Exergy Analysis of Chromatographic Separations in a Fixed-Bed

Ind. Eng. Chem. Res. 2004, 43, 3183-3193

3183

SEPARATIONS Exergy Analysis of Chromatographic Separations in a Fixed-Bed Column M. L. O. Maia, M. Ottens, and L. A. M. van der Wielen* Department of Biotechnology, Delft University of Technology, Julianalaan 67 2628 BC, Delft, The Netherlands

In this work, we evaluate the use of exergy analysis on a single chromatographic column. A sensitivity analysis was done to unravel the relation between several parameters that might influence the quality of separation and the associated exergy loss. From the resulting analysis we concluded that the division of the exergy content in its components is important in revealing the nature of process irreversibilities. In addition, results show that the dilution problem is adequately described by the exergy change of mixing. All in all, results have shown that a minimum exergy loss occurs for a high production rate while keeping purity requirements constant. Consequently, exergy loss can be used as a tool for chromatographic separation optimization. 1. Introduction Chromatography is a powerful separation technique used in pharmaceutical, food, and bulk chemical industries. The separation of xylene isomers and fructose from high fructose corn syrups are examples of the large-scale application of chromatography in bulk chemicals production, whereas resolution of enantiomers and recovery of amino acids, antibiotics, and proteins from complex biological streams are increasing in importance in fine chemicals manufacturing. In all cases, however, the total manufacturing costs and environmental impact of the chromatographic processes must be minimized. Therefore, large-scale chromatographic operations are increasingly performed in so-called simulated moving bed (SMB) systems, where equipment size, resin inventory, and eluent consumption is often considerably reduced relative to fixed-bed (FB) operation.1 The main cost contributors in chromatographic processes are, apart from equipment and resin costs, operational costs such as net eluent consumption, recovery of the products from the solvent streams, regeneration of the eluent, and energy costs related to mechanical work and thermal operations. Optimization of any chromatographic technology is a complex matter because many parameters affect the performance of the system significantly. Therefore, general rules for designing and optimizing chromatographic processes are difficult to establish.1 It is a common practice to group all the parameters in dimensionless numbers to reduce the number of variables and simulations needed. Felinger and Guiochon2 proposed the use of a hybrid cost function that considers the * To whom correspondence should be addressed. Tel.: +31-15-2782332. Fax: +31-15-2782355. E-mail: [email protected].

importance of solvent use and production rate at a given weight factor. However, no price specifications are presented. Instead, production rate is maximized by simultaneously changing loading factor and number of plates (efficiency). The production costs are approximately reflected using a weight that is inversely proportional to the ratio of the capital costs and operating costs. In another contribution, Felinger and Guiochon3 use the product of the production rate and recovery yield as the objective function. They claim that in this case both productivity and recovery yield are optimized. Moreover, this objective function is used in the comparison of different modes of chromatographic separation.4 A detailed cost optimization can be found in Jupke.5 Minimizing a cost objective function, however, inevitably has the drawback of depending on economic parameters that may be arbitrary and/or fluctuating in time. Alternatively, optimization procedures based on thermodynamic concepts provide an understanding of the underlying physical phenomena in a process, and thus can provide physical reasons for process inefficiencies. Exergy analysis is one of these optimization procedures. This sort of analysis has become increasingly important given the considerable energy requirements of the process industry. The efficient use of the energy can be accounted for by combining the First and Second Law of Thermodynamics, that is, using the exergy concept. The identification of the optimal thermal condition of the feed of distillation columns6 is an example of the application of the exergy concept. Smith7 applied exergy analysis for the determination of the optimum pressure for supercritical carbon dioxide extraction. In addition, pressure swing adsorption (PSA) systems studied by Barnerjee,8 where the compressor work is the main concern, were optimized using exergy analysis. A review on the opportunities and limitations of the use of exergy

10.1021/ie030603u CCC: $27.50 © 2004 American Chemical Society Published on Web 05/11/2004

3184 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004

analysis on both process synthesis and optimization can be found elsewhere.9 The goal of exergy analysis is to minimize the exergy loss that is closely related to process’ irreversibilities and thus arises in any real process. Concerning chromatography, sources of irreversibility are related to heat effects, pressure drop, mixing of solutes and eluent, and mass transfer between two phases due to finite driving forces. These sources combined contribute to the exergy loss of this system. The thermodynamic price for these irreversibilities is paid in energy for pumping of fluid phase, eluent use, product dilution, and thus downstream separation costs. Bailly10 (1984) has proved that the reduction of entropy generation during a separation in a single chromatography column parallels the reduction of the solvent consumption. Therefore, an exergy analysis may be of use when designing and/or optimizing chromatographic separations. The optimization and design of simulated moving bed systems is of particular importance. So far, however, no open literature concerning the use of exergy concept in fixed and simulated moving bed chromatographic systems seems available. Due to the fact that SMB systems are constituted of a number of interconnected FB chromatographic columns, the first step toward the understanding of the exergy analysis of an SMB system is to evaluate the use of exergy analysis for one single chromatographic column. The aim of this work is to provide a methodology to perform exergy analysis on a fixed-bed chromatographic column. A sensitivity analysis is used to understand the relation between several parameters that might influence the quality of separation and the associated exergy loss. The separation of a binary ideal mixture was used as a case study.

outflowing and entering the systems boundaries, respectively. The latter is given by

2. Exergy Analysis

3. Fixed-Bed Adsorption Model

An exergy balance (eq 1) for a steady flow in an open system is obtained by combining the First and Second Law of Thermodynamics.11

Chromatography is a separation technique where a multicomponent feed is pulsed into a column packed with a suitable sorbent. The degree of separation depends on the column length and the differences in component affinity for the sorbent. Henley and Seader14 give the convective-dispersion model (eq 5) that describes the separation.

Exin ) Exout + ExQ + W + ∆Exloss

(1)

where

ExQ )

Q*(Tr - T0) Tr

∑

(1a)

represents the exergy transfer due to heat flow (Q) from the system.12 The subscript “0” indicates the reference state, which for exergy analysis is imposed by the somewhat arbitrary environment.11 In this work, the reference temperature and pressure are, respectively, 298.15 K and 1.01 bar. W is the exergy transfer due to shaft work (kJ/s) performed by the system. Finally, the last term in eq 1, ∆Exloss (kJ/s) represents exergy loss due to irreversibilities within the system. Concerning the material flows, the exergy transfer is given by

∑n*exout Exin ) ∑n*exin

Exout )

(1b) (1c)

with n being the molar flow rate (mol/s) and exout and exin (kJ/mol) the specific exergy of the material streams

ex ) (h - h0) - T0(s - s0)

(1d)

Equation 1d can be divided into three terms (eq 2), those being chemical, physical, and mixing exergy.

ex ) exchem + exphys + ∆mixex

(2)

Hinderink13 et al. present a detailed discussion about the definition of each term, together with the computational procedure. For separation processes, the chemical exergy can be neglected because no chemical transformation occurs and the same components are involved for both inflowing and outflowing streams. Thus, only the mixing and physical terms of the total exergy content are relevant. According to Hinderink et al.,13 physical exergy of a multicomponent stream can be computed using pure components enthalpies and entropies. Moreover, when no phase transitions occur on bringing multicomponent material streams to the reference conditions, its exergy content is given by eq 3.

exphys ) ∆actualf0(

∑i xihi - T0∑i xiSi)

(3)

where the subscript “actual” refers to the actual T and P of the mixture and “0” the conditions at the reference state. The mixing exergy is calculated using the enthalpy and entropy change of mixing.13

∆mixex ) ∆mixH - T0∆mixS

∂2ci ∂(uci) ∂ci (1 - ) ∂q ji -DL 2 + + + )0 ∂z ∂t ∂t ∂z

(4)

(5)

with ci and qi being the concentration of species i in the liquid and solid phase, respectively, z the axial coordinate, t the time, and the bed porosity. The first term accounts for the axial dispersion with dispersion coefficient DL, the second allows for the axial variation of the fluid velocity (convective term), the third is the accumulation of the solute in the liquid, and the fourth is the accumulation in the sorbent. The concentration of the liquid and solid are coupled via a mass transfer relation:

∂q ji ) f(q j i,ci) ∂t

(6)

Equation 6 describes the adsorption rate and is a partial differential equation when mass transfer is limited by pore diffusion. In this contribution, we consider the exergy loss that occurs due to the following irreversibilities: mixing of

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3185

Figure 1. Chromatographic separation of binary pulse on FB. In case 1, the boundary for exergy analysis considers constant temperature and pressure. In case 2, the system boundary is extended to include the pump.

eluent and solutes, finite driving force for mass transfer between the phases, and pressure drop. For this purpose, we use a simpler model, which assumes plug flow and neglects axial dispersion effects. Furthermore, the mass transfer is accounted by the common linear driving force relation (LDF).15 The overall mass transfer coefficient is divided into two parts: external mass transfer from the bulk liquid to solid surface and internal pore diffusion within the particle itself. The model used is then given by

ji ∂ci ∂ci (1 - ) ∂q + + )0 ∂z ∂t ∂t

u

(7a)

∂q ji ) MTi ∂t

(7b)

MTi ) ka(qi/ - qi)

(7c)

1 1 1 ) + k ke kintraintra

(7d)

with ke being the film mass transfer coefficient, kintra the intraparticle mass transfer coefficient, k the overall mass transfer coefficient, a the interfacial area, and qi* the solid phase equilibrium concentration. The relation between the concentration of the stationary phase and mobile phase at equilibrium is described by a suitable isotherm, which is often in liquid chromatography the multicomponent Langmuir equation.1

qi* ) qmax

Kici

1+

nc

sorbent is free of solute. The cycle starts when a pulse of the mixture to be separated replaces the eluent stream at the same volumetric flow rate. After the pulse, the eluent stream is resumed to the column until the end of the cycle. The entire cycle lasts for a period of time called elution time (telution), which is large enough to ensure that all solutes leave the column. At the end of this time interval, therefore, the system is back to its initial state. Therefore, its exergy content is unchanged and eq 1 can be applied. The exergy loss occurring on each cycle is then divided by the total amount of product that can be obtained with the required purity. The general procedure to perform the exergy balance on a fixed-bed chromatographic column is described below: 1. Given pulse volume, duration (tpulse), flow rates, and composition, use the convective-dispersion model with a suitable isotherm (eqs 7 and 8) to compute the product composition profile. 2. Compute the specific physical and mixing exergy content of the feed stream using eqs 3 and 4, respectively. During the entire cycle the feed stream is in fact the sum of two streams: pulse and eluent. However, these two streams do not exist simultaneously. Equations 9 and 10 represent the exergy content of the feed stream computed for the entire cycle duration. Moreover, note that the eluent stream contains a single pure component and, therefore, the mixing term is null after the pulse. t)t ∫t)0

Exfeed phys ) ( (8)

Kjcj ∑ j)1

The set of partial differential equations are solved using spatial discretization and fourth-order Runge-Kutta method for integration in time. Moreover, the pressure effects are accounted by using the Ergun relation. 4. Exergy Balance on Fixed-Bed Columns Methodology In this work, the overall exergy balance of the system considers the exergy loss during one isocratic chromatographic cycle (Figure 1). The initial system is filled with eluent, which is pumped through the column, and the

pulse

npulse(t) dt) * expulse phys + (

∫t)tt)t

t)t ∫t)0

∆mixExfeed ) (

elution

pulse

pulse

neluent(t) dt) * exeluent (9) phys

npulse(t) dt) * ∆mixex

(10)

3. Repeat step 2 for the product streams. In this case, the dynamic nature of the process causes the specific exergy content of the process streams to vary with time. Equation 12 describes the mixing exergy content of the product stream.

Exphys prod )

t)t ∫t)0

elution

(n*ex(t)) dt

(11)

∑j xj(t)ln(γjxj(t))

(12)

∆mixExprod ) -R‚T0‚

3186 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 Table 1 (a) Column Characteristics ratio length/ internal diameter (Lc/Di) external void fraction internal void fraction (macroporosity) solid bulk density

19 0.42 0.2 1.4 g/cm3

(b) Diffusion Coefficients and Parameters for the Langmuir Isotherm (Saturation Capacity and Equilibrium Ratios) at 57 °C Dm × 105 (cm2/s)

qmax × 103 (mol/g)

K (L/mol)

ke × 104 (cm/s)

ki × 104 (cm/s)

3.16 3.16 3.61

1.75 1.75 1.75

4.2 24.0 12.0

2.12 2.12 2.32

1.13 1.13 1.30

m-xylene p-xylene toluene

(c) Feed and Eluent Concentration

Figure 2. Chromatogram for the separation of a general binary mixture.

concentration (vol %) feed

composition (product dilution) and is given by eq 15. For case 1, we have a mixing exergy loss.

50% m-xylene 50% p-xylene pure toluene

eluent

∆mixExloss ) ∆mixExin - ∆mixExout

4. Compute overall exergy content of the process streams using eq 2. 5. For the particular case of chromatography, the exergy transfer due to heat flow is neglected because the system temperature is often too close to the reference temperature. Due to the equivalence of exergy and work, the exergy transfer due to shaft work is given by the pump work (eq 13).

W ) φ∆P

(13)

where φ is the volumetric flow rate. Pressure drop (∆P) is computed using the Ergun correlation. 6. Compute the exergy loss using eq 1.

The second condition accounts for the fact that in liquid chromatography both solvent and pulse streams have to be pumped in the column. This process condition is here called case 2. Mixing exergy loss alone does not cover energy degradation throughout the bed due to pressure drop. A realistic optimization of a chromatographic separation process should include this problem, though. From an exergy analysis point of view, the evaluation of physical exergy content of the streams and exergy transfer due to shaft work must be used in order to be able to deal with this problem (case 2). Equation 16 describes the overall exergy balance that now includes the exergy transfer due to shaft work.

∆Exloss ) ∆mixExin - ∆mixExout + W

5. Results To illustrate the exergy analysis on batch chromatography, we use the separation of a binary mixture that can be considered a thermodynamically ideal solution. The relative concentration of the components is constant in all simulations. The details concerning column dimensions and adsorption data16 are presented in Table 1. Several simulations were performed to investigate the contribution to the exergy loss of parameters such as flow rates, pulse volume, and particle diameter. Results are presented as a function of dimensionless parameters such as reduced time (τ), reduced velocity (ν),17 and modified Stanton number (St)5 given by eq 14.

t‚u L

(14a)

dp‚u Dm

(14b)

τ) ν)

6 L Steff,i ) koverall,i‚ ‚ dp u

(14c)

For the sake of an easier understanding and simplicity, exergy analysis is applied to two different conditions. First, we consider the system to be at constant temperature and pressure, here called case 1. In this case, the exergy transfer due to heat and shaft work is not applicable. Thus, the use of eq 1 leads to the conclusion that the exergy loss is only due to change in

(15)

(16)

In our particular case, both inflowing (in) and outflowing (out) streams are at reference pressure P0. Therefore, the pump work is equivalent to the physical part of the total exergy loss, which occurs within the process boundary considered. Physical exergy content of a material stream is, of course, also dependent on the temperature. For the presented case study, the separation is always carried out isothermally and thus the temperature effect cancels out. From now on, we divide the total exergy loss in two parts, namely, mixing exergy loss and physical exergy loss. For batch chromatography a cut strategy (Figure 2) is needed to determine the fractions collection. Accordingly, the exergy output and the exergy loss depend on the strategy used. Of course, for optimization problems, these four cut times are parameters that should be adjusted to obtain maximum productivity, yield, etc. In this contribution, however, two different criteria were used to determine the four cut times (Figure 2). The first (t1) and fourth (t4) cut times are found considering that an arbitrarily chosen amount (1% of the injected volume) of the weaker and stronger binding components, respectively, can be wasted. The second (t2) and third (t3) cut times are found according to the purity specification for both products 1 and 2 (Figure 2). In this work, the product purity required is 98% for both components. As a result of the cut strategy, two useful product streams can be obtained as long as purity requirements are achieved. The fraction of the outflowing stream, which corresponds to the peaks overlap, is called here


Figure 3. Chromatograms for different Stanton number obtained when particle sizes are changed: (A) St ) 3.74 (HPLC); (B) St ) 0.18 (LC). Weaker binding component is represented by points (×) and stronger binding component is represented by a solid line.

waste stream (Figure 2). Concerning exergy analysis, the exergy that is carried away with the outflowing streams can also be divided into two parts: exergy of the useful products and exergy of waste products. The latter represents the undesired exergy loss while the former is here called desired exergy loss. Note that the two losses are due to internal process irreversibilities. Apart from the exergy loss, results concerning product purity and recovery along with solvent usage are presented. In the authors mind, this is a way to link the exergy loss to parameters that will ultimately have an impact on process costs. In other words, exergy analysis indicates potential downstream processing costs. According to the procedure shown in item 4, the elution time (telution) is held constant regardless of the input variable that was changed. To account correctly for the eluent used, therefore, one should not consider the fraction of solvent consumed from the cut time t4 to telution (Figure 2). This fraction is here called unused solvent (Vunused) and is given by eq 17.

Vunused ) φ*(telution - t4)

(17)

5.1. Particle Size. Larger particles are typical for large-scale adsorption processes, whereas the smaller are characteristic for analytical HPLC systems. On one hand, small particles allow for higher mass transfer coefficient and thus less dilution is possibly achieved. On the other hand, mechanical energy consumption (pump work) increases due to a higher pressure drop. In this item, an analysis based on Stanton numbers is performed to evaluate the impact of mass transfer effects on the exergy loss. Only the particle size was changed in the simulations performed here. All other variables such as volumetric flow rate and pulse volume are kept constant. The chromatograms (Figure 3), plotted with reduced time, were obtained for two different particle sizes. In Figure 3A the peaks are much sharper and high concentrations for both components are achieved. In addition, hardly any overlapping can be seen in Figure 3A. Such behavior is solely due to the fact that the particles used in Figure 3B have a radius that is approximately 43 times larger than those used in Figure 3A. In either case, an exergy loss is expected and the reason relies on the fact that the exergy content of the pulse (equimolar binary mixture) is decreased when mixed with the solvent. The average concentration of the products is lower than its initial value (C/CF < 1.0).

Figure 4. Overall mixing exergy loss, together with its two terms, plotted against Stanton number (lines and points): (×) desired exergy loss (useful products), (b) undesired exergy loss (wasted products), and (4) overall mixing exergy loss.

The same observation holds for the solvent, which is initially a pure component. 5.1.1. Case 1: Constant Temperature and Pressure. For the analysis here consider the process boundary shown in Figure 1A. In this case, the exergy transfer through process boundary is due to material streams and the exergy loss is only due to changes in composition, that is, mixing exergy loss (eq 15). Figure 4 shows that decreasing Stanton number leads to an increasing overall mixing exergy loss. Such behavior can be explained by the fact that peak overlap and band broadening are expected to be higher in the range of small Stanton numbers. One limitation of analyzing the overall mixing exergy loss is that no distinction is made between the dilution of products in the solvent and the lack of separation that occurs for overlapping peaks. A further division of the mixing exergy loss in its two parts can help us to overcome this problem. On the basis of the cut strategy (Figure 2), the mixing exergy loss comprises the desired exergy loss, associated with useful products, and the second is the undesired exergy loss, which indicates wasted products. The desired exergy loss is unavoidable, even for a complete separation, and inherent to chromatographic separation. However, it will be larger for broad peaks and lower for sharp peaks. The obvious economic penalty for this dilution is paid in the downstream process to recover the products from the eluent. The second contribution to the overall mixing exergy loss originates from the lack of separation and is strongly related to the cut strategy used. For the separation of a binary mixture, for example, the cut strategy used here considers that the peaks overlap is a waste stream (between point 2 and 3 in Figure 2). This stream leads to low recoveries even when purity requirements are achieved, and could be recycled. No recycling is, however, considered in this approach. The exergy content of this stream is considered as a loss as it means that for this particular cycle the separation was not achieved and thus no useful work was performed. For most of the range of Stanton numbers presented in Figure 4, hardly any change of the desired exergy loss can be observed. However, for the smallest values there is a trend for this loss to decrease. The reason is that the impact of mass transfer, which causes band broadening, has become stronger and therefore the products are highly diluted. Eventually the dilution is such that the products (products 1 and 2) are almost pure eluent, causing the exergy loss to decrease. Con-

3188 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 Table 2. Number of Cycles Depending upon the Pulse Volume Vpulse/V0 (%)

0.187

0.935

1.869

3.738

7.475

9.344

14.950

22.425

28.032

33.638

37.375

number of cycles

200

40

20

10

5

4

3

2

2

2

1

Figure 5. Overall mixing exergy loss plotted against product recovery for 98% purity requirement (lines and points): ([) purity weaker binding, (0) purity stronger binding, and (4) total recovery.

Figure 6. Overall exergy loss (lines and points) caused by masstransfer effects and pressure drop over the bed: (O) mixing exergy loss, (]) physical exergy loss, and (/) total exergy loss.

cerning the undesired exergy loss (waste products), an opposite trend in its behavior can be observed, due to the increasing peaks overlap. This means that the contribution of the lack of separation tends to be higher in the region of bigger particles with low Stanton numbers. However, the dilution problem is reflected in both terms of the overall mixing exergy loss. In this work, no cost calculation is presented. However, one reasonable way to show the impact of high mixing exergy loss on production costs is to plot it against product recovery and purity level achieved when using different particle sizes. Figure 5 shows that low overall mixing exergy loss means high total recovery for a purity level of 98% for both components. Conversely, product recovery decreases dramatically (lower than 30%) when mixing exergy loss increases. Furthermore, it was not possible to achieve the purity specification of the weaker binding component within this region. Summarizing, mixing exergy loss is directly related to product dilution and thus its minimization can lead to more concentrated products as well as high recovery rate while attending purity specification. 5.1.2. Case 2: Considering ∆P. Figure 6 shows the total, physical, and mixing part of the exergy loss for several particle sizes recalculated as Stanton numbers. The former is the summation of the two contributions for exergy loss, namely, mixing and physical. For most of the range used here, mixing exergy loss has a dominant impact on the total loss. As Stanton numbers decrease, physical and mixing exergy loss shows op-

posite behavior. It can be seen in Figure 6 that this physical exergy loss seems to present a minimum. Although larger particle sizes allow for lower pressure drop, the decrease in the production is such that the physical exergy loss, which was here divided by the amount produced, increases. From this point on, only one product achieves purity specification and then the total production is dramatically decreased. Nevertheless, for optimization purpose the total exergy loss is the one to be used. Adding both contributions to the exergy loss, a particle size exists that minimizes the total exergy loss (Figure 6). In other words, larger particles allow for lower physical exergy loss while increasing product dilution, while a smaller dp allows low dilution but at the expense of high physical exergy loss. 5.2. Pulse Volume. Pulse volume is an operating parameter that should be adjusted when optimizing/ designing chromatographic separation carried out on FB. It is important for exergy analysis, and for the sake of a fair comparison, the input exergy to be the same in all simulations. For this reason, we consider a fixed total volume to be separated and then a number of cycles, with different pulse volumes, are used for the separation (Table 2). No recycle of both eluent and waste streams, however, was accounted for. In addition, the pulse volume used can vary on each cycle. In our case, the total amount of feed to be separated corresponds to 37.3% of the total void volume of the column. Therefore, if we start with a pulse volume of 28%, a second pulse of 9.3% will be required. Concerning exergy calculations, one should take into account the number of cycles when computing the exergy loss. For mixing exergy loss, for instance, eq 15 is rewritten as below: Ncycle

∆mixEx

loss

in

) ∆mixEx -

∑ j)1

∆mixExout(j)

(18)

Equation 18 describes the exergy loss, which occurs for the separation of the total amount of feed. In this sense, the exergy content of the feed is constant. Conversely, the exergy content of the outflowing stream is dependent on both the pulse volume and the number of cycles. The chromatograms of the two first and two last pulse volumes used here (Table 2) are shown in Figure 7A,B. For Gaussian distribution (Figure 7A), obtained for a small sample size, all solutes migrate with the same rate and the retention time is independent of the sample size. However, increasing the pulse volume (Figure 7B), we move to the region of nonlinear isotherms and solutes retention time varies with the sample size. 5.2.1. Case 1: Constant Temperature and Pressure. An exergy balance at constant temperature and pressure (case 1, Figure 1) was performed to investigate the influence of the amount injected in the column on the mixing exergy loss. In Figure 8, results are presented as total mixing exergy loss, undesired exergy loss (waste products), and desired exergy loss (useful products). It can be observed that total mixing exergy loss decreases with increasing pulse volumes (Figure 8). To


Figure 7. (A) Chromatogram for the separation of weaker binding (line and points) from stronger binding (lines) obtained for Vpulse/ Vo: 0.93% (open spheres) and 1.8% (open triangles). (B) Chromatogram for the separation of weaker binding (line and points) from stronger binding (lines) obtained for Vpulse/Vo: 33.6% (open spheres) and 37.4% (open triangles).

Figure 8. Exergy loss due to change in pulse volumes at constant temperature and pressure (mixing exergy loss): total loss (4), desired loss (+), and undesired loss (0).

Figure 9. Total recovery for 98% purity of both products for several pulse volumes: purity stronger binding (4), purity weaker binding (×), and total recovery (0).

understand this behavior, we have to analyze the two components of this loss separately. First, it is clear that the total mixing exergy loss is much more strongly influenced by the desired loss than by the undesired one. The chromatograms (Figure 7A,B) show not only that the useful products concentration is increasing for increasing pulse volume but also that peak overlap is not significant. In fact, the Stanton number (St ) 3.74) and reduced velocity (ν ) 7.9), used in all simulations, bring the separation to a zone where purity requirements are possible to be achieved for all pulse volumes used. Peak overlap is the major cause for decrease in the recovery. Figure 9 shows the recovery for 98% purity of both components when different pulse volumes are used. The total recovery means the total amount of the two products obtained regarding the purity specification and

Figure 10. Volume of solvent used per gram of product with the specified purity.

number of cycles needed. It can be seen that, for smaller pulse volume, the recovery is high. Also, the saturation of the column, which occurs for increasing pulse volumes, leads to a decrease in the recovery. Solvent usage is presented in Figure 10. Of course, the number of cycles needed plays an important role. It is particularly interesting to see that when two cycles are needed (Table 2), the amount injected on each cycle is very different from one another. Actually, during the first cycle a large amount of feed is introduced into the column, while at the second one, a much smaller amount is introduced. As a result, we see in Figure 10 that in this range the amount of solvent is practically constant, but decreases when just one cycle is used. We now have to focus our attention in the second contribution for the mixing exergy loss. This term, here called undesired exergy loss, is the exergy content of the waste stream (Figure 2) itself. For very small pulse volumes that require several cycles to separate the total amount, this exergy loss seems to reduce. At this range of pulse volume, the peak overlapping is not significant. Consequently, the production is practically constant (Figure 9). Therefore, the undesired exergy loss at this region is mainly influenced by the number of cycles. However, when the amount injected starts to decrease, the production due to saturation of the column (more impressive peaks overlapping), the undesired exergy loss starts to increase. Eventually, it seems that this term will lead the overall mixing exergy loss to also increase. For this particular case, overall mixing exergy loss for smaller pulse volume is practically only due to the exergy of useful products. As the pulse volume increases, the loss due to the waste stream exergy has a much stronger impact and eventually will dominate the overall loss (Figure 8). From Figure 8 one can conclude that the optimum pulse volume is the highest value. This result seems to be in agreement with classical chromatographic analysis. Together with product recovery, solvent consumption is an important parameter when computing process costs. The conclusion here is that this particular case benefits from higher pulse volumes. It is may be possible that the impact on the process costs of decreasing recovery is not so important as the decrease in solvent usage. 5.2.2. Case 2: Considering ∆P. Another important parameter to be optimized in batch chromatography is the mechanical energy consumption (pump work). In this case, eqs 15 and 16 describe the overall exergy balance.


Figure 11. Exergy loss due to pressure drop.

Figure 13. Chromatograms for the separation of a binary mixture for two different reduced velocities. For ν ) 7.9, product 1: thick line and product 2: thin line. For ν ) 59.7: product 1 (b) and product 2 (0). Stanton number 3.74.

Figure 12. Mixing exergy loss (lines and points) obtained for several reduced velocities (flow rate): (4) total mixing exergy loss, (0) undesired exergy loss (wasted products), and (]) desired exergy loss (useful products).

As mentioned before, the pressure drop over the bed, that is, a function of flow rate, particle size, etc., and not of the pulse volume, contributes to the total exergy loss. Of course, the physical exergy loss per cycle is the same. There is a clear trend in this part of the exergy loss to decrease as the number of cycles decreases (increasing pulse volume). However, when the amount injected requires the same cycle number (Table 1), the physical exergy loss is mainly a function of the amount produced (Figure 11). It is expected, then, that this exergy loss will increase for very small production, which occurs for a saturated column (high pulse volume). Summarizing, the pressure drop for this particular parameter is the same in all simulations. However, the behavior of the physical exergy loss is the compromise between number of cycles and production. 5.3. Flow Rate. Together with pulse volume, flow rate is also one of the operating parameters that has to be adjusted when optimizing batch chromatography. Higher flow rates allow for increasing throughput. On the other hand, the equilibrium between solid and mobile phase is not instantaneous and thus the solute molecules are carried further down the column than would be expected in equilibrium conditions. The effect of nonequilibrium mass transfer becomes worse when the flow rate increases because less time is available for the equilibrium to be approached. The natural consequence of these effects is product dilution and therefore mixing exergy loss occurs. 5.3.1. Case 1: Constant Temperature and Pressure. Several simulations were performed using a wide range of reduced velocities (Figure 12). In this section, we used two particle diameters and varied the flow rates for both cases. Mixing exergy loss (lines and triangles curve) increases with increasing reduced velocity as expected.

Figure 14. Mixing exergy loss (lines and points) obtained for several reduced velocities (flow rate): (4) total mixing exergy loss; (0) undesired exergy loss (wasted products), and (]) desired exergy loss (useful products).

However, a close look in Figure 12 shows that this increase is not significantly high. To further understand the reason for total mixing exergy loss, the two components of this loss are also presented. First, we consider the desired exergy loss. The dilution of the useful products does no significantly differ when reduced velocity is increased. For this particular case, the mass transfer coefficient is high due to the small particle size used. Chromatograms in Figure 13 show sharp and well-resolved peaks can be obtained for the smallest and highest reduced velocity used here. Therefore, the desired exergy loss is not only the most important contributor to the total loss but also practically constant over the range here used. The main effect of the increasing reduced velocity is that the peaks are more overlapping and thus the waste stream exergy loss also increases. However, it is worth noticing that for the range of reduced velocities used the impact of this second contribution is always lower than the first one. In addition, its increase is not significantly high as well. Again this behavior might occur due to the high mass transfer coefficient. An opposite trend, however, is found when mass transfer effects are increased by using larger particles. In this case, for the same range of flow rate used in the previous example the reduced velocities are much larger. The impact of changing reduced velocity (Figure 14) on total mixing exergy loss is also much larger. It is important to address here that for reduced velocity higher than 160 only the stronger biding component is recovered with specified purity but at very low produc-


Figure 15. Chromatograms for the separation of a binary mixture for two different reduced velocities. For ν ) 51.7, product 1: thin line and product 2: thicker line. For ν ) 258.5: product 1 (O) and product 2 (0). St ) 0.18.

Figure 16. Effect of mixing exergy loss: (A) on solvent consumption (0); (B) purity and recovery: (4) purity product 2, ([) purity product 1, (0) recovery product 1, and (×) recovery product 2.

tion rate. As a result, mixing exergy loss seems to change trend but still it increases (dashed line in Figure 14). When we compare the two contributors for the total mixing exergy loss, it is clear that in this case the waste stream is the most important reason for the mixing exergy loss. In Figure 15 it can be seen that the peaks overlap is much larger in this case. Concerning the desired exergy loss (useful products), there is a slight trend for this term to decrease. The reason for this decrease is that, at increased reduced velocities, a high degree of dilution results in a useful product stream with almost pure eluent. Up to now in our analysis we can conclude that the increase in the reduced velocity will lead to higher mixing exergy loss. In addition, we used two different particle sizes and thus we showed that the impact of the reduced velocity is higher when mass transfer effects are stronger (larger particles). It was also shown that there is a need to divide the mixing exergy loss into its two contributions in order to understand the reasons for the exergy loss. We now have to show that the lowest values for the mixing exergy loss occur in the region of maximum recovery and minimum solvent usage (Figure 16). For this, we use the smaller particle size. Minimizing the mixing exergy loss may eventually lead to lower process costs. Figure 16A presents the solvent consumed for the range of reduced velocity used here. Again it can be seen that there is a straightforward relation between mixing exergy loss and solvent used. This is so because the amount of solvent used is directly related to the product dilution when the pulse volume is held constant. More-

Figure 17. Exergy loss for increasing reduced velocity: (0) mixing exergy loss and (4) physical exergy loss.

over, product recovery decreases with increasing mixing exergy loss while keeping the purity requirement constant. 5.3.2. Case 2: Considering ∆P. Fixed-bed chromatographic separation systems require the eluent to be pumped continuously through the column. Increasing flow rates cause the pressure drop and consequently the mechanical energy consumption (pump work) to increase as well. For the particular range of reduced velocities shown in Figure 17, the mixing exergy loss is almost constant and the physical exergy loss increases dramatically. In other words, the changes in the reduced velocity have a much stronger impact on the physical term of the overall exergy loss. 6. Discussion In this contribution, we evaluate the use of exergy analysis on the optimization of a chromatographic separation. In this field of separation, simulated moving bed systems have particularly attracted much attention in recent years. Although SMB is already a powerful separation technique, there is still room for further optimization. The lack of literature concerning the use of exergy analysis of SMB units and the fact that exergy has proved to be a useful tool for energy intensive systems have challenged the authors to perform this study. We decided to start this learning process by applying the exergy balance on one single column based on the fact that SMB equipment is constituted by a series of single chromatographic columns. A sensitivity analysis investigating the influence of traditionally adjusted parameters, namely, particle size, pulse volume, and flow rate, were carried out and then the resulting exergy loss was computed. The gained knowledge is the following: 1. The initial state of the column should be reestablished so that the nonflow exergy (the exergy change of the bed itself) does not play a role in the analysis. 2. Physical exergy loss equals the pump work needed to transport the fluid phase through the column. 3. There is a so-called desired exergy loss occurring, given the mixing of eluent and products. This is an inevitable loss, given the nature of the separation technique. However, the lack of good performance leads to an unnecessary loss, called undesired exergy loss. In our approach to the problem, we studied the two terms of the overall exergy loss during a chromatographic separation, that is, the mixing and physical exergy loss, separately. The former lumps together the effect of mixing of eluent and solutes and finite driving


forces for mass transfer. The latter accounts for the presence of pressure drop. Such division of the exergy loss in its components has shown to be of much importance in revealing the nature of process irreversibilities. Within the framework of chromatography optimization, there are two main concerns, namely, eluent use and mechanical energy (pump work). The issue of reducing eluent use is important not only because it has an impact on process costs but also because it leads to product dilution. Results have shown that when mixing exergy is lower, the dilution is also lower. This result is particularly important when the analysis is broadened to include the downstream process, which recovers the products from the eluent. Product dilution may cause unnecessary primary energy consumption if distillation columns are used. At this point, the traditional optimization of solvent use is not enough any longer. Exergy analysis seems also to be useful when dealing with the problem of mechanical energy (pump work) consumption. It is well-known that mass transfer resistance limits the performance of an SMB unit. To avoid such problems, small particle sizes are recommended. However, the resulting pressure drop can be very high and then mechanical energy becomes an important issue, especially if we are dealing with largescale processes. This latter issue can be adequately handled by evaluating the physical term of the exergy loss. Having said that, the authors think that exergy analysis may be the important tool for the improvement of the entire process. For a proper comparison between the presented new chromatographic process optimization tool in this paper to existing ones (i.e., cost optimization2,5), both should yield chromatographic fixed-bed separation dimensions in terms of column diameter, length, particle diameter, and operational settings such as solvent usage for a fixed productivity at a fixed purity. When giving the same result, confidence in the optimal setup and operation is increased. Given different setup and operational settings, questions arise: which method is more valid and why? Cost optimization uses different objective functions using productivity and recovered yield as a function of loading factor and tray number. Exergy optimization includes mass transfer effects, dilution, convection, adsorption, and heat transfer in one single framework. The actual optimum may change with varying prices in cost optimization; for exergy optimization this is not the case: irrespective of prices a fixed optimum is found. As this paper is reporting on the developed exergy methodology, the actual comparison is made only superficially.

the performance of the separation, a further division of the mixing exergy loss in two parts, that is, undesired and desired exergy loss, was made. In addition, the energy consumption can be minimized looking at the physical exergy loss. A minimum total exergy loss was found in a range of particle sizes that gives high product recovery and minimum energy consumption. Therefore, the operation at minimal exergy loss may lead to a reduction in the process costs.

7. Conclusion

Subscripts

As part of the broader research project that concerns the use of exergy analysis as a tool to optimize chromatographic separations, we presented here the results concerning FB operation mode. Our goal is to demonstrate how to perform the exergy balance on a FB column and above all to learn how to interpret the results obtained. A close relation between mixing exergy loss and the quality of the separation was found. That means that higher product recoveries were achieved for low mixing exergy loss while meeting the purity specification. To obtain some insight concerning the impact of solvent regeneration and product recovery from the solvent on

Acknowledgment This research was financially supported by The Netherlands’ Department of Economic Affairs, the Department of Public Housing, Spatial Planning and Environmental Affairs, and the Department of Education, Culture and Sciences. Nomenclature a ) interfacial area (cm2) c ) concentration of mobile phase (mol/L) d ) diameter (cm) Dm ) diffusion coefficient (cm2/s) Ex ) exergy (kJ) ex ) specific exergy (kJ/mol) H ) enthalpy (kJ) h ) specific enthalpy (kJ/mol) K ) equilibrium constant k ) mass-transfer coefficient (cm/s) L ) length (cm) n ) molar flow rate (mol/min) P ) pressure (bar) q ) concentration stationary phase (mol/g) Q ) heat flow (kJ/min) R ) equivalent radius (cm) qmax ) loading capacity of the adsorbent (mol/g) T ) temperature (K) S ) entropy (kJ/K) St ) Stanton number t ) time instant (min) V ) volume (cm3) u ) velocity (cm/s) W ) shaft work s ) specific entropy (kJ/mol‚K) x ) liquid-phase mole fraction Greek Symbols ) bed porosity η ) pump efficiency τ ) reduced time ∆ ) variation φ ) volumetric flow rate (cm3/min) ν ) reduced velocity 0 ) ambient conditions chem ) chemical e ) external i ) component Int ) interstitial intra ) intraparticle mix ) mixture p ) particle phys ) physical r ) heat reservoir c ) column Superscript * ) equilibrium


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Received for review July 18, 2003 Revised manuscript received March 22, 2004 Accepted March 26, 2004 IE030603U

Exergy Analysis of Chromatographic Separations in a Fixed-Bed

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