Discrete Equilibrium Data from Dynamic Column Breakthrough

Article pubs.acs.org/IECR

Discrete Equilibrium Data from Dynamic Column Breakthrough Experiments Reza Haghpanah,† Arvind Rajendran,*,† Shamsuzzaman Farooq,*,‡ Iftekhar A. Karimi,‡ and Mohammad Amanullah† †

School of Chemical and Biomedical Engineering, Nanyang Technological University, 62 Nanyang Drive, Singapore 637459 Department of Chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576

‡

ABSTRACT: A new method for extracting discrete equilibrium data from a set of dynamic column breakthrough experiments is described. Instead of the classical approach where an isotherm model, i.e., a function, is used to describe the equilibrium, this approach represents the isotherm as a set of discrete points. For a given set of discrete fluid phase concentrations, an optimization method is used to determine the corresponding solid loadings that lead to the best-fit prediction of the experimental breakthrough profile. In this work, we develop the algorithm and validate it using single-component case studies, for a variety of isotherm shapes. The practical use of the method is demonstrated by applying it to experimentally measured breakthrough profiles taken from the literature.

1. INTRODUCTION Chromatography is now routinely used for preparative separation of enantiomers, biological compounds, and other valuable products.1−3 Both single column and a variety of multicolumn processes have been developed and successfully commercialized.4 For a given separation, the economics of a chromatographic process is closely related to the choice of the stationary and mobile phase. For preparative separations it is important to choose them in such a manner that maximum productivity and minimum solvent consumption can be achieved. A rational approach to design and scale-up of these processes invariably involves the characterization of the separation at analytical scales.5 From these experiments, important information about the adsorption equilibrium and mass transfer kinetics are extracted. These scale-independent parameters can be used for scale-up and to make preliminary process feasibility studies. Hence, characterizing the adsorption equilibria of the solute between the mobile and stationary phases is an important step in process design and scale-up. Characterizing a stationary phase in order to design a separation process ideally involves (i) measuring reliable single component equilibrium and kinetic data, (ii) modeling the observed single component behavior, and (iii) developing experimentally verified predictive multicomponent equilibrium and kinetic models that rely on single component parameters. Several techniques, including batch measurements, frontal analysis, elution by characteristic points (ECP), inverse methods, etc., are commonly used.1,5 In frontal analysis, a step change in the fluid phase concentration is introduced at the column inlet. A detector placed at the column outlet measures the breakthrough profile which is completed when the exit concentration is identical to that at the inlet. The detector signal can be integrated to obtain the equilibrium solid loading corresponding to the feed concentration. After this, the feed concentration is altered and the experiment is repeated.5,6 In the case of ECP, either a step or pulse input is made and the elution profile is measured. In both cases, the part of the elution © 2012 American Chemical Society

profile that consists of the simple wave (desorption for a favorable isotherm and adsorption for an unfavorable isotherm) is considered.7,8 The wave velocity of the simple wave is proportional to the local slope of the isotherm and hence by calculating the time at which a particular concentration elutes allows the calculation of the local slope from which the isotherm can be estimated.1 It is important to note that the ECP method is effective only for columns that have a high efficiency.7 Finally, in the inverse method, the elution profile of a pulse/step input is considered for estimating the isotherm.9 In this technique a functional form of the isotherm is assumed, and the parameters are treated as decision variables. A simulation program is then used to determine the values of these parameters that provide the best fit of the experimental elution profile. For columns with low to moderate efficiencies, frontal analysis and the inverse method are more widely used compared to others. There are other techniques to measure isotherms, and the interested reader is referred to more detailed treatments.1,5 While frontal analysis yields accurate data, it is timeconsuming as experiments need to be performed at multiple inlet compositions in order to obtain the complete isotherm. The inverse method is, however, advantageous, as it requires fewer experiments.10 As mentioned earlier, the inverse method is based on estimating the isotherm parameters for an assumed isotherm model, i.e., a continuous functional form. This model is usually chosen by prior experience or knowledge about the adsorption mechanism. In many cases, it is possible that the assumed model may not properly describe the adsorption equilibrium. The ultimate goal of the present study is to develop a methodology to extract unambiguous equilibrium data from a minimum set of breakthrough responses in a way Received: Revised: Accepted: Published: 14834

June 12, 2012 October 11, 2012 October 11, 2012 October 11, 2012 dx.doi.org/10.1021/ie3015457 | Ind. Eng. Chem. Res. 2012, 51, 14834−14844

Industrial & Engineering Chemistry Research

Article

Figure 1. Dynamic column breakthrough for liquid chromatography. (a) Schematic of a typical experimental setup. (b) Typical disturbance introduced at the column inlet. (c) Typical concentration profile measured at the column outlet. Shaded area is proportional to the capacity of the column.

conditions, the mass balance of the solute is given by the transport-dispersive model

that can then be used in process design calculations without any restriction. In this work, we will focus only on single component chromatographic systems where the velocity change due to adsorption/desorption is commonly neglected, while noting that this method can be extended to bulk adsorption systems without loss of generality.

(1 − ε) ∂q ∂(c) ∂c ∂ 2c − = DL 2 − u ∂t ∂z ∂t ε ∂z

where c and q are the concentrations of the solute in the mobile phase and in the solid phase, respectively, and DL is the axial dispersion. Here, owing to the low concentration of the solute in the mobile phase, velocity change due to the adsorption and desorption of the solute is neglected. The accumulation in the solid phase can be described by detailed models that separately recognize the film resistance around the solid particles and macropore/micropore resistances inside the particles. However, in this study the simple linear driving force (LDF) model that lumps all the mass transfer resistances into the mass transfer coefficient, k, was used. The LDF model can be written as

2. DYNAMIC COLUMN BREAKTHROUGH EXPERIMENTS 2.1. Brief Overview of the Experimental Setup and Procedure. The dynamic column breakthrough (DCB) is a versatile experimental system to measure adsorption equilibrium and kinetic information. The schematic for a typical experimental setup is shown in Figure 1a. Here, the system consists of a pair of pumps to deliver the solvent and the solute, the chromatographic column, and a detector. The DCB experiments are performed in the following sequence. The column is initially saturated with solute free solvent. At time t = 0, a known disturbance, typically a step change in solute concentration (cf. Figure 1b), is introduced at the column inlet and the response of the column to this disturbance is measured at the outlet using a suitable detector (cf. Figure 1c). For a positive step change, i.e., an adsorption step, the equilibrium solid phase loading corresponding to the inlet fluid phase concentration for initially clean bed can be determined by simple mass balance on the solute * = qfeed

ε ⎡u⎛ ⎢ ⎜c feedtfeed − (1 − ε) ⎣ L ⎝

∫0

t feed

(2)

∂q = k(q* − q) ∂t

(3)

where q* is the solid phase concentration in equilibrium with the concentration of the solute in the mobile phase. The LDF model is widely used in the dynamic modeling of adsorption and chromatographic separation processes. The mathematical form of the model is physically analogous to a situation where the transport resistance is confined in a thin layer, such as the external fluid film around an adsorbent particle or barrier resistance confined at the mouth of adsorbent pores. However, it has been successfully used to capture breakthrough dynamics of an adsorption column for all forms of diffusional transport in adsorbent pores, namely, macropore and micropore diffusion.11 Further, it also allows distinction among the modes of transport in the macropores, such as molecular, Knudesn, and surface diffusion. Moreover, analytically derived or numerically verified equations have been reported in the literature that allow estimation of the LDF rate constant, k in eq 3, from the appropriate intraparticle diffusional time constant of the adsorbate and available physical characteristics of the adsorbent.11 It has been demonstrated that by choosing operating conditions appropriately it is possible to fully capture the breakthrough response using the LDF model, even in the

⎤ ⎞ c(t ) dt ⎟ − c feed ⎥ ⎠ ⎦ (1)

where q*feed is the equilibrium solid loading corresponding to the fluid phase concentration (cfeed), tfeed is the duration of the feed, u is interstitial velocity, L is column length, and ε is the bed voidage. This yields one point of the isotherm, i.e., (cfeed, q*feed). However, if the complete isotherm is required, experiments at other concentrations need to be performed. 2.2. Modeling Dynamic Column Breakthrough Experiments. The modeling of dynamic column breakthrough experiments is well-established in the literature.1,11 In this study, we consider liquid chromatography where the solute of interest is present in a nonadsorbing solvent. Under these 14835

dx.doi.org/10.1021/ie3015457 | Ind. Eng. Chem. Res. 2012, 51, 14834−14844


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Figure 2. The concept of discrete equilibrium data. (a) Continuous isotherm. (b) Discrete equilibrium data corresponding to continuous isotherm shown in part a. (c) Comparison of elution profiles from continuous isotherm and discrete equilibrium data cfeed = 10 g L−1, tinj = 400 s. (d) Comparison of elution profiles from continuous isotherm and discrete equilibrium data at low concentration range.

micropore controlled intraparticle mass transfer limitation.12 Finally, the equilibrium relationship is given by q* = q*(c)

column was initially clean prior to the introduction of the disturbance, and hence, cinit = 0. The partial differential equations were discretized by the finite difference scheme in the axial direction with 200 grid points, and the resulting system of ordinary differential equations was solved using ode45, an inbuilt solver in MATLAB.

(4)

where q*(c) denotes the continuous functional form of the isotherm. The above equations are solved numerically along with suitable initial and boundary conditions. Typical initial and boundary conditions are as follows c(t = 0, z) = c init

(5)

q(t = 0, z) = q*(c init)

(6)

3. DISCRETE EQUILIBRIUM DATA 3.1. Concept of Discrete Equilibrium Data. An adsorption isotherm describes the equilibrium relationship between the fluid and solid phase concentrations of the solute and is typically represented by a continuous function. If the adsorbent surface is uniform and all sites exhibit identical interaction energies, the functional relationship is rather simple. However, if surface/energetic heterogeneities exist, then the functional relationship can be complex. For a particular sorbent−sorbate system, this relationship is not known in advance. Typically, a functional form, i.e., eq 4, is chosen that best fits the measured experimental equilibrium data. Often the equilibrium data measurement, e.g., by batch static measurements, is bypassed and the chosen functional form of the equilibrium isotherm is regressed directly by fitting the experimental breakthrough profile. However, some caution should be exercised, since different functional forms can yield qualitatively similar breakthrough profiles. Hence, the parameters regressed for a chosen form do not guarantee that the true equilibrium data have been captured. They merely represent

and boundary conditions can be written as ⎧ c(t ) = c init t≤0 ⎪ ⎪ at z = 0: ⎨ c(t ) = c feed 0 ≤ t ≤ t inj ⎪ ⎪ c(t ) = c init t inj ≤ t ⎩ at z = L :

∂c =0 ∂z

(7)

(8)

where cinit and cfeed correspond to the initial and feed concentrations and tinj represents the injection time of a pulse. A large value of tinj, i.e., tinj → ∞, represents a step input. The effect of axial dispersion at the column inlet has been neglected in eq 7. For the cases studied in this paper, the 14836



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Figure 3. Bounds for decision variables for (a) type I (favorable equilibrium), (b) type III (unfavorable equilibrium), and (c) type II (equilibrium with inflection point). The shaded area represents the search domain.

the best fit of the chosen isotherm model form to the true equilibrium data. In order to overcome the above limitation, the concept of “discrete equilibrium data” is introduced. In this scheme, unlike the traditional approach, the equilibrium relationship is not restricted by a functional form. Instead, the equilibrium solid loading (q̃j*) is specified for a set of discrete values of the mobile phase concentration (c̃j) in the range of the feed concentration. Figure 2a shows an example of a continuous isotherm, and the equivalent discrete equilibrium data is shown in Figure 2b. The equilibrium loading, q*, at any concentration c between two adjacent discrete points c̃j and c̃j+1 with c̃j < c < c̃j+1 is obtained by linear interpolation: q* =

qj*̃ + 1 − qj*̃ cj̃ + 1 − cj̃

(c − cj̃ ) + qj*̃

q* =

for

cj̃ ≤ cj̃ + 1

(11)

shown in Figure 2a was represented by a varying number of discrete points. In the case study, the concentration range c = 0−10 g L−1 was discretized into N equidistant points and the corresponding values of the equilibrium loading q̃j* was obtained from eq 11. The effect of the number of discrete points, N, was investigated through a simulation study using the models described in section 2.2. In this study, the elution profiles obtained from the discrete equilibrium data were compared with the one from the known continuous isotherm. Figure 2c,d shows the effect of the number of discrete equilibrium points on the elution profiles. It is clear from Figure 2d that discretizing the isotherm using 10 points leads to significant deviations compared to the elution profile calculated using a continuous isotherm, particularly at low concentrations. This can be explained by the fact that using fewer points in the low concentration region leads to an inaccurate description of the Henry’s constant, which governs the low concentration portion of the desorption profile. In addition, it is clear from Figure 2c that elution profiles from 40 discrete equilibrium points and higher show no discernible difference compared to the elution profile from the continuous isotherm. In order to strike a balance between the number of decision variables, which will lead to longer computational time in optimization (explained in the next section), and the accuracy of describing the equilibrium in a satisfactory manner, we decided to discretize the concentration interval into 50 equidistant points.

(9)

This representation involving discrete data points is equivalent to piecewise linear function rather than an assumed and smooth functional form. For a very large number of discrete equilibrium points, the representation indeed approaches the continuous limit. While the exact trend of equilibrium data is not fixed by a functional form, they are subject to the thermodynamic constraint qj*̃ ≤ qj*̃ + 1

6c 1 + 0.6c

(10)

Since the discrete equilibrium data cover the entire operating range of experiment, eq 9 suffices for numerically simulating the breakthrough experiment. 3.2. Effect of Number of Discrete Points. For an accurate representation of continuous data the number of discrete points is critical. In order to explore this, an arbitrary continuous isotherm

4. DETERMINATION OF DISCRETE EQUILIBRIUM DATA The feasibility of using discrete equilibrium data in breakthrough simulation was verified in the previous section. This section focuses on the estimation of discrete equilibrium data 14837



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specific nature of the equilibrium relationship as described below. For a favorable isotherm (type I), i.e., concave downward without inflection point, since qfeed * corresponds to the maximum possible value of c̃j, this value is set as the upper bound for all decision variables. In addition, it can be safely assumed that the following relationship is satisfied

from DCB experiments. The determination of discrete equilibrium data involves identifying a set of values for the equilibrium loading q̃j*, corresponding to a set of values for the fluid phase concentration c̃j, such that the breakthrough curve calculated using the discrete equilibrium data matches the experimental DCB at the column exit as closely as possible. In order to validate the method, the technique was first tested on synthetic data based on points extracted from a numerically simulated breakthrough curve. This curve was obtained on the basis of an assumed and smooth isotherm function. The breakthrough curves generated in this fashion are termed as “continuous” in the figures. 4.1. Problem Formulation. The determination of the discrete equilibrium data is cast as an optimization problem, where the objective is to choose a set of decision variables, q̃j*, to minimize the difference between the experimental breakthrough profile and that simulated using the discrete equilibrium data. The error between the two profiles, J, is defined as

qj*̃ ≥

out out (t ) − cdisc (t ) | ∑ |cexpt t=0

* qfeed c feed

cj̃

(14)

* cj̃ ≤ qj*̃ ≤ qfeed

(15)

Similar assumptions led to the following bounds for each decision variable for an unfavorable isotherm (type III), i.e., concave upward.

(12)

* qfeed

0 ≤ qj*̃ ≤

where t represents the time instances at which breakthrough concentrations are available. The breakthrough profile obtained using continuous isotherm is the synthetic data that replaces the experimental data in this methodology development phase of the study. For single component experiments, since no enrichment beyond the feed concentration is possible, the maximum fluid phase concentration is that corresponding to the feed. Once this is determined, the interval c = 0, ..., cfeed is discretized into equally spaced intervals and an initial set of q̃j* is chosen. In order to arrive at the optimal results faster, it is important to narrow down the search space by choosing appropriate bounds for the decision variables. These bounds depend on the type of equilibrium, which can be broadly identified from the shape of the breakthrough curve. For example, the response of a system following a favorable isotherm (type I) to a long pulse is characterized by a delayed sharp rise in concentration up to a plateau corresponding to the feed concentration followed by a smooth transition. An opposite trend is observed for a system following an unfavorable isotherm (type III). Breakthrough profiles characteristic of various isotherms are shown in Figure 3. The value of qfeed * corresponding to the feed composition can be calculated from eq 1, which follows from a simple mass balance of the chromatographic column subject to a step change of the input. It is worth noting that the value of qfeed * is required in an approximate manner. The accuracy with which it can be calculated does not affect the success of method described here. The choice of the bounds for the decision variables plays an important role in the solution of the optimization problem. On the one hand, providing large bounds can increase the computational time required to reach the optimum solution. On the other hand, providing a narrow range might limit the search space to find the right solution. For all the cases considered in this paper, the most general bounds can be represented by

* 0 ≤ qj*̃ ≤ qfeed

c feed

Hence, the lower bound for each decision variable is given by eq 14. Therefore, the bound for each decision variable is expressed as

tf

J=

* qfeed

c feed

cj̃

(16)

Finally, for isotherms with an inflection point (type II), the lower and upper bounds are as specified in eq 13. The search regions for the three isotherm types are shown by the shaded area in Figure 3. In order to further restrict the choice of q̃j* in each iteration, for all these types of equilibrium data discussed above, the thermodynamic constraint given by eq 10 is imposed. It is important to point out that in all case studies reported here, the solutions were also achievable without the constraints, i.e., eqs, 15 and 16, although at the expense of additional computational time. In this study, the inbuilt MATLAB optimization routine f mincon with sequential quadratic programming (SQP) algorithm was used. This routine is based on a gradient search and hence requires an initial guess for the decision variables. While, in principle, any initial guess that satisfies the above constraints can be used, we attempted using the following simple initial guess in all the case studies: qj*̃ =

* qfeed c feed

cj̃

(17)

4.2. Recommended Protocol. In order to estimate the equilibrium data, a minimum of one experimental run is required. In order to cover the entire concentration range, this experiment could be performed by injecting a long pulse, i.e., inputs for both adsorption and desorption, and measuring the corresponding response. Since the determination of the search space requires, at least approximately, the value of q*feed, the pulse should be long enough so that the feed concentration is observed in the elution profile. Note that the conversion of the detector signal to concentration units is required and can be performed by using suitable calibration function.13,14 If one expects to obtain the equilibrium data over a different range of the concentration, other experiments can be performed and the results analyzed together as discussed below. The technique described here is essentially a dynamic measurement and like any other technique is sensitive to the accuracy with which parameters such as flow rates and bed voidages can be measured.

(13)

However, in order to reduce the computational time, it is worth narrowing down the search space by exploiting the 14838



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The models required for simulation are described in section 2.2. The section also discusses how the LDF model can be used to describe the mass transfer effects. All the case studies described below assume that the mass transfer can be represented by a lumped mass transfer coefficient, k. However, it is important to clarify that the idea of obtaining discrete equilibrium data by inverting breakthrough response is not dependent on the use of the LDF model. It is entirely possible to include a full concentration dependent diffusion model to the simulator, if necessary.

Table 2. Isotherms Used To Generate Simulated Breakthrough Profiles

5. CASE STUDIES In order to validate the technique, several case studies were considered. First, the effectiveness of the technique was tested using synthetic elution profiles where the experimental elution profiles were substituted with simulated elution profiles from a known isotherm. In all the cases, no information of the known isotherm was provided to the optimizer. The effect of isotherm types (case 1), the effect of initial guess (case 2), and the possibilities of estimating discrete equilibrium data from multiple sets of elution profiles (case 3) and from a short pulse (case 4) were investigated. The values of various parameters used for the case studies are listed in Table 1.

of those from the continuous form. Further, the estimated values of the mass transfer coefficient for types I, II, and III are 0.09998, 0.100005, and 0.09991 s−1, respectively, which are in excellent agreement with the value assumed to generate the continuous elution profiles. It is also worth noting that in all cases convergence was attained with the initial condition provided by eq 17. The exercise was repeated for isotherms with different nonlinearities, and in all cases excellent agreement was seen between the estimated discrete equilibrium data and the assumed continuous isotherm. It has been further confirmed that the proposed inversion method works equally well when there is significant spreading of the mass transfer front due to large mass transfer resistance and axial dispersion. In these studies, the mass transfer and axial dispersion coefficients are also treated as decision variables, and the optimizer is able to accurately determine them, in addition to accurately estimating the discrete equilibrium data. 5.2. Case 2. Effect of Initial Guess. The optimization algorithm used in this study is based on a gradient search technique. Hence, it is important to ensure that it is not sensitive to the initial guess. In order to verify this, the optimization runs for all cases discussed in section 5.1 were repeated by varying the initial guess. Figure 4 shows representative results from the study. For the case of type I and III isotherms, the initial guesses were chosen to have the general shape of the isotherm but also to enforce that they pass through the origin and (cfeed, q̃*feed). From the figure, it is clear that the optimizer converged to the same solution irrespective of initial guess for all the three isotherm types. The case of the type II isotherm (Figure 4e) is worth noting. In this case, although none of the initial guesses had the shape of the eventual solution, the optimizer invariably converged at the correct solution. This demonstrates the robustness of the technique. 5.3. Case 3. Multiple Sets of Elution Profiles. While theoretically it would be sufficient to perform one breakthrough experiment corresponding to the maximum desired concentration, equilibrium data obtained from breakthrough experiments conducted at multiple concentrations can be beneficial for improved data reliability. The optimization technique can be extended to include multiple breakthrough experiments by modifying eq 12 as

isotherm type

parameter

value 25 0.46 0.66 0.0083 4.98 × 10−5 50 200 0.5

type II

5.1. Case 1. Isotherm Types. Three types of isotherms, favorable, unfavorable, and an isotherm with inflection point, were studied. For each simulated experiment, the elution profile for known values of cfeed and tinj was calculated by using a known continuous isotherm. The only parameter that was required to define the search region, the equilibrium loading corresponding to the feed concentration, q*feed, was obtained from the elution profile from eq 1. Once this value was measured, the 50 discrete points were distributed evenly according to eq 17. In addition, the possibility of determining the mass transfer coefficient simultaneously with the equilibrium data was studied. The breakthrough curves obtained from the continuous isotherms were simulated using k = 0.1 s−1. In the optimization problem, in addition to q̃*j , the mass transfer coefficient, k, was treated as an additional decision variable. The bounds of k were

0.001 ≤ k ≤ 1

12c q* = 1 + 1.2c

type III

Table 1. Parameters Used for Case Studies column length, l (cm) column diameter, d (cm) column void fraction, ε flow rate, Q (cm3 s−1) axial dispersion, DL (cm2 s−1) number of discrete equilibrium data, N number of grid points time interval (s)

isotherm for case studies

type I

(18)

n

and a value of k = 1 was chosen as an initial guess for all cases. The isotherms used to generate the synthetic breakthrough profiles are detailed in Table 2. The results of the case study for the three different isotherm shapes is shown in Figure 4. The figure shows the comparison of the elution profiles along with the estimated discrete equilibrium points. It can be seen that the equilibrium data obtained in all three cases are in excellent agreement with the assumed isotherm. The values of q̃j* were always within ±0.5%

J̅ =

⎛

tf

400

q* =

0.16c 1 − 0.08c

2000

q* =

9c (1 − 0.05c)(1 − 0.05c + 0.9c)

1200

⎞

out out ⎟ ∑ λi⎜⎜∑ |cexpt, i(t ) − cdisc, i(t )|⎟ i=1

⎝ t=0

tinj (s)

⎠

(19)

where λi is the weight that is assigned to each experiment. In order to validate the technique, the example of a type I isotherm reported in section 5.1 was chosen. In this case study, three elution profiles obtained by changing the feed concentration at cfeed = 5, 7, 10 g L−1 were considered. A value of λi = 1 was assigned, in order to allocate equal weights 14839



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Figure 4. Effect of initial guess on the estimation of discrete equilibrium data. For panels a, c, and e, the closed and open symbols represent the initial guess and the corresponding estimated discrete equilibrium data, respectively.

connecting the feed state, c = cfeed, with c = 0 is given by the following expression:1,15

for all three experiments. The equilibrium data obtained by simultaneously fitting three elution profiles is shown in Figure 5. These results show that the technique can also be applied to multiple experiments simultaneously. 5.4. Case 4. Short Pulse Inputs. All the previous case studies considered the situation where the pulse was long enough so that the feed state can be measured at the column outlet. In this section, we investigate the effectiveness of the technique for the case of a short pulse input, one in which the feed concentration is not reached in the elution profile. For the case of type I isotherms, the retention time of the sharp front

τ=

⎡ * ⎤ L⎢ 1 − ε qfeed ⎥ 1+ u ⎢⎣ ε c feed ⎥⎦

(20)

In the case of a pulse injection where the feed state is eroded, i.e., if tinj is not long enough, the shock from the adsorption (connecting c = 0 and c = cfeed) step interacts with the simple wave from the desorption (connecting c = cfeed and c = 0) step. This interaction causes the shock to decelerate as it travels 14840



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Figure 5. Determination of discrete equilibrium data from multiple sets of elution profiles tinj = 400 s. Figure 6. Determination of discrete equilibrium data from elution profile with short pulse input.

15,16

through the column. Under these situations, it is possible to obtain a reasonable estimate of the upper bound of q*feed by rearranging eq 20: * = qfeed

εc feed ⎡ uτ ⎤ − 1⎥ ⎢ ⎦ (1 − ε) ⎣ L

6. EXPERIMENTAL VALIDATION OF THE TECHNIQUE The technique developed to obtain discrete equilibrium data from the DCB elution profile has been tested with synthetic breakthrough data in sections 5.1−5.4. The true test of the technique requires experimental validation. In the following, three experimental elution profiles, reported in the literature, are considered. In system 1, the experimental elution profile of 2-phenylethanol (PE) on a reversed-phase material (Kromasil 100−5C18, dp = 10 μm, Eka Nobel, Sweden), reported by Lisec et al., is considered.6 In system 2, the experimental elution profile of phenol on a Gemini-C18 column (Phenomenex, Torrance, CA) is studied.17 The experimental parameters are summarized in Table 3. Due to the sharp breakthrough profiles in these two case studies, an arbitrarily low value of axial dispersion, DL = 4.0 × 10−5 cm2 s−1, was chosen. In addition,

(21)

Note that using the retention time of the sharp front from the pulse response in the above equation will lead to an overestimation of the q*feed. The value hence obtained can be used as an upper bound for the decision variables. The general search space given by eq 13 can be employed to estimate the discrete equilibrium data. In order to illustrate the procedure, a type I isotherm with high nonlinearity is considered, but now with a smaller value of injection time, tinj = 130 s. The results of the estimation and the corresponding elution profile are shown in Figure 6. It is observed that the estimation of the equilibrium data is good up to c ≈ 3.3 g L−1 but shows deviation for higher concentrations. It is worth noting that this compares well with the maximum concentration measured at the column outlet. It is well-known that under these situations the equilibrium data, no matter the technique by which it is measured, can be used reliably only up to the maximum concentration that is observed in the elution profile. It is also worth emphasizing that the same solution can be obtained by using a conservative upper bound, and using a bound provided by eq 21 only helps to minimize computational effort.

Table 3. Parameters for the Experimental Case Studies

14841

parameter

system 16

system 217

system 318

column length, L (cm) column diameter, d (cm) column void fraction, ε flow rate, Q (cm3 s−1) feed concentration, cfeed (g L−1) injection duration, tinj (s)

25 0.46 0.54 0.017 20.3 308.62

15 0.46 0.6978 0.017 160 207.97

25 0.46 0.77 0.017 4.124 60, 30



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Figure 7. Determination of discrete equilibrium data from experimental elution profile reported by Lisec et al. for system 1. Experimental parameters are given in Table 3.

Figure 8. Determination of discrete equilibrium data from experimental elution profile reported by Gritti and Guiochon for system 2. Experimental parameters are given in Table 3.

Figure 9. Determination of discrete equilibrium data from experimental elution profile reported by Cherrak et al. for system 3. Experimental parameters are given in Table 3.

due to the small particle size, 2000 grid points were used in the * was calculated using eq 1 and set simulation. The value of qfeed as an upper bound for all the decision variables. The general search space (Figure 3c) for the optimizer was defined

accordingly. In addition, the mass transfer coefficient was treated as decision variable, and bounds were set as 0.01 ≤ k ≤ 10. The values of the q̃* were determined by the optimizer to minimize J as given by eq 12. The discrete equilibrium data 14842



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ACKNOWLEDGMENTS Funding from A*STAR, under a thematic research program on carbon capture and utilization, is acknowledged. We thank Andreas Seidel-Morgenstern, Otto von Guericke University, for information regarding the experimental aspects reported in section 6.

obtained with the corresponding elution profile for systems 1 and 2 are shown in Figures 7 and 8, respectively, along with the points obtained from multiple frontal analysis experiments as reported in these studies. The value of mass transfer coefficient obtained by the optimizer for systems 1 and 2 are k = 3.38 and 7.76 s−1, respectively. Clearly, discrete equilibrium data that are obtained from each elution profile shows good agreement with the corresponding data obtained from multiple frontal analysis experiments. Figures 7b and 8b also demonstrate that the elution profiles calculated using the discrete equilibrium data show good match with the experimental ones. In system 3, the effect of short pulse while using multiple elution profiles is validated by the experimental elution profile of S-3-chloro-1-phenyl-1-propanol (S-CPP) on Chiracel OB-H cellulose tribenzoate coated on silica gel substrate column (Daicel, Tokyo, Japan), reported by Cherrak et al.18 From this study two experimental elution profiles with different injection volumes were utilized in the optimization problem. The experimental parameters are listed in Table 3. The upper bound for the decision variables is calculated by eq 21 and general search space defined for the optimizer. The same bounds used in systems 1 and 2 were specified for the mass transfer coefficient. The discrete equilibrium data obtained with the corresponding elution profile are shown in Figure 9a. It is important to reemphasize that the discrete equilibrium data are reliable up to the maximum concentration in the elution profile c ≈ 3.4 g L−1. The value of mass transfer coefficient obtained by the optimizer is k = 8.33 s−1. Note that the values of mass transfer coefficient obtained by the optimizer for these three cases are very high, indicating that negligible mass transfer resistance is expected in chromatographic systems employing small-sized particles. The elution profiles from the discrete equilibrium data shown in Figure 9b are in good agreement with the experimental profiles.

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NOMENCLATURE c fluid phase concentration of solute (g L−1) c̃j specified fluid phase concentration (g L−1) DL axial dispersion coefficient (cm2 s−1) d column diameter (cm) J optimization objective function (g L−1) J ̅ optimization objective function with multiple elution profiles (g L−1) k mass transfer coefficient (s−1) L column length (cm) N number of discrete equilibrium data n number of elution profiles Q volumetric flow rate (cm3 s−1) q solid phase concentration of solute (g L−1) q* equilibrium solid phase concentration of solute (g L−1) q̃*j solid phase concentration of solute for a specified value of c̃j (g L−1) t time (s) tinj injection duration (s) u interstitial velocity (cm s−1) z axial coordinate (cm)

Greek Symbols

ε bed voidage λ weighting factor τ breakthrough mean residence time (s)

Subscripts and Superscripts

7. CONCLUSIONS A new technique has been proposed to obtain discrete equilibrium data from dynamic column breakthrough experiments and it has been implemented using an efficient algorithm. It has been tested with both synthetic and experimental data. This method achieves from one adsorption/desorption breakthrough run what requires several runs in frontal analysis chromatography by progressively increasing the sorbate concentration in the feed. The proposed method has the potential to offer a faster alternative to the elaborate and tedious gravimetric and volumetric methods of measuring equilibrium data. Although the application has been limited to isothermal liquid chromatographic system in this study, the technique in principle, can be extended to DCB runs for gas systems with significant velocity change and heat effect due to the adsorption and desorption of adsorbable component. These extensions will be addressed in a future paper.

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expt experiment disc discrete f final feed feed state i number of elution profiles init initial inj injection j discrete points index out outlet

REFERENCES

(1) Guiochon, G.; Felinger, A.; Katti, A. M.; Shirazi, S. Fundamentals of Preparative and Nonlinear Chromatography; Academic Press: Boston, 2006. (2) Guiochon, G. Preparative liquid chromatography. J. Chromatogr. A 2002, 965, 129−161. (3) Rajendran, A. Design of preparative−supercritical fluid chromatography. J. Chromatogr. A 2012, 1250, 227−249. (4) Rajendran, A.; Paredes, G.; Mazzotti, M. Simulated moving bed chromatography for the separation of enantiomers. J. Chromatogr. A 2009, 1216, 709−738. (5) Seidel-Morgenstern, A. Experimental determination of single solute and competitive adsorption isotherms. J. Chromatogr. A 2004, 1037, 255−272. (6) Lisec, O.; Hugo, P.; Seidel-Morgenstern, A. Frontal analysis method to determine competitive adsorption isotherms. J. Chromatogr. A 2001, 908, 19−34. (7) Guan, H.; Stanley, B.; Guiochon, G. Theoretical study of the accuracy and precision of the measurement of single-component isotherms by the elution by characteristic point method. J. Chromatogr. A 1994, 659, 27−41.

AUTHOR INFORMATION

Corresponding Author

*A.R.: tel, +65 6316 8813; fax, +65 6794 7553; e-mail, arvind@ ntu.edu.sg. S.F.: tel, +65 6516 6545; fax, +65 6779 1936; e-mail, [email protected] Notes

The authors declare no competing financial interest. 14843



Article

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Discrete Equilibrium Data from Dynamic Column Breakthrough

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