Transfer by Diffusion

Just as heat transfer refers to the energy transfer within a system or between a system and its surroundings occurring because of a difference in temperature, mass transfer refers to the transfer of mass (i.e., matter) which occurs within a system or between the system and its surroundings due to a difference in the concentration of a particular component between two points which are not in equilibrium. 

In most unit operations it is of considerable importance that material is transferred from one phase to another across a boundary. The transfer of material from a solid phase to a liquid phase (as typically in leaching), or the transfer of material between one liquid phase to another liquid phase (as typically in molten metal and molten slag phases), extraction or between liquid and vapor phases (as typically in distillation) are well-known examples encountered in practice. 

The problems relating to mass transfer may be elucidated out by two clear-cut yet different methods one using the concept of equilibrium stages, and the other built on diffusional rate processes. The selection of a method depends on the type of device in which the operation is performed. Distillation (and sometimes also liquid extraction) are carried out in equipment such as mixer settler trains, diffusion batteries, or plate towers which contain a series of discrete processing units, and problems in these spheres are usually solved by equilibrium-stage calculation. Gas absorption and other operations which are performed in packed towers and similar devices are usually dealt with utilizing the concept of a diffusional process. All mass transfer calculations, however, involve a knowledge of the equilibrium relationships between phases. 

A limit to mass transfer is attained if two phases come to equilibrium and the net transfer of material comes to a halt. For a process in practice, which must have a reasonable production rate, equilibrium must be avoided, as the rate of mass at any point is proportional to the compelling or driving force, which is the departure from equilibrium at that point. In order to evaluate driving forces, a knowledge of equilibria between phase is therefore fundamentally important. Several kinds of equilibria are important in mass transfer. 

To inject a general note it may be pointed out that two very important laws, called Fick s laws, form the basis of diffusion theory. The first law can be expressed in the following form  

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An analogy exists between mass transfer by diffusion and heat transfer by conduction. Each involves coHisions between molecules and a gradient as the driving force which causes flow. Eor diffusion, this is a concentration gradient for conduction, the driving force is an energy gradient. Eourier s... 

Axial diffusion Mass transfer by diffusion along streamlines that occurs at... 

The inverse of the time eonstant tmicro (mieromixing) ean be interpreted as a transfer eoeffieient for mass transfer by diffusion. 

In the previous discussion it has been assumed that the vapour is a pure material, such as steam or organic vapour. If it contains a proportion of non-condensable gas and is cooled below its dew point, a layer of condensate is formed on the surface with a mixture of non-condensable gas and vapour above it. The heat flow from the vapour to the surface then takes place in two ways. Firstly, sensible heat is passed to the surface because of the temperature difference. Secondly, since the concentration of vapour in the main stream is greater than that in the gas film at the condensate surface, vapour molecules diffuse to the surface and condense there, giving up their latent heat. The actual rate of condensation is then determined by the combination of these two effects, and its calculation requires a knowledge of mass transfer by diffusion, as discussed in Chapter 10. 

Total transfer = Transfer by diffusion + Transfer by bulk flow. 

Interest extends from transfer to single particles to systems in which the particles are in the form of fixed or fluidised beds. The only case for which there is a rigorous analytical solution is that for heat by conduction and mass transfer by diffusion to a sphere. 

The liquid evaporating into the gas is transferred by diffusion from the interface to the gas stream as a result of a concentration difference (c0 — < ) where cunit volume) and c is the concentration in (he gas stream. The rate of evaporation is then given by ... 

True equilibrium cannot be established at the interface between two different electrolytes, since ions can be transferred by diffusion. Hence, in thermodynamic calculations concerning such cells, one often uses corrected OCV, % ... 

This model assumes that the air/water interface from the blade to the Wilhelmy plate can be divided into a number of equal small cells. We apply a simple argument that the rate of mass transfer by diffusion is proportional to the difference in concentration between the neighboring cells, while the concentration and the surface pressure within each cell are assumed homogeneous. 

The commonest multiple step control mechanism in use is that of diffusion to the surface of the catalyst combined with one of the adsorption or surface reaction steps. Mass transfer by diffusion is proportional to the difference between partial pressures in the bulk of the gas and at the catalyst surface,... 

The heat transferred by diffusion can be expressed in terms of the latent heat of vaporization of the water ... 

Note that Equation 5.16 gives us the opportunity to see which heat transfer mechanism is dominant. That is, it provides the ratio of the heat transferred by convection to that transferred by diffusion. 

Based on the above-mentioned assumptions, the mass, momentum and energy balance equations for the gas and the dispersed phases in two-dimensional, two-phase flow were developed [14], In order to solve the mass, momentum and energy balance equations, several complimentary equations, definitions and empirical correlations were required. These were presented by [14], In order to obtain the water vapor distribution the gas phase the water vapor diffusion equation was added. During the second drying period, the model assumed that the particle consists of a dry crust surrounding a wet core. Hence, the particle is characterized by two temperatures i.e., the particle s crust and core temperatures. Furthermore, it was assumed that the heat transfer from the particle s cmst to the gas phase is equal to that transferred from the wet core to the gas phase i.e., heat and mass cannot be accumulated in the particle cmst, since all the heat and the mass is transferred by diffusion through the cmst from the wet core to the surrounding gas. Based on this assumption, additional heat balance equation was written. 

This chapter will focus on three types of membrane extracorporeal devices, hemodialyzers, plasma filters for fractionating blood components, and artificial liver systems. These applications share the same physical principles of mass transfer by diffusion and convection across a microfiltration or ultrafiltration membrane (Figure 18.1). A considerable amount of research and development has been undertaken by membrane and modules manufacturers for producing more biocompatible and permeable membranes, while improving modules performance by optimizing their internal fluid mechanics and their geometry. 

For this purpose, the removal procedures are mainly based on membrane separation that ideally should bring free and bound toxins to a nonspedfic adsorption device (ion-exchangers and/or activated charcoal). Blood should not perfuse directly such components, due to bioincompatibity aspeds. Therefore, several processes have been proposed to correctly handle toxins carried by plasma [27]. They are described in the following sections. All of them need a physical barrier between the blood cells and the adsorption system. This physical sieve is always a membrane with adequate properties, through which toxins can be transferred by diffusion or convection. 

When a chemical reaction occurs in a steady-state system, products must be transferred out of the system and reactants must be transferred into the system. To avoid additional entropy generation, we take these transfers as being transfers by diffusion to reservoirs at material equilibrium with the system. The entropy generated in the system by the chemical reaction is then, by Eq. (9), the entropy increase of the reservoirs ... 

For all of the general techniques of Figure 2, the separations are achieved by enhancing the rate of mass transfer by diffusion of certain species relative to mass transfer of all species by bulk movement within a particular phase. The driving force and direction of mass transfer by diffusion is governed by thermodynamics, with the usual limitations of equilibrium. Thus, both transport and thermodynamic considerations are crucial in separation operations. The rate of separation is governed by mass transfer, while the extent of separation is limited by thermodynamic equilibrium. Fluid mechanics also plays an important role, and applicable principles are included in other chapters. 

For liquid pools containing a mixture of liquids of varying volatility, the light components will evaporate first. When evaporation rates are high, the surface concentration of the volatile components will be depleted. At this time the transfer by diffusion of these components from the bulk of the liquid to the surface becomes the limiting factor. The one-dimensional representation of vertical diffusion for a component i through a liquid layer of depth z is given by ... 

The phenomenon of limiting current can appear at anodes as well as at cathodes. When an indifferent electrode is immersed into a solution of ferrous salt or any other anodically oxidizable substance the limiting current density will be attained at a certain terminal voltage. The limiting ourrent density will depend not only on the maximum number of ions (Fe++) transferred by diffusion or even migration in a unit of time from the bulk of the solution to the electrode but also on the maximum transport-rate of the products (Fe+++) in the opposite direction. 

Concentration overvoltage is caused by slow diffusion of the electroactive species. In many cases ions which react at the electrode are not the most mobile ions which transport the current. For example, in the case of electrolysis of cryolite-alumina melts, the Na+ ions transport the current, but at the cathode the Al(in) ions are discharged. In the melt, aluminum form complexes with oxygen and fluoride ions and it is transferred by diffusion. 

These rate determining steps are shown in Figure 5.6. As the reaction is written in equation (5.26), mass transfer in the boundary layer and mass transfer by diffusion in the product layer can be limiting for the reactant gas. A, making its way in from the bulk gas to the unreacted core, or for the product gas, R, making its way out. 

The measurements first reveal an increase in the water level, which can be attributed to moisture transfer from the stopper into the product. The decrease observed subsequently could be due to water expulsion as a result of some product transformation (crystallization of sugar within the substrate, for example). The water would then be adsorbed by the stopper. As a matter of fact, it is well known that the capacity of stoppers to adsorb water depends on their composition [13-15]. Finally, in the third stage, transfer by diffusion through the stopper could account for the slow moisture recovery in each sample. 

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