Chapter 16. Mass Transfer Analysis for Distillation, Absorption, Stripping, and Extraction

Thus far, we have used an equilibrium-stage analysis procedure even in packed columns in which there are no stages. A major advantage of this procedure is that it does not require determination of the mass transfer rate. A major disadvantage is mass transfer is not analyzed.

In packed columns, it is conceptually incorrect to use the staged model even though it works if a correct height equivalent to a theoretical plate (HETP) is used. In this chapter we develop a physically more realistic model for packed columns based on mass transfer between phases. After developing the model for distillation, we discuss mass transfer correlations that allow prediction of the required coefficients for common packings. Next, the analysis is repeated for dilute and concentrated absorbers and strippers. A simple model for mass transfer on a stage is developed for distillation, and stage efficiency is estimated. After a mass transfer analysis of mixer-settler extractors, Section 16.8 and this chapter’s appendix develop the rate model for distillation.

It is assumed that readers have some knowledge of basic mass transfer concepts either from Chapter 15 or from other sources (e.g., Cussler, 2009; Geankoplis, 2003; McCabe et al., 2005).

1. Derive and use mass transfer analysis (HTU-NTU) for packed distillation columns

2. Use HTU-NTU analysis for dilute and concentrated packed absorbers and strippers

3. Use mass transfer correlations to determine the HTU

4. Derive and use HTU-NTU analysis for cocurrent flow

5. Use mass transfer analysis to determine tray efficiency of binary distillation systems

6. Use mass transfer analysis for extraction mixer-settler design

7. Use a computer simulator for rate-based simulation of distillation columns

The mass transfer rate in the gas phase can be written in terms of the individual coefficients, Eq. (15-28a), or the overall coefficients, Eq. (15-29a). For differential height, dz, in the rectifying section the mass transfer rate in terms of the individual coefficients is

where N_A is the flux of A in kmol/(m²·h) [or lbmol/(ft²·h)], a is the m² transfer area/m³ packed volume (or ft²/ft³), and A_mtdz is the column volume available for mass transfer, which in a packed bed is A_cdz. This equation has units of kmol/h or lbmol/h. The mass transfer rate must also be equal to changes in the amount of MVC in liquid and vapor.

L and V are constant molal flow rates. Combining Eqs. (16-1) and (16-2), we obtain

Integrating from z = 0 to z = h, where h is the total height of packing in a section, we obtain

We have assumed that the term V/(k_yaA_c) is constant in the section. The limits of integration for y_A in each section are shown in Figure 16-1. Equation (16-4) is often written as

where the height of a gas-phase transfer unit, H_G, in a packed column is

The more general form for H_G is

where the area for mass transfer, A_mt, may be less than or equal to the column cross-sectional area A_C (e.g., see Section 16.6). The number of gas-phase transfer units, n_G, is

The height of transfer unit terms are commonly known as HTUs and the number of transfer units as NTUs. Thus, the model is often called the HTU-NTU model.

If we substitute Eq. (16-6) in (16-5) and solve for n_G, we obtain

Since hA_c is the volume of this section of the column, (hA_c)/V is a measure of the residence time of the vapor. Thus, the number of transfer units is proportional to (ka) × (residence time). This is true of all definitions of the number of transfer units in Table 16-1.

An exactly similar analysis can be done in the liquid phase by starting with Eq. (15-28b). The result for each section is

which is usually written as

where

In Section 15.4 we noted that although basic Eq. (15-24a) is the same, several different combinations of the mass transfer coefficient and the driving force can be employed to analyze complicated mass transfer systems. If the driving force and the mass transfer coefficient are changed, then the definition of HTU will also change. Table 16-1 lists the most commonly used definitions for driving force, mass transfer coefficient, and HTU.

To integrate to calculate n_G and n_L, interfacial mole fractions y_A,I and x_A,I must be related to bulk mole fractions y_A and x_A. To do this, set Eqs. (15-28a) and (15-28b) equal to each other. After rearrangement, we obtain

The last equality on the right comes from the definitions of H_G and H_L. The left-hand side of this equation can be identified as the slope of a line from the point representing the interfacial mole fractions (y_A,I, x_A,I) to the point representing the bulk mole fractions (y_A, x_A). Since there is no interfacial resistance, the interfacial mole fractions are in equilibrium and must be on the equilibrium curve (Figure 16-2A). The bulk mole fractions y_A and x_A are related by a mass balance through segment dz around either the column top or bottom. This operating line in the rectifying section is

In the stripping section the operating line that relates y_A to x_A is

Since these operating equations are exactly the same as the operating equations for staged systems (Chapter 4), they intersect at the feed line.

A modified McCabe-Thiele diagram can be used to determine x_A,I and y_A,I. From any point (y_A, x_A) on the operating line, draw a line with slope = –k_xa/k_ya. The intersection of this line with the equilibrium curve gives the interfacial mole fractions y_A,I and x_A,I that correspond to y_A and x_A (see Figure 16-2A). After this calculation is done for a series of points, plot 1/(y_A,I – y_A) versus y_A as shown in Figure 16-2B. The area under the curve is n_G. n_L is determined by plotting 1/(x_A – x_A,I) versus x_A. The areas can be determined from graphical or numerical integration, such as Simpson’s rule [see Eq. (9-12) and Example 16-1].

Calculations for stripping and enriching sections should be done separately. For example, in the stripping section,

In the determination of n_G for the stripping section, y_A,in,S is the vapor mole fraction leaving the reboiler. This is illustrated in Figure 16-1 for a partial reboiler. The mole fraction leaving the stripping section, y_A,out,S, can be estimated at the intersection of the operating lines (Figure 16-2A). Note that this estimate makes y_A,out,S = y_A,in,E.

Calculating interfacial mole fractions adds an extra step to the calculation. Since it is often desirable to avoid this step, overall mass transfer coefficients [Eq. (15-29)] are often used. In terms of overall driving force the mass transfer rate corresponding to Eq. (16-1) is

y*_A is mole fraction in vapor in equilibrium with liquid at mole fraction x_A (Figure 16-2A). Setting Eq. (16-18) equal to Eq. (16-2), we obtain

Integration of this equation over a section of the column gives

This equation is usually written as

where the height of an overall gas-phase transfer unit is

and the number of overall gas-phase transfer units is

Exactly the same steps can be done in terms of the liquid mole fractions. The result is

where

and

We can calculate overall HTU from individual HTUs. For example, substituting Eq. (15-31b) into Eq. (16-22) and using definitions in Eqs. (16-5) and (16-11), we obtain

H_OL can be found by substituting Eq. (15-31c) into Eq. (16-25) and using definitions of H_G and H_L:

Obviously, H_OG and H_OL are related:

An advantage of overall transfer heights and units is that y*_A – y_A is easily found from vertical lines, and x_A – x*_A can be found from horizontal lines, as shown in Figure 16-2A. The number of transfer units, n_OG or n_OL, is then determined by a calculation similar to calculation of n_G illustrated in Figure 16-2B. A disadvantage of using overall coefficients is that heights H_OG and H_OL are much less likely to be constant than H_G and H_L. If H_G and H_L are constant, H_OG and H_OL cannot be exactly constant, since m, the slope of the equilibrium curve, varies. The various NTU values must be related, since packing height, h, in Eqs. (16-5), (16-10), (16-21), and (16-24) is the same, but HTU values vary. These relationships are derived in Problem 16.C1.

This approach can easily be extended to the more complex continuous columns discussed in Chapters 4 and 8 and to the batch columns discussed in Chapter 9. Any of these situations can be analyzed by plotting the appropriate operating lines and then proceeding with the HTU-NTU analysis. An alternative procedure is described in Problem 16.G1.

while a general equation for the straight operating line is

With straight equilibrium and operating lines the integral in the definition of n_OG can be evaluated analytically. The difference between the equilibrium and operating lines, y* – y, is

when m = L/V, this becomes

Then Eq. (16-23) becomes

which is

Since h = H_OGn_OG = N × (HETP), we can solve for HETP:

N can be obtained from Eq. (12-12). Comparison of Eqs. (16-31) and (12-12) shows that N = n_OG when mV/L = 1.0. Thus,

The situation when operating and equilibrium lines are straight but not parallel is shown in Figure 16-4B. The difference between equilibrium and operating lines is still given by Eq. (16-30), but the terms with x do not cancel out. By substituting in x from the operating equation, y* – y in Eq. (16-23) can be determined as a linear function of y. After integration and considerable algebraic manipulation, n_OG is

where

The value of HETP can be determined from Eq. (16-32), where N is found from the Kremser equation, Eq. (12-22), and n_OG from Eq. (16-34a). This result is

The use of this result is illustrated in Example 16-2.

This analysis can also be done in terms of liquid mole fractions. The results are

where

and

Equations (16-34a) and (16-34b) are called the Colburn equations.

Although derived for straight operating and equilibrium lines, Eqs. (16-36a) and (16-36b) will be approximately valid for curved equilibrium or operating lines. HETP should be determined separately for each section of the column. If H_OG is approximately constant, then HETP must vary since mV/L varies. For maximum accuracy HETP can be calculated for each stage (Sherwood et al., 1975).

The correlation for H_G is

where ψ is a packing parameter that is given in Figure 16-5 (Bolles and Fair, 1982) for common packings, and other special terms are defined in Table 16-2. Viscosity, density, surface tension, and diffusivities should be defined in consistent units so that the Schmidt number and the ratios of liquid to water properties are dimensionless. The packing height, h_p, is the height of each packed bed; thus, the stripping and enriching sections should be considered separately.

The correlation for H_L is

In Eq. (16-38) ϕ is a packing parameter shown in Figure 16-6, and C_fL is a vapor load coefficient shown in Figure 16-7 (Bolles and Fair, 1982). The value of u_flood in Figure 16-7 is from the packed-bed flooding correlation in Figure 10-27.

Calculated H_G and H_L values can vary from location to location in each section. When this occurs, an integrated mean value should be used. Overall HTU values can be obtained from Eqs. (16-27a) and (16-27b). Even if H_G and H_L are constant, H_OG and H_OL will vary in a distillation column, owing to curvature of the equilibrium curve.

Bolles and Fair (1982) show that there is considerable scatter in modeled HETP data versus experimental HETP data. HETP was calculated from Eq. (16-36). For 95.0% confidence in the results, Bolles and Fair suggest a safety factor of 1.70 in the determination of HETP. They note that this large a safety factor is usually not used, since there are often a number of hidden safety factors such as not including end effects and using nonoptimum operating conditions. However, if a tight design is used, then the 1.70 safety factor is required. This large a safety factor emphasizes that design of packed distillation systems is still largely an art not a science.

where W_G and W_L are the fluxes in lb/(h·ft²), and Sc_v is the Schmidt number for the gas phase. The constants are given in Table 16-3. The expression for H_L (in ft) developed by Sherwood and Holloway (1940) is

where Sc_L is the Schmidt number for the liquid. Units of liquid viscosity, µ, are centipoise. The constants are given in Table 16-3.

The simple correlations are obviously easier to use than Eqs. (16-37) and (16-38) since only the Schmidt number and the viscosity need to be estimated. However, Eqs. (16-40a) and (16-40b) will not be as accurate; thus, they should be used only for preliminary designs. These correlations were developed from absorption data and will be less accurate for distillation.

Water is frequently the solvent in absorption systems. The approximate values for H_OG for water as solvent are listed in Table 16-4 for random plastic packings.

where J_A is the flux with respect to a coordinate system moving at the molar average velocity of the fluid. As shown in Section 15.4.2 this leads to a transfer rate equation that is superficially similar to the previous equations [see Eq. (15-32f)].

The product of (concentrated mass transfer coefficient) × (area/volume) is defined as

where the logarithmic mean mole fraction is defined in the same manner as Eq. (15-32d).

For very dilute systems, (1 – y_A)_lm = 1.0 and .

We will now repeat the analysis of a packed section using Eq. (16-41), including the nonconstant total flow rates. Figure 16-8A is a schematic diagram of an isothermal absorber with plug flow. The rate of mass transfer in a segment of column dz is

Comparison of this equation with Eq. (16-1) shows the sign on the mole fraction difference has been switched, since the direction of solute transfer in absorbers is opposite to that of MVC transfer in distillation. In addition, the modified mass transfer coefficient is used. Solute mass transfer is also related to change in solute flow rates in the gas or liquid streams.

This equation differs from Eq. (16-2) derived for distillation since neither V nor L is constant.

V can be related to the constant flow rate of carrier gas, G, Eq. (12-41a):

Combining Eqs. (16-45), (16-46), and (12-41a), we obtain

After taking the derivative, substituting in Eq. (12-41a), and cleaning up the algebra, we obtain

Integrating this equation we obtain

Substituting in Eq. (16-43) we obtain

The term V/(k_yaA_c) is the height of a gas-phase transfer unit H_G defined in Eq. (16-6).

The variation in H_G can be determined from Eq. (16-37) and Figure 16-5, which are valid for both absorbers and distillation. The term that varies the most in Eq. (16-37) is the weight mass flux of liquid, W_L. H_G depends on W_L to the –0.5 to –0.6 power. In a single section of an absorber, a 20.0% change in liquid flow rate is quite large. This will cause at most a 10.0% change in H_G. k_ya is independent of concentration, since the concentration effect was included in k′_y in Eq. (16-42). Since the variation in H_G over the column section is relatively small, we will treat H_G as a constant. Then Eq. (16-50) becomes

which is usually written as

Note that n_G for concentrated absorption is defined differently from n_G for distillation, Eq. (16-7). The difference in the limits of integration in the two definitions for n_G occurs because the direction of transfer of the component A in distillation is the opposite of the direction in absorption. There are additional terms inside the integral sign in absorption because the mass transfer takes place through a stagnant film and is not equimolar counter transfer as in distillation.

The method for finding the interfacial compositions is similar to that used to develop Eq. (16-13) and Figure 16-2, except that Eq. (16-42) and the corresponding equation in terms of liquid mole fractions are used as the starting point. The procedure is illustrated in Figure 16-8B. Use of this procedure lets us calculate the integrand in Eq. (16-52) at a series of points. The integral in Eq. (16-52) can be found either numerically or graphically.

Often, the integral in Eq. (16-52) can be simplified. The first simplification often employed is to replace the logarithmic mean with an arithmetic average.

When Eq. (16-53) is substituted in Eq. (16-52), n_G can be simplified:

This equation shows that n_G for absorption is essentially the n_G for distillation plus a correction factor. The interfacial mole fraction y_A,I can be determined as shown in Figure 16-8B. The integral in Eq. (16-54) can then be determined graphically or numerically. For very dilute systems 1 – y_A is approximately 1.0 everywhere in the column. Then the correction factor in Eq. (16-54) will be approximately zero. Thus, n_G for dilute absorbers reduces to the same formula as for distillation.

For dilute absorbers and strippers, (1 – y_A)_lm = 1.0. Then a in Eq. (16-41). In this case we can use the overall gas-phase mass transfer coefficient. Following a development that parallels the analysis presented earlier for distillation, Eqs. (15-29a) and (16-18) to (16-23), we obtain for dilute absorbers

where H_OG is defined in Eq. (16-22), and

This n_OG is essentially the same as for distillation in Eq. (16-23).

If the operating and equilibrium lines are straight, n_OG can be integrated analytically. The result is the Colburn equation given in Eqs. (16-31) and (16-34a). An alternative integration gives an equivalent equation.

The development done here in terms of gas mole fractions can obviously be done in terms of liquid mole fractions. The development is exactly analogous to that presented here. The result for liquids is

where H_L is defined in Eq. (16-11), and

Equation (16-59) can often be simplified to

For dilute systems the correction factor in Eq. (16-60) becomes negligible. For dilute systems the analysis can also be done in terms of the overall transfer coefficient.

where H_OL is defined in Eq. (16-25), and

If the operating and equilibrium lines are both straight, n_OL can be integrated analytically. The result is the Colburn equation, Eq. (16-34b), or the equivalent expression

The development of the equations for concentrated systems presented here is not the same as those in Cussler (1997) and Sherwood et al. (1975). Since the assumptions have been different, the results are slightly different. However, the differences in these equations are usually not important, since the inaccuracies caused by assuming an isothermal system with plug flow are greater than those induced by changes in the mass transfer equations. For dilute systems all the developments reduce to the same equations.

If there are multiple solutes transferring, the analysis is significantly more complicated than the analysis shown here (Taylor and Krishna, 1993). These complications are beyond the scope of this chapter but can be solved with the Maxwell-Stefan approach.

A schematic of a cocurrent absorber is shown in Figure 16-9A. The analysis will be done for dilute systems using overall mass transfer coefficients. The system is assumed to be isothermal. Liquid and vapor are assumed to be in plug flow, and total flow rates are constant. The rate of mass transfer in segment dz is

which can be related to the changes in solute flow rates:

Combining these equations we obtain

If V/(k_yaA_c) is constant, Eq. (16-66) can be integrated to give

where H_OG is given in Eq. (16-22), and

This development is similar to the development for countercurrent systems. The analyses differ when we calculate y_A – y*_A. The operating equation is [see Eq. (12-63)]

This operating line and the calculation of are shown in Figure 16-9B. When the operating and equilibrium lines both are straight, n_OG can be obtained analytically. The result corresponding to the Colburn equation is (King, 1980)

where

If a completely irreversible reaction occurs in the liquid phase, y*_A = 0 everywhere in the column. Thus, the equilibrium line is the x axis, and the integration of Eq. (16-68) is straightforward.

Exactly the same result is obtained for cocurrent and countercurrent columns with irreversible reactions, but cocurrent columns can have higher liquid and vapor flow rates.

H_OG is related to the individual coefficients by Eq. (16-27a). Equations (16-37) and (16-38) should not be used to determine values for H_L and H_G for cocurrent columns because these correlations are based on data in countercurrent columns at lower gas rates than used in cocurrent columns. Mass transfer coefficients can be considerably higher in cocurrent systems (Reiss, 1967). Gianetto et al. (1973) operated with a 15-fold velocity increase and observed a 40-fold increase in k_L when liquid-phase resistance controlled. They recommend cocurrent operation for absorption with chemical reaction. Harmen and Perona (1972) did an economic comparison of cocurrent and countercurrent columns. For the absorption of CO₂ in carbonate solutions in which reaction is slow they conclude that countercurrent operation is more economical. For CO₂ absorption in monoethanolamine (MEA), in which reaction is fast, they conclude that countercurrent is better at low liquid fluxes, whereas cocurrent is preferable at high liquid fluxes.

A schematic diagram of a tray is shown in Figure 16-10. The column is operating at steady state. A mass balance will be done for the mass balance envelope indicated by the dashed outline. The vapor above the trays is assumed well mixed; thus, the inlet vapor mole fraction does not depend on the position along the tray, . The vapor leaving the balance envelope has not yet had a chance to be mixed, and its composition is a function of position . The rising vapor bubbles are assumed to perfectly mix the liquid vertically. Thus, x does not depend on the vertical position z, but the vapor fraction y does depend on z. The liquid mole fraction can be a function of the distance, , along the tray measured from the start of the active region, = 0, to the end of the active region, = _a. At steady state a solute or MVC mass balance for the vapor phase is

If we use the overall gas-phase mass transfer coefficient, K_y, this equation is

where is the vapor mole fraction in equilibrium with the liquid of mole fraction, x. On trays the area for mass transfer, A_mt, is the active area, A_active, which is where vapor-liquid contact occurs on the tray. Both A_active and V are assumed constant. Dividing Eq. (16-73b) by Δz and taking the limit as Δz goes to zero, we obtain

This equation can be integrated from z = 0 to z = h. The boundary conditions are

After algebraic manipulation, the solution to Eqs. (16-74) and (16-75) is

The point efficiency, E_pt, is defined in Eq. (10-5). Comparing this equation to Eqs. (16-21) and (16-22), we obtain two alternative representations:

We would like to relate the point efficiency to the Murphree vapor efficiency given by Eq. (10-2). This relationship depends on the liquid flow conditions on the tray. There are two limiting flow conditions that allow us to simply relate E_pt to E_MV. The first of these is a tray where the liquid is completely mixed. This means that x is a constant and equal to x_out, so that and y = y_out. Therefore E_MV = E_pt, and for a completely mixed stage

The second limiting flow condition is plug flow of liquid with no mixing along the tray. By assuming that each packet of liquid has the same residence time, we can derive the relationship between E_MV and E_pt (Lewis, 1936; King, 1980; Lockett, 1986):

where m is the local slope of the equilibrium curve, Eq. (15-30b). Since plug flow is often closer to reality than a completely mixed tray, Eq. (16-78) is more commonly used than Eq. (16-77).

Real plates often have mixing somewhere in between these two limiting cases. These situations are discussed elsewhere (King, 1980; Lockett, 1986).

In these equations x_D is the weight or mole fraction of solute A in the dispersed phase, and x*_D is the weight or mole fraction that would be in equilibrium with the continuous-phase fraction of A, x_C. Note that x_D may refer to either raffinate or extract, which is different than the notation in Table 13-2. Determination of a, the area/volume (m² dispersed phase/m³ total liquid volume in mixer) for mass transfer, is discussed in Section 16.7.3. The units of K_O–ED are (kg solute A in dispersed phase)/[s (m² dispersed phase)(mass fraction solute in dispersed phase)] or the equivalent in molar units. Because mass fraction A in dispersed phase = (kg solute A in dispersed phase)/(total kg of dispersed phase), the units of K_O–ED can also be written as (total kg dispersed phase)/[s (m² dispersed phase)].

The overall mass transfer coefficient is related to the individual coefficients for the continuous and dispersed phases with a sum of resistances model similar to that developed in Eq. (15-31c).

where the term m_CD is the average slope of the equilibrium curve

If equilibrium is linear and the extract phase is continuous, then m_CD is constant and equals the distribution coefficient defined in Eq. (13-6), K_d = y/x. If raffinate is continuous, then m_CD is the inverse of the distribution coefficient, 1/K_d. Using an incorrect value of m_CD is a common error.

The extractor height is determined from h = (NTU)(HTU). Simplified equations for the number of extraction transfer units for the dispersed phase, n_OE–D, and the continuous phase, n_OE–C, defined in the same way as n_OL in Eq. (16-26) are

For packed and countercurrent columns (Figure 13-2), Eq. (16-81a) can be integrated by the methods developed in Sections 16.1 and 16.4. Very convenient analytical solutions are obtained when the equilibrium and operating equations are both linear. Since the dispersed phase can be either extract or raffinate, we write these equations as n_O–Ex for raffinate and n_O–Ey for extract. If solvent is the dispersed phase, then n_O–ED = n_O–Ey. These results, forms of the Colburn equation, are

These four equations are valid if R / (mS) ≠ 1.0. The grouping (mS) / R is the extraction factor introduced in Chapter 13. To understand some of the differences between a staged analysis and a mass transfer analysis, these results should be compared with the Kremser Eqs. (13-11a) and (13-11b) (see Problem 16.A1).

The height of the transfer unit is defined in a similar way as in Eq. (16-22), which is

Although the exact form used depends on the equipment and whether K_Overalla is in mass or molar units, a typical definition in columns for H_O–raf in molar units is

The term (Q_raf /A_c) is the superficial velocity of the raffinate phase in the column, and (ρ_raf / MW_raf) is a conversion factor because (K_O–rafa) is in molar units.

It is also often very useful to note that because n_O–ED = h / H_O–ED, we can write n_O–ED as

Q_D is the volumetric flow rate (m³/s) of the dispersed phase, and V_mixer is the total mixer volume (m³). The term V_mixer / Q_D is the dispersed-phase residence time based on the dispersed-phase superficial velocity. Since the value of n_O–ED can often be determined from integration of Eq. (16-81a), Eq. (16-83a) provides a method for determining (K_O–EDa) from experimental data.

It would probably make more sense to define

H_O–ED = (Q_D/ϕ_dA_mixer)(ρ_D/MW_D)/(K_O–EDa)

and

n_O–ED = (V_mixer ϕ_d)(K_O–EDa)/[Q_D(ρ_D/MW_D)]

(V_mixerϕ_d / Q_D) is the residence time of the dispersed phase, and ϕ_d is the volumetric fraction of the dispersed phase in the mixer (see Section 13.4.1). However, we follow the standard approach so that the values of (K_O–EDa) from the literature agree with our formulation.

Because extraction mixer-settlers typically operate at stage efficiencies above 80.0% and often in the range from 95.0% to 100.0%, the equilibrium-stage analysis in Section 13.14 is often used with an assumed value for stage efficiency. However, a more accurate design will result if a mass transfer analysis is used to estimate stage efficiency. The purpose of the analysis is to estimate the value of the dispersed-phase Murphree stage efficiency, E_MD.

As usual, x_D,out* is the mole fraction of solute in the dispersed phase that is in equilibrium with the actual mole fraction of solute in the continuous phase, x_C,out.

There are two different analyses for mixer-settler mass transfer in the literature. The simpler analysis (Seader et al, 2011; Treybal, 1980) assumes that both the continuous phase and the dispersed phase are well mixed, and the dispersed phase has the same average residence time everywhere in the mixer. Thus, x_D, x_C, and x_D* are constant and equal to their values at the mixer outlet. Then Eq. (16-81a) simplifies to

We can manipulate Eq. (16-84) for the Murphree dispersed-phase stage efficiency so that it can be written in terms of n_OE–D. First,

which when compared to Eq. (16-85) can be recognized as

Once we determine the mass transfer coefficient from the appropriate correlations, n_O–ED can be determined from Eq. (16-83b) and E_MD from Eq. (16-86). Alternatively, if the concentrations are measured, then E_MD and/or n_OE–D can be determined from Eqs. (16-84) and (16-85), respectively, and K_O–EDa from Eq. (16-83b).

The second analysis procedure uses a differential equation approach (Frank et al., 2008) that parallels the analysis in Section 16.6. The continuous phase takes the role of the liquid phase in Figure 16-10, and the dispersed phase takes the part of the vapor. The derivation follows the change in concentration of a packet of dispersed phase from z = 0 to z = h but then requires mixing at z = h. This requires the assumption that the continuous phase, but not the dispersed phase, is well mixed. Continuous and dispersed phases exit together instead of separately as in Figure 16-10, but this does not change the analysis. The steady-state dispersed-phase mass balance is

In the notation used for extraction with the same geometry as in Figure 16-10, this is

Q_D/A_mixer is the superficial linear velocity of the dispersed phase in the mixer. Dividing this equation by Δz and taking the limit as Δz goes to zero,

We can integrate from z = 0 to z = h with the boundary conditions

The result is the point efficiency

For a mixer with a well-mixed continuous phase (but the dispersed phase is not well mixed), x*_D, = constant = x*_D,out, since it is in equilibrium with the continuous phase, which is the same everywhere. In addition, if the dispersed phase has the same average residence time everywhere in the mixer, the same amount of solute is transferred in or out of the dispersed phase, and x_D, = constant = x_D,out. Thus, the Murphree efficiency equals the point efficiency, E_MD = E_pt. Substituting in V_mixer = hA_mixer, the Murphree efficiency is

Equation (16-92) is equivalent to Eq. (16-77) derived for a distillation tray. Equation (16-92) can be used either to estimate the value of E_MD if K_O–EDa is known or to estimate K_O–EDa if concentrations are measured and E_MD is determined from Eq. (16-84). Frank et al. (2008) replace Q_D in Eq. (16-92) with (Q_D + Q_C)ϕ_d, which is the flow rate of dispersed phase in the mixer. If ϕ_d = ϕ_d,feed, then (Q_D + Q_C)ϕ_d = Q_D, and the results are identical.

Equations (16-86) and (16-92) are not identical because they are based on different models. Both models assume the continuous phase is well mixed, but their assumptions about the dispersed phase are different. These differences emphasize the comments in Chapter 15 that analysis of mass transfer remains a topic for research and discussion. With identical values of K_O–EDa, Eq. (16-92) always predicts a higher value for E_MD than does Eq. (16-86). This is illustrated in Example 16-5. Another way of looking at this is that with the same E_MD, a larger K_O–EDa value will be back-calculated from Eq. (16-86b) than from Eq. (16-92). Both equations predict that n_O–ED will be large and E_MD will approach 1.0 if (K_O–EDa) × (residence time) is large. Residence time is large if the mixer volume is large compared to the total liquid flow rate.

Which equation is correct? If the continuous phase is not well mixed, then neither model is appropriate. If the mixer has good mixing of the continuous phase, then the appropriate model to use depends on mixing within the dispersed phase. In very clean systems there is often considerable internal circulation, and the drops tend to be well mixed. The well-mixed model is appropriate, and most of the data in the literature were analyzed with this model. In dirty systems with dust, crud, or surfactants (e.g., soap or proteins) present, the drop surface can be rigid, and internal circulation is suppressed—then the differential model is more appropriate. Mass transfer coefficients obtained with clean systems do not apply to dirty systems, and vice versa.

Unfortunately, the choice of model to use depends on how the value of K_O–EDa was determined. The same raw data (concentrations) will result in different K_O–EDa values for the two models. Using one model to determine K_O–EDa and then using a different model with these K_O–EDa values will not give correct answers. This is illustrated in Example 16.5. Thus, use the same model that was used to determine K_O–EDa.

The geometric equations for spheres in Table 15-5 are useful for estimation of a. For n spherical drops in a liquid

and the surface area per volume of the mixer is

Since volume of n drops = (nπ/6)(average drop diameter)³, the volume fraction dispersed phase is

Solving for volume of vessel and substituting into Eq. (16-96b), the surface area per volume is

Methods for estimating ϕ_D in mixers are discussed in Section 13.14.1. The surface area per volume, a, of mixers can be estimated once the average diameter of the drops is known.

Several correlations for the average drop diameter in mixers have been published. Treybal (1980) recommends the following equation for baffled vessels:

The term Δρ is the absolute value of (ρ_C – ρ_D), and d_p^o is obtained from

Treybal notes that the terms in parentheses are dimensionless.

A few results for individual drops are presented in Section 16.7.4.1. In practice, drops are in swarms, and the correlations in Section 16.7.4.2 must be used. A summary of a conservative (safe) design procedure for mixers is outlined in Section 16.7.4.3. Additional correlations for mass transfer coefficients are available (e.g., Frank et al., 2008; Slater, 1994; Treybal, 1980; Wankat and Knaebel, 2008).

For creeping flow in clean systems, the Sherwood number can be approximated as

The drop Reynolds number Re_drop = (ρ_Cdu_t/µ_C) assumes the drop is at its terminal velocity u_t, usually estimated from Stokes’s law, Eq. (13-53). The continuous-phase Schmidt number Sc_C = [µ_C/(ρ_CD_AC)] includes the effect of molecular diffusivity. Equation (16-98b) simplifies to Eq. (16-98a) as u_t and Re_drop → 0. At higher Reynolds numbers (10 < Re_drop < 1200, 190 < Sc_C < 241,000, and 1000 < Pe_C < 10⁶) Steiner’s empirical results for circulating drops are

where the continuous-phase Peclet number Pe_C = (d u_t/D_AC).

For rigid drops, which often occur in less clean conditions, Steiner obtained

The 0.0103 Re_drop term is a correction for the effect of wakes and is usually quite small. For conditions in between fully circulating and rigid, Steiner recommends

For clean systems, Eq. (16-100b) is preferred, while for dirty systems in which the drops are often rigid, use Eq. (16-100a), which predicts lower values of the mass transfer coefficient (Slater, 1994).

Although mass transfer for individual drops is time dependent, if drop life is fairly long, then a pseudo-steady state is reached, and there is an asymptotic value of k_D. If interfacial tension is low and the drop fairly large, there will be significant internal circulation, and the rate of mass transfer will be much larger than that predicted by molecular diffusion alone. Internal circulation results in a significant increase in dispersed-phase mass transfer coefficient k_D; however, the presence of small amounts of surfactants or dirt that collect at the interface reduce internal circulation markedly and may introduce a resistance to mass transfer at the interface that is not included in Eq. (16-80b). Because industrial plants are not always scrupulously clean, dispersed-phase mass transfer coefficients in plant operations often are significantly lower than values obtained in scrupulously clean laboratories.

For large drops with toroidal internal circulation, Handlos and Baron solved the flow and mass transfer equations. A simplified form of their result is in reasonable agreement with experimental results obtained under clean conditions for large drops (Slater, 1994):

Note that with internal circulation (typical of large drops in clean systems), there is no dependence on the molecular diffusivity D_AD. For rigid drops (typically small drops or dirty interfaces) with no circulation, a limiting solution at long times is (Slater, 1994)

As expected, the molecular diffusivity of solute in dispersed phase, D_AD, is important when there is no internal circulation. Equation (16-101b) is conservative (predicted k_D is low).

In extractions in which feed is the continuous phase and equilibrium favors transfer into the raffinate [m_CD in Eq. (16-80c) is small], the overall mass transfer resistance is dominated by the continuous phase. In this case, the correlation of Skelland and Moeti is recommended for mixer design (Frank et al., 2008; Slater, 1994; Wankat and Knaebel, 2008).

where d_i is the impeller diameter in meters and ω is the impeller speed in 1/s. This equation is restricted to low dispersed-phase holdup, ϕ_d < 0.06. For rigid drops, correlations developed for solid particles can be used to provide conservative values for k_C (Treybal, 1980).

There are few studies of dispersed-phase mass transfer coefficients in mixers. Frank et al. (2008) recommend Skelland and Xien’s (1990) correlation for transfer from dispersed phase to continuous phase. They studied batch extraction in a baffled mixer with six-flat-blade turbines.

In this equation, ρ_M is defined in Eq. (13-47), µ_M is defined in Eq. (13-48), N is the impeller speed in rps, t_o is the initial time (s) at which the dispersed phase is injected, and t_F,95 is the time (s) at which 95.0% of mass transfer has occurred. In a continuous mixer (t_F,95 – t_o) can be considered the residence time of the dispersed phase in the mixer that will result in 95.0% extraction of solute.

However, obtaining data on the exact solutions to be used in the plant is often not possible, and a prediction of stage efficiency must be made. The following approach results in a conservative estimate (stage efficiency is low) because every equation is considered conservative, and the additional mass transfer that occurs in the settler is neglected.

1. Estimate the fraction of the dispersed phase, ϕ_D, and power, P, as in Example 13-6.

2. Use Eqs. (16-97a) and (16-97b) to estimate the drop diameter, d_p.

3. Use Eq. (16-96d) to estimate a.

Assume that in the plant the extractor contains particulates and surface active agents; thus, the drops are rigid.

4. Use Eq. (16-103) to estimate k_C.

5. Use Eq. (16-104) to estimate k_D.

6. Use Eq. (16-80b analog) (in Example 16-5) to estimate the overall mass transfer coefficient k_LD in m/s.

7. Use Eq. (16-93a) to estimate K_O–ED.

8. Use Eq. (16-83b) to estimate n_O–ED.

9. Use the stirred tank model Eq. (16-86) to determine the stage efficiency E_MD.

10. Check the results for any assumptions made during the calculation, and repeat the calculation with a better assumption if necessary.

This approach is illustrated in Problem 16.D21.

Since a Fickian mass transfer analysis can lead to logical inconsistencies when extended to three or more components, a fundamental rate analysis of multicomponent distillation must be based on the Maxwell-Stefan mass transfer model for nonideal multicomponent systems (Section 15.7.8). These calculations require significant detail beyond the scope of an introductory textbook; therefore; the methods are summarized in enough detail to explain what the commercial simulator does (Lab 13 in this chapter’s appendix) but not in enough detail to write a program. Readers interested in complete details are referred to Taylor and Krishna (1993) and Aspen Plus (2010).

The detailed rate model of distillation starts with material and energy balances for the vapor and liquid on each stage. If there are no reactions, the bulk vapor and liquid-phase component material balances are (Taylor and Krishna, 1993; Aspen Plus, 2010)

Transfer to the vapor from the liquid, , is arbitrarily considered positive. Transfer terms across the film are determined from a generalized matrix form of the Maxwell-Stefan equations (Krishna and Standart, 1976). Energy balances are required for both bulk vapor and bulk liquid phases. For the bulk vapor phase, this equation is

where Q_j^v is the external heat load to the vapor, and E^v_j is the energy transfer rate from the bulk liquid. There is a similar equation for the bulk liquid phase. The rate of energy transfer to the vapor from the liquid across the film is given by a rate equation:

In this equation, a¹_j is the interfacial area for heat transfer, is the vapor-phase heat transfer coefficient, T_j^v is the temperature bulk vapor phase, is the temperature of the interface, and is the partial molar enthalpy of component i, all on stage j. There is a similar equation for energy transfer from the vapor to the liquid. Since the film is assumed to have no accumulation of mass or energy, the film interface is at equilibrium:

These equations are all written in matrix form.

Mass transfer coefficients and interfacial area per volume a are obtained from correlations based on experimental data. Heat transfer coefficients are obtained from the Chilton-Colburn analogy, Eq. (15-63), using the experimentally determined mass transfer correlations. The most commonly used correlation for interfacial area per volume a is the Zuiderweg (1982) correlation. The correlation depends on the regime of operation of the sieve plates. In the spray regime, the correlation is

where A_active is the active area of the tray, A_hole is the total hole area of a tray in m², Q_L and Q_V are the volumetric flow rates of liquid and vapor in m³/s, σ is the surface tension in N/m, FP is the flow parameter defined in Eq. (10-9), l_w is the weir length in m, and h_L is the calculated clear-liquid height in m on the tray. In the froth regime, the correlation is

The clear liquid height h_L for these correlations is calculated from

where p is the pitch of the sieve plate holes in m. Zuiderweg notes that these should be considered as apparent interfacial areas because they are based on the mass transfer coefficients back-calculated from the operation of relatively large-diameter Fractionation Research Institute distillation columns based on the somewhat improbable assumption that the mass transfer coefficients are independent of flow regime and velocity. Thus, the effects of flow regime and velocity have been lumped into the calculation of a. Despite these difficulties, Eqs. (16-107a) and (16-107b) are widely used.

Zuiderweg (1982) also determined correlations for the liquid and vapor mass transfer coefficients (both in m/s):

In this equation, D_L is in m²/s.

The equation for the vapor-phase mass transfer coefficient is unique in that it does not depend on the diffusivity of the vapor and is the same for all components. Zuiderweg’s correlation essentially assumes that liquid-phase resistances control; thus, it should not be used if vapor-phase and liquid-phase resistances are the same order of magnitude or the vapor-phase resistance controls.

Chan and Fair (1984) used the AIChE (1958) correlation to determine k_L,ia. This correlation with k_L,ia in (m/s)(m²/m³) is

The liquid diffusivity D_L,i is in cm²/s, U_a = superficial vapor velocity in the active area of the tray, m/s, and ρ_V is the vapor density, kg/m³. The Chan and Fair correlation for k_V,ia is

The vapor diffusivity, D_V,i, is in cm²/s, k_V,i is in m/s, a is in m²/m³, f is the fractional approach to flooding, and h_L is the liquid holdup on the plate in cm calculated from the correlation of Bennett et al. (1983). Note that different correlations use parameters in different units.

A more recent mass transfer correlation was developed by Chen and Chuang (1993). They recommend using the clear liquid height calculated from Eq. (16-107c). Their correlations for mass transfer coefficients are

In these equations, σ is the interfacial surface tension, N/m, and t_V = (froth height)/U_A and t_L ≈ ρ_Lt_V /ρ_V are the average residence times per pass for the vapor and liquid, s. They recommend a correlation of Stichlmair for the interfacial area per volume a, but Aspen Plus (2010) recommends using the Zuiderweg (1982) correlations in conjunction with the Chen and Chuang mass transfer correlation.

All of the correlations provide mass transfer coefficients at a point on the plate. To determine the overall amount transferred, a flow model for the stage is required. Chan and Fair (1984) recommend calculating the Peclet number to determine which flow model is appropriate:

The distance traveled, z_l, in m is from the exit of the downcomer to the overflow weir. The eddy diffusivity (in m²/s) is determined from the Barker and Self correlation:

The liquid holdup h (in m) can be estimated from the correlation of Bennett et al. (1983), although if the only use of the calculation is to estimate the eddy diffusivity, the weir height can be substituted for h_L. U_active is the vapor velocity in the active area of the plate, m/s. The residence time t_L,residence can be estimated from

In this equation, a_active is the active area of the plate in m², Q_L is the volumetric flow rate of the liquid in m³/s, and the liquid holdup height, h_L, is in m.

Which mass transfer correlation should be used? One can simulate the distillation with each of these three correlations plus the other mass transfer correlations supported by the simulator. If any of the correlations predict results that are clearly outliers, they probably should not be used. If a conservative design is desired, then use the correlation that predicts the least separation. If the column is operating outside the range of validity of a correlation, the correlation should be used with great caution. If a company has had good results using a particular correlation, they will probably keep using it. Obviously, all of the correlations for mass transfer coefficients and for interfacial area per volume depend on the geometry of the plates, downcomers, and columns. Thus, the design requires the specification of these variables.

Which flow model should be used? If the Peclet number is low, use the completely mixed model. This is the simplest flow model, and the mass transfer coefficients are the same everywhere on the plate. Small-diameter columns usually are well mixed and have low values of Pe. Large columns with a large value of z_l often have large Pe values, and a plug flow model is appropriate. The rate-based model with a plug flow occasionally predicts better separation than the equilibrium-staged model (see Section 10.2). If a conservative result is desired, use the mixed model. The mixed model should always have results that show less separation than the equilibrium model.

It is highly recommended that the column be designed with equilibrium stages first (Chapter 6), including design of internals with a tray-rating simulation (Lab 10), since with this starting point, the rate-based design is more likely to converge.

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A1. Compare Colburn Eq. (16-81d) to the equivalent Kremser Eq. (13-11b), and compare Eq. (16-81e) to (13-11a). If we relate n_O–Ey to N and set S = E, y_N+1 = y_in, y₁ = y_out, and y₁* = y_out*, what terms are similar, and what terms are different? [Note that Eq. (13-11a) is inverted compared to Eq. (16-81e).]

A2. Mass transfer models include transfer in only the packed region. Mass transfer also occurs in the column ends where liquid and vapor are separated. Discuss how these end effects affect a design. How could one experimentally measure end effects?

A3. Are stages with well-mixed liquids less or more efficient than stages with plug flow of liquid (assume K_Ga are the same)? Explain your result with a physical argument.

A4.

a. The Bolles and Fair (1982) correlation indicates that H_G is more dependent on liquid flux than on gas flux. Explain this on the basis of a simple physical model.

b. Why do H_G and H_L depend on the packing depth?

c. Does H_G increase or decrease as µ_G increases? Does H_G increase or decrease as µ_L increases?

A5. Why is the mass transfer analysis for concentrated absorbers considerably more complex than the analysis for binary distillation or for dilute absorbers?

A6. Construct your key relations chart for this chapter.

A7. While designing a mixer-settler extraction system, you obtain a mass transfer correlation from a book. Unfortunately, the book does not explain which model was used. Which model would you use to determine the stage efficiency? Why?

A8. To design a mixer-settler extraction system you obtain a mass transfer coefficient from the Internet, but units of the mass transfer coefficient are not listed. What should you do?

A9. Why are mass transfer coefficients from clean drops higher than mass transfer coefficients in dirty systems? What is the practical significance of this?

A10. The rate design method for distillation columns is less likely to converge, takes more time to set up, and requires more data than the equilibrium model. When would you decide you should use the rate design model?

B. Generation of Alternatives

B1. Develop contactor designs that combine advantages of cocurrent, crossflow, and countercurrent cascades.

C. Derivations

C1. Derive the relationships among the different NTU terms for binary distillation.

C2. Derive Eq. (16-94).

C3. Derive the following equation to determine n_OG for distillation at total reflux for systems with constant relative volatility:

C4. The correlation for mass transfer of rigid drops in swarms, Eq. (16-103), consists of the diffusion-only result added to the effect of stirring on the mass transfer rate.

a. Show that Eq. (16-103) becomes Eq. (16-98a) when the stirrer is turned off.

b. Show that under normal mixer operating conditions (e.g., in Example 16-5) the stirring term [second term on RHS of Eq. (16-103)] >> 2.0. The value 2.0 is the diffusion-limited term.

c. When the second term on RHS of Eq. (16-103) >> 2.0, we can neglect the 2.0. Thus, if we have a mass transfer coefficient for one condition and change only one variable (e.g., particle diameter), we can find the mass transfer coefficient at the new condition by writing the equation for both sets of conditions, taking the ratio of the two equations, and noting that everything except k_C and d_p divides out. Show that the final result is

k_C,new = k_C,old(d_p,new/d_p,old)^–0.17333

C5. Extraction is almost invariably a ternary mass transfer problem instead of binary because of partial miscibility of diluent and solvent. Typically, as solute is removed from diluent, solvent is less soluble in diluent and must also be transferred to the extract phase. As the extract phase gains more solute, diluent becomes more soluble in the extract, and there will be transfer of diluent into the extract phase. For systems with low partial miscibility (e.g., water-chloroform-acetone in Table 13-5) the amount of additional mass transfer is small, and binary analysis is probably accurate. For systems with significantly more miscibility (e.g., methylcyclohexane-toluene-ammonia at the higher temperatures in Table 13-15) significant amounts of diluent and solvent can be transferred. Set up, but do not solve the Maxwell-Stefan difference equations for a ternary extraction. Since formation of two liquid layers occurs in nonideal systems, the model needs to use the nonideal form of the Maxwell-Stefan equations (see Problem 15.C10). Show how you would calculate Δx_A, Δx_B, , , , ρ_m. Unfortunately, activity coefficient and diffusivity or mass transfer data required are often not available.

C6. Derive the following expression for determining K_ya from the measurement of E_MV in a distillation column if the flow pattern is plug flow.

K_ya = –[V/(hA_active)]ln{1 – [L/(mV)]ln[E_MV(mV/L) + 1]}

D. Problems

*Answers to problems with an asterisk are at the back of the book.

D1.* For Examples 16-1 and 16-2, estimate an average H_OG in the enriching section. Then calculate n_OG and h_E = H_OG,avg n_OG.

D2.* If 1.0-in. metal Pall rings are used instead of 2.0-in. rings in Example 16-2:

a. Recalculate flooding velocity and required diameter.

b. Recalculate H_G and H_L in the enriching section.

D3. In part E of Example 16-2 a HETP value of 2.15 ft is calculated for the top of the enriching section. Since the average error in individual mass transfer coefficients k_y and k_x can be ±24.4% (Wankat and Knaebel, 2008), calculate the range of HETP values at the top of the column (m = 0.63) and for a geometric average of m over the enriching section (m = 0.577). Determine the safety factor that should be used compared to the 2.15 ft originally calculated.

D4.* A distillation column is separating a two-phase feed that is 60.0% liquid, 40.0 mol% methanol, and 60.0 mol% water. Distillate product is 92.0 mol% methanol, and bottoms is 4.0 mol% methanol. A total reboiler and a total condenser are used. Reflux is a saturated liquid. Operation is at 101.3 kPa. Assume CMO, and use L/D = 0.9. Under these conditions H_G = 1.3 ft and H_L = 0.8 ft in both the enriching and stripping sections. Determine the required heights of both the enriching and stripping sections. Equilibrium data are given in Table 2-7.

D5. A column with 6.0-ft of packing can be operated as a stripper with liquid feed, as an enricher with vapor feed, or at total reflux. We are separating methanol from isopropanol at 101.3 kPa. Equilibrium can be represented by constant relative volatility, α = 2.26.

a. At total reflux vapor mole fraction methanol entering column y_in = 0.650, and vapor mole fraction methanol leaving y_out = 0.956. Determine n_OG and the average value of H_OG.

b. We operate the system as an enricher with L/D = 2.0. The vapor mole fraction methanol entering the column y_in = 0.783 (this is feed), and vapor mole fraction methanol leaving y_out = 0.940. Determine n_OG and the average value of H_OG.

c. This problem was generated with a constant HETP. Why do estimates in parts a and b differ?

D6.* A distillation column operating at total reflux is separating methanol from ethanol. Average relative volatility is 1.69. Operation is at 101.3 kPa. We obtain methanol mole fractions of y_out = 0.982 and y_in = 0.016.

a.* If there is 8.01 m of packing, determine the average H_OG using the result of Problem 16.C3.

b. Check your results for part a using Eq. (16-23) with values of y – y* determined from Eq. (2-22b) using y = x for total reflux.

D7. A distillation column operating at total reflux is separating acetone and ethanol at 1.0 atm. The height of packing is 2.0 m. The column has a partial reboiler and a total condenser. The bottoms composition is x = 0.10, and distillate composition is 0.90. Equilibrium data are in Problem 4.D7. Estimate the average value of H_OG.

D8.* We wish to strip SO₂ from water using air at 20.0°C. Inlet air is pure. Outlet water contains 0.01 mol% SO₂, and inlet water contains 0.11 mol% SO₂. Operation is at 855 mm Hg, and L/V = 0.9(L/V)_max. Assume H_OL = 0.84 m and that the Henry’s law constant is 22,500 mm Hg/mole fraction SO₂. Calculate the packing height required.

D9. Calculate stage efficiency for the mixer in an extraction system for values of n_O–ED varying from 0.1 to 100.0 for both the completely mixed staged model and the differential equation model. Compare the results.

D10.* A packed tower is used to absorb ammonia from air using aqueous sulfuric acid. Gas enters the tower at 31.0 lbmol/(h-ft²) and is 1.0 mol% ammonia. Aqueous 10.0 mol% sulfuric acid is fed at a rate of 24.0 lbmol/(h-ft²). Equilibrium partial pressure of ammonia above a solution of sulfuric acid is zero. We desire an outlet ammonia composition of 0.01 mol% in gas.

a. Calculate n_OG for a countercurrent column.

b. Calculate n_OG for a cocurrent column.

c. What is the importance of gas and liquid flow rates?

D11. Water originally saturated with carbon tetrachloride (CCl₄) at 25.0°C and 1.0 atm is stripped with pure air at 25.0°C and 745 mm Hg. Exit water has 1.0 ppm (mole) of CCl₄. Operation is at L/V = 0.8(L/V)_max. If H_OL = 0.62 m, find the height of packing. Data are in Table 12-2.

D12.* We wish to absorb ammonia into water at 20.0°C. At this temperature H = 2.7 atm/mole fraction. Pressure is 1.1 atm. Inlet gas is 1.3 mol% NH₃, and inlet water is pure water.

a. In a countercurrent system we wish to operate at L/G = 15.0(L/G)_min. A y_out = 0.00004 is desired. If H_OG = 0.75 ft at V/A_c = 5.7 lbmol air/(h-ft²), determine the height of packing.

b. For a cocurrent system a significantly higher V/A_c can be used. At V/A_c = 22.8, H_OG = 0.36 ft. If the same L/G is used as in part a, what is the lowest y_out that can be obtained? If y_out = 0.00085, determine the packing height.

D13. Repeat Example 16-3 to determine n_OG, except use a cocurrent absorber.

a. Same conditions as Example 16-3. If specifications can be met, find y_out and n_OG. If specifications cannot be met, explain why not.

b. Same conditions as Example 16-3, except x_out = 0.002. If specifications can be met, find y_out and n_OG. If specifications cannot be met, explain why not.

c. Same conditions as Example 16-3, except x_out = 0.0003 and L/V = 40.0. If specifications can be met, find y_out and n_OG. If specifications cannot be met, explain why not.

D14.* We are separating methanol and water in a staged distillation column at total reflux to determine Murphree efficiency. Pressure is 101.3 kPa. The column has a 2.0-in. head of liquid on each well-mixed stage. Molar vapor flux is 30.0 lbmol/(h-ft²). Near the top of the column, when x = 0.8 we measure E_MV = 0.77. Near the bottom, when x = 0.16, E_MV = 0.69. Equilibrium data are given in Table 2-7.

a. Calculate k_xa and k_ya.

b. Estimate E_MV when x = 0.01.

D15. The large-scale column in Example 16-4 has a saturated liquid with feed mole fraction z = 0.5, and separation is essentially complete (x_dist ∼ 1.0 and x_bot ∼ 0). The Murphree vapor efficiency is often approximately constant in columns. Assume the value calculated in Example 16-4, E_MV = 0.97, is constant in the large-scale column (plug flow trays). Calculate E_pt and K_ya in the stripping section at x = 0.10 and x = 0.30 and in the enriching section at x = 0.9 and at x = 0.7 (shown in Example 16-4) as a check on your procedure.

D16. Although the largest errors in calculating the height of a packed column are errors in 1) mass transfer coefficients and 2) VLE data, calculation errors can also be significant because calculation of n_G and n_OG both require subtracting y values that are close to each other and taking the inverse of this difference. Suppose that all values of y_AI – y_A in the table in Example 16-1 are too low by 0.001 (thus, for y_A = 0.8 the value of y_AI – y_A should be 0.013). Recalculate the area for n_G,enriching and the required height of packing in the enriching section.

D17. Errors in mass transfer coefficients obviously affect the value of H_G and hence the height of the packed section. These errors also affect calculation of y_AI and thus calculation of n_G and the height of the packed section. Return to the calculation in Example 16-1. We want to calculate the change in y_AI – y_A and in 1/(y_AI – y_A). We will do this calculation for a value of y_A = 0.8. Assume that the equilibrium values given in Table 2-1 follow a straight line between x = 0.7472 and x = 0.8943, and determine the equation for this straight line.

a. Calculate the range of values of 1/(y_AI – y_A) for a value of y_AI = 0.8 if H_G varies by ±24.4% compared to the 0.4054 m used in Example 16-1, but H_L does not vary from the 0.253-m value. Then, assuming the percent of error in 1.0/(y_AI – y_A) is constant throughout the enriching section, estimate range in values of n_G and in height of packing (remember that H_G varies).

b. Calculate the range of values of 1/(y_AI – y_A) for a value of y_A = 0.8 if both H_G and H_L vary by ±24.2% compared to 0.4054 m and 0.253 m used in Example 16-1 for H_G and H_L, respectively. Then, assuming the percent of error in 1.0/(y_AI – y_A) is constant throughout the enriching section, estimate the range in values of n_G and in the height of packing (remember that H_G varies).

D18. For extraction of benzoic acid from water into toluene with toluene the dispersed phase, we measure the following mole fractions of benzoic acid: x_D,in = 0, x_D,out = 0.00023, and x_C,out = 1.99 × 10^–6. The mixer is 0.75-m tall and has a 0.75-m diameter. Flow rate of the dispersed phase is Q_D = 0.0012 m³/s and Q_C = 0.097 m3/s. Data for density, equilibrium, and molecular weight are in Example 16-5.

a. Determine stage efficiency E_{MD,mole_frac}. Note: The unit conversion in Eq. (16-94) is required to calculate m for equilibrium, y = mx, in mole fractions.

b. Calculate value of k_O–EDa using the completely mixed staged model.

c. Calculate value of k_O–EDa using the differential equation model.

d. If by accident the value of k_O–EDa calculated using the completely mixed staged model is used to predict the stage efficiency using the differential equation model, what (incorrect) value of E_{MD,mole_frac} is obtained?

D19. For extraction of benzoic acid from water into toluene with toluene the dispersed phase, we measure the following concentrations of benzoic acid: C_D,in = 0, C_D,out = 0.00023, and C_C,out = 0.00536 with concentrations in mol/m³. The mixer is 0.75-m tall and has a 0.75-m diameter. Flow rate of the dispersed phase is Q_D = 0.0012 m³/s and Q_C = 0.097 m³/s. ϕ_d = 0.09. Data for density, equilibrium, and molecular weight are in Example 16-5.

a. Determine stage efficiency E_MD,Conc.

b. Calculate value of k_O–EDa in concentration units using the completely mixed staged model.

c. Calculate value of k_O–EDa in concentration units using the differential equation model.

d. If by accident the value of k_O–EDa calculated using the completely mixed staged model is used to predict stage efficiency using the differential equation model, what (incorrect) value of E_MD,Conc is obtained?

e. This problem is a clone of Problem 16.D18 but with a different value for continuous-phase concentration and in different units.

D20. If 1-in. metal Raschig rings are used instead of 2-in. rings in Example 16-2:

a. Recalculate the flooding velocity and the required diameter.

b. Recalculate H_G and H_L in the enriching section.

D21.* Estimate average particle diameter, mass transfer coefficients, and mixer stage efficiency for extraction of benzoic acid from water into toluene for Example 13-6.

D22. A small distillation column with a partial reboiler, a total condenser, and a liquid-liquid separator is separating 100.0 kmol/h of saturated liquid feed that is 19.0 mol% water and 81.0 mol% n-butanol. Operation is at 1.0 atm. Distillate product (from liquid-liquid separator) is 97.5 mol% water. Bottoms product is 4.0 mol% water. Boilup ratio = 0.50. Sieve trays in the column have approximately a 5.0-cm height of liquid. Column diameter is 1.0 m. In the enriching section when x_W = 0.48, E_MV was measured as 0.68. In the stripping section when x_W = 0.16, E_MV was 0.59. Assume trays are well mixed.

a. Calculate k_xa and k_ya.

b. Estimate E_MV when x_W = 0.36.

c. Suggestion: Plot equilibrium data on a y-x diagram, and find m by drawing tangent lines.

D23. Assume trays are plug flow (see Problem 16.C6), and repeat Problem 16.D22 parts a and b. In addition calculate E_pt for the three mole fractions x_W = 0.48, 0.36, and 0.16. Compare your E_pt values and the E_MV value. Explain what is occurring.

D24. Use the Peclet number [Eq. (16-111a)] to determine which model (completely mixed or plug flow) is appropriate for the distillation column calculation at x_W = 0.48 in Problems 16.D22 and 16.D23. To determine z₁ use the tray geometry discussed in Section 10.5.

Data: MW butanol = 74.12, MW water = 18.0, assume ideal gas.

Density butanol water mixtures at 298.15 K: x_but (mole fraction) = 0.46527, ρ = 0.83283 g/cm³.

x_but (mole fraction) = 0.53444, ρ = 0.82205 g/cm³.

Source: Egorov and Makarov (2012).

E. More Complex Problems

E1. The value of E_MD = 0.927 in Example 16-5 is lower than the 0.95 value your supervisor expects. You are tasked to see if the current mixer with the current feed rate can be coaxed into operating at E_MD = 0.95 by reducing solvent flow rate S. Your immediate answer is no, but your supervisor wants proof.

a. Find solvent flow rate that gives E_MD = 0.95.

b. Calculate new values of x_D,out and x_C,out.

c. Find new value of ϕ_D.

d. Estimate new values of d_p and a.

e. Estimate new values of k_C and K_O–EDa.

f. Estimate new values of n_O–ED and E_MD.

g. Suggest operating changes that might work.

Data are listed in Example 16-5, and although the mixer is a different size, the physical property data in Example 13-6 also apply to this system. The controlling resistance is continuous-phase mass transfer. Assume drops are rigid.

G. Computer Simulation Problems

G1. Aspen Plus is not programmed to use a mass transfer approach (HTU-NTU) for binary packed-bed distillation. However, you can obtain accurate values of (y* – y) and m for distillation of binary mixtures from Aspen Plus. Then, Eq. (16-19) can be integrated without assuming that H_OG = V/(K_yaA_c) is constant. If we assume H_L and H_G in Eq. (16-27a) are constant in each column section and we expand Eq. (16-19) before integrating, we obtain

The integrals can be integrated numerically external to Aspen Plus once appropriate values have been determined. Note that vapor and liquid leaving a stage in Aspen Plus results are in equilibrium (y₃ value is y₃* in equilibrium with x₃). Vapor stream y₄ is a passing stream with liquid stream x₃ (they are on operating line). For example, if we pick y₄ in Aspen’s tray compositions in results, x₃ is on the operating line with y₄, and y₃ is y* value needed to calculate y* – y in Eq. (16-21a) [in other words, (y* – y) for this example = Aspen values for (y₃ – y₄)]. Then m is the slope of the equilibrium curve at y (= y₄ from Aspen). One method to find m is to run Analysis in Aspen with 100 points. Then numerically determine

slope = (y_n+1 – y_n)/(x_n+1 – x_n) if y is near the center of (y_n+1 – y_n) or as

slope = (y_n+1 – y_n–1)/(x_n+1 – x_n–1) if y is near the value of y_n.

Then m is the average of the slopes at x (corresponds to y_A*) and x₁ (corresponds to y_A,1); however, x₁ is unknown. To simplify the calculations, calculate m from equilibrium at the average value, y_A,avg = (y_A + y_A*)/2.0.

Use Aspen Plus with NRTL for VLE to simulate the distillation problem in Example 16-1. First, find the optimum feed plate and the total number of stages that give x_D = 0.80 ± 0.0012 and x_B = 0.02 ± 0.0004. Then, use Eq. (16-21a) and the values obtained from Aspen Plus to determine the height of the stripping section. Compare to Example 16-1.

G2. Continue the calculation in Problem 16.G1, but determine the height of the enriching section.

G3. (Fairly involved problem.) We wish to distill 80.0 mol/s of a saturated vapor feed at 15.0 atm. The feed is 10.0 mol% ethane, 30.0 mol% propane, 50.0 mol% n-butane, and 10.0 mol% n-pentane. The column operates at 15.0 atm, has a partial condenser, and produces a vapor distillate. A kettle-type reboiler is used. Our goal is to design a column using mass transfer rate analysis that will have a maximum n-butane mole fraction in the vapor distillate of y_D,C4,max = 0.00875 and a maximum mole fraction of propane in the liquid bottoms of x_{Bot, C3,max} = 0.005833.

a. Since this feed is the same as in Lab 13 except that the feed is a saturated vapor, start with N (Aspen notation) = 35 and the feed on stage 16. Increase reflux ratio to L/D = 5.0, and distillate flow rate = 32.0 mol/s. Do an equilibrium run with Tray Sizing with one section to see if one section works.

b. Then, find (L/D)_min with equilibrium runs, and operate at L/D = 1.2(L/D)_min. Find the optimum feed stage, the number of stages needed, and the column diameter at 75.0% flooding calculated with Fair’s method for an equilibrium-staged model.

c. Convert to a rate model at 75.0% flood. Use default values for tray and downcomer design parameters. With Aspen Plus use the Chen and Chuang (1993) mass transfer rate correlations, the Chilton and Colburn correlation for heat transfer, the Zuiderweg (1982) interfacial area correlation, and the VPLUG flow model. In the Design tab of Tray Rating click on the box labeled “Design mode to calculate column diameter.” Check that the DC backup/Tray spacing and weir loading are okay. Find the optimum feed stage with the rate-based model and the number of stages that just gives the desired purities.

d. Do a rate model again at 75.0% flood. Use default values for tray and downcomer design parameters. With Aspen Plus use the Chen and Chuang (1993) mass transfer coefficient correlation, the Chilton and Colburn correlation for heat transfer, the Zuiderweg (1982) interfacial area correlation, and the Mixed-flow model. Use the design mode in tray-rating design to calculate the column diameter. Check that the DC backup/Tray spacing and weir loading are okay. Find the optimum feed stage with the rate-based model and the number of stages that just gives the desired purities.

e. Compare and explain results in parts b, c, and d.

H. Spreadsheet Problems

H1. Use a spreadsheet program with a sixth-order polynomial fit (see Appendix 2B [at the end of Chapter 2]) for the ethanol-water VLE to determine n_OG, H_OG, and the height of the enriching section for Problem 16.D1.

H2. Use a spreadsheet program with a sixth-order polynomial fit (see Appendix 2B [at the end of Chapter 2]) for the ethanol-water VLE to determine n_OG, H_OG, and the height of the stripping section for Examples 16-1 and 16-2.

Lab 13. Rate-Based Modeling of Distillation

Goal:

Use Aspen Plus simulation to explore detailed rate-based analysis of multicomponent distillation.

Preparation:

• Review computer labs as needed.

• Do Lab 10, or find your results from when you did Lab 10.

• Read Sections 15.7 and 16.8

I. Problem Setup. Set up and run the following distillation problem* on Aspen Plus.

1. We wish to distill 80.0 mol/s of a feed at 25.0°C and 15.0 atm. The feed is 10.0 mol% ethane, 30.0 mol% propane, 50.0 mol% n-butane, and 10.0 mol% n-pentane. The column operates at 15.0 atm, has a partial condenser, and produces a vapor distillate. A kettle-type reboiler is used. Our goal is to design a column using mass transfer rate analysis that will have a maximum n-butane mole fraction in the vapor distillate of y_D,C4,max = 0.00875 and a maximum mole fraction of propane in the liquid bottoms of x_{Bot, C3,max} = 0.005833.

Preliminary analysis with an equilibrium model using the Peng-Robinson VLE was done in Lab 10. Because the flow rate in Lab 10 is significantly higher, the diameter is scaled to the reduced feed rate of this problem using the ratio

After doing this scaling, Lab 10 shows that the following variables are a reasonable starting point: N (Aspen notation) = 35, and the feed is on stage 16. Reflux ratio is L/D = 2.5, and distillate flow rate = 32.0 mol/s. The column has two sections, section 1 is stages 2 to 15, and section 2 is stages 16 to 34. Each section has one pass. For initial diameters use Dia₁ = 1.36 m and Dia₂ = 1.745 m. In both sections use sieve plates with the Fair flooding calculation method with flooding at 70.0%. Choose tray spacing = 0.60 m, weir height = 0.0508 m, hole diameter = 0.0127 m, fraction sieve hole area to active area = 0.12, and downcomer clearance = 0.0381 m. Leave other variables in Tray Rating blank, which uses the default numbers for the variables in both sections.

2. If you have not already done Lab 10, you should perform a number of equilibrium runs to find reasonable operating conditions and diameters for the column sections. If you have done Lab 10, this step can be skipped.

3. Draw the column using RadFrac. The Setup, Components, Properties, and Streams are input in exactly the same way as for other Aspen Plus simulations. In the Block (for your distillation column)→Setup→Configuration Tab→Calculation Type menu, select Rate-Based. Complete the other items in the Configuration and other tabs as normal.

4. Now click Tray Rating. In the Object Manager, click New, and call this section 1. In the Specs tab, input the stages for this section (2 to 15), pick Sieve from the Tray Type menu, and input the other variables as listed in item 1. In the Design/Pdrop tab, pick the Fair flooding calculation method, and do NOT check Update section pressure drop. In the Layout tab, the tray type is sieve, and use the defaults for the other items. At this point do not put any numbers into the Downcomers tab (this means default values will be used).

5. In the menu on the left side, you will see Tray Rating, Section 1.Click on the arrow next to Section 1. Below Setup there will be Rate-Based. Click Rate-Based. In the Rate-Based tab, click the box labeled Rate-based calculations (Aspen Plus allows you to have sections with equilibrium calculations as long as at least one section is done rate based). Clicking this box activates the menus below. Use the default values for Calculation Parameters. Select the Mixed-flow model (called Mixed-Mixed in the report), which assumes that vapor and liquid are well mixed so that the bulk properties are the same as the exit properties. This model is appropriate for trays (not packing) and is used in Section 16.6 to derive Eq. (16-77) for binary distillation. The effect of flow model is looked at in item 10. Select Film (not Filmrxn) for both Liquid and Vapor in the Film Resistance section, and select No for both nonideality corrections.

6. In the Correlations tab there are three menus. Unfortunately, there is no single best choice for the Mass transfer coefficient method correlation to use. We will first use the Chen and Chuang (1993) correlation and try others later. For the Heat transfer coefficient method, the Chilton and Colburn correlation is standard, and for the Interfacial area method, the Zuiderweg (1982) correlation is most commonly used.

7. In the Design tab, click the box labeled Design mode to calculate column diameter. This means a different diameter than you specified will be used. The Base stage is usually selected as the stage that floods first—from the equilibrium runs, this is stage 4. Base flood is 0.7, since we are designing at 70.0% of flooding.

8. Repeat steps 4 to 7 but for section 2. Return to and click Tray Rating, and in the Sections tab click New, and call it section 2. In the tabs, section 2 is from stages 16 to 34 and has a different diameter (1.745 m) than section 1. Other values are the same as in section 1. Items 5 and 6 are the same as for section 1. In item 7 the Base Stage for section 2 is stage 34.

9. Click Next, run the simulation, and look at the Report (View→Report→check the box for Block IDs). Look at the distillate (y values for stage 1) and bottoms (x values for stage 35) mole fractions to be sure the specifications are met. Note that the report now contains a Tray Rating section and that under the heading Rating Results different diameters have been calculated for each section. Look at the Downcomer (DC) backup for both sections. Both sections should be OK (DC backup/Tray spacing < 1/2). Next check the weir loading. It should be < 70.0 m²/h = 0.01944 m²/s (Torzewski, 2009). Both of the sections should be OK. Also look at the velocity in the downcomers. With a minimal foaming system it should be less than 0.21 m/s in both sections. Record values of parameters needed for comparison in step 10.

10. We will now look at the effect of changing the flow model. Go to Tray Rating, section 1, and click Rate-Based (if Rate-based is not shown, click the arrow next to section 1). Then in the Rate Based tab under Calculation parameters in the menu for Flow model, select VPLUG (called Mixed-Plug in the report). Repeat these steps for section 2. In the VPLUG model the vapor is assumed in plug flow on the tray while the liquid is mixed. The vapor concentration is calculated as an average of the plug flow concentrations. VPLUG can be used with either trays or packing. Click Next, and run the simulation. Compare the results (y_D,C4, x_Bot,C3, DC backup/Tray spacing, and weir loading) with the results from item 9. You should see very little difference in DC backup/Tray spacing and weir loading and a better separation with the VPLUG model. Data on the trays (either direct observation data of flow patterns or comparison of results with models) can be used to determine which model is more appropriate. Otherwise, the Peclet number can be estimated from Eq. (16-111a) and used to determine if a mixed or plug flow model is more appropriate. The conservative approach is to select the Mixed model.

11. We will now look at other mass transfer correlations. Use the Mixed model for both sections.

a. In both sections select the Chan and Fair (1984) mass transfer correlation. Click Next, and run the simulation. Compare y_D,C4, x_Bot,C3, DC backup/Tray spacing, and weir loading with the runs from items 9 and 10.

b. In both sections select the Zuiderweg (1982) mass transfer correlation. Click Next, and run the simulation. Compare y_D,C4, x_Bot,C3, DC backup/Tray spacing, and weir loading with the runs from items 9, 10, and 11a.

The DC backup/Tray spacing and weir loading should be acceptable for all three runs. For a conservative design, select the worst separation (highest values of y_D,C4 and x_Bot,C3). Check if this separation meets the specifications.

12. The column is a workable design but is not optimized. Choose the Mixed model and the mass transfer correlation with the worst separation (this will be most conservative and hence a safe design) and partially optimize. Find the optimum feed stage based on lowest values of y_D,C4 and x_Bot,C3 and then find the minimum number of stages that will just give the desired separation or slightly better. Report your partially optimized design (N, N_F, y_D,C4, x_Bot,C3, Q_c, Q_R, and for both sections the diameters, values of DC backup/Tray spacing, and weir loadings). Note: As the feed stage and the total number of stages are changed, the starting and ending stages for the sections in Tray Rating have to be adjusted accordingly. When changing the total number of stages, it may be necessary to reinitialize to obtain convergence.