Mass transfer resistances in series

The paracellular pathway consists of the tight junction (TJ) in series with the tortuous lateral space (LS) (Figs. 6 and 7), i.e., mass transfer resistances in series ... 

The Biot number Biwo in this case is defined as (Vp/Sex)ks/De where l/ks represents an overall mass transfer resistance which consists of the liquid side gas-to-liquid and liquid-to-solid mass transfer resistances in series as given below ... 

This equation expresses the two-film concept of Whitman (40), the numerator being a concentration driving force and the denominator being the sum of two mass transfer resistances in series. Analogous expressions may be written for two liquid film resistances in series. Figure 1 illustrates qualitatively the concentration profiles through two liquid films. 

The effect of the adsorbent shape on mass transfer is much more complex. There are several mass transfer resistances in series and/or parallel, each one of which may be controlling for a particular set of conditions. However, in general, mass transfer rates will be larger for particles with larger specific surface... 

The right-hand side of Eqnation 6.99 represents two mass transfer resistances in series. When k- kt, the mass transfer is controlled by the interfacial resistance, and when k, the mass transfer is controlled by the conventional convective diffusive transport and local equilibrium applies at the interface. 

A heterogeneous model is being developed by taking three mass transfer resistances in series for both ethanol and sugar these are as follows ... 

The model equations are developed for the floes with three mass transfer resistances in series for sugar and ethanol. The present floe model is more... 

Transfer of oxygen and carbon dioxide between alveoli of the lung and blood capillaries. There are a total of six major mass transfer resistances in series. 

This chapter discusses mass transfer coefficients for dilute solutions extensions to concentrated solutions are deferred to Section 9.5. In Section 8.1, we give a basic definition for a mass transfer coefficient and show how this coefficient can be used experimentally. In Section 8.2, we present other common definitions that represent a thicket of prickly alternatives rivaled only by standard states for chemical potentials. These various definitions are why mass transfer often has a reputation with students of being a difficult subject. In Section 8.3, we list existing correlations of mass transfer coefficients and in Section 8.4, we explain how these correlations can be developed with dimensional analysis. Finally, in Section 8.5, we discuss processes involving diffusion across interfaces, a topic that leads to overall mass transfer coefficients found as averages of more local processes. This last idea is commonly called mass transfer resistances in series. 

RELATIVE CONTRIBUTION (PERCENTAGE BASIS) OF MASS TRANSFER RESISTANCE IN THE MOBILE AMD STATIONARY PHASE TO COLUMN PLATE HEIGHT FOR A SERIES OF 0.32 mm I.D. OPEN TUBULAR COLUMNS USING UMDECAME AT 130 C AS THE TEST SOLUTE... 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

S. Yuan and H.G. Schwartzberg, Mass Transfer Resistance in Cross Membrane Evaporation into Air, Recent Advances in Separation Science, AIChE Symposium Series Number 68, AIChE, New York, NY, Vol. 120, p. 41 (1972). 

Mass transfer resistance in a continuous-contact separation device is the inverse of the mass transfer coefficient. In membrane contactors, the total resistance could be expressed as three resistances in series. These include the individual resistances in each flowing phase and the membrane resistance (Figure 2.4). For a liquid-gas contact system Equation 2.2 could be written for each diffusing species ... 

Mass transfer occurs only by diffusion across the immobilized phase in the pores. The direction of mass transfer of any molecular species depends on the concentration driving force maintained across the membrane for that species. The presence of the stationary phase in the membrane pore creates an extra diffusional mass-transfer resistance [6], However, it can be shown that in many cases, the membrane resistance is negligible and that in most cases, the highly active mass-transfer area created inside a membrane contactor more than compensates for any additional mass-transfer resistance [15,16], Mass-transfer resistance in a continuous-contact separation device is the inverse of the mass-transfer coefficient. In membrane contactors, the total resistance could be expressed as three resistances in series. These include the individual resistances in each flowing phase and the membrane resistance. For a liquid-gas contact system. Equation 4.2 could be written for each diffusing species [6] ... 

Provided chemical reaction does not occur simultaneously with the diffusion processes in an adsorbent particle, analysis of the response to a pulse input of an adsorbate to a column packed with an adsorbent provides a convenient experimental method of deducing the separate contributions of inter- and intraphase mass transfer and diffusion to the overall resistance to adsorption. This is because each one of the resistances to mass transfer is in series and thus linearly additive. 

The mathematics of mass diffusion within a single phase has thus been well estab-Ushed. Diffusion between phases such as air and water was not fully understood until 1923 when Whitman proposed the two-film theory in which transfer is expressed using two mass transfer coefficients in series, one for each phase (Whitman, 1923). This concept has been rediscovered in pharmacology and probably in other areas and is now more correctly termed the two-resistance theory. ... 

Expressions similar to equations 6 and 7 may be derived in terms of an overall Hquid-phase driving force. Equation 7 represents an addition of the resistances to mass transfer in the gas and Hquid films. The analogy of this process to the flow of electrical current through two resistances in series has been analyzed (25). 

For particles with a bidispersed pore structure, the mass-transfer parameter in the LDF approximation (column 2 in Table 16-12) can be approximated by the series-combination of resistances... 

The rectangular isotherm has received special attention. For this, many of the constant patterns are developed fully at the bed inlet, as shown for external mass transfer [Klotz, Chem. Revs., 39, 241 (1946)], pore diffusion [Vermeulen, Adv. Chem. Eng., 2,147 (1958) Hall et al., Ind. Eng. Chem. Fundam., 5,212 (1966)], pore diffusion with film resistance in series [Weber and Chakraborti, AIChE J. 20, 228 (1974)], the... 

If enzymes are immobilized by copolymerization or microencapsulation, the intraparticle mass-transfer resistance can affect the rate of enzyme reaction. In order to derive an equation that shows how the mass-transfer resistance affects the effectiveness of an immobilized enzyme, let s make a series of assumptions as follows ... 

The mass transfer resistance between the bubble and emulsion phases can be described in terms of bubble-to-cloud and cloud-to-emulsion resistances in series. Thus, we have... 

Modeling of H F contactors is in most papers based on a simple diffusion resistance in series approach. In many systems with reactive extractants (carriers) it could be of importance to take into account the kinetics of extraction and stripping reactions that can influence the overall transport rate, as discussed in refs. [30,46], A simple shortcut method for the design and simulation of two-phase HF contactors in MBSE and MBSS with the concentration dependent overall mass-transfer and distribution coefficients taking into account also reaction kinetics in L/L interfaces has been suggested [47]. 

Ariga et al. [48] have investigated the behavior of the monolith reactor in which Echerichia coli with P-galactosidase or Saccharomyces cerevisiae was immobilized within a thin film of K-carragcenan gel deposited on the channel wall. The effects of mass transfer resistance and axial dispersion on the conversion were studied. Those authors found that the monolith reactor behaved like the plug-flow reactor. The residence-time distribution in this reactor was comparable to four ideally mixed tanks in series. The influence of gas evolution on liquid film resistance in the monolith reactor was also investigated. It was shown that at low superficial gas velocities, the gas bubble may adhere to the wall, which decreases the effective surface area available for the reaction. The authors concluded that the reactor was very effective in the reaction systems accompanied by gas evolution, such as fermentations. 

In general, when molecules transport through the epithelium cell layer directly into the mesenteric blood draining the small intestine, the total mass transfer resistance may be described by the sum of resistance for barriers (aqueous boundary layer in front of the membrane, the membrane itself, and the aqueous boundary layer in blood side) in series ... 

In analogy with electrical conductance and as shown by Liss and Slater (1974) and as discussed by Mackay and Leinonen (1975) and Mackay (1991) the reciprocal of the mass transfer velocity is a measure of resistance to the gas transfer and is composed of two resistances in series, the liquid phase resistance, r , and the gas phase resistance, r ... 

Tertiary distributions apply when Laplace s equation is replaced by a series of n equations of the form (5.59) coupled with electroneutrality (5.62) where n represents the number of ionic species in the system. Thus, tertiary distributions relax the assumption that concentrations are uniform. Ohmic, kinetic, and mass-transfer resistances all play a role. 

Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, Diffusion-Reaction Interactions in Catalyst Pellets, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor r, can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction ... 

If mass transfer in the film and diffusion inside the pores are taken into account the effective mass transfer coefficient is given as a series connection of the internal (1/kpore) and external (l/kf,im) mass transfer resistance (Eq. 6.138) ... 

The water flux achieved in OD can be described in terms of an overall mass transfer co-efficient, K, and the water vapor pressure gradient between the bulk feed and strip streams [Eq. (10)]. The total resistance to mass transfer, given by l/K, is the sum of three separate resistances in series [Eq. (11)]. Here, l/Kf, l/Kj and l/K are the resistances imposed by the feed-side boundary layer, the membrane, and the strip-side boundary layer respectively. 

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