Equation phase concentration

In these equations, the first term is a correction for finite liqiiid-phase concentrations, and the integral term represents the numbers of transfer units required for dilute solutions. It would be very unusual in practice to find an example in which the first (logarithmic) term is of any significance in a stripper design. 

Methanol is frequently used to inhibit hydrate formation in natural gas so we have included information on the effects of methanol on liquid phase equilibria. Shariat, Moshfeghian, and Erbar have used a relatively new equation of state and extensive caleulations to produce interesting results on the effeet of methanol. Their starting assumptions are the gas composition in Table 2, the pipeline pressure/temperature profile in Table 3 and methanol concentrations sufficient to produce a 24°F hydrate-formation-temperature depression. Resulting phase concentrations are shown in Tables 4, 5, and 6. Methanol effects on CO2 and hydrocarbon solubility in liquid water are shown in Figures 3 and 4. 

In Table 10.2, this correlation is shown, comparing solid phase concentration calculated from the retention times of the fronts, and using the adsorption isotherm equation. 

Equation (10.12) is the simplest—and most generally useful—model that reflects heterogeneous catalysis. The active sites S are fixed in number, and the gas-phase molecules of component A compete for them. When the gas-phase concentration of component A is low, the k a term in Equation (10.12) is small, and the reaction is first order in a. When a is large, all the active sites are occupied, and the reaction rate reaches a saturation value of kjkd-The constant in the denominator, is formed from ratios of rate constants. This makes it less sensitive to temperature than k, which is a normal rate constant. 

Note that [ A]heterogeneous has uuits of mol/(m s) but remains a function of gas-phase concentrations. The composite term of Chapter 9 and Equation... 

Equation (11.1) replaces the liquid-phase concentration with an equivalent gas-phase concentration. It is obviously possible to do it the other way, replacing the gas-phase concentration with an equivalent liquid-phase concentration. Then... 

Solution The initial liquid-phase concentration of oxygen is 0.219mol/m as in Example 11.1. The final oxygen concentration will be 1.05 mol/m. The phase balances. Equations (11.11) and (11.12), govern the dynamic response. The flow and reaction terms are dropped from the liquid phase balance to give... 

In Eq. (1.5) the surface coverage is given by 9c, and 9c is related to parameter X of Eq. (1.7). Equation (1.5) can be rewritten to show explicitly its dependence on gas-phase concentration. Equation (1.17a) gives the result. This expression can be related to practical kinetic expressions by writing it as a power law as is done in Eq. (1.18b). Power-law-type rate expressions present the rate of a reaction as a function of the reaction order. In Eq. (1.17b) the reaction order is m in H2 and —n in CO. 

The diffusion system. Figure 8.31(B), is a useful and simple apparatus for preparing mixtures of volatile and moderately volatile vapors in a gas stream [388]. The method is based on the constant diffusion of a vapor from a tube of accurately known dimensions, producing a gas phase concentration described by equation (8.12). 

The equations for effectiveness factors that we have developed in this subsection are strictly applicable only to reactions that are first-order in the fluid phase concentration of a reactant whose stoichiometric coefficient is unity. They further require that no change in the number of moles take place on reaction and that the pellet be isothermal. The following illustration indicates how this idealized cylindrical pore model is used to obtain catalyst effectiveness factors. 

In the special case of an ideal single catalyst pore, we have to take into account that diffusion is quicker than in a porous particle, where the tortuous nature of the pores has to be considered. Hence, the tortuosity r has to be regarded. Furthermore, the mass-related surface area AmBEX is used to calculate the surface-related rate constant based on the experimentally determined mass-related rate constant. Finally, the gas phase concentrations of the kinetic approach (Equation 12.14) were replaced by the liquid phase concentrations via the Henry coefficient. This yields the following differential equation ... 

Liquid phase concentration calculated by Henry s law (Equation 12.13). ... 

The definition of the particle effectiveness factor 77 involves the intrinsic rate of reaction, ( rA)int> for reaction A - products, at the exterior surface conditions of gas-phase concentration (cAs) and temperature (Ts). Thus, from equation 8.55,... 

Write an equation of the form of Equation (8) for each reversible reaction. To obtain the initial fluid phase concentration mj, it is necessary to know the analytical concentrations in the solution at the start of the heterogeneous reaction as well... 

Most early theories were concerned with adsorption from the gas phase. Sufficient was known about the behaviour of ideal gases for relatively simple mechanisms to be postulated, and for equations relating concentrations in gaseous and adsorbed phases to be proposed. At very low concentrations the molecules adsorbed are widely spaced over the adsorbent surface so that one molecule has no influence on another. For these limiting conditions it is reasonable to assume that the concentration in one phase is proportional to the concentration in the other, that is ... 

Although Rs values of high Ks compounds derived from Eq. 3.68 may have been partly influenced by particle sampling, it is unlikely that the equation can accurately predict the summed vapor plus particulate phase concentrations, because transport rates through the boundary layer and through the membrane are different for the vapor-phase fraction and the particle-bound fraction, due to differences in effective diffusion coefficients between molecules and small particles. In addition, it will be difficult to define universally applicable calibration curves for the sampling rate of total (particle -I- vapor) atmospheric contaminants. At this stage of development, results obtained with SPMDs for particle-associated compounds provides valuable information on source identification and temporal... 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

If [X] is the concentration of AH in the aqueous phase (Figure 8), the concentration of AH in the octanol is P times this. The aqueous-phase concentration of the ion is [X] times the degree of dissociation. Multiplying this product by the ion-pair partition coefficient Pj (or P -) gives the concentration of the ion pair in the octanol. The actual amount of species in a phase is given by its concentration times the volume of the phase. At the pK, the amount of neutral species equals the amount of ionized species. Setting the sum of the two terms in the left quadrants equal to the sum of the two terms on the right, one can derive Equation 20. The equivalent e ression for bases is Equation 21. 

The distribution of the solute between the mobile and the stationary phases is continuous. A differential equation that describes the travel of a zone along the column is composed. Then the band profile is calculated by the integration of the differential mass balance equation under proper initial and boundary conditions. Throughout this chapter, we assume that both the chemistry and the packing density of the stationary phase are radially homogeneous. Thus, the mobile and stationary phase concentrations as well as the flow velocities are radially uniform, and a one-dimensional mass balance equation can be considered. 

The equilibrium models of chromatography are given by the mass balance equation given in Equation 10.8 and a proper isotherm equation, q = f(C), should be used to relate the mobile phase and stationary phase concentrations. 

In the lumped kinetic model, various kinetic equations may describe the relationship between the mobile phase and stationary phase concentrations. The transport-dispersive model, for instance, is a linear film driving force model in which a first-order kinetics is assumed in the following form ... 

Note that this kinetic equation is rather similar to Equation 10.15. The major difference between Equations 10.15 and 10.19 is that the general rate and the lumped pore models assume that adsorption takes place from the stagnant mobile phase within the pores, while the lumped kinetic model assumes that the mobile phase concentration is the same in the pores and between the particles. 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

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