Mass transfer rate assumptions

Bakowsld [B/ Chem. Eng., 8, 384, 472 (1963) 14, 945 (1969)]. It is based on tbe assumption that tbe mass-transfer rate for a component moving to tbe vapor phase is proportional to tbe concentration of tbe component in tbe liquid and to its vapor pressure. Also, tbe interfacial area is assumed proportional to liquid depth, and surface renewal rate is assumed proportional to gas velocity. The resulting general equation for binaiy distillation is... 

Comparison of Eq. (184) with Eq. (183) shows the effect of size distribution for the case of fast chemical reaction with simultaneous diffusion. This serves to emphasize the error that may arise when one applies uniform-drop-size assumptions to drop populations. Quantitatively the error is small, because 1 — is small in comparison with the second term in the brackets [i.e., kL (kD)112). Consequently, Eq. (184) and Eq. (183) actually give about the same result. In general, the total average mass-transfer rate in the disperser has been evaluated in this model as a function of the following parameters ... 

Ammonia is absorbed in a falling film of water in an absorption apparatus and the film is disrupted and mixed at regular intervals as it flows down the column. The mass transfer rate is calculated from the penetration theory on the assumption that all the relevant conditions apply. It is found from measurements that the muss transfer rate immediately before mixing is only 16 pet cent of that calculated from the theory anil the difference has been attributed to the existence of a surface film which remains intact and unaffected by the mixing process. If the liquid mixing process lakes place every second, what thickness of surface film would account for the discrepancy, ... 

Explain the basis of the penetration theory for mass transfer across a phase boundary. What arc the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed (Do not present a solution of the differential equal ion.) Obtain the age distribution function for the surface ... 

The convective heat transfer coefficient may be approximated as that due to heat transfer without the presence of mass transfer. This assumption is acceptable when the evaporation rate is small, such as drying in normal air, and for conditions of piloted ignition, since XL is typically small. Mass transfer due to diffusion is still present and can be approximated by... 

At any height z above the bottom of the bed, the mass transfer rate per unit time, on the assumption of piston flow of gas, is given by ... 

The same assumptions and equations used in developing an expression for kialob for single-screw extruders can be used to describe mass transfer rates in the type of corotating twin-screw extruder considered here. In this regard. Fig. 14 is relevant, and ki a/ob will be given by Eq. (53) ... 

The assumption of transfer by a purely turbulent mechanism in the Handlos-Baron model leads to the prediction that the internal resistance is independent of molecular diffusivity. However, such independence has not been found experimentally, even for transfer in well-stirred cells or submerged turbulent jets (D4). In view of this fact and the neglect of shape and area oscillations, models based upon the surface stretch or fresh surface mechanism appear more realistic. For rapid oscillations in systems with Sc 1, mass transfer rates are described by identical equations on either side of the drop surface, so that the mass transfer results embodied in Eqs. (7-54) and (7-55) are valid for the internal resistance if is replaced by p. Measurements of the internal resistance of oscillating drops show that the surface stretch model predicts the internal resistance with an average error of about 20% (B16, Yl). Agreement of the data for drops in liquids with Eq. (7-56) considerably improves if the constant is increased to 1.4, i.e.. 

The model provides a good approach for the biotransformation system and highlights the main parameters involved. However, prediction of mass transfer effects on the outcome of the process, through evaluation of changes in the mass transfer coefficients, is rather difficult. A similar mass transfer reaction model, but based on the two-film model for mass transfer for a transformation occurring in the bulk aqueous phase as shown in Figure 8.3, could prove quite useful. Each of the films presents a resistance to mass transfer, but concentrations in the two fluids are in equilibrium at the interface, an assumption that holds provided surfactants do not accumulate at the interface and mass transfer rates are extremely high [36]. 

A convention used in most literature on ozone mass transfer and in the rest of this book is to define the mass transfer coefficient as the one that describes the mass transfer rate without reaction, and to use the enhancement factor E to describe the increase due to the chemical reaction. Furthermore, the simplification that the major resistance lies in the liquid phase is used throughout the rest of the book. This is also based on the assumption that the mass transfer rate describes physical absorption of ozone or oxygen, since the presence of a chemical reaction can change this. This means that KLa - kLa and the concentration gradient can be described by the difference between the concentration in equilibrium with the bulk gas phase cL and the bulk liquid concentration cL. So the mass transfer rate is defined as ... 

A knowledge of the velocity profiles within falling films under various flow conditions would be of very great value, making it possible to calculate the rates of convective heat and mass transfer processes in flowing films without the need for the simplified models which must be used at present. For instance, the analyses of Hatta (H3, H4) and Vyazovov (V8, V9) indicate clearly the differences in the theoretical mass-transfer rates due to the assumption of linear or semiparabolic velocity profiles in smooth... 

With additional assumption of analogy between mass and heat transfer, which is valid for low mass transfer rate processes, similar approach has ever been proposed to model the particle—fluid heat transfer (Hou and Li, 2010). The overall heat transfer, however, may... 

State the assumptions made in the penetration theory for the absorption of a pure gas into a liquid. The surface of an initially solute-free liquid is suddenly exposed to a soluble gas and the liquid is sufficiently deep for no solute to have time to reach the far boundary of the liquid. Starting with Fick s second law of diffusion, obtain an expression for (i) the concentration, and (ii) the mass transfer rate at a time t and a depth y below the surface. 

Mathematical models for mass transfer at the NAPL-water interface often adopt the assumption that thermodynamic equilibrium is instantaneously approached when mass transfer rates at the NAPL-water interface are much faster than the advective-dispersive transport of the dissolved NAPLs away from the interface [28,36]. Therefore, the solubility concentration is often employed as an appropriate concentration boundary condition specified at the interface. Several experimental column and field studies at typical groundwater velocities in homogeneous porous media justified the above equilibrium assumption for residual NAPL dissolution [9,37-39]. 

The flow is hydrodynamically fully developed at the entrance of the reactor. As was stated in Section II, this assumption results in a negligible error in the prediction of the pressure drop, and therefore the influence of the flow development on the mass transfer rate is also negligible. Because the flow is fully developed, the gas velocity will only have one component, namely, in the axial (z-) direction, u., ... 

The enhancement of the mass transfer rate in the first part of the reactor, resulting from the developing concentration profile, is negligible. It was stated in Section III that for relative pitches larger than 1.1 this assumption is valid, and that even for smaller relative pitches the effect of the enhanced mass transfer rate is small, except when film layer mass transfer is the limiting factor in the reactor performance. 

According to Sirkar s assumption [84], diffusion mass transfer rate through a membrane having a solvent-filled pore (hydrophobic), or an aqueous solution-filled pore (hydrophilic or ion exchange) may be expressed through the diffusion coefficients of the solute in the respective interface layers ... 

The simplification in Eqs. (2) and (3) is equivalent to saying that k is much smaller than the mass transfer rate D/5, which is of the order of 5s. From Chapter 1, this implies that the R/P redox couple gives a chemically reversible cyclic voltammogram at a scan rate corresponding to the same mass transfer rate. From a dimensionless analysis and Table 5 in Chapter 1, this corresponds to a scan rate v such as Fv/RT D/5", that is, v of the order 0.lVs Thus, observation for identical or close experimental conditions of a chemically reversible cyclic voltammogram at v 0.1 V s" for the redox couple of interest is sufficient to prove the validity of the assumptions leading to the homogeneous equivalent rate laws in Eqs. (7) and (8),... 

To derive the overall kinetics of a gas/liquid-phase reaction it is required to consider a volume element at the gas/liquid interface and to set up mass balances including the mass transport processes and the catalytic reaction. These balances are either differential in time (batch reactor) or in location (continuous operation). By making suitable assumptions on the hydrodynamics and, hence, the interfacial mass transfer rates, in both phases the concentration of the reactants and products can be calculated by integration of the respective differential equations either as a function of reaction time (batch reactor) or of location (continuously operated reactor). In continuous operation, certain simplifications in setting up the balances are possible if one or all of the phases are well mixed, as in continuously stirred tank reactor, hereby the mathematical treatment is significantly simplified. 

The explicit method of Krishna (1979d, 1981b) is most successful if the are close together and, therefore (or for other reasons), the total flux is low. At high rates of mass transfer, the assumption of constant (or of [/3][B] ) is a poor one, particularly in... 

In Section 12.2.2 we derived an expression that allows us to calculate the average molar fluxes in a vertical slice of froth on a tray under the assumptions that the vapor rises through the froth in plug flow and the liquid in the vertical slice is well mixed. Extend the treatment and derive an expression for the average mass transfer rates for the entire tray if the liquid is in plug flow. Some clues as to how to proceed may be found in Section 13.3.3. 

The instantaneous mass transfer rate is expressed as a function of time. In order to calculate an average mass transfer coefEcient we need to average the instantaneous coefEcient over the total exposure time period. To do this we need to know the age distribution function, which represents the fraction of elements having ages between t and t + dt t the surface. In the penetration theory, it is assumed that all the elements reside at the interface for a time period of the same length. As a consequence of this assumption the age distribution function is [6] ... 

According to assumptions 1-4, the overall mass-transfer rate may be derived through the sums of individual step resistances and measured bulk phase concentrations [Sp], [Sg], (for BLM), and [Sr], (for details, see Section 2.4.3). 

A check on the column height may be carried out based on mass transfer rate. This requires the availability of mass transfer coefficient data. Depending on appropriate assumptions. Equation 15.22 or 15.24, or their equivalent counterparts may then be used to calculate the column height, as discussed in Section 15.3. 

The assumption that stages in a distillation column are in equilibrium allows calculations of concentrations and temperatures without detailed knowledge of flow patterns and heat and mass transfer rates. This assumption is a major simplification. 

Assuming constant molar overflow, LjV is constant and the assumption of equimolar counterdiffusion is valid, so that the flux of one component across the vapor-liquid interface is equal and opposite to the flux of the other component = -Nb)-For a diffeential height dz in a packed column, the mass transfer rate is ... 

It was previously indicated that the analysis of the previous sections of this chapter could apply equally well to heat transfer problems or to the single-solute mass transfer problem with 0 interpreted as the dimensionless concentration, 9 = (c — Coo)/(co — Coo), with c representing the mass fraction of solute and the Schmidt number substituted for the Prandtl number. The key assumption in this assertion is that the interfacial velocity generated that is due to the transfer of solute to or from the body surface is small enough to play a neghgible role in both the fluid motion and as a convective contribution to the mass transfer rate. In this section, we consider how the problem changes if this assumption is not satisfied.4... 

Asai et al. (1994) have developed a reaction model for the oxidation of benzyl alcohol using hypochlorite ion in the presence of a PT catalyst. Based on the film theory, they develop analytic expressions for the mass-transfer rate of QY across the interface and for the inter-facial concentration of QY. Recently, Bhattacharya (1996) has developed a simple and general framework for modeling PTC reactions in liquid-liquid systems. The uniqueness of this approach stems from the fact that it can model complex multistep reactions in both aqueous and organic phases, and thus could model both normal and inverse PTC reactions. The model does not resort to the commonly made pseudo-steady-state assumption, nor does it assume extractive equilibrium. This unified framework was validated with experimental data from a number of previous articles for both PTC and IPTC systems. 

To increase the mass transfer rate, Tokuda et al. [7] carried out normal and differential pulse voltammetry at micropipettes and extracted the rate constant values within the range from 0.009 to 0.2 cm/s for facilitated transfers of Li+, Na+, Ca2+, Sr2+, and Ba2+ to nitrobenzene (NB) with two different crown ethers (DB18C6 and DB24C8). The assumption of a = 0.5 for all IT reactions and the use of TR-drop compensation may have affected the accuracy of those results. The upper limit for the measurable rate constant was about 0.5 cm/s, too slow to probe facilitated transfer of potassium ions. 

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