Overall mass transfer coefficients, example

The values of the mass transfer coefficient will be different on each side of the boundary. For example, when a gas dissolves in a liquid, feg in the gas film will be different from k[ in the liquid film. However, the concentration at the interface is not always known and this leads to the use of overall mass transfer coefficients in conjunction with overall driving forces. The following argument shows how these are related to the individual film coefficients. 

Individual mass transfer coefficients of the solute (for example, titanium compound) can be experimentally determined in every layer and membrane. Thus, overall mass transfer coefficients of the feed ( fcp) and strip sides of the OHLM system may be calculated from Equations 13.1a, 13.1b, 13.2a, and 13.2b. 

It is the overall mass transfer coefficient, however, that ultimately controls the rate of removal of a substance by air stripping. For example, dichloroethane, which has a lower Henry s constant than trichloroethylene, has been found easier to remove by air stripping, owing to its higher mass transfer coefficient (10). The mass transfer coefficient for a specific substance in a specific air-stripping system may be calculated by (9) ... 

Equation 12.1.16 is based on the assumptions that may be considered constant over the height of the formation zone and that the total flux is zero. If the total flux is not zero the right-hand side of Eq. 12.1.16 must be multiplied by the bootstrap coefficient and the calculation of the overall mass transfer coefficient modified (see Example 12.1.2). 

When the microporous membrane is in the form of a hollow fiber (see Fig. 2.7), the interfacial areas on the two sides of the hollow fiber are different. The overall mass-transfer coefficient may be defined based on the surface area calculated using either the inside diameter (ID) or the outside diameter (OD) of the hollow fiber. For calculating an overall mass-transfer coefficient, the interfacial area should be based on the diameter where the aqueous-organic phase interface is located. Consider, for example, the aqueous feed and strip phases in hydrophobic fiber lumen (tube side) and organic LM phase on the shell side. The rate of solute extraction per unit fiber length with the aqueous-organic interface located on the fiber ID ... 

Equations (64 )-(66 ) are equal to those used by other researchers [3-20, 28, 35-37]. These authors obtained Eqs (64 )-(66 ) by considering the basic Stokes-Einstein equation. We obtained the same equations as a particular case from Eq. (6), based on kinetics of irreversible processes (nonequihb-rium thermodynamics). In Table 5.3 are presented examples of individual and overall mass-transfer coefficients, obtained at titanium(IV) ion transport through the BOI ILM system [1]. 

Overall mass-transfer coefficients Kp/p on the feed side and Kp/R on the strip side were calculated by Eqs (8) and (9). Dependence of cadmium, copper, and zinc overall mass-transfer coefficients on the feed, carrier, and strip solutions flow rates is shown in Table 6.3. Flow rates varied in the range 0.5—1.5 crn /s. As an example, this dependence on the feed flow rate variations is shown in Fig. 6.3. 

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... 

For mass transfer across the hollow-fiber membrane contactors described in Example 2.14, the overall mass-transfer coefficient based on the liquid concentrations, Kf, is given by (Yang and Cussler, 1986)... 

The transfer of QX from the aqueous phase to the organic phase (or of QY from the organic phase to the aqueous phase) can be quantified by a differential equation (Wang and Yang, 1991a), which when solved gives a simple correlation for calculating the overall mass transfer coefficient. For example, (or QY we can derive... 

Using the same data as in Example 10.4-1, calculate the overall mass-transfer coefficient K y, the flux, and the percent resistance in the gas and liquid films. Do this for the case of A diffusing through stagnant B. 

Ans- (a) z = 2.362 m (7.75 ft) Tower Height Using Overall Mass-Transfer Coefficient. Repeat Example 10.6-2, using the overall liquid mass-transfer coefficient K a to calculate the tower height. 

The concept of overall mass-transfer coefficients is in many ways similar to that of overall heat-transfer coefficients in heat-exchanger design. And, as is the practice in heat transfer, the overall coefficients of mass transfer, are frequently synthesized through the relationships developed above from the individual coefficients for the separate phases. These can be taken, for example, from the correlations of Chap. 3 or from those developed in later chapters for specific types of mass-transfer equipment. It is important to recognize the limitations inherent in this procedure [6]. 

The introductory Section 3.1.2.5 in Chapter 3 identifies the negative chemical potential gradient as the driver of targeted separation, and the relevant species flux expression is developed in Section 3.1.3.2 (see Example 3.1.9 also). Section 3.1.4 introduces molecular diffusion and convection and basic mass-transfer coefficient based flux expressions essential to studies of distillation and other phase equilibrium based separation processes. Section 3.1-5.1 introduces the Maxwell-Stefan equations forming the basis of the rate based approach of analyzing distillation column operation. After these fundamental transport considerations (which are also valid for other phase equilibrium based separation processes), we encounter Section 3.3.1, where the equality of chemical potential of a species in all phases at equilibrium is illustrated as the thermodynamic basis for phase equilibrium (Le. = /z ). Direct treatment of distillation then begins in Section 3.3.7.1, where Raouit s law is introduced. It is followed by Section 3.4.1.1, where individual phase based mass-transfer coefficients are reiated to an overall mass-transfer coefficient based on either the vapor or liquid phase. 

Although the relations given have been developed for any gas-liquid system, they are also valid for any vapor-liquid system. The only difference is that Raoult s law is to be used instead of Henry s law as the equilibrium relation between the phases. For example, consider transfer of species i from vapor phase to liquid phase, Xigb to Xub, with the overall mass-transfer coefficient being expressed in terms of the vapor phase, i.e. Kxg. 

At the beginning of this section, it was indicated that the route of overall mass-transfer coefficients was a convenient way of avoiding the difficulties created by the difficult-to-obtain interfacial concentrations Ca/ Pav Xau, etc. We have found it to be so. It is also possible to solve for the interfacial concentration Cab or JCAgi/ otc. from either equations (3.4.3) (3.4.4) if k i, k g and the equilibrium relation are known for given bulk concentrations. For example, from equation (3.4.3),... 

Equations (3.4.18) and (3.4.19) provide the desired relations between the individual phase mass-transfer coefficients kAw, kAo and the overall mass-transfer coefficients Kaw and Kao-Although these results are derived for transfer from the aqueous to the organic phase, they are equally valid for transfer in the opposite direction. As before, the overall mass-transfer resistance is the sum of the resistances of the individual phases, the aqueous and the organic. Further, under particular conditions, either the aqueous- or the organic-phase resistance controls. For example, if kao >> 1 (the solute prefers the organic phase strongly over the aqueous phase), then the aqueous phase resistance is in control ... 

Example 3.4.1 Calculate the overall mass-transfer coefficient for the extraction of diethylamine (A) from its dilute solution in water into toluene. The following data on this extraction have been obtained from Treybal (1963, p. 498) Ao = 0.735 kAw = 0.761b mol/hr-ft -(lb mol/ft ) = 0.1291b mol/hr-ft -(lb mol/tf). Calculate both ICaw and ICao in units of of cm/s. 

The outline of the presented material is as follows. The mass transfer that occurs at the NAPL-water interfaces at the pore scale is generally approximated using a linear model based on film theory. In extending this formulation to the representative elemental volume (REV) scale in porous media, it is necessary to define an overall mass transfer rate coefficient. The theoretical development of phenomenological models that are used to estimate these overall mass transfer coefficients is presented. Methods to up-scale these REV scale models to field scale are presented. A set of examples based on intermediate-scale laboratory tests is presented to demonstrate the use of these methods. 

Example 8.5-4 Overall mass transfer coefficients in a packed tower We are studying gas absorption into water at 2.2 atm total pressure in a packed tower containing Berl saddles. From earlier experiments with ammonia and methane, we believe that for both gases the mass transfer coefficient times the packing area per tower volume is 18 Ibmol/hrft for the gas side and 530 Ibmol/hrft for the liquid side. The values for these two gases may be similar because methane and ammonia have similar molecular weights. However, their Henry s law constants are different 75 atm for ammonia and 41,000 atm for methane. What is the overall gas-side mass transfer coefficient for each gas ... 

The overall mass transfer coefficient A often involves diffusion across interfaces. For example, in the lung, oxygen is transported from the air in the alveolus to the alveolus wall, across that wall, and into the blood. These different steps are often described as different mass transfer resistances. Thus, (1/A) is the overall resistance to mass transfer, and equals the sum of the other mass transfer resistances. If this idea is new or not completely clear, then you may wish to review Section 8.5. 

Example 11.2-2 Toxin removal vs. dialysate flow A hollow fiber dialyzer is 30 cm long, 3.8 cm in diameter, and contains a volume fraction of hollow fibers f of 0.2 which are 200 pm in diameter. The overall mass transfer coefficient k in these fibers is 3.6 10 " cm/ sec, and can be assumed independent of blood and dialysate flows. 

Thus the limiting resistance to mass transfer depends most critically on m. If m is much less than one, which is normally what we seek in an extraction solvent, then the rate of extraction will be controlled by the mass transfer in the raffinate phase. If m is much more than one, then the extraction rate will be dominated by mass transfer in the extract phase. This situation is different than that in absorption, where the mass transfer in the gas is usually so fast that it doesn t affect the overall mass transfer coefficient. We explore these ideas further in the following example. 

At the same time, the overall heat transfer coefficient is simpler than the overall mass transfer coefficient developed in Section 8.5. Both coefficients are related to a sum of resistances, but the mass transfer case also involves weighting factors that are often confusing. These factors relate the concentrations on different sides of the interface. In the heat transfer case, the interfacial temperature in, for example, the hot fluid at the wall equals the interfacial temperature of the solid wall in contact with the hot fluid. This equality means no weighting factors and a simpler mathematical form. 

When it is known that Hqg varies appreciably within the tower, this term must be placed inside the integr in Eqs. (5-277) and (5-278) for accurate calculations of hf. For example, the packed-tower design equation in terms of the overall gas-phase mass-transfer coefficient for absorption would be expressed as follows ... 

Solution of the required column height is achieved by integrating the two component balance equations and the heat balance equation, down the column from the known conditions Xi , yout and TLin, until the condition that either Y is greater than or X is greater than Xqui is achieved. In this solution approach, variations in the overall mass transfer capacity coefficient both with respect to temperature and to concentration, if known, can also be included in the model as required. The solution procedure is illustrated by the simulation example AMMON AB. 

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