Transfer Principles Concentrated Systems

Mass-Transfer Principles Concentrated Systems When solute concentrations in the gas and/or liquid phases are large, the equations derived above for dilute systems no longer are applicable. The correct equations to use for concentrated systems are as follows  

The factors yBM and xBM arise from the fact that, in the diffusion of a solute through a second stationary layer of insoluble fluid, the resistance to diffusion varies in proportion to the concentration of the insoluble stationary fluid, approaching zero as the concentration of the insoluble fluid approaches zero. See Eq. (5-198). 

The factors yBM and xBM cannot be justified on the basis of mass-transfer theory since they are based on overall resistances. These factors therefore are included in the equations by analogy with the corresponding film equations. 

In dilute systems the logarithmic-mean insoluble-gas and nonvolatile-liquid concentrations approach unity, and Eq. (5-274) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-197). 

Substitution of Eqs. (5-275) through (5-278) into Eq. (5-274) results in the following simplified formula  

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Mass-Transfer Principles Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-hquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equihbrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a hquid are brought into contact. 

We use the very simple case of a first-order irreversible liquid-phase reaction A — B where the rate of reaction is given by r = k Ca in mol/(l sec), k is the reaction rate constant in sec-1 and Ca is the concentration of component A in mol/l. Later we will show how the same principles can be applied to distributed system and also for other rates like rate of mass transfer for heterogeneous systems with multiple phases. 

Electron transfer is usually carried out in bulk, condensed matter. In the gas phase, the lower concentrations of the donor and acceptor reduce the chance of an encounter between them in comparison with condensed phases. Furthermore, in the absence of a solvent, no stabilization of the separated ions by solvation is possible, enhancing the chance of charge recombination. The volume of published papers in this field is therefore much smaller for gaseous systems than for condensed matter. Nonetheless, gas-phase systems are in principle simpler to analyze and comparison with theory is more straightforward. The analysis of electron transfer in condensed systems usually starts from the (sometimes experimentally inaccessible) gaseous system. Therefore, efforts to study electron transfer in the gas phase continue, and have indeed shed much light on the mechanism of the process. 

The course of a surface reaction can in principle be followed directly with the use of various surface spectroscopic techniques plus equipment allowing the rapid transfer of the surface from reaction to high-vacuum conditions see Campbell [232]. More often, however, the experimental observables are the changes with time of the concentrations of reactants and products in the gas phase. The rate law in terms of surface concentrations might be called the true rate law and the one analogous to that for a homogeneous system. What is observed, however, is an apparent rate law giving the dependence of the rate on the various gas pressures. The true and the apparent rate laws can be related if one assumes that adsorption equilibrium is rapid compared to the surface reaction. 

Thermodynamic principles govern all air conditioning processes (see Heat exchange technology, heat transfer). Of particular importance are specific thermodynamic appHcations both to equipment performance which influences the energy consumption of a system and to the properties of moist air which determine air conditioning capacity. The concentration of moist air defines a system s load. 

In principle, the catalytic converter is a fixed-bed reactor operating at 500—620°C to which is fed 200—3500 Hters per minute of auto engine exhaust containing relatively low concentrations of hydrocarbons, carbon monoxide, and nitrogen oxides that must be reduced significantly. Because the auto emission catalyst must operate in an environment with profound diffusion or mass-transfer limitations (51), it is apparent that only a small fraction of the catalyst s surface area can be used and that a system with the highest possible surface area is required. 

On the basis of the principle of grafted TLC, reversed-phase (RP) and normal-phase (NP) stationary phases can also be coupled. The sample to be separated must be applied to the first (2.5 cm X 20 cm) reversed-phase plate (Figure 8.16(a)). After development with the appropriate (5ti 5yi) mobile phase (Figure 8.16(b)), the first plate must be dried. The second (20 cm X 20 cm) (silica gel) plate (Figure 8.16(c)) must be clamped to the first (reversed-phase) plate in such a way that by use of a strong solvent system (Sj/, SyJ the separated compounds can be transferred to the second plate (Figure 8.16(d)). Figure 8.16(e) illustrates the applied, re-concentrated... 

Pneumatic Conveying Pneumatic conveying systems can generally be scaled up on the principles of dilute-phase transport. Mass and heat transfer can be predicted on both the slip velocity during acceleration and the slip velocity at full acceleration. The slip velocity increases as the solids concentration is increased. 

Notice that the right-hand side of Eq. (34) is equal to the ratio of the transformed concentration at the second measurement point to the transformed concentration at the first measurement point. In the terminology of control engineering, this quantity is the transfer function of the system between Xo and Xm- The Laplace-transform method is possible because the diffusion equation is a linear differential equation. Thus, the right-hand side of Eq. (34) could in principle be used in a control-system analysis of an axial-dispersion process. 

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