Concentration driving force

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

Oxygen transfer rate (OTR) The product of volumetric oxygen transfer rate kj a and the oxygen concentration driving force (C - Cl), (ML T ), where Tl is the mass transfer coefficient based on liquid phase resistance to mass transfer (LT ), a is the air bubble surface area per unit volume (L ), and C and Cl are oxygen solubility and dissolved oxygen concentration, respectively. All the terms of OTR refer to the time average values of a dynamic situation. 

It was shown in Chapter 3 that supersaturation, or concentration driving force, is essential for any crystallization. In a batch crystallizer supersaturation can be generated in several ways, either solely or in combination ... 

X is distance along tube, AC is concentration driving force... 

As an example, it may be supposed that in phase 1 there is a constant finite resistance to mass transfer which can in effect be represented as a resistance in a laminar film, and in phase 2 the penetration model is applicable. Immediately after surface renewal has taken place, the mass transfer resistance in phase 2 will be negligible and therefore the whole of the concentration driving force will lie across the film in phase 1. The interface compositions will therefore correspond to the bulk value in phase 2 (the penetration phase). As the time of exposure increases, the resistance to mass transfer in phase 2 will progressively increase and an increasing proportion of the total driving force will lie across this phase. Thus the interface composition, initially determined by the bulk composition in phase 2 (the penetration phase) will progressively approach the bulk composition in phase 1 as the time of exposure increases. 

In general, the concentration of the reactant will decrease from CAo in the bulk of the fluid to CAi at the surface of the particle, to give a concentration driving force of CAo - CAi)-Thus, within the pellet, the concentration will fall progressively from CAi with distance from the surface. This presupposes that no distinct adsorbed phase is formed in the pores. In this section the combined effects of mass transfer and chemical reaction within the particle are considered, and the effects of external mass transfer are discussed in Section J 0.8.4. 

When the resistance to mass transfer to the external surface of the pellet is significant compared with that within the particle, part of the concentration driving force is required to overcome this external resistance, and the concentration of reacting material at the surface of the pellet Cm is less than that in the bulk of the fluid phase Cao- In Sections 10.7.1-10.7.3, the effect of mass transfer resistance within a porous particle... 

AT is intended to include any and all of the effects of the sorption rate of monomer on the surface, steric arrangement of active species, the addition of the monomer to the live polymer chain, and any desorption needed to permit the chain to continue growing. We assume a steady state in which every mole of propylene that polymerizes is replaced by another mole entering the shell from the gas, so that all of the fluxes are equal to Ny gmol propylene reacted per second per liter of total reactor volume. The following set of equations relates the molar flux to each of the concentration driving forces. 

Mass Transfer Rates. Mass transfer occurs across the interface. The rate of mass transfer is proportional to the interfacial area and the concentration driving force. Suppose component A is being transferred from the gas to the liquid. The concentration of A in the gas phase is Ug and the concentration of A in the liquid phase is u . Both concentrations have units of moles per cubic meter however they are not directly comparable because they are in different phases. This fact makes mass transfer more difficult than heat transfer since the temperature is the temperature regardless of what phase it is measured in, and the driving force for heat transfer across an interface is just the temperature difference Tg—Ti. For mass transfer, the driving force is not Ug—ai. Instead, one of the concentrations must be converted to its equivalent value in the other phase. 

Supersaturation can be expressed as the concentration driving force Ac, the supersaturation ratio, S, or the relative supersaturation, a ... 

Note that the transfer rate equation is based on an overall concentration driving force, (X-X ) and overall mass transfer coefficient, Kl. The two-film theory for interfacial mass transfer shows that the overall mass transfer coefficient, Kl, based on the L-phase is related to the individual film coefficients for the L and G-phase films, kL and ko, respectively by the relationship... 

The next task is to find a value for x that causes the three experimental data sets to exhibit overlay. As shown in Figure 50.3b, overlay is found for x 0.5. We also note that at this value of x, the curves become close to straight lines, meaning that y 1. Thus a reasonable description of the reaction s concentration driving forces is given by eq. (8) withx 0.5 andy i. 

The relationship between the overall height of a transfer unit and the individual film transfer units H, and HG, which are based on the concentration driving force across the liquid and gas films, is given by ... 

The driving force of the mass-transfer process now can be related to the concentration gradient of the reacting species, or to the concentration difference between electrode and bulk solution. Mass-transfer rates then can be related in a general way to the concentration driving force. For example, if... 

Depending on the driving force we choose to employ in our analysis, there are several definitions of mass transfer coefficients that may be considered appropriate for use. If we consider an arbitrary interface between a fluid and the external surface of a catalyst particle, we might choose to define a mass transfer coefficient based on a concentration driving force (kc) as... 

Assuming the same concentration driving force in both fixed and fluidized beds, use... 

In the above theory, the interfadal concentrations CGi and C, are not measurable directly and are therefore of relatively little immediate use. In order to overcome this apparent difficulty, overall mass transfer rate equations are defined by analogy to the film equations. These are based on overall coefficients of mass transfer, KG and KL, and overall concentration driving forces, where... 

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