Chapter 15: Mass Transfer Analysis

Up to now we have used an equilibrium stage analysis procedure even in packed columns where there are no stages. A major advantage of this procedure is that it does not require determination of the mass transfer rate.

In packed columns, it is conceptually incorrect to use the staged model even though it works if the correct height equivalent to a theoretical plate (HETP) is used. In this chapter we will develop a physically more realistic model for packed columns that is based on mass transfer between the phases. After developing the model for distillation, we will discuss mass transfer correlations that allow us to predict the required coefficients for common packings. Next, we will repeat the analysis for both dilute and concentrated absorbers and strippers and analyze cocurrent absorbers. Finally, a simple model for mass transfer on a stage will be developed, and the estimation of stage efficiency will be considered.

Although they are reviewed in section 15.1, it is assumed that readers have some previous knowledge of basic mass transfer concepts (e.g., Cussler, 1997; Geankoplis, 2003; McCabe et al., 2005; Taylor and Krishna, 1993).

15.1 BASICS OF MASS TRANSFER

Some basic concepts of mass transfer were considered on Section 1.3. More details will be presented here. The basic mass transfer equation [a repeat of Eq. (1-4)] is:

(15-1)

The driving force is the concentration difference that drives the mass transfer and can be represented as a difference in mole fracs, a difference in partial pressures, a difference in concentrations, and so forth. The mass transfer coefficient includes the effect of diffusivity and flow conditions, and its units depend upon the units used for the other terms. We will typically use a driving force in mole fracs and will have lb moles/hr-ft² or kg moles/hr-m² for the units of the mass transfer coefficient. Other definitions of the mass transfer coefficient are outlined in Table 15-1.

TABLE 15-1. Definitions of mass transfer coefficients and HTUs

For equilibrium-staged separations we are interested in mass transfer from one phase to another. This is illustrated schematically in Figure 15-1 for the transfer of component A from the liquid to a vapor phase. x_I and y_I are the interfacial mole fracs. For dilute absorbers and strippers and for distillation where there is equimolar countertransfer of the more volatile and less volatile components, the mass transfer equation becomes

(15-2a)

or

(15-2b)

where k_y and k_x are the individual mass transfer coefficients for the vapor and liquid phases, respectively. These equations define k_y and k_x. Unfortunately, there are two major problems with these equations when they are applied to vapor-liquid and liquid-liquid contactors. First, the interfacial area A₁ between the two phases is very difficult to measure. This problem is usually avoided by writing Eq. (15-2) as

(15-3a)

(15-3b)

where a is the interfacial area per unit volume of the column (m²/m³ or ft²/ft³). Since a is often no easier to measure than A_I, we usually measure and correlate the products k_ya and k_xa. Typical units for ka are kg moles/hr-m³ or lb moles/hr-ft³ (see Table 15-1).

Figure 15-1. Mass transfer at interface

The second problem is that the interfacial mole fracs are also very difficult to measure. To avoid this problem, mass transfer calculations often use a driving force defined in terms of hypothetical equilibrium mole fracs.

(15-4a)

(15-4b)

These equations [which are a repeat of Eq. (1-5)] define the overall mass transfer coefficients K_y and K_x. is the vapor mole frac which would be in equilibrium with the bulk liquid of mole frac x_A, and is the liquid mole frac in equilibrium with the bulk vapor of mole frac y_A.

To obtain the relationship between the overall and individual coefficients, we begin by assuming that there is no resistance to mass transfer at the interface. This assumption implies that x_I and y_I must be in equilibrium. The mole frac difference in Eq. (15-4a) can be written as

(15-5a)

where m is now the average slope of the equilibrium curve at x_A and x_AI.

(15-5b)

Combining Eq. (15-5a) with Eqs. (15-3) and (15-4a), we obtain

which leads to the result

(15-6a)

Similar manipulations starting with Eq. (15-4b) lead to

(15-6b)

This sum-of-resistances model shows that the overall coefficients will not be constant even if k_x and k_y are constant if the equilibrium is curved and m varies. Equation (15-6) also shows the effect of equilibrium on the controlling resistance. If m is small, then from Eq. (15-6a) K_y ~ k_y and the gas-phase resistance controls. If m is large, then Eq. (15-6b) gives K_x ~ k_x and the liquid-phase resistance controls. If there is a resistance at the interface (for example, if a surface-active agent is present), then this resistance must be added to Eq. (15-6).

15.2 HTU-NTU ANALYSIS OF PACKED DISTILLATION COLUMNS

Consider the packed distillation tower shown in Figure 15-2. Only binary distillation with constant molal overflow (CMO) will be considered. Let A be the more volatile component and B the less volatile component. In addition to making L/V constant and satisfying the energy balances, CMO automatically requires equimolal counterdiffusion, N_A = −N_B. Thus, CMO simplifies the mass balances, eliminates the need to solve the energy balances, and simplifies the mass transfer equations. We will also assume perfect plug flow of the liquid and vapor. This means that there is no eddy mixing to reduce the separation.

Figure 15-2. Packed distillation column

The mass transfer can be written in terms of the individual coefficients Eqs. (15-3) or overall coefficients Eqs. (15-4). For the differential height dz in the rectifying section, the mass transfer rate is

(15-7)

where N_A is the flux of A in kg moles/m²-hr or lb moles/ft²-hr and A_c is the column cross-sectional area in m² or ft². This equation then has units of kg moles/hr or pounds moles/hr. The mass transfer rate must also be equal to the changes in the amount of the more volatile component in the liquid and vapor phases.

(15-8)

where L and V are constant molal flow rates. Combining Eqs. (15-7) and (15-8), we obtain

(15-9)

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

(15-10)

We have assumed that the term V/(k_yaA_c) is constant. The limits of integration for y_A in each section are shown in Figure 15-2. Equation (15-10) is often written as

(15-11)

where the height of a gas-phase transfer unit H_G is

(15-12a)

and the number of gas-phase transfer units n_G is

(15-12b)

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.

An exactly similar analysis can be done in the liquid phase by starting with Eq. (15-3b). The result for each section is

(15-13)

which is usually written as

(15-14)

where

(15-15a,b)

In order to do the integrations to calculate n_G and n_L we must relate the interfacial mole fracs y_AI and x_AI to the bulk mole fracs y_A and x_A. To do this we start by setting Eqs. (15-3a) and (15-3b) equal to each other. After simple rearrangement, this is

(15-16)

where 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 fracs (y_AI, x_AI) to the point representing the bulk mole fracs (y_A, x_A). Since there is no interfacial resistance, the interfacial mole fracs are in equilibrium and must be on the equilibrium curve (Figure 15-3A). The bulk mole fracs are easily related by a mass balance through segment dz around either the top or the bottom of the column. This operating line in the rectifying section is

(15-17)

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

(15-18)

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

Figure 15-3. Analysis of number of transfer units; A) determination of equilibrium or interfacial values, B) graphical integration of Eq. (15-12b) shown for stripping section of Example 15-1

We can now use a modified McCabe-Thiele diagram to determine x_AI and y_AI. From any point (y_A, x_A) on the operating line, draw a line slope −k_xa/k_ya. The intersection of this line with the equilibrium curve gives the interfacial mole fracs that correspond to y_A and x_A (see Figure 15-3A). After this calculation is done for a series of points, we can plot 1/(y_AI − y_A) vs y_A as shown in Figure 15-3B. The area under the curve is n_G. n_L is determined by plotting 1/(x_A − x_AI). The areas can be determined from graphical integration or numerical integration such as Simpson’s rule [see Eq. (9-12) and Example 15-1].

It will be most accurate to do the calculations for the stripping and enriching sections separately. For example, in the stripping section,

(15-19a,b)

In the determination of n_G for the stripping section, y_A,in,S is the vapor mole frac leaving the reboiler. This is illustrated in Figure 15-2 for a partial reboiler. Mole frac y_A,out,S is the mole frac leaving the stripping section. This mole frac can be estimated at the intersection of the operating lines. This is shown in Figure 15-3A. Note that this estimate makes y_A,out,S = y_A,in,E.

Calculating the interfacial mole fracs adds an extra step to the calculation. Since it is often desirable to avoid this step, the overall mass-transfer coefficients in Eq. (15-4) are often used. In terms of the overall driving force the mass transfer rate corresponding to Eq. (15-7) is

(15-20)

Setting this equation equal to Eq. (15-8), we obtain

(15-21)

Integration of this equation over a section of the column gives

(15-22)

This equation is usually written as

(15-23)

where the height of an overall gas phase transfer unit is

(15-24a)

and the number of overall gas phase transfer units is

(15-24b)

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

(15-25)

where

(15-26a,b)

The advantage of this formulation is that is easily found from vertical lines shown in Figure 15-3A. The value can be found from horizontal lines as shown in the figure. The number of transfer units, n_OG or n_OL, is then easily determined. Calculation of n_OG is similar to the calculation of n_G illustrated in Figure 15-3B. The disadvantage of using the overall coefficients is that the height of an overall transfer unit, H_OG or H_OL, is much less likely to be constant than H_G or H_L. This is easy to illustrate, since we can calculate the overall HTU from the individual HTUs. For example, substituting Eq. (15-6a) into Eq. (15-23) and rearranging, we obtain

(15-27a)

H_OL can be found by substituting Eq. (15-6b) into Eq. (15-26a)

(15-27b)

Obviously, H_OG and H_OL are related:

(15-27c)

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 in the column. The various NTU values must be related, since h in Eqs. (15-11), (15-14), (15-23), and (15-25) is obviously the same, but the HTU values vary. These relationships are derived in Problem 15-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.

EXAMPLE 15-1.Distillation in a packed column

We wish to repeat Example 4-3 (distillation of ethanol and water) except that a column packed with 2-inch metal pall rings will be used. F = 1,000 kg moles/hr, z = 0.2, T_F = 80° F, x_D = 0.8, x_B = 0.02, L/D = 5/3, and p = 1 atm. Use a vapor flow rate that is nominally 75% of flooding. In the enriching section H_G = 1.33 feet and H_L = 0.83 feet, and in the stripping section H_G = 0.93 feet and H_L = 0.35 feet (see Example 15-2).

Solution

A. Define. Determine the height of packing in the stripping and enriching sections.

B and C. Explore and plan. The solution obtained in Example 4-3 can be used to plot the operating lines and the feed line. These are exactly the same as in Figure 4-13. Since the ethanol-water equilibrium is very nonlinear, the design will be more accurate if an individual mass transfer coefficient is used. Thus, use Eqs. (15-11) and (15-12) for the enriching and stripping sections separately. The term (y_AI − y_A) can be determined as illustrated schematically in Figure 15-3A and for this example in Figure 15-4A. n_G can be found for each section as shown in Figures 15-3B and 15-4B.

D. Do it. The equilibrium and operating lines from Example 4-3 are plotted in Figure 15-4A. In the stripping section, Eq. (15-16) gives a slope of

where from Example 4-3 or from mass balances. Lines with a slope = −5.37 are drawn in Figure 15-4A from arbitrary points on the stripping section operating line to the equilibrium curve. Values of y_A are on the operating line, while y_AI values are on the equilibrium line. The following table was generated.

Figure 15-4. Solution to Example 15-1; A) determination of y_AI = y_EI, B) graphical integration for enriching section

From this table 1/(y_AI − y_A) vs. y_A is easily plotted, as shown in Figure 15-3B. n_G is the area under this curve from y_A,in,S = 0.17 to y_A,out,S = 0.442. y_A,in,S is the vapor mole frac leaving the partial reboiler. Determination of y_A,in,S is shown in Figure 15-4. y_A,out,S is the vapor mole frac at the intersection of the operating lines. The area in Figure 15-3B can be estimated from Simpson’s rule (although the area will be overestimated since the minimum in the curve is not included) or other numerical integration schemes.

(15-28a)

where

(15-28b)

Note that Simpson’s rule uses the end and middle points. For the stripping section, this is

And the height of packing in the stripping section is

In the enriching section the slope is

Arbitrary lines of this slope are shown on Figure 15-4A. The following table was generated.

The plot of 1/(y_AI − y_A) vs. y_A is shown in Figure 15-4B. An approximate area can be found using Simpson’s rule, Eqs. (15-28a) and (15-28b), for areas A₁ and A₂. For area A₁ the initial point is selected as the maximum point, and the middle point y_A = 0.7625 with f = 107 was calculated.

n_G,E is the total area = 19.6. Then the height in the enriching section is

h_E = H_G,E n_G,E = (1.33)(19.6) = 26.1 feet

E. Check. The operating and equilibrium curves were checked in Example 4-3. The areas can be checked by counting squares in Figures 15-3B and 15-4B. More accuracy could be obtained by dividing Figure 15-3B into two parts. The HTU values will be estimated and checked in Example 15-2.

F. Generalize. The method illustrated here can obviously be used in other distillation systems. Since the curve for n_G can be very nonlinear, it is a good idea to plot the curve as shown in Figures 15-3B and 15-4B before doing the numerical integration.

15.3 RELATIONSHIP OF HETP AND HTU

In simple cases the HTU-NTU approach and the HETP approach discussed in Chapter 10 can be related with a derivation similar to that used for the Kremser equation (section 12.5). If the operating and equilibrium curves are straight and parallel, mV/L = 1, we have the situation shown in Figure 15-5A. The equilibrium equation is

(15-29a)

Figure 15-5. Calculation of y* − y with linear equilibrium and operating lines; A) mV/L = 1, B) mV/L ≠ 1

while a general equation for the straight operating line is

(15-29b)

Now the integral in the definition of n_OG can easily be evaluated analytically. The difference between the equilibrium and operating lines, y* − y, is

15-30

when m = L/V, this becomes

Then Eq. (15-24b) becomes

which is easily integrated.

(15-31)

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

(15-32)

N can be obtained from Eq. (12-20). Comparison of Eqs. (15-31) and (12-20) shows that N = n_OG when mV/L = 1. Thus,

(15-33)

If the operating and equilibrium lines are straight but not parallel, then we have the situation shown in Figure 15-5B. The difference between equilibrium and operating lines is still given by Eq. (15-30), but the terms with x do not cancel out. By substituting in x from the operating equation, y* − y in Eq. (15-24) can be determined as a linear function of y. After integration and considerable algebraic manipulation, n_OG is found to be

(15-34a)

where

(15-35a)

This analysis can also be done in terms of liquid mole fracs. The result is

(15-34b)

where

(15-35b)

Equations (15-34a) and (15-34b) are known as the Colburn equations. The value of HETP can be determined from Eq. (15-32), where N is found from the Kremser Eq. (12-30) and n_OG from Eq. (15-34a). This result is

(15-36)

The use of this result is illustrated in Example 15-2.

Although it was derived for a straight operating line and straight equilibrium lines, Eq. (15-36) will be approximately valid for curved equilibrium or operating lines. HETP should be determined separately for each section of the column, since mV/L is not usually the same in the enriching and stripping sections. For maximum accuracy the HETP can be calculated for each stage in the column (Sherwood et al., 1975).

15.4 MASS TRANSFER CORRELATIONS FOR PACKED TOWERS

In order to use the HTU-NTU analysis procedure we must be able to predict the mass transfer coefficients or the HTU values. There has been considerable effort expended in correlating these terms (see Wang et al., 2005, for an extensive review). Care must be exercised in using these correlations since HTU values in the literature may be defined differently. The definitions given here are based on using mole fracs in the basic transfer equations (see Table 15-1). If concentrations or partial pressures are used, the mass transfer coefficients will have different units, which will lead to different definitions for HTU although the HTU will still have units of height. In working with these correlations, terms must be expressed in appropriate units.

15.4.1 Detailed Correlations for Random Packings

We will use the correlation of Bolles and Fair (1982), for which HTUs are defined in the same way as here. The Bolles-Fair correlation is based on the previous correlation of Cornell et al., (1960a, b) and a data bank of 545 observations and includes distillation, absorption, and stripping. This model and variations on it remain in common use (Wang et al., 2005).

The correlation for H_G is

(15-37)

where ψ is a packing parameter that is given in Figure 15-6 (Bolles and Fair, 1982) for common packings, and other special terms are defined in Table 15-2. 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.

Figure 15-6. Packing parameter ψ for H_G calculation (Bolles and Fair, 1982) excerpted by special permission from Chemical Engineering, 89 (14), 109 (July 12, 1982), copyright 1982, McGraw-Hill, Inc., New York, N.Y. 10020

TABLE 15-2. Terms for Eqs. 15-37) and (15-38)

The correlation for H_L is

(15-38)

In this equation φ is a packing parameter shown in Figure 15-7, and C_fL is a vapor load coefficient shown in Figure 15-8 (Bolles and Fair, 1982). The value of u_flood in Figure 15-8 is from the packed bed flooding correlation in Figure 10-25.

Figure 15-7. Packing parameter φ for H_L calculation (Bolles and Fair, 1982) excerpted by special permission from Chemical Engineering, 89 (14), 109 (July 12, 1982), copyright 1982, McGraw-Hill, Inc., New York, N.Y. 10020

Figure 15-8. Vapor load coefficient C_fL for H_L calculation (Bolles and Fair, 1982) excerpted by special permission from Chemical Engineering, 89 (14), 109 (July 12, 1982), copyright 1982, McGraw-Hill, Inc., New York, N.Y. 10020

The calculated H_G and H_L values can vary from location to location in the column. When this occurs, an integrated mean value should be used. The overall HTU values can be obtained from Eqs. (15-27). Even if H_G and H_L are constant, H_OG and H_OL will vary owing to the curvature of the equilibrium curve.

Bolles and Fair (1982) show that there is considerable scatter in modeled HETP data vs. experimental HETP data. HETP was calculated from Eq. (15-36). For 95% 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 distillation systems is an art not a science.

EXAMPLE 15-2. Estimation of H_G and H_L

Estimate the values of H_G and H_L for the distillation in Examples 4-3 and 15-1 using 2-inch metal pall rings.

Solution

A. Define.We want to find H_G and H_L in both the stripping and enriching sections. This will be done as if we had completed Example 4-3 but not Example 15-1. Thus, we know the number of equilibrium stages required but we have not estimated packing heights.

B and C. Explore and Plan. We will use the Bolles and Fair (1982) correlation shown in Eqs. (15-37) and (15-38) and Figures 15-6 to 15-8. Obviously, we need to estimate the physical properties required in this correlation. We will do this for the striping and enriching sections separately. This estimation is easiest if a computer physical properties package is available. We will illustrate the estimation using values in the literature. The packing height, h_p, must be estimated for each section. These heights will be estimated from the number of stages in each section multiplied by an estimated HETP. Flow rates will be found from mass balances and then will be converted to weight units. A diameter calculation will be done to determine the actual percent flooding.

D. Do it. We will do calculations at the top of the column and assume that these values are reasonably accurate throughout the enriching section. Estimation of properties at the bottom of the column will be used for the stripping section. External balances give D = 230.8 kg moles/hr, B = 769.2 kg moles/hr. Flooding at top.

From the ideal gas law,

where T = 78.4°C from Figure 4-14.

Liquid density. 80 mole % ethanol is 91.1 wt %. From Perry and Green (1984), ρ_L = 0.7976 g/mL at 40° C and ρ_L = 0.82386 at 0° C. By linear interpolation, ρ_L = 0.772 g/mL at 78.4° C. At 78.4° C, ρ_w = 0.973 g/mL.

For the flooding curve in Figure 10-25 the abscissa is

Ordinate (flooding) = 0.197. Then

The (62.4)² converts ρ_L and ρ_G to lb/ft³. µ_L is estimated as 0.52. Then,

The molar vapor flow rate is

V = (L/D + 1)D = 615.4 kg moles/hr

which allows us to find the column cross-sectional area.

Round this off to 5 feet, Area = 19.6 ft². This roundoff reduces the % flooding. Actual fraction flooding = 0.75 (17.2/19.6) = 0.66

A repeat of the calculation at the bottom of the column shows that the column will flood first at the top since the molecular weight is much higher.

Estimation at top.

Liquid diffusivities. From Reid et al. (1977, p. 577), for very dilute systems D_EW = 1.25 × 10⁻⁵ cm²/s and D_WE = 1.132 × 10⁻⁵ at 25° C. The effect of temperature can be estimated since the ratio D_L µ_L/T ~ constant. At the top we want D_L at 78.4° C = 351.5 K.

Estimating viscosities from Perry and Green (1984) using 95% ethanol:

The liquid surface tension can be estimated from data in the Handbook of Chemistry and Physics.

σ_L(78.4) = 18.2 dynes/cm

σ_W(78.4) = 62.9 dynes/cm

For vapors the Schmidt number can be estimated from kinetic theory (Sherwood et al., 1975, pp. 17-24). The equation is

where the collision integrals Ω_D, Ω_v and the Lennard-Jones force constants σ_AB and σ_B are discussed by Sherwood et al. (1975). At the top of the column the result is Sc_v = 0.355.

The liquid flow rate at the top is

L = (L/D)D = 384.6 kg moles/hr.

The liquid flux W_L is

In Eq. (15-37) D′_col = 2, ψ = 141 from Figure 15-6 at 66% flood, b₁ = 1.24 and b₂ = 0.6. We can estimate h_p as (No. stages) × (HETP), where an average HETP is about 2 feet. Then

h_p = (11)(2) = 22 feet

Equation (15-37) is then

For H_L we calculate W_L as 1724 lb/hr-ft² and φ = 0.07 from Figure 15-7. C_fl = 0.81 from Figure 15-8. Then from Eq. (15-38),

Note that µ_L in Sc_L is in poise (0.01 P = 1 cP).

These calculations can be repeated for the bottom of the column. The results are: H_G,S = 0.93 feet and H_L,S = 0.35 feet.

E. Check. One check can be made by estimating HETP using Eq. (15-36). At the top of the column the slope of the equilibrium curve is m ~ 0.63. This will vary throughout the column. Then from Eq. (15-27a), at the top

Note that H_OG will vary in the enriching section since m varies. From Eq. (15-36),

This is close to our estimated HETP, so our results are reasonable. The packing heights calculated in Example 15-1, h_S = 1.66 and h_E = 26.1, differ from our initial estimates. A second iteration can be done to correct H_G and H_L. For example,

which is a 6% correction. Changing H_G and H_L will change the slopes of the lines used to calculate y_AI; thus, n_G will also change.

F. Generalize. This calculation is long and involved because of the need to estimate physical properties. This part of the problem is greatly simplified if a physical properties package is available on the computer. In this example m is close to one. Thus, both terms in Eq. (15-6) are significant and neither resistance controls. Thus, H_G and H_L are the same order of magnitude.

Models for structural packings are reviewed by Wang et al. (2005).

15.4.2 Simple Correlations

The detailed correlation is fairly complex to use if a physical properties package is not available. Simplified correlations are available but will not be as accurate (Bennett and Myers, 1982; Greenkorn and Kessler, 1972; Perry and Green, 1984; Sherwood et al., 1975; Treybal, 1955): For H_G the following empirical form has been used (Bennett and Myers, 1982; Greenkorn and Kessler, 1972; Treybal, 1955):

(15-39)

where W_G and W_L are the fluxes in lb/hr-ft², and Sc_v is the Schmidt number for the gas phase. The constants are given in Table 15-3. The expression for H_L developed by Sherwood and Holloway (1940) is

(15-40)

where Sc_L is the Schmidt number for the liquid. The constants are given in Table 15-3.

TABLE 15-3. Constants for determining H_G and H_L from Eqs. (15-39) and (15-40); range of W_L in Eq. (15-40) is 400 to 15,000

The correlations are obviously easier to use than Eqs. (15-37) and (15-38) since only the Schmidt number and the viscosity need to be estimated. However, Eqs. (15-39) and (15-40) will not be as accurate; thus, they should only be used 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 15-4 for random packings.

TABLE 15-4. Approximate H_OG values for absorption in water (Reynolds, et al, 2002); the H_OG for ceramic packing is approximately twice the H_OG for plastic packing.

15.5 HTU-NTU ANALYSIS OF ABSORBERS AND STRIPPERS

The HTU-NTU analysis for concentrated absorbers and strippers with one solute is somewhat more complex than for distillation because total flow rates are not constant and solute A is diffusing through a stagnant film with no counterdiffusion, N_B = 0. We will assume that the system is isothermal. When there is appreciable mass transfer through a stagnant film the basic mass transfer Eq. (15-2) must be modified. With N_B = 0 the mass transfer flux with respect to a fixed axis system is (Bennett and Myers, 1982; McCabe et al., 2005; Sherwood et al., 1975)

(15-41)

where J_A is the flux with respect to an axis moving at the molar average velocity of the fluid. This leads to a transfer rate equation that is superficially similar to the previous equations.

(15-42)

Now the mass transfer coefficient is defined as

(15-43)

where the logarithmic mean mole frac is defined as

(15-44)

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

We will now repeat the analysis of a packed section using Eq. (15-41) and including the nonconstant total flow rates. Figure 15-9A is a schematic diagram of an absorber. The absorber is assumed to be isothermal, and plug flow is assumed. The rate of mass transfer in a segment of the column dz is given by

(15-45)

Figure 15-9. Absorber calculation: A) schematic of column, B) calculation of interfacial mole fracs; slope, s = −k′_x/k′_y

Comparison of this equation with Eq. (15-7) shows that the sign on the mole frac difference has been switched, since the direction of solute transfer in absorbers is opposite to that of transfer of the more volatile component in distillation. In addition, the modified mass transfer coefficient is used. The solute mass transfer can also be related to the change in solute flow rates in the gas or liquid streams.

(15-46)

This equation differs from Eq. (15-8) derived for distillation since neither V nor L is constant.

The variations in V can be related to the constant flow rate of carrier gas, G.

(15-47)

which is the same as Eq. (12-12). Combining Eqs. (15-45) to (15-47), we obtain

(15-48a)

After taking the derivative, substituting in Eq. (15-47), and cleaning up the algebra, we obtain

(15-48b)

Integrating this equation we obtain

(15-49)

Substituting in Eq. (15-43), we obtain

(15-50)

The term V/k_yaA_c is the height of a gas-phase transfer unit H_G defined in Eq. (15-12a).

The variation in H_G can be determined from Eq. (15-37) and Figure 15-6, which are valid for both absorbers and distillation. The term that varies the most in Eq. (15-37) 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% change in liquid flow rate would be quite large. This will cause at most a 10% change in H_G. k_ya is independent of concentration, since the concentration effect was included in in Eq. (15-42). Since the variation in H_G over the column section is relatively small, we will treat H_G as a constant. Then Eq. (15-50) becomes

(15-51)

which is usually written as

(15-52)

Note that n_G for concentrated absorption is defined differently from n_G for distillation, Eq. (15-12b). The difference in the limits of integration in the two definitions for n_G occurs because the direction of transfer of component A in distillation is the negative 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 countertransfer as in distillation.

The method for finding the interfacial compositions is similar to that used to develop Eq. (15-16) and Figure 15-3 except that Eq. (15-42) and the corresponding equation in terms of liquid mole fracs are used as the starting point. The procedure is illustrated in Figure 15-9B. Use of this procedure lets us calculate the integrand in Eq. (15-52) at a series of points. The integral in Eq. (15-52) can be found either numerically or graphically.

Often, the integral in Eq. (15-52) can be simplified. The first simplification often employed is to replace the logarithmic mean with an arithmetic average.

(15-53)

When Eq. (15-53) is substituted into Eq. (15-52), n_G can be simplified.

(15-54)

This equation shows that n_G for absorption is essentially the n_G for distillation plus a correction factor. The interfacial mole frac y_AI can be determined as shown in Figure 15-9B. The integral in Eq. (15-54) can then be determined graphically or numerically. For very dilute systems 1 − y_A is approximately 1 everywhere in the column. Then the correction factor in Eq. (15-54) 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. Then in Eq. (15-41). 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. (15-4a) and (15-20) to (15-24), we obtain for dilute absorbers

(15-55)

where H_OG was defined in Eq. (15-24a) and

(15-56)

This n_OG is essentially the same as for distillation in Eq. (15-24b).

If the operating and equilibrium lines are straight, n_OG can be integrated analytically. The result is the Colburn equation given in Eqs. (15-31) and (15-34a). An alternative integration gives an equivalent equation.

(15-57)

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

(15-58)

where H_L was defined in Eq. (15-15a) and

(15-59)

Equation (15-59) can often be simplified to

(15-60)

For dilute systems the correction factor in Eq. (15-60) becomes negligible. For dilute systems the analysis can also be done in terms of the overall transfer coefficient.

(15-61)

where H_OL is defined in Eq. (15-26a) and

(15-62)

If the operating and equilibrium lines are both straight, n_OL can be integrated analytically. The result is the Colburn Eq. (15-34b), or the equivalent expression,

(15-63)

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 will usually not be 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.

EXAMPLE 15-3. Absorption of SO₂

We are absorbing SO₂ from air with water at 20°C in a pilot-plant column packed with 0.5-in. metal Raschig rings. The packed section is 10-feet tall. The total pressure is 741 mmHg. The inlet water is pure. The outlet water contains 0.001 mole frac SO₂, and the inlet gas concentration is y_in = 0.03082 mole frac. L/V = 15. The water flux W_L = 1,000 lb/hr-ft². The Henry’s law constant is H = 22,500 mmHg/mole frac SO₂ in liquid. Estimate H_OL for a 10-feet high large-scale column operating at the same W_L and same fraction flooding if 2-inch metal Pall rings are used.

Solution

A. Define. Calculate H_OL for a large-scale absorber with 2-inch metal Pall rings.

B. Explore. We can easily determine n_OL for the pilot plant. Then H_OL = h/n_OL for the pilot plant. Since the Henry’s law constant H is large, m is probably large. This will make the liquid resistance control, and H_L ~ H_OL. Then Eq. (15-38) can be used to estimate H_L = H_OL for the large-scale column. Only φ varies, and it can be estimated from Figure 15-7.

C. Plan. First calculate m = H/P_tot = 22,500/741 = 30.36. This is fairly large, and from Eq. (15-6b) the liquid resistance controls. For the pilot plant we can calculate n_OL from the Colburn Eq. (15-34b) since m is constant and L/V is approximately constant. Then H_L = H_OL = h/n_OL. The variation in φ with the change in packing can be determined from Figure 15-7, and H_OL ~ H_L in the large column can be estimated from Eq. (15-38).

D. Do it. From Eq. (15-35b), x^*_out = (y_in − b)/m, so

L/mV = 15/30.36 = 0.4941. From Eq. (15-34b) with x_in = 0 and x_out = 0.001,

Then H_L ~ H_OL = 10 ft/7.012 = 1.426 feet. From Figure 15-7 at W_L = 1,000, φ (0.5-in Raschig rings) = 0.32, while φ (2-inch Pall rings) = 0.62. Then from Eq. (15-38),

since all other terms in Eq. (15-38) are constant.

E. Check. These results are the correct order of magnitude. A check of n_OL can be made by graphically integrating n_OL.

F. Generalization. This method of correlating H_L or H_G when packing size or type is changed can be used for other problems. The large value of m in this problem allowed the assumption of liquid-phase control. This assumption simplifies the problem since H_OL ~ H_L. If liquid-phase control is not valid, this problem becomes significantly harder.

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.

15.6 HTU-NTU ANALYSIS OF CO-CURRENT ABSORBERS

In Chapter 12 we noted that co-current operation of absorbers was often employed when a single equilibrium stage was sufficient. Co-current operation has the advantage that flooding cannot occur. This means that high vapor and liquid flow rates can be used, which automatically leads to small-diameter columns.

A schematic of a co-current absorber is shown in Figure 15-10A. The analysis will be done for dilute systems using overall mass transfer coefficients. The system is assumed to be isothermal. The 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

(15-64)

which can be related to the changes in solute flow rates

(15-65)

Figure 15-10. Co-current absorber; A) schematic of column, B) calculation of y − y*

Combining these equations we obtain

(15-66)

If V/(k_yaA_c) is constant, Eq. (15-66) can be integrated to give

(15-67)

where H_OG is given in Eq. (15-24a) and

(15-68)

This development follows the development for countercurrent systems. The analyses differ when we look at the method for calculating y_A − . The operating equation is [see Eq. (12-63)]

(15-69)

This operating line and the calculation of y_A − are shown in Figure 15-10B. When the operating and equilibrium lines are both straight, n_OG can be obtained analytically. The result corresponding to the Colburn equation is (King, 1980)

(15-70)

where

(15-71)

If a completely irreversible reaction occurs in the liquid phase, = 0 everywhere in the column. Thus, the equilibrium line is the x axis, and the integration of Eq. (15-68) is straightforward.

(15-72)

Exactly the same result is obtained for co-current and countercurrent columns, but co-current columns can have higher liquid and vapor flow rates.

H_OG is related to the individual coefficients by Eq. (15-27a). Unfortunately, it is dangerous to use Eqs. (15-37) and (15-38) to determine the values for H_L and H_G for co-current columns because the correlations are based on data in countercurrent columns at lower gas rates than those used in co-current columns. Reiss (1967) reviews co-current contactor data and notes that the mass transfer coefficients can be considerably higher than in countercurrent systems. Harmen and Perona (1972) did an economic comparison of co-current and countercurrent columns. For the absorption of CO₂ in carbonate solutions where the reaction is slow they concluded that countercurrent operation is more economical. For CO₂ absorption in monoethanolamine (MEA), where the reaction is fast, they concluded that countercurrent is better at low liquid fluxes whereas co-current was preferable at high liquid fluxes.

15.7 MASS TRANSFER ON A TRAY

How does mass transfer affect the efficiency of a tray column? This is a question of considerable interest in the design of staged columns. We will develop a very simple model following the presentations of Cussler (1997), King (1980), Lewis (1936), and Lockett (1986).

A schematic diagram of a tray is shown in Figure 15-11. 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 to be well mixed; thus, the inlet vapor mole frac 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 upon the vertical position z, but the vapor fraction y does depend on z. The liquid mole frac 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, . At steady state a solute or more volatile component mass balance for the vapor phase is

(15-73a)

Figure 15-11. Schematic of tray

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

(15-73b)

where is the vapor mole frac in equilibrium with the liquid of mole frac . A_active is the active area for vapor-liquid contact on the tray. Both A_active and V are assumed to be constant. Dividing Eq. (15-73b) by Δz and taking the limit as Δz goes to zero, we obtain

(15-74)

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

(15-75a)

(15-75b)

After algebraic manipulation, the solution to Eqs. (15-74) and (15-75) is

(15-76a)

The point efficiency E_pt was defined in Eq. (10-5). Comparing this equation to Eqs. (15-23) and (15-24a), we obtain two alternative representations.

(15-76b)

(15-76c)

We would like to relate the point efficiency to the Murphree efficiency given by Eq. (10-2). This relationship depends upon 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 is a constant and is equal to x_out, so that and = y_out. Therefore E_MV = E_pt, and

(15-77)

for a completely mixed stage.

The second limiting flow condition is plug flow of liquid with no mixing along the tray. If we assume 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):

(15-78)

where m is the local slope of the equilibrium curve, Eq. (15-5b). Since plug flow is often closer to reality than a completely mixed tray, Eq. (15-78) is more commonly used than Eq. (15-77).

Real plates often have mixing somewhere in between these two limiting cases. These situations are discussed elsewhere (AIChE, 1958; King, 1980; Lockett, 1986).

EXAMPLE 15-4. Estimation of stage efficiency

A small distillation column separating benzene and toluene gives a Murphree vapor efficiency of 0.65 in the rectifying section where L/V = 0.8 and x_benz = 0.7. The tray is perfectly mixed and has a liquid head of 2 inches. The vapor flux is 25 lb mole/hr-ft². (a) Calculate K_ya. (b) Estimate E_MV for a large-scale column where the trays are plug flow and the liquid head h becomes 2.5 inches. Other parameters are constant.

Solution

a. From Eq. (15-77) assuming that the active area of the tray equals the area available for flow, A_active = A_flow (this is probably off by a few percent), we obtain,

(15-79)

since V/A_c = 25, h = 2/12 ft, and from Eq. (10-20c) A_active ~ A_c (1-2η) with η = 0.1 this is

b. In the large-diameter system E_pt is given by Eq. (15-76a). Since h = 2.5/12 and V/A_active = 25,

Increasing the liquid pool height increases the efficiency since the residence time is increased.

The Murphree vapor efficiency for plug flow is found from Eq. (15-78). The slope of the equilibrium curve, m, can be estimated. Since the equilibrium is

the slope is

With α = 2.5 and x = 0.7, m = 0.595. Then Eq. (15-78) is

The plug flow system has a significantly higher Murphree plate efficiency than a well-mixed plate where E_MV = E_pt = 0.73. Note that K_ya is likely to vary throughout the column since m varies (see Problem 15.D15). E_MV is also dependent upon m and will change from stage to stage. The effect of concentration changes can be determined by calculating K_y from Eq. (15-6a).

15.8 SUMMARY—OBJECTIVES

At the end of this chapter you should be able to satisfy the following objectives:

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

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

3. Use the mass transfer correlations to determine the HTU

4. Derive and use HTU-NTU analysis for co-current flow

5. Use mass transfer analysis to determine tray efficiency

REFERENCES

AIChE, Bubble Tray Design Manual, AIChE, New York, 1958.

Bennett, C.O. and J.E. Myers, Momentum, Heat and Mass Transfer, 3rd ed., McGraw-Hill, New York, 1982.

Bolles, W.L. and J.R. Fair, “Improved Mass Transfer Model Enhances Packed-Column Design,” Chem. Eng., 89 (14), 109 (July 12, 1982).

Cornell, D., W.G. Knapp, H.J. Close, and J.R. Fair, “Mass Transfer Efficiency-Packed Columns. Part II,” Chem. Eng. Prog., 56(8), 48 (1960).

Cornell, D., W.G. Knapp, and J.R. Fair, “Mass Transfer Efficiency-Packed Columns. Part I,” Chem. Eng. Prog., 56(7), 68 (1960).

Cussler, E.L., Diffusion. Mass Transfer in Fluid Systems, 2nd ed., Cambridge Univ. Press, Cambridge, UK, 1997.

Geankoplis, C. J., Transport Processes and Separation Process Principles, 4^th ed., Prentice Hall, Upper Saddle River, New Jersey, 2003.

Greenkorn, R.A. and D.P. Kessler, Transfer Operations, McGraw-Hill, New York, 1972.

Harmen, P. and J. Perona, “The Case for Co-Current Operation,” Brit. Chem. Eng., 17, 571 (1972).

Hines, A.L. and R.N. Maddox, Mass Transfer Fundamentals and Applications, Prentice-Hall, Upper Saddle River, New Jersey, 1985.

King, C.J., Separation Processes, 2nd ed., McGraw-Hill, New York, 1980.

Lewis, W.K. Jr., “Rectification of Binary Mixtures. Plate Efficiency of Bubble-Cap Columns,” Ind. Eng. Chem., 28, 399 (1936).

Lockett, M. J., Distillation Tray Fundamentals, Cambridge University Press, Cambridge, UK, 1986.

McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations in Chemical Engineering, 7th ed., McGraw-Hill, New York, 2005.

Mickley, H.S., T.K. Sherwood, and C.E. Reed, Applied Mathematics in Chemical Engineering, 2nd ed., McGraw-Hill, New York, 1957, pp. 187-191.

Perry, R.H. and D. Green, Perry’s Chemical Engineer’s Handbook, 6th ed., McGraw-Hill, New York, 1984.

Reid, R.C., J.M. Prausnitz, and T.K. Sherwood, The Properties of Gases and Liquids, 3rd ed., McGraw-Hill, New York, 1977.

Reiss, L.P., “Co-current Gas-Liquid Contacting in Packed Columns,” Ind. Eng. Chem. Process Design Develop., 6, 486 (1967).

Reynolds, J., J. Jeris and L. Theodore, Handbook of Chemical and Environmental Engineering Calculations, Wiley, New York, 2002.

Sherwood, T.K. and F.A.L. Holloway, “Performance of Packed Towers-Liquid Film Data for Several Packings,” Trans. AIChE, 36, 39 (1940).

Sherwood, T.K., R.L. Pigford, and C.R. Wilke, Mass Transfer, McGraw-Hill, New York, 1975.

Taylor, R. and R. Krishna, Multicomponent Mass Transfer, Wiley, New York, 1993.

Treybal, R.E., Mass-Transfer Operations, McGraw-Hill, New York, 1955, p. 239.

Treybal, R.E., Mass Transfer Operations, 3rd ed., McGraw-Hill, New York, 1980.

Wang, G. Q., X. G. Yuan and K. T. Yu, “Review of Mass-Transfer Correlations for Packed Columns,” Ind. Engr. Chem. Res., 44, 8715 (2005).

HOMEWORK

A. Discussion Problems

A1. What is a controlling resistance? How do you determine which resistance is controlling?

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

A3. Is a stage with a well-mixed liquid less or more efficient than a stage with plug flow of liquid across the stage (assume K_Ga is 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.

B . Generation of Alternatives

B1. Develop contactor designs that combine the advantages of co-current, cross-flow, and countercurrent cascades.

C . Deriviations

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

C2. Derive an equation analogous to Eq. (15-36) to relate HETP to H_OL.

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

D . Problems

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

D1.* For Examples 15-1 and 15-2, estimate an average H_OG in the enriching section. Then calculate n_OG and h_E = H_OG,avg n_OG.

D2.* If 1-in metal Pall rings are used instead of 2-inch rings in Example 15-2:

a . Recalculate the flooding velocity and the required diameter.

b . Recalculate H_G and H_L in the enriching section.

D3. If 1-in metal Raschig rings are used instead of 2-inch Pall rings in Example 15-2:

a . Recalculate the flooding velocity and the required diameter.

b . Recalculate H_G and H_L in the enriching section.

D4. * A distillation column is separating a feed that is 40 mole % methanol and 60 mole % water. The two-phase feed is 60% liquid. Distillate product should be 92 mole % methanol, and bottoms 4 mole % 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 feet and H_L = 0.8 feet 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. We have a column that has a 6-foot section of packing. The column can be operated as a stripper with liquid feed, as an enricher with a vapor feed or at total reflux. We are separating methanol from isopropanol at 101.3 kPa. The equilibrium can be represented by a constant relative volatility, α = 2.26.

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

b . We operate the system as an enricher with L/D = 2. The vapor mole frac methanol entering the column, y_in = 0.783 (this is the feed) and the vapor mole frac methanol leaving, y_out = 0.940. Determine n_OG, the average value of H_OG, and the value of HETP.

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

D6. * A distillation column operating at total reflux is separating methanol from ethanol. The average relative volatility is 1.69. Operation is at 101.3 kPa. We obtain methanol mole fracs of y_out = 0.972 and y_in = 0.016.

a . If there is 24.5 feet of packing, determine the average H_OG using the result of Problem 15.C3.

b . Check your results for part a, using a McCabe-Thiele diagram.

D7. A distillation column operating at total reflux is separating acetone and ethanol at 1 atm. There is 6.8 feet of packing in the column. The column has a partial reboiler and a total condenser. We measure the bottoms composition in the partial reboiler as x = 0.10 and the liquid composition in the total condenser as x = 0.9. Equilibrium data is in Problem 4.D7. Estimate the average value of H_OG.

D8. * We wish to strip SO₂ from water using air at 20° C. The inlet air is pure. The outlet water contains 0.0001 mole frac SO₂, while the inlet water contains 0.0011 mole frac SO₂. Operation is at 855 mmHg, and L/V = 0.9 (L/V)_max. Assume H_OL = 2.76 feet and that the Henry’s law constant is 22,500 mmHg/mole frac SO₂. Calculate the packing height required.

D9. Carbon tetrachloride (CCl₄) will be stripped from water using pure air at 25°C and 745 mm Hg pressure. The inlet water is saturated with carbon tetrachloride and the exit water should contain 1 ppm (mol). Assume that H_OL = 1.93 feet. Data are available in Table 12-2. Find the height of packing required if operation is at (L/V) = 0.8 (L/V)_max.

D10. * A packed tower is used to absorb ammonia from air using aqueous sulfuric acid. The gas enters the tower at 31 lb mole/hr-ft² and is 1 mole % ammonia. Aqueous 10 mole % sulfuric acid is fed at a rate of 24 lb mole/hr-ft². The equilibrium partial pressure of ammonia above a solution of sulfuric acid is zero. We desire an outlet ammonia composition of 0.01 mole % in the gas stream.

a. Calculate n_OG for a countercurrent column.

b. Calculate n_OG for a co-current column.

D11. An air stream containing 50 ppm (mole) of H₂S is to be absorbed with a dilute NaOH solution. The base reacts irreversibly with the acid gas H₂S so that at equilibrium there is no H₂S in the air. An outlet gas that contains 0.01 ppm (mole) of H₂S is desired. L/V = 0.32.

a. Calculate n_OG for a co-current system.

b. Calculate n_OG for a countercurrent system.

D12. * We wish to absorb ammonia into water at 20°C. At this temperature H = 2.7 atm/mole frac. Pressure is 1.1 atm. Inlet gas is 0.013 mole frac NH₃, and inlet water is pure water.

a. In a countercurrent system we wish to operate at L/G = 15 (L/G)_min. A y_out = 0.00004 is desired. If H_OG = 0.75 ft at V/A_c = 5.7 lb moles air/hr-ft², determine the height of packing required.

b. For a co-current 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 required.

D13. Repeat Example 15-3 to determine n_OG except use a co-current absorber.

a. Same conditions as Example 15-3. If specifications can be met, find y_out and n_OG. If specifications cannot be met, explain why not.

b. Same conditions as Example 15-3 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 15-3 except x_out = 0.0003 and L/V = 40. If specifications can be met, find y_out and n_OG. If specifications cannot be met, explain why not.

D14. * We are operating a staged distillation column at total reflux to determine the Murphree efficiency. Pressure is 101.3 kPa. We are separating methanol and water. The column has a 2-inch head of liquid on each well-mixed stage. The molar vapor flux is 30 lb moles/hr-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 15-4 has a feed that is a saturated liquid with a feed mole frac z = 0.5, and separation is essentially complete (x_dist ~ 1 and x_bot ~ 0). The Murphree vapor efficiency is often approximately constant in columns. Assume the value calculated in Example 15-4, 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. Repeat the enriching section calculation at x = 0.7 (shown in Example 15-4) as a check on your procedure.