Gradients in concentration

Pore dijfusion in fluid-filled pores. These pores are sufficiently large that the adsorbing moleciile escapes the force field of the adsorbent surface. Thus, this process is often referred to as macropore dijfusion. The driving force for such a diffusion process can be approximated by the gradient in mole fraction or, if the molar concentration is constant, by the gradient in concentration of the diffusing species within the pores. 

Another possibility of constructing a chiral membrane system is to prepare a solution of the chiral selector which is retained between two porous membranes, acting as an enantioselective liquid carrier for the transport of one of the enantiomers from the feed solution of the racemate to the receiving side (Fig. 1-5). This system is often referred to as membrane-assisted separation. The selector should not be soluble in the solvent used for the elution of the enantiomers, whose transport is driven by a gradient in concentration or pH between the feed and receiving phases. As a drawback common to all these systems, it should be mentioned that the transport of one enantiomer usually decreases when the enantiomer ratio in the permeate diminishes. Nevertheless, this can be overcome by designing a system where two opposite selectors are used to transport the two enantiomers of a racemic solution simultaneously, as it was already applied in W-tube experiments [171]. 

We turn now to the numerical solution of Equations (9.1) and (9.3). The solutions are necessarily simultaneous. Equation (9.1) is not needed for an isothermal reactor since, with a flat velocity profile and in the absence of a temperature profile, radial gradients in concentration do not arise and the model is equivalent to piston flow. Unmixed feed streams are an exception to this statement. By writing versions of Equation (9.1) for each component, we can model reactors with unmixed feed provided radial symmetry is preserved. Problem 9.1 describes a situation where this is possible. 

To improve the mixing quality in the tubular reactor, Kenics type in-line static mixer reactor was employed. The in-line static mixers were designed to mix two or more fluids efficiently since an improved treinsport process such as flow division, radial eddying, flow constriction, and shear reversal eliminated the gradients in concentration, velocity and temperature. However, only 70 % conversion was achieved with one Kenics mixer unit. As shown in Table 2, five mixer units were required to achieve the maximum conversion. 

Diffusion is the movement of mass due to a spatial gradient in chemical potential and as a result of the random thermal motion of molecules. While the thermodynamic basis for diffusion is best apprehended in terms of chemical potential, the theories describing the rate of diffusion are based instead on a simpler and more experimentally accessible variable, concentration. The most fundamental of these theories of diffusion are Fick s laws. Fick s first law of diffusion states that in the presence of a concentration gradient, the observed rate of mass transfer is proportional to the spatial gradient in concentration. In one dimension (x), the mathematical form of Fick s first law is... 

Mass transfer involving electrolytes may be influenced by gradients in electrical potential as well as by gradients in concentration or pressure. For example, in... 

Diffusion Mass transfer driven by a gradient in concentration. 

Fick s laws Equations that describe the relationship between gradients in concentration and the rate of diffusion. See Eqs. (13) and (16). 

In Section 21.5, the treatment is based on the pseudohomogeneous assumption for the catalyst + fluid system (Section 21.4). In this section, we consider the local gradients in concentration and temperature that may exist both within a catalyst particle and in the surrounding gas film. The system is then heterogeneous. We retain the assumptions of... 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

Once formed, the protons diffuse through the platinum layer and enter deep into the layer of semi-permeable membrane. They travel from the left-hand side of the membrane to its right extremity in response to a gradient in concentration. (Movement caused by a concentration gradient will remind us of dye diffusing through a saucer of water, as described on p. 129.)... 

Fick s law states that if a gradient in concentration of species A exists, say (dnA/dy), a flow or flux of A, say jA, across a unit area in the y direction will be proportional to the gradient so that... 

Diffusion That form of mass transport in which motion occurs in response to a gradient in concentration or composition, itself caused by a gradient of the chemical potential fi. Diffusion is ultimately an entropy-driven process. 

The physical mechanism of membrane water balance and the formal structure of modeling approaches are straightforward. Under stationary operation, the inevitable electro-osmotic flux has to be compensated by a back flux of water from cathode to anode, driven by gradients in concentration, activity, or liquid pressure of water. The water distribution in PEMs that is generated in response to these driving forces decreases from cathode to anode. With increasing/o, the water distribution becomes more nonuniform. the water content near the anode falls below the percolation threshold of proton conduction, X < X. This leaves only a small conductivity due to surface transport of water. As a consequence, increases dramatically this can lead to failure of the complete cell. 

The types of water transport that can alter the major ion concentrations without significantly affecting their relative abimdances are listed in Table 3.8. Note that these are all physical phenomena. In other words, the spatial gradients in concentrations of conservative substances are largely controlled by physical, rather than chemical, processes. 

The timing of events that have subsequently affected the ores can also reveal when elements such as radiogenic Pb have been mobilized from the deposits and moved into the surrounding environment. These elements would elevate element concentrations in the surrounding environment, with gradients in concentrations as vectors to the deposits (Hoik et al. 2003, Kister et al. 2004). 

Therefore, any result that follows from considerations of the form of Fick s second law applies to evolution of heat as well as concentration. However, the thermal and mass diffusion equations differ physically. The mass diffusion equation, dc/dt = V DVc, is a partial-differential equation for the density of an extensive quantity, and in the thermal case, dT/dt = V kVT is a partial-differential equation for an intensive quantity. The difference arises because for mass diffusion, the driving force is converted from a gradient in a potential V/u to a gradient in concentration Vc, which is easier to measure. For thermal diffusion, the time-dependent temperature arises because the enthalpy density is inferred from a temperature measurement. 

The derivation of Equation (2.60) might be seen as a long-winded way of arriving at a trivial result. However, this derivation explicitly clarifies the assumptions behind this equation. First, a gradient in concentration occurs within the membrane, but there is no gradient in pressure. Second, the absorption of a component into the membrane is proportional to its activity (partial pressure) in the adjacent gas, but is independent of the total gas pressure. This is related to the approximation made in Equation (2.52), in which the Poynting correction was assumed to be 1. 

A maximum reactor temperature of 500 K is used in this study. This maximum temperature occurs at the exit of the adiabatic reactor under steady-state conditions. Plug flow is assumed with no radial gradients in concentrations or temperatures and no axial diffusion or conduction. 

According to Fick s firstlaw of diffusion [56], the mass flux density due to molecular diffusion is proportional to the gradient in concentration. Formally,... 

As the second application of irreversible thermodynamics we consider the Soret effect (1893) for a two-component system a flow of particles under the influence of a temperature gradient produces a gradient in concentration. We are ultimately interested in the magnitude of this effect under steady state conditions. Let J0, Jlt J2 be the entropy and particle fluxes... 

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