Y -factor for mass transfer

In defining a y-factor (jd) for mass transfer there is therefore good experimental evidence for modifying the exponent of the Schmidt number in Gilliland and Sherwood s correlation (equation 10.225). Furthermore, there is no very strong case for maintaining the small differences in the exponent of Reynolds number. On this basis, the y-factor for mass transfer may be defined as follows ... 

The mass- and heat-transfer coefficients kc and h can be written in terms of the Colburn y-factors for mass and heat transfer jo and jn, respectively ... 

This leads to rate equations with constant mass transfer coefficients, whereas the effect of net transport through the film is reflected separately in thej/gj and Y factors. For unidirectional mass transfer through a stagnant gas the rate equation becomes... 

Note Since the model is linear for the special case considered, the same equation is also satisfied by the other three variables.) The following observations may be made from Eq. (98) that expresses the dimensionless dispersion coefficient A (i) The first term describes dispersion effects due to velocity gradients when adsorption equilibrium exists at the interface. We note that this expression was first derived by Golay (1958) for capillary chromatography with a retentive layer, (ii) The second term corresponds to dispersion effects due to finite rate of adsorption (since this term vanishes if we assume that adsorption and desorption are very fast so that equilibrium exists at the interface), (iii) The effective dispersion coefficient reduces to the Taylor limit when the adsorption rate constant or the adsorption capacity is zero, (iv) As is well known (Rhee et al., 1986), the effective solute velocity is reduced by a factor (1 + y). (v) For the case of irreversible adsorption (y — oo and Da —> oo), the dispersion coefficient is equal to 11 times the Taylor value. It is also equal to the reciprocal of the asymptotic Sherwood number for mass transfer in a circular... 

Individual heat-transfer coefficient, W/m -°C or Btu/ft -h-°F Mass flux relative to a plane of zero velocity, kg mol/m -s or lb mol/ft -h J, Jg, of components A and B, respectively average value of component A, caused by turbulent action Colburn factor for heat transfer, hlCpG) Cgfifkfl, dimensionless Colburn j fector for mass transfer, (ky MlG lpD Y, dimensionless Overall mass-transfer coefficient in gas phase, kg mol/m -s-unit mole fraction or lb mol/ft -h-unit mole fraction Ky, new value in Example 21.5... 

Another correlation used for predicting rates of mass transfer in fixed and fluidized beds is that of Chu, Kalil and Wetteroth (1953). The y-factor for... 

Average transport coefficients for transfer between the bulk-fluid and particle surface can be correlated in terms of dimensionless groups that characterize the flow conditions. It is common practice to correlate experimental data in terms of y -factors. Usually, the mass transfer coefficient is obtained from the j factor for mass the heat transfer coefficient is obtained from j factor analogy. There have been many experimental studies of mass transfer in fixed-beds and summaries and analyses of the results are available (Whitaker 1972 Dwivedi and Upadhay 1977). For Reynolds numbers greater than 10, the following relationship (Dwivedi and Upadhay 1977) between jo and the Reynolds number represents available data ... 

The chemical method used to estimate the interfacial area is based on the theory of the enhancement factor for gas absorption accompanied with a chemical reaction. It is clear from Equations 6.22-6.24 that, in the range where y > 5, the gas absorption rate per unit area of gas-liquid interface becomes independent of the liquid phase mass transfer coefficient /cp, and is given by Equation 6.24. Such criteria can be met in the case of absorption with... 

The gassed condition is important in mass transfer calculations. In general, the ungassed- and gassed-power are related by the equation PG = P. The value of P is calculated from the power correlation for Rushton turbines [16]. The correction factor, y, is calculated from the following equations ... 

Wiebelt J.A., Ruo S.Y. (1963) Radiant-interchange configuration factors for finite right circular cylinder to rectangular plane. International Journal of Heat Mass Transfer 6, 143-146. 

Thus assumption of the same value for interfacial area in physical and chemical absorption leads to uncertainty, especially if the mass transfer coefficient is deduced from k a measured by physical absorption or desorption and from a in chemical absorption. The effective interfacial area in the case of a fast-reaction system where the absorbing capacity is increased by a chemical reactant is substantially larger than the effective interfacial area for physical absorption or desorption, as pointed out by Joosten and Danckwerts (JIO). These authors introduced a correction factor y, the ratio between the increase in liquid absorption capacity and the increase in mass transfer due to chemical reaction ... 

This example also shows the effects of mass- and enei y-transfer resistances within the catalyst pellet. The temperature increases toward the center of the pellet and increases the rate, but the oxygen concentration goes down, tending to reduce the rate. The global value of 49.8 x 10" is the resultant balance of both factors. Hence the net error in using the bulk conditions to evaluate the rate would be [(49.8 — 43.6)/49.8] (100), 12.5%. In this case the rate increase due to external and internal thermal effects more than balances the adverse effect of internal mass-transfer resistance. The procedure for calculating the effects of internal gradients on the rate is presented in Chap. 11. 

In order to achieve the best possible catalyst conversion efficiency at a constant volume, while minimizing the power drain due to excessive pressure drop through the converter, one would maximize the heat and mass transfer with respect to the pressure drop. In other words, in the graph shown in Figure 7 for 100% open frontal area, where the Heat Mass Transfer Factor is on the x-axis and the Pressure Drop Factor is on the y-axis, the slope of the curve should to be as shallow as possible. All of the channel shapes evaluated here tend to fall close to the same hne. However, as the open frontal area decreases from 100%, the Pressure Drop Factor increases while the Heat Mass Transfer Factor remains constant so that the relative attractiveness of some channel structures will be improved as the OFA is taken into account. 

Interstitial mass transfer in fixed beds is frequently a significant factor in adsorption dynamics. Sherwood et al. (1975) developed an equation for both gases and liquids that employs the Colbum-Chilton y-factor ... 

The effect of one-way diffusion in the gas film is to increase the mass-transfer rate for the gas film by the factor 1/(1 — y)j, as shown by Eq. (21.39), so the effective overall coefficient K a is somewhat larger than the normal value of K a ... 

Gamson et al have successfully used the y-factor method to cocrelate titeir experimental results for heat and mass transfer between a bed of granular solids and a gas stream. 

The mass transfer equation could have been extracted from the rectangular coordinate entry in Table B.ll of Bird et al. (2002, p. 851) by setting = vg and Vy = Vr. Except for the factor of sin 0, the equation of continuity could have been extracted from the rectangular coordinate entry in Table B.4 of Bird et al. (2002, p. 846) by setting = ve and Vy = Vr. Hence, the locally flat description of this problem is justified, and the only remaining influence of spherical coordinates is the factor of sin 9 in the equation of continuity. The boundary conditions for Ca(y, x) are repeated here for completeness ... 

There have been several studies of mass and heat transfer to wire screens, and the work by Shah and Roberts [19] covers the range of Reynolds numbers typical of commercial ammonia oxidation. They studied the decomposition of H2O2 on Ag or Pt screens of different mesh size and presented an empirical correlation for the jjj factor For 5 < Re, y < 245,... 

Ca, is the fluid reactant concentration in the pore, Rp the pore radius. D,p in this model may be a harmonic mean of the bulk and Knudsen diflusion coefficient with real geometries it would be a true effective difTusivity including the tortuosity factor and an internal void fraction. D p is an effective diffiisivity for the mass transfer inside the solid and is a correction factor accounting for the restricted availability of reactant surface in the region where the partially reacted zones interfere. For Jt(y) < LJ2 (shown in Fig. 4.5-2) or j>2 < J f e factor ( = 1 for L/ > R y) > L/2 or >i < y < yj the factor = 1 — (40/x) where tgB = (2/L) Ji (y) - (L/2) for y < yi the factor C 0, where R(y) is the radial position of the reaction front. It is clear from Eq. 4.S-1 that no radial concentration gradient of A is considered within the pore. 

Of course, no mathematical model is completely rigorous. Each mathematical model will always contain some empirical parts. As the degree of empiricism decreases, the model reliability increases. As an example, in the development of mathematical models for packed beds, the mass and heat transfer resistances are usually computed using the empirical y-factor correlations for mass and heat transfer between the solid surface and the bulk fluid. Having some of the components of the mathematical models as empirical relations does not make the model empirical it is only that some of the parameters involved in the quantitative description of the processes taking place within the system are determined from empirical relations obtained experimentally. 

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