PROCESS FUNDAMENTALS AND BASIC CALCULATION METHODS
RATE-BASED CALCULATIONS
This section reviews the basics of the mass-transfer coefficient and mass-transfer unit approaches to modeling extraction performance. These methods have been used for many years and continue to provide a useful basis for the design of extractors and extraction processes. Additional discussions of these and other rate-based methods are given in the books edited by Godfrey and Slater [Liquid-Liquid Extraction Equipment (Wiley, 1994)] and by Thornton [Science and Practice of Liquid-Liquid Extraction, vol. 1 (Oxford, 1992)]. For discussions of more mechanistic methods that include characterization of drop breakage and coalescence rates, drop size distributions, and drop pop- ulation balances, see Leng and Calabrese, Chap. 12 in Handbook of Industrial Mixing, Paul, Atiemo-Obeng, and Kresta, eds. (Wiley, 2004); Goodson and Kraft, Chem. Eng. Sci., 59, pp. 3865–3881 (2004); Attarakih, Bart, and Faqir, Chem. Eng. Sci., 61, pp. 113–123 (2006); and Schmidt et al., Chem. Eng. Sci., 61, pp. 246–256 (2006). These methods are the subject of current research. Also see the discussion of general approaches to analyzing dispersed-phase systems given by Ramkrishna, Sathyagal, and Narsimhan [AIChE J., 41(1), pp. 35–44 (1995)]. For dis- cussions of the effect of contaminants on mass-transfer rates, see Saien et al., Ind. Eng. Chem. Res., 45(4), pp. 1434–1440 (2006); and Dehkordi et al., Ind. Eng. Chem. Res., 46(5), pp. 1563–1571 (2007).
Solute Diffusion and Mass-Transfer Coefficients For a binary system consisting of components A and B, the overall rate of mass transfer of component A with respect to a fixed coordinate is the sum of the rates due to diffusion and bulk flow:
NA= −DAB + NA (15-58)
where NA= flux for component A (moles per unit area per unit time)
DAB= mutual diffusion coefficient of A into B (area/unit time)
z= dimension or direction of mass transfer (length) C= total concentration of A and B (mass or mole per unit
volume)
CA= concentration of A (mass or mole per unit volume)
Equation (15-58) is written for steady-state unidirectional diffusion in a quiescent liquid, assuming that the net transfer of component B is negligible. For transfer of component A across an interface or film between two liquids, it may be rewritten in the form
NA= (CA− CiA) (15-59)
where (1 − xA)m= mean mole fraction of component B
Ci
A= concentration of component A at interface CA= concentration of component A in bulk
For steady-state counter diffusion where NA+ NB= 0, the flux equa-
tion simplifies to NA=DAB(CA− CiA) (15-60) ∆z DAB ∆z(1 − xA)m CA C ∂CA ∂z Zt N
The flux also may be written in terms of an individual mass-transfer coefficient k
NA= k(CA− CiA) (15-61)
where k= (15-62)
In Eqs. (15-58) to (15-62), the flux is expressed in terms of mass or moles per unit area per unit time, and the concentration driving force is defined in terms of mass or moles per unit volume. The units of the mass-transfer coefficients are then length per unit time. Other definitions of the flux and resulting mass-transfer coefficients also are used. When mass-trans- fer coefficients are used, it is important to understand their definition and how they were determined; they need to be used in the same way in any subsequent calculations. Additional discussion of mass- transfer coeffi- cients and mass-transfer rate is given in Sec. 5. Also see Laddha and Degaleesan, Transport Phenomena in Liquid Extraction (McGraw-Hill, 1978), Chap. 3; Skelland, Diffusional Mass Transfer (Krieger, 1985); Skel- land and Tedder, Chap. 7 in Handbook of Separation Process Technology, Rousseau, ed. (Wiley, 1987); Curtiss and Bird, Ind. Eng. Chem. Res., 38(7), pp. 2515–2522 (1999); and Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed. (Wiley, 2002). Available correlations of molecular dif- fusion coefficients (diffusivities) are discussed in Sec. 5 and in Poling, Prausnitz, and O’Connell, The Properties of Gases and Liquids, 5th ed. (McGraw-Hill, 2000). The prediction of diffusion coefficients is discussed by Bosse and Bart, Ind. Eng. Chem. Res., 45(5), pp. 1822–1828 (2006).
Mass-Transfer Rate and Overall Mass-Transfer Coefficients In transferring from one phase to the other, a solute must overcome cer- tain resistances: (1) movement from the bulk of the raffinate phase to the interface; (2) movement across the interface; and (3) movement from the interface to the bulk of the extract phase, as illustrated in Fig. 15-25. The two-film theory first used to model this process [Lewis and Whitman, Ind. Eng. Chem., 16, pp. 1215–1220 (1924)] assumes that motion in the two phases is negligible near the interface such that the entire resistance to transfer is contained within two laminar films on each side of the inter- face, and mass transfer occurs by molecular diffusion through these films. The theory further invokes the following simplifying assumptions: (1) The rate of mass transfer within each phase is proportional to the difference in concentration in the bulk liquid and the interface; (2) mass-transfer resis- tance across the interface itself is negligible, and the phases are in equi- librium at the interface; and (3) steady-state diffusion occurs with negligible holdup of diffusing solute at the interface. Within a liquid- liquid extractor, the rate of steady-state mass transfer between the dis- persed phase and the continuous phase (mass or moles per unit time per unit volume of extractor) is then expressed as
RA= = kda(Cd,i− Cd)= kca(Cc− Cc,i) (15-63)
where Ci= concentration at interface (mass or moles per unit volume)
C= concentration in bulk liquid (mass or moles per unit volume) kc= continuous-phase mass-transfer coefficient (length per
unit time)
kd= dispersed-phase film mass-transfer coefficient (length
per unit time)
a= interfacial area for mass transfer per unit volume of extractor (length−1)
Subscripts d and c denote the dispersed and continuous phases. The concentrations at the interface normally are not known, so the rate expression is written in terms of equilibrium concentrations assuming that the rate is proportional to the deviation from equilibrium:
RA= = koda(Cd∗− Cd)= koca(Cc− Cc∗) (15-64)
where the superscript * denotes equilibrium, and kocis an overall mass-
transfer coefficient given by
= + (15-65)
Continuous Dispersed phase resistance phase resistance
1 m dc volk d 1 k c 1 k oc dC dt dC dt DAB ∆z(1 − x A)m
{
{
Similarly, the overall mass-transfer coefficient based on the dispersed phase is given by
= + (15-66)
Dispersed Continuous phase resistance phase resistance
Assuming mass-transfer coefficients are constant over the range of conditions of interest, Eq. (15-64) may be integrated to give
= exp(−kocaθ) ≈ (15-67)
whereθ is the contact time.
In Eqs. (15-65) and (15-66), mdcvol= dCd⁄dCcis the local slope of the
equilibrium line, with the equilibrium concentration of solute in the dispersed phase plotted on the ordinate (y axis), and the equilibrium concentration of solute in the continuous phase plotted on the abscissa (x axis). Note that mdcvolis expressed on a volumetric basis
(denoted by superscript vol), i.e., in terms of mass or mole per unit volume, because of the way the mass-transfer coefficients are defined. The mass-transfer coefficients will not necessarily be the same for each solute being extracted, so depending upon the application, mass- transfer coefficients may need to be determined for a range of differ- ent solutes. As noted earlier, other systems of units also may be used as long as they are consistently applied.
The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k-Dn. For two-film theory, n = 1 as discussed
above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460–1467 (1951)] allow for surface renewal or pen- etration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0; the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with inter- nal circulation, or as oscillating drops. These theories are discussed by Skelland [“Interphase Mass Transfer,” Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)].
In the design of extraction equipment with complex flows, mass- transfer coefficients are determined by experiment and then corre- lated as a function of molecular diffusivity and system properties. The available theories provide an approximate framework for the data. The correlation constants vary depending upon the type of equipment and operating conditions. In most cases, the dominant mass-transfer resistance resides in the feed (raffinate) phase, since
Cc Cc,initial Cc− Cc∗ Cc,initial− Cc∗ mdcvol k c 1 k d 1 k od
the slope of the equilibrium line usually is greater than unity. In that case, the overall mass-transfer coefficient based on the raffinate phase may be written
= + ≈ for large mervol (15-68)
where mervolis defined by the usual convention in terms of concentration
in the extract phase over that in the raffinate phase, mervol= dCi,extract/
dCi,raffinate. This approximation is particularly useful when the extraction
solvent is significantly less viscous than the feed liquid, so the solute diffusivity and mass-transfer coefficient in the extract phase are rela- tively large.
Mass-Transfer Units The mass-transfer unit concept follows directly from mass-transfer coefficients. The choice of one or the other as a basis for analyzing a given application often is one of pref- erence. Colburn [Ind. Eng. Chem., 33(4), pp. 450–467 (1941)] pro- vides an early review of the relationship between the height of a transfer unit and volumetric mass-transfer coefficients (kora). From a
differential material balance and application of the flux equations, the required contacting height of an extraction column is related to the height of a transfer unit and the number of transfer units
Zt=
Xin
Xout = Hor× Nor (15-69)
where Vris the velocity of the raffinate phase, a is the interfacial area
per unit volume, and the superscript * denotes the equilibrium con- centration. The transfer unit model has proved to be a convenient framework for characterizing mass-transfer performance.
Thus, mass-transfer units are defined as the integral of the differen- tial change in solute concentration divided by the deviation from equi- librium, between the limits of inlet and outlet solute concentrations:
Nor=
Xin
Xout (15-70)
When the equilibrium and operating lines are linear, the solution to Eq. (15-70) can be expressed as
Nor= E= m′ , E≠ 1 (15-71) S′ F′ ln
X X ′ ′fr− − Y Y ′′s s m m ′′1− E1
+ E1 1− E1 dX X− X∗ dX X− X∗ Vr k ora 1 k r 1 m er volk e 1 k r 1 k or Cd* Cc* Cc Cc Cd Cd Cd,i Cd,i Cc,i Cc,i Slope: mdc = (Cd/Cc)*
FIG. 15-25 Two-film mass transfer.
where Noris the number of overall mass-transfer units based on the
raffinate phase. The units are the same as those used previously for the KSB equation [(Eq. 15-48)]. Rearranging Eq. (15-71) gives
= (15-72)
Note that Eq. (15-71) is the same as the KSB equation except in the denominator. Comparing these equations shows that the number of overall raffinate phase transfer units is related to the number of theo- retical stages by
Nor= N × (15-73)
The difference becomes pronounced when values of the extraction factor are high. When E= 1, the number of mass-transfer units and number of theoretical stages are the same:
Nor= N = − 1 for E= 1 (15-74)
As with the KSB equation, in the special case where E< 1, the maxi-
mum performance potential is represented by
max≈ for E< 1 and large Nor (15-75) Equation (15-71) often is referred to as the Colburn equation. Although commonly used to represent the performance of a differen- tial contactor, it models any steady-state, diffusion-controlled processes with straight equilibrium and operating lines. As with the KSB equa- tion, the operating line is straight even when solute concentration changes significantly as long as Bancroft coordinates are used, and both the KSB and Colburn equations can be used to model applications involving a highly curved equilibrium line by dividing the analysis into linear segments. With these approaches, these equations often can be used for applications involving high concentrations of solute.
Solutions to the Colburn equation are shown graphically in Fig. 15-26. Note the contrast to the KSB equation solutions shown in Fig. 15-24. The KSB equations are best used to model countercurrent contact devices where the separation is primarily governed by equilibrium limitations, such as extractors involving discrete stages with high stage efficiencies. The Colburn equation, on the other hand, better represents the perfor- mance of a diffusion rate-controlled contactor because performance approaches a definite limit as the extraction factor increases beyond E= 10 or so, corresponding to a diffusion rate limitation where addition of extra solvent has little or no effect. Note that in Eq. (15-71) the extraction factor always appears as 1/E, and this is how a finite diffusion rate is taken into account. The KSB equation can be misleading in this regard because it predicts continued improvement as the extraction factor increases without limit. Rate-based models most often are utilized for applications with no discrete stages; however, even staged equipment may be mod- eled best by the number of mass-transfer units when the extraction fac- tor is higher than about 3, especially when stage efficiencies are low.
The height of an overall mass-transfer unit based on raffinate phase compositions Horis the total contacting height Ztdivided by the num-
ber of transfer units achieved by the column.
Hor= (15-76)
The value of Horis the sum of contributions from the resistance to
mass transfer in the raffinate phase (Hr) plus resistance to mass trans-
fer in the extract phase (He) divided by the extraction factor E:
Hor= Hr+ (15-77) He E Zt N or 1 1− E X′f− Y′s m′ X′ r− Y′s m′ X′f− Y′s m′ X′ r− Y′s m′ ln E 1− 1 E exp [Nor(1− 1 E)] − 1 E
1− 1 E
X′f− Y′s m′
X′
r− Y′s m′
The individual transfer unit heights are given by
Hr= (15-78)
He= (15-79)
where Q = volumetric flow rate Acol= column cross-sectional area
k = film mass-transfer coefficient (length per unit time) a = interfacial mass-transfer area per unit volume of extractor and subscripts r and e denote the raffinate and extract phases, respec- tively. As discussed earlier, the main resistance to mass transfer gener- ally resides in the feed (raffinate) phase.
The lumped parameter Horoften is employed for design of extrac-
tion columns. Its value reflects the efficiency of the differential con- tactor; higher contacting efficiency is reflected in a lower value of Hor.
It deals directly with the ultimate design criterion, the height of the column, and reliable values often can be obtained from miniplant experiments and experience with commercial units. For processes with discrete contacting stages, mass-transfer efficiency may be expressed as the number of transfer units achieved per actual stage. For applications involving transfer of multiple solutes, the value of Hor
or Norper actual stage may differ for each solute, as discussed earlier
with regard to stage efficiencies and mass-transfer coefficients. EXTRACTION FACTOR AND GENERAL