Article pubs.acs.org/IECR
Finite Separating Degree in Feasibility Analysis Methods of Reactive Distillation Process Jianchu Ye, Yong Sha,* Yale Xu, and Daowei Zhou Department of Chemical Engineering and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, Fujian, China ABSTRACT: A ∞/∞ analysis method was developed to evaluate the effect of feasibility analysis methods on designing the reactive distillation (RD) process. Its mathematic derivation revealed that results from the reactive flash-cascade model were the same as results from the ∞/∞ analysis method only if the predicted column discharge was a pure compound or a common azeotrope under chemical equilibrium, but those two results were different when the predicted column discharge was a reactive azeotrope. The same phenomenon also appeared in comparison between the singular point equation method and the reactive flash-cascade model rebuilt by a single mole-based parameter. In the case of the reactive azeotrope appearing in the column discharge, both the theoretical discovery and the case study suggested that the separating degree of a RD column predicted by the feasibility analysis was finite, and it may be broken through in reality by means of imposing greater separating power, such as more theoretical trays or a bigger reflux ratio. Results from the feasibility analysis should be checked carefully because appropriate operation parameters can change the predicted separating performance of the RD column.
1. INTRODUCTION Reactive distillation (RD) is a highly integrated unit operation and has shown its significant advantages in chemical industry due to the combination between reaction and distillation separation.1,2 So far, lots of works have been carried out for the purpose of both fundamental understanding and industrial application of RD. Generally, the coupled kinetic-thermodynamic feasibility analysis is the first step to develop a new RD process.3−6 Besides the feasibility of the process, it can also provide necessary information for further conceptual design and experimental verification while the process is feasible.7−9 Based on the different simplification and abstraction of the real RD process, various feasibility analysis methods and models were proposed and developed to acquire as much information as possible, for example, shortcut design methods,10,11 the continuous multiphase reactor model,12,13 the reactive flashcascade model,14 and the singular point equations established by Qi et al.15 In general, a feasibility analysis was carried out with various Damköhler numbers Da as the parameter, and it can influence the existence and location of fixed points in the residue curve map. Therefore, feasible separations can be classified as a function of the Damkohler number. As an important result from the feasibility analysis, the possible product composition at the bottom and top of a RD column can be predicted theoretically for a fixed Da. It is worthy of noting that these methods had no need for preset operation parameters of a RD column: for example, the reflux ratio and the theoretical tray number. Prediction of the possible product compositions was based on the basic kinetic-thermodynamic properties of the RD system. It is well-known that a feasibility analysis of a conventional distillation process, such as residue curves or distillation curves, can accurately give the discharge composition of the distillation column under the infinite separating power. Unlike the conventional distillation, feasibility analysis of a RD process should be examined carefully. For example, with respect to the © 2015 American Chemical Society
esterification of cyclohexene and formic acid to cyclohexyl formate, the feasibility analysis of Steyer et al. indicated that the column top discharge was a ternary mixture containing cyclohexyl formate.16 However, the rigorous design work of Katariya et al. indicated that cyclohexyl formate hardly existed in the top discharge even under total reflux.17 Therefore, it was possible for this process that the results from the feasibility analysis could lead to a wrong judgment. Meantime, it also indicated that the operation parameters of a RD column, such as the reflux ratio, can result in a deviation between the actual performance and the prediction from the feasibility analysis. In general, results from the feasibility analysis of the RD process play an important role in the consequent RD design, and the predicted production composition is expected to be an essential guide. However, there was no systematic discussion of the correspondence of the results from feasibility analysis with the actual performance of a RD process, though it is very helpful to re-evaluate the effects of feasibility analysis on designing reactive distillation processes. Comparison of these methods can show quantitative difference among methods, and understanding of the difference is helpful for choosing feasibility analysis methods for a RD process. It is necessary to evaluate the accuracy of the product composition predicted by feasibility analysis methods. In this work, a ∞/∞ analysis method for the RD process was established, and it evolved from the distillation curve method of conventional distillation processes and can give the possible discharge compositions of a RD column with infinite separating power. The comparison between the current common feasibility analysis models and the ∞/∞ analysis model was Received: Revised: Accepted: Published: 12607
June 13, 2014 April 27, 2015 November 12, 2015 November 12, 2015 DOI: 10.1021/ie504842h Ind. Eng. Chem. Res. 2015, 54, 12607−12614
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Industrial & Engineering Chemistry Research J
carried out by mathematical analysis and a case study. The effects of feasibility analysis on designing RD processes were evaluated.
∑ υT
ln − 1 − ln +
j
j=1
kj kf ,j
Dacol , jR n , j = 0
(1 ≤ n ≤ N) (1)
The species mass balance of the n tray (1 ≤ n ≤ N) was simplified from eq A5 to th
2. ∞/∞ ANALYSIS METHOD FOR THE RD PROCESS The configuration of the RD column for ∞/∞ analysis is shown in Figure 1, and notations used in the subsequent
ln − 1(xi , n − 1 − xi , n) + yi , n + 1 − yi , n J
+
∑ (υi ,j − υT xi ,n) j
j=1
kj kf ,j
Dacol , jR n , j (2)
=0
In eq 2, xi,0 = yi,1 and yi,N+1 = xi,N. With the phase equilibrium relation eq A6 between yi,n and xi,n, the compositions of vapor and liquid phases in the RD column with infinite separating power can be solved by eqs 1 and 2 by this ∞/∞ analysis. Therefore, this ∞/∞ analysis method was also considered as a feasibility analysis method, in fact. It needed only Da as the parameter, and it also can predict the discharge composition of a RD column. The acquired composition profiles could be called “the reactive distillation curve”. Actually, when Da = 0, it becomes “the distillation curve”, which can be used to analyze the feasibility of the conventional distillation process. The ∞/∞ analysis method can always give the maximum separating degree for any reactive system, since its configuration implies that it has infinite separating power, i.e. the total reflux and infinite trays. Results from the ∞/∞ analysis further approach the actual situation and can be regarded as a basic reference. Therefore, it is convenient to investigate feasibility analysis methods by means of deviations and comparisons between results from the ∞/∞ analysis method and other current feasibility analysis methods. By means of results from the ∞/∞ analysis method as the reference, results from different feasibility analysis methods for the RD process were investigated in the following section by means of rigorous mathematic analysis and a case study.
Figure 1. Configuration of a RD column for ∞/∞ analysis.
derivation are marked in this figure. Like the conventional distillation column, the RD column in Figure 1 had no feed or discharge, so the top and bottom compositions were not influenced by the feed composition. Moreover, it was imposed to be operated with infinite reflux ratio and infinite trays, and this meant that the separating power of this RD column was infinite. As a RD column, reaction was supposed to exist at every tray except the nonreactive reboiler and condenser. This ∞/∞ analysis method was different from other current feasibility analysis methods because the enthalpy balance was considered. It was supposed that each component had a constant enthalpy value so that further simplification can be carried out. This lead to the result that the ∞/∞ analysis further approached the actual situation because the restraint of the enthalpy balance on the RD column exists in reality. At the steady state of this RD column, the equations of the overall and species mass balance and enthalpy balance are given in Appendix A. Because of the unequal relation hl ≠ hv for the enthalpies of the vapor and liquid components, the comparison between the overall mass balance equation eq A2 and the enthalpy balance equation eq A4 showed that the vapor flow rate Vn should be a constant, which is noted as V. The Damköhler number of each tray is defined as Dacol,n,j = kf,jmcat,n/ V (1 ≤ n ≤ N). While the catalyst amount mcat,n on each tray remained the same, Dacol,n,j was constant, which was noted as Dacol,j. This was an important feature of the ∞/∞ analysis method developed in this work. This suggested that a ∞/∞ analysis can be carried out under a specific Damköhler number Da, and it can provide reasonable comparison between the ∞/∞ analysis method and other feasibility analysis methods. The relation between the liquid flow rates can be acquired as below from eqs A2 and A4, where ln = Ln/V and l0 = lN = 1.
3. RESULTS AND DISCUSSION 3.1. Finite separating degree in the reactive flashcascade model. The ∞/∞ analysis method was based on the single mole-based parameter. For the convenience of comparison, the reactive flash-cascade model equations in Chadda et al.’s work,14 which used Da and the mass-based vapor fraction β as two parameters for a nonequimolar reaction case, were rebuilt with one mole-based parameter in Appendix B. As the fixed-point criteria of the reactive flash-cascade process, eqs 3 and 4 can be obtained when the left terms of eqs B5 and B6 in Appendix B were equal to zero. For convenience, the subscript “n” was removed. J
xi − yi +
∑ (υi ,j − υT xi) j
j=1 J
yi − xi +
∑ (υi ,j − υT xi) j
j=1
kj kf ,j kj kf ,j
Da1s , jR j = 0 (3)
Da1r , jR j = 0 (4)
In eqs 3 and 4, the parameter Da integrated two original parameters Da and β in the reactive flash-cascade process. and Da1s,n,j = Da*s,n,j/β1s,n and Da1r,n,j = Da*r,n,j/(1 − β1r,n). Obviously, the possible discharge composition from the RD column bottom and top should meet eqs 3 and 4, respectively. 12608
DOI: 10.1021/ie504842h Ind. Eng. Chem. Res. 2015, 54, 12607−12614
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Industrial & Engineering Chemistry Research For an actual RD process, if it was assumed that the flashcascade method can achieve the same prediction as the ∞/∞ analysis method, it suggested that the possible column bottom composition should simultaneously meet eq 2 from the ∞/∞ analysis method and eq 3 from the flash-cascade method. The two equations were rewritten with the new subscript under a fixed Damköhler number as below. J
∑ (υi ,j − υT xi ,N )
xi , N − yi , N +
j
j=1
kj kf ,j
DajRN , j = 0
J
xi , N − 1 − yi , N +
∑ (υi ,j − υT xi ,N − 1) j
j=1
analysis is also helpful to determine the feeding ratio of the entrainer. For a kind of reactive system, such as methyl acetate production by esterification,18,19 while the predicted column discharge corresponded to a pure compound or a common azeotrope under chemical equilibrium, eq 9 can be satisfied. For this kind of RD process, it is suggested that the flash-cascade method can also predict the same results as those from the ∞/∞ analysis method with an infinite separating power. However, for other reactive systems, while their stable singular point was a reactive azeotrope resulting from coupling between reaction and liquid−vapor phase equilibrium, failure of eq 9 indicated the composition predicted by the flash-cascade analysis model was nearly impossible to be the same as those by the ∞/∞ analysis method. The ∞/∞ analysis method can provide information on the composition at the bottom or top of the RD column with an infinite separating power. If the stable singular point of a reactive system was a reactive azeotrope, the inconsistency between results from the flash-cascade analysis model and the ∞/∞ analysis method indicated that the composition at the RD column bottom or top predicted by the flash-cascade analysis model resulted from a finite separating power, and the separating degree of the RD column was finite. In other words, results from the reactive flash-cascade model were unspecific, and it represented the finite separating degree in comparison with the maximum separating degree predicted by the ∞/∞ analysis method. Obviously, for the case of the reactive azeotrope, the finite separating degree predicted by the reactive flash-cascade model could be unfavorable, since it could result in the wrong judgment for further development of the RD process, but it also could be favorable because the prediction could be broken through by means of changing separating power, i.e. changing tray numbers and the reflux ratio. 3.2. Universality of finite separating degree. Although the analysis above was obtained merely by comparison between the reactive flash-cascade model and the ∞/∞ analysis method, it can be extended to other current feasibility analysis methods, for example, Venimadhavan et al.’s method20 and Qi et al.’s method.15 These methods had the same composition equation of singular points as eq 3 at the column bottom. The same expression form indicated that the possible bottom products predicted by these two methods were also related to a finite separating degree. In Qi et al.’s method,15 the possible discharge composition at the column top was calculated by eq 10, which had a different expression form from eq 4.
(5)
kj kf ,j
DajRN , j = 0 (6)
After subtracting eq 5 by eq 6, eq 7 can be obtained. ⎛ (xi , N − xi , N − 1)⎜⎜1 − ⎝
J
⎞ ⎟ Da R j N , j⎟ = 0 j kf ,j ⎠
∑ υT j=1
kj
(7)
Based on the mass balance equation (eq 1) of the RD column, 1 − ∑Jj=1υTj(kj/kf,j)DajRN,j = lN−1, the value of the term lN−1 was always greater than 0, so 1 − ∑Jj=1υTj(kj/kf,j)DajRN,j ≠ 0. Therefore, it can be concluded that the relation xi,N−1 = xi,N = xi was the only solution of eq 7. In addition, the tray number N of the RD column approached infinity in the ∞/∞ analysis, so the liquid compositions of several trays approaching the column bottom should also be equivalent or approximately equivalent due to the continuity of the calculated results. This suggested that the relation xi,N−2 = xi,N−1 = xi,N = xi was true near the RD column bottom. Similarly, near the RD column bottom, RN−2,j = RN−1,j = RN,j = Rj and yi,N−2 = yi,N−1 = yi,N = yi. In this situation, the following eq 8 can be obtained from eq 2. J
∑ (υi ,j − υT xi) j
j=1
kj kf ,j
DajR j = 0 (8)
Therefore, the relation yi − xi = ∑Jj=1(υi,j − υTjxi)(kj/kf,j) Dacol,jRj can be obtained on the base of eq 5 or eq 6. Accordingly, eq 9 was finally derived as the condition that the flash-cascade analysis and the ∞/∞ analysis can give the same predicted result. The same equation describing the possible discharge composition at the RD column top can also be derived from eqs 1−2 and eq 4. J
xi − yi =
∑ DajR j = 0 j=1
J
yi − xi +
(9)
∑ (υi ,j − υT yi ) j
j=1
Obviously, eq 9 always held true for the nonreactive system, and it suggested that the reactive flash-cascade model can always predict the maximum separating degree of a nonreactive distillation operation. It was consistent with common sense. As the intrinsic phase equilibrium characteristics of the conventional distillation system under a fixed pressure, this maximum separating degree does not change with some operating parameters such as the reflux ratio and the theoretical tray number. Therefore, it is easy to determine whether the separation target can be achieved through a distillation operation or not by means of the feasibility analysis. In addition, for an azeotrope distillation process, the information about the maximum separating degree from the feasibility
kj kf ,j
DaREACL , jR j = 0 (10)
In eq 10, DaREACL,j = kf,jmcat/L and DaREACL,j = kf,jmcat/L. Qi et al.’s method also had the finite separating degree discussed above when it was used to predict the possible top product. In order to prove this finite separating degree, it was necessary to establish the mathematical relationship between the results of Qi et al.’s model and the reactive flash-cascade model. As shown in Appendix B, the mass balance equations of the rectifying section of the reactive flash-cascade process can be simplified in two different ways. Therefore, the fixed-point criterions of the reactive flash-cascade process can also be described by eqs 11 and 12, which are different expression forms from eqs 3 and 4. 12609
DOI: 10.1021/ie504842h Ind. Eng. Chem. Res. 2015, 54, 12607−12614
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Industrial & Engineering Chemistry Research J
xi − yi +
∑ (υi ,j − υT yi ) j
j=1 J
yi − xi +
∑ (υi ,j − υT yi ) j
j=1
kj kf ,j kj kf ,j
These thermodynamic parameters were obtained by means of the software Aspen Plus.
Da 2s , jR j = 0 (11)
ln γi =
Da 2r , jR j = 0
∑j xjτjiGji ∑k xkGkj
+
∑ j
⎛ ∑ x τ G ⎞ ⎜⎜τij − m m mj mj ⎟⎟ ∑k xkGkj ⎠ ∑k xkGkj ⎝ xjGij
(12)
(14)
In eqs 11 and 12, Da2s,n,j = Da*s,n,j/(1 − β2s,n) and Da2r,n,j = Da*r,n,j /β2r,n. Equation 12 can be utilized to calculate the possible discharge composition at the RD column top, and it obviously had the same equation form as eq 10 in Qi et al.’s method. The relation between β1r,n and β2r,n in eq B10 suggested that Qi et al.’s method and the rebuilt flash-cascade model can predict the same discharge composition at the column top when the parameters in these two methods met eq 13.
Table 2. Parameters of the Antoine Equation log(psat) = a − b/(T + c) (T in K and psat in Pa) a b c
A
B
C
22.3688 3112.67 −83.263
20.9923 3150.52 −51.020
21.2235 3367.73 −56.420
(13)
Based on the feasibility analysis methods investigated in this work, the possible bottom and top compositions of a RD column with a specific Damköhler number were calculated and presented in Figure 2. In Figure 2, the x-axis and y-axis,
In other words, the compositions of the singular points predicted by eq 10 of Qi et al.’s method and eq 4 of the rebuilt flash-cascade model should have the same changing path with the increase of Da, and the difference between the results from the two models was only due to different definitions of the model parameter Da. Therefore, the finite separating degree also existed in Qi et al.’s method. 3.3. Discussion with case study. In this section, a case study was presented to illustrate the conclusions derived above clearly. A hypothetical ternary reaction-separation A/B/C system was utilized in the case study because it was difficult to find a suitable existing system in reality. The hypothetical reaction was A + B ⇔ C. The reaction rate constant k = (1.0 × 1011) exp(−75000/RgT) kmol/(kg·s), and the dimensionless rate of the hypothetical reaction was expressed as R = xAxB − xC/Ka with Ka = (5.0 × 10−4) exp(30000/RgT). These hypothetical kinetic data resulted from rounding of the kinetic parameter of the reaction where TAME was produced from 2M1B and methanol.21,22 The hypothetical phase equilibrium thermodynamics data of the A/B/C system resulted from the isopropanol (A)/toluene (B)/cyclopentanone (C) system because this system had a relatively ideal thermodynamic behavior: the reactants A and B formed the only azeotrope, and the product C was the heaviest component. The system had no saddle point and distillation boundary because there was no existence of the additional azeotrope. The vapor phase of the hypothetical ternary system was assumed to be ideal, and the nonideality of the liquid phase was described by the NRTL equation (eq 14) with Gij = exp(cijτij) and τij = aij + bij/T. For this hypothetical system, the binary interaction parameters are shown in Table 1. The saturated vapor pressures of the components were calculated by the Antoine equation with the parameters shown in Table 2.
Figure 2. Effect of the Damköhler number Da on the composition of possible discharge compositions at the (a) bottom and (b) top of the column.
DaREACL , j =
Da1r , j J
kj
1 + ∑j = 1 υTj k Da1r , jR j f ,j
Table 1. Parameters of the NRTL Method
aij aji bij bji cij/cji
A/B
B/C
C/A
5.4034 −8.9671 −1696.1053 3572.991900 0.3
0.0 0.0 410.4028 −307.5687 0.3
0.0 0.0 227.1068 82.8049 0.3
respectively, represented the mole content of A and B in singular points, and Da was in the range 0 to 1.0 with a difference of 0.1 between two consecutive points. In order to present the relationship between calculated results from different feasibility analysis models, some singular points obtained under the same Da, such as 0.1, were circled with the Da value. 12610
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Industrial & Engineering Chemistry Research Results from the ∞/∞ analysis are also shown in Figure 2 for the convenience of comparison. With regard to the infinite separating power, the tray number N of the RD column for the ∞/∞ analysis was set to 200 in the calculation. Calculated results showed that more column trays had no influence on the discharge composition. In order to locate the calculated composition of stable singular points in the same region, Da values in the calculation for the ∞/∞ analysis were in the range of 0 to 0.05 with a difference of 0.01 between two consecutive points. In all calculations, the pressure was fixed at 101.3 kPa, and the reference temperature of 360 K was chosen to calculate the reference reaction rate constant. It should be noted especially that all of those points presented in Figure 2b were vapor compositions, since the top discharge of the distillation column was the vapor from the first tray actually, not the liquid. Figure 2 showed that, with the increase of Da, the changing paths of the possible discharge composition were the same for Qi’s model and the flash-cascade model, although the calculated compositions at the same Da were not always the same. For the top discharge composition, ∑Jj=1υTjRj was less than 0 due to forward reaction near the column top, and this led to the later appearance of singular points calculated by Qi et al.’s method in comparison with those by eq 12 from the rebuilt flash-cascade model. However, as shown in Figure 2b, the difference between results from those two models was obvious when Da was moderate, and the reasons were that the composition of the singular point was close to the chemical equilibrium and Rj was close to zero while Da was large. For the hypothetical ternary system in this work, these two current common feasibility analysis methods failed to meet eq 9. As shown in Figure 2, their results were obviously different from results from the ∞/∞ analysis. This was attributed to the higher separating degree of the ∞/∞ configuration. In comparison with two current common feasibility analysis methods, the ∞/∞ analysis results showed that the content of the heavy component C was lower at the top discharge of the RD column with the infinite separating power, and the content of the light component A was also lower in the bottom discharge. Results showed the same trend: at the bottom of the column, the concentration of C decreased while the concentration of B increased with the increase of Da; at the top of the column, the amount of A increased when Da increased. The comparison among calculated results confirmed that the current feasibility analysis methods for the RD process corresponded to a finite separating degree if eq 9 was unsatisfied. In order to further show this finite separating degree of the current feasibility analysis methods, two approximately real RD columns with different separating power were simulated: one with higher separating power and another with lower separating power. In the calculation, the enthalpy of each component was still supposed to be a constant, so modeling equations of the real RD column can be simplified as eqs 15 and 16, where l0 = 1 − d and lN = 1 + b. The liquid and vapor compositions (xi,n and yi,n, 1 ≤ n ≤ N) in the column with the certain feed information and operating parameters can be calculated by these equations. In calculation, the flow rate of the distillate was set to a very low value to avoid the influence of the column feed on the discharge composition as much as possible. All of the parameter setup is summarized in Table 3. As shown in Figure 2, the calculated results from the higher separating power were
Table 3. Feed Information and Operating Parameter of the Real Column Calculateda N = 30; r = 2; Da = 0:0.01:0.1 N = 10; r = 0.1; Da = 0:0.1:1 a
b = f/1000
d = f/1000
“real_higher” in Figure 2a “real_lower” in Figure 2a
“real_higher” in Figure 2b “real_lower” in Figure 2b
F = 1 kmol/s; xfA = 0.4; xf B = 0.3; nf = N/2.
marked by real_higher, and calculated results from the lower separating power were marked by real_lower. J
ln − 1 − ln +
kj
∑ υT
j
kf ,j
j=1
Dacol , jR n , j = 0
(1 ≤ n ≤ N) (15a)
J
ln − 1 + f − ln +
∑ υT
j
j=1
kj kf ,j
Dacol , jR n , j = 0
(n = nf ) (15b)
ln − 1(xi , n − 1 − xi , n) + yi , n + 1 − yi , n J
+
∑ (υi ,j − υT xi ,n) j
j=1
kj kf ,j
Dacol , jR n , j = 0
(1 ≤ n ≤ N ) (16a)
ln − 1(xi , n − 1 − xi , n) + f (xfi − xi , n) + yi , n + 1 − yi , n J
+
∑ (υi ,j − υT xi ,n) j
j=1
kj kf ,j
Dacol , jR n , j = 0
(n = nf ) (16b)
As shown in Figure 2, the separating degree of this real RD column was always less than that predicted by the ∞/∞ analysis when Da ≠ 0. The less the separating power of the real RD column was, the farther the changing path of the discharge composition was away from that of the ∞/∞ analysis. However, there was no obvious relationship among the results of the feasibility analyses shown in Figure 2. The comparison among these calculated results further confirmed that the separating degree of the current feasibility analysis methods was finite. Beside the example presented in this work, some calculated results in the literature also supported the conclusion acquired in this work from another viewpoint. For example, in the work of Qi et al.,15 the feasibility analysis for the IPOAC production process by esterification showed that the transformed compositions of HOAC and IPA in the top discharge were 0.425 and 0.656, respectively, at Da = 0.33. This discharge composition did appear in a real RD column with single feed and a Da number of 0.33 when the separation degree of this RD column was finite, and its reflux ratio and tray number of the column were 0.7 and 15, respectively. The predicted result from the feasibility method failed to meet the critical condition (eq 9), so it was actually corresponding to a specific and finite separating degree other than a maximum separating degree. However, it was difficult to specify this finite separating degree related to prediction by the feasibility method unless rigorous simulation or experimental work was carried out. This finite separating degree suggested that results from current feasibility analysis methods needed to be examined 12611
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carefully before further design of the RD process, especially for the case of a reactive azeotrope. However, this finite separating degree could be broken through in a real RD column by means of higher reflux ratio or more theoretical trays, since it did not accomplish the maximum separating degree. It should be noted that the ∞/∞ analysis method implied the infinite separating power so as to maximize the separating degree, and other current feasibility analysis methods had no information about separation power: i.e. tray numbers and flux ratio. The ∞/∞ analysis method could be attractive because it can give important reference information, such as the discharge composition of a RD column with infinite separating power and maximum separating degree.
Vn + 1yi , n + 1 + Ln − 1xi , n − 1 − Vnyi , n − Lnxi , n +
∑ υi ,jkjmcat , nR n, j j=1
=0
(1 ≤ n ≤ N)
(A5)
Phase equilibrium equations
■
yi , n = K i , nxi , N
(A6)
APPENDIX B: REBUILDING THE REACTIVE FLASH-CASCADE MODEL OF CHADDA ET AL. (2001) The overall and component mass balances for the nth stage in the stripping cascade and in the rectifying cascade shown in Figure B1 are expressed as
4. CONCLUSIONS As an evaluation tool, the ∞/∞ analysis method for the RD system was established in this work, and it utilized Da as the only parameter. It can predict the discharge composition of a RD column with maximum separating degree. The compositions of the singular points predicted by Qi et al.’s method and the rebuilt flash-cascade model were similar, and they had the same changing path with the change of Da. The difference between the results from these two models were attributed to different definitions of the model parameter Da. The ∞/∞ analysis method gave the different prediction from that of the current common feasibility methods while the predicted bottom or top product was neither a pure compound nor a common azeotrope under chemical equilibrium. While the composition of the stable singular point corresponded to a reactive azeotrope, the composition predicted by current feasibility analysis models did not correspond to the composition of the RD column with an infinite separating power, and it was actually achieved in a finite separating degree. It may be broken through in reality with the help of a greater separating power, such as more theoretical trays and higher reflux ratio. This phenomenon was obviously different from designing a non-RD process, where feasibility analysis can give the exact limitation of the conventional distillation.
J
Ln − 1 +
∑ υT kjmcat ,nR j = Vn + Ln j
(B1)
j=1 J
Ln − 1xi , n − 1 +
∑ υi ,jkjmcat ,nR j = Vnyi ,n + Lnxi ,n j=1
(B2)
J
Vn − 1 +
∑ υT kjmcat ,nR j = Vn + Ln j
(B3)
j=1 J
Vn − 1yi , n − 1 +
∑ υi ,jkjmcat ,nR j = Vnyi ,n + Lnxi , n j=1
(B4)
■
APPENDIX A: MESH EQUATIONS FOR ∞/∞ ANALYSIS At the steady state, the balance equations and phase equilibrium equations in a RD column for the ∞/∞ analysis are given as below. Overall mass balance equations L0 − V1 = 0
(Condenser)
Figure B1. Reactive stripping and rectifying cascade process.14
(A1) J
Vn + 1 + Ln − 1 − Vn − Ln +
From the above equations, the composition difference of the liquid phase between two consecutive stages (n − 1, n) in the stripping or rectifying section can be derived by eliminating the specific parameter Ln:
∑ υT kjmcat ,nR n,j = 0 j
j=1
(1 ≤ n ≤ N)
LN − VN + 1 = 0
(A2)
(Reboiler)
xi , n − xi , n − 1 = β1s , n(xi , n − yi , n )
(A3)
J
+
Enthalpy balance equations
∑ (υi ,j − υT xi ,n) j
j=1
J
Vn + 1hv + Ln − 1hl − Vnhv − Lnhl + hl ∑ υTjkjmcat , nR n , j = 0
kf ,j
Das*, n , jR j (B5)
yi , n − yi , n − 1 = (1 − β1r , n)(yi , n − xi , n)
j=1
J
(1 ≤ n ≤ N)
kj
(A4)
+
∑ (υi ,j − υT xi ,n) j
j=1
Species balance equations 12612
kj kf ,j
Dar*, n , jR j (B6)
DOI: 10.1021/ie504842h Ind. Eng. Chem. Res. 2015, 54, 12607−12614
Article
Industrial & Engineering Chemistry Research where the Damköhler number Da*s,n,j = kf,jmcat,n/Ln−1 and the mole-based vapor fraction β1s,n = Vn/Ln−1 in the stripping section, Da*r,n,j = kf,jmcat,n/Vn−1 and β1r,n = Vn/Vn−1 in the rectifying section. In addition, eqs B2 and B4 can also be respectively simplified by eliminating Vn to eqs B7 and B8, which are different expression forms from eqs B5 and B6.
■
+
∑ (υi ,j − υT yi ,n ) j
j=1
*Tel.: +86 13606052265. Fax: +86 592 2183054. E-mail:
[email protected].
kj kf ,j
Das*, n , jR j
Notes
The authors declare no competing financial interest.
(B7)
■
yi , n − yi , n − 1 = β2r , n(yi , n − xi , n) J
+
∑ (υi ,j − υT yi ,n ) j
j=1
kj kf ,j
J
kj
∑ υT
j
j=1
J
β1r , n + β2r , n = 1 +
∑ υT
j
j=1
kf ,j
kj kf ,j
REFERENCES
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Dar*, n , jR j (B8)
In these two equations, β2s,n = Ln/Ln−1 and β2r,n = Ln/Vn−1. The relationships among the phase fractions can be easy to be derived from eqs B1 and B3: β1s , n + β2s , n = 1 +
AUTHOR INFORMATION
Corresponding Author
xi , n − xi , n − 1 = (1 − β2s , n)(xi , n − yi , n ) J
s, r = reactive stripping and rectifying cascade process, respectively sat = saturated vapor pressure
Das*, n , jR j (B9)
Dar*, n , jR j (B10)
Notation
b, d = dimensionless discharge flow rate of column bottom and top, respectively Da = Damköhler number Da* = the original Damköhler number in the flash-cascade process F = feed flow rate, kmol/s f = dimensionless feed flow rate h = enthalpy, kJ/kmol J, N = number of reactions and stages, respectively k = forward reaction rate constant, kmol/(kg·s) kf = forward reaction rate constant at the reference temperature, kmol/(kg·s) K = thermodynamic equilibrium constant Ka = chemical equilibrium constant l = dimensionless liquid flow rate mcat = catalyst amount, kg p = pressure, Pa r = reflux ratio R = dimensionless reaction term Rg = universal gas constant, = 8.314 kJ/(kmol·K) T = temperature, K V, L = vapor and liquid mole flow rate, respectively, kmol/s y, x = mole fraction of vapor and liquid phase, respectively xf = mole fraction of feed
Greak Letters
β = phase fraction of the reactive flash vessel υ = stoichiometric coefficient of component υT = mole change of reaction
Subscripts
1s, 1r = reactive flash-cascade model built by eliminating Ln 2s, 2r = reactive flash-cascade model built by eliminating Vn col = column i, j, n = component, reaction and stage, respectively REACL = reactive condenser built by Qi et al. (2004) 12613
DOI: 10.1021/ie504842h Ind. Eng. Chem. Res. 2015, 54, 12607−12614
Article
Industrial & Engineering Chemistry Research (20) Venimadhavan, G.; Malone, M. F.; Doherty, M. F. Bifurcation study of kinetic effects in reactive distillation. AIChE J. 1999, 45, 546− 556. (21) Luyben, W. L. Comparison of Pressure-Swing and ExtractiveDistillation Methods for Methanol-Recovery Systems in the TAME Reactive-Distillation Process. Ind. Eng. Chem. Res. 2005, 44, 5715− 5725. (22) Luyben, W. L.; Yu, C. C. Reactive Distillation Design and Control; John Wiley & Sons: Hoboken, NJ, 2008.
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DOI: 10.1021/ie504842h Ind. Eng. Chem. Res. 2015, 54, 12607−12614
