Evaluation of the Mass-Transfer Coefficient

It is clear from Equation 10 that the UF flux is determined largely by the mass-transfer coefficient (K). With proteins and other polyelectrolytes, we can modify the gel concentration (Cg) by altering the pH and/or ionic strength of the medium. However, for most process streams, that option is not permissable. Therefore, the optimization of flux is largely effected through the parameters that effect the mass transfer coefficient. 

The mass transfer-heat transfer analogies well known in the chemical engineering literature make possible an evaluation of the mass transfer coefficient (K) and provide insight into how membrane geometry and fluid-flow conditions can be specified to optimize flux.24 

Laminar Flow. The Graetz or Leveque solutions25 26 for convective heat transfer in laminar flow channels, suitably modified for mass transfer, may be used to evaluate the mass transfer coefficient where the laminar parabolic velocity profile is assumed to be established at the channel entrance but where the concentration profile is under development down the full length of the channel. For all thin-channel lengths of practical interest, this solution is valid. Leveque s solution26 gives  

Sherwood Number = Kdh/D Reynold s Number = Udh/i Schmidt Number = v/D equivalent hydraulic diameter channel length average velocity of fluid kinematic viscosity = ju/p viscosity of fluid density of fluid 

A review of Equations 14-18 shows that the flux (or mass transfer coeffi 

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Shah (1979) evaluation of the mass transfer coefficients in various types of equipment. Also experimental devices for finding these coefficients. 

When the surface reaction is fest, k determines the overall kinetics, and since mass transfer is a first-order process, this dependence is observed instead of the true order of the surface reaction. This is a serious mistake if intrinsic kinetic data are sought, but if the intrinsic kinetics is already known, this strategy can be used for an experimental evaluation of the mass transfer coefficient, km-... 

Instead of using a constant value of the mass transfer coefficient k at each pressure given by Harwell (Walley et al., 1973), Levy and Healzer (1980) developed an entrainment parameter p. GF and P are evaluated by solving the following two equations simultaneously ... 

Experimental determination of the mass transfer coefficient is based on the appropriate mass balance on the specific reactor used (Figure B 1-2). The simpler the reactor system is, the simpler the mass balance model for evaluating the experimental results can be. For example, if mixing in the reactor deviates too far from ideality, kL is no longer uniform throughout the reactor. Neither method as described below can then be used. Instead a more complicated model of the mixing zones in the reactor would be necessary (Linek, 1987 Stockinger, 1995). 

Cas-Liquid Mass Transfer Gas-liquid mass transfer normally is correlated by means of the mass-transfer coefficient K a versus power level at various superficial gas velocities. The superficial gas velocity is the volume of gas at the average temperature and pressure at the midpoint in the taiik divided by the area of the vessel. In order to obtain the partial-pressure driving force, an assumption must be made of the partial pressure in equihbrium with the concentration of gas in the liquid. Many times this must be assumed, but if Fig. 18-26 is obtained in the pilot plant and the same assumption principle is used in evaluating the mixer in the full-scale tank, the error from the assumption is limited. 

Several empirical correlations are available for the evaluation of external mass transfer coefficients [74,98]. 

The above derivatives must be evaluated at the interface composition before use in computing the Jacobian elements. This additional complexity in evaluating the derivatives of the vapor-phase mass transfer rate equations arises because we have used mass fluxes and mole fractions as independent variables. If we had used mass fractions in place of mole factions the derivatives of the rate equations would be simpler, but the derivatives of the equilibrium equations would be more complicated. For simplicity, we have ignored the dependence of the mass transfer coefficients themselves on the mixture composition and on the fluxes. 

Initially, we investigated the influence of the mass transfer coefficient, k, on the permeate flux, assuming all the activity coefficients were unity. However, as shown in Fig. 4.8A, for 0.21 M TOABr in toluene, (dashed lines), the data could not be described in this way, no matter what the mass transfer coefficient values were. Even when the mass transfer coefficient value -> oo (line 4 on Fig. 4.8 A), the model predicts an osmotic pressure of around 6 bar, at a concentration of 0.21 M, which is not observed experimentally. Activity differences could be responsible for this difference. Further evaluation of the influence of model parameters confirmed that the activity coefficient of toluene in the boundary layer does indeed have an important effect in this system. Even a very small change in the activity coefficient has a significant effect on the permeate flux. 

The characterisation of mass transfer is essential for the design of the micro-reactor. In liquid-liquid flows most studies have focused on the estimation of overall mass transfer coeflicients, while no model based on theory has been developed so far. The overall volumetric mass transfer coeflicient (kua) is a characteristic parameter of a system used to evaluate the performance of the contactors, and is a combination of the mass transfer coefficient (kp), which depends mainly on the difiusivity of solute, characteristic diffusion length and interfacial hydrodynamics, and of the specific interfacial area (a), which depends on the flow pattern. The prediction of the overall volumetric mass transfer coeflicient remains difficult due to secondary phenomena, tike interfacial instabilities. 

It was further shown that individual transport coefficients could be combined into overall mass transfer coefficients to represent transport across adjacent interfacial layers. The underlying concept is referred to as two-film theory. Chapter 1 has been confined to simple applications of the mass transfer coefficient which is either assumed to be known, or is otherwise evaluated numerically in simple fashion. 

Two sources of error, which may have affected the accuracy of the results reported in refs. 13-15, were identified in later studies. One of them is the lack of silanization of the outer pipet wall. The formation of a thin aqueous film on the hydrophilic glass surface may have resulted in the true ITIES area significantly larger than that evaluated from the diffusion limiting current (see Section 1.2.2.2). This should result in overestimated values of the mass-transfer coefficient and standard rate constants calculated from the dimensionless parameter X = k /m. Another source of error— the uncertainty in fitting experimental IT voltammograms to the theory—is discussed below. 

Some of the exponents in Eqs. (50) and (52) can be evaluated from experimental data. For example, Calderbank and Moo-Young (C4) investigated several chemical systems and found that the mass-transfer coefficient per unit area was a function of the Schmidt number to the power of from 0.50 to 0.67 this would also be the value of B6. In addition, they found that agitation had no effect on KL therefore, s is equal to zero. 

It was shown later that a mass transfer rate sufficiently high to measure the rate constant of potassium transfer [reaction (10a)] under steady-state conditions can be obtained using nanometer-sized pipettes (r < 250 nm) [8a]. Assuming uniform accessibility of the ITIES, the standard rate constant (k°) and transfer coefficient (a) were found by fitting the experimental data to Eq. (7) (Fig. 8). (Alternatively, the kinetic parameters of the interfacial reaction can be evaluated by the three-point method, i.e., the half-wave potential, iii/2, and two quartile potentials, and ii3/4 [8a,27].) A number of voltam-mograms obtained at 5-250 nm pipettes yielded similar values of kinetic parameters, = 1.3 0.6 cm/s, and a = 0.4 0.1. Importantly, no apparent correlation was found between the measured rate constant and the pipette size. The mass transfer coefficient for a 10 nm-radius pipette is > 10 cm/s (assuming D = 10 cm /s). Thus the upper limit for the determinable heterogeneous rate constant is at least 50 cm/s. 

The RHSE has the same limitation as the rotating disk that it cannot be used to study very fast electrochemical reactions. Since the evaluation of kinetic data with a RHSE requires a potential sweep to gradually change the reaction rate from the state of charge-transfer control to the state of mass transport control, the reaction rate constant thus determined can never exceed the rate of mass transfer to the electrode surface. An upper limit can be estimated by using Eq. (44). If one uses a typical Schmidt number of Sc 1000, a diffusivity D 10 5 cm/s, a nominal hemisphere radius a 0.3 cm, and a practically achievable rotational speed of 10000 rpm (Re 104), the mass transfer coefficient in laminar flow may be estimated to be ... 

The mass transfer coefficient can be found along with other constants from appropriate rate data, or it can be evaluated from an independent known correlation of mass transfer data, of which several are available. In most of... 

The expression for the enhancement factor E, eq. (35), has first been derived by van Krevelen and Hof-tijzer in 1948. These authors used Pick s law for the description of the mass transfer process and approximated the concentration profile of component B by a constant Xb, over the entire reaction zone. It seems worthwhile to investigate whether the same equation can be applied in case the Maxwell-Stefan theory is used to describe the mass transfer process. To evaluate the Hatta number, again an effective mass transfer coefficient given by eq. (34), is required. The... 

Now some expressions for the quantities involved in the mass transfer coefficient will be obtained [96]. The velocity at the interface is evaluated by using the following two approximations (1) The x velocity component, although dependent on x, is replaced by its average U over the path of length x0. (2) The velocity distribution in the vicinity of the interface is approximated for each path of length x by that valid for the motion of a plate moving with the velocity V over a semiinfinite liquid whose velocity is E7S. 

Then, to evaluate the intrinsic reaction coefficient, we need the mass transfer coefficient in the liquid film, which can be calculated by means of the correlation of Dharwadkar and Sylvester (fw = 1) (eq. (3.433)) ... 

Model application in the pulsing-flow regime The mass transfer coefficient in the liquid-solid film is evaluated by means of the Dhai wadkar and Sylvester correlation (eq.3.433), and is found to be 0.45 s. Then, the several parameters of the model eq. (5.379) are shown in Table 5.18. 

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]. 

The Sherwood number can be determined from the solution of the nondimensional problem by evaluating the nondimensional mass-fraction gradients at the channel wall and the mean mass fraction, both of which vary along the channel wall. With the Sherwood number, as well as specific values of the mass flow rate, fluid properties, and the channel geometry, the mass transfer coefficient hk can be determined. This mass-transfer coefficient could be used to predict, for example, the variation in the mean mass fraction along the length of some particular channel flow. 

Evaluate the mass transfer by determining the mass transfer coefficient for a range of operating conditions using the same water that will be oxidized later if possible. 

Most of the properties change somewhat from one end to the other of industrial columns for effecting separations, so that the mass transfer coefficients likewise vary. Perhaps the property that has the most effect is the mass rate of flow which appears in the Reynolds number. Certainly it changes when there is a substantial transfer of material between the two phases in absorption or stripping and even under conditions of constant molal overflow in distillation processes, the mass rate of flow changes because of differences of the molecular weights of the substances being separated. As a practical expedient, however, mass transfer coefficients are evaluated at mean conditions in a column. 

As the concentration of EtOH increased from 0 to 10%, the effective steady-state mass transfer coefficient declined from 0.17 1/hr to 0.11 1/hr, which was due in part to the change in Darcy velocity. Using correlations developed for Ke,ss as a function of Darcy velocity and alcohol concentration (Taylor, 1999), the effect of EtOH concentration can be evaluated at a single, representative Darcy velocity. For example, using a Darcy velocity of 4.0 cm/hr, the value of Ke,ss would be 0.14, 0.13, and 0.13 for 4% Tween 80, 4% Tween 80 + 5% EtOH and 4% Tween 80 + 10% EtOH, respectively. Thus, the addition of EtOH to 4% Tween 80 had no discemable influence on the effective steady-state mass transfer coefficient. It should be recognized, however, that although the mass transfer coefficient remained essentially unchanged, the steady-state concentration of PCE in the column effluent (C") and the cumulative PCE mass recovery increased substantially as a result of EtOH addition (Table 2). This behavior can be explained by the fact that the equilibrium solubility of PCE (C" sat) increased by more than 50%, from 26,900 mg/L to 42,300 mg/L, with the addition of 10% EtOH. 

It is clear from Fig. 16.32 that for the quantitative evaluation of convective mass transfer, the product of the two quantities - mass transfer surface and mass transfer coefficient - is crucial. As a result, a constant product of mass transfer coefficient and mass transfer surface under the conditions of a constant fluidization mass flow has a constant number of transfer units (NTU). 

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