously play an important role in IGC experiments and must be accounted for properly. An extensive computational program is planned to simulate the individual processes and to assess their influence on chromatographic behavior. In a recent paper, simulated behavior of three types of system was described (9). In the simplest case, only diffusion in the gas phase was operative. This case corresponds to elution of an ideal marker. Simultaneous effects of gaseous diffusion and partitioning of the probe between the phases were simulated next, assuming an instantaneous... 

Diffusion coefficients in binary liquid mixtures are of the order 10 m /s. Unlike the diffusion coefficients in ideal gas mixtures, those for liquid mixtures can be strong functions of concentration. We defer illustration of this fact until Chapter 4 where we also consider models for the correlation and prediction of binary diffusion coefficients in gases and liquids. 

The ideal Thiele-Damkohler theory also assumes that mass transfer in the particle occurs exclusively by diffusion. In a gas reaction, however, the volume of the reacting mixture expands if the mole number increases, and contracts if the mole number decreases. If it expands, forced convection out of the particle counteracts reactant diffusion into it and thereby slows the reaction down. If the volume contracts, forced convection sucks reactant into the particle and speeds the reaction up [16,28]. 

Equimolar Counterdiffusion in Binary Cases. If the flux of A is balanced by an equal flux of B in the opposite direction (frequently encountered in binary distillation columns), there is no net flow through the film and like is directly given by Fick s law. In an ideal gas, where the diffusivity can be shown to be independent of concentration, integration of Fick s law leads to a linear concentration profile through the film and to the following expression where (P/RT)y is substituted for... 

Equimolar Counterdiffusion. Just as unidirectional diffusion through stagnant films represents the situation in an ideally simple gas absorption process, equimolar counterdiffusion prevails as another special case in ideal distillation columns. In this case, the total molar flows and are constant, and the mass balance is given by equation 35. As shown eadier, noj/g factors have to be included in the derivation and the height of the packing is... 

General Situation. Both unidirectional diffusion through stagnant media and equimolar diffusion are idealizations that ate usually violated in real processes. In gas absorption, slight solvent evaporation may provide some counterdiffusion, and in distillation counterdiffusion may not be equimolar for a number of reasons. This is especially tme for multicomponent operation. 

For the special case of steady-state unidirectional diffusion of a component through an inert-gas film in an ideal-gas system, the rate of mass transfer is derived as... 

For an ideal gas, the total molar concentration Cj is constant at a given total pressure P and temperature T. This approximation holds quite well for real gases and vapours, except at high pressures. For a liquid however, CT may show considerable variations as the concentrations of the components change and, in practice, the total mass concentration (density p of the mixture) is much more nearly constant. Thus for a mixture of ethanol and water for example, the mass density will range from about 790 to 1000 kg/m3 whereas the molar density will range from about 17 to 56 kmol/m3. For this reason the diffusion equations are frequently written in the form of a mass flux JA (mass/area x time) and the concentration gradients in terms of mass concentrations, such as cA. 

Temperature and enthalpy are not the only conditions that determine whether a change is favourable. Consider the process shown in Figure 7.5. A closed valve links two flasks together. The left flask contains an ideal gas. The right flask is evacuated. When the valve is opened, you expect the gas to diffuse into the evacuated flask until the pressure in both flasks is equal. You do not expect to see the reverse process—with all the gas molecules ending up in one of the flasks—unless work is done on the system. 

However, this porosity takes into account all the open pores—even those that are not connected between each other, which are useless in fuel cell operation. Therefore, the effective porosity, which counts only the interconnected pores, is more critical when determining the optimal diffusion layer in a fuel cell. This porosity can be determined by using volume filtration techniques. For example, a porous sample is immersed in a liquid that does not enter inside the pores (e.g., mercury at low pressures) and then the total volume of the material can be determined. Next, the specimen is put inside a container of known volume that contains an inert gas, and the changed pressure is recorded. After this, a second evacuated chamber of known volume is connected to the system, and the new pressure is recorded. With these pressures and the ideal gas law, the volume of open pores and thus the effective porosity can be determined [195]. 

For single-component gas permeation through a microporous membrane, the flux (J) can be described by Eq. (10.1), where p is the density of the membrane, ris the thermodynamic correction factor which describes the equilibrium relationship between the concentration in the membrane and partial pressure of the permeating gas (adsorption isotherm), q is the concentration of the permeating species in zeolite and x is the position in the permeating direction in the membrane. Dc is the diffusivity corrected for the interaction between the transporting species and the membrane and is described by Eq. (10.2), where Ed is the diffusion activation energy, R is the ideal gas constant and T is the absolute temperature. 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

A plug flow or tubular flow reactor is tubular in shape with a high length/diameter (1/d) ratio. In an ideal case (as in the case of an ideal gas, this only approached reality) flow is orderly with no axial diffusion and no difference in velocity of any members in the tube. Thus, the time a particular material remains within the tube is the same as that for any other material. We can derive relationships for such an ideal situation for a first-order reaction. One that relates extent of conversion with mean residence time, t, for free radical polymerizations is ... 

---