Chapter 29. Adsorption and Chromatography

Adsorption is a mass transfer process by which a chemical component from a fluid phase is transferred and attached to a solid phase. This can be an effective way of removing trace contaminants from either a gas or liquid stream. The species being adsorbed is called the adsorbate while the solid on which it is adsorbed is called the adsorbent. The adsorption process depends on the selective partitioning of the adsorbate in the solid phase. Thus the equilibrium distribution of the adsorbent between the fluid and solid phase is an important thermodynamic property needed in the design of adsorption equipment. We therefore discuss first the various types of adsorption isotherms. Since the species is transferred from the fluid to the solid phase, a number of mass transfer steps are involved in the process. These steps are discussed next. This is followed by models for batch and continuous adsorption processes. For batch adsorbers the fluid phase is modeled using a macroscopic model while for continuous adsorption in a packed column, the mesoscopic model is useful. These models are coupled suitably with the differential model for the adsorbent particle as illustrated in this chapter. Models are needed to calculate the time of operation and design and scale-up of an adsorption column using laboratory data.

Chromatography is a related process where two or more components are separated due to the difference in adsorption property of these species over a solid. One model is to assume that equilibrium exists at all points in the column. This generates the first-level model for the system, the so-called ideal separation model. However, mass transfer effects are important and one of the goals is to examine their effect on the extent of separation and show the deviation from ideal behavior.

The quantity adsorbed is proportional to the surface area of the adsorbent and efficient adsorption needs materials with a large area. This can be achieved only if they are highly porous solids. Activated carbon is one such common adsorbent and has a surface area of 1000 m²/g or more. The process of activation is used to create this high area and consists of two steps: charging and heating of a carbon source such as coconut shells in the range of 500 to 700 °C in a gas phase with a low oxygen concentration, and treating the charred material to an oxidizing gas such as CO or steam to volatilize the tarry material and thereby create a large surface area. A network of micro- and macropores is created by this process. Macropores provide less area but provide pathways to transport to micropores. The micropores provide the large area where most of the adsorption occurs. Hence both macro- and micropore volumes are important. The relative proportion of these can be adjusted by choosing suitable starting materials. For example, coconut shells, which are dense, produce activated carbon with more micropores while bituminous coal as a starting material produces an adsorbent more macropores.

Other common materials are silica gel and the zeolites. Silica gel is a porous form of silicon dioxide with typical pore sizes in the 4–7 nm range and a surface area of around 700 m²/g. It has strong adsorption properties for water and is commonly used as a desiccant. It is also commonly used as a packing in chromatography columns. Zeolites are a class of compounds belonging to the alumina-silicate family of chemicals. They are also known as molecular sieves due to their ability to selectively sort molecules based primarily on a size exclusion process. This is due to a very regular pore structure of molecular dimensions. The maximum size of the molecular or ionic species that can enter the pores of a zeolite is controlled by the dimensions of the channels. For example, zeolite 5° A has a nominal pore diameter of 0.5 nm and can separate n-paraffins from branched ones. Zeolites in general have wide applications both as catalysts (e.g., catalytic cracking in the petroleum industry) and as adsorbents (e.g., oxygen separation from nitrogen).

At equilibrium the two processes balance each other. Therefore

k_aC_A(1 – θ) = k_dθ

Let the ratio of the two rate constants be designated as K_A, an equilibrium constant. Thus let K_A be defined as k_a/k_d. Using this in the above equation and solving for θ we get

θ=KACA1+KACA

The adsorbed concentration of A,_qA, is defined as q_maxθ, where q_max is the concentration at complete coverage corresponding to θ equal to 1. (A monolayer coverage of the sites is implied in the Langmuir model.) Hence

which is referred to as the Langmuir adsorption isotherm. This is often rearranged and expressed as

qA=qmaxCAK+CA

where K is the reciprocal of K_A. A plot of 1/q_A versus 1/C_A would be linear if the Langmuir isotherm holds and such a plot is useful to estimate the two constants needed to characterize the model.

where K_A,linear = q_maxK_A. The linear model is useful when the solute concentration is low, for example, applications such as pollutant removal. The nonlinear Langmuir model (Equation 29.1) is needed for higher concentrations, for example, for adsorption-based separation processes.

Generally the adsorption capacity K_A decreases with temperature. This is consistent with Le Chatelier’s principle since adsorption is an exothermic process.

A similar equation holds for species B:

where q_i,max is the maximum amount of adsorption for species i at complete coverage of the all active sites. The model is often called the extended Langmuir model. It ignores the interaction between adsorbed A and B and simply includes the reduction of vacant sites for adsorption of, say A, due to the simultaneous adsorption of B.

Here K_A,f and n are the two parameters needed in this model. The range of n is usually from 1 to 5. n = 1 is the linear case. The Freundlich isotherm can be derived by assuming a heterogeneous surface with a non-uniform distribution of the heat of adsorption over the surface.

Type I is the standard Langmuir isotherm and shows a monotonic approach to q_max; it applies for the case where the adsorption is restricted to a monolayer. Type II occurs when the monolayer adsorption occurs at low pressures and a multilayer condensation type of phenomena occurs at higher pressures. Type III is a case where the condensation type of phenomena is the dominant mechanism. Adsorbents showing this pattern are not useful in practice because the adsorbed concentration is low except at very high concentrations or pressures.

Types IV and V are systems showing a hysteresis phenomena, which is due to capillary condensation. These are capillary condensation versions of Type II and III, respectively.

Loss of solute in the liquid phase = Gain of mass in the solid phase

This can be written mathematically as

where q¯ is the average adsorbed concentration in the solid. C_A is the concentration in the bulk liquid. The average concentration needs a transient diffusion model for the particle. This particle-level model is common to both the batch slurry adsorber discussed in this section and the fixed bed adsorber discussed in Section 29.4.

Note: The notation K is used here for the linear adsorption relation as an abbreviation for the K_A,linear in Equation 29.2. This should not be confused with the K_A in the Langmuir model. Also q_A will be abbreviated as q and C_Ap is abbreviated as C_p.

The average concentration in the pores (a spherical particle is assumed here) is given as

Using the local equilibrium relation, this can be written in terms of the pore level concentration C_p as:

The variation of C_p as a function of r can be modeled by the transient diffusion model:

The two terms on the left-hand side represent the mass capacity in the pores and the mass capacity on the solid. Usually K is large and the second term, the mass capacity over the solid, is much larger and hence in further discussion the first term is dropped. Also note that the surface diffusion (diffusion of adsorbed species along the surface) is ignored here and only the pore diffusion is included.

Since q = KC_p (assuming a linear isotherm here) the PDE can also be written as

Equation 29.10 requires two boundary conditions and one initial condition. The center condition at r = 0 is the symmetry condition while the boundary condition at r = R depends on whether external mass transfer (liquid to solid) is important or not. Thus we can use the Dirichlet at the surface if the liquid to solid mass transfer is relatively fast compared to internal diffusion. This provides the following condition for the case of no external resistance:

If the external mass transport is included, a balance of flux at the solid surface leads to the following Robin condition:

The initial condition is taken as zero concentration, that is, a fresh adsorbent particle.

The solution of Equation 29.10 can be used in Equation 29.8 to get q¯ and ∂q¯/∂t, which in turn is needed in Equation 29.6. This is the exact procedure. It is useful to solve the particle model numerically since the analytical solution is cumbersome in spite of the fact that the PDE is linear. The solution can be done using orthogonal collocation, which appears to be the most efficient for this system. This method is discussed in Section 29.3.5. An approximate analytical solution is based on the linear driving force model, which is presented in the following section.

The quadratic profile assumption can be used to derive an approximate lumped transfer coefficient for internal diffusion. Equivalently its reciprocal is the internal diffusional resistance. The external mass transfer leads to an additional drop in concentration from bulk to the surface. The overall resistance is then the sum of the external solid–liquid resistance and the internal diffusion resistance. The overall driving force is represented as CAL–q¯/K. The rate of transfer is represented as a transfer coefficient, k₀, times the driving force. Thus the uptake per particle volume is modeled as

The exchange resistance 1/k₀ is modeled as the reciprocal of the sum of two resistances.

An approximate formula for this exchange coefficient is

The first term on the right-hand side is the resistance to solid–liquid mass transfer and the second term is resistance to internal diffusion.

The equivalent representation in terms of the resistance is presented in Figure 29.4. This figure captures the essence of the linear driving force model.

(Solute in liquid + solute in solid) at time t = Solute in liquid at time zero

Using this material balance, the following equation can be derived:

Here C_Ai is the initial concentration of the solute in the bulk liquid.

Using the above relation for q¯A in Equations 29.6 and 29.13, we can show that

An adsorption factor α is defined as

α=VLVSK

Using this factor Equation 29.16 is written as

Integration of Equation 29.16 provides the concentration profile in the liquid as a function of time. The resulting expression is

The final concentration in the liquid is obtained in the limit t tending to infinity:

This can also be verified by an overall mass balance from time 0 to ∞.

The final concentration depends only on α, which can be obtained by changing the volume of the liquid to that of the adsorbent, V_L/V_S. The measured value of the final composition is used to find the parameter α using Equation 29.19, which in turn provides the equilibrium constant, K. The assumption of linear equilibrium can also be tested by this set of experiments since K should be independent of the final concentration for a linear isotherm.

Note also that for each final concentration obtained by changing the V_L/V_S ratio, the corresponding adsorbed concentration is given by Equation 29.15:

This equation is independent of the type of isotherm since it is purely based on mass balance. Hence q_A versus C_A data can be obtained in a batch slurry experiment by merely changing the solid loading progressively and allowing each set to reach the steady state. The data can then be used to fit a suitable isotherm.

Equation 29.10 is first made dimensionless in distance by defining ξ = r/R, leading to

where K′ = K(1 – ∊_p) is used as an abbreviated notation. Note that the time is not dimensionless here. The right-hand side is the Laplacian defined in dimensionless coordinates and can be approximated using the B matrix given in Section 18.6. This is applied at all the interior points leading to a set of equations for dC_pi/dt:

Here N is the total number of collocation points and C_pi is the concentration of A in the pores at collocation point i.

The point N is the surface of the catalyst (r = R) and the concentration at this point, C_pN , is assigned by using the the boundary condition given by Equation 29.12. Using the A matrix for the derivative, the collocation version of this boundary condition is

Here the parameter Bi is defined as = k_mR/D_e. Rearranging this equation, the surface concentration is given as

Note that if Bi is large, the Dirichlet condition holds and C_pN can be simply approximated as C_AL. This is a case where the film mass transfer is rapid compared to internal diffusion.

An additional equation for dC_AL/dt can be obtained from Equation 29.6. This needs an average adsorbed concentration, q¯. The relation q¯=KC¯p is used. The integral to calculate the average concentration in the pore, C¯p, is approximated using the w values, the quadrature weights, and therefore

Using this in Equation 29.6 and rearranging we get

where α is defined as V_L/V_SK′.

The initial value equations (Equations 29.22 and 29.26) can be readily integrated, for example, using ODE45. The time evolution of the concentration in the interior points in the pore and in the bulk liquid can then be calculated simultaneously. Further details are left as an exercise.

The model is similar to the tracer response curve in a packed bed studied in Chapter 14. Modeling can be done on many levels of progressive complexity and the approach is discussed in the following.

Models where the concentration profiles in the solid are not solved explicitly are called pseudo-homogeneous models. If the profiles in the solid are also included the model is a called heterogeneous model. The pseudo-homogeneous type of model can again be developed at various levels of detail.

The first category of model is based on local equilibrium and provides an approximate estimate of the column size needed and also the effect of the adsorption isotherm on the outlet concentration. Here no mass transfer resistances are considered nor is the effect of axial dispersion.

A second level of model keeps the local equilibrium assumption but introduces the axial dispersion effect.

The third level includes the mass transfer resistances in addition to dispersion. But the particle-scale model is not solved in detail and instead lumped into an overall mass transfer coefficient based on the LDF concept.

Models where the solid concentration profile is also included and the gas and solid are treated as separate phases are called heterogeneous models. The gas phase is modeled using dispersion and the solid phase is modeled using a transient diffusion-adsorption model; the two are coupled and solved simultaneously.

in – out = accumulation

In = u_GA(C_A)_x where u_G is the superficial velocity of the gas. Similarly, out = u_GA(C_A)_x+Δx.

The accumulation is the time derivative of moles of A in the system. Moles of A are present in the gas phase in the void space plus that adsorbed on the solids. The adsorbed concentration is in equilibrium with the gas concentration. Hence the adsorbed concentration is equal to K_AC_A for linear adsorption. Combining the two, the following expression is obtained for the moles of A in the differential volume:

Moles of A = [∊_B + K_A(1 – ∊_B)] C_AAΔx

Here ∊_B is the bed voidage, that is, the fraction of the bed that is not occupied by the solid.

Putting all these together into the gas phase balance, we can derive the following differential equation for the concentration variation of A in the system:

A wave speed w_s can be defined as

Using this Equation 29.27 can be expressed in a simpler form as

This is the simplest PDE of the hyperbolic type. The equation requires an inlet condition C_Ai(0, t) (which can be any a specified function of time) and an initial condition (taken as zero here). The solution is simply a time delay:

Here H is the Heaviside step function.

The inlet value simply moves along the bed with the wave velocity. The inlet response therefore appears at position x after an elapse of time of L/w_s for a step input.

The time constant for a plug flow, also called the retention time (which is also the first moment of the system for a pulse input) is given by

The second moment can be shown to be equal to zero and therefore there is no spread or dispersion. A pulse introduced at the inlet will appear at the exit intact after the elapse of this retention time. Similarly a square wave input at the inlet will appear at the exit as a square wave after this time. A step input will appear as a step input at the exit after the elapse of the retention time.

The exit response is shown schematically in Figure 29.5. (Note that the ∊_B term in Equation 29.31 is normally dropped as done in this figure.)

The exit response to a step input is shown in Figure 29.6. For a continuous flow of solute with the inlet gas, no solute appears in the exit for time less than the retention time. After this time, the exit concentration is equal to the inlet concentration. Hence for a given length of the bed the maximum time of operation is equal to the retention time.

The steepness of the breakthrough curve shows how much of the bed is underutilized. If it is very steep a large part of the bed is utilized, approaching an ideal case. Thus prediction of the shape of the breakthrough curve is quite important. It is also useful to know what effects the bed length, bed diameter, and particle diameter have on the shape of the breakthrough curve. The mass transfer models are useful to do these calculations.

The spread due to dispersion is modeled by adding an additional contribution to axial dispersion, but other assumptions in the ideal model are retained. Thus film diffusion and internal diffusion are not accounted for and a local equilibrium is assumed at any location in the adsorber. This, for example, could apply for small particles where the mass transfer coefficients are large and the internal diffusion resistance is also small.

The gas balance now reads

where the first term on the right-hand side reflects the spread due to axial dispersion. The model now needs one initial condition and two boundary conditions. This is similar to the dispersion model studied in Section 14.3. The time constant is simply adjusted to account for the accumulation in the solid. The response curve would look similar to that plotted in Figure 14.2.

The dispersion coefficient is an important parameter in quantifying the pulse spread. Correlations to calculate these were reviewed in Section 14.6 and are useful to simulate adsorption and chromatographic columns.

The gas balance equation is now modeled with the accumulation in the particle added as an additional term. The q¯A=KCA assumption is no longer used:

where q¯A is the average concentration in the solid:

The variation of q_A as a function of r can be modeled by the transient diffusion model. This part of the model is given in Section 29.3.2.

The solution of this model needs a coupled computational scheme. At each location x in the bed, the particle model has to be solved to provide the q¯A information. This provides the required coupling for the gas phase balance. A number of numerical methods can be used. Collocation methods are again suitable but need some additional refinements. Since there are two spatial variables, one in the bed x and the second internal to the particle r, two separate collocation discretizations are needed. The method is sometimes called double collocation. Sampath et al. (1975) provide details of this method. Although this paper is for a reacting solid problem it has the same mathematical structure as the transient adsorption simulation model.

Here ξ is the dimensionless axial distance, defined as

and τ is dimensionless time corrected for the motion of the fluid, defined as

Here k₀ is given by the LDF model (Equation 29.14).

tstl=(dpsdpl)2−α

where t_s is the time for estimated breakthrough in a small-scale column and t_L is the value for the large-scale column. d_p is the particle diameter used.

The exponent of the particle diameter α has a value of 0 for a constant diffusion coefficient and is equal to 1 for diffusivity proportional to concentration. The latter value is representative if surface diffusion is the dominant pore transport mechanism. Spread by internal diffusion is the main controlling mechanism implied in this scale-up rule. Since the time constant for the constant diffusivity is proportional to the square of the diameter, the exponent is two for the constant diffusivity case.

The matching of the Reynolds number is used to determine the operating velocity in the small columns. Hence

υsυl=dlds

Particle to column diameter should be chosen as 20 or more to avoid channeling near the walls.

We first look at the process assuming plug flow and then correct the model for the effect of axial dispersion. The pulse of A will appear intact after the elapse of retention time if there are no factors that cause the broadening of the pulse. The retention time for A is given as

t¯A=LuG(εB+(1−εB)KA)

Note that the mean retention time depends on the adsorption constant, as observed from the above equation. If the two solutes have different mean retention times, they appear in the exit stream at different times, causing a separation. This is the principle of chromatography. If the column is modeled as an ideal system, the two components will appear as two sharp peaks at the corresponding mean retention time. In reality, there are dispersion effects, which cause a spread; this is referred to as pulse broadening. An illustrative response curve is shown in Figure 29.8.

If the injected sample is small, the pulse becomes Gaussian and the concentration distribution can be modeled as a Gaussian distribution around the mean:

The variance depends on the extent of axial dispersion, solid–liquid mass transfer, and the internal diffusion. Each term contributes in an additive manner to the variance. Equation 14.45 can be used to estimate the contribution of the above three factors to the variance. The spread around the mean retention time is about 4σ, as shown in Figure 29.8 for a Gaussian distribution.

The area under the curve is a measure of the amount of that component. Overlapping peaks are often noticed. This depends on the σ parameter, which causes the spread. If t¯A and t¯B are somewhat close and simultaneously if σ is also high, then overlap will result. Any factor that decreases σ will help in reducing the overlap. For example, increasing the column length can be helpful. Change in temperature, temperature programing, change in carrier gas composition, or change in column material are other tools which can be used to increase the resolution between adjacent peaks.

29.2 How should the experimental data of q versus C_A be plotted in order to test the Langmuir model?

29.3 State the Freundlich isotherm and also the units of the equilibrium constant used in the model.

29.4 How should the experimental data of q versus C_A be plotted in order to test the Freundlich model?

29.5 If a plot of 1/q versus 1/C_A is linear, what do the slope and intercept represent?

29.6 If a log-log plot of q versus C_A is linear, what do the slope and intercept represent?

29.7 What can cause hysteresis behavior in an adsorption isotherm, that is, are the q versus C_A plots different for adsorption compared to that for desorption?

29.8 What is the LDF approximation?

29.9 What is the assumption used to derive the LDF approximation?

29.10 What are the differences between the pseudo-homogeneous model and heterogeneous model for packed columns?

29.11 What is the ideal pulse velocity in an adsorption or chromatographic column?

29.12 What is the breakthrough curve?

29.13 What principle is used in chromatographic separation?

29.14 What factors cause pulse broadening in a chromatographic column?

q, STP cm3/gm	40	165	350	545	760	910	970
p, atm	45.5	91.5	113	121	125	126	126

Fit a suitable isotherm model for the this data. Deterimine which isotherm (Langmuir or Freundlich) provides a better fit to the data.

29.2 Batch adsorber calculations. An aqueous solution of phenol with a concentration of 0.01 mol/L is treated with activated carbon in a batch adsorber. An adsorbent loading of 10 g/L is used. The equilibrium constant K has a value of 100. The external mass transfer coefficient is 5 × 10^–5 m/s. The internal diffusion coefficient is 2 × 10^–9 m²/s and the particle size used is 1.5 mm diameter. Find the α parameter and use this to find the final concentration of phenol in the liquid. Find the k₀ parameter and use this to track the concentration versus time profile of phenol in the liquid.

29.3 Laplace transform method. Obtain the solution to Equation 29.29 in the Laplace domain. Find the first and second moments and verify the expression for the time constant is given in Equation 29.31 and also that the variance is zero. Look up books that provide a table of inverse transforms and verify the time domain solution given in the text.

29.4 Ideal adsorption model. An aqueous solution of 3 g/cm³ each of glucose and fructose is fed into an adsorption column packed with an ion exchange resin. The adsorption isotherm is modeled as a linear equation q = KC_A with K = 0.26 for glucose and K = 0.40 for fructose. The units are g per 100 cm³. The superficial velocity is 0.031 cm/s and the bed void fraction is 0.4. If the feed is introduced as a sharp pulse in a column of length 200 cm, determine the time when the two solutes appear at the exit. If a square wave feed pulse lasting 500 seconds is introduced, sketch the exit concentration of both the solutes. Assume the pulse travels as a wavefront with the wave velocity given by Equation 29.28.

29.5 Wave speed for nonlinear equilibrium. A wave speed w_s was shown to be given by Equation 29.28 for a linear adsorption isotherm. Show that for a nonlinear isotherm the wave speed can be calculated as

ws=uGεB+(1−εB)(dq/dC)

29.6 Klinkenberg model for breakthrough. Air at 25 °C and 1 atm containing 0.9% benzene is fed to an adsorption tower that is 30 cm in diameter and 2 m long. The air flow rate is 5 kg/min. The bed is packed with silica gel with an effective diameter of 2.6 mm. The bed void fraction is 0.5. The isotherm is linear with K equal to 5.2. The mass transfer coefficient k₀ was found from separate experiments to have a value of 0.2 min^–1. Calculate the retention time. Use the Klinkenberg model and plot the breakthrough curve. Find the time at which 5% benzene appears in the exit. Find the fraction of the bed that is not utilized. Repeat the calculation if the bed were 8 m long. Determine by what factor the pressure drop would increase. Note that the cost of pumping has to be balanced with the need for more frequent regeneration.

29.7 Packed bed: size specifications. A packed column is used to separate a solute from an air stream of 1 L/min flow. A superficial velocity of 4 cm/s is suggested and the column will operate for 4 hours before regeneration. The equilibrium constant is 500. The exit stream should contain no more than 5% of the incoming solute. The particle size is 3 mm and the bed void fraction 0.4. The diffusion coefficient of the solute in air is 0.16 cm²/s. Calculate the solid–liquid mass transfer coefficient and the k₀ parameter. Find the column height needed. Find the pressure drop in the column. Examine the sensitivity of your results by changing the superficial velocity and the suggested operation time.

29.8 Scale-up of a packed adsorber. A small-scale unit is operating with a particle radius of 0.1 mm at a velocity of 25 m/hour and the breakthrough was observed after 24 sec. Based on this what should be operating velocity if a pilot scale adsorber will operate at 5 m/hour? What would be breakthrough time in this column?

29.9 Chromatographic separation. Consider separation of two components with the following properties: K_A = 1; K_B = 2; D_A = 2 × 10^–4 m²/s; D_B = 8 × 10^–4 m²/s. The eluting fluid has a flow rate of 1 cm³/min and the column of diameter of 5 mm. The length of the column is 2 m. The particle size is 0.5 mm.

Find σ and test if there is an overlapping of the peaks. Assume that the spread is due to internal diffusion only.

Sketch and plot the exit concentrations of the two solutes as a function of time.