Mass Transfer in One Phase

Postulating that n is dependent on the turbulence in the system, Dobbins (1956) proposed that under sufficiently turbulent conditions, n approaches 0.5 (surface renewal or penetration theory), while under laminar or less turbulent conditions n approaches 1.0 (film theory). Thus, the selection of the value for n to predict the mass transfer coefficient should depend on the degree of turbulence in the system  

The mass transfer flux N out of one phase is the product of the film coefficient and the concentration gradient in the film, and is equal to the flux into the second phase  

The concentrations of the diffusing material in the two phases immediately adjacent to the interface cLi, cGi are generally unequal, but are usually assumed to be related to each other by the laws of thermodynamic equilibrium (see Section B 3.1.3). 

In order to calculate the specific mass transfer rate into the liquid, mass per unit time and unit volume, the specific surface area, a, defined as transfer surface area per volume of liquid, is needed in addition to kL. 

The transfer interface produced by most of the mass transfer apparatus considered in this book is in the form of bubbles. Measuring the surface area of swarms of irregular bubbles is very difficult. This difficulty in determining the interfacial area is overcome by not measuring it separately, but rather lumping it together with the mass transfer coefficient and measuring kLa as one parameter. 

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Lewis and Whitman (1924) proposed that this resistance to mass transfer across an interface is the sum of the resistances in each phase. They called this concept the two-film theory. As Treybal (1968) pointed out, their two-film theory does not depend on which model is used to describe the mass transfer in each phase, therefore, the two-resistance theory would be a more appropriate name. It would also cause less confusion, since the names film theory (mass transfer in one phase) and two-film theory (mass transfer between... 

Mass transfer in one phase to the interface (mainly diffusion) ... 

The kinetics of mass transfer in one phase is usually described by the following equation that is based on the assumption that mass transfer in first approximation is a first order process ... 

Mass transfer in one phase and reaction in the other phase... 

Transfer of material between phases is important in most separation processes in which two phases are involved. When one phase is pure, mass transfer in the pure phase is not involved. For example, when a pure liqmd is being evaporated into a gas, only the gas-phase mass transfer need be calculated. Occasionally, mass transfer in one of the two phases may be neglec ted even though pure components are not involved. This will be the case when the resistance to mass transfer is much larger in one phase than in the other. Understanding the nature and magnitudes of these resistances is one of the keys to performing reliable mass transfer. In this section, mass transfer between gas and liquid phases will be discussed. The principles are easily applied to the other phases. 

Mass-Transfer Principles Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-hquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equihbrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a hquid are brought into contact. 

The factors determining the appearance of ordered cell-like motions were first investigated by Sternling and Scriven (S33) who considered the two-dimensional stability of a plane interface separating two immiscible semi-infinite fluid phases with mass transfer occurring between the phases. This system was shown to be unstable for mass transfer in one direction, but stable for transfer in the opposite direction. For an interfacial tension-lowering solute, instability... 

Ideally, biphasic catalysis is performed in such a way that mass-transfer from one phase to the other does not restrict the rate of the reaction. An elegant solution to overcome this potential limitation is reversible two phase-single phase reaction conditions. An example of a temperature-controlled reversible ionic liquid-water partitioning system has been demonstrated for the hydrogenation of 2-butyne-l,4-diol, see Figure 3.2.1251... 

However, Eq. (13.166) can be used only when there is no resistance to mass transfer in one of the phases. For resistance in both phases, the individual HTU values can be combined ... 

The resistance to mass transfer according to (1.221) and (1.223) is made up of the individual resistances of the gas and liquid phases. Both equations show how the resistance is distributed among the phases. This can be used to decide whether one of the resistances in comparison to the others can be neglected, so that it is only necessary to investigate mass transfer in one of the phases. Overall mass transfer coefficients can only be developed from the mass transfer coefficients if the phase equilibrium can be described by a linear function of the type shown in eq. (1.217). This is normally only relevant to processes of absorption of gases by liquids, because the solubility of gases in liquids is generally low and can be described by Henry s law (1.217). So called ideal liquid mixtures can also be described by the linear expression, known as Raoult s law. However these seldom appear in practice. As a result of all this, the calculation of overall mass transfer coefficients in mass transfer play a far smaller role than their equivalent overall heat transfer coefficients in the study of heat transfer. 

The difference between the thermal behavior in predominantly distillative and predominantly absorptive processes is a consequence of the different types of mass transfer in each case. In the former, mass is transferred from the liquid phase to the vapor phase and vice versa at approximately the same molar rate. The net material transfer between the phases is therefore small and the ratio of liquid flow to vapor flow (L/V) in a column section is nearly constant. In absorption or stripping columns, there is a net mass transfer in one direction and the L/V ratio is not constant. 

The two-film model is a simple example of this approach. A system of two fluids exists, with a distinct interface between the two (gas/liquid or two immiscible liquids). For purposes of this example, we assume a gas/liquid interface Figure 3.33 illustrates the region near the interface. There will be a film (or boundary layer) on each side of the interface where, due to mass transfer from one phase to the second (gas to liquid in the figure), the concentration of A is changing from its value in the bulk phase, in gas and Ca.s in liquid. The thickness of the boundary layer on each side of the interface will typically be different and a function of the fluid and flow conditions in each phase. At steady-state, the flux of A can be described as ... 

The chemical potential p, has an important function in the system s thermodynamic behavior analogous to pressure or temperature. A temperature difference between two bodies determines the tendency of heat to pass from one body to another while a pressure difference determines the tendency for bodily movement. We will show that a difference in chemical potential can be viewed as the cause for chemical reaction or for mass transfer from one phase to another. The chemical potential p greatly facilitates the discussion of open systems, or of closed systems that undergo chemical composition changes. 

For equilibrium staged and sorption separations, we are interested in mass transfer from one phase to another. This is illustrated schematically in Figure 15-5 for the transfer of component A from the liquid to a vapor phase. Xj and yj are the interfacial mole fractions. For dilute absorbers and strippers and for distillation where there is equimolar countertransfer of the more volatile and less volatile components, the mass-transfer Eq. flS-26bl can be written for each stage in the following different forms ... 

After all, a transfer works between two states of a substance, either two locations separated by space (two sites) or two amounts separated by the time course but on the same site, as explained in the beginning of this chapter. In the simplest case, one of the states/sites is free to adapt its content under the effect of the transfer, while the other is maintained fixed, either because the site is far from the region of space perturbed by the transfer (bulk) or because it is the initial state (before the transfer is active). A less simple case is met when the mass transfer is embedded into a sequence of steps composing a reactional chain in a heterogeneous mechanism. Other mass transfers (in different phases), chemical reactions or adsorption steps onto a surface, electron transfer steps, etc. may work in series (or for some of them in parallel) with the mass transfer, making both concentrations vary on the two states/sites. 

Classification of the separation techniques according to those involving phase change or mass transfer from one phase to another, known as diffusional operations, and those that are useful in the separation of solid particles or drops of a liquid and that are generally based in the application of an external physical force, known as mechanical separations. 

It is also useful to plot H against 1 /m (Figure 2.21). In both presentations (Figures 2.20, 2.21) the intercept of the linear portion of the H plot will equal Tkd. Thus, if the particle size dp is known, can be calculated and a measure of packing regularity obtained. From the slope of the linear part of the H-u curve one also can estimate the film thickness rff, if A and k are known (resistance to mass transfer in liquid-phase term). 

Interface computation It aims at predicting the influence of interfacial effect on the mass transfer, such as Marangoni convection and Rayleigh convection. Such effects may lead to the increase in the separation efficiency. Besides, the investigation of interfacial behaviors is also the basic step to understand the details of mass transferred from one phase to the other. This part of computation is described in the last two chapters of this book. 

In the mass transfer processes, the volume and density of each phase are changing due to the mass transferred from one phase to the other. 

Abstract In this chapter, an exothermic catalytic reaction process is simulated by using computational mass transfer (CMT) models as presented in Chap. 3. The difference between the simulation in this chapter from those in Chaps. 4,5, and 6 is that chemical reaction is involved. The source term in the species conservation equation represents not only the mass transferred from one phase to the other, but also the mass created or depleted by a chemical reaction. Thus, the application of the CMT model is extended to simulating the chemical reactor. The simulation is carried out on a wall-cooled catalytic reactor for the synthesis of vinyl acetate from acetic acid and acetylene by using both c — Sc model and Reynolds mass flux model. The simulated axial concentration and temperature distributions are in agreement with the experimental measurement. As the distribution of lx shows dissimilarity with Dj and the Sci or Pri are thus varying throughout the reactor. The anisotropic axial and radial turbulent mass transfer diffusivities are predicted where the wavy shape of axial diffusivity D, along the radial direction indicates the important influence of catalysis porosity distribution on the performance of a reactor. 

Abstract The mass transferred from one phase to the adjacent phase must diffuse through the interface and subsequently may produce interfacial effect. In this chapter, two kinds of important interfacial effects are discussed Marangoni effect and Rayleigh effect. The theoretical background and method of computation are described including origin of interfacial convection, mathematical expression, observation, theoretical analysis (interface instability, on-set condition), experimental and theoretical study on enhancement factor of mass transfer. The details of interfacial effects are simulated by using CMT differential equations. 

Notice that the multiplier of the first differential in Eq.9.4.5, the temperature, is the potentisd for heat transfer and that of the second one, the pressure, is the potential for P-V work. By analogy, the multiplier of each of the remaining differentials must be also a potential. It is indeed the potential for mass transfer from one phase to an other, or for the feasibility of chemical reactions in a mixture of compounds, and it is called, appropriately, the Chemical Potential of component . 

To determine how the height of a theoretical plate can be decreased, it is necessary to understand the experimental factors contributing to the broadening of a solute s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theoretical plate is determined by four contributions multiple paths, longitudinal diffusion, mass transfer in the stationary phase, and mass transfer in the mobile phase. 

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