Transfer, mass

Mass transfer is a main concern in chemical and bioprocess engineering since most industrial processes uses mass transfer operations not only to mix components but also to separate them The design of these operations is a core part of a chemical and bioprocess engineer curriculum and is based on knowing material balances, energy balances, equilibrium, and mass transfer. 

Here we present briefly the mechanism underlying mass transfer. In mass transfer we differentiate the various contributions to it from diffusion, convection, and dispersion. 

Diffusion. Diffusion corresponds to mass transfer due to movement of molecules (Fig. 6.17) between the surrounding media. This is the most relevant mechanism for mass transfer in solids. It also occurs in liquids and gases but usually in such cases convection and dispersion become more relevant Since the rate of diffusion depends on the kinetic energy of the molecules and the degree of cohesion between them, it occurs more easily in gases, then in liquids, and, finally, in solids. For the same reason diffusion tends to occur faster as the temperature increases. 

The rate of mass transfer per area (flux AC), depends on the concentration gradient AC/AX, the surface involved in the process (A) is quantified using the following expression, known as Fick s first law of diffusion  

If you look closely at this equation and Fourier s law of heat transfer, they look very similar. This similarity is not casual because Pick s observations, based on mass transfer, gave similar results to those observed for heat transfer. This similarity among heat, mass, and momentum transfer are very relevant for chemical and bioprocess engineering and makes it possible to group them into the so-called discipline of transport phenomena. 

Heat transfer is directly related to the concept of mass transfer. Q is used to heat products of the reaction, which, on a molecular level, means tiiat the greater Q is, the more kinetic energy is transferred to the product molecules. This Hnetic energy can be transferred to other molecules via coUision. However, for that transfer to occur to key molecules such as those found in the vaporized candle wax, the energy has to be delivered, via fast-moving molecules, to the right place. That movement of mass is called mnss transfer. 

Finally, in cases involving a poured liquid accelerant, mass transport of the fuel occurs in a lateral direction, controlled by the characteristics of the surface. Gasoline on a nonporous surface like concrete will diffuse easily, while gasoline on a porous surface like wood or carpet will tend to be absorbed. As a result, porous and semiporous surfaces should be sampled in depth, since the chances of finding residual accelerants is increased in such cases. 

Rates of gas-liquid, liquid-liquid, and solid-liquid mass transfer are important, and often control the overall rates in bioprocesses. For example, the rates of oxygen absorption into fermentation broths often control the overall rates of aerobic fermentation. The extraction of some products from a fermentation broth, using an immiscible solvent, represents a case of liquid-liquid mass transfer. Solid-liquid mass transfer is important in some bioreactors using immobilized enzymes. 

In various membrane processes (these will be discussed in Chapter 8), the rates of mass transfer between the liquid phase and the membrane surface often control the overall rates. 

Transport processes are involved when a current is passed through a fuel cell. Ions and neutral species that participate in the electrochemical reactions at the anode or cathode have to be transported to the respective electrode surfaces. In Section 1.3.2, we introduced the charge transfer kinetics-controlled electrode reactions in 

1 Fast-Speed Electrode Reaction Rearrangement of Equation 1.81 gives 

When the exchange current density is very large compared with the electrode reaction current density, that is, /, the left side of Equation 1.105 is approximately equal to zero. Then Equation 1.105 can be written as 

Equation 1.109 has the same form as the Nemst equation. It indicates that the surface concentrations of species involved in the Faradaic process are related to the electrode potential by an equation of the Nemst form when the exchange current density is very large. Such electrode reactions are often called reversible or Nemstian, because the principal species obey thermodynamic relationships at the electrode surface. 

For a reversible electrode reaction, the electrode potential is related to the surface concentrations of species by the Nemst equation. The net rate of the electrode reaction,, is then governed completely by the rate at which the electroactive 

The net heat input into the particle cloud is thus obtained as 64.05 kW/m2. It is interesting to note that without particles the net heat flux between the two plates is only 8.54 kW/m2. 

The rate at which metastable phases dissolve or are replaced is an important problem in carbonate diagenesis. Carbonate mineral assemblages persist metastably in environments where they should have altered to stable assemblages. The question is what are the time scales of these alterations They are certainly variable ranging from a few thousand to a few hundreds of millions of years. Even calcites in very old limestones show chemical and structural heterogeneities, indicating that the stabilization of these phases is not complete. Unfortunately, it is difficult, but not impossible, to apply directly the lessons learned about carbonate mineral dissolution and precipitation in the laboratory to natural environments. 

Because of all the data available concerning Bermudian limestone diagenesis, several authors (Lafon and Vacher, 1975 Plummer et al., 1976 Vacher, 1978 Vacher et al., 1989) have attempted to determine rates of carbonate mineral stabilization in Bermudian calcarenites and the mass transfer involved in the process. 

One approach used to date (Plummer et al., 1976 Halley and Harris, 1979 Budd, 1984, 1988) is to assume that the rate of conversion of aragonite to calcite (dA/dt) in the phreatic freshwater zone is directly proportional to the mass of aragonite present at a given time, and that the surface area of aragonite per unit mass of aragonite present in the phreatic zone does not change with time  

Solution of equation 7.9 for Bermuda, where dA/dt = 7.2 cm3 nr3 y-1, and A = 80,000 cm3 nr3 gives a value of k = 9 x 10 5 y-1. This k applied to the aragonite stabilization rate of calcarenite in Bermuda and Joulters Cay gives a half-life of  

In contrast, Lafon and Vacher (1975) estimated the half-life of aragonite in the vadose zone of Bermuda as about 230,000 years, and the half-life of high-magnesian calcite as only 60,000 years. Minor corrections need to be made to these estimates, because of new estimates of limestone ages for Bermuda, but the fact still remains that stabilization in the vadose freshwater zone is much slower than in the phreatic zone (Land, 1970 Steinen and Matthews, 1973 Steinen, 1974 Vacher et al 1989). 

This chapter considers the vapor-liquid equilibrium of mixtures, conditions for bubble and dew points of gaseous mixtures, isothermal equilibrium flash calculations, the design of distillation towers with valve trays, packed tower design. Smoker s equation for estimating the number of plates in a binary mixture, and finally, the computation of multi-component recovery and minimum trays in distillation columns. 

A common operation in the CPI for the separation of fluid mixtures into their components is distillation. The distillation process is a separation technique that depends on the difference between the compositions of liquid and vapor phases at equilibrium. This implies that the temperatures and pressures of the phases must be the same, and no composition change occurs with time. Equilibrium can be achieved after a long period of thorough mixing and contact between the two phases. An ideal gas is, by definition, one that follows the ideal gas laws  

The conditions under which a given component or mixture approaches ideal behavior depend on the critical temperature and critical pressure. Other principles of gas behavior are Dalton s law of additive pressures and Amagat s law of additive volumes. These are  

These laws are correct when conditions are such that each component and the mixture obey the ideal gas law. The equilibrium between two phases can be related to the equality of the chemical potential and defined in terms of the Gibbs free energy as 

Considering a pure fluid of component i and substituting Equation 7-1 into Equation 7-5, 

The removal of water or solvents is a three-step process  

Heat is first transferred to the material that naturally contains water, such as milk, tobacco, carrots, or to which liquid water or solvent was added in a preceding process (such as for forming or coating). The heat is necessary to evaporate the liquid to a vapor form for easy removal (mass transfer). 

The driving force that causes the liquid to migrate to the surface of the material or piece being dried is the difference in vapor pressure between the inside and the surface of the pieces being dried. 

Similarly, the driving force causing the liquid to vaporize and causing the vapor to migrate away from the surface is the same difference in vapor pressure that caused (b). 

The practical way to create and maintain an appreciable difference in vapor pressure to continually force rapid mass transfer is to move a stream of hot poc and air to constantly wipe the wet surface (i.e., convection heating). Neither radiant burners nor electric elements are as effective unless accompanied by circulating fans. Convection burners provide a circulating (wiping, mass transfer) effect. 

Equations (5,22), (5.24), and (5.26) are most useful for ordinary purposes, and moderate variations in c are usually taken care of by use of an average, 

Illustration 1. Calculate the rate of diffusion of ethanol across a film of water solution 0.2 cm. thick at 20°C., when the concentrations on either side of the film are 14 and 9.6 wt. per cent ethanol. Under these conditions, the diffusivity of the ethanol may be taken as 0.74 X 10 sq. cm./sec. 

Unsteady-state Conditions. Arnold (2) has integrated the Maxwell-Stefan equation for gaseous diffusion in the case of the semi-infinite column, or diffusion from a plane at which the concentrations are kept constant into a space filled with gas extending to infinity, both for vaporization of a liquid into a gas and absorption of a gas by a liquid. It is possible that the resulting equations could be applied successfully to liquid diffusion for similar circumstances, provided that an assumption analogous to Dalton s law for gases can be made and that D is assumed to remain constant. The direct application to extraction operations of such equa- 

Empirical Estimation of Diffusivities for Nonelectrolytes. Dilute Solutions. Two reasonably successful approaches have been made to the problem of estimation of diffusivities in the absence of measured data, based on an extension of the kinetic theory to liquids and on the theory of absolute reaction rates. 

Kinetic theory approach. By application of the kinetic theory of gases to the liquid phase, Arnold (1) obtained an expression of D paralleling in form that obtained previously for gases  

A slab of porous solid 1/2-in. thick is soaked in pure ethanol [1]. The void space in the solid occupies 50% of its volume. The pores are fine, so that molecular diffusion can take place through the liquid in the passages there is no convective mixing. The effective diffusivity of the system ethanol-water in the pores is one-tenth that in the free liquid. 

If the slab is placed in a large well-agitated reservoir of pure water at 77°F, how long will it take for the mass fraction of ethanol at the center of the slab to fall to 0.009 Assume that there is no resistance to mass transfer in the water phase and that the concentration of ethanol in the water, and thus at the surface of the slab, is constant at zero. 

Since the densities of alcohol and water differ by only 20%, then one may assume that the total density remains constant such that 

This system of equations was solved in the context of heat transfer previously (Example 7.1). There the solution was given as 

Note that use is made of the solution from aprevious problem that was solved in dimensionless form. Also, the analogy between heat conduction and molecular diffusion can best be exploited when dimensionless variables are utilized. 

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However, a note of caution should be added. In many multiphase reaction systems, rates of mass transfer between different phases can be just as important or more important than reaction kinetics in determining the reactor volume. Mass transfer rates are generally higher in gas-phase than liquid-phase systems. In such situations, it is not so easy to judge whether gas or liquid phase is preferred. 

The process of mass transfer is repeated over and over to synchronize time intervals, used. 

To speed up the process of attainment of the temperature steady value one can use special operations calculation without a kiln rotation, using large time intervals and calculation in two-dimensional R-tp geometry without regard for heat and mass transfer along an axis The program for realization of discussed simulation algorithms enables to calculate temperature in cells, a total number of which can not exceed 130 thousands A circular kiln structure can contain up to three layers. 

Basu., B., Date, A.W. Int. J. Heat Mass Transfer, 33 (1990) Nr. 6, S.l 149 Haferkamp, H., Gerken, J., Stegemann, D., Reichert, Ch. In-situ Untersuchung des Hartstofftransports beim Laserstrahl-Dispergieren mittels Hochgeschwindigkeits-Radioskopie, Metall, Nr. 3/96... 

The effect can be important in mass-transfer problems (see Ref. 57 and citations therein). The Marangoni instability is often associated with a temperature gradient characterized by the Marangoni number Ma ... 

The high rate of mass transfer in SECM enables the study of fast reactions under steady-state conditions and allows the mechanism and physical localization of the interfacial reaction to be probed. It combines the usefid... 

Though the case of constant matrix elements and the example investigated by Hite are the only situations for which Che stoichiometric relations have been fully established in pellets of arbitrary shape, it is worth mentioning situations in which these relations are known not to hold. When the composition and pressure at the surface of the pellet may vary in an arbitrary way from point to point it seems unlikely on intuitive grounds that equations (11.3) will be satisfied, and Hite and Jackson [77] confirmed by direct computation that there are, indeed, simple situations in which they are violated. Less obviously, direct computation [75] has also shown them to be violated even when the pressure and composition of the environment are the same everywhere, in the case where finite resistances to mass transfer exist at the surface of Che pellet. 

A proper resolution of Che status of Che stoichiometric relations in the theory of steady states of catalyst pellets would be very desirable. Stewart s argument and the other fragmentary results presently available suggest they may always be satisfied for a single reaction when the boundary conditions correspond Co a uniform environment with no mass transfer resistance at the surface, regardless of the number of substances in Che mixture, the shape of the pellet, or the particular flux model used. However, this is no more than informed and perhaps wishful speculation. 

N. Satterfield, Mass Transfer in Heterogeneous Catalysis, M.I.T. Press, 1970. 

Kovats retention indices RI Mass transfer coefficient h... 

Caffeine is extracted from beverages by a solid-phase microextraction using an uncoated fused silica fiber. The fiber is suspended in the sample for 5 min and the sample stirred to assist the mass transfer of analyte to the fiber. Immediately after removing the fiber from the sample it is transferred to the gas chromatograph s injection port where the analyte is thermally desorbed. Quantitation is accomplished by using a C3 caffeine solution as an internal standard. 

Schematics illustrating the contributions to band broadening due to (a) multiple paths, (b) longitudinal diffusion, and (c) mass transfer.

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. 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

To minimize the multiple path and mass transfer contributions to plate height (equations 12.23 and 12.26), the packing material should be of as small a diameter as is practical and loaded with a thin film of stationary phase (equation 12.25). Compared with capillary columns, which are discussed in the next section, packed columns can handle larger amounts of sample. Samples of 0.1-10 )J,L are routinely analyzed with a packed column. Column efficiencies are typically several hundred to 2000 plates/m, providing columns with 3000-10,000 theoretical plates. Assuming Wiax/Wiin is approximately 50, a packed column with 10,000 theoretical plates has a peak capacity (equation 12.18) of... 

Kovat s retention index (p. 575) liquid-solid adsorption chromatography (p. 590) longitudinal diffusion (p. 560) loop injector (p. 584) mass spectrum (p. 571) mass transfer (p. 561) micellar electrokinetic capillary chromatography (p. 606) micelle (p. 606) mobile phase (p. 546) normal-phase chromatography (p. 580) on-column injection (p. 568) open tubular column (p. 564) packed column (p. 564) peak capacity (p. 554)... 

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