Chapter 19. Introduction to Adsorption, Chromatography, and Ion Exchange

In the first 14 chapters, we looked at separation techniques such as distillation and extraction that are often operated as steady-state, equilibrium-staged separations. Exceptions were batch distillation in Chapter 9 and batch extraction in Section 13.6. In Chapter 17 we studied both batch and continuous crystallization. Outlet concentrations can be found from equilibrium calculations, but crystal size distributions required population balances. In Chapter 18 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 mass balances and rate expressions for more complex flow patterns. Since membrane processes are normally operated at steady state, analysis is usually straightforward. In this chapter we study three closely related processes that are rarely operated or analyzed as steady-state, equilibrium-staged systems. These sorption 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 adsorbent surface. 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 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 components in a mixture), but operation is devised to separate components from each other. You may have analyzed composition of samples with analytical chromatography (gas or liquid) in labs. In ion exchange the solid contains charged groups that interact with charged ions in the liquid. 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 equilibrium-staged processes. They are often used for chemical analysis, separation of dilute mixtures, and separation of difficult mixtures where 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 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 (it sounds like it would be good to eat, but it is not). The most common equipment for sorption processes is a stationary packed bed of solid. The solid in sorption systems directly causes separation, which is different than packed beds for equilibrium separations where solid is inert but increases interfacial area and mass transfer coefficients between gas and liquid. If feed is introduced continuously into a sorption packed bed, the bed will eventually saturate (e.g., approach feed concentration) and separation will cease. Much of the art and expense of designing these systems is in the regeneration step that removes component from solid. Regeneration is so important that different processes are often named on the basis of the regeneration method used.

This chapter is a simplified introduction to the fascinating and valuable sorption separation methods. Three-fourths of the chapter rely on equilibrium analysis—mass transfer is introduced in Section 19.6. 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; Ruthven et al., 1994; Yang, 1987; 2003).

Note: A nomenclature list for this chapter is included in the front matter of this book.

1. Determine equilibrium constants from data and use equilibrium equations in calculations

2. Explain in your own language how different sorption processes (e.g., elution chromatography, temperature swing adsorption [TSA], pressure swing adsorption [PSA], simulated moving beds [SMB], monovalent-monovalent ion exchange, and water softening) work

3. Explain the meaning of each term in development of solute movement equations and use this theory for both linear and nonlinear isotherms to predict outlet concentration and temperature profiles for a variety of different operations, including elution chromatography, TSA, PSA, SMB, and ion exchange

4. Explain the meaning of each term in column mass and energy balances and in mass and heat transfer equations

5. Use Lapidus and Amundson solution plus superposition to determine outlet concentration profiles for linear adsorption and chromatography problems

6. Use theory of linear chromatography with very small pulses to analyze chromatography systems

7. Use length of unused bed (LUB) theory in combination with experimental data to design columns

8. Use Aspen Chromatography to simulate adsorption and chromatography systems.

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

Porosities are dimensionless.

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

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 solvent dissolves. In effect, ε_p = 0. Macroporous ion exchange resins have permanent pores and ε_p > 0, but often K_d,i < 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 19-1. Superficial velocity, which is easy to measure, is average velocity volumetric flow of fluid would have in an empty column. Thus

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

v_inter ε_e A_c = v_super A_c

which gives

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

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

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

Bulk and particle densities are also in kg/m³. If fluid is a gas with ρ_f << ρ_p,

Unfortunately, it is often unclear which density is being referred to. By comparing values given by the manufacturer to approximate values listed in Table 19-1, one may be able to determine which density is being referred to.

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

Note that τ is dimensionless. Equation (19-4) 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 diffusion is hindered by walls or at pore mouth. Thus we will treat τ and D_effective as empirical (experimentally measured) quantities. Molecular diffusivity can be determined experimentally, but there are also a number of well-known methods to predict D_molecular (see Section 15.3).

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 porous structure and to oxidize the surface. Additional chemical treatments such as with iodine can be used to produce specialty carbons (Yin et al., 2007). Carbons are produced for both liquid and gas separations. Because starting materials and chemical treatments vary, different activated carbons can have very different properties. Thus, average values reported in Table 19-1 should be used only for very preliminary calculations and approximate designs. Experimental data on a particular brand, grade, and size of activated carbon should 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. Since activated carbon is commonly used 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 in which a hydrocarbon is cracked or polymerized on the surface to create desired uniform pore size (Ruthven et al., 1994). Because of 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. Design of kinetic adsorption processes is highly specialized (e.g., see Ruthven et al., 1994).

Zeolite molecular sieves are crystalline aluminosilicates with 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 number of water molecules per unit cell. Since zeolites are crystals, pores have exact dimensions (Ruthven, 1984; Ruthven et al., 1994; 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, 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 nitrogen in air. The major steric exclusion application is separation of straight-chain hydrocarbons (used for biodegradable detergents) from branched-chain hydrocarbons. Zeolite properties are reported in Table 19-1.

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 material during one stage of its production [Reynolds et al., 2002].) Silica gel is commonly used for drying gases and liquids because it has a high affinity for water. Silica gel is complementary 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 to as low water content. Columns with a layer of silica gel at feed end and a layer of zeolite at product end are commonly used for drying, since they combine best properties of both adsorbents. Note that silica gel can be damaged by liquid water. Silica gel properties are reported in Table 19-1.

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. Activated alumina properties are listed in Table 19-1.

A large number of other materials are used commercially as adsorbents. These include an uncharged form of organic polymer resins commonly used for ion exchange (see Section 19.5). Although considerably more expensive, these resins compete with activated carbon for 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 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 stationary phase, often a C₈ or a C₁₈ compound, is chemically bonded to inert solid, which is usually silica gel. Equilibrium behavior of these specialized packings is usually similar to gas-liquid absorption or liquid-liquid extraction, but because of presence of inert solid, equipment and operating principles are similar to adsorption-chromatography.

Typical isotherm data for gas adsorption of single components are shown in Figures 19-2A and 19-2B. Figure 19-2A shows adsorption of different gasses. Hydrogen is least strongly adsorbed, followed by methane. Adsorption is strongest for ethane and ethylene, and then maximum adsorption is less for propane, although its adsorption is stronger at low partial pressures. Each adsorbent has an optimum size molecule for maximum adsorption. Figure 19-2B compares adsorption of ethylene on different adsorbents.

At low concentrations or partial pressures, isotherms are often linear. As gas partial pressure increases, isotherms become nonlinear—they curve. Equilibrium data for adsorption of single gases are often fit with the Langmuir isotherm,

where q_A is amount of species A adsorbed and q_A,max is maximum amount of species A that can adsorb (kg/kg adsorbent or mol/kg adsorbent), p_A is partial pressure of species A (mm Hg, kPa, or other pressure units), and K_A is adsorption equilibrium constant in suitable units. Note that for very small partial pressures, K_A,p p_A << 1.0 and Eq. (19-5a) simplifies to a linear form:

while at very high partial pressures, K_A,p p_A >> 1.0, Eq. (19-5a) simplifies to

This is a horizontal line that represents saturation of adsorbent.

For liquids isotherms are usually written in terms of liquid concentration c_A (mol/m³ or kg/m³). For example, the Langmuir isotherm for a liquid is

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

Adsorption equilibrium constants K_A,c and K′_A,c are in different units for liquid systems than for gas systems. Equilibrium constants for several systems are listed in Table 19-2.

Isotherm data at different temperatures invariably shows that there is less material adsorbed as temperature increases. Adsorption equilibrium constants often follow an Arrhenius form:

where K_Ao is a pre-exponential factor, ΔH is heat of adsorption (e.g., in J/kg), R is gas constant, and T is absolute temperature (e.g., in K). Since adsorption is invariably exothermic, ΔH is negative. If 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 whether points are on a straight line. Typically, q_A,max slowly decreases as temperature increases.

Langmuir used a simple kinetic argument (e.g., Wankat, 1990) to derive Eqs. (19-5a) and (19-6a). When this argument is used, q_A,max is coverage obtained with a monolayer. Langmuir’s isotherm can also be derived with a statistical mechanics argument (e.g., Ruthven, 1984). Data are often correlated with a Langmuir isotherm 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

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 Langmuir isotherm data plot as a straight line (see Example 19-1 and homework Problem 19.D1).

The Langmuir isotherm is thermodynamically correct for single-component systems. It has been used to develop a variety of other isotherms, such as BET isotherm (multiple adsorbate layers), Langmuir-Freundlich isotherm, and linear-Langmuir isotherm (add a linear isotherm and Langmuir isotherm). Langmuir isotherms are also commonly extended to adsorption of multicomponent mixtures. For example, for simultaneous adsorption of components A and B, Eq. (19-6c) becomes

This equation correctly predicts that two adsorbates compete for adsorption sites. That is, if concentration of B increases, amount of A adsorbed decreases. Unfortunately, very few systems follow Eqs. (19-8) exactly, and if a_A does not equal a_B, Eq. (19-8) is not thermodynamically consistent (LeVan and Vermeulen, 1981). Despite these problems Eq. (19-8) 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, ideal adsorbed solution (IAS) theory, uses Langmuir isotherms for single-component isotherms. Details of IAS and additional isotherms are discussed by Do (1998), Ruthven (1984), Valenzuela and Myers (1989), and Yang (1987; 2003).

Complete analysis of sorption processes requires computer simulation with a rather complex simulation program to solve coupled algebraic and partial differential equations. Unfortunately, simulators often do not provide a physical picture of why 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 to understand results from experiments or simulations. The role of solute movement analysis in sorption processes is analogous to the role of McCabe-Thiele diagrams in distillation, absorption, and extraction. Solute movement theory is used to understand separations and for troubleshooting, not for final design.

This fraction can be calculated by considering distribution of an incremental change in solute concentration Δc (e.g., in mol/m³) and its corresponding change in the amount sorbed Δq (e.g., in mol/kg adsorbent). Amount in each location can be determined by doing an inventory of moles.

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. (19-11a) and (19-11b) are in moles adsorbate if c is mol/m³.

The K_d term is not included in Eq. (19-11c), since we assume Δq is based on a measurement that automatically includes any steric hindrance. Equation (19-11c) will also be in moles adsorbate. If q and c are in different units than in this derivation, the mass balances will be slightly different [compare Eqs. (19-14a) through (19-14c) to Eq. (19-13)].

Inserting Eq. (19-11) into Eq. (19-10), we obtain

After Eq. (19-12) is substituted into Eq. (19-9), the result can be simplified to

The exact form of Eq. (19-13) depends on units of equilibrium data. For example, if isotherm expression is q = f(x) with q in kg solute/kg solid and x in mass fraction (kg solute/kg fluid), then there must be a ρ_F (kg fluid/m³) term in Eqs. (19-11a) and (19-11b), and Δx replaces Δc in these equations. Then Eq. (19-13) becomes

If q and c are both in mol/m³, equation for u_s is obtained by eliminating ρ_s from Eq. (19-13).

For gas systems equilibrium is often expressed in terms of partial pressure, p_A [e.g., Eqs., (19-5a) and (19-5b)]. Solute velocity for gases is then

where ρ_m,f is fluid molar density. For an ideal gas, ρ_m,f = p_tot/RT and p_tot/ρ_m,f = RT. Thus, for ideal gases,

In some ion exchange and biochemical systems c and q are both in g/L or mol/L. Then Δq is also in g/L or mol/L, and the ρ_s term in Eqs. (19-11c), (19-12), and (19-13) does not appear. The result is

Equations (19-13) and (19-14) allow us to calculate average velocity of solute if we can calculate (Δq/Δc), (Δq/Δx), or (Δq/Δp_A). Since detailed derivations of Eqs. (19-14a) to (19-14d) were not presented, I strongly encourage you to do at least one part of Problem 19.C2. Understanding derivations will be helpful in Sections 19.2.2 to 19.5 when we insert appropriate isotherms into Eqs. (19-13) and (19-14). In the vast majority of commercial sorption separations, solute velocity depends only on equilibrium behavior (Δq/Δc), (Δq/Δx), or (Δq/Δp_A) and packed column properties, not on mass transfer rates. Mass transfer is critically important to determine how solute spreads from the average (see Sections 19.6 to 19.8).

When Eq. (19-6b) is valid, and Eq. (19-13) becomes

for any adsorbate i. Terms do not depend on concentration of adsorbate i. Thus, at low concentrations where linear isotherms are valid, solute velocity is constant. Solute velocity u_s,i depends on temperature, since depends on temperature [e.g., following Eq. (19-7a)]. If we have a number of different solutes with different values of equilibrium constant, the weakest sorbed solute (lowest value of ) moves fastest, and the strongest sorbed solute (highest value of ) moves slowest. Since they move at different speeds, they can be separated. A single-porosity form of this equation is also commonly used [see Problem 19.C5, which includes Eq. (19-15b)].

To visualize what this looks like, we start with an initially clean (c_A = 0) packed column. At time t₀, we start adding a feed with concentration c_A,F at a known interstitial velocity. Equation (19-15a) can be used to calculate solute velocity u_s,A (the numerical calculation procedure is illustrated in Example 19-2). In Figure 19-4A 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 slope of the characteristic line shown in the figure. The procedure will probably be easiest to understand after you study Example 19-2. Concentrations in the column are shown at four times in Figures 19-4B through 19-4E. Solute moves upward in a wave at a constant velocity u_s,A. If adsorption is strong, the wave moves slowly, whereas if adsorption is weak, it moves quickly. Waves for nonlinear systems are shown later in Figure 19-15.

For systems with weight fraction units, or where q is in kg solute/kg adsorbent and x or y is in kg solute/kg fluid (weight fraction), Eq. (19-14a) becomes

For ideal gases with Eq. (19-14c) becomes

With linear isotherms the denominator in Eqs. (19-15a) through (19-15d) is independent of concentration. Thus, for purposes of algebraic manipulation, it is convenient to write these equations as

Although very convenient, linear isotherms find limited applications in industrial processes. To decrease column diameters and thus reduce costs, most industrial separations are done at high concentrations or very close to solubility limits. Since isotherms are probably nonlinear and use of linear isotherms will lead to large errors, one needs to use nonlinear theories (Section 19.6) for concentrated systems. However, linear isotherms are appropriate for many chromatography applications, since solute concentrations are often very low.

Purge cycles are commonly used in elution chromatography, particularly in analytical chemistry. Elution chromatography involves input of a feed pulse into a packed column followed by co-flow (Figure 19-5A) of an inert solvent or carrier gas. (If solvent or carrier gas also adsorbs, the process can become gradient or displacement chromatography, which are discussed later.) Columns can be packed with adsorbents or chromatography packings mentioned previously. If solutes have different equilibrium isotherms, solutes will move at different velocities and will be separated (Figure 19-5C). Both gas and liquid chromatography are commonly operated in elution mode to determine compositions of unknown samples. Large-scale elution chromatography systems are also becoming more common, particularly in pharmaceutical and fine chemical industries. Detailed design considerations for large-scale biochromatography systems are discussed in detail by Ladisch (2001). Dilution that is inherent in purge operations makes these processes expensive for large-scale separation, but with complicated feeds there may be no better alternative.

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

There is an analogy that may be useful in understanding solute movement analysis. Problems are similar to your high school algebra problems where two trains leave a station at 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 feed time, t_F) leaves the second station.

In actual practice it is much easier to calculate exit times, as shown in step E of Example 19-2, than to draw a solute movement diagram exactly to scale. Calculation of exit times can be done with a spreadsheet for much more complex processes. However, a freehand sketch of solute movement diagram should always be made, since it will guide calculations and provide a visual check on them.

In real systems 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, predictions for the simple elution chromatography system shown in Figure 19-6B would spread and solute peaks would overlap, as shown in Figure 19-5C. This calculation is illustrated later in Example 19-10. If high product purities are required, zone spreading will force us to either shorten the feed step and/or lengthen the column to move 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 slow peak with fast peak from the next feed pulse. Net result is elution chromatography can be expensive for large-scale applications. Simulated moving bed systems are often less expensive for binary separations and are analyzed in Section 19.3.3.

A number of variations of elution chromatography have been developed. In flow programming flow rate is increased to more rapidly elute the late components in Figure 19-5C. This technique does not change volume of elutant required, but reduces time of operation. Temperature programming increases temperature of the entire column during elution. Increased temperature decreases adsorption of solutes so that they exit sooner and are more concentrated. The related temperature gradient method increases temperature of 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 solvent to decrease sorption of slow solutes. Common changes in solvent are to change ionic strength, polarity, pH, fraction of organic solvent, and 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 solvent gradient is produced with a chemical that is more strongly sorbed than all feed components, the process is called displacement chromatography (see extensive review by Cramer and Subramanian [1990]).

A major commercial problem is 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 19.3.2) also require too much purge gas for strongly adsorbed materials. Thermal cycles (Section 19.3.1) 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, nomenclature is confusing, since displacement cycles for adsorption have a desorbent that can adsorb less or more than adsorbate, whereas in displacement chromatography the desorbent is the most strongly adsorbed compound.) Cycles will be basically the same as the counterflow cycle shown in Figure 19-5B. Since displacement adsorption requires that desorbent be recovered from product—often by distillation—they are relatively expensive processes that are used only when other adsorption processes fail. Both purge and displacement cycles are commonly used in SMB systems (Section 19.3.3).

Regeneration is based on the large reduction in equilibrium observed in most adsorption systems when temperature is increased, Eq. (19-7), and simultaneous reduction in partial pressure or concentration with addition of purge gas. Both of these effects lead to removal of adsorbate. Since large increases in 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 have been developed (Natarajan and Wankat, 2003). These modifications are explored as homework problems.

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

To study TSA systems with solute movement analysis we must determine (1) effect of temperature changes on solute waves, (2) rate at which a temperature wave moves in column, and (3) effect of temperature changes on concentration. The first of these is easy. As temperature increases equilibrium constants, K_A and K′_A both decrease, often following an Arrhenius relationship, as shown in Eq. (19-7). If effect of temperature on equilibrium constants is known, new values of equilibrium constants can be calculated and new solute velocities can be determined.

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

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

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

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

where W is column weight per length (kg/m), and ΔT_pf, ΔT_s, and ΔT_w are changes in pore fluid, solid, and wall temperatures induced by the change in fluid temperature. If we divide numerator and denominator of Eq. (19-19) by ΔT_f, we have ratios of ΔT_pf, ΔT_s, and ΔT_w to ΔT_f. If heat transfer is very rapid, the system is in thermal equilibrium, T_f = T_pf = T_s = T_w. This equality requires that changes in temperature all be equal:

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

As a first approximation, thermal wave velocity is independent of concentration and temperature. Temperature is constant along lines with slope = u_th.

In TSA processes temperature is increased to remove adsorbate from adsorbent. This happens when thermal and solute waves intersect. 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 concentration and adsorbent loading to change to c₂ and q₂ (currently both unknown). Since solute movement theory assumes local equilibrium, c₂ and q₂ are in equilibrium at T₂. The control volume shown in Figure 19-7B (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. Thermal waves are assumed to move faster than solute waves, which is true for most dilute liquid and some dilute gas systems. For a differential slice of column of arbitrary height Δz, temperature of the slice will change from initial temperature T₁ to final temperature T₂ if Δt = Δz/u_th. The mass balance for the differential slice over time interval Δt is

Since Δt = Δz/u_th, this simplifies to

Equation (19-23) and the isotherm equation can be solved simultaneously for the unknowns c₂ and q₂. For a linear isotherm, Eq. (19-6b), simultaneous solution is

Example 19-3 illustrates that for liquid systems with u_th > u_s, use of a hot feed liquid (T₂ > T₁) will increase outlet concentration.

If solute waves move faster than thermal waves, which may occur in dilute gas systems, a mass balance equation and solution similar but subtly different from Eqs. (19-22) to (19-24) can be derived. In this situation concentrated solute exits ahead of the thermal wave instead of behind it, as predicted by Eqs. (19-22) to (19-24). 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 not included in this introductory treatment, solute concentrates, or focuses, at the temperature boundary (Wankat, 1990).

A number of different thermal cycles are used commercially. Figure 19-9 shows an alternate TSA cycle commonly used for recovery of solvents (typically volatile organic compounds (VOC) of intermediate molecular weight ∼45 to 200) from drying and curing operations (Basmadjian, 1997; Fulker, 1972; Wankat, 1986). Activated carbon is used as adsorbent, and steam is used as regeneration gas. Horizontal beds with a depth of 1 to 2 meters are often employed, since strong adsorption of solvent on 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 2 hours. Because 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, typical design capacity used for activated carbon is about 25% to 30% of maximum carbon capacity. Bed capacity can be increased by reducing relative humidity of feed gas to less than 50%.

In ideal applications of this process (e.g., removing small amounts of toluene from air), peak mole fractions of toluene are close to 1.0 (Basmadjian, 1997) and toluene is almost completely immiscible with water. Thus adsorbate can be recovered from the exiting regeneration vapor by condensing the vapor and allowing liquid to separate into an organic layer and a water layer. If adsorbate is miscible with water (e.g., ethanol), the condenser/settler shown in Figure 19-9 must be replaced with a distillation column, which greatly increases capital and operating costs.

Note that there can be safety hazards in operation of activated carbon solvent recovery equipment (Figure 19-9). If solvent being recovered is flammable, care must be taken to prevent a fire. If feed gas is air, then concentration of solvent in feed gas must be kept below lower explosion limit, and is often kept below one-quarter of lower explosion limit to provide a safety margin. This requirement invariably means that feed gas must be quite dilute and flow rates are large. If feed gas is hot, it is often cooled before the adsorber to increase safety and to increase adsorption capacity. An alternative is to operate at much higher concentrations using nitrogen or carbon dioxide as carrier gas, but then carrier gas must be recovered and recycled. If hot activated carbon can catalyze a reaction with feed, a cooling step is added. Sometimes a drying step is added before cooling, since water may interfere with adsorption or react with feed. If feed gas is concentrated, adsorbent can become quite hot because of the large heat of adsorption. Unfortunately, carbon beds occasionally catch fire when this happens. Fire can be prevented by significant cooling of feed gas, incorporating a cooling step in the cycle, or replacing air with an inert gas.

Since several companies provide package units for activated carbon solvent recovery, new engineers are more likely to be involved in 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, feed may last for several months. Regeneration of activated carbon is difficult and is usually done by removing carbon from the column and sending it to a kiln to burn off adsorbates. In small units (e.g., those used to purify tap water in homes) carbon is discarded after use. Activated carbon is commonly used in bottling plants to remove chlorine from water by reacting with carbon to produce HCl (Wankat, 1990). The slightly acidic water should be used immediately after treatment, since it no longer contains chlorine to stop microbial growth.

A major industrial application of liquid adsorption is drying of organic solvents, shown in Figure 19-10 (Basmadjian, 1984). Upward flow is used during refilling and feed steps to avoid trapping gas in the bed. Since water content in the organic is usually low, the feed step may be relatively long. Once breakthrough occurs (substantial amounts of water appear in exiting solvent), 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 cost of energy to evaporate residual liquid and desorb water. Adsorptive drying usually is less expensive than distillation when water concentrations in solvent are low.

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

Typical range for heat of adsorption |ΔH_ads| is from 1000 to 4000 kJ/kg (average ∼2500), and typically 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) states an isothermal analysis can be used if predicted maximum temperature increase is less than 1°C or 2°C. Situations with large temperature increases are economically important, but 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.

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

where γ typically is between 1.15 and 1.5. Because of volumetric expansion of product gas from p_h to purge gas at p_L, significantly fewer moles of purge gas are needed than feed gas. Dropping pressure also reduces partial pressure, which helps desorb adsorbate. The final step in the Skarstrom cycle is column repressurization. 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 19-11. 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 19-11B. PSA systems with from 1 to 12 columns are used commercially. PSA has fast cycles—a minute or two is common in industry, and some cycles are as short as a few seconds. Short cycles lead to high productivity and hence relatively small adsorbers.

Figure 19-12 illustrates a simple vacuum swing cycle. Feed enters at high pressure, which may be essentially atmospheric pressure. If p_h is significantly above atmospheric pressure, a short optional blowdown step can be included. A vacuum pump is used to reduce pressure to very low pressures. At very low pressures, partial pressure is very low and very little adsorbate can be adsorbed (see Figure 19-2). 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 column repressurization. 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). For example, it is common to operate PSA purge at a pressure less than atmospheric.

Simple Skarstrom PSA cycles (Figure 19-11A) have 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 a weakly adsorbed carrier gas is being processed, and equilibrium is linear over the concentration range of interest. If mass transfer is very rapid, then solute movement theory can be applied. Since system is very dilute, the system is assumed to be isothermal and gas velocity is constant. In more concentrated PSA systems neither of these assumptions is true, and more complicated theories or complete simulations must be used (Ruthven et al., 1994).

During isobaric periods (feed at p_h and purge at p_L), solute moves at a velocity u_s. For an ideal gas and a linear isotherm in partial pressure units, solute velocity is given by Eq. (19-15d). Normally, K_d,i = 1.0 and all adsorption sites are accessible to small gas molecules.

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

This parameter is ratio of the amount of weakly adsorbed to the amount of strongly adsorbed adsorbate in a column segment. If weakly adsorbed component does not adsorb, then in Eq. (19-27). Since , β_strong < 1.0. The shift in mole fraction of strongly adsorbed species A when pressure changes is

Equation (19-28a) 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

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

Application of solute movement theory is illustrated in Example 19-4. Before doing this activity, we note that axial dispersion is normally significant in gas systems. Thus we expect solute movement theory will overpredict separation that occurs. Alternatively, γ required in Eq. (19-26) for a given product purity will be larger in a real system than predicted by solute movement theory (γ = 1 for linear systems). For separations based on differences in equilibrium isotherms, if solute movement theory predicts a separation is not feasible or will not be economical, more detailed calculations will rarely improve results.

This section illustrated PSA calculations for the simplest possible case—local equilibrium theory for trace components for an isothermal system. If feed mole fraction of strongly adsorbed component is higher, velocity will vary in the column, and the isotherm is probably nonlinear. 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 are required. Ruthven et al. (1994) provide an advanced treatment. In addition, PSA has spawned a large number of inventive cycles to accomplish different purposes. White and Barkley (1989) and White (2008) discuss practical aspects of PSA design, such as pressure drops, velocity limit to prevent bed fluidization, retaining heat of adsorption in the bed, and start-up.

The most efficient approach to separating a binary mixture by adsorption appears to be to have a countercurrent process similar to extraction. In addition, it is convenient to regenerate adsorbent within the device. True moving bed (TMB) systems (Figure 19-14A) were invented to achieve these two goals. Zones 2 and 3 separate the two solutes. (Zones are packed regions between inlet and outlet ports.) Zone 1, which is optional but almost always included, adsorbs weakly adsorbed solute A onto solid so that desorbent D can be recycled. Zone 4 regenerates solid by removing strongly adsorbed solute B using a purge or displacement with desorbent D. Figure 19-14A is loosely analogous to distillation: A is light key, B is heavy key, zones 2 and 3 are enriching and stripping sections, respectively; and zones 1 and 4 are analogous to a total condenser and a total reboiler, respectively. TMB systems would work if chemical engineers had technology to build large-scale systems that could move a solid countercurrent to a fluid with no axial mixing. Since this goal has proved to be elusive, UOP developed the SMB.

An SMB system is shown in Figure 19-14B for four different time steps. SMB columns are arranged in a continuous loop with inlet and outlet ports. After a set time period, switching time t_sw, port locations are all advanced by one column. From the viewpoint of an observer at the extract port, solid moves downward when the switch is done while fluid continues to move continuously upwards. Thus an intermittent, countercurrent movement of solid and liquid has been simulated. Port switching is continued indefinitely. In Figure 19-14B 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 SMBs have been developed. Most common is to have two or more columns per zone instead of one column per zone, shown in Figure 19-14B. This makes operation closer to a TMB and improves the product purity.

The SMB system shown in Figure 19-14B is quite a complicated system, particularly if compared to simple elution chromatography shown in Figure 19-5B. Chin and Wang (2004) discuss practical aspects of SMB systems, such as pump placement, pressure drops, and valve requirements. SMBs are used in industry for high-purity separations of binary feeds, since much less desorbent and adsorbent are required. Solute movement analysis helps to explain how this complicated process works.

Each column in Figure 19-14B is a chromatographic column. Port switching changes flow rate and boundary conditions for each column. Solute velocity in each SMB column can be determined in the same way as for elution chromatography. Thus solute velocity in any SMB column is given by Eqs. (19-15). Chromatography and SMB processes differ only in column coupling, which are boundary conditions!

To separate solutes A and B completely, SMBs must meet the following conditions:

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

u_A,1 t_sw ≤ L

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

Equations are written as equalites instead of inequalities, since it is easier to manipulate equations.

Zones 2 and 3. To separate A and B we want net movement of A upward and net movement of B downward in the column. Since velocity is higher in zone 2, the condition on solute B controls.

In zone 3 upward movement of solute A controls.

These conditions require that solute A break through and solute B not break through from zones 2 and 3 during a switching period.

Zone 4. To regenerate, all solute B must be removed in zone 4; thus solute B must have net upward movement.

Equations (19-29a) to (19-29d) can be simultaneously satisfied by changing flow rates of feed, desorbent, and two product streams in Figure 19-14B to change velocities in each zone. For example, since A product stream is withdrawn between zones 1 and 2, v₁ < v₂. By proper selection of velocities and port velocity, we can satisfy these four equations (see Example 19-5).

Usually desorbent must be removed from A and B product streams. Increasing desorbent flow rate increases cost for removal and will also increase column diameters and require more adsorbent. Thus desorbent to feed ratio, D/F, often controls SMB system cost. For an ideal system with no zone spreading (no axial dispersion and very fast mass transfer rates), solute movement theory can be used to calculate D/F by solving Eqs. (19-29a) to (19-29d) simultaneously with Eq. (19-15) and mixing mass balances with constant density.

where v_Feed = F/(ε_e A_c) is the interstitial velocity that feed would have in the column. F is volumetric flow rate of feed, and A_c is the cross-sectional area of columns with similar definitions for desorbent and product velocities. If F and A_c are known, we can first solve for u_port, v₁, v₂, v₃, and v₄, then for v_A,product, v_B,product, and v_D, and finally D/F = v_D /v_Feed. These calculations are developed in Problem 19.C10. Minimum D/F ratio, (D/F)_min can be calculated by setting all M_i = 1.0. This minimum has 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 practice there is considerable spreading due to mass transfer resistances, axial dispersion, and mixing in transfer lines and valves. If an SMB is operated at (D/F)_min, raffinate and extract products will not be pure. To obtain higher purities, D/F is usually increased; however, SMBs are more complicated than binary distillation. The additional desorbent must be distributed throughout four zones to give optimum velocities in each zone. One optimization approach is to pick values of multipliers M₁ < 1, M₂ < 1, M₃ > 1, and M₄ > 1. Then velocities and D/F can be determined by solving the solute movement equations (see Problem 19.C10 for equation for D/F). The experiment or simulation is then run again with these new flow rates. The procedure is repeated until desired purities are achieved.

To complete the design we need a value for F. Then velocities and pressure drops can be calculated (e.g., with Ergun equation). If pressure drop is too large, F or column diameter need to be adjusted. Because of dispersion and mass transfer resistances, products will not be 100% pure. Actual purities can be determined by experiment or detailed simulation (see Aspen Chromatography Lab AC7 in this chapter’s appendix). If more separation is needed, one can reduce M₁ and M₂ while increasing M₃ and M₄. TMBs (Figure 19-14A) are also of interest and can be analyzed with solute movement theory and by detailed simulations.

Equations (19-14), which were derived for any type of isotherm, are starting points for analysis of solute movement with nonlinear isotherms. We need to substitute a specific nonlinear equilibrium expression to determine Δq/Δc. This insertion differs from inserting Δq/Δc = K′ for linear isotherms because the resulting expression is concentration dependent, and two separate forms result depending on initial column and feed concentrations. There are a huge number of expressions for nonlinear isotherms. To be specific, we focus on the Langmuir isotherm, Eqs. (19-5a), (19-6a), and (19-6c), which are popular forms.

We can now calculate this derivative for any desired nonlinear isotherm [for linear isotherms dq/dc = K′, and the result is Eq. (19-15)]. Specifically for Langmuir isotherm in Eq. (19-6c),

and solute wave velocity is

Note that for nonlinear isotherms solute wave velocity depends on concentration. For Langmuir isotherms, as solute concentration increases, the denominator in Eq. (19-32b) decreases and solute wave velocity increases. Another way to look at this is that dq/dc is the local slope or tangent of the isotherm. Figure 19-2 shows that for a Langmuir isotherm dq/dc is largest and thus velocity u_s is smallest as c approaches zero. Values for u_s can be calculated from Eq. (19-32c) for a number of specific concentrations, and the diffuse wave can be plotted as shown in Figure 19-16A. Times at which these solute waves, which are at known concentrations, exit column (at z = L) can be determined (see Figure 19-16A), and the outlet concentration profile can be plotted as in Figure 19-16B. Outlet wave varies continuously and is diffuse (not sharp). This result agrees with experiments that show zone spreading is proportional to column length and waves have a smooth, even spread in concentration. Analysis procedure is illustrated in Example 19-6.

where subscripts “before” and “after” refer to conditions immediately before and immediately after the shock wave. Fluid and solid before the shock wave (c_A,before and q_A,before) are assumed to be in equilibrium, as are values of c_A,after and q_A,after after the shock wave. For a general Langmuir isotherm, Eq. (19-6c), shock wave velocity is

Shock wave velocity depends on concentrations on both sides of the shock wave. The resulting shock wave is shown in Figure 19-17B, and outlet concentration is shown in Figure 19-17C. Superficially, the outlet concentration profile in Figure 19-17C looks like results from a linear isotherm (concentration jumps to c_feed). However, for linear isotherms this outlet step occurs at t = L/u_s, which is constant regardless of feed and initial concentrations. The shock wave outlet step occurs at t = L/u_sh, which depends on both feed and initial concentrations. In addition, shock waves are self-sharpening, a concept explored in Example 19-7. Waves in systems with linear isotherms are not self-sharpening.

In experiments (Figure 19-15A) outlet concentration profiles are not sharp, as shown in Figures 19-17B and 19-18B. Instead, finite mass transfer rates and finite amounts of axial dispersion spread waves, while the isotherm effect (illustrated in Example 19-7) counteracts this spreading. The final result is a dynamic equilibrium where waves spread a certain amount and then stop spreading. Once formed, this constant pattern wave has a constant width regardless of column length.

Interaction of shock and diffuse waves can be analyzed with the theory presented in this section (see Problem 19.E1). If there are two or more adsorbates that compete for sites [e.g., with multicomponent Langmuir isotherms, Eq. (19-8)], interactions often occur between shock waves and diffuse waves from different components (e.g., Ruthven, 1984; Yang, 1987). Theories for two or more interacting solutes are beyond the scope of this introductory chapter.

The most common materials used for ion exchange are polymer resins with charged groups attached (Anderson, 1979; Dechow, 1989; LeVan et al., 1997; Wankat, 1990). Cation exchange resins have fixed negative charges, whereas anion exchange resins have fixed positive charges. Resin is called strong if it is fully ionized and weak if it is not fully ionized. Most commonly used strong resins are based on polystyrene (see Figure 18-3E) cross-linked with divinylbenzene (DVB) with benzene-sulfonic acid groups attached to polystyrene. Most common strong anion exchange resins have a quaternary ammonium structure. Cross-linked polystyrene resins are commonly used for water treatment and have very good chemical resistance, although they are attacked by chlorine. Copolymers of DVB and acrylic or methacrylic acid are used for weak acid resins. No single type of weak base resin is dominant, although use of a tertiary amine group on a polystyrene-DVB resin is common. Although capacity of weak exchangers is lower than for strong resins, they also require less regenerant. Weak resins tend to be less robust than strong resins and need to be protected from chemical attack. Table 19-4 lists typical properties of ion exchange resins.

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

In this equation R^– represents fixed negative charges such as SO₃– on resin. Hydrogen and sodium ions that are exchanging are called counter-ions. Chloride ion, which has same charge as the fixed SO₃– groups, is called the co-ion. Although chloride ion does not directly affect reaction, at high concentrations it does affect equilibrium. 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).

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

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

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

A⁺ + RB + X^– = AR + B⁺ + X^–

Illustrated in Eq. (19-36), the mass action equilibrium expression simplifies to

where local concentrations of co-ion X^– in the resin interior are identical (and very low) in numerator and denominator and hence cancel. (These concentrations are very low in resin because of a phenomenon known as Donnan exclusion—the very high concentrations of fixed charges on resin exclude free anion X^– from the resin.) Equilibrium constant K_AB isn’t really constant (e.g., see Wankat, 1990), but in dilute solutions it will be close to constant. Solving Eq. (19-40a) for y_A,

The order of subscripts is important, K_BA = 1/K_AB. These equations can be applied to any monovalent exchange by substituting in appropriate symbols for exchanging ions. Experimental values for equilibrium constants are required. A few representative values are given in Table 19-5. If we know equilibrium constants K_AB and K_CB, we can calculate value of K_CA from

This result expands the usefulness of Table 19-5. However, since values in Table 19-5 are approximate (e.g., Dechow [1989] lists the following values for strong acid resins: H⁺ = 1.26, Na⁺ = 1.88, , K⁺ = 2.63, Cs⁺ = 2.91, and Ag⁺ = 7.36), they should not be used for detailed design calculations.

The equilibrium expression for a divalent ion exchanging with a monovalent ion [e.g., reaction in Eq. (19-37)] is not as simple. If D represents divalent calcium ion and B monovalent sodium ion, reaction is

D⁺² + RB₂ + 2X^– = RD + 2B⁺ + 2 X^–

and equilibrium constant from the mass action expression is

Selected values of divalent-monovalent equilibrium constants are given in Table 19-5. If K_DB and K_AB are known, then the desired constant K_DA is

Substituting summation Eqs. (19-39) into Eq. (19-41), we obtain the equilibrium expression,

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

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

Co-ions (ions with the same charge as ions fixed to resin) are excluded, and they have K_DE = 0. Exchanging ions are not excluded, and K_DE = 1. Note that (1 –ε_e) ρ_s does not appear in Eq. (19-44) because volumetric units are commonly used for ion concentrations in solution and on resin.

Equation (19-44) 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 ion velocity is

The derivative (dy_i/dx_i) can be determined from the appropriate equilibrium expression, such as Eq. (19-40b) or Eq. (19-42b). These derivatives are

for monovalent-monovalent exchange and

for divalent-monovalent exchange.

Shock waves occur when conditions are 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 ion, a shock wave would be expected. The shock wave equation is

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

Note that ion velocities depend on the ratio (c_RT/c_T) regardless of the isotherm form. This agrees with our physical intuition. If resin capacity c_RT is high while 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 total ion concentration. When feed total ion concentration is changed, an ion wave passes through the column. For relatively dilute solutions resin is already saturated with counter-ions, and more ions cannot be retained. For a total ion balance, total ions are excluded, K_DE = 0, and Eq. (19-44) becomes

The same result is obtained for co-ions. Total ion and co-ion waves move rapidly through the column. A total ion wave affects counter-ion (u_A or u_D) velocities for all ion exchange systems.

For exchange of ions of equal charges (e.g., monovalent-monovalent), equilibrium is not affected by a total ion wave and x_i,after = x_i,before and y_i,after = y_i,before. Equilibrium for exchange of ions with different charges (e.g., divalent-monovalent or trivalent-monovalent) is changed. A mass balance on a segment of column (see Problem 19.C9) shows that

For monovalent-monovalent exchange,

while for divalent-monovalent exchange,

but x_{after_total_ion_wave} is in equilibrium with y_{after_total_ion_wave}.

The calculation of these effects is illustrated in Example 19-8.

This introductory presentation on ion exchange has been restricted to dilute solutions with exchange of two ions. A variety of more complex situations, including complex equilibria and mass transfer, partial ion exclusion, resin swelling (see Table 19-4), and backwashing to remove dirt, often occur in practice but are beyond the scope of this introductory section. Information about a variety of phenomena, different equipment, equilibrium theory, and mass transfer are discussed by Helfferich (1962) and Dechow (1989).

Fluid flowing in external void volume in Figure 19-1 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 radial distance from the center of the bed. During the course of separation, sorbates are first transferred down the column by bulk transfer. Sorbates then transfer across the external film and diffuse in pores until they reach 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 bulk fluid. Desorption is favored during regeneration by increasing temperature or dropping sorbate concentration.

Since the first step in separation, bulk transport, is assumed to be rapid enough to keep bulk concentration (at any given value of z, t) constant, we start with an analysis of film mass transfer.

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

The left-hand side of this equation is accumulation of solute on solid and in fluid within the pores. The right-hand side is mass transfer rate across the film. Since amount adsorbed in particles and concentration of pore fluid are functions of r, terms and are defined as average amount adsorbed in particle and 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_e = ε in Eq. (19-48b).

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

For ordinary or Fickian diffusion D_effective is related to diffusivity in free solution through the tortuosity factor, Eq. (19-4). Equation (19-50) assumes the mean free paths of molecules are significantly less than the radius of the pores. This is the usual case with liquids and with gases at high pressures.

Knudsen diffusion occurs when the mean free paths of molecules are significantly greater than the pore radius. In this case instead of colliding with other molecules, a molecule collides with pore walls. The same diffusion equation can be used, but now D_effective is determined from Eq. (19-4) with D_K, Knudsen diffusivity replacing D_molecular. Knudsen diffusivity can be estimated from (Yang, 1987)

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

This value of D is then used instead of D_molecular in Eq. (19-4) to estimate D_effective. Use of Eqs. (19-51) and (19-52) is often required for adsorption of gases in adsorbents with small pores.

In surface diffusion 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. Surface diffusion flux is

where surface diffusion coefficient D_s depends strongly on 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 diffusion terms are available in books by Do (1998), Ruthven (1984), and Yang (2003).

The first three terms are accumulation in fluid between particles, within pore fluid with an average concentration and sorbed on solid, respectively. The fourth term represents convection, and the fifth term is axial dispersion. Coefficient E_D is axial dispersion coefficient due to both eddy and molecular effects. can be related to c_i and through Eqs. (19-48b) and (19-50). A slightly simpler equation is obtained with a single-porosity model:

Solution of an appropriate set of equations—equilibrium, Eqs. (19-48b) or (19-49), diffusion Eq. (19-50) [with diffusivities calculated from D_molecular, Eq. (19-51), or (19-52) inserted into Eq. (19-4) and/or use of Eq. (19-53)], and with column mass balance Eq. (19-54) or (19-55) plus a suitable set of boundary conditions—is very difficult. This calculation is so difficult that even with detailed simulators a simplified procedure is often employed.

Equation (19-56a) is very similar to Eq. (19-48b) except that c* replaces concentration of pore fluid c_pore and lumped parameter mass transfer coefficient k_m,c replaces film coefficient k_f. As expected, lumped parameter expressions using a single-porosity model are simpler:

Equations (19-56a) and (19-56b) or (19-57a) and (19-57b) can be converted into each other if equilibrium is linear (k_m,q = k_m,c/K′). Since both Eqs. (19-57a) and (19-57b) are commonly used, it is necessary to be clear which lumped parameter coefficient is reported—unfortunately, authors usually don’t put a subscript denoting driving force on k. Look for their driving force equation to determine which form they used.

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

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

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

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

where the Reynolds number was defined in Eq. (19-60) and the Peclet number is defined as

In many systems, particularly liquid systems, resistance due to pore diffusion is much more important than resistance due to film mass transfer [D_effective is small and hence the second term on the right-hand side of Eq. (19-58a) is much larger than the first term]. Then,

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 kinetics and mass transfer in adsorbents can be found in books by Do (1998), Ruthven (1984), and Yang (2003).

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

Transfer of energy from fluid to solid can often be represented as a lumped parameter expression:

The first term is accumulation of energy in solid, the second term is heat transfer rate from fluid to solid, and the last term is 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 fire.

Since solid and fluid are in equilibrium, q is related to c and T through the isotherm. After assuming solid properties (ε_e, ε_p, K_d, and ρ_s) are constant, applying chain rule and simplifying, Eq. (19-65a) becomes

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. (19-65b) 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. (19-14).

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 negligible heat of adsorption. The result is that temperature is constant along lines of constant thermal velocity where thermal velocity is given by Eq. (19-21).

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

Local equilibrium analysis can be extended to systems with variable interstitial velocity (concentrated gases), interacting solute isotherms such as Eq. (19-8), or finite heats of adsorption (concentrated gases), but these extensions are beyond this chapter’s scope (e.g., see Ruthven, 1984; Yang, 1987).

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

The zone spreading observed in solutions of Eqs. (19-54) and (19-55) for linear isotherms looks identical regardless of whether it is caused by mass transfer resistances, axial dispersion, or both. Thus, from an experimental result it is impossible to determine if 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 behavior of more complex systems.

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

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

For sufficiently long columns, solution for each solute is

where u_s,i is the single-porosity form of the solute velocity, Eq. (19-15b) (included in Problem 19.C5). The term erf is the error function, which is definite integral:

Since the error function is a definite integral, for any value of argument (value within brackets) the error function is a number. Error function values 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, including Excel. A brief tabulation of values is presented in Table 19-7. Spreadsheets are a convenient method for solving linear problems with the Lapidus and Amundson solution. These are explored in Problem 19.H1.

Note that when z = u_s,i t, which is solute movement solution, argument of the error function is 0, erf is 0, and c_i/c_F,i = ½. Thus, solute movement theory predicts the center of the spreading wave.

Equation (19-69) is the solution for the breakthrough curve for linear isotherms. Example 19-9 illustrates 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).

An effective axial dispersion constant E_eff,i is employed in Eqs. (19-67) and (19-69), since it includes 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 E_D predicted from Chung and Wen correlation Eq. (19-61), wave spreading will be significantly underpredicted by Eq. (19-69). How do we determine an effective axial dispersion coefficient? Dunnebier et al. (1998) compared solutions that included mass transfer and axial dispersion to results of the Lapidus and Amundson solution. The effective axial dispersion coefficient can be estimated from

where we have assumed that q_i and c_i are in same units and K′_i is dimensionless. If q_i and c_i are in different units, then appropriate density (or densities) need to be included with K′_i [procedure is similar to that used to derive Eqs. (19-14a) and (19-14b)]. Value of k_m,c,i can be estimated from Eqs. (19-58) to (19-60), since k_m,c,i = K′_i k_m,q,i. For linear systems Lapidus and Amundson solution with E_eff.i gives identical results as solutions of Eqs. (19-55) and (19-57a).

We have already employed superposition in some examples. In Example 19-2 we solved a linear chromatography system for separation of anthracene and naphthalene. This solution was derived by obtaining solution for anthracene and solution for naphthalene and then superimposing these two solutions (Figure 19-6). We inherently assumed the two adsorbates are independent. This type of procedure cannot usually be applied to nonlinear systems, since solutes interact [e.g., as shown by the multicomponent Langmuir isotherm, Eq. (19-8)].

Figure 19-6 also illustrates another aspect of superposition. The first step increase for naphthalene in Figure 19-6 is the breakthrough solution (a feed of concentration c_F is introduced to an initially clean column) using solute movement theory for a system with a linear isotherm. If we define X_{breakthrough,i} as solution for breakthrough with linear isotherms, then for a column that initially has no solute, solution is

Solution to a step-down in feed concentration from c_F,i to zero is feed concentration minus breakthrough solution. Based on superposition this solution is loaded column concentration [c_i (z,t) = c_F,i] minus breakthrough solution X_{breakthrough,i} started at time t = t_elution. Thus at outlet (z = L),

Although solutions including mass transfer and/or dispersion are more complicated than simple solute movement solutions shown in Figure 19-6, the superposition principle remains valid for any linear system. Thus Eq. (19-73) is valid for any linear system if X_{breakthrough,i} is any solution for breakthrough with linear isotherms, including solution in Figure 19-6 or in Eq. (19-69). For example, the Lapidus and Amundson solution for elution of a column fully loaded at a fluid concentration of c_F,i using pure solvent to remove adsorbate can be determined from Eqs. (19-69) and (19-73):

Superposition provides this result 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 stopping the pulse. Solution for this pulse is breakthrough solution (step-up from t = t_start) minus a breakthrough solution (step-down from t = t_end):

The period of this pulse is t_F = t_end – t_start. Any solution for breakthrough for linear isotherms, such as Eq. (19-69), can be substituted into Eq. (19-75).

where t is time after pulse is fed to the column. Equation (19-76), Gaussian solution for linear chromatography, is extensively used to predict analytical chromatography outcomes. Outlet concentrations are determined by setting z = L. Maximum peak outlet concentration occurs at z = L and t = L/u_s,i

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. (19-76) shows that number of stages N_i is

where Pe_L,i is the Peclet number based on the column length.

N_i is also related to L through height of an equilibrium plate (HETP_i).

These results allow us to calculate the value of N_i or HETP_i if E_eff is known, or vice-versa, calculate E_eff if N_i or HETP_i is known. This conversion is useful, since easy methods to estimate N_i from chromatographic results are available (see Example 19-10 and homework problems). However, the staged model is awkward, since the number of stages N_i and HETP_i for each component can be different. Since chromatography is usually used for relatively difficult separation of similar compounds, the effective axial dispersion coefficients from Eq. (19-71) are often quite close to each other. Thus use of average values results in little error.

The Gaussian solution can be written in shorthand notation as

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

and in length units

where t_R,i is the molecule’s retention time, t_R,i = L/u_s,i.

In linear systems the variances (σ_i²) 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 (19-79) and (19-80a) can be used to analyze experimental peaks, which are essentially plots of concentration vs. time, to determine u_s,i and N_i. From Eq. (19-79) the peak maximum must occur when x_t,i = 0. Since the outlet concentration profile is being measured at x = L, from Eq. (19-80a), the peak maximum occurs when t_max,i = t_R,i = L/u_s,i, which is identical to solute movement result. Thus u_s,i and an experimental value for equilibrium constant K_i′ can be determined from time the peak maximum exits the column. This procedure is illustrated in Example 19-10.

N_i can be determined from (Bidlingmeyer and Warren, 1984; Giddings, 1965; Jönsson, 1987; Wankat, 1990)

where peak maximum and width are measured in time units. The constant depends on 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 width to use experimentally is pulse width at the half height (c_i = c_max,i/2):

The standard deviation σ_t,i of an experimental Gaussian peak can be estimated from

There are a number of other methods for determining N_i and σ_t,i of Gaussian peaks (Bidlingmeyer and Warren, 1984; Giddings, 1965; Jönsson, 1987; Wankat, 1990) that all give essentially the same results, although the method presented here is less sensitive to peak asymmetry than some of the other methods (Bidlingmeyer and Warren, 1984). The use of these equations is illustrated in Example 19-10.

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 peaks. If two peak maxima are separated by more than the spreading of the two peaks, 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)

where t_R,A and t_R,B are retention times when peak maxima exit. When R = 1.0, the two peak maxima 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.

Resolution can be predicted by substituting expressions for retention times and standard deviations in Eq. (19-82). Assuming 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

This equation indicates how we can increase resolution if separation is inadequate. For example, increasing column length, which linearly increases N according to Eq. (19-78a), increases resolution by a factor of L^1/2. More effect can be obtained by changing equilibrium isotherms (Schoenmakers, 1986) to increase distance between peaks (see Problem 19.B3) or by decreasing particle diameter, which according to Eq. (19-71) decreases E_eff,i, thereby increasing N in Eq. (19-78b).