Mass molecular diffusion

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Neither the penetration nor the surface renewal theory can be used to predict mass transfer coefficients directiy because T and s are not normally known. Each suggests, however, that mass transfer coefficients should vary as the square root of the molecular diffusivity, as opposed to the first power suggested by the film theory. 

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

The rate of mass transfer (qv) depends on the interfacial contact area and on the rate of mass transfer per unit interfacial area, ie, the mass flux. The mass flux very close to the Hquid—Hquid interface is determined by molecular diffusion in accordance with Pick s first law ... 

For weU-defined reaction zones and irreversible, first-order reactions, the relative reaction and transport rates are expressed as the Hatta number, Ha (16). Ha equals (k- / l ) where k- = reaction rate constant, = molecular diffusivity of reactant, and k- = mass-transfer coefficient. Reaction... 

Neglecting flow nonuniformities, the contributions of molecular diffusion and turbulent mixing arising from stream sphtting and recombination around the sorbent particles can be considered additive [Langer et al., Int. ]. Heat and Mass Transfer, 21, 751 (1978)] thus, the axial dispersion coefficient is given by ... 

An important mixing operation involves bringing different molecular species together to obtain a chemical reaction. The components may be miscible liquids, immiscible liquids, solid particles and a liquid, a gas and a liquid, a gas and solid particles, or two gases. In some cases, temperature differences exist between an equipment surface and the bulk fluid, or between the suspended particles and the continuous phase fluid. The same mechanisms that enhance mass transfer by reducing the film thickness are used to promote heat transfer by increasing the temperature gradient in the film. These mechanisms are bulk flow, eddy diffusion, and molecular diffusion. The performance of equipment in which heat transfer occurs is expressed in terms of forced convective heat transfer coefficients. 

Its molecular diffusivity should be low and it should be conserved (i.e., a mass balance on it must be possible). 

Volumetric mass transfer coefficient, kLa The proportionality coefficient reflecting both molecular diffusion, turbulent mass transfer, and specific area for mass transfer. 

The problems that arise when experiments are carried out in a greatly reduced scale can be overcome if the Reynolds number is high and the flow pattern is governed mainly by fully developed turbulence. It is possible to ignore the Reynolds number, the Schmidt number, and the Prandtl number because the structure of the turbulence and the flow pattern at a sufficiently high level of velocity will be similar at different supply velocities and therefore independent of the Reynolds number. The transport of thermal energy and mass by turbulent eddies will likewise dominate the molecular diffusion and will therefore also be independent of the Prandtl number and the Schmidt number. 

The mass transfer effect is relevant when the chemical reaction is far faster than the molecular diffusion, i.e. Ha > 1. The rapid formation of precipitate particles should then occur spatially distributed. The relative rate of particle formation to chemical reaction and/or diffusion can as yet be evaluated only via lengthy calculations. 

The liquid-liquid extraction process is based on the specific distribution of dissolved components between two immiscible fluids, for instance, between aqueous and organic liquids. The process refers to a mass exchange processes in which the mass transport of component (j) from phase (1) to phase (2) by means of convection or molecular diffusion acts to achieve the chemical potential (p) equilibrium (134) ... 

The term mass transfer is used to denote the transference of a component in a mixture from a region where its concentration is high to a region where the concentration is lower. Mass transfer process can take place in a gas or vapour or in a liquid, and it can result from the random velocities of the molecules (molecular diffusion) or from the circulating or eddy currents present in a turbulent fluid (eddy diffusion). 

In many practical mass transfer processes, unsteady state conditions prevail. Thus, in the example given in Section 10.1, a box is divided into two compartments each containing a different gas and the partition is removed. Molecular diffusion of the gases takes place and concentrations, and concentration gradients, change with time. If a bowl of liquid... 

Kishinev ski/23 has developed a model for mass transfer across an interface in which molecular diffusion is assumed to play no part. In this, fresh material is continuously brought to the interface as a result of turbulence within the fluid and, after exposure to the second phase, the fluid element attains equilibrium with it and then becomes mixed again with the bulk of the phase. The model thus presupposes surface renewal without penetration by diffusion and therefore the effect of diffusivity should not be important. No reliable experimental results are available to test the theory adequately. 

In many applications of mass transfer the solute reacts with the medium as in the case, for example, of the absorption of carbon dioxide in an alkaline solution. The mass transfer rate then decreases in the direction of diffusion as a result of the reaction. Considering the unidirectional molecular diffusion of a component A through a distance Sy over area A. then, neglecting the effects of bulk flow, a material balance for an irreversible reaction of order n gives ... 

In a gas absorption process, the solute gas A diffuses into a solvent liquid with which it reacts. The mass transfer is one of steady state unidirectional molecular diffusion and the concentration of A is always sufficiently small for bulk flow to be negligible. Under these conditions the reaction is first order with respect to the solute A. 

At a depth l below the liquid surface, the. concentration of A has fallen to one-half of the value at the. surface. What is the. ratio of the. mass transfer rate at this depth t to the. rate, at the surface Calculate the numerical value of the ratio when l /k/D = 0.693, where. D is the molecular diffusivity and k the first-order rate constant. 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

An open bowl, 0.3 m in diameter, contains water at 350 K evaporating into the atmosphere, if the air currents are sufficiently strong to remove the water vapour as it is formed and if the resistance to its mass transfer in air is equivalent to that of a 1 mm layer for conditions of molecular diffusion, what will be the rate of cooling due to evaporation The water can be considered as well mixed and the water equivalent of the system is equal to 10 kg. The diffusivity of water vapour in air may be taken as 0.20 ctn2/s and the kilogram molecular volume at NTP as 22.4 in3. 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

A pure gas is absorbed into a liquid with which it reacts. The concentration in the liquid is sufficiently low for the mass transfer to be governed by Pick s law and the reaction is first order with respect to the solute gas. It may be assumed that the film theory may be applied to the liquid and that the concentration of solute gas falls from the saturation value to zero across the film. Obtain an expression for the mass transfer rate across the gas-liquid interface in terms of the molecular diffusivity, 1), the first order reaction rate constant k. the film thickness L and the concentration Cas of solute in a saturated solution. The reaction is initially carried our at 293 K. By what factor will the mass transfer rate across the interface change, if the temperature is raised to 313 K7... 

A soluble gas is absorbed into a liquid with which it undergoes a second-order irreversible reaction. The process reaches a steady-state with the surface concentration of reacting material remaining constant at (.2ij and the depth of penetration of the reactant being small compared with the depth of liquid which can be regarded as infinite in extent. Derive the basic differential equation for the process and from this derive an expression for the concentration and mass transfer rate (moles per unit area and unit time) as a function of depth below the surface. Assume that mass transfer is by molecular diffusion. 

From the penetration theory, the mass transfer rate per unit area N, is given in terms of the concentration difference AC, between the interface and the bulk fluid, the molecular diffusivity D and the age t of the. surface clement by ... 

The devolatilization of a component in an internal mixer can be described by a model based on the penetration theory [27,28]. The main characteristic of this model is the separation of the bulk of material into two parts A layer periodically wiped onto the wall of the mixing chamber, and a pool of material rotating in front of the rotor flights, as shown in Figure 29.15. This flow pattern results in a constant exposure time of the interface between the material and the vapor phase in the void space of the internal mixer. Devolatilization occurs according to two different mechanisms Molecular diffusion between the fluid elements in the surface layer of the wall film and the pool, and mass transport between the rubber phase and the vapor phase due to evaporation of the volatile component. As the diffusion rate of a liquid or a gas in a polymeric matrix is rather low, the main contribution to devolatilization is based on the mass transport between the surface layer of the polymeric material and the vapor phase. 

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