Mass transfer diffusional resistance

Given the availability of hollow fiber membranes adequately permeable to substrates and products, and the control of fluid flow all around the fibers in the bundle in order to assure uniform flow distribution and to avoid stagnation (in order to reduce mass transfer diffusional resistances), the technique offers several advantages. Enzyme proteins can be easily retained within the core of the fibers with no deactivation due to coupling agents or to shear stresses, and the enzyme solution can be easily recovered and/or recycled. 

As illustrated ia Figure 6, a porous adsorbent ia contact with a fluid phase offers at least two and often three distinct resistances to mass transfer external film resistance and iatraparticle diffusional resistance. When the pore size distribution has a well-defined bimodal form, the latter may be divided iato macropore and micropore diffusional resistances. Depending on the particular system and the conditions, any one of these resistances maybe dominant or the overall rate of mass transfer may be determined by the combiaed effects of more than one resistance. 

Fig. 6. Concentration profiles through an idealized biporous adsorbent particle showing some of the possible regimes. (1) + (a) rapid mass transfer, equihbrium throughout particle (1) + (b) micropore diffusion control with no significant macropore or external resistance (1) + (c) controlling resistance at the surface of the microparticles (2) + (a) macropore diffusion control with some external resistance and no resistance within the microparticle (2) + (b) all three resistances (micropore, macropore, and film) significant (2) + (c) diffusional resistance within the macroparticle and resistance at the surface of the...

Although they are termed homogeneous, most industrial gas-phase reactions take place in contact with solids, either the vessel wall or particles as heat carriers or catalysts. With catalysts, mass diffusional resistances are present with inert solids, the only complication is with heat transfer. A few of the reactions in Table 23-1 are gas-phase type, mostly catalytic. Usually a system of industrial interest is liquefiea to take advantage of the higher rates of liquid reactions, or to utihze liquid homogeneous cat ysts, or simply to keep equipment size down. In this section, some important noncatalytic gas reactions are described. 

The effectiveness factor accounts for the diffusional resistances in the liquid-filled catalyst pores. It does not account for the mass transfer resistance between the liquid and gas phases. This is the job of the ki and kg terms. 

Gas-liquid reactors. Gas-liquid reactors are quite common. Gas-phase components will normally have a small molar mass. Consider the interface between a gas and a liquid that is assumed to have a flow pattern giving a stagnant film in the liquid and the gas on each side of the interface, as illustrated in Figure 7.2. The bulk of the gas and the liquid are assumed to have a uniform concentration. It will be assumed here that Reactant A must transfer from the gas to the liquid for the reaction to occur. There is diffusional resistance in the gas film and the liquid film. 

Let us systematically delineate the transport pathways of the nondissociated and protonated species of the P-blockers by applying Eq. (82). The insignificance of the mass transfer resistance of the ABL on the overall transport process, as evidenced by the lack of influence of stirring on Pe, indicates that the passive diffusional kinetics are essentially controlled by the cell monolayer and filter. Therefore, Eq. (82) simplifies to... 

Comparison of As and A reveals substantial resistance to diffusional mass transfer. 

The resistance to mass transfer of reactants within catalyst particles results in lower apparent reaction rates, due to a slower supply of reactants to the catalytic reaction sites. Ihe long diffusional paths inside large catalyst particles, often through tortuous pores, result in a high resistance to mass transfer of the reactants and products. The overall effects of these factors involving mass transfer and reaction rates are expressed by the so-called (internal) effectiveness factor f, which is defined by the following equation, excluding the mass transfer resistance of the liquid film on the particle surface [1, 2] ... 

The resistance in each phase is made up of two parts the diffusional resistance in the laminar film and the resistance in the bulk fluid. All current theories on mass transfer, i. e. film, penetration, and surface renewal assume that the resistance in the bulk fluid is negligible and the major resistance occurs in the laminar films on either side of the interface (Figure 3-2). Fick s law of diffusion forms the basis for these theories proposed to describe mass transfer through this laminar film to the phase boundary. 

There are three distinct mass-transfer resistances (1) the external resistance of the fluid film surrounding the pellet, (2) the diffusional resistance of the macropores of the pellet, and (3) the diffusional resistance of the zeolite crystals. The external mass-transfer resistance may be estimated from well-established correlations (4, 5) and is generally negligible for molecular sieve adsorbers so that, under practical operating conditions, the rate of mass transfer is controlled by either macropore diffusion or zeolitic diffusion. In the present analysis we consider only systems in which one or other of these resistances is dominant. If both resistances are of comparable importance the analysis becomes more difficult. 

By using the first two moments it is, therefore, possible to calculate the mass-transfer coefficients which characterize the diffusional resistances of a particle with a bipore system. To calculate these moments it is necessary to integrate the experimental values of C(L, t) as a function of time at different values of U. 

The MSSR presents the same advantages as BSCR, such as high efficiency of heat-and mass-transfer and minimal intraparticle diffusional resistance, and are convenient for use in batch processes. For these reasons, the slurry-agitated reactors are also suitable for kinetic studies in the laboratory. Some of their major drawbacks are large power requirement for mechanical agitation,... 

If the rate constants for the sorption-desorption processes are small equilibrium between phases need not be achieved instantaneously. This effect is often called resistance-to-mass transfer, and thus transport of solute from one phase to another can be assumed diffusional in nature. As the solute migrates through the column it is sorbed from the mobile phase into the stationary phase. Flow is through the void volume of the solid particles with the result that the solute molecules diffuse through the interstices to reach surface of stationary phase. Likewise, the solute has to diffuse from the interior of the stationary phase to get back into the mobile phase. 

Many investigations have been conducted of the mass transfer coefficient at the external surfaces of particles and of other diffusional mechanisms. Some of the correlations are discussed in Chapters 13 and 17. A model developed by Rosen [Ind. Eng. Chem. 46, 1590 (1954)] takes into account both external film and pore diffusional resistance to mass transfer together with a linear isotherm. A numerical example is worked out by Hines and Maddox (1985, p. 485). 

The system is heterogeneous, external mass and heat transfer between the pellet and the bulk gas is negligible, but the intraparticle diffusional resistance is considerable. 

However, nonmonotonic kinetics alone will not produce multiplicity. Such kinetics have to be coupled with a diffusion process, either in the form of a mass-transfer resistance between the catalyst pellet surface and the bulk gas, or within the pores of the pellets. If the flow conditions and the catalyst pellets size are such that diffusional resistances between the bulk gas phase and the catalytic active centers are negligible and the... 

FIGURE 3.1 Concentration profiles in a passive sampling device. The driving force of accumulation is the difference in chemical potentials of the analyte between the bulk water and the sorption phase. The mass transfer of an analyte is governed by the overall resistance along the whole diffusional path, including contributions from the individual barriers (e.g., aqueous boundary layer, biofilm layer, and membrane). 

The plate theory assumes that an instantaneous equilibrium is set up for the solute between the stationary and mobile phases, and it does not consider the effects of diffusional effects on column performance. The rate theory avoids the assumption of an instantaneous equilibrium and addresses the diffusional factors that contribute to band broadening in the column, namely, eddy diffusion, longitudinal diffusion, and resistance to mass transfer in the stationary phase and the mobile phase. The experimental conditions required to obtain the most efficient system can be determined by constructing a van Deemter plot. 

If the equilibrium is linear, exact analytical solutions for the column response can be obtained even when the rate expression is quite complex. In most of the published solutions, axial dispersion is also neglected, but this simplification is not essential and a number of solutions including both axial dispersion and more than one diffusional resistance to mass transfer have been obtained. Analytical solutions can also be obtained for an irreversible isotherm with negligible axial dispersion, but the case of an irreversible isotherm with significant axial dispersion has not yet been solved analytically. 

A practical approach is to use the venturi tube (see Fig. 10.13) with a swarm of representative droplets of a certain diameter to estimate an effective mass transfer coefficient, including diffusional and reactive resistances. The value obtained would be... 

Some of this theoretical thinking may be utilized in reactor analysis and design. Illustrations of gas-liquid reactors are shown in Fig. 19-26. Unfortunately, some of the parameter values required to undertake a rigorous analysis often are not available. As discussed in Sec. 7, the intrinsic rate constant kc for a liquid-phase reaction without the complications of diffusional resistances may be estimated from properly designed laboratory experiments. Gas- and liquid-phase holdups may be estimated from correlations or measured. The interfacial area per unit reactor volume a may be estimated from correlations or measurements that utilize techniques of transmission or reflection of light, though these are limited to small diameters. The combined volumetric mass-transfer coefficient kLa, can be also directly measured in reactive or nonreactive systems (see, e.g., Char-pentier, Advances in Chemical Engineering, vol. 11, Academic Press, 1981, pp. 2-135). Mass-transfer coefficients, interfacial areas, and liquid holdup typical for various gas-liquid reactors are provided in Tables 19-10 and 19-11. 

The model presented in this paper assumes that mass transfer and chemical kinetics are consecutive steps. Primary resistance for the transfer of toluene from the organic phase into the acid phase lies in the organic phase film. This mass transfer step is followed by homogeneous reaction uniformly over the entire acid phase. If the nitrator is run such that the toluene content of the organic phase is high, diffusional resistance as well as the chemical reaction will be confined to an acid film and the model proposed by Hanson and co-workers (11) will be more applicable. 

From a kinetic standpoint (4), mass transfer per unit volume in distillation is limited only by the diffusional resistances on either side of the vapor-liquid interface in turbulent phases, with no inerts present, In almost every other separation process, there are inert solvents or... 

Vogelpohl (193) and Medina et al. (203) applied the diffusional interaction method for predicting ternary distillation composition profiles using binary data. They achieved this by eliminating the first two steps and assuming that all the mass transfer resistance is in the vapor. This procedure was shown to give excellent agreement with experimental data for dissimilar components. Biddulph and Kalbassi (194), however, report some discrepancies between prediction and experiment due to this assumption. 

Although multiplicities of the effectiveness factor have also been detected experimentally, these are of minor importance practically, since for industrial processes and catalysts, Prater numbers above 0.1 are less common. On the contrary, effectiveness factors above unity in real systems are frequently encountered, although the dominating part of the overall heat transfer resistance normally lies in the external boundary layer rather than inside the catalyst pellet. For mass transfer the opposite holds the dominating diffusional resistance is normally located within the pellet, whereas the interphase mass transfer most frequently plays a minor role (high space velocity). 

The true intrinsic kinetic measurements require (1) negligible heat and mass transfer resistances by the fluids external to the catalyst (2) negligible intraparticle heat and mass transfer resistances and (3) that all catalyst surface be exposed to the reacting species. The choice of the reactor among the ones described in this section depends upon the nature of the reaction system and the type of the required kinetic data. Generally, the best way to determine the conditions where the reaction is controlled by the intrinsic kinetics is to obtain rate per unit catalyst surface area as a function of the stirrer speed. When the reaction is kinetically controlled, the rate will be independent of the stirrer speed. The intraparticle diffusional effects and flow uniformity (item 3, above) are determined by measuring the rates for various particle sizes and the catalyst volume, respectively. If the reaction rate per unit surface area is independent of stirrer speed, particle size, and catalyst volume, the measurements can be considered to be controlled by intrinsic kinetics. It is possible... 

It is interesting to note the dependence of slope in the just-described plot on dp. The term psdp/6ks oc dp7-20, whereas 1 /fc, would either change with dp (if intraparticle resistance is significant) or be independent of dp (if intraparticle diffusional resistant is absent). Thus, a study of the exponent dependency of the slope on dp gives insight into the controlling step for the reaction. For small dp, when the slope is independent of dp, both intraparticle and liquid-solid mass-transfer resistances may be assumed to be negligible. 

Figure 3 Electrical equivalent circuit model commonly used to represent an electrochemical interface undergoing corrosion. Rp is the polarization resistance, Cd] is the double layer capacitance, Rct is the charge transfer resistance in the absence of mass transport and reaction intermediates, RD is the diffusional resistance, and Rs is the solution resistance, (a) Rp = Rct when there are no mass transport limitations and electrochemical reactions involve no absorbed intermediates and nearly instantaneous charge transfer control prevails, (b) Rp = Rd + Rct in the case of mass transport limitations.

This methods depends on the implicit assumption that the uptake rate is controlled entirely by intracrystalline diffusion in an isothermal system, with all other resistances to either mass or heat transfer negligible. This is a valid approximation if diffusion is sufficiently slow or if the zeolite crystals are sufficiently large but the dominance of intracrystalline diffusional resistance should not be assumed without experimental verification. In many practical systems, particularly with small commercial zeolite crystals, the external heat and mass transfer resistances are in fact dominant. A detailed discussion of such effects has been given by Lee and Ruthven(5-7). 

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