Mass transfer problems

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 ... 

Tank Cells. A direct extension of laboratory beaker cells is represented in the use of plate electrodes immersed into a lined, rectangular tank, which may be fitted with a cover for gas collection or vapor control. The tank cell, which is usually undivided, is used in batch or semibatch operations. The tank cell has the attraction of being both simple to design and usually inexpensive. However, it is not the most suitable for large-scale operation or where forced convection is needed. Rotating cylinders or rotating disks have been used to overcome mass-transfer problems in tank cells. An example for electroorganic synthesis is available (46). 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

Wetted-waU or falhng-film columns have found application in mass-transfer problems when high-heat-transfer-rate requirements are concomitant with the absorption process. Large areas of open surface are available for heat transfer for a given rate of mass transfer in this type of equipment because of the low mass-transfer rate inherent in wetted-waU equipment. In addition, this type of equipment lends itself to annular-type coohng devices. 

Air is commonly run with tube-side feed. The permeate is run countercurrent with the separating sldn in contact with the permeate. (The feed gas is in contact with the macroporous back side of the membrane.) This configuration has proven to be superior, since the permeate-side mass-transfer problem is reduced to a minimum, and the feed-side mass-transfer problem is not limiting. 

The [nimary objective of the VOC-condensation system is to meet mass-recovery objectives. However, heat is a key element in realizing the mass objectives. Hence, the mass and heat interactions of the problem have to be identified and reconciled. This can be achieved by converting the VOC-recovery task from a mass-transfer problem to a heat-transfer duty. This can be accomplished by relating the... 

This chapter sets out to provide a means of handling these types of interphase mass transfer problems taking into consideration their fundamental characterizing variables, the conservation of mass, and appropriate constitutive relationships. 

Certain hydrodynamical problems, as well as mass-transfer problems in the presence of surface-active agents, have been investigated theoretically under steady-state conditions (L3, L4, L10, R9). However, if we take into account the fact that in gas-liquid dispersions, the nonstationary term must appear in the equation of mass- or heat-transfer, it becomes apparent that an exact analysis is possible if a mixing-contacting mechanism is adopted instead of a theoretical streamline flow around a single bubble sphere. 

While PLB were introduced first (14,15) more recently small PB 5-15 M diameter) have become of major interest. This is a result of the higher separation speeds found with such particles. Not only is the "stagnant" mobile phase mass transfer problem reduced, as in PLB, but solute mixing in the flowing stream is enhanced as a result of the smaller distance between the particles. The performances achieved with the small particle columns are equivalent to those obtained with capillary columns in gas chromatography (13), Examples illustrating the separation speed of such columns will be presented in the applications section of this paper. 

Some of the important dimensionless groups pertinent to heat and mass transfer problems are listed in Table 3.5. 

So you say wow But MATLAB can do much more and fancier than that. We try one more example with Bessel functions, which you can come across in heat and mass transfer problems with cylindrical geometry. 

To consider the convective mass transfer problem of a rotating hemisphere electrode, we assume that sufficient inert salts are present in the electrolyte that the migrational... 

The theories that have been developed to describe mass transfer arise from the law of conservation of mass, which states that mass can be neither created nor destroyed. According to this law, the total mass in a particular region in space can increase only by the addition of mass from the surroundings and can decrease only by the loss of mass back to them. Processes such as radioisotope decay and nuclear fission are exceptions to this law, since they involve the interconversion of matter and energy. In the absence of nuclear decay, however, the law of conservation of mass holds and is broadly applicable to mass transfer problems. 

Note that the equation includes rates of gain and loss rather than total amounts. As a result, the mathematical form will be a differential equation rather than an algebraic one. The differential form is preferred in almost all mass transfer problems, because variations in the rates with position and time can be incorporated accurately. Each term in the equation will take on a specific functional form depending on the parameters and mass transfer characteristics of the problem of interest. 

Note that in the component mass balance the kinetic rate laws relating reaction rate to species concentrations become important and must be specified. As with the total mass balance, the specific form of each term will vary from one mass transfer problem to the next. A complete description of the behavior of a system with n components includes a total mass balance and n - 1 component mass balances, since the total mass balance is the sum of the individual component mass balances. The solution of this set of equations provides relationships between the dependent variables (usually masses or concentrations) and the independent variables (usually time and/or spatial position) in the particular problem. Further manipulation of the results may also be necessary, since the natural dependent variable in the problem is not always of the greatest interest. For example, in describing drug diffusion in polymer membranes, the concentration of the drug within the membrane is the natural dependent variable, while the cumulative mass transported across the membrane is often of greater interest and can be derived from the concentration. 

In many problems, the system ultimately will reach a time-invariant state in which the total mass and/or the masses of the components are no longer changing. This condition is called the steady state. In mass transfer problems, the math-... 

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

Many heterogeneous catalytic organic reactions are run in the liquid-phase, and liquid phase reactions present special mass transfer problems. Diffusion barriers exist between the gas and the liquid and between the liquid and the solid, so there are gas-liquid-solid diffusion barriers. When these barriers are too large, the true chemical rate at the surface is not observed. 

Mass transfer problems can be addressed in several simple ways by varying the stirring rate, varying the amount of catalyst, varying the temperature, grinding, and poisoning titration. Several of these are discussed in an article by Roberts.48... 

Yet another way to detect mass transport problems is with a newly developed poisoning technique.24,26,49,50 This technique works for liquid-phase hydrogenations and possibly for other reactions that are poisoned by CS2. It takes advantage of the fact that CS2 poisons Pd and Pt linearly until all reaction stops. If mass transfer problems exist, the initial linear decrease in rate occurs at a slope less steep than the slope of the chemically controlled rate (Fig. 1.7). If no mass transport problems exist, the rate decreases linearly from the start with no change in slope. Therefore a plot of rate versus amount of CS2 reveals the existence or absence of mass transport problems 49... 

These do not contain the variable t (time) explicitly accordingly their solutions represent equilibrium configurations. Laplace s equation corresponds to a natural equilibrium, while Poisson s equation corresponds to an equilibrium under the influence ofg(x, y). Steady heat-transfer and mass-transfer problems are elliptic. 

Squires, A. M., Applications of Fluidized Beds in Coal Technology, lecture, Intern. School on Heat and Mass Transfer Problems in Future Energy Production, Dubrovnik, Yugoslavia (1975a)... 

So far only propene and butene are hydroformylated commercially using the RCH/RP process. A reason which has been postulated for this is the decreasing solubility in water with increasing number of C atoms in both the starting alkene and the reaction products (Figure 5.4) and the associated mass-transfer problems in the relatively complicated gas-liquid-liquid, three-phase reaction. 

For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a boundary value problem . The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism ... 

In order to keep the mild conditions, hydroxycarbonylation has been performed in biphasic media, maintaining the catalyst in the aqueous phase thanks to water-soluble mono- or diphosphine ligands. In the presence of the sodium salt of trisulfonated triphenylphosphine (TPPTS), palladium was shown to carbonylate efficiently acrylic ester [19], propene and light alkenes [20,21] in acidic media. For heavy alkenes the reduced activity due to the mass transfer problems between the aqueous and organic phases can be overcome by introducing an inverse phase transfer agent, and particularly dimeihyl-/-i-cyclodextrin [22,23]. Moreover, a dicationic palladium center coordinated by the bidentate diphosphine ligand 2,7-bis(sulfonato)xantphos (Fig. 2) catalyzes, in the presence of tolylsulfonic acid for stability reasons, the hydroxycarbonylation of ethylene, propene and styrene and provides a ca. 0.34 0.66 molar ratio for the two linear and branched acids [24],... 

The solubility of some simple terminal alkenes in water is listed in Table 8.2. As the length of the alkyl group increases the solubility of the alkene rapidly decreases. Even with rapid mixing, mass transfer problems due to the low solubility of substrates can occur. As such, alternative solvents to water in biphasic processes are required. 

The limitations of hydroformylation reactions in water are the same as those of hydrogenation reactions, i.e. the poor solubility of the substrates (see Section 8.2.1). While aqueous-organic biphasic hydroformylation works well for alkenes with chain lengths up to C7, the solubility of longer chain alkenes is too low for viable processes. Although simple alkenes are poorly soluble, many functional alkenes have solubilities in water that are sufficiently high to avoid mass transfer problems, but at the same time this can impede separation. 

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