Chapter 17: Introduction to Adsorption, Chromatography, and Ion Exchange

In the first 15 chapters we looked at separation techniques such as distillation and extraction that are often operated as equilibrium-staged separations, and even when they are not operated this way can be analyzed as equilibrium-staged separations. In Chapter 16 we studied membrane separations that are not operated as equilibrium processes; however, well-mixed membrane separators are analogous to flash distillation, and staged models are useful for integrating the mass balances and rate expressions for more complex flow patterns. Since the membrane processes are normally operated at steady state, the analysis is usually straightforward. In this chapter we will study three closely related processes that are rarely operated or analyzed as steady state, equilibrium-staged systems. These processes are usually operated in packed columns in a cycle that includes feed and regeneration steps; thus, as normally operated, these processes are inherently unsteady state.

Adsorption (note the “d” not a “b”) involves contacting a fluid (liquid or gas) with a solid (the adsorbent). One or more of the components of the fluid are attracted to the surface of the adsorbent. These components can be separated from components that are less attracted to the surface. Adsorption is commonly used to clean fluids by removing components from the fluid or to recover the components. Many homes and apartments use a carbon “filter” (actually an adsorber) for water purification. Chromatography is a similar process that uses a solid packing material (an adsorbent or other solid that preferentially attracts some of the components in the mixture), but the operation is devised to separate components from each other. You may have analyzed the composition of samples with analytical chromatography (gas or liquid) in chemistry or engineering labs. In ion exchange the solid contains charged groups that interact with charged ions in the liquid. The best-known application of ion exchange is water softening to remove calcium and magnesium ions and replace them with sodium ions. These separation methods are complementary to the equilibrium-staged processes. They are often used for chemical analysis, separation of dilute mixtures, and separation of difficult mixtures where the equilibrium-staged separations either do not work or are too expensive.

The three separation techniques studied in this chapter are similar since a solid phase causes the separation. When we want to lump them together we will call them sorption processes. The general term for an adsorbent, ion exchange resin or chromatographic packing is sorbent. The most common equipment for sorption processes is a stationary packed bed of the solid. The solid in sorption systems directly causes the separation, which is different than packed beds for the equilibrium separations studied earlier where the solid was used to increase the interfacial area and mass transfer coefficients between the gas and liquid. If feed is introduced continuously into the packed bed, the bed will eventually saturate (e.g., approach the feed concentration) and separation will cease. Much of the art and expense of designing these systems is in the regeneration step that removes the component from the packing material. Regeneration is so important that different processes are often named based on the regeneration method used.

This chapter is a simplified introduction to the fascinating and valuable sorption separation methods. The development is similar, but at a more introductory level, to that in Wankat (1986; 1990). Once you understand this chapter, you will be able to discuss these techniques with experts and will be prepared to begin more detailed explorations of these methods in more advanced books (e.g., Do, 1998; Ruthven, 1984; Yang, 1987; 2003).

17.1 SORBENTS AND SORPTION EQUILIBRIUM

Since sorption and sorbents are quite different from the equilibrium-staged processes and the membrane separations studied earlier, we need to first carefully define the terms needed to study and design these systems. After a short description of the different sorbents, the equilibrium behavior of sorbent systems will be introduced. Then the last fundamental piece required is the mass transfer characteristics of sorbents and sorption processes.

17.1.1 Definitions

The most common contacting device for adsorption, chromatography, and ion exchange is the packed bed shown schematically in Figure 17-1. The particles are packed in the cylindrical column of cross-sectional area A_c and the length of the packed section is L. Some type of support netting or frit is used at the bottom of the packed section and a hold down device such as a net or frit is used at the top of the packed section. Figure 17-1 illustrates a number of important variables. The external porosity ε_e is the fraction of the column volume that is outside the particles. To some extent the value of ε_e depends on the shape of the particles (e.g., ε_e is smaller for spheres than for irregular shaped particles) and the packing procedure used. It is important to have a uniformly packed column with a constant value of ε_e. The internal porosity ε_p is the fraction of the volume of the pellets that consists of pores and thus, is available to the fluid. The total volume available to the fluid is

(17-1a)

Figure 17-1. Schematic of adsorption column and particle (Wankat, 1986), reprinted with permission, copyright 1986, Phillip C. Wankat

From this equation we can define the total porosity ε_T as the sum of all the voids.

(17-1b)

Porosities are dimensionless quantities.

Although the fluid can fit into all the pores, molecules such as proteins may be too big to fit into some or all of the pores. This size exclusion can be quantified in terms of a dimensionless parameter K_d where K_d = 1.0 if the molecule can penetrate all of the pores and K_d = 0 if the molecule can penetrate none of the pores. The value of K_d,i for a given molecule i can also be between 0 and 1 since the pores are not of uniform size. The volume available to a molecule is,

(17-1c)

This picture is useful but does not match all adsorbents. Gel-type ion-exchange resins have no permanent pores. Instead they consist of a tangled network of interconnected polymer chains into which the solvent dissolves. In effect, ε_p = 0. Macroporous ion-exchange resins have permanent pores and ε_p > 0, but often K_d < 1.0 for large molecules. Many activated carbons have both macropores and micropores; thus, there are two internal porosities. Molecular sieve zeolite adsorbents are used as pellets that are agglomerates of zeolite crystals and a binder such as clay. In this case, there is an interpellet porosity (typically, ε_e ~ 0.32), an intercrystal porosity (ε_p1 ~ 0.23) and an intracrystal porosity (ε_p2 ~ 0.19), which has K_d,i ≤ 1.0 (Lee, 1972).

Two different velocities are typically defined for the column shown in Figure 17-1. The superficial velocity, which is easy to measure, is the average velocity the volumetric flow of fluid would have in an empty column. Thus,

(17-2a)

where Q is the volumetric flow rate (e.g., in m³/s), the cross sectional area is in m² and v_super is in m/s. The interstitial velocity v_inter (also in m/s) is the average velocity the fluid has flowing in between the particles. Since the cross sectional area actually available to the fluid is ε_e A_c, a mass balance on the flowing fluid is

v_inter ε_e A_c = v_super A_c

which gives

(17-2b)

Since ε_e is less than 1.0, v_inter > v_super. Very large molecules that are totally excluded from the pores and are not adsorbed move at an average velocity of v_inter.

There are also different densities of interest. The first, the fluid density ρ_f (e.g., in kg/m³) is familiar. The second density is the structural density ρ_s (e.g., also in kg/m³) of the solid. This is the density of the solid if it is crushed and compressed so that there are no pores and all of the air is removed. The particle or pellet density ρ_p is the average density of the particles consisting of solid plus the fluid in the pores.

(17-3a)

Manufacturers often report a bulk density ρ_b of the adsorbent. This density is the weight of the adsorbent as delivered, which includes fluid in the pores and between the particles, divided by the volume of the container. The bulk density can be calculated from the other densities.

(17-3b)

The bulk and particle densities will also be in kg/m³. If the fluid is a gas with ρ_f << ρ_p,

(17-3c)

Unfortunately, it is often unclear as to which density is being referred to. By comparing the values given by the manufacturer to the approximate values listed in Table 17-1, one may be able to determine which density is being referred to.

Table 17-1. Properties of common adsorbents (Humphrey and Keller, 1997; Reynolds et al., 2002; Ruthven et al., 1994; Wankat, 1990; Yang, 1987)

One last useful definition is the tortuosity τ. The tortuosity relates the effective diffusivity in the pores D_effective to the molecular diffusivity in free solution, D_molecular

(17-4)

Note that τ is dimensionless. Equation (17-4) was originally derived from geometric considerations using a simple geometric model. However, measured values of τ are often much larger than expected from purely geometric arguments because the diffusion is hindered by the walls or at the pore mouth. Thus, we will treat τ and D_effective as empirical (experimentally measured) quantities. The molecular diffusivity can be determined experimentally, but there are also a number of well-known methods to predict D_molecular (Perry and Green, 1997, pp. 5-48 to 5-55).

17.1.2 Sorbent Types

A variety of sorbents are used commercially for separations. We will first present a very short introduction to commercially used adsorbents. If more detail is required, refer to Yang (2003) who presents very detailed analyses. The commercially important adsorbents are highly porous and have high surface areas per gram. This high surface area greatly increases the capacity for adsorption at the surface. Typically, 98% of the adsorption occurs in the pores inside the particles and only 2% on the external surface. Molecules that adsorb are called the adsorbate. Adsorbed molecules typically have a density close to that of a liquid. Thus, there are major density and volume changes when gases are adsorbed, but very little change when liquids are adsorbed.

Activated carbon is a very porous adsorbent with a carbon backbone but a number of other species such as oxides of carbon on the surface. Since activated carbon is inexpensive, strongly adsorbs organic compounds, and has a large number of applications, it is the most commonly used adsorbent (Bonsal et al., 1988; Faust and Aly, 1987). It is produced by carbonizing a material such as wood, coke, or coconut shells. Activation is typically done with carbon dioxide or steam to create the porous structure and to oxidize the surface. Additional chemical treatments such as with iodine can be used to produce specialty carbons. Carbons are produced for both liquid and gas separations. Because the starting materials and the chemical treatments vary, different activated carbons can have very different properties. Thus, the average values reported in Table 17-1 should be used only for very preliminary calculations and approximate designs. Experimental data on the particular brand, grade, and size of activated carbon must be used for more detailed designs.

Because activated carbon has essentially a nonpolar surface, water is adsorbed weakly, often by capillary condensation in gas systems (Yang, 1987). Thus, many organic compounds are much more strongly adsorbed than water. This makes activated carbon the usual adsorbent of choice for processing aqueous solutions and humid gases. Activated carbon is commonly used for pollution control to remove organic compounds from water. Equilibrium isotherms have been measured for a large number of compounds (Dobbs and Cohen, 1980). Activated carbon is also frequently used to adsorb small amounts of organics from gases, process sugar, purify alcohol, provide personal protection as part of the complex mixture of adsorbents included in gas masks, and many other applications.

Carbon molecular sieves (CMS) have very tightly controlled pore structures. They are prepared in a manner that is similar to activated carbon except there is often an additional step where a hydrocarbon is cracked or polymerized on the surface to create the desired uniform pore size (Ruthven et al., 1994). Because of the extra care required in processing and because of patent protection, CMS are significantly more expensive than activated carbon. Currently, CMS are commonly used for producing pure nitrogen from air. Unlike the vast majority of commercial adsorbents, CMS can separate based on different diffusion rates instead of different equilibrium behavior. The design of these kinetic adsorption processes is highly specialized and is beyond the scope of this chapter (e.g., see Ruthven et al., 1994).

Zeolite molecular sieves are crystalline aluminosilicates with the general formula

M_x/n[(AlO₂)_x (SiO₂)_y]z H₂O

where M represents a metal cation such as lithium, sodium, potassium or calcium of valence n; x and y are integers (y ≥ x); and z is the number of water molecules per unit cell. Since zeolites are crystals, the pores have exact dimensions (Ruthven, 1984; Ruthven et al., 1994; Sherman, 1999; Yang, 1987, 2003). Thus, zeolites can be used to separate based partially on steric exclusion although separations where steric exclusion is not employed are more common. A large number of synthetic and naturally occurring zeolites are known (Vaughan, 1988) although not all are used commercially as adsorbents. Commercial applications of zeolites as adsorbents include drying air and natural gas, drying organic liquids, removal of carbon dioxide, separation of ethanol and water to break the azeotrope, and separation of oxygen from the nitrogen in air. The major steric exclusion application is the separation of straight-chain hydrocarbons (used for biodegradable detergents) from branched-chain hydrocarbons. Some of the properties of zeolites are reported in Table 17-1.

Silica gel is an amorphous solid made up of colloidal silica SiO₂ that is normally used in a dry granular form. [The name “gel” arises from the jellylike form of the material during one stage of its production. (Reynolds et al., 2002).] Silica gel is commonly used for drying gases and liquids since it has a high affinity for water. Silica gel is complimentary to zeolites since it is cheaper and has a higher capacity at water vapor pressures greater than about 10 mm Hg (Humphrey and Keller, 1997) but cannot dry the fluid to as low a water content. Columns with a layer of silica gel at the feed end and a layer of zeolite at the product end are commonly used for drying since they can combine the best properties of both adsorbents. Note that silica gel can be damaged by liquid water. Some of the properties of silica gel are reported in Table 17-1.

Activated alumina Al₂O₃ is also commonly used for drying gases and liquids and is not damaged by immersion in liquid water. It is produced by dehydrating aluminum trihydrate, Al (OH)₃ by heating. Activated alumina has properties that are similar to silica gel although it is physically more robust. It competes with silica gel in drying applications although its capacity is a bit lower at water vapor pressures greater than about 1 mm Hg (Humphrey and Keller, 1997). Activated alumina is also used in water treatment to selectively remove excess fluoride. Some of the properties of activated alumina are listed in Table 17-1.

A large number of other materials have been used commercially as adsorbents. These include an uncharged form of the organic polymer resins commonly used for ion exchange (see Section 17.5). Although considerably more expensive, these resins compete with activated carbon for the recovery of organics. Activated bauxite is an impure form of activated alumina. A number of clays are used for purification of vegetable oils. Bone char, which has both adsorptive and ion exchange capacities, is used in purification of sugar.

When the common adsorbents are used in chromatography applications (Cazes, 2005), they are used as much smaller particles with a much tighter particle size distribution. A number of specialized packing materials have been developed for chromatographic applications. In gas-liquid chromatography a high-boiling, nonvolatile liquid (the stationary phase) is coated onto an inert solid such as diatomaceous earth. A similar method called liquid-liquid chromatography coats an immiscible liquid on an inert solid. This packing is now often replaced with bonded packing where the stationary phase, often a C₈ or a C₁₈ compound, is chemically bonded to the inert solid, which is usually silica gel. The equilibrium behavior of these specialized packings is usually similar to gas-liquid absorption or liquid-liquid extraction, but because of the presence of the inert solid, the equipment and operating principles are similar to adsorption-chromatography.

17.1.3 Adsorption Equilibrium Behavior

Equilibrium behavior of adsorbents is usually determined as constant temperature isotherms. Valenzuela and Myers (1984, 1989) present the most extensive compilations of isotherm data, and are the best entry point into the very large literature on adsorption equilibrium measurements and theories. Basmadjian (1986) has extensive data on water isotherms from gases and liquids. Dobbs and Cohen (1980) and Faust and Aly (1987) present adsorption data for common pollutants.

Relatively typical isotherm data for gas adsorption of single components is shown in Figures 17-2A and 17-2B. Figure 17-2A shows the effect of changing the gas. Hydrogen is least strongly adsorbed, followed by methane. Adsorption peaks for ethane and ethylene and then the maximum is less for propane although its adsorption is stronger at low partial pressures. Each adsorbent has an optimum size molecule for peak adsorption. Figure 17-2B compares the adsorption of ethylene on different adsorbents.

Figure 17-2. Adsorption isotherms for pure gases: A) gases on Columbia grade L activated carbon at 310.92 K (Ray and Box, 1950; Valenzuela and Myers, 1989), B) Ethylene on different adsorbents at varying temperatures (Valenzuela and Myers, 1989). Key: 1 = Columbia grade L activated carbon at 310.92 K. 2 = Taiyo Kaken Co., Japan, attrition resistant activated carbon at 310.95 K. 3 = Pittsburgh Chemical Co. BPL activated carbon at 301.4 K. 4. Union Carbide 13X Linde zeolite at 298.15 K. 5 = Davison Chemical Co. silica gel at 298.15 K.

At low concentrations or partial pressures isotherms are often linear. As the partial pressure of the gas increases the isotherm becomes nonlinear, that is, it curves. Equilibrium data for adsorption of single gases is often fit with the Langmuir isotherm,

(17-5a)

where q_A is the amount of species A adsorbed and q_A,max is the maximum amount of species A that can adsorb (kg/kg adsorbent or gmoles/kg adsorbent), p_A is the partial pressure of species A (mm Hg, kPa, or other pressure units), and K_A is the adsorption equilibrium constant in suitable units. Note that for very small partial pressures, K_A,p p_A << 1.0 and Eq. (17-5a) simplifies to the linear form

(17-5b)

while at very high partial pressures, K_A,p p_A >> 1.0, Eq. (17-5a) simplifies to

(17-5c)

This is a horizontal line that represents saturation of the adsorbent.

For liquid systems the isotherm is usually written in terms of the liquid concentration c_A (gmoles/m³ or kg/m³),

(17-6a)

In the linear limit when K_A c_A << 1.0, this simplifies to

(17-6b)

The adsorption equilibrium constants K_A,c and will be in different units for liquid systems than for gas systems. Equilibrium constants for several systems are listed in Table 17-2.

Table 17-2. Equilibrium Constants

Isotherm data at different temperatures invariably shows that there is less material adsorbed as the temperature increases. The adsorption equilibrium constant often follows an Arrhenius form

(17-7a)

where K_Ao is a pre-exponential factor, ΔH is the heat of adsorption (e.g., in J/kg), R is the gas constant and T is the absolute temperature (e.g., in K). Since adsorption is invariably exothermic, ΔH is negative. If the Arrhenius form is followed, a plot of ln K_A vs. 1/T will be a straight line with a slope of −ΔH/R. Don’t automatically assume that data follow an Arrhenius form. Plot the data and check if it is on a straight line. Typically, q_A,max slowly decreases as temperature increases, perhaps in a linear fashion.

Langmuir used a simple kinetic argument (e.g., Wankat, 1990) to derive Eqs. (17-5a) and (17-6a). When this argument is used, q_A,max is the coverage obtained with a monolayer. Langmuir’s isotherm can also be derived with a statistical mechanics argument (e.g., Ruthven, 1984). The Langmuir isotherm is used to correlate data even when there is reason to believe that the mechanism postulated by Langmuir is incorrect. For liquid systems it is common to write the Langmuir isotherm as

(17-6c)

and there is no implication that a or b have physical interpretations. Correlation of data is best done by multivariable regression techniques, but is often done by plotting c/q vs. c (or p/q vs. p). In these plots the Langmuir isotherm plots as a straight line (see Example 17-1 and homework Problem 17.D1).

The Langmuir isotherm is thermodynamically correct for single component systems. It has also been used as the basis for a variety of other isotherms such as the BET isotherm (multiple layers of adsorbate), the Langmuir-Freundlich isotherm, and the linear-Langmuir isotherm (add a linear isotherm and a Langmuir isotherm Eq. (17-6c) with different values of a). The Langmuir isotherm is also commonly extended to the adsorption of multicomponent mixtures. For example, for the simultaneous adsorption of components A and B Eq. (17-6c) becomes

(17-8a,b)

This equation correctly predicts that the two adsorbates compete for adsorption sites. That is, if the concentration of B increases the amount of A adsorbed will decrease. Unfortunately, very few systems follow Eqs. (17-8) exactly, and if a_A does not equal a_B Eq. (17-8) is not thermodynamically consistent (LeVan and Vermeulen, 1981). Despite these problems Eq. (17-8) and its extensions to more components are commonly used for theories for multicomponent adsorption because this is the simplest form that shows competition.

A thermodynamically correct approach, the Ideal Absorbed Solution (IAS) theory, uses the Langmuir isotherm as the basic single component isotherm for the adsorption of mixtures. Since the details for these additional isotherms are beyond the scope of this book, readers who need to use more complex isotherms are referred to Do (1998), Ruthven (1984), Valenzuela and Myers (1989), and Yang (1987; 2003).

EXAMPLE 17-1. Adsorption equilibrium

Experimental equilibrium data for the adsorption of methane on Calgon Carbon Corp. PCB activated carbon are listed in Table 17-3. Determine if the Langmuir isotherm, Eq. (17-5a), is a good fit to the data at T = 373 K and if the adsorption equilibrium constant K_A,p follows the Arrhenius form, Eq. (17-7). The other values for K_A,p are K_A,p (296 K) = 2.045 × 10⁻³ (kPa)⁻¹ and K_A,p (480 K) = 1.888 × 10⁻⁴ (kPa)⁻¹ (see Problem 17-D1).

Solution

A. Define. Find the best-fit parameter values for K_A,p and q_A,max in Eq. (17-5a) for the 373 K data. Then plot the isotherm data and the Langmuir isotherm to determine if this is a good fit. Finally, determine if the K_A,p data satisfies Eq. (17-7) and find ΔH.

B and C. Explore and plan. Equation (17-5a) can be rearranged so that it will be a straight line. Multiply both sides by (1 + K_A,pp_A) and divide by q_A

(17-5c)

If the Langmuir isotherm is valid, a plot of p_A/q_A vs. p_A will be linear. Direct nonlinear fitting of the raw data can also be done instead of linearization.

Take the natural log of both sides of the Arrhenius relationship, Eq. (17-7a)

(17-7b)

A plot of ln K_A,p vs. (¹/_T) will be a straight line if the Arrhenius equation is followed. We can also check to see if q_max follows an Arrhenius form.

D. Do it. The values of p/q at 373 K are listed in Table 17-3. The plot of p/q vs. p is shown in Figure 17-3A.

Intercept = 360 =¹/(q_maxK_A)

Thus,

and the Langmuir isotherm at 373 K is

The plot of this isotherm is shown in Figure 17-3B. Agreement between the Langmuir curve and the data are quite good.

The values for the Arrhenius plot are,

TABLE 17-3. Equilibrium data for methane on Calgon PCB activated carbon (Ritter and Yang, 1987; Valenzuela and Myers, 1989)

The plots of ln K_A,p vs.¹/_T and ln q_max vs.¹/_T are both shown in Figure 17-3C. Clearly the data for both plots are well fit by straight lines and the Arrhenius relation is satisfied for both K_A,p and q_max. Then from Eq. (17-7b) for the K_A,p plot, slope = −ΔH/R

Slope = 1840

E. Check. Figure 17-3B is a check on the fit of the Langmuir constants. Since the agreement between the curve and the data points is quite good, the analysis is confirmed. The close agreement of the Arrhenius plot is another check on the analysis procedure.

F. Generalize. The fit to the straight line in Figure 17-3C is closer than for many other adsorption systems. Remember that although the amount adsorbed generally decreases as temperature increases, adsorption does not always follow an Arrhenius relationship. Note that the values of q_max are even less likely to follow the Arrhenius relationship although this system does.

17.2 SOLUTE MOVEMENT ANALYSIS FOR LINEAR SYSTEMS: BASICS AND APPLICATIONS TO CHROMATOGRAPHY

Packed columns similar to Figure 17-1 are the most common contacting devices used for adsorption and chromatography. Although there are exceptions, they are usually operated vertically with the flow parallel to the axis of column. Adsorption, chromatography, and ion exchange in packed columns are inherently unsteady state or batch type processes. Since the sorbent is stationary, it will saturate at the feed concentration if feed enters the column continuously. Thus, there must also be a regeneration step that removes most of the sorbate from the packing. The commonly used regeneration methods are to use an inert purge stream, change the temperature, change the pressure, and use a desorbent. After the regeneration step, there may be an optional cooling or drying step. Then the next cycle starts with the feed step. These processes will be analyzed in Sections 17.2 to 17.8 using increasingly complex analysis procedures. In this section we will start with the simplest theory, solute movement theory for linear isotherms, applied to elution chromatography, the simplest process to analyze.

Figure 17-3. Plots of equilibrium data for Example 17-1; A) plot to give straight line for Langmuir isotherm, B) Langmuir isotherm, C) check on Arrhenius relationship

Complete analysis of sorption processes requires computer simulation with a rather complex simulation program to solve the coupled algebraic and partial differential equations. Unfortunately, simulators often do not provide a physical picture of why the separation occurs and once a result is obtained simulators don’t tell what to do to improve the separation. A relatively simple tool that is based on physical arguments and can be solved with pencil and paper (or a spreadsheet) will prove to be very useful even if it is not completely accurate. Solute movement analysis is a tool that allows engineers to use physical reasoning and understanding so that they can understand the results from experiments or simulations. The role of solute movement analysis in sorption processes is thus, analogous to the role of McCabe-Thiele diagrams in distillation, absorption, and extraction. Solute movement theory is used to understand the separations and for troubleshooting, not for final design.

17.2.1 Movement of Solute in a Column

Solute or sorbate within the packed section of a column can be in one of three locations. The solute can be in the interstitial void space ε_e and be moving at the interstitial velocity v_inter. If the solute is in the intraparticle voids (1 − ε_e)ε_p or sorbed to the stationary solid it will have a net axial velocity of zero. In other words, the solute molecules are either scooting forward axially at a high velocity (remember that v_inter is greater than v_super), or they aren’t moving at all. The average solute velocity u_s is

(17-9)

This fraction can be calculated by considering the distribution of an incremental change in solute concentration Δc (e.g., in kmol/m³) and its corresponding change in the amount sorbed Δq (e.g., in kmole/kg adsorbent).

(17-10)

The amount in each location can be determined by calculating the inventory of mass.

(17-11a)

(17-11b)

where K_d is the fraction of pores that molecules can squeeze into. This term becomes important in size exclusion chromatography which separates molecules based on size and ideally has no adsorption, Δq = 0 (Wu, 2004).

Both Eqs. (17-11a) and (17-11b) are in kg moles adsorbate if c is kg moles/m³.

(17-11c)

The K_d term is not included in Eq. (17-11c) since we assume Δq is based on a measurement that automatically includes any steric hindrance. Equation (17-11c) will also be in kg moles adsorbate. If q and c are in different units than in this derivation, the mass balances will be slightly different (compare Eqs. 17-14a, b and c to 17-13).

Inserting Eq. (17-11) into Eq. (17-10) one obtains,

(17-12)

After Eq. (17-12) is substituted into Eq. (17-9), the result can be simplified to (see homework Problem 17.C3)

(17-13)

The exact form of Eq. (17-13) depends upon the units of the equilibrium data. For example, if q is in kg solute/kg solid and x is in kg solute/kg fluid so that the isotherm expression is q = f(x), then there must be a ρ_F (kg fluid/m³) term in Eqs. (17-11a) and (17-11b) and Δx replaces Δc in these equations. Then Eq. (17-13) becomes

(17-14a)

If q and c are both in gmoles/m³, the equation for u_s is obtained by eliminating ρ_s from Eq. (17-13).

For gas systems equilibrium is usually expressed in terms of the partial pressure, p_A (e.g., Eqs., (17-5a) and (17-5b)). The solute velocity for gases is then

(17-14b)

where is the molar density. For an ideal gas and . Thus, for ideal gases,

(17-14c)

Equations (17-13) and (17-14) allow us to calculate the average velocity of the solute if we can calculate (Δq / Δc), (Δq / Δx), or (Δq / Δp_A). If we assume that mass transfer is very rapid so that the solid and fluid are locally (at each z value) in equilibrium, the solute velocity depends only upon the equilibrium behavior (Δq / Δc), (Δq / Δx), or (Δq / Δp_A) and the properties of the packed column, not upon mass transfer rates. Mass transfer will be critically important to determine how the solute spreads from the average (see Sections 17.4 to 17.6). In Sections 17.2.2, 17.3 and 17.6 we will insert the appropriate isotherm into Eqs. (17-13) and (17-14).

17.2.2 Solute Movement Theory for Linear Isotherms

The theory becomes simplest when the linear isotherm, Eq. (17-5b) or (17-6b), is used. Since almost all equilibrium data becomes linear at low enough concentrations or partial pressures, there are a number of real applications of linear isotherms.

When Eq. (17-6b) is valid, , and Eq. (17-13) becomes

(17-15a)

for any adsorbate i. Note that none of the terms depend upon the concentration of adsorbate i. Thus, at low concentrations where the linear isotherm is valid the solute velocity becomes constant. The solute velocity u_s,i depends upon temperature since depends upon temperature (e.g., following Eq. (17-7)). If we have a number of different solutes with different values of the equilibrium constant, the weakest sorbed solute (lowest value of ) moves the fastest, and the strongest sorbed solute (highest value of ) moves the slowest. Since they move at different speeds, they can be separated. A single-porosity form of this equation is also commonly used (see Problem 17.C5, which includes Eq. (17-15b).

To visualize what this looks like, we will start with a packed column that is initially clean (c_A = 0). At time t₀, we start adding a feed with a concentration c_A,F at a known interstitial velocity. Equation (17-15a) can be used to calculate the solute velocity u_s,A (the numerical calculation procedure is illustrated in Example 17-2). In Figure 17-4A the solute movement or characteristic diagram for this process is plotted. Solute starts at z = 0 at time t₀, and moves upward at velocity u_s,A, which is the slope of the characteristic line shown in the figure. The procedure will probably be easiest to understand after you study Example 17-2. The concentrations in the column are shown at four times in Figures 17-4B, C, D, and E. The solute moves upward in a wave at a constant velocity u_s,A. If adsorption is strong the wave moves slowly while if adsorption is weak it moves quickly. Waves for nonlinear systems are shown later in Figure 17-15.

Figure 17-4. Wave movement for step change in feed concentration. A) Solute movement diagram for linear isotherm. B, C, D, E are concentrations in column at t = t₀, t₁, t₂, and t₃, respectively. Since t₃ = breakthrough time t_br = L/u_s,A, entire column and outlet are at c_A,F.

For systems in wt frac, , or (q in kg solute/total kg solid and x and y in kg solute/kg fluid), Eq. (17-14a) becomes

(17-15c)

For ideal gases with Eq. (17-14c) becomes

(17-15d)

With linear isotherms the denominator in Eq. (17-15a) through Eq. (17-15-d) is independent of concentration. Thus, for purposes of algebraic manipulation it is convenient to write these equations as

(17-15e)

Although very convenient, linear isotherms find limited applications in industrial processes. To decrease the diameter of the columns and thus, reduce costs, most industrial separations are done at high concentrations or very close to the solubility limit. Since the isotherms are probably nonlinear and use of linear isotherms can lead to large errors, one needs to use the nonlinear theories in Section 17.6 for concentrated systems.

17.2.3 Application of Linear Solute Movement Theory to Purge Cycles and Elution Chromatography

The simplest regeneration cycle is a purge cycle using an inert carrier gas for gas systems or an inert solvent for liquid systems. This cycle can be operated co-flow (Figure 17-5A) or counterflow (Figure 17-5B). When the purge gas or liquid enters the column, the partial pressure or concentration of the adsorbate will drop since it is being diluted. This causes adsorbate to desorb (see Figure 17-2), thus, allowing it to be flushed from the column. The ideal carrier gas or solvent has the following properties: easy to separate from the adsorbate, easy to remove from the bed, nontoxic, nonflammable, available, and inexpensive. For purge gas systems nitrogen and hydrogen are close to ideal as carrier gases. Since the adsorbate is diluted, purge cycles are not commonly used for large-scale commercial adsorption processes.

FIGURE 17-5. Purge systems. A) co-flow (e.g., elution chromatography), B) counterflow, C) outlet concentrations for elution chromatography of aspartic acid (asp), alanine (ala), and phenylalanine (phe) on cation exchange resin (Agosto et al., 1989). Reprinted with permission from Ind. Eng. Chem. Research, 28, 1358 (1989), copyright 1989 American Chemical Society.

Purge cycles are commonly used in elution chromatography, particularly in analytical chemistry. Elution chromatography involves input of a feed pulse into the packed column followed by co-flow (Figure 17-5A) of an inert solvent or carrier gas. (If the solvent or carrier gas also adsorbs, the process can become gradient or displacement chromatography, which are discussed later.) The column can be packed with any of the adsorbents or chromatography packings mentioned previously. If the solutes have different equilibrium isotherms, the solutes will move in the column at different velocities and will be separated (Figure 17-5C). Both gas and liquid chromatography are commonly operated in this elution mode to determine the compositions of unknown samples. Large-scale elution chromatography systems are also becoming more common, particularly in the pharmaceutical and fine chemical industries. Detailed design considerations for large-scale biochromatography systems are discussed in detail by Bonnerjea and Terras (1994), Ladisch (2001), and Rathore and Velayudhan (2004). The dilution that is inherent in purge operations tends to make these processes expensive for large-scale separation, but with complicated feeds there may be no better alternative separation method.

EXAMPLE 17-2. Linear solute movement analysis of elution chromatography

A 1-meter column is packed with activated alumina. The column initially contains pure liquid cyclohexane solvent. At time t = 0 a feed pulse that contains 0.0001 gmole/L anthracene and 0.0002 gmoles/L naphthalene in cyclohexane is input for 10.0 minutes. The superficial velocity is 20 cm/min for both feed and solvent steps. Use the solute movement theory to predict the outlet concentrations.

Data: Bulk density (fluid is air) = 642.6 kg/m³, ε_p = 0.51, ε_e = 0.39, ρ_f (cyclohexane) = 0.78 kg/L. adsorbent and adsorbent where A = anthracene and N = naphthalene. K_d = 1.0 for both anthracene and naphthalene.

Solution

A. Define. The apparatus is sketched in Figure 17-5a. We want to find when the anthracene and naphthalene pulses exit the column.

B. Explore. Equation (17-15) can be used to determine numerical values u_s,A and u_s,N. Lines drawn on a solute movement diagram with these slopes from the start (t = 0 min) and end (t = 10 min) of the feed pulse outline the movement of average molecules of anthracene and naphthalene. Since , u_s,N > u_s,A, and the naphthalene should exit the column first.

C. Plan. Calculate u_s,N and u_s,A. Plot the solute movement diagram for a 10 minute feed pulse.

D. Do it. The interstitial velocity can be determined from Eq. (17-2b),

v_inter = v_super / ε_e = (0.2 m/min)/0.39 = 0.513 m/min.

The structural density can be determined by rearranging Eq. (17-3c),

ρ_s = ρ_b /[(1 − ε_e)(1 − e_p)] = (642.6 kg/ m³)/(0.61)(0.49) = 2150 kg/m³

The solute velocities can now be calculated from Eq. (17-15)

Note that we had to convert from L/kg to m³/kg. Then,

These solute velocities are the slopes for each solute on the solute movement diagram (Figure 17-6A). The lines for each solute can be drawn from the start and finish of the feed pulse. The resulting solute pulses or waves exit the column at z = L = 1.00 m as shown in Figure 17-6B. The naphthalene exits from 54.95 to 64.95 minutes and the anthracene exits from 74.07 to 84.07 minutes. The naphthalene and anthracene peaks are both at their feed concentrations, 0.0002 and 0.0001 gmole/L, respectively.

FIGURE 17-6. Results for Example 17-2 for elution chromatography; A) solute movement diagram, B) outlet concentration profiles

E. Check. Naphthalene input at t = 0 will exit at t = L/u_s,N = 1.0 m/(0.0182 m/min) = 54.95 minutes. Similarly the anthracene input at t = 0 exits at t = 74.07 minutes. Since the peaks both last 10 minutes the naphthalene exits at 64.95 minutes and the anthracene at 84.07 minutes. The peak centers start at t_feed/2 = 5 minutes, and are at 59.95 and 79.07 minutes. These results agree with the graph in Figure 17-6A.

F. Generalize. Since no spreading phenomena are included in the simple solute movement theory, the outlet peaks are predicted to be square waves (Figure 17-6B). When mass transfer and axial dispersion are included, the curves are spread out more as was illustrated in Figure 17-5C. The solute movement theory correctly predicts the behavior of an average molecule. Thus, the time for the center of the peaks is correctly predicted. Note that the dominant term in the denominator for both solutes is the adsorption term. This will be the case when there is relatively strong adsorption.

Basmadjian (1997) reports that a typical linear velocity for the feed step for adsorption in liquid systems is 0.001 m/s, which is 6 cm/min. The purge step may be roughly ten times faster. Thus, the flow rates in this problem are reasonable.

There is an analogy that may be useful in understanding this solute movement analysis. The problems are similar to algebra problems where two trains start to leave a station at the same time, but with different velocities (u_A and u_B in chromatography). You want to calculate when each train arrives at a second station (a distance L away) and when the tail end of each train (analogous to the feed time, t_F) leaves the second station.

In actual practice it is much easier to calculate the exit times as shown in step E of Example 17-2 rather than drawing the solute movement diagram exactly to scale. This can easily be done with a spreadsheet even for much more complex processes. However, a sketch of the solute movement diagram should always be made since it will guide the calculations and provide a visual check on the calculations.

In real systems the results predicted by solute movement theory are spread considerably by axial dispersion, mass transfer resistances and mixing in column dead volumes, valves and pipes. Thus, the predictions for the simple elution chromatography system shown in Figure 17-6B would spread and the two solute peaks would overlap as shown in Figure 17-5C. This calculation will be illustrated later in Example 17-10. If high product purities are required, this zone spreading requires either a very short feed step and/or a long column to move the peaks apart. In addition, the next feed pulse must be delayed for a considerable time or zone spreading will result in too much overlap of the slow peak with the fast peak from the next feed pulse. The net result is that elution chromatography can be very expensive for large-scale applications. Simulated moving bed (SMB) systems are often less expensive, and are analyzed in Section 17.3.3.

A number of variations of elution chromatography have been developed. In flow programming the flow rate is increased to more rapidly elute the late components in Figure 17-5C. This technique does not change the volume of elutant required, but reduces the time of operation. Temperature programming increases the temperature of the entire column during the elution. Increased temperature decreases the adsorption of the solutes so that they exit both sooner and more concentrated. The related temperature gradient method increases the temperature of the fluid entering an adiabatic column. Temperature changes usually have more effect in gas systems than in liquid systems. In liquid systems it is common to use a solvent gradient to change the solvent to decrease the sorption of the solutes. Common changes in the solvent are to change ionic strength, polarity, pH, the fraction of organic solvent, and the addition of a strongly sorbed material. Gradients, which can be done as continuous changes or as step functions, are commonly used in bioseparations (Ladisch, 2001). If a step change is made in the concentration of a chemical that is more strongly sorbed than all the components of the feed, the process is called displacement chromatography [see Cramer and Subramanian (1990) for an extensive review].

A major commercial problem is the purification of strongly adsorbed species such as moderate molecular weight (C₁₀ to C₂₀) straight-chain hydrocarbons (Ruthven, 1984). Purge systems require excessive purge gas or solvent for these strongly adsorbed species. Pressure swing cycles (Section 17.3.2) also require too much purge gas for strongly adsorbed materials. Thermal cycles (Section 17.3.1) can cause excessive thermal decomposition since very high temperatures are required. Displacement cycles using a desorbent (e.g., n-pentane, n-hexane, and ammonia) that is adsorbed have proven to be effective for this otherwise intractable problem (Wankat, 1986). (Unfortunately, the nomenclature is confusing since displacement cycles for adsorption have a desorbent that can adsorb less or more than the adsorbate while in displacement chromatography the desorbent is the most strongly adsorbed compound.) The cycles will be basically the same as the counterflow cycle shown in Figure 17-5B. Since displacement adsorption requires that the desorbent be recovered from the product—often by distillation—they are relatively expensive processes that are only used when other adsorption processes fail. Both purge and displacement cycles are commonly used in SMB systems (Section 17.3.3).

17.3 SOLUTE MOVEMENT ANALYSIS FOR LINEAR SYSTEMS: THERMAL AND PRESSURE SWING ADSORPTION AND SIMULATED MOVING BEDS

Since purge cycles use large amounts of solvent, other regeneration methods have been developed. These methods and their analysis with the solute movement theory is the topic of this section.

17.3.1 Temperature Swing Adsorption

Temperature Swing Adsorption (TSA) is commonly used for gas systems particularly for the recovery or removal of trace components that are strongly adsorbed (Ruthven, 1984; Yang, 1987). Typical applications are removal of pollutants such as volatile organic compounds (VOC) (Fulker, 1972; Reynolds et al., 2002) and drying gases (Basmadjian, 1984, 1997). The basic cycle using counterflow of the hot regenerant gas is shown in Figure 17-7A. The cycle can be thought of as a purge operation with a hot purge gas. The feed step can be done either with upward or downward flow. If the feed gas flows continuously, two or more units are operated in parallel. Although the pure gas is usually the desired product, in some cases the concentrated adsorbate is the valuable product. After the regeneration step, an optional cooling step may be inserted. Cooling is required if the feed gas can react with the hot adsorbent. Insertion of the cooling step tends to produce a purer product, but lowers the productivity (kg feed processed)/(hour × kg adsorbent).

Figure 17-7. Thermal swing adsorption; A) counterflow cycle for gas systems, B) differential control volume for mass balances when temperature changes u_th > u_s. Part B is modified from Wankat (1986) with permission, copyright 1986, Phillip C. Wankat.

The basis for this regeneration method is the large reduction in the equilibrium constant observed in most systems when the temperature is increased, Eq. (17-7), and the simultaneous reduction in the partial pressure or concentration with the addition of the purge gas. Both of these effects lead to removal of the adsorbate from the column. Since large increases in the adsorbate concentration can occur, TSA systems are also used to concentrate dilute gas streams. When cooling is not required to prevent chemical reactions, a number of modifications of the basic cycle have been developed (Natarajan and Wankat, 2003). These modifications are explored as homework problems.

One major disadvantage of the TSA system shown in Figure 17-7A is that large amounts of pure regeneration gas may be required to heat the adsorption column and adsorbent. This occurs because at normal pressures the volumetric heat capacity of the gas is quite low compared to the volumetric heat capacity of the adsorbent and the metal shell of the column. Thus, regeneration may be relatively slow, expensive, and not produce the desired concentrated adsorbate product. This disadvantage tends to be minor when the feed gas is quite dilute and the adsorption is strong. Since the feed step will be quite long, the relative amount of hot regenerant gas used is reasonable. However, if the feed gas is concentrated (above a few percent) the adsorbent will saturate fairly quickly and the feed step will be relatively short. Since the same amount of hot regenerant gas is required to heat the column, the ratio (hot regenerant gas/feed gas) becomes excessive.

To study TSA systems with the solute movement analysis we must determine the effect of temperature changes on the solute waves, the rate at which a temperature wave moves in the column, and the effect of temperature changes on concentration. The first of these is easy. As temperature increases the equilibrium constants, K_A and , both decrease, often following an Arrhenius type relationship as shown in Eq. (17-7). If the effect of temperature on the equilibrium constants is known, new values of the equilibrium constants can be calculated and new solute velocities can be determined.

Changing the temperature of the feed to a sorption column will cause a thermal wave to pass through the column. The velocity of this thermal wave can be calculated by a procedure analogous to that used for solute waves. The thermal wave velocity will be the fraction of the change in thermal energy in the mobile phase multiplied by the interstitial velocity,

(17-16)

Since changes in the energy contained in the fluid in the pores, in the solid, and the walls are stagnant, this fraction is,

(17-17)

In this derivation we are assuming a pure thermal wave with no adsorption, no reactions and no phase changes. Thus, energy changes are totally due to specific heat. For example, the amount of energy change in the mobile phase is

(17-18)

where C_P,f is the heat capacity of the fluid and ΔT_f is the change in fluid temperature. Substituting in the appropriate terms, the fraction of energy change in the mobile phase is

(17-19)

where W is the weight of the column per length (kg/m),and ΔT_pf, ΔT_s and ΔT_w are the changes in pore fluid, solid and wall temperatures induced by the change in fluid temperature.

If we divide the numerator and denominator of Eq. (17-19) by ΔT_f, we will have the ratios of ΔT_pf, ΔT_s and ΔT_w to ΔT_f. If heat transfer is very rapid, the system will be in thermal equilibrium, T_f = T_pf = T_s = T_w. This equality requires that the changes in temperature all be equal

(17-20)

and the ratios of changes in temperatures are all one. Combining Eqs. (17-17), (17-19) and (17-20), the resulting thermal wave velocity is

(17-21)

As a first approximation, and solute movement theory is a first approximation, the thermal wave velocity is independent of concentration and temperature. Temperature is constant along the lines with a slope equal to the numerical value of u_th.

In TSA processes the purpose of increasing the temperature is to remove the adsorbate from the adsorbent. This happens when the thermal wave intersects the solute wave. Assume that the column is initially at a uniform temperature T₁, concentration c₁, and adsorbent loading q₁. Fluid at temperature T₂ is fed into the column. This temperature change causes the concentration and adsorbent loading to change to c₂ and q₂ (currently both unknown). Since the solute movement theory assumes local equilibrium, c₂ and q₂ are in equilibrium at T₂. The control volume shown in Figure 17-7B (Wankat, 1986) will be used to develop the mass balance for this temperature change. Initially, the thermal wave is at the bottom of the control volume. The thermal wave is assumed to move faster than the solute wave, which is true for most dilute liquid and some dilute gas systems. For a differential slice of column of arbitrary height Δz, the temperature of the slice will change from the initial temperature T₁ to the final temperature T₂ if Δt = Δz/u_th.

The mass balance for the differential slice over the time interval Δt is,

(17-22)

Since Δt = Δz/u_th, this simplifies to,

(17-23)

Equation (17-23) and the appropriate isotherm equation can be solved simultaneously for the unknowns c₂ and q₂. For a linear isotherm, Eq. (17-6b), the simultaneous solution is

(17-24)

For a liquid system where u_th > u_s, the use of a hot feed liquid (T₂ > T₁) will increase the outlet concentration. This is illustrated in Example 17-3.

EXAMPLE 17-3. Thermal regeneration with linear isotherm

Use a thermal swing adsorption process to remove traces of xylene from liquid n-heptane using silica gel as adsorbent. The adsorber operates at 1.0 atm. The feed is 0.0009 wt frac xylene and 0.9991 wt frac n-heptane at 0°C. Superficial velocity of the feed is 8.0 cm/min. The absorber is 1.2 meters long and during the feed step is at 0°C. The feed step is continued until xylene breakthrough occurs. To regenerate use counterflow of pure n-heptane at 80°C, and continue until all xylene is removed. Superficial velocity during purge is 11.0 cm/min.

Data: At low concentrations isotherms for xylene: q = 22.36x @ 0°C, q = 2.01x @ 80°C, q and x are in g solute/g adsorbent and g solute/g fluid, respectively (Matz and Knaebel, 1991). ρ_s = 2100 kg/m³, p_f = 684 kg/m³, C_ps = 2000 J/kg°C, C_pf = 1841 J/kg°C, ε_e = 0.43, ε_p = 0.48, K_d = 1.0.

Assume: Wall heat capacities can be ignored, heat of adsorption is negligible, no adsorption of n-heptane, and system is at cyclic steady state. Using the solute movement theory

a) Determine the breakthrough time for xylene during the feed step.

b) Determine the time for the thermal wave to breakthrough during both the feed and purge steps.

c) Determine the xylene outlet concentration profile during the purge step.

Solution

A. Define. The process is similar to the sketch in Figure 17-7A but without the optional cooling step. The breakthrough time for solute is the time that xylene first appears at the column outlet, z = L. The thermal wave breakthrough times occur when the temperature starts to decrease (during the feed step) or increase (during the purge step). The desired outlet concentration profile is xylene concentration vs. time.

B. Explore. Since operation is at cyclic steady state (each cycle is an exact repeat of the previous cycle), the column will be hot when the cold feed is started. The cold feed causes a cold thermal wave during the feed step. We expect that this wave will move faster than the xylene wave (this expectation will be checked while doing the calculations); thus, the waves are independent. When the flow direction is reversed, the xylene wave concentration is unchanged until the xylene wave intersects the thermal wave. This causes a period when the regenerated liquid exiting the bottom of the column is at the feed concentration (study Figure 17-8 to understand this.) When the two waves intersect the temperature changes, the isotherm parameter changes, and the xylene wave velocity changes. At the same time, xylene is desorbed and the xylene concentration in the fluid increases.

C. Plan. Since mass fractions are used in the equilibrium expression, we use Eq. (17-15c) to calculate the velocity of the solute at both 0 and 80°C. The thermal wave velocity is determined from Eq. (17-21) with W = 0. The effect of the temperature change on the fluid concentration can be determined either from a mass balance over one cycle or from Eq. (17-24).

D. Do it.

To calculate solute velocity,

Use Eq. (17-15c),

Where K′ (0°C) = 22.36.

And xylene breakthrough time is,

The thermal wave velocity from Eq. (17-24) with W = 0 is

And thermal breakthrough time is,

For the purge step u_th,purge = −6.466cm/min. It is negative since the flow direction is reversed. This breakthrough time is

Note that the thermal wave moves considerably faster than the solute wave and breakthrough is quicker. After the temperature change K′ (80°C) = 2.01, and the solute velocity and breakthrough time during purge are

The xylene mass balance on one cycle at cyclic steady-state is, In = Out.

The outlet stream shown in Figure 17-8, consists of one part at x_F and one part that is concentrated.

Set In = Out, divide out A_c and ρ_f (assumed to be constant), and solve for x_conc,

The xylene exiting the bottom of the column is at x_out = x_F = 0.0009 for the first 18.56 minutes of the regeneration step. Then from 18.56 to 27.629 minutes of the regeneration x_out = x_conc = 0.02095. If regeneration continues for times longer than 27.629 minutes, x_out = 0. The average wt frac during regeneration is x_out,avg = 0.00748.

E. Check. Equation (17-24) can be used to check the outlet xylene wt frac. This equation is applied at the point where the adsorbent changes temperature. From Figure 17-8, this occurs during the purge step; thus, all velocities should be calculated at the purge velocity. Since solute velocity is directly proportional to temperature,

u_s(v_purge, T=0) = u_s(v_F, T=0) (v_purge/v_F)= (0.3799)(-11.0/8.0) = -0.5224cm/min.

and x_xy (T = 80°C) = (23.277)(0.0009) = 0.02095, which agrees with the mass balance result.

Figure 17-8. Solute movement solution for counterflow TSA in Example 17-3

F. Generalize. This large increase in the solute concentration during thermal regeneration is a general phenomenon for strongly adsorbed solutes if the feed is dilute. The energy required to concentrate the dilute xylene in the n-heptane by adsorption is significantly less than the energy required to do the same concentration by distillation. (Going from 0.0009 to 0.02095 wt frac xylene may not seem like much change, but removal of a very large amount of pure n-heptane is necessary to obtain this amount of concentration.)

If the solute waves move faster than the thermal wave, which may occur in dilute gas systems, a mass balance equation and solution similar but subtly different than Eqs. (17-22) to (17-24) can be derived (see homework Problem 17.C9). In this situation the concentrated solute exits ahead of the thermal wave instead of behind it as predicted by Eqs. (17-22) to (17-24). One other case that can occur but is rare in dilute systems is when u_s(T_hot)> u_th > u_s(T_cold). In this case, which is beyond the scope of this introductory treatment, the solute concentrates, or focuses, at the temperature boundary (Wankat, 1990).

A number of different thermal cycles are used commercially. Figure 17-9 shows an alternate TSA cycle commonly used for recovery of solvents (typically VOC of intermediate molecular weight ~45 to 200) from drying and curing operations (Basmadjian, 1997; Fulker, 1972; Wankat, 1986). Activated carbon is used as the adsorbent and steam is used as the regeneration gas. Horizontal beds with a depth of one to two meters are often employed since strong adsorption of the solvent on the activated carbon allows for quite short beds and large gas flows require a large cross-sectional area to avoid excessive pressure drop. If feed gas needs to be treated continuously, two or more adsorbers are used in parallel, with a typical feed time of approximately two hours. Because the latent heat of steam is high, a large amount of energy can be rapidly transferred into the adsorber heating it quickly. A “heel” of leftover solvent is usually left in the bed since complete regeneration of the bed would require excessive amounts of steam. Because of incomplete regeneration and competition with water vapor for adsorption sites, the typical design capacity used for the activated carbon is about 25% to 30% of the maximum capacity of the carbon. Bed capacity can be increased by reducing the relative humidity of the feed gas to less than 50%.

Figure 17-9. Solvent recovery with activated carbon and steam regeneration

In ideal applications of this process (e.g., removing small amounts of toluene from air) the peak mole fractions of toluene are close to 1.0 (Basmadjian, 1997) and the toluene is almost completely immiscible with water. Thus, the adsorbate can be recovered from the steam by condensing the concentrated adsorbate stream and allowing the liquid to separate into an organic layer and a water layer. If the adsorbate is miscible with water (e.g., ethanol), the condenser/settler shown in Figure 17-9 must be replaced with a distillation column, which greatly increases capital and operating costs.

Note that there can be safety hazards in the operation of the activated carbon solvent recovery equipment shown in Figure 17-9. If the solvent being recovered is flammable, care must be taken to prevent a fire. If the feed gas is air, then the concentration of the solvent in the feed gas must be kept below the lower explosion limit, and is often kept below ¼ of the lower explosion limit to provide a safety margin. This requirement invariably means that the feed gas must be quite dilute and the flow rates are large. If the feed gas is hot, it is often cooled before the adsorber to increase safety and to increase the adsorber’s capacity. An alternative is to operate at much higher concentrations using nitrogen or carbon dioxide as the gas for the drying or curing operation, but then the carrier gas must be recovered and recycled. If the hot activated carbon can catalyze a reaction with the feed, a cooling step is added to the process. Sometimes a drying step is added before the cooling step since water may interfere with the adsorption or react with the feed. If the feed gas is concentrated, the adsorbent can become quite hot because of the large heat of adsorption. Unfortunately, carbon beds occasionally catch on fire when this happens. This can be prevented by significant cooling of the feed gas, incorporating a cooling step in the cycle, or replacing air in the process with an inert gas.

Since several companies provide package units for activated carbon solvent recovery, new engineers are more likely to be involved in the purchase and installation of a unit than in designing a new unit. The more you know about solvent recovery with activated carbon, the better choice of unit and better bargain you will be able to make for your company.

Various thermal cycles are also employed for liquid systems although they tend to be somewhat different than those used for gases. The largest application of liquid adsorption is the use of activated carbon to treat drinking water and wastewater (Faust and Aly, 1987). Since contaminant levels are very low and adsorption tends to be very strong, the feed portion of the cycle may last for several months. Regeneration of the activated carbon is difficult and is usually done by removing the carbon from the column and sending it to a kiln to burn off the adsorbates. In small units (e.g., those used to purify tap water in homes) the carbon is discarded after use. Activated carbon is commonly used in bottling plants to remove chlorine from water by reacting with the carbon to produce HCl (Wankat, 1990). The slightly acidic water should be used immediately after use since it no longer contains chlorine to stop microbial growth.

A major industrial application of liquid adsorption is the drying of organic solvents (Basmadjian, 1984). A typical process is shown in Figure 17-10. Upward flow is used during the refilling and feed steps to avoid trapping gas in the bed. Since the water content in the organic is usually low, the feed step may be relatively long. Once breakthrough occurs (substantial amounts of water appear in the exiting solvent), the feed is turned off and the column is drained. Regeneration is done with downward flow of hot gas and is usually followed by a cooling step. Adsorptive drying competes with drying by distillation (Chapter 8). Operating expenses for adsorptive drying are dominated by the cost of energy to evaporate residual liquid and desorb water. Adsorptive drying usually has an economic advantage compared to distillation when the water concentrations in the solvent are low.

Figure 17-10. Drying liquid solvents by adsorption

In concentrated systems the energy generated by adsorption can be as large or significantly larger than the sensible heat from the temperature change. This causes a coupling of the concentration and temperature waves, and they often travel together. Basmadjian (1997) presents a simple way to estimate the maximum temperature rise in the system.

(17-25)

The typical range for the heat of adsorption |ΔH_ads| is in the range from 1000 to 4000 kJ/kg (average ~ 2500) and typically the gas heat capacity C_P,f is approximately 1.0 kJ/kg. This estimate gives a maximum temperature rise of approximately 25°C for a feed containing 1.0 wt % adsorbate and a maximum temperature rise of 1.25°C for a feed containing 0.05 wt % adsorbate. Basmadjian (1997) recommends that an isothermal analysis can be used if the predicted maximum temperature increase is less than 1°C or 2°C. Situations with large temperature increases obviously need to be designed for and are economically important, but the detailed theoretical treatment is beyond the scope of this introductory chapter. Interested readers should consult Basmadjian (1997), LeVan et al. (1997), Ruthven (1984) or Yang (1987). These more concentrated systems can also be simulated with commercial simulators.

17.3.2 Pressure Swing Adsorption

Pressure swing adsorption (PSA) and vacuum swing adsorption (VSA) cycles are alternatives to thermal cycles for gas systems. They are particularly useful for more concentrated feeds and/or adsorbates that are not strongly adsorbed. Figure 17-11A shows the steps in the basic Skarstrom cycle (named after Charles Skarstrom, the inventor of the process) for PSA (Ruthven et al., 1994; Wankat, 1986). Usually, the desired product is the pure product gas after adsorbate removal. Typical applications include drying gases, purifying hydrogen, and producing oxygen or nitrogen from air.

Figure 17-11. Pressure swing adsorption; A) steps for single column in Skarstrom cycle, B) Use of two columns in parallel for continuous feed and product. Period of feed step = period of blowdown + purge + repressurization steps.

Following the feed step at the higher pressure, the pressure in the column is reduced to the lower pressure by counterflow blowdown. At this reduced pressure the column is purged (counterflow to the feed) using part of the pure product gas. Since pure product gas is used as a purge, the purge product (or waste) gas contains both adsorbate and carrier gas. The volume of purge gas required for the Skarstrom cycle is,

(17-26)

where γ typically is between 1.15 and 1.5. Because of the volumetric expansion of the product gas from p_h to purge gas at p_L, significantly less moles of purge gas are needed than feed gas. Dropping the pressure also reduces the partial pressure, which helps desorb the adsorbate. The final step in the Skarstrom cycle is repressurization of the column. This step was originally done with fresh feed gas although with concentrated systems it is now much more common to use high-pressure product gas as shown in Figure 17-11. Usually two or more columns are operated in parallel, but with cycles out of phase so that one column is producing product when the other needs to be purged or repressurized. One method of doing this is shown in Figure 17-11B. PSA systems with from one to twelve columns are used commercially. PSA has the advantage that cycles can be very fast—a minute or two is common in industry and some cycles are as short as a few seconds. These short cycles lead to high productivity and hence relatively small adsorbers.

Figure 17-12 illustrates a simple vacuum swing cycle. The feed enters at the high pressure, which may be essentially atmospheric pressure. If p_h is significantly above atmospheric pressure, a short optional blowdown step is included. Then a vacuum pump is used to reduce the pressure to very low pressures. At very low pressures the partial pressure is very low and very little adsorbate can be adsorbed (see Figure 17-2). Unfortunately, this step is slow and productivities of VSA systems are low. However, VSA has the advantage that a relatively pure product gas and a relatively pure adsorbate product can be produced. For example, VSA units can separate air into an oxygen product and a nitrogen product. The final step is repressurization of the column. VSA units are usually operated with several columns in parallel. A large number of variations of PSA, VSA, and combinations of PSA and VSA cycles have been invented (Kumar, 1996; Ruthven et al., 1994; Tondeur and Wankat, 1985). For example, the purge step in a PSA cycle may be operated at a pressure less than atmospheric pressure.

Figure 17-12. Basic vacuum swing adsorption cycle

The simple Skarstrom cycle for PSA shown in Figure 17-11A has constant pressure (isobaric) periods and periods when pressure is changing. We will assume that a very dilute gas stream containing trace amounts of adsorbate A in an weakly adsorbed carrier gas is being processed and that over the concentration range of interest the linear isotherm, Eq. (17-5b), is accurate. If mass transfer is very rapid, then the solute movement theory can be applied. Since the system is very dilute, the gas velocity is constant and the system is assumed to be isothermal. In more concentrated PSA systems neither of these assumptions are true, and a more complicated theory must be used (Ruthven et al., 1994).

During the isobaric periods (feed at p_h and purge at p_L) the solute moves at a velocity u_s. For a gas that can be assumed to be ideal and an isotherm in partial pressure units, the solute velocity is given by Eq. (17-15d). Normally, K_d,i = 1.0 and all of the adsorption sites are accessible to the small gas molecules.

During blowdown the mole fraction of adsorbate in the gas increases because of desorption as the pressure drops. During repressurization the opposite occurs and the adsorbate mole fraction in the gas decreases. When the pressure changes, the solute waves also shift location. Determining the mole fraction changes and shifts in location for these steps requires solution of the partial differential equations for the system (Chan et al., 1981). The results for linear isotherms are relatively simple. Define parameter β_strong for the strongly adsorbed component as

(17-27)

This parameter measures the ratio of the amount of weakly adsorbed to the amount of strongly adsorbed adsorbate in a segment of the column. If the weakly adsorbed component does not adsorb, then in Eq. (17-27). . Then the shift in mole fraction of the strongly adsorbed species A is

(17-28a)

Equation (17-28a) predicts an increase in mole fraction y_A for a decrease in pressure (try it to convince yourself). The shift in location of solute waves can be found from

(17-28b)

where the axial distance z must be measured from the end of the column that is closed (this can vary during the PSA cycles). Note that if z_before = 0, then z_after = 0 also. Solute waves at the closed end of the column cannot shift.

Application of the solute movement theory will be illustrated in Example 17-4. Before doing this, we note that axial dispersion is normally significant in gas systems. Thus, we expect that the solute movement theory will over-predict the separation that occurs. Alternatively, the value of γ required in Eq. (17-26) for a given product purity will be larger in a real system than predicted by the solute movement theory (γ = 1 for linear systems). For separations based on differences in equilibrium isotherms, if the solute movement theory predicts that a separation is not feasible or will not be economical, more detailed calculations will rarely improve the results.

EXAMPLE 17-4 PSA system

A 0.50 m. long column is used to remove methane (M) from hydrogen using Calgon Carbon PCB activated carbon. The feed gas is 0.002 mole fraction methane. Superficial velocity is 0.0465 m/s during the feed step. The high pressure is 3.0 atm while the low pressure is 0.5 atm. A standard 2-column Skarstrom cycle is used. The symmetric cycle is:

Repressurize with feed	0 to 1 sec.
Feed step at pH	1 to 30 sec.
Blowdown	30 to 31 sec.
Purge at pL	31 to 60 sec.

The operation is at 480 K. Use a pure purge gas with a purge to feed ratio of γ = 1.1. Carbon properties: ρ_s = 2.1 g/cc, K_d = 1.0, ε_p = 0.336, ε_e = 0.43. Equilibrium data are available in Table 17-3 and has been analyzed in Example 17-1. Draw the characteristic diagram for the first cycle assuming the bed is initially clean at 0.5 atm and predict the outlet concentration profile.

Solution

A. Define. Plot the movement of solute during the four steps shown in Figure 17-11A and use this diagram plus appropriate equations to predict outlet mole fractions.

B. Explore. The Table in Example 17-1 gives K_M = 1.888 × 10⁻⁴ (kPa)⁻¹ and q_max = 3.84 m mol/g at 480 K. Since the feed mole fraction is very low, the isotherm will be in the linear range where . Since hydrogen does not adsorb, . We will assume operation is isothermal.

C. Plan. Start by repressurizing with the feed. We can calculate β_M from Eq. (17-27) and the distance the wave moves in the column can be determined from Eq. (17-28b). During the feed step at 3.0 atm the methane travels at a constant solute velocity given by Eq. (17-15d). There will be two waves as shown in Figure 17-13A. During blowdown Eq. (17-28a) is used to determine the new mole fraction. Waves during the purge step again follow Eq. (17-15d) but with v_purge = γv_feed.

Figure 17-13. Solute movement solution for PSA system in Example 17-4; A) solute movement diagram, B) outlet concentration profiles

D. Do it. Repressurization Step: From Eq. (17-27)

The units in the last term in the denominator are a little tricky. The gas constant used is . If a different gas constant is used, the units on the other terms have to be adjusted.

Equation (17-28b) is used for the shift in location of solute waves. Because the top of the column in Figure 17-11A is the closed end, z is measured for this step from that end. Then the feed end is z = 0.50 m. From Eq. (17-28b)

This is 0.50 − 0.3415 = 0.1585 m from feed end of column (point 1 in Fig. 17-13A).

The mole fraction methane at this location can be determined from Eq. (17-28a)

where y_{M, before} = 0.002 is the feed mole fraction from the bottom of the column which shifts during repressurization to point 1. The mole fraction y_{M, after} is lower since methane is adsorbed as the column is pressurized.

Feed Step: Equation (17-15d) is used to determine the methane solute velocity u_M,

(17-15d)

(Note the denominator is the same as the denominator for β_M).

Since ε_ev_inter = v_super = 0.0465 m/s,

In the 29 seconds of the feed step the methane waves can move 0.462 m. Thus, one of the waves breaks through while the other does not (see points 2 and 3 on Figure 17-13A).

The wave that breaks through travels 0.3415 m, which requires (0.3415 m)/(0.01592 m/s) = 21.45 s. Including the 1.0 seconds for repressurization this is 22.45 s after the start of the cycle. Point 3 is at z = 0.462 m.

Blowdown: Point 3 will shift according to Eq. (17-28b) with the closed end again at the top. Thus, measuring from the top we have

z_before = 0.5 − 0.462 = 0.038 m

and from Eq. (17-28b)

or 0.50 − 0.056 = 0.444 m from the bottom. With y_before = y_F = 0.002, Eq. (17-28a) gives

As expected, methane mole fraction increases as methane desorbs.

Purge Step. The methane velocity during the purge step is again given by Eq. (17-15d); however, the interstitial velocity is increased since γ > 1.0

Then

There are two waves, (from top of column, point 5, and from point 4). They can both travel a distance 0.508 m in 29 s; thus, they both exit the column.

Wave from Point 4 exit time: 0.444 m/0.01751 m/s + 31 s = 56.36 s (Point 8)

Wave from Point 5 exit time: 0.5 m/0.01751 m/s + 31 s = 59.57 s (Point 9)

Outlet Mole Fraction Profile: At the top of the column

0 to 1 s, No product

1 to 22.45 s (point 2), y = 0

At point 2, y = 0.000488

At point 6, to estimate mole fraction, follow solute back to point 7 at t = 1 second (end of blowdown).

The final mole fraction at point 7 follows solute that enters during repressurization at a specific, but unknown, pressure between p_L = 0.5 and p_H = 3.0 atm. Equation (17-28) can be employed with z_after = 0.462, z_before = 0.50, p_after = 3.0, β_M = 0.2128 to calculate this unknown pressure, p_before. Then

(17-28c)

Since the feed entered at y_{M, before} = 0.002, Eq. (17-28a) can be used to estimate y_{M, after}.

Since concentrations are constant along the trace of the solute movement, this is also the mole fraction at point 6.

A similar procedure can be used (see Problem 17-D10) to find intermediate point 10 (shown at 26.126 s and y_M = 0.000876). The outlet profile is not linear.

During blowdown, gas exits (at bottom of column) initially at y_F = 0.002 and increases to y_{after, BD} = 0.0082. The exact shape can be estimated by the procedure used above. The mole fraction is constant at 0.0082 until gas from point 4 exits at Point 8 at 56.36 s. Gas mole fraction drops to y_out = 0 and the column is completely regenerated at point 9 (59.57 s). The intermediate point 11 shown in Figure 17-13B is estimated (see Problem 17-D10).

E. Check. Because the flow rates vary in unknown ways during repressurization and blowdown steps, a complete mass balance check is not possible. However, an approximate check balancing methane flows in the feed and purge steps can be done.

where the molar density is for an ideal gas and A_c = cross-sectional area.

The integrals can be estimated by assuming the variation in y_M is linear.

The inlet and outlet amounts are reasonably close.

F. Generalization Notes: 1. The ratio of moles gas fed to the purge gas used

is and (ignoring repressurization and blowdown) the ratio of product gas (hydrogen) to feed gas is

PSA produces a significant amount of high-pressure pure product because the gas is expanded before it is used for purging.

2. This design is inappropriate if pure hydrogen is desired during the entire feed step. Breakthrough can be prevented by changing the design (see Problem 17-B2).

3. Repressurization with feed causes the methane to penetrate the bed a significant distance during this step. Repressurization with product works better (see Problem 17-D11).

4. This example uses complete regeneration (the column is clean at the end of the cycle). Incomplete regeneration (leaving a heel) allows for more production of pure product and is employed in industrial systems.

This section illustrates PSA calculations for the simplest possible case—the local equilibrium theory for trace components for an isothermal system. If the mole fraction of the strongly adsorbed component is higher in the feed, the isotherm is likely to be nonlinear and the velocity will vary along the length of the column. In addition, operation is much more likely to be adiabatic instead of isothermal. It is also common to have both components adsorb or to have more than two components. If dispersion and mass transfer resistances are important, detailed simulations will be required. In addition, PSA has spawned a large number of inventive cycles to accomplish different purposes. If you need to understand any of these situations, Ruthven et al. (1994) provide an advanced treatment. White and Barkley (1989) discuss practical aspects of PSA design such as pressure drops, the velocity limit to prevent fluidization of the bed and start-up.

17.3.3 Simulated Moving Beds

Most adsorption processes remove all adsorbed solutes from a nonadsorbed or weakly adsorbed carrier gas or solvent. Elution chromatography, on the other hand, was developed to separate a number of solutes from each other with all of the products containing the carrier gas or solvent. Simulated moving bed (SMB) technology is a melding of purge or displacement adsorption and chromatographic methods that was developed by UOP in the late 1950s and early 1960s (Broughton and Gerhold, 1961; Broughton et al., 1970). SMBs are currently extensively used for binary separations. Although gas-phase systems have been studied, all current commercial applications are liquid-phase (see Problem 17-A6). Common industrial applications of SMBs are separation of aqueous solutions of glucose from fructose to make sweeteners (heavily used in soft drinks), separation of p-xylene (used to make polyesters) from m-xylene, and the separation of optical isomers in the pharmaceutical industry. The SMB process is relatively expensive since the products must be separated from the solvent or desorbent usually by evaporation or distillation (Ruthven, 1984; Wankat, 1986). Chin and Wang (2004) discuss practical aspects of SMB systems such as pump placement, pressure drops, and valve requirements.

The most efficient approach to separating a binary mixture by adsorption appears to be to have a counter-current process similar to extraction. Since it is convenient to regenerate the adsorbent within the device, the result is the true moving bed (TMB) system (Figure 17-14A). Zones 2 and 3 do the actual separation of the two solutes. (Zones are packed regions between inlet and outlet ports.) Zone 1, which is optional but almost always included, adsorbs the weakly adsorbed solute A onto the solid so that desorbent D can be recycled. Zone 4 regenerates the solid by removing strongly adsorbed solute B using a purge or displacement with desorbent D. This figure is loosely analogous to a distillation unit: A is the light key, B is the heavy key, zones 2 and 3 are the enriching and stripping sections, respectively; and zones 1 and 4 are analogous to a total condenser and a total reboiler, respectively. The TMB system would work if chemical engineers had the technology to build large-scale systems that could move a solid counter-current to a fluid with no axial mixing. Since this goal has proved to be elusive, UOP developed the SMB.

Figure 17-14. Separation of binary mixtures by adsorption; A) TMB, B) SMB showing complete cycle with four time steps

An SMB system is shown in Figure 17-14B for four different time steps. The system is arranged in a continuous loop with inlet and outlet ports. After a set time period, the switching time t_sw, the port locations are all advanced by one column. From the viewpoint of an observer at the extract port the solid moves downward when the switch is done while the fluid continues to move continuously upwards. Thus, an intermittent, counter-current movement of the solid and liquid has been simulated. The port switching is continued indefinitely. In Figure 17-14B with four columns the cycle repeats after every four switches. If there are N columns, the cycle repeats after N switches. A large number of modifications to SMB systems have been developed. The most common is to have two or more columns per zone instead of the one column per zone shown in Figure 17-14B. This makes the operation closer to the TMB and improves the purity of products.

The SMB system shown in Figure 17-14B is quite a complicated system, particularly if compared to the simple elution chromatographic system shown in Figure 17-5B. The SMB is used in industry for high purity separations of binary feeds since much less desorbent and adsorbent are required. The solute movement analysis helps to explain how this complicated process works.

Each column in Figure 17-14B acts as a chromatographic column that undergoes a series of steps. The solute velocity in each column of the SMB can be determined in the same way as for an elution chromatography system. Thus, the solute velocity in any column of the SMB is given by Eqs. (17-15). The chromatography and SMB processes differ only in the way columns are coupled, which are their boundary conditions!

In order for the SMB in Figure 17-14B to separate solutes A and B completely the following conditions must be met:

Zone 1. To produce pure desorbent for recycle, solute A must not breakthrough (that is, appear in the outlet of zone 1) during the switching period, t_sw. Thus,

u_A,1 t_sw ≤ L

Since the average port velocity is defined as, u_port = L/t_sw, this condition becomes

(17-29a)

The equation is written as an equality instead of as an inequality since it is easier to manipulate equations than inequalities.

Zone 2. To separate A and B we want net movement of A up the column and net movement of B down the column. Thus,

(17-29b,c)

These conditions require that solute A will breakthrough and solute B will not breakthrough from zone 2 during the switching period.

Zone 3. To separate A from B we again want net upward movement of A and net downward movement of B. Thus,

(17-29d,e)

Zone 4. To regenerate the column, all solute B must be removed in zone 4. This requires that solute B have a net upward movement.

(17-29f)

Equations (17-29a) to (17-29f) can all be simultaneously satisfied by changing the flow rates of feed, desorbent and the two product streams in Figure 17-14B to change the velocities in each zone. For example, since the A product stream is withdrawn between zones 1 and 2, v₁ < v₂. By proper selection of the velocities and the port velocity, we can satisfy these six equations. Note that since v₂ > v₃, Eqs. (17-29b) and (17-29e) are automatically satisfied if M_3A = M_2A, M_3B = M_2B and Eqs. (17-29d) and (17-29c) are satisfied. Thus, we need to simultaneously solve Eqs. (17-29a), (17-29c), (17-29d), and (17-29f).

Usually the desorbent must be removed from the A and B product streams. Increasing the amount of desorbent will increase the cost for this removal and will also increase the diameters of the columns requiring more adsorbent. Thus, the ratio of desorbent to feed, D/F, often controls the cost of SMB systems. For an ideal system with no zone spreading (no axial dispersion and very fast mass transfer rates) the solute movement theory can be used to calculate D/F by solving Eqs. (17-29a), (17-29c), (17-29d), and (17-29f) simultaneously with Eq. (17-15) and the velocity relationships

(17-30a,b,c,d)

where v_Feed = F/(ε_e A_c) is the interstitial velocity the feed would have in the column. F is the volumetric flow rate of feed and A_c is the cross-sectional area of the columns with similar definitions for the desorbent and product velocities. If F and A_c are known, then we can first solve for u_port, v₁, v₂, v₃, and v₄, then for v_A,product, v_B,product, and v_D. Then D/F = v_D / v_Feed. These calculations are developed in Problem 17-C11. The minimum D/F ratio, (D/F)_min can be calculated by setting all M_i = 1.0. This minimum has a significance similar to that of (L/D)_min in distillation. For linear systems (D/F)_min = 1.0, which is also the thermodynamic minimum.

In actual practice there is considerable spreading due to mass transfer resistances, axial dispersion, and mixing in the transfer lines and valves. If the SMB is operated at (D/F)_min, the raffinate and extract products will not be pure. To obtain higher purities D/F is usually increased; however, the SMB is more complicated than binary distillation. The additional desorbent must be distributed throughout the four zones to give the optimum velocities in each zone. One approach to this optimization is to pick values of the multipliers M₁ < 1, M_2B < 1, M_3A > 1, and M₄ > 1. Then the value of velocities and D/F can be determined by solving the solute movement equations (see Problem 17.C11 for the equation for D/F). The experiment or simulation is then run again with these new flow rates. The procedure is repeated until the desired purities are achieved.

Example 17-5. SMB system

Ching and Ruthven (1985) found that the equilibrium of fructose and glucose on ion exchange resin in the calcium form was linear for concentrations below 5 g/100 ml. Their equilibrium expressions are: q_gluc = 0.51 c_gluc, q_fruc = 0.88 c_fruc, at 30°C where both q and c are in g/liter. For this resin, ε_p = 0 and ε_e = 0.4.

We want to design an SMB system to separate fructose and glucose. If the switching time t_sw = 5 min and D_col = 0.4743 m, design an SMB system with one column per zone for this separation at the minimum D/F = 1.0.

Solution

A. Define. To design the system we need to determine L, and all feed and product flow rates. Since feed flow rate F is not specified, we can find flow rates as functions of F.

B. Explore. Since q and c are in mass/volume, u_s,i is obtained from Eq. (17-15a) by removing ρ_s. Thus, in terms of Eq. (17-15e) the solute velocity constant is

where and . The resulting velocities can be used in Eq. (17-29), (17-30) and u_port = L/t_sw. The value of u_port (see Problem 17.C.11) is

(17-31a)

where C_B is for the solute with stronger adsorption (fructose).

C. Plan. We can calculate C_gluc and C_fruc (fructose is more strongly adsorbed). Then u_port can be found as a function of v_F or F. Equations (17-29) and (17-30) can be used to calculate all other flow rates and L = u_portt_sw.

D. Do it. From the expression for C_i

The feed velocity if F is in m³/min.

With M_i = 1.0

L = u_portt_sw = (25.478 F m/min) (5 min.) = 127.39 F (in m).

From Eqs. (17-29c) and (17-15e) we obtain

v₂ = M_2B u_port/C_fruc = (1.0) (25.478 F)/0.4310 = 59.113 F

From Eq. (17.30b)

v₃ = v₂ − v_fruc = 59.113 F − 14.147 F = 44.966 F

Similarly,

v₁ = M_1A u_port/C_gluc = (1.0)(25.478 F)/0.5666 = 44.966 F = v_recycle

v₄ = M_4B u_port/C_fruc = (1.0)(25.478 F)/0.4310 = 59.113 F

Then, from Eq. (17-30a), Eq. (17-30c), and Eq. (17-30d)

vgluc, prod = vA, prod = v2 − v1 = 14.147 F
vfruc, prod = vB, prod = v4 − v3 = 14.147 F
vD = v4 − v1 = 14.147 F	and	D/F = vD/vF = 1.0.

E. Check. As expected, D/F = 1.0 since all the M_i = 1.0. Also A_prod/F = v_{A, prod}/v_F = 1.0 and B_prod/F = 1.0.

F. Generalization. The equal values for feed, desorbent and products; and v₁ = v₃, v₂ = v₄ occurs only for all M_i = 1. Usually, v₄ is the highest velocity. Further development of equations is done in Problem 17.C11.

To complete the design we need a value for F in m³/min. Then the velocities and pressure drops can be calculated (e.g., with the Ergun equation). If pressure drop is too large, F or D_col need to be adjusted. Because of dispersion and mass transfer resistances the two products will not be 100% pure. The actual purities can be determined by experiment or detailed simulation. If more separation is needed, one can reduce M_1A and M_2B while increasing M_3A and M_4B (see Problem 17.D13).

The TMB shown in Figure 17-14A is also of interest, and can be analyzed using solute movement theory. This analysis is explored in Problem 17.C12.

17.4 NONLINEAR SOLUTE MOVEMENT ANALYSIS

Since most adsorption and ion exchange separations of commercial significance operate in the nonlinear region of the isotherm, the previous analysis needs to be expanded to nonlinear systems. Nonlinear behavior is distinctly different than linear behavior since one usually observes shock or constant pattern waves during the feed step and diffuse or proportional pattern waves during regeneration. Experimental evidence for constant pattern waves is shown in Figure 17-15A. The isotherm for carbon dioxide on activated carbon is a Langmuir-type shape. During loading we expect shock or constant pattern waves. This is clearly shown in the top figure since the waves can be moved along the time axis and easily be superimposed on each other. During desorption (elution) a diffuse or proportional pattern wave is expected (Figure 17-15B). These waves cannot be superimposed on each other. The width of proportional pattern waves is directly proportional to the distance the wave travels in the column.

Figure 17-15. Adsorption and desorption of CO₂ on activated carbon; A) adsorption breakthrough curves illustrating constant pattern behavior, B) desorption (elution) curves illustrating proportional pattern behavior, v_super = 4.26 cm/s. Reprinted from Weyde and Wicke (1940).

Equation (17-14), which was derived for any type of isotherm, is the starting point for the analysis of movement of solutes with nonlinear isotherms. We now need to substitute the desired nonlinear equilibrium expression into the last term in the denominator for Δq/Δc. This insertion differs from inserting Δq/Δc = K′ in the linear case because two separate forms of the equation result depending on the operation and because the result will be concentration dependent. There are a huge number of expressions for nonlinear isotherms. To be specific, we will focus on the Langmuir isotherm, Eq. (17-5a), Eq. (17-6a), and Eq. (17-6c), which are probably the most popular forms.

17.4.1 Diffuse Waves

For an isotherm of Langmuir shape, if the column is initially loaded at some high concentration, c_high, and is fed with a fluid of low concentration, c_low, the result is a diffuse or proportional pattern wave. Since the derivative of q with respect to c exists,

(17-32a)

We can now calculate this derivative for any desired nonlinear isotherm (for linear isotherms dq/dc = K′ and the result is Eq. (17-15)). Specifically for the Langmuir isotherm in Eq. (17-6c)

(17-32b)

and the solute wave velocity is

(17-32c)

Note that for nonlinear isotherms the solute wave velocity depends upon the concentration. For Langmuir isotherms as the solute concentration increases the denominator in Eq. (17-32b) will decrease and the solute wave velocity increases. Another way to look at this is that dq/dc is the local slope or tangent of the isotherm. Figure 17-2 shows that for a Langmuir isotherm dq/dc is largest and thus, the velocity u_s is smallest as c approaches zero. Values for u_s can be calculated from Eq. (17-32c) for a number of specific concentrations and the diffuse wave can be plotted as shown in Figure 17-16A. The times at which these solute waves, which are at known concentrations, exit the column (at z = L) can be determined (see Figure 17-16A) and the outlet concentration profile can be plotted as in Figure 17-16B. Note that the outlet wave varies continuously and is diffuse. This result agrees with experiments that show zone spreading is proportional to the column length and have a smooth even spread in concentration. The analysis procedure is illustrated in Example 17-6.

Figure 17-16. Diffuse wave analysis; A) solute movement graph for Example 17-6, B) predicted outlet concentration profile

EXAMPLE 17-6. Diffuse wave

A 100.0 cm long column is packed with activated alumina. The column is initially totally saturated at c = 0.011 gmole/L anthracene in cyclohexane solvent. It is then eluted with pure cyclohexane solvent (c = 0) at a superficial velocity of 30.0 cm/min. Predict and plot the outlet concentration profile using solute movement theory.

Data: ε_e = 0.42, ε_p = 0, K_d = 1.0, ρ_f (cyclohexane) = 0.78 kg/L, ρ_p = 1.465 kg/L, Equilibrium where q = gmole/kg and c = gmoles/L.

Assume operation is isothermal.

Solution

A. Define. Find the values of the outlet concentration at different times using the solute movement theory.

B. Explore. The most important decision is whether a diffuse or a shock wave (Section 17.4.2) will result. With a Langmuir isotherm a diffuse wave results when a more concentrated solution (c = 0.011) is eluted with a dilute solution (c = 0.0). If this is incorrect, the analysis will show us that there is an error.

C. Plan. Since ε_p = 0, the solute velocity (Eq. 17-32c) for a diffuse wave becomes,

Substituting in the parameter values this becomes

Since u_s depends upon the concentration, we can select arbitrary values of the concentration ranging from 0.011 to zero, calculate u_s, and determine t_out = L/u_s.

D. Do it. The values are tabulated below for selected values of concentration.

The solute movement solution using these values is shown in Figure 17-16A and the outlet concentration profile plotted from the tabulated values is shown in Figure 17-16B.

E. Check. A check can be made with a mass balance over the entire elution time.

−Outlet − Accumulation = 0

where the outlet concentration = ∫ A_c v_super c_out(t), and accumulation = A_c ρ_s [q(c=0) – q(c=0.011)] with q determined from the isotherm.

F. Generalize. The shape shown in Figure 17-16B, particularly the strong tailing at very low concentrations is typical of elution behavior with highly nonlinear isotherms. Complete removal of anthracene with an isothermal purge step will take a large amount of solute.

The width of the wave (in time units) at z = L is easily determined as

(17-33)

Since the width is proportional to the column length L, this result agrees with experimental observations.

17.4.2 Shock Waves

For an isotherm with a Langmuir shape, if the column is initially loaded at some low concentration, c_low, (c_low = 0 if the column is clean) and is fed with a fluid of a higher concentration, c_high (see Figure 17-17A), the result will be a shock wave. The feed step in adsorption processes usually results in shock waves. Experiments show that when a shock wave is predicted the zone spreading is constant regardless of the column length (a constant pattern wave). With the assumptions of the solute movement theory (infinitely fast rates of mass transfer and no axial dispersion), the wave becomes infinitely sharp (a shock) and the derivative dq/dc does not exist. Thus, the Δq/Δc term in the denominator of Eq. (17-14) must be retained as discrete jumps in q and c, and the shock wave velocity is,

(17-34)

Figure 17-17. Shock wave analysis: A) inlet concentration; B) shock wave following Eq. (17-34); C) outlet concentrations with solid line predicted by solute movement theory, and dashed line representing experimental result (modified from Wankat, 1986). Reprinted with permission, copyright, 1986 Phillip C. Wankat.

where the subscripts “before” and “after” refer to the conditions immediately before and immediately after the shock wave. The fluid and solid before the shock wave (c_before and q_before) are assumed to be in equilibrium as are the values of c_after and q_after after the shock wave. For a general Langmuir isotherm, Eq. (17-6c), the shock wave velocity is

(17-35)

Now the shock wave velocity depends upon the concentrations on both sides of the shock wave. The resulting shock wave is shown in Figure 17-17B and the outlet concentration is shown in Figure 17-17C.

Superficially, the outlet concentration profile in Figure 17-17C looks like the result from a linear isotherm (concentration jumps to c_feed). However, for the linear isotherm this outlet step occurs at t = L/u_s, which is constant regardless of the feed and initial concentrations. The shock wave outlet step occurs at t = L/u_sh, which depends upon both the feed and initial concentrations. In addition, shock waves are self-sharpening, a concept explored in Example 17-7. Waves in systems with linear isotherms are not self-sharpening (see Problem 17.D3).

EXAMPLE 17-7. Self-sharpening shock wave

A 100.0 cm long column is packed with activated alumina. The column is initially filled with pure cyclohexane solvent (c = 0.0 gmole/L anthracene). At t = 0 a feed containing 0.0090 gmole/liter anthracene in cyclohexane solvent is input. At t = 10 minutes a feed containing c = 0.011 gmole/L anthracene in cyclohexane solvent is input. Superficial velocity is 20.0 cm/min. Predict and plot the outlet concentration profile using solute movement theory.

Data: ε_e = 0.42, ε_p = 0, K_d = 1.0, ρ_f (cyclohexane) = 0.78 kg/L, ρ_p = 1.465 kg/L, Equilibrium q = 22c/(1 + 375c) where q = gmole/kg and c = gmoles/L.

Solution

A. Define. Find the values of the outlet concentration at different times using the solute movement theory.

B. Explore. The most important decision is whether a diffuse or a shock wave (Section 17.4.2) will result for each feed step. With a Langmuir isotherm a shock wave results when a concentrated feed (first c = 0.009 then c = 0.011) is fed to a column that is initially more dilute (first c = 0.0 then c = 0.009). Thus, we expect a first shock wave followed after 10 minutes by a second, which is faster. If they intersect, there will be a third shock wave (feed c_F = 0.011 and initial c = 0.0). (Realizing that there could be a third shock wave is probably the hardest part of this problem.)

C. Plan. The shock velocity can be calculated from (Eq. 17-35) with ε_p = 0. With ε_p = 0, the solid density ρ_s is equal to the particle density ρ_p. The interstitial velocity is

v_inter = v_super/ε_e = 20.0/0.42 = 47.62 cm/min

Substituting in the values of parameters, except for Δq/Δc, the shock wave velocity is

where

For the first shock wave, since c_initial = 0, q (c_initial) = 0. Since c_F = 0.009, we can calculate q_sh1 (c_F) from the Langmuir isotherm, which allows us to calculate Δq/Δc and u_sh1. A similar calculation can be done to calculate u_sh2. Then we can calculate at what distance the two shock waves intersect. If this occurs before the end of the column, we can determine the velocity of the third shock wave.

D. Do it.

Shock wave 1 (c_initial = 0, c_F = 0.009)

Shock wave 2 (c_initial = 0.009, c_F = 0.011). q_2initial = 0.04526

The solute movement diagram is shown in Figure 17-18A. The intersection of the two shock waves occurs at z_intersect, t_intersect. The first shock wave travels

z_intersect = u_sh,1 t_intersect

while the second shock wave travels

z_intersect = u_sh,2 (t_intersect − 10)

Setting these equal and solving for t_intersect

Then

z_intersect = (4.262)(13.608 min) = 58.0 cm

After this distance there is a third shock wave with c_initial = 0.0 (the initial column concentration at t = 0) and c_F = 0.011 (feed concentration after 10 minutes).

This shock exits at

The complete solute movement diagram is shown in Figure 17-18A and the outlet concentration profile is in Figure 17-18B.

E. Check. As expected u_sh,2 > u_sh,3 > u_sh,1. We can also check the mass balance until breakthrough occurs

Inlet − Accumulation = 0

The inlet consists of feed 1 (c_F = 0.009) for ten minutes and feed 2 (c_F = 0.011) until breakthrough at t_out = 22.146 minutes. The accumulation = Δq ρ_s (πD²/4) L where the change in the amount adsorbed, Δq = q(c = 0.011) – q(c=0) = 0.0472. This mass balance is satisfied.

F. Generalize. The single shock wave that results when two steps are input occurs because of the self-sharpening behavior of shock waves. Diffuse waves can also be sharpened if they are followed with a shock wave. This phenomenon is used in commercial cycles. The regeneration cycle will require much less purge if an adsorbate tail (called a heel) is left in the column (see Figure 17-16B). This diffuse wave is then sharpened when the next feed step forms a shock wave.

The wave interactions shown in Figure 17-18A are a very important part of the study of nonlinear systems. Shock waves and diffuse waves can also interact (Wankat, 1990, Chapter 07). If there are two or more adsorbates that compete for sites (e.g., with the multicomponent Langmuir isotherm, Eq. (17-8)), interactions often occur between shock waves and diffuse waves from the different components (e.g., Ruthven, 1984; Yang, 1987). The theories for two or more interacting solutes are beyond the scope of this introductory chapter.

Figure 17-18. Analysis and results for Example 17-7: A) solute movement diagram showing intersection of two shock waves, B) outlet concentration profile

In experiments (Figure 17-15A) the outlet concentration profiles are not sharp as shown in Figures 17-17B and 17-18B. Instead the finite mass transfer rates and finite amounts of axial dispersion spread the wave while the isotherm effect (illustrated in Example 17-7) counteracts this spreading. The final result is a dynamic equilibrium where the wave spreads a certain amount and then stops spreading. Once formed, this constant pattern wave has a constant width regardless of the column length.

17.5 ION EXCHANGE

Ion exchange is a unit operation in which ions held on a solid resin are exchanged for ions in the feed solution. For most people the most familiar ion exchange system is water softening, which replaces calcium and magnesium ions (“hard” water ions) in the feed water with sodium ions (“soft” water ions). If the feed water containing calcium and magnesium is continued, eventually the resin will become saturated with these ions and no additional exchange occurs. To produce soft water the resin needs to be regenerated, which can be done with a concentrated solution of sodium chloride salt. A number of other ion exchange separations are done commercially.

The most common materials used for ion exchange are polymer resins with charged groups attached (Anderson, 1979; Dechow, 1989; Dorfner, 1991; LeVan et al., 1997; Wankat, 1990). Cation-exchange resins have fixed negative charges while anion-exchange resins have fixed positive charges. The resin is called strong if the resin is fully ionized and weak if the resin is not fully ionized. The most commonly used strong resins are based on polystyrene (see Figure 16-3E) cross-linked with divinyl benzene. The most common strong cation exchange resin uses benzene-sulfonic acid groups while the most common strong anion-exchange resins have a quaternary ammonium structure. These resins are commonly used for water treatment and have very good chemical resistance although they are attacked by chlorine. Copolymers of divinylbenzene and acrylic or methacrylic acid are used for weak acid resins. No single type of weak base resin is dominant although the use of a tertiary amine group on a polystyrene-DVB resin is common. Although the capacity of the weak exchangers is lower than for strong resins, they also require less regenerant. The weak resins tend to be less robust than the strong resins and need to be protected from chemical attack. Typical properties of ion-exchange resins are listed in Table 17-4.

TABLE 17-4 . Properties of common ion-exchange resins (LeVan et al., 1997; Wankat, 1990)

Ion exchange involves a reversible reaction between ions in solution and ions held on the resin. An example of monovalent cation exchange is the removal of sodium ions from the resin using hydrochloric acid.

(17-36)

In this equation R⁻ represents the fixed negative charges such as on the resin. The hydrogen and sodium ions that are exchanging are called counter ions. The chloride ion, which has the same charge as the fixed groups, is called the co-ion. Although the chloride anion does not directly affect the reaction, at high concentrations it does affect the equilibrium characteristics. Exchange of a divalent cation with a monovalent cation is also common, and is exemplified by removal of calcium ions from water and replacement with sodium ions (water softening).

(17-37)

where X⁻ is any anion. Of course, there are a variety of other possibilities.

Standard practice is to define the equivalent fractions of ions in solution x_i and on the resin y_i

(17-38)

where c_i is the concentration of ion i in solution (e.g., in equivalents/m³), c_T is the total concentration of ions in solution, c_Ri is the concentration of ion i on the resin (in volume units such as equivalents/m³), and c_RT is the total concentration of ions on the resin. The total concentration of ions on the resin c_RT, the resin capacity, is a constant equal to the concentration of fixed negative sites set when the resin is made. One advantage of using equivalent fractions is they must sum to one.

(17-39)

17.5.1 Ion Exchange Equilibrium

For a simplified view of ion exchange equilibrium for binary ion exchange assume the equilibrium constant can be determined by the law of mass action. Let A represent the hydrogen ion and B the sodium ion. For monovalent ion exchange

A⁺ + RB + X⁻ = AR + B⁺ + X⁻

illustrated in Eq. (17-36), the mass action equilibrium expression simplifies to

(17-40a)

where we have assumed the local concentrations of co-ion X⁻ in the interior of the resin are both identical (and very low) in the numerator and denominator and hence cancel. (These concentrations are very low in the resin because of a phenomenon known as Donnan exclusion—the very high concentrations of fixed charges on the resin exclude the free anion X⁻ from the resin.) The equilibrium constant K_AB isn’t really constant (e.g., see Wankat, 1990), but in dilute solutions it will be very close to constant. Note that this equation is similar to Eq. (2-22a) for constant relative volatility. Solving Eq. (17-40a) for y_A,

(17-40b)

These equations can be applied to any monovalent exchange by substituting in the appropriate symbols for the exchanging ions. Experimental values for the equilibrium constant are required. A few representative values are given in Table 17-5. If we know the equilibrium constants K_AB and K_CB, we can calculate the value of K_CA from (see Problem 17.C13),

(17-40c)

TABLE 17-5 . Approximate equilibrium constants for ion exchange (Anderson, 1997)

This result expands the usefulness of Table 17-5 (e.g., try Problem 17.D17a). However, since the values in Table 17-5 are approximate, they should not be used for detailed design calculations.

The equilibrium expression for a divalent ion exchanging with a monovalent ion (e.g., the reaction in Eq. (17-37)) is not as simple. If we let D represent the divalent calcium ion and B the monovalent sodium ion, the reaction is

D⁺² + RB₂ + 2X⁻ = RD + 2B⁺ + 2 X⁻

and the equilibrium constant is

(17-41)

Selected values of divalent-monovalent equilibrium constants are given in Table 17-5. If K_DA and K_BA are known, then the desired constant K_DB is

(17-42)

Substituting the summation Eqs. (17-39) into Eq. (17-41b), we obtain

(17-43)

This equation can conveniently be solved for y_D for any specified value of x_D using the formula for solution of quadratic equations. Note that the effective equilibrium parameter in Eq. (17-43) is (K_DB c_RT /c_T). Since the total concentration in the fluid can easily be changed, this effective equilibrium parameter can be changed. This behavior is illustrated in Example 17-8 and Figure 17-19.

Figure 17-19. Equilibrium for copper-sodium exchange for Example 17-8; c_T = 0.01 N, c_T = 2.5 N

The equilibrium parameters are temperature dependent. However, ion exchange is usually operated at a constant temperature near the ambient temperature.

17.5.2 Movement of Ions

The solute movement theory developed in Sections 17.3. and 17.4. is easily extended to ion movement. For gel-type ion-exchange resins, which are most popular, there are no permanent pores and ε_p = 0. The development of the solute movement theory from Eqs. (17-10) to (17-14) is modified by setting ε_p = 0, expressing concentrations in terms of the equivalent fractions x and y, and including a Donnan exclusion factor K_DE. The result is

(17-44)

Co-ions (ions with the same charge as the ions fixed to the resin) are excluded and they have K_DE = 0. Exchanging ions are not excluded and K_DE = 1. Note that the solid density ρ_s does not appear in Eq. (17-44) because volumetric units are commonly used for ion concentrations in solution and on the resin.

Equation (17-44) can be applied to either diffuse or shock waves. Diffuse waves occur if a column that is concentrated in ion A (or D) is fed a solution of low concentration A (or D) and K_AB >1.0 [or (K_DB c_RT /c_T) > 1.0]. If K_AB <1.0 [or (K_DB c_RT /c_T) < 1.0] diffuse waves occur when a column that has a dilute amount of ion A (or D) is fed a solution that is concentrated in A (or D). For diffuse waves the ion velocity is

(17-45)

The derivative (dy_i/dx_i) can be determined from the appropriate equilibrium expression such as Eq. (17-40b) or Eq. (17-42b).

Shock waves occur when the conditions are the opposite of those for diffuse waves. For example, if K_AB >1.0 [or (K_DB c_RT /c_T) > 1.0] and a solution concentrated in A (or D) is fed to a column containing a dilute solution of the ion, a shock wave would be expected. The shock wave equation is

(17-46)

The equivalent concentrations of ion A (or D) in solution x_i and on the resin y_i are assumed to be in equilibrium both before and after the shock wave.

The ion velocities depend upon the ratio (c_RT/c_T) regardless of the form of the isotherm. This agrees with our physical intuition. If the resin capacity c_RT is high while the concentration of ion in solution c_T is low, we would expect that waves would move slowly.

Because changes in the total ion concentration can affect both equilibrium and ion velocities, we need to balance the total ion concentration. When the total ion concentration in the feed is changed, an ion wave passes through the column. For relatively dilute solutions the resin is already saturated with counter-ions, and more ions cannot be retained. For a balance on all ions, the total ions are excluded, K_DE = 0, and Eq. (17-44) becomes

(17-47)

This is the same result that is obtained for co-ions. The total ion and co-ion waves move rapidly through the column. The total ion wave affects counter-ion (u_A or u_D) velocities for all ion exchange systems and equilibrium for the exchange of ions with different charges (e.g., divalent-monovalent or trivalent-monovalent). The calculation of these effects is illustrated in Example 17-8.

EXAMPLE 17-8. Ion movement for divalent-monovalent exchange

An ion exchange column is filled with a strong acid resin (c_RT = 2.0 equivalents/L, ε_e = 0.40). The column initially is at a total ion concentration of c_T = 0.01N with x_Na = 0.90, x_Cu = 0.10. Chloride is the co-ion. At t = 0 we feed a 2.5 N aqueous solution of NaCl (x_Na = 1.0). The selectivity constants can be calculated from Table 17-5. The column is 50 cm long. The counter ions are not excluded (K_E = 1.0). The superficial velocity throughout the experiment is 20.0 cm/min.

Predict:

a. Equilibrium behavior at c_T = 0.01 N and at c_T = 2.5 N

b. The time the total ion wave exits

c. The values of xCu and yCu after the total ion wave exits

d. The time and shape of the exiting sodium wave

Solution

A. Define. We first want to find the equilibrium parameters at c_T = 0.01 N and at c_T = 2.5 N, and then plot the equilibrium results. Then, find the breakthrough time for the total ion wave, the equivalent fractions of copper at c_T = 2.5 N, and the outlet concentration profile for sodium.

B. Explore. The determination of the equilibrium behavior, the ion wave, and the values of x_Cu and y_Cu follows the equations, but the order can be a bit confusing. One reason for showing this example is to clarify how the calculations proceed. Sometimes, the biggest challenge is determining if the sodium wave is a shock or diffuse wave.

C. Plan. The selectivity can be found from Eq. (17-42) and the equilibrium parameter is (K_DB c_RT /c_T). The equilibrium curves can be found at arbitrary x values from Eq. (17-43). The velocity of the total ion wave is equal to the interstitial velocity. The equivalent fractions of copper can be found by solving Eq. (17-43) with x_Cu,before = 0.10 and c_T = 0.01 N to find y_Cu,before. When the total ion wave passes, set y_Cu,after = y_Cu,before, and solve Eq. (17-43) with this value of y_Cu,after and c_T = 2.50 N for x_Cu,after. Finally, the sodium breakthrough time can be calculated from either Eq. (17-45) or Eq. (17-46) after we decide if it is a diffuse or shock wave, respectively. To do this we will look at the shape of the isotherm for this decrease in copper concentration in the feed.

D. Do it.

First, find equilibrium curve at c_T = 0.01 N. Since copper is divalent and sodium is monovalent, use Eqs. (17-42) and (17-43). From Eq. (17-42)

Then Eq. (17-43) becomes

At x_Cu= 0.1 (the initial concentration)

Solving for y_Cu with a spreadsheet, we obtain y_Cu = 0.814. The equilibrium table below at c_T = 0.01 N was generated using this spreadsheet.

When and Eq. (17-42) is

and the equilibrium results are

These two tables are plotted in Figure 17-19. Note that with c_T = 0.01 N a very favorable isotherm results while with c_T = 2.5 N the isotherm is unfavorable.

The velocity of the total ion wave is u_{total_ion} = v_super/ε_e = 20.0/0.4 = 50 cm/min, and the breakthrough time of this wave is t_br = L/u_{total_ion} = 50.0/50.0 = 1.0 min.

From the table of equilibrium values at c_T = 0.01 N, when x_Cu,before = 0.10, y_Cu,before = 0.814. Then, when the total ion wave passes (now c_T = 2.5 N), y_Cu,after = y_Cu,before = 0.814. At c_T = 2.5 N and y_Cu,after = 0.814, we can solve Eq. (17-41c) for x_Cu,after.

The result is x_{Cu, after} = 0.836.

Note the large increase in copper equivalent fraction in the liquid since it is desorbed from the resin at the higher total ion concentration.

When the 2.5 N sodium chloride solution is fed to the column, the copper equivalent fraction, and concentration in the feed drop. Since the column is at 2.5 N when the sodium wave reaches any part of the column, we use that equilibrium curve in Figure 17-19. This is an unfavorable isotherm for copper; thus, with a drop in copper concentration in the feed a shock wave results. Then use Eq.(17-46) to calculate the shock wave velocity.

where Δy_Cu = y_{Cu, after NaCl} − y_{Cu, before NaCl}.

Δy_cu = 0.0 − 0.814 = −0.814, Δx_cu = 0 − 0.836 = −0.836, and

This shock wave exits at t_NaCl = L/u_sh = 50/16.96 = 2.95 minutes

The solute movement diagram is plotted in Figure 17-20.

Figure 17-20. Solute movement diagram for Example 17-8

E. Check. A copper mass balance on the entire cycle can be used as a check, Cu in − Cu out = Cu accumulation

From t = 0 to t = 2.95 minutes there is no copper flowing into the system. The outlet copper amount consists of one minute at (c_T = 0.01 N, x_Cu = 0.10) and 1.95 minutes at (c_T = 2.5 N, x_Cu = 0.836) as shown in Figure 17-20. Since we know the conditions at the beginning of the cycle (c_T = 0.01 N, x_Cu,before = 0.10, y_Cu,before = 0.814) and at the end of the cycle (c_T = 2.5 N, x_{Cu,after_NaCl} = 0.0, y_{Cu,after_NaCl} = 0.0), we can calculate the accumulation term for the cycle. The mass balance then becomes

This is certainly within the accuracy of the calculations.

F. Generalization. The change in equilibrium behavior from a favorable isotherm to an unfavorable isotherm when c_T is increased only occurs for exchange of ions with unequal charges. This change can result in shock waves during both the feed and regeneration steps. Since shock waves show significantly less spreading than diffuse waves, significantly less regenerant is required. This phenomenon is used to advantage in water softeners that exchange “hard” Ca⁺⁺ and Mg⁺⁺ ions (they precipitate when heated and foul cooking utensils or heat exchangers and interfere with soap) at very low c_T values with Na⁺ ions. Regeneration is done with a concentrated salt (NaCl) solution at high c_T.

This introductory presentation on ion exchange has been restricted to dilute solutions with the exchange of two ions. A variety of more complex situations including complex equilibria and mass transfer, partial ion exclusion, swelling of the resin (typically 40 to 50% for strong-acid resins), and the need for backwashing to remove dirt often occur in practice, but are beyond the scope of this introductory section. Information about the variety of phenomena, different equipment, and equilibrium theory is discussed by Helferrich (1962), Dechow (1989), and Tondeur and Bailly (1986) respectively.

17.6 MASS AND ENERGY TRANSFER

For detailed predictions and understanding of sorption separations we need to do a detailed analysis of diffusion rates, mass and energy transfer, and mass and energy balances in the column. In order to make the results somewhat tractable, we will make all the usual assumptions listed in Table 17-6.

TABLE 17-6 . Assumptions for mass and energy transfer analysis

The fluid flowing in the external void volume in Figure 17-1 is usually assumed to have a constant concentration (or partial pressure) at each axial distance z; thus, this bulk concentration is not a function of the radial distance from the center of the bed. During the course of separation the sorbates are first transferred down the column by bulk transfer. The sorbates then transfer across the external film and diffuse in the pores until they reach the sorbent sites. At these sites they are sorbed in a usually rapid step. For an equilibrium system, some sorbates will always be desorbing and diffusing back into the bulk fluid. Desorption is favored during the desorption step by increasing temperature or dropping the concentration of the sorbate.

Since the first step in the separation process, bulk transport, is assumed to be rapid enough to keep the bulk concentration (at any given value of z, t) constant, our first step will be to analyze film mass transfer.

17.6.1 Mass Transfer and Diffusion

Mass transfer across the film (see Figure 17-1) occurs by a combination of diffusion and convection. As is usual in film mass transfer, the driving force is assumed to be (c-c_pore) where the concentration in the pores c_pore is calculated at the surface of the particles. The film transfer term becomes −k_f a_p (c-c_pore) where k_f is the film mass transfer coefficient (m/s) and a_p is the surface area of the particles per volume (m⁻¹). For spherical particles

(17-48a)

With porous particles the film equation is a bit more complicated than in Chapter 15 since the accumulation of mass in the particle is distributed between the pores and the solid. The resulting equation is

(17-48b)

The left hand side of this equation is the accumulation of solute on the solid and in the fluid within the pores. The right hand side is the mass transfer rate across the film. Since the amount adsorbed in the particles and the concentration of the pore fluid are functions of r, and are the average amount adsorbed in the particle and the average pore fluid concentration, respectively.

A more familiar looking version of the film transport equation is obtained for a single-porosity model. This result can be formally obtained by setting ε_p = 0 and ε = ε_e in Eq. (17-48b).

(17-49)

Since mass transfer rates in the film are usually quite high compared to diffusion in the particles, film resistance is rarely important in commercial processes (Basmadjian, 1997).

After passing through the film, solute then diffuses in the pores by normal diffusion (large pores), Knudsen diffusion (small pores), or surface diffusion. In polymer resins where there are no permanent pores, the solute diffuses in the polymer phase. For spherical particles with a radial coordinate r, the diffusion equation in pores is

(17-50)

For ordinary or Fickian diffusion D_effective is related to the diffusivity in free solution through the tortuosity factor, Eq. (17-4). Equation (17-50) assumes that the mean free paths of the molecules are significantly less than the radius of the pores. This is the usual case with liquids and gases at high pressures.

Knudsen diffusion occurs when the mean free path of the molecules is significantly greater than the radius of the pores. In this case instead of colliding with other molecules, a molecule collides with the pore walls. The same diffusion equation can be used, but now D_effective is determined from Eq. (17-4) with D_K, the Knudsen diffusivity replacing D_molecular. The Knudsen diffusivity can be estimated from (Yang, 1987)

(17-51)

where r_p the pore radius is in cm, the absolute temperature T is in Kelvin, M is the molecular weight of the solute, and D_K is in cm²/s. If the mean free path of the molecules is the same order of magnitude as the pore diameter, both ordinary and Knudsen diffusion mechanisms are important. The diffusivity can be estimated as

(17-52)

This value of D is then used instead of D_molecular in Eq. (17-4) to estimate D_effective. The use of Eqs.(17-51) and (17-52) is often required for the adsorption of gases in adsorbents with small pores.

In surface diffusion the adsorbate does not desorb from the surface but instead diffuses along the surface. This mechanism can be important in gas systems and when gel-type (nonporous) resins are used. The surface diffusion flux is

(17-53)

where the surface diffusion coefficient D_s depends strongly on the surface coverage, q. Currently, values of D_s must be back-calculated from diffusion or adsorption experiments. Ordinary, Knudsen, and surface diffusion may occur simultaneously. More detailed descriptions of the diffusion terms are available in the books by Do (1998), Ruthven (1984) and Yang (2003).

17.6.2 Column Mass Balances

The equations for film diffusion and diffusion inside the particle both tell us what is happening at a given location inside the column. To determine what is happening for the entire column we need a mass balance on the solid and fluid phases. In order to write reasonably simple balance equations, we usually make a number of “common-sense” assumptions such as those listed in Table 17-6. The resulting equation for the two-porosity model is

(17-54)

The first three terms are the accumulation in the fluid between particles, within the pore fluid with an average concentration , and sorbed on the solid, respectively. The fourth term represents convection while the fifth term is axial dispersion. Coefficient E_D is the axial dispersion coefficient due to both eddy and molecular effects. The value of in Eq. (17-54) can be related to c_i and through Eqs. (17-48a) and (17-50). A slightly simpler equation is obtained using the single-porosity model,

(17-55)

Solution of the appropriate set of equations: equilibrium, 17-48a or 17-49, 17-50 (with diffusivities calculated from D_molecular, 17-51, or 17-52 inserted into 17-4 and/or use of 17-53), and either 17-54 or 17-55 plus a suitable set of boundary conditions is very difficult. This calculation is so difficult that even with detailed simulators a simplified procedure is usually employed.

17.6.3 Lumped Parameter Mass Transfer

One very common simplification is to assume that the film diffusion and diffusion in the particles can be lumped together in a lumped parameter mass transfer expression. In this form the total of all the mass transfer is assumed to be proportional to the driving force caused by the concentration difference or by the driving force caused by the difference in amount adsorbed . The value is the concentration that would be in equilibrium with and is the amount adsorbed that would be in equilibrium with fluid of concentration c_i. Note that neither nor actually exist in the column—they are hypothetical constructs. The two resulting equations are

(17-56a)

(17-56b)

Note that Eq. (17-56a) is very similar to Eq. (17-48b) except that c* replaces the concentration of the pore fluid c_pore and the lumped parameter mass transfer coefficient k_m,c replaces the film coefficient k_f. As expected, the lumped parameter expressions using the single-porosity model are simpler.

(17-57a)

(17-57b)

Equations (17-56a) and (17-56b) or (17-57a) and (17-57b) can be converted into each other if the equilibrium is linear. Since both Eqs. (17-57a) and (17-57b) are commonly used, it is necessary to be clear which lumped parameter coefficient is reported—unfortunately, authors usually don’t put a subscript denoting the driving force on this term. Look for their driving force equation to determine which form they used.

The lumped parameter models are useful because they simplify the theory and the coefficients can often be estimated with reasonable accuracy. The most common way to determine k_m,q is with a sum of resistances approach (Ruthven et al., 1994) that is similar to the approach used in Section 15.1.

(17-58a)

The value of a_p, the mass transfer area per volume, is usually estimated as 6/d_p. A number of correlations have been developed to estimate the film coefficient. The Wakoa and Funazkri (1978) correlation appears to be quite accurate

(17-59)

where the Sherwood, Schmidt and Reynolds numbers are defined as,

(17-60)

A final value needed to solve the complete set of equations is the eddy dispersion coefficient, E_D. The Chung and Wen (1968) correlation is commonly used to determine E_D.

(17-61)

where the Reynolds number was defined in Eq. (17-60) and the Peclet number is defined as

(17-62)

In many systems, particularly liquid systems, the resistance due to diffusion in the pores is much more important than the resistance due to mass transfer across the film (D_effective is small and hence the second term on the right hand side of Eq. (17-58) is much larger than the first term). Then,

(17-58b)

Thus, when pore diffusion controls the mass transfer coefficient is independent of fluid velocity and proportional to (1/d_p)². Basmadjian (1997) suggests that initial estimates can be made with k_m,ca_p values of 10^-1 1/s for gases and 10^-2 to 10^-3 1/s for liquids. More detailed discussions of the kinetics and mass transfer in adsorbents can be found in the books by Do (1998), Ruthven (1984), and Yang (2003).

17.6.4 Energy Balances and Heat Transfer

Since there are usually significant heat effects in gas systems, energy balances will be required. For the single-porosity model the energy balance (based on the assumptions in Table 17-6) for the fluid, particles, and column wall is

(17-63)

The first two terms and the last term represent accumulation of energy in the fluid, the particle, and the column wall. The third term is convection of energy while the fourth term is the axial dispersion of energy. The fifth term (first term on right hand side) represents the heat transfer from the column walls. Because industrial scale systems have a small ratio of wall area to column volume, the fifth term is often negligible (the column is adiabatic), and the sixth term is often negligible because the mass of the column wall is small compared to the mass of adsorbent.

The transfer of energy from the fluid to the solid can often be represented as a lumped parameter expression of the following form:

(17-64)

The first term represents accumulation of energy in the solid, the second term is the heat transfer rate from the fluid to the solid, and the last term is the heat generated by adsorption. This last term can be quite large. Increases in gas temperature of over 100°C can occur in gas adsorption systems, and if oxygen is present activated carbon beds can catch on fire.

17.6.5 Derivation of Solute Movement Theory

The solute movement equations can be derived rigorously by solving the mass and heat transfer equations with a set of limiting assumptions. Start with the column balance on fluid and solid, Eq. (17-54). Assuming that mass transfer is very rapid, the bulk fluid and solid will be in equilibrium. Thus, c = c* which is in equilibrium with q = q*, and the lumped parameter expression, Eq. (17-56a) or (17-56b) is not required. In addition, assuming that axial dispersion is negligible, Eq. (17-54) becomes

(17-65a)

Since the solid and fluid are in equilibrium, q is related to c and T through the isotherm. After assuming that solid (ε_e, ε_p, K_d, and ρ_s) properties are constant, applying the chain rule and simplifying, Eq. (17-65a) becomes

(17-65b)

The total derivative dT/dt = 0 for isothermal systems, systems with instantaneous temperature changes, and systems with square wave changes in the temperature. With these simplifications Eq. (17-65b) can be solved by the method of characteristics (e.g., Ruthven, 1984; Sherwood et al., 1975). The result for constant interstitial velocity (valid for liquids, exchange adsorption and dilute gases) is that concentration is constant along lines of constant solute velocity where the solute velocity is given by Eq. (17-14).

A similar analysis can be applied to the energy balance equation by assuming very rapid energy transfer, negligible axial thermal diffusion, constant solid and fluid properties (e.g., densities and heat capacities), constant interstitial velocity and the heat of adsorption can be neglected. The result is that temperature is constant along lines of constant thermal velocity where the thermal velocity is given by Eq. (17-21).

The solute movement analysis is thus a physically based analysis that can be derived rigorously with appropriate limiting assumptions. If mass transfer is slow and the velocity is high or the column is short, the solute may not have sufficient residence time in the column to diffuse into the solid. The solute then skips the separation mechanism (equilibrium between solid and fluid) and exits with the void volume of the fluid. In this situation the predictions of solute movement are not useful. Basmadjian (1997) states that one of the following conditions must be satisfied to avoid this “instantaneous breakthrough,”

(17-66)

Local equilibrium analysis can be extended to systems with variable interstitial velocity (concentrated gases), interacting solute isotherms such as Eq. (17-8), or finite heats of adsorption (most important for concentrated gases), but this extension is beyond the scope of this chapter (e.g., see Ruthven, 1984; Yang, 1987).

17.6.6 Detailed Simulators

Simultaneous solution of the combined mass and energy balances and the equilibrium expressions is a formidable task, particularly for multicomponent, nonlinear systems. Until the 1990s solution of this set of differential, algebraic equations (DAE) was typically a task done by Ph.D. students for their thesis or by a few industrial experts who devoted their careers to simulation of sorption processes. This situation changed with the development of fairly general solvers for DAEs such as SPEEDUP, gPROMS, and Aspen Custom Modeler, and later the development of simulators such as ADSIM and Aspen Chromatography designed specifically for adsorption, chromatography and ion exchange. Current versions of the DAE solvers and simulators are reasonably user friendly. Currently, almost all designs of distillation and absorption processes are designed using simulators. In the future the design of sorption systems will follow down the simulation path, and most designs will be based on simulation programs. Teaching the use of these simulation packages is discussed by Wankat (2006), and the introductory computer laboratories for Aspen Chromatography are included in the appendix to this chapter.

In the next two sections the mass and energy transfer equations will be used to obtain realistic solutions for a variety of simplified adsorption problems.

17.7 MASS TRANSFER SOLUTIONS FOR LINEAR SYSTEMS

The solution of differential equations is much simpler when the equations are linear. The various sets of differential equations for mass transfer discussed in Section 17.6 are all linear if the equilibrium isotherm is linear and the system is isothermal. (Note that nonisothermal operation introduces the Arrhenius relationship, Eq. (17-7), which is decidedly nonlinear.) This section is limited to isothermal operation of systems with linear isotherms Eqs. (17-5b) or (17-6b).

One characteristic of the solutions for Eqs. (17-54) and (17-55) for linear isotherms is mass transfer resistances and axial dispersion both cause zone spreading that looks identical if the mass transfer parameters or axial dispersion parameters are adjusted. Thus, from an experimental result it is impossible to determine if the spreading was caused solely by mass transfer resistances, solely by axial dispersion, or by a combination of both. This property of linear systems allows us to use simple models to predict the behavior of more complex systems.

17.7.1 Lapidus and Amundson Solution for Local Equilibrium with Dispersion

In a classic paper Lapidus and Amundson (1952) studied liquid chromatography for isothermal operation with linear, independent isotherms when mass transfer is very rapid, but axial dispersion is important. Although the two-porosity model can be used (Wankat, 1990), the solution was originally obtained for the single-porosity model. Starting with Eq. (17-55), we substitute in the equilibrium expression Eq. (17-6a) to remove the variable q (solid and fluid are assumed to be in local equilibrium). Since the fluid density is essentially constant in liquid systems, the interstitial fluid velocity v_inter can be assumed to be constant. The resulting equation for each solute is

(17-67)

An effective axial dispersion constant E_eff has been employed in Eq. (17-67) since it includes the effects of both mass transfer and axial dispersion. Under most experimental conditions mass transfer resistances are important and axial dispersion effects are rather small. If we use the value of E_D predicted from the Chung and Wen correlation Eq. (17-61), the spreading of the wave will be significantly under-predicted. How do we determine the effective axial dispersion coefficient? Dunnebier et al. (1998) compared solutions that included mass transfer and axial dispersion to the results of the Lapidus and Amundson solution. The effective axial dispersion coefficient can be estimated from

(17-68)

where we have assumed that q and c are in the same units and K′ is dimensionless. If q and c are in different units, then appropriate density(s) need to be included with K′ [the procedure is similar to that used to derive Eqs. (17-14a) and (17-14b)]. The value of k_M,c can be estimated from Eqs. (17-58) to (17-60). For linear systems the Lapidus and Amundson solution with E_eff gives identical results as solving Eqs. (17-55) and (17-57a).

For a step input from c=0 to c=c_F, the boundary conditions used by Lapidus and Amundson were

c = cF for z = 0 and t > 0

c = 0 for t = 0 and z > 0

c = 0 as z approaches infinity and t > 0.

The last boundary condition, the infinite column boundary condition, greatly simplifies the solution. It is approximately valid for long columns.

For sufficiently long columns the solution for each solute is

(17-69)

where u_s is the single-porosity form of the solute velocity, Eq. (17-15b) (included in Problem 17.C5). The term erf is the error function, which is the definite integral

(17-70)

Since the error function is a definite integral, for any value of the argument (the value within the brackets) the error function is a number. The values of the error function can be calculated from the normal curve of error available in most handbooks, are tabulated in Wankat (1990), and are available in many computer and calculator packages. A brief tabulation of values is presented in Table 17-7.

Table 17-7. Values for error function

Note that when z = u_st, which is the solute movement solution, the argument of the error function is zero, erf is zero, and c/c_F = ½. Thus, the solute movement theory predicts the center of the spreading wave.

Equation (17-69) is the solution for the breakthrough curve for linear isotherms. Example 17-9 will illustrate that the result is an S-shaped curve, which matches experimental results. S-shaped breakthrough curves are also predicted by other solutions for linear sorption systems (e.g., Carta, 1988; Rosen, 1954).

17.7.2 Superposition in Linear Systems

Another characteristic of linear systems is that superposition is valid. In other words, solutions can be added and subtracted to give the solution for a combined process. Note that superposition is not valid for nonlinear isotherms. This was illustrated in Example 17-7 where two shock waves combined to form a single shock wave. For linear systems the two waves remain separate and form a staircase type arrangement.

We have already employed superposition in some examples. In Example 17-2 we solved a linear chromatography system for the separation of anthracene and naphthalene. This solution was derived by obtaining the solution for anthracene and the solution for naphthalene and then superimposing these two solutions (Figure 17-6). We inherently assumed that the two adsorbates are independent. This type of procedure cannot usually be applied to nonlinear systems since the solutes interact (e.g., as shown by the multicomponent Langmuir isotherm, Eq. (17-8)).

Figure 17-6 also illustrates another aspect of superposition. The first step increase for naphthalene in Figure 17-6 is the breakthrough solution (a feed of concentration c_F is introduced to an initially clean column) using the solute movement theory for a system with a linear isotherm. The solution to a step-down in feed concentration from c_F to zero is the feed concentration minus the breakthrough solution. Although the solutions including mass transfer and/or dispersion are more complicated than the simple solute movement solutions shown in Figure 17-6, the superposition principle remains valid for any linear system.

Consider an adsorption column that initially contains no solute. At t = t_feed a feed with a concentration c_F is introduced. The breakthrough solution, which is the behavior of c_out, is

(17-72)

where X_breakthrough is any solution for breakthrough with linear isotherms including the solution in Figure 17-6 or in Eq. (17-69). Now suppose that we want the solution for elution of a column fully loaded at a fluid concentration of c_F using pure solvent to remove the adsorbate. Based on superposition this solution is the solution for the loaded column (c (z, t) = c_F) minus the breakthrough solution started at time t = t_elution (we now have a step down in concentration instead of a step up). Thus, at the outlet (z = L)

(17-73)

If we use the Lapidus and Amundson breakthrough solution, Eq. (17-69), then the elution solution is

(17-74)

The advantage of using superposition is this result is obtained with little effort.

As another example, suppose a pulse of feed is input at t = t_start and stopped at t = t_end and pure solvent is fed to the column after the pulse is stopped. The solution for this pulse is the breakthrough solution (step-up from t = t_start) minus a breakthrough solution (step-down from t = t_end).

(17-75)

The period of this pulse is t_F = t_end − t_start. Any solution for breakthrough for linear isotherms, such as Eq. (17-69) can be substituted into Eq. (17-75).

EXAMPLE 17-9. Lapidus and Amundson solution for elution

A column is packed with ion exchange resin in Ca⁺² form. The column is initially saturated with glucose at a concentration of 10.0 g/liter. It is then eluted with pure water starting at t = 0 at a velocity, v_inter = 20 cm/min. Column: L = 75.0 cm, D_col = 4.0 cm. Properties: ε_e = 0.39, ε_p = 0, E_eff = 5.0 cm/min, and K_d,i = 1.0. Determine the elution curve.

A. Define. Plot c_out vs. time.

B. and C. Explore and plan. Note that this problem is not ion exchange, but is a chemical complexing of the glucose with the Ca⁺² on the resin. Equilibrium data are given in Table 17-2, q = 0.51c. For step down (elution) the Lapidus and Amundson solution is given by Eq. (17-74) where u_s is given by Eq. (17-15a) but without the ρ_s team (because q and c are in same units).

D. Do it. The solute velocity is,

From Eq. (17-74) the argument where t_elution=0.

Thus,

(A)

Given a Table of Values, the easiest solution method is to pick a value of the argument “a” listed in the table and then calculate both c_out and t for this value.

For example, if we select a = 0, which is in Table 17-7,

The equation for time becomes

For the general case (a ≠ 0) we can solve for t either directly or by multiplying both sides of Eq. (A) by (t^1/2) and rearranging

3.3354(t^1/2)² − a(t^1/2) − 22.486 = 0

For example, if a = 0.4 [erf (0.4) = 0.4284 is listed in Table 17-7], and we find t = 6.437 min. Then at this time

If a = -0.4, t = 7.060 min, c_out = 2.858g/l.

If this is done for other values of “a”, we can generate the table below and the curve shown in Figure 17-21.

E. Check. The center of the pattern (c=c_F/2 for this symmetric curve) should occur at the time calculated by the solute movement solution.

Thus, the overall mass balance checks.

F. Generalization. The symmetric S-shaped curve shown in Figure 17-21 is characteristic of linear systems. In linear systems, the elution curve (Fig. 17-21) and the breakthrough curve (feed to a clean column) are also symmetric. The shape of the breakthrough curve can be seen if the page is flipped over and you look through the backside of the paper. This transformation comes from comparison of the elution solution, Eq. (17-74), with the breakthrough solution, Eq. (17-69).

17.7.3 Linear Chromatography

In analytical applications of elution chromatography very small feed pulses are used. Thus, concentrations are invariably very low and the isotherms are almost always linear. If the pulse is differential (t_F is very small compared to the time required to elute the components), Eq. (17-75) for the Lapidus and Amundson solution applied to a differential pulse can be simplified to (Wankat, 1990)

(17-76)

Figure 17-21. Outlet concentration profile for Example 17-9

where t is the time after the pulse is fed to the column. Equation (17-76) is the Gaussian solution for linear chromatography, which in various forms is extensively used to predict outcomes for analytical chromatography. Outlet concentrations are determined by setting z = L. The maximum outlet concentration of the peak occurs at z = L and t = L/u_s

(17-77)

The classic paper by Martin and Synge (1941) on liquid-liquid chromatography used an equilibrium-staged model with linear isotherms for the chromatographic column. Comparison of staged solutions with Eq. (17-76) shows that the number of stages N is

(17-78a)

where N is also related to L through the height of an equilibrium plate (HETP).

(17-78b)

and Pe_L is the Peclet number based on the column length. These results allow one to calculate the value of N or HETP if E_eff is known, or vice-versa calculate E_eff if N or HETP is known. This conversion is useful since easy methods to estimate N from chromatographic results are available (see Example 17-10 and homework problems).

The Gaussian solution can be written in shorthand notation as

(17-79)

where x is the deviation from the location of the peak maximum and σ is the standard deviation. The terms for x and σ must be in the same units—time, length, or volume. For example in time units

(17-80a)

and in length units

(17-80b)

where t_R is the molecule’s retention time.

In linear systems the variances (σ²) from different sources add. This is equivalent to stating that the amount of zone spreading from different sources is additive. Mathematically, this ability to add variances is the reason we can use an effective diffusion coefficient to model a system where mass transfer resistances are important.

Equations (17-79) and (17-80a) can be used to analyze experimental peaks, which are essentially plots of concentration vs. time, to determine the value of u_s and N. From Eq. (17-79) the peak maximum must occur when x_t = 0. Since the outlet concentration profile is being measured at x = L, the peak maximum t_max occurs when t_max = L/u_s, which is identical to the solute movement result. Thus, u_s and hence an experimental value for the equilibrium constant K′ can be determined from the time the peak maximum exits the column. This procedure is illustrated in Example 17-10.

The standard deviation σ_t of an experimental Gaussian or close to Gaussian peak can be estimated as ¼ of the width (measured in time units). The value of N can be determined from (Bidlingmeyer and Warren, 1984; Giddings, 1965; Jönsson, 1987; Wankat, 1990),

(17-81a)

where the peak maximum and width are measured in time units. The value of the constant depends upon what width is used. The easiest derivation uses the width as the distance between intersections of the two tangent lines with the base line, which is 13.4% of the total height. The simplest to use experimentally is the width of the pulse at the half height (c = c_max/2).

(17-81b)

Combining the estimate σ_t = 0.25 (width of peak in time units) with Eq. (17-81a) written twice—once for each pair in Eq. (17-81b), we can obtain an easy to measure estimate,

(17-81c)

All of the methods give the same results for Gaussian peaks, but measurements made relatively low in the peak (e.g., the 4 σ_t width) are more sensitive to asymmetry (Bidlingmeyer and Warren, 1984). The use of these equations is illustrated in Example 17-10.

The purpose of chromatography is to separate different compounds. Separation occurs because compounds travel at different solute velocities. At the same time axial dispersion and mass transfer resistances spread the peaks. If two peak maxima are separated by more than the spreading of the two peaks then they are said to be resolved. As a measure of how well the peaks are separated, chromatographers use resolution, defined as (Giddings, 1965; Jönsson, 1987; Wankat, 1990),

(17-82)

where t_R,A and t_R,B are the retention times when the peak maxima exit, and w_A and w_B are the widths of the two peaks (in time units) measured with the tangents. When R = 1.0, the maxima of the two peaks are separated by 2(σ_t,A + σ_t,B) ≈ 4σ_t, and there is about a 2 % overlap in the two peaks. An R = 1.5 is considered to be complete baseline resolution of the two peaks.

The resolution can also be predicted by substituting in the expressions for retention times and the standard deviations. Assuming that the N values are the same for the two components (a reasonable assumption since resolution is usually calculated for similar compounds) the resulting fundamental equation of chromatography (Giddings, 1965; Wankat, 1990) is

(17-83)

This equation indicates how we can increase resolution if the separation is inadequate. For example, increasing column length, which increases N according to Eq. (17-78b), increases resolution, but only as L^1/2. More effect can be obtained by changing the equilibrium isotherms (Schoenmakers, 1986) to increase the distance between the two peaks (see Homework Problem 17.B3) or increasing N by decreasing the particle diameter, which decreases H in Eq. (17-78b) (see Problem 17. A11).

EXAMPLE 17-10. Determination of linear isotherm parameters, N, and resolution for linear chromatography

A chromatogram is run in a preparative chromatographic system to separate acetonaphthalene (AN) from dinitronaphthalene (DN). The results are shown in Figure 17-22. Find , , N and the resolution. The K′ values should be in units kg solute/kg adsorbent.

Figure 17-22. Analysis of Gaussian chromatography peaks for Example 17-10

Data: L = 50.0 cm, Solvent flow rate = 100.0 cm³/min, Pulse time = 0.02 minutes, feed concentration AN = 2.0 g/liter, feed concentration DN = 1.0 g/liter, Internal diameter of column = 2.0 cm, ε_e = 0.4, ε_p = 0.46, K_d,i = 1.0, ρ_s = 2222 kg/m³.

Solution

A. Define: Find , , N and the Resolution.

B and C. Explore and plan. We can use the peak maxima in Figure 17-22 to find the times for the two peak maxima, t_max,i. The solute velocities can then be determined as u_s,i = L/t_max,i. This allows us to determine and by solving Eq. (17-15a) for the K′ values. This equation is

(17-84)

The value of N for each solute can be calculated from Eqs. (17-81a) and (17-81b). Since the width at the half height is easier to measure, we will use this approach.

The resolution can be determined from Eq. (17-82) using Eq. (17-81c) to estimate the values of σ_AN and σ_DN.

D. Do it. The values of the peak maximum concentrations and the half height values are given in Figure 17-22.

The solute velocity is

u_s,AN = L/t_max,AN = 50.0 cm/ 4.46 min = 11.21 cm/min. The interstitial velocity is, v_inter = v_super /ε_e = volumetric flow rate/(πD²/4)/ε_e = (100 cm³/min)/(π(2.0 cm)²/4)/0.4 = 79.58 cm/min

Then can be determined from Eq. (17-84)

Combining Eqs. (17-81a,b), we have

From Eq. (17-81c)

σ_AN = 0.425 (4.96 – 3.99) = 0.412

Similar calculations give , and σ_AN = 0.429

The resolution can be calculated from Eq. (17-82)

E. Check: The values K′ of determined are within 2 % of the values in the literature, and .

F. Generalization: This is a very low resolution, which agrees with Figure 17-22 since the peaks are clearly not separated. To obtain better resolution in an analytical system, much smaller particles would probably be used to drastically increase N. An alternative is to use different chromatographic packing that has a higher selectivity. In a preparative system these methods could be employed and the column length would probably be increased.

17.8 LUB APPROACH FOR NONLINEAR SYSTEMS

In general, we cannot obtain analytical solutions of the complete mass and energy balances for nonlinear systems. One exception to this is for isothermal systems when a constant pattern wave occurs. Constant pattern waves are concentration waves that do not change shape as they move down the column. They occur when the solute movement analysis predicts a shock wave.

Experimental results (Figure 17-15) and the shock wave analysis showed that the wave shape for constant pattern waves is independent of the distance traveled. This allows us to decouple the analysis into two parts. First, the center of the wave can be determined by analyzing the shock wave with solute movement theory. Second, the partial differential equations for the column mass balance can be simplified to an ordinary differential equation by using a variable = t − z/u_sh that defines the deviation from the center of the wave. This approach is detailed in more advanced sources (e.g., Ruthven, 1984; Sherwood et al., 1975; Wankat, 1990).

A simplified analysis procedure called the Length of Unused Bed (LUB), or Mass Transfer Zone (MTZ), method that uses experimental data to design columns during constant pattern operation is used in industry. This method is based on the work of Michaels (1952). The constant pattern wave inside the bed is shown schematically in Figure 17-23A. After being fully developed, the pattern does not change as it moves through the bed. The width of this pattern (called the length of the MTZ, L_MTZ) is usually arbitrarily measured from 0.05 c_F to 0.95 c_F. The reason for not using zero or the feed concentration is it is very difficult to determine exactly when these values are left or attained. During operation, the feed step is stopped at t_br when the outlet concentration reaches a predetermined level, usually 5% of the feed concentration. Note that a fraction of the bed is not fully used for adsorption since the feed step was stopped before the bed was fully saturated.

Figure 17-23. Schematic of constant pattern profiles and unused portion of bed; A) inside column, B) outlet concentration profile

Of course, it is difficult to measure what is happening inside the bed; however, if we run a column to saturation and measure the outlet concentrations we can infer what happened inside the bed. The outlet concentration profile is shown schematically in Figure 17-23B. The width of the MTZ t_MTZ (which is again arbitrarily measured from 0.05 c_F to 0.95 c_F) is now easy to measure. The length of the MTZ inside the bed L_MTZ can be calculated as,

(17-85a)

The shock wave velocity can be calculated from Eq. (17-34) or from experimental data.

(17-85b)

where t_center is the time the center of the pattern, 0.5 (c_F – c_initial), exits the column.

All of the bed up to the MTZ is fully utilized for adsorption. Within the MTZ the fraction of bed not used is (Area unused bed in MTZ)/(Total area of MTZ). This ratio can be determined from Figures 17-23A, or from the measurements shown in Figure 17-23B. Thus, the frac bed use is

(17-86a)

Measuring the ratio of the areas isn’t necessary if the adsorption system produces a symmetric breakthrough curve (e.g., as shown in Figure 17-23B). For symmetric breakthrough curves the ratio of areas is always one half. Thus, for symmetric breakthrough curves the frac bed use is

(17-86b)

For symmetric breakthrough curves if L/L_MTZ = 1.0, frac bed use = 0.5; if L/L_MTZ = 2.0, frac bed use = 0.75; if L/L_MTZ = 3.0, frac bed use = 0.833; if L/L_MTZ = 4.0, frac bed use = 0.875, and so forth. The optimal bed length for adsorption is often between two to three times the L_MTZ. If a frac bed use is chosen, the column length can be determined.

(17-86c)

Once the frac bed use is known, we can find the bed capacity.

(17-86d)

where q_F is the amount adsorbed at the feed concentration in the appropriate units. Of course, q_F can be determined from the equilibrium isotherm for an isothermal adsorber or from the experimentally determined value of u_sh (see Problem 17.C.16).

We often measure the pattern velocity u_sh and the width of the MTZ t_MTZ experimentally with a laboratory column. This measurement needs to be done with the design values for the initial and final concentrations. It is most convenient if the measurement is done with the same velocity and same particle sizes as in the large-scale unit; however, if pore diffusion controls, we can adjust the results for changes in velocity and particle diameter. For constant pattern systems the width of the MTZ t_MTZ is inversely proportional to k_m,qa_p (Wankat, 1990),

(17-87a)

If pore diffusion controls k_m,qa_p can be estimated from Eq. (17-58b). The result is,

(17-87b)

The use of these equations in the LUB analysis is illustrated in Example 17-11.

EXAMPLE 17-11. LUB approach

A laboratory column that is 25.0 cm long is packed with 0.10 cm BPL activated carbon. The operation is at 1.0 atm and 25°C. The column initially contains pure hydrogen. At t = 0 we introduce a feed gas that is 5.0 vol % methane and 95.0% hydrogen. The inlet superficial velocity is 25.0 cm/sec. We measure the outlet wave and find that the center exits at 18.1 s and the width (from 0.05 c_F to 0. 95 c_F) is 9.6 s. The breakthrough curve appears to be symmetric. Assume the hydrogen does not adsorb and that pore diffusion controls methane mass transfer.

a. Determine the shock velocity u_sh, L_MTZ, and frac bed use in the lab unit.

b. We now want to design a larger unit with the same particle size. The superficial velocity will be increased to 50.0 cm/s and the frac bed use will be increased to 0.90. Determine the new column length and new breakthrough time (when c = 0.05 c_F).

Solution

Part a. This part is straightforward.

From Eq. (17-85b) the pattern velocity is

u_sh = L/t_center = 25.0 cm/18.1 s = 1.38 cm/s

Then from Eq. (17-85a)

L_MTZ = u_sht_MTZ = (1.38 cm/s)(9.6 s) = 13.3 cm

And for a symmetrical breakthrough curve the frac bed use can be obtained from Eq. (17-86b)

Frac. bed use = 1 − 0.5 L_MTZ/L = 1 − 0.5 (13.3 cm)/25.0 cm = 0.73

Part b.

B. and C. Explore and plan. We need to relate L_MTZ to velocity. Starting with Eq. (17-85a), we can substitute in Eqs. (17-85c) and (17-87b) to obtain

(17-88)

The effective pore diffusion constant is not a function of velocity. Taking the ratio of the flow rates, we can find both u_sh and L_MTZ for the large-scale unit. Since the desired frac bed use is known, Eq. (17-86c) can be solved for length L.

The breakthrough time can be calculated from the time the center exits and t_MTZ. Referring to Figure 17-23B, for a symmetric breakthrough curve,

(17-89)

The required values for the large-scale system can now all be calculated.

D. Do it. The values of u_sh and L_MTZ for the large-scale unit are obtained from the values of the laboratory unit by multiplying the lab scale values by the ratio of velocities.

ush,large scale = ush,lab (vsuper,large scale/vsuper,lab) = (1.38 cm/s)(50/25) = 2.76 cm/s

LMTZ,large scale = LMTZ,lab (vsuper,large scale/vsuper,lab) = (13.3 cm)(50/25) = 26.6 cm

From Eq. (17-85c),

L = 0.5 LMTZ/(1 − frac bed use) = (0.5) (26.6 cm)/(1 − 0.9) = 133.0 cm

Equation (17-89) can be used to find the breakthrough time by using Eq. (17-85a) to solve for t_MTZ and (17-84b) to solve for t_center.

tMTZ = LMTZ / ush = (26.6 cm)/(2.76 cm/s) = 9.6 s (unchanged)

tcenter = L/ush = (133.0 cm)/(2.76 cm/s) = 48.2 s

tbr = tcenter – 0.5 tMTZ = 48.2 – (0.5)(9.6) = 43.4 s

E. Check. As expected, both L_MTZ and the value of t_center scale proportionally to the velocity ratio. The values for L and t_br are difficult to check independently, but the values are reasonable.

F. Generalization. If the frac bed use had been kept constant and only the velocity was changed, the column length would double in the large-scale system (this comes from Eq. (17-86c) since L_MTZ doubles). The large increase in the required bed length is mainly caused by increasing the frac bed use in the large-scale column. Since t_MTZ is independent of velocity when pore diffusion controls, there was no change in the value of t_MTZ; however, the breakthrough time did change, but not proportionally to the velocity change.

The LUB approach is used for the adsorption step. During desorption a proportional pattern (diffuse) wave usually results as shown in Figure 17-15B (monovalent-divalent ion exchange can be an exception to this—see Example 17-8). Since the shape of the pattern changes with length, the LUB approach cannot be used for desorption. However, the results of the solute movement theory for diffuse waves are often quite accurate. Thus, the diffuse wave predictions can be used for preliminary design. The desorption step should be checked with experimental data.

17.9 CHECKLIST FOR PRACTICAL DESIGN AND OPERATION

There are always practical considerations in the design of separation systems that may not be obvious based on the theories. Since the practical considerations for adsorption, chromatography, and ion exchange are very different than for the other separations considered in this book, they have been collected here.

1. Broadly speaking, the sorption (feed) step makes money, and the regeneration step costs money. The optimum sorbent is often a trade-off between these two steps (relatively strong sorption to process the feed, but not so strong that regeneration is not feasible).

2. Regeneration (broadly defined) is always a key cost and is often the controlling cost. These costs are: TSA—energy for heating, PSA—energy for compression, Chromatography and SMB—solvent or desorbent recovery (ultimately energy if distillation or evaporation are used), IEX—regeneration chemicals.

3. Sorption methods of separation always compete with other separation methods such as distillation or gas permeation. Thus, always need to consider alternatives.

4. Compression costs are significant in large-scale adsorption systems for gases; thus, pressure drop is critically important since it controls compression costs.

5. Theories all assume that the column is well packed. If it isn’t, it won’t work well. Want D_col/d_p > ~30. This limit is often important in lab columns.

Special equipment is needed to pack small particles. Packing large diameter columns with small particles needs to be done by experts.

If a wet column is allowed to dry out, the packing will often crack, which will cause channeling.

Packed beds are efficient depth filters. To prevent clogging, feeds containing particulates must be filtered. It is common, particularly in IEX, to use upward flow wash steps to remove particulates from the column.

Movement of the bed will cause attrition of brittle packings such as activated carbon and zeolites that will result in fines that can clog the bed or frits. Use a hold down plate, frit, or net to help prevent attrition.

Soft packing materials (e.g., Sephadex and agarose) require different packing procedures than rigid packings. The swelling and contracting of polymer packings, particularly ion-exchange resins, must be designed for.

6. Simulations and other solutions to the theories can only include phenomena that were built into the model. Experiments are usually needed to find the unexpected.

7. If the fluid velocity is high and the mass transfer rate is low, there may not be enough residence time for much of the solute to diffuse into the sorbent. This solute, which bypasses the packing, does not undergo separation and exits at the feed concentration. If separation problems are observed, try reducing the fluid velocity by one or more orders of magnitude.

8. Many adsorbents, particularly activated carbon, show a very high initial adsorption capacity. After regeneration, this capacity is not fully regained. When testing adsorbents do extensive cleaning and/or washing first, and then do several complete cycles. Do not use initial results for design of cyclic processes.

9. Slow decay of adsorbents due to irreversible adsorption of trace components or thermal deactivation of active sites is also common. When this occurs, operating conditions must be adjusted accordingly. Because of this poisoning, adsorption processes, which use surface phenomena, are often much more sensitive to trace chemicals than distillation and other separation techniques that rely on bulk properties. An occasional wash step or extreme regeneration step may be needed. A short life for the sorbent, which can be a problem in biological operations, often makes the process uneconomical. Long term pilot plant tests with the actual feed from the plant are useful to determine the seriousness of these problems.

11. The surface properties and surface morphology of sorbents is critically important. Thus, different activated carbons are different adsorbents and are not interchangeable. Different batches of what is supposed to be the same sorbent may differ significantly. Thus, batches must be sampled and tested before being used on a large scale.

12. Pressure and flow spikes can be very detrimental if they cause the bed to shift since this can result in channeling or attrition. Unfortunately, these spikes naturally occur when concentrated feeds are adsorbed. They are greatly reduced if the concentrated feed is introduced in steps (e.g., go from 0% to 45% and then to 90% in two steps instead of a single step from 0% to 90%).

13. Temperature increases must be controlled. Adsorbates may be thermally sensitive and some adsorbents such as activated carbon readily burn. Hot adsorbents are also more likely to catalyze unwanted reactions.

14. Beds in series are often treated as if they were equivalent to a single long bed. However, their transient behavior is different and depends on the connecting pipes and valves.

15. Personnel must always wear respirators when entering chromatography or adsorption columns. Many adsorbents adsorb oxygen and others may desorb toxic gases.