Diffusion pressure-gradient

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

It must be emphasized that equation (10.14) is based on the assumption of strictly isobaric diffusion. In practice, we may ask, how closely must the pressures on the two sides of the diffusion cell be balanced if undue error is not to be associated with the use of (10.14) The answer, of course, is that pressure gradients must be small enough that the term in grad p on the right hand side of equation (10.4) is negligible coranared with the term in grad Since the viscous term dominates the coeffi-... 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

In pressure diffusion, a pressure gradient is estabUshed by gravity or in a centrifugal field. The lighter components tend to concentrate in the low pressure (center) portion of the fluid. Countercurrent flow and cascading extend the separation effect. 

A second type of apparatus based on the pressure diffusion effect is the separation nozzle. Pressure gradients in a curved expanding jet produce an isotopic separation similar to that in a centrifuge. The separation effect obtained with a single jet is relatively small, and separation nozzle stages, similar to gaseous diffusion stages, must be used in a cascade to realize most of the desired separations. 

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

When clay or similar material is dried, often a pressure gradient is developed by the forces of repulsion between particles as shrinkage brings the particles close together (25). This gradient forces Hquid toward the surface and the resulting moisture profile resembles that characteristic of Hquid diffusion. 

The structure of the solid determines the mechanism for which internal liquid flow may occur. These mechanisms can include (1) diffusion in continuous, homogeneous solids, (2) capillary flow in granular and porous sohds, (3) flow caused by shrinkage and pressure gradients, (4) flow caused by gravity, and (5) flow caused by a vaporization-condensation sequence. 

Diffusion-blading loss. This loss develops because of negative velocity gradients in the boundary layer. Deceleration of the flow increases the boundary layer and gives rise to separation of the flow. The adverse pressure gradient that a compressor normally works against increases the chances of separation and causes significant loss. 

The outlet diffuser is used to eonvert the high absolute veloeity leaving the exdueer into statie pressure. If this eonversion is not done, the effieieney of the unit will be low. This eonversion of the flow to a statie head must be done earefully, sinee the low-energy boundary layers eannot tolerate great adverse pressure gradients. 

Maxwell,02) postulated that the partial pressure gradient in the direction of diffusion for a constituent of a two-component gaseous mixture was proportional to ... 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

The following factors affect net diffusion of a substance (1) Its concentration gradient across the membrane. Solutes move from high to low concentration. (2) The electrical potential across the membrane. Solutes move toward the solution that has the opposite charge. The inside of the cell usually has a negative charge. (3) The permeability coefficient of the substance for the membrane. (4) The hydrostatic pressure gradient across the membrane. Increased pressure will increase the rate and force of the collision between the molecules and the membrane. (5) Temperature. Increased temperature will increase particle motion and thus increase the frequency of collisions between external particles and the membrane. In addition, a multitude of channels exist in membranes that route the entry of ions into cells. 

When Ap=0 (there is no pressure gradient in the system), LJtJl = 0 (diffusion of ions through the membrane can be neglected), and if the conductivities of the two membranes are identical, then in view of the second of Eqs (6.2.18) this condition can be expressed as... 

In many diffusion problems of practical importance in the pharmaceutical sciences, such as intrinsic dissolution studies and drug release from solid dosage forms, the medium under consideration is not at rest. In addition to concentration changes due to diffusion, there are concentration changes by convection. External forces, such as pressure gradients and temperature differences, can cause convective flows. Although convection can also be caused by diffusion itself, our discussion is limited to convection caused by external forces, since convection produced by diffusion is negligible (less than 10%) for most pharmaceutical problems. 

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