Standing Wave Design and Optimization of Nonlinear Four-Zone

Article pubs.acs.org/IECR

Standing Wave Design and Optimization of Nonlinear Four-Zone Thermal Simulated Moving Bed Systems Nicholas Soepriatna, N. H. Linda Wang,* and Phillip C. Wankat School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, Indiana 47906, United States S Supporting Information *

ABSTRACT: The standing wave design (SWD) and optimization method are developed for the first time for nonlinear thermal simulated moving bed (SMB) systems with significant mass-transfer effects. The design method was verified using computer simulations for two systems. The adsorption isotherms of fructose/glucose and p-xylene/toluene systems have weak and strong temperature dependence, respectively. The results show that nonlinear thermal SMB systems can simultaneously produce pure solvent as well as concentrated, high-purity products from dilute feeds. The condition that allows for production of pure solvent is determined from local equilibrium analysis. Optimization of ten decision variables to achieve minimum solvent consumption and maximum productivity takes less than 5 min using a laptop computer. The fructose/glucose thermal SMB system products are increased by >25% with solvent consumption decreased 2-fold. The p-xylene/toluene thermal SMB system product concentrations can be increased 10-fold while simultaneously producing pure solvents.

1. INTRODUCTION AND LITERATURE REVIEW Simulated moving bed (SMB) is an efficient separation process. It originates from the true moving bed (TMB) concept, which involves counter-current movement of both the solid phase (adsorbent) and the fluid phase. The SMB consists of a series of fixed beds connected to form a loop in which countercurrent movement of the two phases is simulated by intermittent switching of the inlet and outlet ports to follow the migrating bands (Figure 1a). The most common SMB system is a four-zone SMB. Each zone has a different role. Zone I regenerates the sorbent using a desorbent. Zones II and III separate A and B by adsorbing the slow moving B and desorbing fast moving A. Zone IV regenerates the desorbent that is then recycled back to zone I. The SMB mimics a TMB such that the counter-current flow in the SMB confines the mass-transfer zones (MTZ) inside the columns in the loop (Figure 2a). Two high-purity products (A and B) are drawn from the raffinate and extract product ports in regions where the two migrating bands do not overlap. SMB has a much higher sorbent utilization than batch chromatography because the total column packing length is similar to the mass-transfer zone length.1 Only partial separation of the migrating bands is needed to obtain high-purity products with high yield. Overlapping of the bands in the loop results in more concentrated products, higher sorbent productivity, and less solvent required for the separation compared to batch chromatography. Furthermore, feed input and product withdrawals are continuous and the sorbent and desorbent are automatically recycled.2,3 SMB has been extensively used in the petrochemical industry since its first successful commercialization by the petrochemical company UOP. The first use of SMB systems by UOP was for hydrocarbon separations in the Molex process to separate nparaffin from branched and cyclic hydrocarbons.4 The largest SMB application is UOP’s Parex process, separating p-xylene from its C8 aromatic isomers.5 Besides petrochemicals, SMB © 2015 American Chemical Society

systems are used for sugar separations including separations of sucrose from beet or cane molasses, 6 fructose from carbohydrate syrup, 7 and separation of fructose from glucose.8−12 SMB is also extensively used for purification of chiral pharmaceuticals and fine chemicals13−18 In current SMB processes for large-scale production, the desorbent composition and temperature are fixed. Many recent studies including gradient SMB operations aim to improve the SMB’s performance, such as product purity (Pu), yield (Y), productivity (PR), and solvent consumption (D/F). Gradient operations can be incorporated in SMB systems to vary the adsorption and desorption strength of the solutes via variation of temperature, pressure, or solvent composition. Such operations are more efficient because each SMB zone is operated under conditions optimal for its respective role, such as adsorption or desorption of solute. The focus of this study is on SMB systems with temperature gradients, or thermal SMB for short (Figure 1b). Compared to other gradient operations, temperature gradients have the advantage of not requiring the addition of other chemicals, which may become an issue when solvent recovery is expensive. Because the solute adsorption strength is usually weaker at higher temperature, the regeneration and desorption zones (I and II) should have higher temperatures to facilitate desorption, whereas the adsorption zones (III and IV) should have lower temperatures to facilitate adsorption. The few studies on thermal SMB systems show greatly improved SMB efficiency.19−24 The early studies of thermal SMB used jacketed columns to create temperature gradients19,20 but showed that the separation was limited by heat Special Issue: Doraiswami Ramkrishna Festschrift Received: Revised: Accepted: Published: 10419

April 6, 2015 June 28, 2015 July 2, 2015 July 2, 2015 DOI: 10.1021/acs.iecr.5b01296 Ind. Eng. Chem. Res. 2015, 54, 10419−10433

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Industrial & Engineering Chemistry Research

Figure 1. Cycle operation of a four-zone (a) isothermal SMB and (b) nonisothermal SMB. A is the fast moving solute (raffinate), and B is the slow moving solute (extract).

transfer. The use of heat exchangers to change the temperature outside the adsorbent bed solved the heat-transfer problem and showed that thermal SMB can reduce solvent consumption (D/ F) by 20−50% and enrich products by 2-fold with the same productivity.21,22 Thermal SMB concentrators have also been extensively studied.23,24 The most recent study on thermal four-zone SMB is a new thermal SMB system operation, called TSMB-FC.23 The withdrawal of pure solvent instead of the addition of desorbent to regenerate the adsorbent was introduced for linear systems. For feeds that consist of dilute solutes dissolved in solvent, this operation produces pure, concentrated extract and raffinate streams plus pure solvent. However, linear isotherm systems apply only to very low concentrations and can predict an unrealistic infinite product concentration. Several design methods to determine the operating parameters for isothermal SMBs have been developed. The most commonly used design method is to couple local equilibrium theory (triangle theory) with detailed process simulations. The local equilibrium theory or triangle theory was first developed for isocratic SMB systems and has since been used to search for the optimal operating parameters for thermal SMB.20 Local equilibrium theory delineates the region of operating parameters which can produce pure products for systems without any mass-transfer effects. When mass-transfer effects are significant, many simulation trials are required to determine the optimum operating parameters. Consequently, SMB design and optimization is both time-consuming and challenging. Another SMB design method is the standing wave design (SWD), which uses TMB assumptions for mass balances in the mobile and solid phases.16,17,23,25−30 Solving these mass balances at steady-state gives the optimum operating parameters (zone velocities and step time) for the SMB system. The SWD method developed for linear isotherms and nonlinear isotherms under isocratic conditions has been

Figure 2. Concentration profiles for nonlinear p-xylene/toluene fourzone SMB at the end of a step under (a) isothermal condition (Tj = 20 °C) and (b) nonisothermal condition (TI = 40 °C, TII = 36 °C, TIII = 34 °C, TIV = 30 °C).

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DOI: 10.1021/acs.iecr.5b01296 Ind. Eng. Chem. Res. 2015, 54, 10419−10433

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ideal systems are used as the initial guess to speed up the optimization for nonideal systems by 40-fold. Optimization of 10 decision variables (four zone temperatures, four zone velocities, step time, and feed flow rate) for minimum solvent consumption and the maximum productivity for the optimal zone temperature can be done within 5 min using a laptop computer.

extensively tested using computer simulations and experiments.25−30 If the temperature, yields, equipment parameters, and material properties are fixed, the SWD method finds, without any simulation, the optimum operating parameters (zone velocities, step time) for maximum productivity and minimum solvent consumption. The equipment parameters include column length, zone lengths, extra-column dead volume, and pressure limit. The material properties include size exclusion factors, particle size, bed void fraction, particle porosity, adsorption isotherms, and intraparticle diffusivities. The SWD method guarantees high product purity and high yield for systems with mass transfer effects. The SWD method was also successfully developed and experimentally tested for SMB systems involving three or more solutes.16,31−33 The SWD equations of thermal SMB systems with linear isotherms were recently developed and validated with computer simulations.23 However, most large-scale production operates with high loading where adsorption isotherms are nonlinear. Therefore, it is important to develop an efficient design and optimization method for nonlinear thermal SMB systems. There are three major objectives of this study. 1. Derive, solve, and verify SWD equations for nonlinear, thermal SMB systems. 2. Develop an optimization method based on the SWD solutions to determine the optimum zone temperatures that minimize solvent consumption (D/F) and maximize adsorbent productivity for the optimum temperatures. 3. Determine the effects of isotherm parameters and their temperature dependence on retention factors (δi,j), solvent consumption (D/F), and product enrichment (EF). In this study, the SWD equations are derived from the steady-state solute mass balance in the mobile and solid phases based on TMB assumptions and then solved for systems with no mass-transfer effects (ideal) and systems with mass-transfer effects (nonideal). If the yield of each solute, the feed flow rate, four zone temperatures, equipment parameters, and material parameters are specified, solutions of the SWD equations determine the five operating parameters (zone velocities and step time) that give the minimum D/F. The SWD solutions for thermal SMB systems are verified using Aspen Chromatography simulations. The results show that SWD and Aspen Chromatography give the same D/F and product concentrations for both isothermal and thermal SMB systems. The method for finding the optimal zone temperatures to achieve minimum solvent consumption (D/F), at a fixed feed flow rate, is developed by incorporating the verified SWD equations for thermal SMB systems into a temperature optimization algorithm. The SWD optimization solutions for ideal systems are used as the initial guesses for the temperature optimization for nonideal systems. SWD solutions for ideal systems are also used to find the conditions which allow the thermal SMB to produce pure solvent. For the optimal zone temperatures, the maximum productivity (PR) and the trade-off curve between D/F and PR are obtained using the productivity optimization algorithm. Three major findings are worth noting. First, the SWD and optimization method for nonlinear thermal SMB systems are developed for the first time and verified using detailed simulations. Second, the conditions allowing a thermal SMB to produce pure solvents and produce pure concentrated products are found. Third, SWD optimization solutions for

2. THEORY 2.1. Nonlinear Adsorption Isotherms with Temperature Dependence. Unlike linear isotherm systems where dilute feeds are used and adsorption of solutes is independent of concentrations, most large-scale production operates at high loading solute concentrations. Assuming a monolayer adsorption, two solutes in a binary system compete for the adsorption sites and the concentrations of the solutes in the adsorbed phase and in the solution phase at equilibrium are dependent on the concentrations of both solutes. The multicomponent Langmuir isotherm for competitive adsorption is qi =

qmax, iK (T )i , j Ci B

1 + ∑A K (T )i , j Ci

;

i = A, B (1)

where the subscripts A and B denote the fast moving and the slow moving solutes, respectively; qmax,i is the maximum adsorption capacity of solute i, Ci the concentration of solute i in the liquid phase, and K(T)i,j the temperature-dependent equilibrium constant of solute i at the temperature in zone j, which follows the Arrhenius relation ⎛ ΔH ⎞ ⎟⎟ K (T )i , j = K∞ , i exp⎜⎜ − ⎝ RTj ⎠

(2)

Although the multicomponent Langmuir isotherm is known to violate thermodynamic constraints if qmax,A ≠ qmax,B,34 it is commonly used for SMB analysis. 2.2. Standing Wave Design Overview for a Four-Zone Thermal SMB. The derivation of the SWD equations for nonlinear (multicomponent Langmuir) isotherms is similar to that of the linear isotherms23 except for the added complexity from the nonlinear isotherm equations. The SWD equations for systems with multicomponent Langmuir isotherms are derived in detail starting from the ideal systems (negligible masstransfer and dispersion effects) and then proceeding to nonideal systems by Soepriatna.35 2.2.1. Ideal Systems (with No Mass-Transfer Effects). The four-zone thermal SMB setup can be referred to Figure 1b.23 The main idea of the standing wave design concept is to set the port velocity equal to a key solute wave velocity in each zone such that the concentration waves appear to be standing to an observer moving with the ports.25,26 u w,j i = us,j i − u port = 0

(3)

where ujw,i is the concentration wave velocity. The desorption waves of the slow (B) and fast (A) moving solute can be standing in zones I and II, respectively. Similarly, the adsorption waves for solutes B and A are standing in zones III and IV, respectively. From this analysis, the design constraints of the solute velocity in each zone is defined as follows: I us,B = u port

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(4a) DOI: 10.1021/acs.iecr.5b01296 Ind. Eng. Chem. Res. 2015, 54, 10419−10433

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Industrial & Engineering Chemistry Research II us,A = u port

(4b)

III us,B = u port

(4c)

IV us,A = u port

uoIV = [1 + PδA,IV ]u port

where δi,j is the retention factors defined by δ B,I = εp + (1 − εp)qmax,BK (T )B,I

(4d)

From eq 4, the standing concentration waves in the fourzone thermal SMB are desorption of solute B in zone I, desorption of solute A in zone II, adsorption of solute B in zone III, and adsorption of solute A in zone IV. These standing wave profiles for both an isothermal and nonisothermal four-zone SMB with mass-transfer and dispersion effects are shown in Figure 2. The concentration profile of the thermal SMB system is slightly different from the isothermal system in which the concentration wave plateaus (Cp,A and Cp,B) around the extract and raffinate ports are higher in thermal SMB. This is expected because a thermal SMB system enables one to acquire a set of operating conditions where the desorbent flow rate is less than the feed flow rate (D/F < 1). The standing wave profiles for a non-ideal system (Figure 2b) show that the solute B desorption wave is confined in zone I, the plateau concentration of solute B and the solute A desorption wave are in zone II, the adsorption wave of solute B and the plateau concentration of solute A are in zone III, and the adsorption wave of solute A is in zone IV. By confining the key waves in the respective zones, one can control the purities of the raffinate and the extract. To obtain high purity products, the key waves should not spread beyond the respective zones. The mass balance equations, which control the wave velocities, are ∂C bji ∂ 2C j ∂C j = E b,j i 2bi − uo̅ j bi − Pke,j i(C bji − Ci*) ∂t ∂x ∂x

(5a)

uo̅ j = uoj − u port

(5b)

εp

(7d)

δA,II = εp + (1 − εp)

uoII = [1 + PδA,II]u port

(7b)

uoIII = [1 + Pδ B,III]u port

(7c)

1 + K (T )B,II Cp,B

(8b)

qmax,BK (T )B,III 1 + K (T )A,III Cs,A + K (T )B,III Cs,B (8c)

δA,IV = εp + (1 − εp)

qmax,A K (T )A,IV 1 + K (T )A,IV Cp,A

(8d)

The port velocity is determined using the mass balance around the feed port, and the result is u port =

QF SεeP(δ B,III − δA,II)

(9)

Solving for the zone velocities in eq 7 is straightforward if the plateau concentrations are known (Cs,A, Cs,B, Cp,A, and Cp,B). However, the plateau concentrations are not known a priori as their values depend on the zone velocities and feed concentrations. Therefore, an iterative procedure is required. This procedure is similar to the ones developed in the literature26−28 except that a simpler procedure for estimating the new plateau concentrations, Cp,A and Cp,B, is introduced. From Figure 2, the outlet concentrations of the raffinate and extract ports roughly corresponds to the plateau concentrations Cp,A and Cp,B, respectively. Thus, the Cp,A and Cp,B are estimated using a simple mass balance between the feed port and its respective outlet port (raffinate for Cp,A and extract for Cp,B). The step-by-step iterative procedure is described in detail in section S1 in the Supporting Information. 2.2.2. Nonideal Systems (with Mass-Transfer Effects). Under nonideal conditions, the mass-transfer and dispersion terms in eqs 5 and 6 cannot be ignored because these terms cause concentration wave spreading. The nonideal SWD equations are basically an ideal SWD but with the addition of correction terms to compensate for the wave spreading. After considerable analysis, the nonlinear, nonideal SWD equations are

where Cjbi and C*i are the concentration of solute i in the mobile and solid phase, respectively; Ejb,i is the axial dispersion coefficient of solute i in zone j; ujo̅ is the interstitial velocity of the mobile phase in the axial direction in zone j; ujo is the velocity in zone j that controls the propagation of the concentration waves relative to the solid phase; P is the bed phase ratio defined by (1 − εe)/εe; εe is the bed void fraction; kje,i is the mass-transfer coefficient of solute i in zone j; uport is the port velocity; and εp is the intraparticle void fraction. For ideal systems, the mass-transfer and dispersion terms in eqs 5 and 6 are ignored. Assuming that the solute concentration in the mobile phase is in equilibrium with that in the solid phase, the steady-state solutions for the zone velocities with ideal systems are (7a)

qmax,A K (T )A,II

δ B,III = εp + (1 − εp)

∂q * ∂C * ∂Ci* + (1 − εp) i = ke,j i(C bji − Ci*) + u portεp i ∂t ∂x ∂t ∂qi* + (1 − εp)u port (6) ∂x

uoI = [1 + Pδ B,I]u port

(8a)

⎡ EI Pδ B,I 2u port 2 ⎤ b,B ⎥ uoI = [1 + Pδ B,I]u port + βBI⎢ I + I I ⎢⎣ L ke,B L ⎥⎦

(10a)

⎡ E II PδA,II 2u port 2 ⎤ b,A ⎥ uoII = [1 + PδA,II]u port + βAII⎢ II + II II ⎢⎣ L ⎥⎦ ke,A L

(10b)

⎡ E III Pδ B,III 2u port 2 ⎤ b,B ⎥ uoIII = [1 + Pδ B,III]u port − βBIII⎢ III + III III ⎢⎣ L ⎥⎦ ke,B L (10c)

⎡ E IV PδA,IV 2u port 2 ⎤ b,A ⎥ uoIV = [1 + PδA,IV ]u port − βAIV ⎢ IV + IV IV ⎢⎣ L ⎥⎦ ke,A L (10d) 10422


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Industrial & Engineering Chemistry Research where δi,j is defined in eq 8 and βji is dependent on the plateau concentrations ⎛ C* ⎞ s,A ⎟⎟ βAII = ln⎜⎜ C ⎝ E,A ⎠

⎛ C E,B ⎞ ⎟⎟ βBI = ln⎜⎜ ⎝ C I,B ⎠

(11a,b)

⎛ CR ,A ⎞ ⎟⎟ βAIV = ln⎜⎜ ⎝ C IV,A ⎠

⎛ Cs,B ⎞ ⎟⎟ βBIII = ln⎜⎜ ⎝ CR ,B ⎠

larger than needed and will ensure the design achieves the targeted purity and yield. The axial dispersion coefficient is determined using the Chung−Wen correlation.37 The port velocity, uport, equation is determined using the mass balance between the feed port, zone II, and zone III as the control volume, and solving for uport. QF εeS

(11c,d)

C E,A = (1 − YA )

YAC F,AQ F

C R,A =

QR

C IV,A =

* = Cs,A

QI Q IV

C E,A

QE

QE

C I,B =

Q IV QI

C F,BQ F

II II ke,A L

C R,B

+

βBIIIPδ B,III 2 III III ke,B L

(13c)

b = −P[δ B,III − δA,II] (12c,d)

c=

* = Cs,B

βAIIPδA,II 2

a=

(12a,b)

QR

(13b)

where

YBC F,BQ F

C R,B = (1 − YB)

Cs,AQ III − C F,AQ F Q II

C E,B =

(13a)

au port 2 + bu port + c = 0

The concentrations in eq 11 are determined using the solute mass balances as follows:28 C F,AQ F

= uoIII − uoII

(12e,f)

QF εeS

+ βAII

II E b,A

LII

(13d)

+ βBIII

III E b,B

LIII

(13e)

εeSujo,

The step time is defined by tsw = LC/uport, Qj = and Lj is the zone length defined by Lj = NjLC, where Nj is the number of columns per zone. In the nonideal systems, the SWD equations have meaningful solutions only when

Cs,BQ III − C F,BQ F Q II

⎡ β IIPδ 2 β IIIPδ B,III 2 ⎤ A,II P 2[δ B,III − δA,II]2 − 4⎢ A II II + B III III ⎥ ⎢⎣ ke,AL ⎥⎦ ke,BL

(12g,h)

where the subscripts F, E, and R represent the feed, extract, and raffinate ports while subscripts I−IV denote the zone number; Cs,i * is the concentration of i before the feed port (Figure 2b). Note that there can be deviations of the system and operating parameters due to external disturbances or errors in the estimated parameters. Deviations in system parameters may include bed void fraction, extra-column dead volume, intraparticle diffusivity, isotherm constants, etc., while those in operating parameters include zone flow rates, step time, and zone temperatures fluctuations. A more robust SWD method which accounts for these deviations has been developed previously for isothermal SMBs.17,36 A similar approach can be applied to thermal SMBs. A systematic analysis can be performed to study the effects of the deviations on the wave velocities in each zone. Additional corrections for the zone velocities can be used to control the key wave velocities such that when the worst combination of parameter deviations occurs, the desired purity and yield can still be achieved. However, this analysis is beyond the scope of this study. The overall mass-transfer coefficient, kje,i, in eq 10 is the lumped mass-transfer coefficient based on the ΔC linear driving force. As the solute concentration increases, the viscosity increases, resulting in an increase in dispersion coefficient and a decrease in the overall lumped mass-transfer coefficient. On the other hand, an increase in temperature results in the opposite for the dispersion and mass-transfer coefficients. If the correlations between viscosity, dispersion, and mass-transfer coefficient versus solute concentrations or temperature are known, such dependence can be easily incorporated in the overall mass-transfer terms in the SWD equations. If they are not known, the mass-transfer coefficient and dispersion coefficient based on the lowest zone temperature can be used in the SWD equations to obtain conservative operating parameters. Because the zone velocity corrections are based on a smaller mass-transfer coefficient and a larger dispersion coefficient than the actual values, the velocity corrections are

⎡Q E II E III ⎤ ⎢ F + βAII b,A + βBIII b,B ⎥ ≥ 0 ⎢⎣ εeS LII LIII ⎥⎦

(14)

The plateau concentrations, Cs,A, Cs,B, Cp,A, and Cp,B, along with the decay coefficients, βji, depend on the zone flow rates, Qj, and feed concentrations. Thus, an iterative procedure is developed for solving the SWD equations for thermal SMB systems and is described in detail in section S2 in the Supporting Information. 2.3. Purity, Productivity, Solvent Consumption, and Enrichment Factor. The purity (Pu) of the outlet port concentrations is related to the specified yields of its respective solute by the following equations: PuE,B = PuR,A =

C E,B C E,B + C E,A C R,A C R,A + C R,B

=

=

YBC F,B YBC F,B + (1 − YA )C F,A

(15a)

YAC F,A YAC F,A + (1 − YB)C F,B

(15b)

If the feed concentrations and yield of both solutes are the same, as in the examples in this study, the product purity has the same values as the yield. Productivity (PR) is defined as the amount of product mass (in kilograms) that can be produced per volume of adsorbent per unit of time. The solvent consumption is denoted as D/F and is defined as the volume of solvent required per volume of feed. PR =

YQ i FC F, i SLC(NI + NII + NIII + NIV )

(u I − u IV ) D = oIII o II F (uo − uo ) 10423

(16)

(17) DOI: 10.1021/acs.iecr.5b01296 Ind. Eng. Chem. Res. 2015, 54, 10419−10433

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Industrial & Engineering Chemistry Research where Yi is the yield of solute i, QF the feed flow rate, CF,i the feed concentration of solute i, S the column cross-sectional area, LC the column length, and Nj the number of columns in zone j. The product enrichment factors (EF) are defined as the concentration ratios between the enriched product and the feed concentrations. EFR,A =

EFE,B =

C R,A C F,A C E,B C F,B

= YA

= YB

QF QR QF QE

= YA

= YB

equations. Substituting eq 7 into eq 17 and rearranging, the resulting equation is δ B,I − δA,IV D = F δ B,III − δA,II

D/F can be reduced by either reducing the difference (δB,I − δA,IV) in the numerator or increasing the difference (δB,III − δA,II) in the denominator. The retention factors (δi,j) are dependent on zone temperature because they are directly related to the adsorption equilibrium constant (K(T)i,j) (eqs 2 and 8). The δB,I is small for high zone I temperature (TI), and δA,IV is large for a low zone IV temperature (TIV). The denominator term, on the other hand, is maximized when zone III temperature (TIII) is low or zone II temperature (TII) is high. Therefore, the D/F of thermal SMBs is strongly affected by the zone temperatures and can be minimized by setting TI and TII at temperatures higher than TIII and TIV. The effect of zone temperatures on D/F is described in detail in section S4 in the Supporting Information. Aside from the zone temperatures (Tj), solvent consumption D/F is also affected by the isotherm nonlinearity. The nonlinear effects on D/F are caused by the (1+K(T)i,jCi) terms embedded in the definition of the retention factors (eq 8). As the nonlinear terms (1+K(T)i,jCi) increase, the retention factors in zones II, III, and IV become smaller. This is especially true for the retention factor in zone III (δB,III), which contains the nonlinear terms of two solutes in the binary system. As K(T)i, or the feed concentration CF,i increases, the retention factor δB,III and the denominator in eq 21 decrease, resulting in a smaller feed flow rate (F). The δA,IV in the numerator of eq 21 also decreases, resulting in a larger solvent flow rate (D). Therefore, the solvent consumption D/F increases with increasing K(T)i or CF,i. Increasing mass-transfer effects (or nonideality) also increases D/F. Equation 10 shows that mass-transfer effects IV cause uIo and uIIo to increase, and uIII o and uo to decrease, resulting in increasing D/F, according to eq 17. The D/F and EF are related to each other in thermal SMB systems. As D/F is decreased, less solvent is used and there is less product dilution, resulting in an increase in product concentration or EF. A more detailed discussion for the relationship between D/F and EF is described in section S5 in the Supporting Information. 2.5. Temperature Optimization. For an isothermal SMB with fixed yields, equipment parameters, and material parameters, the important variables that need optimization are feed flow rate, four zone velocities, and step time, which totals to six decision variables. In thermal SMB systems, there are four more major decision variables, the four zone temperatures, resulting in 10 major decision variables for optimization. Therefore, a fast and efficient method for optimizing all the decision variables is needed. Each zone in thermal SMB systems can be operated at the temperature that is optimal for its respective role. For a system with exothermic adsorption (ΔH < 0), zones I and II in Figure 2 have the role of regenerating the adsorbent and recovering the extract product by desorption of solutes. Thus, the temperatures of zones I and II are higher than those of zones III and IV, which have the role of adsorption. The SWD greatly simplifies SMB optimization because it allows direct determination of the zone velocities and step time that give a minimum D/F. SWD solves the optimization problem for five of the operating parameters (four zone

QF εeS(uoIII − uoIV )

(18a)

QF εeS(uoI − uoII)

(18b)

The solvent consumption (D/F) is assigned as the objective function for the optimization procedure. Because PR is directly proportional to the feed flow rate, QF, when the column specifications are constant, eq 16, an increase in QF results in an increase in PR. The maximum QF can be estimated by setting eq 14 equal to zero and rearranging:

Q F,max

⎡ ⎤ 2 II III ⎥ ⎢ P 2[δ E E b,B B,III − δA,II] b,A =⎢ − βAII II − βBIII III ⎥εeS βBIIIPδ B,III 2 ⎤ ⎢ ⎡ βAIIPδA,II2 L L ⎥ + III III ⎥ II II ⎢⎣ 4⎣⎢ ke,A ⎥⎦ L ke,BL ⎦ (19)

Equation 19 gives the maximum QF and thus the maximum PR, which is limited by the mass-transfer efficiency. If there is a pressure drop limitation, the maximum productivity may be smaller than the value given in eq 19. Furthermore, both the β and the δ terms in eq 19 vary with zone velocities and zone temperatures (eqs 11 and 12). Thus, eq 19 is incorporated in an iteration procedure (Figure S2 in the Supporting Information) to find the maximum feed flow rate, which is described in detail in section S3 in the Supporting Information. 2.4. Analysis of Key Factors Affecting Solvent Consumption or Production. For a fixed QF, D/F is a linear function of the difference between the zone I and zone IV velocities. The adsorbent in zone I can be regenerated with less solvent in a thermal SMB than in an isothermal SMB. In thermal SMB, it is possible to withdraw high-purity solvent from the desorbent port, which occurs when D/F is negative (D/F < 0).23 The SWD solutions for ideal systems can be useful for estimating the condition for producing pure solvent from thermal SMB. According to eq 17, D/F < 0 can be I achieved when uIV o > uo. Because the denominator term is constant for a given QF, the condition needed for the thermal SMB to produce pure solvent (D/F < 0) is uoI − uoIV < 0

(20a)

Equation 20a can be expressed in terms of the retention factor parameters using eq 7. δ B,I − δA,IV < 0

or

δ B,I δA,IV

(21)

1), and the feed concentration is relatively low. The equipment and operating parameters, as well as the adsorption parameters and optimization bound for the fructose/glucose and p-xylene/toluene systems, are listed in Tables 1 and 2, respectively.38,39 In this study, the column configuration is fixed at two columns per zone as this is the standard configuration of fourzone SMB for low-pressure SMB systems. The SWD has been

4. RESULTS AND DISCUSSION 4.1. SWD Benchmarked with Aspen Chromatography. Many assumptions were used in deriving the SWD for nonlinear thermal SMB systems. The initial guesses of the 10425


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Industrial & Engineering Chemistry Research Table 2. System, Operating, Adsorption, and Optimization Bound Parameters for p-Xylene−Toluene System Based on Matz and Knaebel39 System Parameters column

solid phase (silica gel)

liquid phase (n-heptane)

Input Parameters feed concentration yield Adsorption Parameters p-xylene

toluene

Optimization Bound initial feed flow rate temperature range

LC DC εe no./zone dp εp ρS Cps ρL CpL kf,PX kf,tol

30 cm 10 cm 0.43 2 0.0335 cm 0.5 1.05 g/mL 0.92 J/g/K 0.684 g/mL 2.2439 J/g/K 4.712 min−1 5.565 min−1

CF,p‑xylene CF,toluene Y

0.05g/L 0.05g/L 0.99

qmax K∞ ΔH R qmax K∞ ΔH R

51.72 g/L 1.84 × 10−6 L/g −30227 J/mol 8.314 J/K/mol 45.45 g/L 1.57 × 10−5 L/g −24323 J/mol 8.314 J/K/mol

QF Tj

10 mL/min 20−70 °C

Figure 3. SWD versus Aspen Chromatography simulation benchmark for fructose/glucose system: (a) isothermal SMB and (b) nonisothermal SMB (QF = 3.5 mL/min; TII = TIII = TIV = 30 °C).

new plateau concentrations, Cp,i, in eq S1 in the Supporting Information were estimated based on the Cs,i concentrations in zone III. The component mass balances, eq S3 in the Supporting Information, were used to improve the Cp,i, values. The EF values depend on the estimated C p,i and C s,i concentrations as indicated in eq 8 and eq S9 in the Supporting Information. To check the validity of the assumptions, the SWD-calculated purity (obtained from yield) and EF of the thermal SMB process were compared with the Aspen Chromatography results for both the fructose/glucose and pxylene/toluene cases. The feed concentrations of both solutes (CF,i) for the fructose/glucose and the p-xylene/toluene systems were 40 g/L and 0.05 g/L, respectively. These feed concentrations were chosen because the isotherm nonlinearity becomes apparent at these concentrations. Because the feed concentrations and yield of both solutes in the two examples are the same, the purity value is the same as the specified yield (eq 15). The purity (Pu) values from the SWD and Aspen Chromatography simulations agree to within 1% (data shown in Table S2 of section S9 in the Supporting Information). The EF benchmark results for the fructose/ glucose and the p-xylene/toluene systems are shown in Figures 3 and 4, respectively. The EF values for SWD and the Aspen Chromatography also agree to within 1% for both isothermal and nonisothermal operations. The results indicate that the SWD equations are reliable for designing the operating parameters and predicting the product concentrations and the EF values.

Under isothermal operation, it is expected that the EF value is less than 1 because the products are diluted by the solvent used to regenerate the adsorbent in zone I. On the other hand, the EF values for the extract products for both examples increase under the nonisothermal operation. In Figure 3b, the temperatures TII, TIII, and TIV are kept constant at 30 °C as TI increases. In Figure 4b, the temperature TII is 38 °C and the temperatures TIII and TIV are 34 °C. As TI increases, the slow moving solute B adsorbed in zone I needs less solvent to regenerate the adsorbent, resulting in higher extract product concentrations. The enrichment of the fast solute in the raffinate product is smaller than that of the slow solute in the extract because the majority of the fast solute is in zones III and IV, which have the same temperature for the examples in Figures 3 and 4. 4.2. Optimization Results and Analysis. The key results for both the fructose/glucose and the p-xylene/toluene systems are discussed in this section. First, eq 20b,c, the conditions allowing the production of pure solvent, is verified by changing the K∞ and ΔH values and examining the D/F values obtained from the SWD equations for thermal SMB systems. Second, at a fixed feed flow rate, the optimum zone temperatures for both examples are found using the temperature optimization procedure and the results are analyzed. Third, the maximum productivity (PR) is determined using the productivity optimization procedure and the trade-off between PR and D/ F is analyzed. Fourth, the effects of increasing the feed concentration on PR and D/F are shown and explained. 10426


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Figure 5. Effects of changes in isotherm shape (K∞ and ΔH) on adsorption temperature dependency (δB,I/δA,IV) and D/F performance for fructose/glucose thermal SMB system. ω is the multiplier constant for K∞ or ΔH (ω = 1 for the base case). Operating parameters: TI = 70 °C, TII = TIII = TIV = 30 °C, QF = 3.5 mL/min.

Figure 4. SWD versus Aspen Chromatography simulation benchmark for p-xylene/toluene system: (a) isothermal SMB and (b) nonisothermal SMB (QF = 10 mL/min; TII = 38 °C, TIII = TIV = 34 °C).

factors are controlled by zone temperatures and affect the zone velocities in the SMB system. As mentioned previously, the effects of zone temperatures on D/F for the fructose/glucose system is described in detail in section S4 in the Supporting Information. Figure 6 shows the optimization result that achieves the minimum D/F at a feed flow rate of 3.5 mL/min. The maximum and minimum temperatures for the optimization were set at 70 and 30 °C, respectively, because the isotherms reported in the literature are in this temperature range.38 The optimum zone temperatures are found to be as follows: TI = 70 °C and TII = TIII = TIV = 30 °C. The desorption wave of the

4.2.1. Fructose/Glucose System. In thermal SMB systems, the temperature dependence of the adsorption isotherm is most important. A lower D/F can be obtained with stronger adsorption temperature dependence. Two factors that control the adsorption isotherm shape are the pre-exponential factor (K∞) and the adsorption enthalpy (ΔH). According to eq 20b,c, an adsorption isotherm has a strong temperature dependence when δB,I/δA,IV < 1. The effects of these two factors on the isotherms for fructose/glucose system are shown in Figure S5 of section S8 in the Supporting Information. The effects of δB,I/δA,IV with changing K∞ and ΔH are shown in Figure 5. The symbol ω in Figure 5 is the multiplier constant which reduces or increases K∞ and ΔH. For the original adsorption isotherm obtained from the literature (base case), ω = 1.38 For ω > 1, K∞ or ΔH is increased from the base case, and K∞ or ΔH is decreased from the base case for ω < 1. The zone temperatures in Figure 5 are TI = 70 °C and TII = TIII = TIV = 30 °C with QF = 3.5 mL/min. As shown later, these temperatures are the optimum zone temperatures that give the minimum D/F for the fructose/glucose system. As expected, decreasing K∞ or ΔH results in a lower δB,I/δA,IV or stronger temperature dependence, leading to a decrease in D/F. Notice that the δB,I/δA,IV values remain relatively constant when ω < 1 in Figure 5b. Further decrease in ω results in the reversal of the selectivity of fructose versus glucose (not shown). The minimum δB,I/δA,IV value is greater than 1; thus, the system is unable to produce high-purity solvent. Figure 5 indicates that the retention factors play a very important role in D/F of thermal SMB systems. Based on eqs 7, 8, and 21, the retention

Figure 6. Temperature algorithm optimization result for fructose/ glucose thermal SMB system. QF = 3.5 mL/min; TI = 70 °C, TIV = 30 °C. 10427


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conversely, a decrease in zone IV velocity, resulting in an increase in D/F as expected from eq 17. For isothermal systems, the optimal zone temperature for minimal D/F is the limiting temperature 30 °C (Table S2a in Supporting Information). The results are compared with those of the optimal thermal SMB systems. As expected, the optimal isothermal SMB has a D/F higher than that of the optimal thermal SMB. The difference in D/F between ideal and nonideal isothermal SMB is larger than that between ideal and nonideal thermal SMB, indicating larger mass-transfer effects in isothermal systems. Notice that the D/F for the ideal, nonlinear isothermal SMB is about 1.6, which is larger than that for the ideal, linear isothermal system, for which D/F = 1. The difference in D/F performance is due to the nonlinear effects, which reduce the retention factors. Figure 7b shows the effect of nonlinearity on D/F by increasing CF from 40 g/L to 80 g/L. As CF increases, adsorption becomes more nonlinear, leading to a decrease in the retention factors, especially zone III retention factor (δB,III) because it contains the plateau concentrations (Cs,i) of both solutes and thus has the most nonlinear effects compared to the other zones. As a result, D/F increases with increasing CF. Furthermore, comparison of the best isothermal case versus the best thermal case shows that thermal SMB operation improves the D/F performance by ∼2-fold with the same PR. 4.2.2. p-Xylene/Toluene System. The effect of K∞ and ΔH on the D/F is shown in Figure 8. The zone temperatures in Figure 8 are TI = 68 °C, TII = 67 °C, TIII = 37 °C, and TIV = 34 °C with QF = 10 mL/min. These are the optimized zone

slow moving solute (fructose) is confined in zone I while that of the fast moving solute (glucose) is confined in zone II. As expected from eq 21, to minimize D/F, the temperature in zone I has to be the highest (70 °C) to minimize δB,I and the temperatures of zones III and IV have to be the lowest (30 °C) to maximize δB,III and δA,IV. Because glucose adsorption is temperature-independent, the lowest temperature in zone II (30 °C) can facilitate adsorption of fructose and increase the fructose concentration in the extract when a column in zone II is shifted into zone I after port switching. Increasing the feed flow rate can increase productivity, as expected from eq 16. The maximum feed flow rate is reached when the PR algorithm converges using eq 19, see Figure S2 in the Supporting Information. The maximum feed flow rate is QF = 12 mL/min with PR = 2.1 kg of product/h/m3 of adsorbent. However, as PR increases, D/F also increases (Figure 7). For an

Figure 7. Comparison of D/F and PR performance obtained from the optimal temperatures of isothermal SMB and thermal SMB for fructose/glucose system and its effects of (a) mass-transfer effects (nonideality) and (b) CF on the performance. Isothermal SMB optimal temperatures: Tj = 30 °C. Thermal SMB optimal temperatures: TI = 70 °C, TII = TIII = TIV = 30 °C.

ideal system, by contrast, D/F is independent of the feed flow rate and it is always lower than that for a nonideal system (Figure 7a). The difference in D/F between the ideal and the nonideal systems increases as feed flow rate increases because mass-transfer resistance becomes more important. As QF is increased, the zone velocities and the port velocity increase, which corresponds to a decrease in step time. To overcome wave spreading while still maintaining product yield and purity, a larger correction term in the zone velocities (the second term on the right-hand side in eq 10) is required. This increase in correction terms leads to an increase in zone I velocity and,

Figure 8. Effects of changes in isotherm shape (K∞ and ΔH) on adsorption temperature dependency (δB,I/δA,IV) and D/F performance for p-xylene/toluene thermal SMB system. ω is the multiplier constant that reduces K∞ or ΔH from base case (ω = 1). Operating parameters for ω = 1: TI = 68 °C, TII = 67 °C, TIII = 37 °C, and TIV = 34 °C; QF = 10 mL/min. 10428


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Figure 9. Temperature optimization results for p-xylene/toluene system for a specified zone 4 temperature (T4). (a) TIV = 20 °C; (b) TIV = 30 °C; (c) TIV = 34 °C; (d) TIV = 40 °C; (e) zone IV temperature (TIV) overall optimization results for p-xylene/toluene system; and (f) effects of ΔT34 on retention factors. QF = 10 mL/min.

fixed at 20 °C while zones I, II, and III temperatures (TI, TII, and TIII) are optimized to find the minimum D/F. Figure 9a shows how D/F varies when TIII increases. This procedure is repeated for different TIV, and some example results are shown in Figure 9b−d. The minimum D/F values for all the TIV cases are then compared to find the global minimum D/F (Figure 9e). The optimum zone temperatures are as follows: TI = 68 °C, TII = 67 °C, TIII = 37 °C, and TIV = 34 °C. The minimum D/F value is −0.85, meaning that pure solvent is produced from the thermal SMB, which is consistent with eq 20b,c. This condition was not achieved in the fructose/glucose system because δB,I/δA,IV > 1. To explain the trend of the D/F values in Figure 9a−d, a plot of the individual retention factor values with respect to ΔT34 is shown in Figure 9f. In this figure, the zone IV temperature is fixed at 34 °C while the other zone temperatures are optimized

temperatures as discussed next. As expected, lower D/F is achieved at a lower δB,I/δA,IV value, which is obtained when the adsorption isotherm temperature dependence is stronger. This trend is similar to that of the fructose/glucose. In other words, a lower K∞ or ΔH is more advantageous in reducing D/F. Moreover, Figure 8 confirms the initial estimate in eq 20b,c that D/F < 0 is obtained when δB,1/δA,4 < 1 and that the feed concentration is low enough and the adsorption temperature dependency of the p-xylene/toluene system is strong enough to enable the production of pure solvent. More detail on the effects of zone temperatures on D/F for p-xylene/toluene is available in section S4 in the Supporting Information. The temperature optimization results for a feed flow rate of 10 mL/min are shown in Table S4 in the Supporting Information. The temperature range for the optimization is between 20 and 70 °C. First, the zone IV temperature (TIV) is 10429


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Industrial & Engineering Chemistry Research to obtain the minimum D/F. The four zone temperatures can be found in Table S4 in the Supporting Information. Because the temperatures of zones III and IV are lower than those of zones I and II, the retention factors of A and B in these two zones are higher than those in zones I and II. The δA,IV value is not constant, although the temperature of zone IV is fixed because the retention factor values in nonlinear systems are dependent on the concentrations of both solutes. When the temperature of zone III is low, the δA,IV increases according to eq 8d and eq S1A in the Supporting Information. The values of δB,III and δA,IV decrease and eventually level off with increasing zone III temperature (ΔT34). On the other hand, the values of δB,I and δA,II slightly decrease and then exponentially increase with increasing zone III temperature. When the temperature of zone III is relatively low (ΔT34 < 3 °C), the numerator of eq 21, (δB,I − δA,IV), is relatively constant while the denominator, (δB,III − δA,II), is decreasing, resulting in overall a larger negative value of D/F. In this region, D/F decreases with increasing zone III temperature. When the zone III temperature is relatively high (ΔT34 > 3 °C), the values of δB,III and δA,IV are relatively constant while the values of δB,I and δA,II are increasing with increasing temperature. According to eq 21, the D/F value increases with increasing zone III temperature in this region. The maximum feed flow rate found for the optimum zone temperatures is QF = 110 mL/min with PR = 0.0173 kg of product/h/m3 of adsorbent. Figure 10 shows the D/F versus PR comparison between the best isothermal SMB temperatures and the best of the thermal SMB. The optimal temperatures for the isothermal SMB is 20 °C (Table S2b in Supporting Information). The performance comparison between the ideal and nonideal case (Figure 10a,b) and the effects of increasing CF (Figure 10c) are similar to that of the fructose/glucose case. As expected, the ideal case is independent of feed flow rate and always has D/F performance better than that of the nonideal case. Increasing the feed concentration increases the nonlinearity of the system, which leads to increasing D/F. Figure 10 also shows that there is a trade-off between productivity and solvent consumption. Comparison between the best isothermal case versus that of the thermal case for the p-xylene/toluene system shows that there is a significant performance improvement for nonisothermal operation. In the isothermal SMB, the best performance has D/F = 1.31, meaning there is product dilution. By contrast, the best thermal SMB performance has D/F = −0.85, meaning that 85% of the solvent in the feed is recovered as pure solvent, while the extract and raffinate products are 99% pure and concentrated more than 10-fold.

Figure 10. Comparison of D/F and PR performance obtained from the optimal temperatures of isothermal SMB and thermal SMB for pxylene/toluene system and its effects of (a) mass-transfer effects (nonideality), (b) expanded ordinate of panel a, and (c) CF on the performance. Isothermal SMB optimal temperatures: Tj = 20 °C. Thermal SMB optimal temperatures: TI = 68 °C, TII = 67 °C, TIII = 37 °C, TIV = 34 °C.

5. SUMMARY AND CONCLUSIONS The standing wave design method is developed for the first time for nonlinear thermal SMB systems with significant masstransfer effects. An efficient optimization procedure based on the SWD is developed to optimize 10 decision variables, which include the four zone temperatures, the four zone velocities, the step time, and the feed flow rate, to obtain the minimal D/F and its corresponding maximal PR. Other system parameters, such as particle size, column length, and number of columns per zone, are kept constant in this study. The computational time for this procedure requires only a few minutes on a laptop computer. This design and optimization method does not require experimental or process simulation trials. Two examples were discussed in this study: fructose/glucose and p-xylene/toluene. The two examples have different

retention factors (δi,j), temperature dependence of the solute adsorption (δB,I/δA,IV), and feed concentrations (CF). The fructose/glucose system has low retention factors (δi,j < 1), weak adsorption temperature dependence (δB,I/δA,IV > 1), and higher CF. The p-xylene/toluene system has high retention factors (δi,j > 1), strong adsorption temperature dependence (δB,I/δA,IV < 1), and low CF. The results obtained with the nonlinear, nonideal SWD method for the thermal four-zone SMB system for both examples are benchmarked with Aspen Chromatography 2006 simulations. The results indicate SWD provides accurate predictions of the optimum extract and raffinate concentrations and purities. The results support the validity of the nonlinear SWD equations for use in the temperature and productivity optimization procedure. 10430


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performance is better for dilute solutions and when the adsorption isotherms have stronger temperature dependence.

The effects of temperature, mass transfer, and isotherm nonlinearity were studied for thermal SMB system. The temperature effects were studied by investigating the isotherm shape and the adsorption isotherm dependence (δB,I/δA,IV). The isotherm shape is governed by the pre-exponential factor (K∞) in the adsorption equilibrium constant (K(T)i,j) and the heat of adsorption (ΔH). Decreasing either K∞ or ΔH reduces retention factors (δi,j) and δB,I/δA,IV values, which means the adsorption isotherm is more temperature dependent. Because the adsorption isotherm has stronger temperature dependence, the thermal SMB will have a better performance with a lower D/F. It was also determined in this study that δB,I/δA,IV serves as an indicator of whether a specific system has the potential to save solvent using thermal SMB. If δB,I/δA,IV < 1, the system can produce pure solvent (D/F < 0) while producing concentrated, high-purity products. The mass-transfer effect was studied by comparison of the ideal and nonideal systems. Ideal SMB systems will always have lower D/F values compared to the corresponding nonideal systems. A larger mass-transfer effect was observed with increasing feed flow rate. Nonlinear effects increase with increasing feed concentration. Thus, D/F increases significantly with increasing feed concentration. It was also shown that there is a trade-off between D/F and productivity (PR) for optimal zone temperatures. In the fructose/glucose system, extract port product enrichment (EFE,B = 1.13) and minimum raffinate port product dilution (EFR,A = 0.97) are achieved using thermal SMB operation for a low feed flow rate (QF = 3.5 mL/min). Because δB,I/δA,IV > 1 for the fructose/glucose system, the thermal SMB system is not able to produce pure solvent. Lower D/F is obtained at higher TI and lower TIV. As the feed flow rate is increased, PR increases, but D/F also increases.. The optimum zone temperatures are TI = 70 °C and TII = TIII = TIV = 30 °C. At these temperatures, the maximum feed flow rate is QF = 12 mL/min with PR = 2.1 kg product/h/m3 of adsorbent and D/F = 1.2. In comparison with the best case for the isothermal SMB, the thermal SMB is able to increase the product concentration by 25% and 75% for extract and raffinate products, respectively, with a decrease in D/F by 2-fold. In the p-xylene/toluene system, because δB,I/δA,IV < 1, the thermal SMB process for this system is able to produce pure solvent (D/F < 0). Therefore, significant product enrichment for both outlet ports is achieved (EF > 10-fold). In addition, higher product enrichment is observed at lower outlet flow rates. Lower D/F is obtained at higher TI. The minimum D/F values can be obtained when the system is operated under the conditions where, for a fixed TIV, the temperature difference between each zone is at the maximum values for enriching the products. Thus, the higher the temperature difference, the better the thermal SMB performance. The trend of feed flow rate (QF) versus PR or D/F is similar to that of the fructose/ glucose case. The optimum zone temperatures for the pxylene/toluene system are TI = 68 °C, TII = 67 °C, TIII = 37 °C, and TIV = 34 °C. The maximum feed flow rate achieved is QF = 110 mL/min with PR = 0.0173 kg product/h/m3 of adsorbent and D/F = −0.821. The thermal SMB is able to increase the product concentration by more than 10-fold for both extract and raffinate products while producing high-purity solvent. In comparison, the best isothermal SMB at the same productivity has a D/F= 3.8, the products are diluted, and EF = 0.38 and 0.27 for the raffinate and extract products, respectively. The comparison of the two examples shows that the thermal SMB

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ASSOCIATED CONTENT

S Supporting Information *

Nonlinear, Ideal SWD Iterative Procedure (section S1); Nonlinear, Non-Ideal SWD Iterative Procedure (section S2); Thermal SMB Temperature Optimization Procedure (section S3); Effects of Zone Temperatures on Retention Factors (section S4); Effects of Zone Temperatures on EF Performance (section S5); Aspen Chromatography Mass and Energy Balance (section S6); Aspen Chromatography Verification with the Experiments from Ching and Ruthven (section S7); Effects of Pre-exponential Factor (K∞) and the Adsorption enthalpy (ΔH) on the Adsorption Isotherm Shape (section S8); Detailed additional data involving SWD validation using Aspen Chromatography and temperature optimization results (section S9). The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.iecr.5b01296.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +1-765-494-4081. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support from Purdue University is gratefully acknowledged. The authors thank Dr. Anand Venkatesan and Mr. George Weeden Jr. from Purdue University for their helpful discussions and suggestions.

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NOTATION Cjbi = bulk concentration of solute i in the mobile phase in zone j, g L−1 CE,i = concentration of solute i at the extract port, g L−1 CF,i = feed concentration of the solute i, g L−1 Ci = concentration of solute i in the liquid phase, g L−1 Cj,i = concentration of solute i in zone j, g L−1 Ci* = average pore phase concentration of solute in zone i, g L−1 Cp,A = plateau concentration of solute A in zone IV port, g L−1 Cp,B = plateau concentration of solute B in zone II, g L−1 CR,i = concentration of solute i at the raffinate port, g L−1 CpL = heat capacity of the liquid phase, J g−1 K−1 CpS = heat capacity of the solid phase, J g−1 K−1 Cs,A = plateau concentration of solute A in zone III port, g L−1 Cs,b = plateau concentration of solute A in zone III port, g L−1 Ejbi = axial dispersion coefficient of solute i in zone j, cm2 min−1 DC = column diameter, cm D/F = amount of desorbent use over the feed flow rate EFE,i = enrichment factor of solute i at extract port EFR,i = enrichment factor of solute i at raffinate port kjei = effective mass-transfer coefficient of solute i in zone j, min−1 K(T)i,j = temperature-dependent equilibrium constant of solute i in zone j, L g−1 DOI: 10.1021/acs.iecr.5b01296 Ind. Eng. Chem. Res. 2015, 54, 10419−10433

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Industrial & Engineering Chemistry Research K∞ = pre-exponential factor of the Arrhenius equation, L g−1 Lj = zone j length, cm LC = column length, cm Nj = number of columns in zone j P = bed phase ratio PuE,i = product purity of solute i in the extract port, % PuR,i = product purity of solute i in the raffinate port, % PR = productivity, (kg product) (L adsorbent)−1 (h) −1 qi = concentration of solute i in the solid phase, g L−1 qmax,i = maximum adsorption capacity of solute i in the solid phase, g L−1 QE = volumetric flow rate of the extract port, mL min−1 QF = volumetric flow rate of the feed inlet, mL min−1 Qj = volumetric flow rate in zone j, mL min−1 QR = volumetric flow rate of the raffinate port, mL min−1 R = gas constant, J K−1 mol−1 S = column cross-sectional area, cm2 tsw = step time, min Tj = temperature of zone j, K ujo = zone velocity in zone j, cm min−1 uport = port velocity, cm min−1 us,i,j = solute wave velocity of solute i in zone j, cm min−1 uth,i = thermal wave velocity in zone i, cm s−1 uw,i,j = concentration wave of solute i in zone j relative to feed port, cm min−1 Yi = yield of solute i ΔH = enthalpy heat of adsorption, J mol−1 ΔT = temperature difference between zones, °C or K βji = natural log of the ratio between the highest and lowest concentration of solute i in zone j δi,j = retention factor of solute i in zone j εe = interparticle void fraction εp = intraparticle void fraction ω = multiplier constant. K∞ = ω × K∞,base or ΔH = ω × ΔHbase ρL = density of the liquid phase, g mL−1 ρS = density of the solid phase, g mL−1

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Standing Wave Design and Optimization of Nonlinear Four-Zone

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