Rate driving force

Equations 3.12, 3.13, and the final Equation 3.14 are all forms of the classic engineering expression [Rate = (Driving force)/Resistance] where the driving force is expressed as concentration differences. The overall resistance (1 /K ) can be controlled by a low value of either individual coefficient. The mass transfer coefficient (kd) controls crystallization when the reaction is very rapid relative to diffusion, but the reaction coefficient ( r) controls crystallization when diffusion is much more rapid than reaction. In such cases the overall coefficient K may be approximated by the smaller k value. However, the concentrations in the driving force remain measurable (c) or calculable (ceq) rather than non-measureable (c ). 

The Washburn equation has most recently been confirmed for water and cyclohexane in glass capillaries ranging from 0.3 to 400 fim in radii [46]. The contact angle formed by a moving meniscus may differ, however, from the static one [46, 47]. Good and Lin [48] found a difference in penetration rate between an outgassed capillary and one with a vapor adsorbed film, and they propose that the driving force be modified by a film pressure term. 

Electron transfer reaction rates can depend strongly on tire polarity or dielectric properties of tire solvent. This is because (a) a polar solvent serves to stabilize botli tire initial and final states, tluis altering tire driving force of tire ET reaction, and (b) in a reaction coordinate system where the distance between reactants and products (DA and... 

Mass transfer rates may also be expressed in terms of an overall gas-phase driving force by defining a hypothetical equiHbrium mole fractionjy as the concentration which would be in equiHbrium with the bulk Hquid concentration = rax ) ... 

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. 

Log arithmic-Mean Driving Force. As noted eadier, linear operating lines occur if all concentrations involved stay low. Where it is possible to assume that the equiUbrium line is linear, it can be shown that use of the logarithmic mean of the terminal driving forces is theoretically correct. When the overall gas-film coefficient is used to express the rate of absorption, the calculation reduces to solution of the equation... 

This rate equation must satisfy the boundary conditions imposed by the equiUbrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to 2ero at equiUbrium. It maybe a linear driving force expression of the form... 

The Driving Force for Mass Transfer. The rate of mass transfer increases as the driving force, (7 — (7, is increased. can be enhanced as follows. From Dalton s law of partial pressures... 

Oxygen transfer rate (OTR) is estimated by the foUowiag standard procedure (12), where the rate of mass transfer per unit volume of Hquid is taken to be directly proportional to the driving force of the system... 

Nonporous Dense Membranes. Nonporous, dense membranes consist of a dense film through which permeants are transported by diffusion under the driving force of a pressure, concentration, or electrical potential gradient. The separation of various components of a solution is related directiy to their relative transport rate within the membrane, which is determined by their diffusivity and solubiUty ia the membrane material. An important property of nonporous, dense membranes is that even permeants of similar size may be separated when their concentration ia the membrane material (ie, their solubiUty) differs significantly. Most gas separation, pervaporation, and reverse osmosis membranes use dense membranes to perform the separation. However, these membranes usually have an asymmetric stmcture to improve the flux. 

Gas Separation. During the 1980s, gas separation using membranes became a commercially important process the size of this appHcation is stiH increasing rapidly. In gas separation, one of the components of the feed permeates a permselective membrane at a much higher rate than the others. The driving force is the pressure difference between the pressurized feed gas and the lower pressure permeate. 

This system utilizes specific membranes, between which the dmg reservoir is enclosed (Fig. 4). A tiny ehiptical disk, inserted into the cul-de-sac of the eye, releases pilocarpiae steadily. The dmg is deUvered through selected polymeric membranes. The dmg reservoir maintains a saturated solution between the membranes which acts osmoticaHy as the driving force for the dmg to diffuse through the rate-limiting membranes. 

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