External mass transfer correlations

TABLE 16-9 Recommended Correlations for External Mass Transfer Coefficients in Adsorption Beds (Re = evdp/v. Sc = v/D)... 

FIG. 16"10 Sherwood mimher correlations for external mass-transfer coefficients in packed beds for e = 0.4 (adapted from Siiziild, gen. refs.). 

Correlations of heat and mass-transfer rates are fairly well developed and can be incorporated in models of a reaction process, but the chemical rate data must be determined individually. The most useful rate data are at constant temperature, under conditions where external mass transfer resistance has been avoided, and with small particles... 

The determination of the external mass transfer coefficient of CO, k, (see Equation 3) deserves brief comments. Since the complex geometry and flow characteristics in the reactor cell precluded a reliable estimation of k based on correlations given in the literature, the CO oxidation activities of the catalyst... 

In order to solve the mathematical model for the emulsion hquid membrane, the model parameters, i. e., external mass transfer coefficient (Km), effective diffu-sivity (D ff), and rate constant of the forward reaction (kj) can be estimated by well known procedures reported in the Hterature [72 - 74]. The external phase mass transfer coefficient can be calculated by the correlation of Calderback and Moo-Young [72] with reasonable accuracy. The value of the solute diffusivity (Da) required in the correlation can be calculated by the well-known Wilke-Chang correlation [73]. The value of the diffusivity of the complex involved in the procedure can also be estimated by Wilke-Chang correlation [73] and the internal phase mass transfer co-efficient (surfactant resistance) by the method developed by Gu et al. [75]. 

In trickle beds, the gas-to-liquid, kigaGL, and liquid-to-particle, kfaLS, coefficients are used to represent the effect of the external mass transfer resistances. The interfacial areas aGl and <2ls refer to the effective mass transfer surface per unit volume of empty reactor. Due to the fact that the coefficients kig and klL cannot be easily estimated independently from the corresponding interfacial areas aGL and aLS respectively, by simple experimental techniques, correlations are normally reported for the products kigaGL and k,a]S (Smith, 1981). 

To evaluate the rate coefficient km, we need the external mass transfer coefficient. Using the following correlation (eq. (3.352)) ... 

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. 

In the cases above, a two-parameter model well represents the data. A model with more parameters would be more flexible, but by using a partition constant, K, or a desorption rate constant ka and k, , for the mass-transfer coefficients, the data are well described (see Figs. 3.4-15 and 3.4-13). While K would be a value experimentally determined, kp can be estimated from eqn. (3.4-97) with the external mass-transfer coefficient, km, estimated from the correlation of Stiiber et al. [25] or from that of Tan et al. [27], and the effective diffusivity from the Wakao Smith model [36], Typical values of kp obtained by fitting the data of Tan and Liou are shown in Fig. 3.4-16. As expected, they are below the usual mass-transfer correlations, because internal resistance diminishes the global mass transfer coefficient. These data correspond to the regeneration of spent activated carbon loaded with ethyl acetate, using high-pressure carbon dioxide, published by Tan and Liou [45]. 

The mass transfer factor has also been correlated as a function of the Reynolds number only and thus taking account only of hydrodynamic conditions. If e is the voidage of the packed bed and the total volume occupied by all of the catalyst pellets is Vp, then the total reactor volume is Vp/(l - e). Hence the rate of mass transfer of component A per unit volume of reactor is NASx(l - e)/Vp. If we now consider a case in which only external mass transfer controls the overall reaction rate we have ... 

External mass transfer rates are generally correlated in terms of a linear driving force expressions,... 

Solution of the shrinking core model at zero time (t=0) depends only on two parameters the solubility of solute in SC CO2 and the external particle to fluid mass transfer coefficient Kq. Hence, knowing the solubility, measurements of the initial extraction rates allow to determine the values of K(j. Detailed discussion on the evaluated mass transfer coefficients are given in [7].These authors found that the overall mass transfer from particles to fluid depends upon both free and forced convection mechanism. Figure 2 illustrates a parity plot of die experimental values of Sh number (evaluated by zero-time solution of the shrinking core model) and the calculated Sh number (using an appropriate mass transfer correlation). 

Figure 2. Correlation between external mass transfer coefficient and agitation rate...

To calculate the external mass transfer coefficient, the Thoenes—Kramers correlation is used. 

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

A plot of H/(2mq) versus 1/Ug is a straight line with a slope equal to Di and an ordinate equal to 3f + S )/Sq. The coefficient of external mass transfer is estimated using one of the several correlations available for it (see Chapter 5, subsection 5.2.5, correlation of Wilson and Geankoplis [62], Kataoka et al. [87], or the penetration theory [88]). Correcting for the contribution due to the external mass transfer resistance gives the last term in the plate height equation, 5, hence the intraparticle diffusion coefficient, Dg. 

The number of transfer units for each mechanism can be estimated from known parameters and mass transfer correlations (4). For example, for a column with particles 0.01 cm in diameter, a superficial velocity of 0.01 cm/sec, and a solute bulk diffusivity of 7 x 10-7 cm2/sec, the estimated number of transfer units in a packed bed of length L for the four mechanisms, axial dispersion, external fluid film mass transfer, pore diffusion, and solid homogeneous particle diffusion,are... 

Although funher discussion of heat transfer correlations is no doubt worthwhile, it will not help us to determine the mass transfer coefficient and the mass flux from the bulk fluid to the external pellet surface. However, the preceding discussion on heat transfer was not entirely futile, because, for similar geometries, the heat and mass transfer correlations are analogous. If a heat transfer correlation for the Nusselt number exists, the mass transfer coefficient can be estimated by replacing the Nusselt and Prandtl numbers in this correlation by the Sherwood and Schmidt numbers, respectively ... 

The reactant molecules diffuse from the bulk of the fl iiid phase to the surface of the catalyst pellets. This process is usually described by mass transfer rate over a hypothetical external mass transfer resistance. The mass transfer coefficient is usually calculated using J-factor correlations (e.g. Hill, 1977). This step is dependent upon the properties of the gas mixture and the flow conditions around the catalyst pellets as well as the size and shape of the catalyst pellets. This step is usually referred to as external mass transfer of reactant molecules. 

External mass transfer, such as diffusion to particles or to the outside of pipes or cylinders, requires different correlations from those for internal mass transfer, because there is boundary-layer flow over part of the surface, and boundary-layer separation is common. The mass-transfer coefficients can be determined by studying evaporation of liquid from porous wet solids. However, it is not easy to ensure that there is no effect of internal mass-transfer resistance. Complications from diffusion in the solid are eliminated if the solid is made from a slightly soluble substance that dissolves in the liquid or sublimes into a gas. This method also permits measurement of local mass-transfer coefficients for different points on the solid particle or cylinder. 

FLOW OUTSIDE TUBES PARALLEL TO AXIS. Some membrane separators have bundles of hollow fibers in a shell-and-tube arrangement with liquid or gas flowing parallel to the tube axis on the outside of the tubes. The external flow passages are irregular in shape and not uniform, since the fibers are not held in position as are the tubes in a heat exchanger. Empirical correlations for the external mass-transfer coefficient have been proposed using an equivalent diameter to calculate the Reynolds number. For a bundle of fibers with diameter d packed in a shell with c void fraction, the equivalent diameter is... 

When the internal mass transfer/reaction step is rate limiting, an effectiveness factor, I , is usually introduced related to dimensionless parameters characteristic of the reacting system as a Thiele modulus.109 It is worthwhile noting that most of the available correlations are based upon theoretical models assuming diffusion as the only mass transfer pattern. Hence, effects related to external mass transfer resistances are neglected. 

Reactors.- Stirred-type reactors are not common in experiments correlating selectivity to deactivation high flow rates are used to ensure the absence of external mass transfer resistance. Of the bed-type reactors, a number of flow patterns are used. 

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