Determination of Catalyst Loading and Shortcut Design for Binary

Ind. Eng. Chem. Res. 2010, 49, 11517–11529

11517

Determination of Catalyst Loading and Shortcut Design for Binary Reactive Distillation† Jian-Kai Cheng, Jeffrey D. Ward,* and Cheng-Ching Yu Department of Chemical Engineering, National Taiwan UniVersity, Taipei 106, Taiwan

Determination of the catalyst mass is a challenging problem in the conceptual design of reactive distillation systems. In this work, we use the concept of countercurrent cascaded vapor-liquid reactors (CCRs) to develop a shortcut method for the design of binary reactive distillation columns. An analytical expression for the theoretical minimum catalyst loading can be derived as the number of CCRs approaches infinity. On the basis of this theoretical catalyst loading, we present a calculation procedure to obtain the catalyst mass and other basic process parameters, for example, number of reactive stages and number of separation stages, without a detailed model. Three real binary systems are used to illustrate this shortcut method, and the results show that the estimated shortcut designs are similar to the optimal designs. 1. Introduction Multifunctional process units are an integral part of process intensification and reactive distillation (RD) is one of the most common examples of a multifunctional processes. Luyben and Yu1 presented an updated survey indicating that there were 1105 publication and 814 U.S. patents between 1971 and 2007, where 236 reaction systems were studied. Despite recent progress in the feasibility analysis, design, and, in some cases, control of reactive distillation, shortcut design of reactive distillation is not well understood, especially for the design of the reaction zone inside a column. Several authors have proposed methods for feasibility analysis and also for the calculation of the minimum reflux ratio.2,3 Some authors use McCabe-Thiele-type diagrams to analyze binary reactive distillation systems.4-7 Gadewar et al.8,9 apply the attainable region methodology to a countercurrent cascade of vapor-liquid continuous stirred-tank reactors (CSTRs) and explore the feasible region for reaction-separation processes. Lee et al.3 describe a method to estimate the minimum catalyst loading based on the nonreactive rectification body method, the critical composition profiles, the analysis of eigenvectors at pinch points, and geometrical design insights. Subawalla and Fair10 present a method to determine minimum catalyst requirements by simulating a series of isothermal PFRs and ideal separators in series. The outlet of each reactor is fed to an ideal separator to remove product and unreacted reactant is fed to the next PFR. The procedure is repeated until the desired overall conversion is achieved. Then, this minimum catalyst amount is multiplied by 1.2-1.3 to estimate the actual catalyst mass. Other than this, however, the quantitative description of catalyst loading and distribution in RD columns is rarely discussed. For the design of a conventional distillation column, a tradeoff between the capital cost and the operating cost can often be seen. Reflux ratio is usually a dominant design variable to indentify the most economical design. Figure 1 shows a typical design example for a conventional distillation column. Heuristically, the optimal design is usually at reflux ratio approximately * To whom correspondence should be addressed. Tel: +886-2-33663066. Fax: +886-2-2362-3040. E-mail: [email protected]. † J. K. Cheng and J. D. Ward dedicate this manuscript to the memory of Cheng-Ching Yu (1956-2009) who was the principle investigator on this project.

equal to 1.2 times the minimum reflux ratio and the number of stages approximately equal to 2 times the minimum number of stages.11 The purpose of this work is to use a similar method to find dominant design variables and then develop a shortcut method for the design of a reactive distillation column. A binary system with an isomerization reaction is considered. Our proposed shortcut method is the following. First, a minimum reflux ratio is identified. Next, an analytical expression for theoretical minimum catalyst loading as the number of reactive stages approaches infinity is employed to identify the minimum catalyst mass. On the basis of these, a near optimal reflux ratio and the corresponding total catalyst loading can be estimated using the proposed methodology. Finally, the column design can be obtained. The shortcut method is illustrated using three real binary systems.

Figure 1. Trade-off between capital cost and operating cost in the design of a conventional distillation column.

10.1021/ie101291f  2010 American Chemical Society Published on Web 10/07/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 22, 2010

FzF + Lx0 - Vy1 + νξ ) 0

(6)

where zF is the feed composition equal to zero in this case (pure A); νT is the total sum of stoichiometric coefficients, which is zero for this reaction; ν is the stoichiometric coefficient that is one for component B; ξ is the accumulated sum of the reaction amount in all of the reactors

∑ k ((1 - x ) - K1

NCCRs

ξ ) Wcat

F

xn

n

EQ

n)1

)

(7)

where NCCRs is the number of reactors. If the specification is given that Fχ of A is converted to B where χ is conversion, we have the following relationship

Figure 2. Configuration of countercurrent cascaded reactors.

2. Process Studied

ξ ) Fχ

Consider a binary system (A and B) with the reversible isomerization reaction

The reaction occurs in the liquid phase and the reaction rate can be expressed in terms of mole fractions (xi) and the catalyst mass (Wcat)

(

1 Ri ) νiWcatkF xA x KEQ B

)

(2)

where Ri is the reaction rate of component i (mol/s), νi is the stoichiometric coefficient which takes a negative value for the reactants, Wcat is the catalyst holdup (kg), kF is the forward reaction rate constant (mol/kg · sec) and KEQ is the equilibrium constant. Here, kF and KEQ are constant and independent of temperature. To simplify the notation, the component indices are eliminated by setting xB ) x and xA ) 1 - x

(

)

1 Ri ) νiWcatkF (1 - x) x KEQ

(3)

Constant relative volatility is assumed y)

Rx 1 + (R - 1)x

(4)

where x is the liquid mole fraction of component B, y is the vapor mole fraction, and R is the relative volatility. In this case, component B is lighter than component A. For such a process, the countercurrent cascade of vapor-liquid reactors in Figure 2 is studied. The outlet vapor flow (V1) can be further enriched by nonreactive distillation and the inlet liquid flow (L0) can be recycled from the separation. The reactor number is counted from the top down. To reduce the complexity of the simultaneous reaction and separation system, the following additional assumptions are made: (1) There is equal molar overflow. (2) There is a thermoneutral reaction (heat of reaction ) 0). (3) Catalyst is uniformly distributed. (4) The bottom reactor is heated to boil the vapor up. (5) There is no liquid outlet from the bottom reactor. 2.1. Modeling of Countercurrent Cascaded Reactors (CCRs). The reactive zone in a reactive distillation column is equivalent to the series of countercurrent cascaded reactors (CCRs) shown in Figure 2.8,9 The overall material balance and component balance equations around the set of reactors (see envelope I in Figure 2) are V ) F + L + ν Tξ

Combining eqs 5, 6, and 8 gives Lx0 - Vy1 + Fχ ) 0

(1)

ASB

(5)

(8)

(9)

The material balance equation up to the jth reactor (see envelope II in Figure 2) Lx0 + Vyj+1 - Vy1 - Lxj + ξj ) 0

(10)

where ξj is the accumulated sum of the reaction amount from the first reactor to the jth reactor and the corresponding equation is j

ξj )

∑W

(

catkF

n)1

(1 - xn) -

1 x KEQ n

)

(11)

where j e NCCRs. Substituting eq 9 into 10, we obtain eq 12 yj+1 )

ξj L F x + χV j V V

(12)

Given R, kF, and KEQ, eq 12 is a reactive operating line to model CCRs. Note that it can be a nonreactive operating line if ξj ) 0 for all j. Parameters of the base case are listed in Table 1. Now consider the situation at the top of the column, above the reaction section. Figure 3 shows three important system properties, xEQ, RREQ, and xint. xEQ is the composition at chemical equilibrium and defined as below xEQ )

KEQ KEQ + 1

(13)

RREQ is the corresponding reflux ratio when the nonreactive operating line intersects the reaction equilibrium composition (xEQ) RREQ )

χ - yEQ yEQ - xEQ

(14)

where yEQ is the vapor composition corresponding to xEQ. RREQ is a minimum reflux ratio for a binary reactive distillation, below which the design is infeasible because the reactive section cannot achieve a concentration greater than the equilibrium composition. Table 1. System Parameters for the Base Case parameters

value

F R kF KEQ χ xEQ

12.6 4 0.04 2 0.95 0.667

unit mol/s mol/kg · s mol/mol mole fraction


11519

Figure 3. Representation of xEQ, xint, and RREQ.

xint is the composition at the intersection of the actual nonreactive operating line and the equilibrium line, which can be found by combining eq 4 and eq 12 with ξj ) 0 to give xint ) (RR - χ)(R - 1) + R - √((RR - χ)(R - 1) + R)2 - 4RR(R - 1)χ 2RR(R - 1)

(15) If the reflux ratio, number of reactors, and desired conversion are specified, the required catalyst amount in each reactor (Wcat) can be determined by adjusting the value until the desired conversion is achieved. The total catalyst amount can then be determined by multiplying the number of reactors by the amount of catalyst in each reactor. On the basis of the above procedure, the relationship between RR, x1, NCCRs, and total catalyst amount can be explored. First, we look at the modeling of CCRs. According to eq 12, the reactive operating lines will move down because of the reaction in each reactor and are parallel because of the assumption of equal molar overflow. A McCabe-Thiele type diagram can be used to describe the composition changes and the movement of the reactive operating lines, as shown in Figure 4A. In this case, the number of reactors is 10. In particular, if the number of reactors approaches infinity, the composition change in each reactor is so small that the composition follows the equilibrium line very closely, as shown in Figure 4B. Now consider the relationship between the total catalyst amount and the number of reactors. Figure 5 shows that the total catalyst amount decreases when NCCRs increases for fixed x1 and RR. Total catalyst amount reaches a minimum value when NCCRs goes to infinity. This minimum value is a theoretical minimum catalyst amount (Wmin), below which the specification cannot be achieved. The results also indicate that a larger reflux ratio leads to a smaller theoretical minimum catalyst amount. Next, we look at the effect of x1 on the theoretical minimum catalyst amount for fixed RR and as NCCRsf∞. As mentioned above, x1 should be between xint and xEQ, and Figure 6 indicates there is only a slight difference in the theoretical minimum catalyst amount as x1 varies. The reason is that number of reactors above xint is small compared to the number below xint, as shown in Figure 4B. Therefore, the composition change above xint is dominated by the separation. In addition, the catalyst amount in each reactor is very small because NCCRsf∞. Therefore, x1 has little effect on the theoretical minimum catalyst amount. In other words, the theoretical minimum catalyst amount is a function only of RR if χ is specified. For the purpose of process design, this theoretical minimum catalyst amount can be used as a reference to estimate the required catalyst amount and the number of reactive stages. For this purpose it would be useful to have an analytical expression for the theoretical minimum catalyst holdup.

Figure 4. The application of the McCabe-Thiele method for the CCR with (A) NCCRs ) 10; (B) NCCRsf∞. (In panel B, most reactive operating lines are omitted for readability).

Figure 5. Total catalyst amount versus the number of CCRs.

2.2. Derivation of Analytical Expression for Theoretical Minimum Catalyst Loading (Wmin). When the number of singlephase CSTRs approaches infinity, we have a packed-bed reactor (PBR), as shown in Figure 7A. The catalyst loading of a PBR with a reversible reaction can be derived from the plug-flow reactor design equation12 Wcat,total ) F

∫

x1

0

(

)

KEQ + 1 dx -F KEQ ln 1 ) x RB kF KEQ + 1 KEQ 1 (16)

By analogy, when the number of CCRs goes to infinity, we have a packed reactive column, as shown in Figure 7B. Therefore, a design equation for CCRs can be determined in a similar way.

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As seen in Figure 4B, the liquid composition at the bottom reactor (xbot) approaches zero when NCCRs f∞. Before integrating eq 21, some assumptions are made: (1) xbot is assumed to be zero. (2) w ) 0 is at bottom. (3) The integral of the liquid composition will stop at xint where w ) Wmin. (For the cases with fixed RR and x1 > xint, the theoretical catalyst amount remains almost constant.) Therefore, the integral of eq 21 is taken from w ) 0 (x ) 0) to w ) Wmin (x ) xint), and we have a closed form expression for the theoretical minimum catalyst amount (Wmin) Figure 6. The total catalyst amount versus x1 at fixed RR and NCCRf∞. (Note that xint ) 0.6 and xEQ ) 0.667.)

Wmin )

Wmin )

{

∫

Wmin

0

Here, we start from the material balance on the jth reactor (see envelope III in Figure 2) and let Wcat ) ∆w. Then

(

L(xj-1 - xj) + V(yj+1 - yj) + kF (1 - xj) -

)

1 x ∆w ) 0 KEQ j (17)

Equation 17 is divided by ∆w to give L

(

)

(yj - yj+1) (xj-1 - xj) 1 -V + kF (1 - xj) x )0 ∆w ∆w KEQ j (18)

When NCCRsf∞ and ∆wf0, the equation changes from the discrete form to the continuous form and it can be written as L

(

)

dy dx 1 -V + kF (1 - x) x )0 dw dw KEQ

(19)

To replace the term dy, the expression for the vapor-liquid equilibrium (eq 4) is differentiated with respect to x Rdx dy ) [1 + (R - 1)x]2

(20)

Substituting eq 20 into 19, we obtain

dx ) dw

(

kF (1 - x) -

)

1 x KEQ

RV -L [1 + (R - 1)x]2

(21)

xint

(

0

)

(

(22)

)

xint RxEQ(RR + 1) F RRxEQ ln 1 + × kF xEQ (x (R - 1) + 1)2 EQ xint -ln 1 + ln(xint(R - 1) + 1) + xEQ xEQxint(R - 1)2 + xint(R - 1) xint(R - 1) + 1

[ (

Figure 7. Infinite number of the reactors. (A) Single-phase reactor and (B) two-phase reactor.

∫

dw )

RV -L [1 + (R - 1)x]2 dx 1 kF (1 - x) x KEQ

)

]}

(23)

where xEQ ) KEQ/(KEQ + 1), L ) F × RR, and V ) F(RR + 1). The theoretical catalyst amount depends on R, kF, KEQ, χ, RR, and F. When those system parameters vary, Figure 8 shows the theoretical minimum catalyst weight generated by eq 23 is in excellent agreement with that obtained by the simulation when NCCRsf∞. In addition, eq 16 for a single-phase plug flow reactor (PFR) is recovered for the case where R ) 1, RR ) 0 and V ) F. This derivation of the analytical solution for the minimum catalyst weight can also be extended to the reaction with A(LK)TB(HK), which is the opposite case where the feed is introduced into the top of the countercurrent cascaded reactors and there is no vapor outlet from the top reactor. Here, the reflux ratio (RR) is replaced by the boilup ratio (BR), so that BRest ) s × BREQ is used for the calculation shortcut design. An analytical expression for the theoretical minimum catalyst amount can be found by analogy with the system A(HK)TB(LK). Because component A is the light component, the component indices are eliminated by setting xA ) x and xB ) 1 - x. Then

(

Ri ) νiWcatkF x -

1 (1 - x) KEQ

)

(24)

The differential equation for the system with A(LK)TB(HK) becomes

(

)

1 (1 - x) K dx EQ ) dw RV -L [1 + (R - 1)x]2 kF x -

(25)

We make the following assumptions that are analogous to those made in the previous development: (1) The liquid composition of the top reactor (xtop) is assumed to be one. (2) w ) 0 is at top. (3) The integral of the liquid composition will stop at xint where w ) Wmin.


11521

Figure 9. Binary RD column (A) with the bottom reactive section and (B) with the top reactive section.

Figure 8. Comparison between CCRs with NCCRsf∞ and the analytical equation for the minimum catalyst weight. (A) Effect of kF and R, (B) effect of kF and KEQ, and (C) effect of kF and χ. (RR ) 2 and x1 ) 0.4 are fixed.)

Therefore, the integral of eq 25 is taken from w ) 0 (x ) 1) to w ) Wmin (x ) xint), and we have the closed form solution of the theoretical catalyst amount (Wmin)

Wmin )

∫

Wmin

0

dw )

∫

xint

1

{

RV -L [1 + (R - 1)x]2 dx 1 kF x (1 - x) KEQ

(

( [(

)

)

(26)

xint F 1 -BR(1 - xEQ)ln + kF 1 - xEQ K xint RKEQxEQ(BR + 1) 1 ln 2 1 x K (xEQ(R - 1) + 1) EQ EQ (R + KEQ)(R - 1)(1 - xint) (R - 1)xint + 1 + ln R R(KEQ + 1)[(R - 1)xint + 1] (27)

Wmin )

(

)

)

]}

where xEQ ) 1/(KEQ + 1), L ) F(BR + 1), and V ) F × BR. 3. Application of Theoretical Minimum Catalyst Loading In this section, we use the theoretical minimum catalyst weight to design a reactive distillation column. The vapor outlet from the top reactor in the countercurrent reactor cascade will

be introduced into a rectifying section and the liquid inlet to the top reactor will be recycled from the rectifying section, as shown in Figure 9A. To design a reactive distillation column, we are interested in how many stages are required in the reactive section (NRX) and rectifying section (NR), and how much energy is consumed (VS). A reflux ratio and total catalyst weight are required to obtain those design parameters. To determine a near optimal reflux ratio, we can follow the example of the shortcut design of a conventional distillation column and approximate the optimal reflux ratio as a certain factor of the minimum reflux ratio (RRopt = r × RRmin, a typical value of r is 1.2).11 For binary reactive distillation, RREQ is a minimum reflux ratio, below which the design is infeasible because the reactive section cannot achieve a concentration greater than the equilibrium composition. Therefore, we can use RREQ as a reference to estimate the optimal reflux ratio, that is, RRest ) r × RREQ. After that, the corresponding theoretical minimum catalyst weight (Wmin) can be calculated, which is also a good candidate to be a reference to estimate the total catalyst weight, that is, West ) w × Wmin. Our shortcut design procedure depends critically on the values of r and w. In Section 3.1, we show how a shortcut design can be completed if values of r and w are known. In Section 3.2, we use the design procedure “in reverse” to determine suitable values for r and w by averaging the values that give the best design over a range of process parameters. 3.1. Shortcut Design Procedure. Recall that our assumption for the derivation of the theoretical minimum catalyst weight is that kF and KEQ are independent of temperature. In a real column, the temperature range is usually between the boiling points of two components (TB,LK and TB,HK) so a reference temperature must be selected for the calculation of RREQ and Wmin. To guarantee that the values of r (RRest/RREQ) and w (West/ Wmin) are greater than 1, RREQ and Wmin should be as small as possible, and these parameters depend on KEQ and kF. Therefore, it is desired that the estimated KEQ and kF, denoted kF,est and KEQ,est, should be as large as possible. For an endothermic reaction, TB,HK is used to calculate kF,est and KEQ,est. For an exothermic reaction, TB,LK is used for calculation of KEQ,est and TB,HK for calculation of kF,est. Once the system parameters (R, kF,est, and KEQ,est) and the product specification are given, the shortcut design can be quickly obtained by the following procedure: (1) Find the largest kF,est and the largest KEQ,est for temperature between TB,LK and TB,HK. (2) Calculate xEQ and RREQ. (3) Calculate RRest ) r × RREQ and the corresponding xint. (4) Calculate Wmin from eq 23.

11522


(5) Calculate West ) w × Wmin. (6) Calculate other design parameters (VS,est, DC,est, NR,est, and NRX,est) according to the following calculation procedure. If the reflux ratio and total catalyst weight are known, and then the calculation procedure of Subawalla and Fair10 can be employed to estimate design parameters as follows (1) Assuming an F factor of 1 in engineering units, calculate the diameter of the column (DC) using the equation13 DC ) 1.735 × 10

( )

-2

MWT P

0.25

(29)

Wcat,total Fpacking

(30)

(3) Determine the height of reactive zone from the calculated catalyst volume and estimated diameter Vcat ) π 2 D 4 C

(32)

(5) In the rectifying section, the distillate composition is specified as xd and the bottom composition should be between xEQ and xint (xint < xEQ). For a more conservative estimation, xd and xint are used to calculate the minimum number of stages in the rectifying section (NR,min) by the Fenske equation and 2 × NR,min is used to represent the number of stages in the rectifying section

(

NR ) 2 ×

xd 1 - xint 1 - xd xint log R

2 0.1-10 0.05

forward (EF) backward (EB)

17000 16000

LK

HK

10.693 3862

10 3862

Wmin. In this work, two factors are considered for the determination of r and w. First, the shortcut method should lead to a design that is near to a feasible actual design, that is, if the shortcut method predicts values NR,est, NRX,est, RRest and West, then if the process is simulated rigorously using the vales NR,est and NRX,est, the values of RR and W for which the simulation converges, RRact and Wact should be close to RRest and West, respectively. Here, the difference in RR is used as the measurement, which is defined as follows ∆RR )

|RRest - RRact | × 100% RRact

(34)

Second, the actual design with NR,est and NRX,est should be near the optimal design. Here, the total annual cost (TAC) is used as the objective function, so the difference between them is defined as follows ∆TAC )

hcat ) HETPcat

log

AVP BVP

R KEQ,HK forward (kF,HK)

(31)

(4) Determine the number of equivalent reactive stages by dividing the height of reactive zone by the catalytic packing HETP NRX

vapor pressure constants

(28)

(2) Estimate the catalyst volume from the total catalyst weight and maximum packing catalyst density

hcat

relative volatility equilibrium constant at 386.2 K (THK) specific rate constant at 386.2 K (THK) (mol/kg · s) activation energy (cal/mol)

ln PSi ) AVP,i - BVP,i/T

VS0.5

VS ) D(RR + 1)

Vcat )

Table 2. Physical Properties for the Process Studied

)

(33)

Note that NRX,est and NR,est are rounded up to the nearest integer. The reaction with A(LK)TB(HK) is the opposite case where the reaction occurs in the top of the column and there is a nonreactive stripping section, as shown in Figure 9B. The shortcut design can be completed in an analogous manner, changing RRest ) r × RREQ to BRest ) s × BREQ and eq 23 to eq 27 for Wmin. BREQ is a boilup ratio where the operating line crosses the equilibrium composition. So far, the values of r and w have been not specified. The issue of how to determine them is addressed next. 3.2. Determination of the Values of r and w. Lek et al.11 revisited heuristics for the optimization of the conventional distillation columns by examining the optimal designs of 16 examples over the range 1.1 e RRopt/RRmin e 1.6. For the specific case of RRopt/RRmin ) 1.2, the total annual cost of a column designed using shortcut calculations (followed by rigorous simulation, sizing, and costing) is on average 14% more than the minimum. In the same spirit, we desire to identify values of r and w to determine the shortcut values of RR and W given RRmin and

TACact - TACopt × 100% TACopt

(35)

The best choice of r and w should minimize ∆RR and ∆TAC. Therefore, different values of r and w should be evaluated to minimize ∆RR and ∆TAC simultaneously. However, systems with different parameters (e.g., R, KEQ,HK, kF,HK) give different solutions for r and w that minimize ∆RR and ∆TAC. Therefore, we examine systems with different system parameters and evaluate the corresponding ∆RR and ∆TAC. Then, for each set of values of r and w we average the results of system designs with different values of R, KEQ,HK, kF,HK to determine ∆RRavg and ∆TACavg. Our objective is to find a range of r and w to give reasonable shortcut designs which ∆RRavg and ∆TACavg are less than 10% simultaneously. Note that KEQ,HK and kF,HK are the equilibrium constant and the forward reaction rate constant at the boiling point of the heavy component. We start with an ideal binary reactive distillation column with a bottom reactive section. The process description and details of the design procedure are listed in Appendix A. First, each system with different system parameters is optimized based on the total annual cost (TACopt). Once r and w are specified, the shortcut design of each system is generated, which includes RRest, West, NRX,est, and NR,est. Next, NRX,est and NR,est are transferred to the rigorous simulation to obtain the corresponding actual reflux ratio (RRact), total catalyst weight (Wact), and the corresponding TACact. Now we look at the effect of KEQ,HK on the shortcut design with r ) 1.2 and w ) 2. Systems with R ) 2, kF,HK ) 0.05 and KEQ,HK ) 0.1-10 are studied, and kinetic and physical property data are given in Table 2. Figure 10 shows the calculation results of the optimal design, the shortcut design and the actual design as KEQ,HK varies from 0.1 to 10. ∆RR is less than 10% for KEQ,HK below 1, while ∆RR is larger than 10% for KEQ,HK above 1. The reason for this larger deviation is that the required reflux ratio of the actual design (RRact) decreases as KEQ,HK increases. On average, ∆RR


11523

Figure 12. Contour plots of ∆RRavg with respect to r and w as R and kF,HK vary. Figure 10. Effect of KEQ,HK on the shortcut designs (r ) 1.2 and w ) 2), actual design and the optimal design at R ) 2, kF,HK ) 0.05, and KEQ ranging from 0.1 to 10. (EST: shortcut design. ACT: actual design. OPT: optimal design.)

Figure 13. Contour plots of ∆TACavg with respect to r and w as R and kF,HK vary.

Figure 11. (A) ∆RRavg profile with respect to r and w and the corresponding contour plot. (B) ∆TACavg profile with respect to r and w and the corresponding contour plot. ∆RRavg and ∆TACavg are computed by taking average ∆RR and ∆TAC for KEQ ranging from 0.1 to 10 at R ) 2 and kF,HK ) 0.05. (Dash line on the contour plots is the range with r ) 1.2-1.4 and w ) 2.0-3.2.)

for KEQ,HK ranging from 0.1 to 10 is equal to 13.7%, which is denoted ∆RRavg. Next, we look at the comparison between the optimal design and the actual design. TAC of the actual designs is very close to that of the optimal designs as KEQ,HK varies. The tradeoff comes from capital cost (affected by W and RR) and operating cost (affected by RR). Compared to the optimal designs, RR is overestimated and W is underestimated, so the errors in capital cost and operating cost offset each other. On average, ∆TAC for KEQ,HK ranging from 0.1 to 10 is equal to 3.6%, which is denoted ∆TACavg. In the same way, different values of r and w are evaluated and the corresponding ∆RRavg and ∆TACavg are computed, as shown in Figure 11. Note that ∆RRavg and ∆TACavg are computed by taking average ∆RR and ∆TAC for KEQ,HK ranging from 0.1 to 10 at R ) 2 and kF,HK ) 0.05. Figure 11 shows that the shortcut

designs given by the range of r ) 1.2-1.4 and w ) 2.0-3.2 have about 10% difference in both of RRavg and ∆TACavg. Next, kF,HK ranging from 0.01 to 0.1 and R ranging from 1.5 to 2.5 are examined to check if this range of values of r and w is still suitable if these parameters change. The results of ∆RRavg and ∆TACavg for each system are presented in Figures 12 and 13, respectively. To find a suitable range of r and w, which can cover wider ranges of system parameters, we take the average of ∆RRavg and ∆TACavg from the nine cases. As can been seen in Figure 14, the range of r ) 1.2-1.4 and w ) 2.0-3.2 gives acceptable shortcut designs, in which the average values of both of ∆RRavg and ∆TACavg are less than about 10%. Note that these ranges of r and w are based on the averaged effect of different system parameters. Consequently, for some cases they may lead to a shortcut design that deviates from the corresponding actual design and optimal design by more than 10%. 3.3. Comparison of Catalyst-Loading Calculation Method. Subawalla and Fair10 present an alternative method to determine minimum catalyst requirements by simulating a series of isothermal PFRs and ideal separators in series, as shown in Figure 15. Each PFR is assumed to be isothermal and contains just enough catalyst to achieve 99% of the equilibrium conver-

11524


Figure 16. Comparison of our proposed method (r ) 1.2 and w ) 2.4), method of Subawalla and Fair (1.2 × Wmin), and the optimal design in total catalyst amount as KEQ varies. System parameter are R ) 2, kF,HK ) 0.05 and KEQ ranging from 0.1 to 10.

where Fn is the feed flow rate to the nth reactor: Fn ) F1(1 - pχEQ)n-1

(38)

The minimum catalyst amount (Wmin,SF) using this method is: N

Wmin,SF )

∑W

n

)

j)1

Figure 14. Contour plots with respect to r and w (A) ∆RRavg, and (B) ∆TACavg. In this case, ∆RRavg and ∆TACavg are computed by taking average ∆RR and ∆TAC for R ranging from 1.5 to 2.5, kF,HK ranging from 0.001 to 0.1, and KEQ,HK ranging from 0.1 to 10. (Dash line on the contour plots is the range with r ) 1.2-1.4 and w ) 2.0-3.2.)

Figure 15. Scheme of a series of isothermal PFRs and ideal separators in series.

sion. The effluent from the first reactor is introduced into an ideal separator that perfectly separates product from reactant. Then, unconverted reactant is fed to the next reactor. The procedure is repeated until the desired overall conversion is achieved. For the binary system we have considered, the required number of reactors (N) can be expressed as follows (Appendix B) N)

ln(1 - χ) +1 ln(1 - pχEQ)

(36)

where χ is overall conversion, p is percentage of equilibrium conversion, and χEQ is equilibrium conversion. From the design equation for a PFR, the catalyst requirement in each reactor (Wn) can be obtained Wn )

Fn KEQ 1 ln kF KEQ + 1 1 - p

(

)

(37)

F1χ 1 ln kFp 1-p

(

)

(39)

where F1 is the feed flow rate to the first PFR. The calculation is carried out assuming the highest possible temperature to maximize the reaction rates and minimize the catalyst amount. For more realistic operation, for example, adiabatic operation and imperfect separation, this minimum catalyst amount is multiplied by 1.2-1.3 to estimate the actual catalyst mass. Comparison between results from our proposed method, the method Subawalla and Fair, and the optimal design is made as shown in Figure 16. The results show that our proposed method has a somewhat better prediction than the method of Subawalla and Fair. The minimum catalyst loading (Wmin,SF) determined by the method of Subawalla and Fair is greater than the minimum loading determined by our method (Wmin). In that sense, it is not truly the minimum loading but rather a reference value. Furthermore, from eq 39 in the method of Subawalla and Fair the minimum catalyst amount depends on feed flow rate, forward reaction rate constant (the specified temperature), overall conversion, and percentage of equilibrium conversion but not equilibrium conversion (equilibrium constant). However, it may be expected that the required catalyst loading is only weakly dependent on the equilibrium constant because a system with a larger equilibrium conversion requires fewer reactors (N) to meet overall conversion but each reactor requires larger catalyst mass (Wn) so the two effects offset each other. Despite this, the catalyst amount given by the method of Subawalla and Fair is not far from the optimal value. 4. Application to Real Systems In this section, we will apply the shortcut method to the design of several real systems. Three systems are studied, which are the following: (1) The isomerization of 2,3-dimethyl-2-butene (DMB2) to 2,3-dimethyl-1-butene (DMB1).


Table 5. Calculation Result of the Optimal Design and Shortcut Design for Three Systems

Table 3. Thermodynamic Model and Kinetic Model for Three Isomerization Systems system (i)DMB1

(

1 r ) W × kF xDMB2 x KEQ DMB1 -57694.1 kF ) 3.33 × 107 exp RT -8476.92 KEQ ) 2.245 exp RT

(

(

)

NRTLb

)

1 x KEQ DCL3B -35746 kF ) 874.3 exp RT 4131 KEQ ) 0.532 exp RT

r ) W × kF xDCL2B -

(

)

)

NRTLb

(

1 x KEQ EB -35503 kF ) 350.8 exp RT -9100 KEQ ) 1.943 exp RT

r ) W × kF xOX -

(

(

) )

)

DMB2TDMB1 a

13.85 849 187.1 3.06 23.5 2.3 932 -6.4c

method EST

a

DCL2BTDC3B a

a

OXTEB

ACT

a

a

a

OPT

EST

ACT

OPT

EST

ACTa OPTa

14.79 1107 187.9 3.07 24 3 922 1.9d

14.27 1427 181.8 3.02 23 4 905

0.99 1367 25.1 1.17 8.7 25.5 255 7.6c

0.88 1358 20.1 1.15 9 26 211 4.0d

0.97 996 21.2 1.19 11 18 203

39.74 4889 513.3 5.34 62.6 4.4 3554 12.3c

35.39 5686 461.4 5.38 63 5 3156 0.5d

34.90 6719 454.7 5.34 60 6 3140

a EST: Shortcut design. ACT: Actual design with NR,est and NRX,est. OPT: Optimal design. b NR does not include a condenser. NRX includes a reactive reboiler. c ∆RR ) (RREST - RRACT)/RRACT. d ∆TAC ) (TACACT - TACOPT)/TACOPT.

( )

(iii)EB

systems

RR Wcat Vs Dc NRb NRXb TAC ∆(%)

)

(

(ii)DCL3B

thermodynamic model

kinetic modela

11525

c

UNIQUAC

Table 6. Process Characteristics of Three Systems with Reverse Reactions systems

DMB1TDMB2

DCL3BTDCL2B

EBTOX

kF,est KEQ,est

0.560 9.899

0.021 0.584

0.090 7.459

Table 7. Calculation Result of Shortcut Design, Actual Design, and Optimal Design for Three Systems with Reverse Reactions

a R ) 8.314 [kJ/kmol/K], r [mol/s], W [kg], kF [kmol/kg · s], xi [mole fraction]. (i) kF and KEQ are taken from Gangadwala et al.14 (ii) kF is fabricated and KEQ is calculated by Aspen Plus. (iii) kF is fabricated and KEQ is regressed from data given by Chirico and Steele.15 b The binary parameters are predicted by UNIFAC method. c The binary parameters are built in Aspen Plus.

Table 4. Process Characteristics of Three Systems systems

DMB2 T DMB1

DCL2B T DCL3B

OX T EB

LK HK TB,LK TB,HK R ∆Hvap,avga λb kF,est KEQ,est

DMB1 DMB2 328.76 346.35 1.80 28390 3891 0.065 0.118

DCL3B DCL2B 388.00 425.65 2.70 37023 -4480 0.0353 1.91

EB OX 409.35 417.58 1.24 36194 10840 0.010 0.141

a Average heat of vaporization at the boiling point [kJ/kmol]. of reaction [kJ/kmol].

b

Heat

(2) The isomerization of 1,4-dichloro-2-butene (DCL2B) to 3,4-dichloro-1-butene (DCL3B). (3) The isomerization of o-xylene (OX) to ethyl-benzene (ZB). In all three systems the reactant is the heavy component, so the reaction occurs in the bottom of the column, as shown in Figure 9A. The thermodynamic models and kinetic models of the three systems are listed in the Table 3. Using those models, the process characteristics (R, kF,est, and KEQ,est) for the calculation of the shortcut design are obtained, as shown in Table 4. The specification and assumption for each system is the following: (1) A saturated reactant feed with feed flow rate of 12.6 mol/s is assumed. (2) Top pressure in RD column is 1 atm. (3) Ninety-nine perscent conversion of reactant is specified. First, the design of each system is optimized, and all rigorous simulations are performed in Aspen Plus. The optimal design of each system is shown in Table 5. Then, the values r ) 1.2 and w ) 2.4 are used to generate the shortcut design for each system, as shown in Table 5. After that, the values of NR,est and NRX,est generated by the shortcut method are input into Aspen

systems

DMB1TDMB2

DCL3BTDCL2B

EB T OX

method ESTa ACTa OPTa ESTa ACTa OPTa ESTa ACTa OPTa BR Wcat Vs Dc NS b NRXb TAC ∆(%)

1.58 109 19.9 1.00 19.4 2.9 160 -22.7c

2.04 55 25.7 1.15 20 3 150 12.9d

1.67 252 21.0 1.03 21 6 133

3.87 3237 48.7 1.63 10.6 31.0 496 6.7c

3.63 3803 45.7 1.74 11 32 472 0.1d

3.66 3717 46.1 1.75 12 31 472

5.32 632 67.0 1.93 51.9 4.3 545 -10.5c

5.94 927 74.8 2.17 52 5 588 5.0d

5.26 1478 66.3 2.05 56 9 560

a EST: Shortcut design. ACT: Actual design with NR,est and NRX,est. OPT: Optimal design. b NS includes a reboiler. NRX includes a reactive condenser. c ∆BR ) (BREST - BRACT)/BRACT. d ∆TAC ) (TACACT TACOPT)/TACOPT.

Plus (followed by rigorous simulation, sizing, and costing) and the simulation results are listed in the Table 5. The results show that the shortcut design for each system is close to the actual design and also near the optimal design. The average value of ∆RR for the three systems is 8.7%, and the average value of ∆TAC for the three systems is 2.1%. Now this shortcut method is applied to the opposite case where the reaction occurs in the top of the column and there is a nonreactive stripping section. The reverse reactions of the three systems are considered, so the reactants of three systems are the light component. The shortcut designs for three systems are generated by using s ) 1.2 and w ) 2.4, where kF,est and KEQ,est are listed in Table 6. The calculation results of the shortcut design, the actual design and the optimal design are summarized in Table 7. The average value of ∆RR for the three systems with the reverse reactions is 13.3%, and the average value of ∆TAC for the three systems is 6.0%. Note that for the systems with DMB1TDMB2 and ZBTOX, the Fenske equation underestimates the number of stages in the stripping section (NS,est), so Underwood’s general method is used. Details of the calculation procedure are listed in Appendix C. The reason is that the KEQ,est of the both systems is very large so that the corresponding xint is closer to the product purity. In this case, the Fenske equation underestimates the number of stages. Therefore, it is suggested that Underwood’s general method be used for systems with a larger equilibrium constant. However, as seen in Table 5 and Table 7, total catalyst weight given by the shortcut design is always lower than that of the

11526


Table 8. Re-Estimated Total Catalyst Weight systems

re-estimated West (kg)

Wact (kg)

DMB2TDMB1 DCL2BTDCL3B OXTEB DMB1TDMB2 DCL3BTDCL2B EBTOX

1107 1395 5556 112 3340 730

1169 1326 6099 155 3826 927

actual design. The difference comes in part from the fact that NRX,est in the actual design is an integer but not in the shortcut design. Therefore, to obtain more precise total catalyst weight, total catalyst weight estimated by the shortcut method should be divided by the original NRX,est and then multiplied value of NRX,est rounded up to the nearest integer. Re-estimated total catalyst weight for systems are listed in Table 8, and the average deviation is 12%.

Table 9. Calculation Result of Shortcut Design, Actual Design, and Optimal Design for Three Systems (The Simplified Analytical Equation of the Theoretical Minimum Catalyst Is Used)

5. Simplification of the Analytical Equation for Wmin In this section, some limiting conditions are considered to simplify the analytical expression for Wmin. For systems with a small equilibrium constant, the equilibrium composition is small. In this condition, the vapor-liquid equilibrium line can be assumed to follow Henry’s law, as shown as below y ) kHx

(40)

where kH is assumed to be equal to yEQ/xEQ. First consider countercurrent cascaded reactors with A(HK)TB(LK). Then, take liquid mole fraction (x) derivative of eq 40 and substitute it into eq 19 to obtain

dx ) dw

(

1 x KEQ kHV - L

kF (1 - x) -

)

(41)

Therefore, the integral of eq 41 is taken from w ) 0 (x ) 0) to w ) Wmin,S (x ) xint), and we have a simplified analytical equation of the minimum catalyst amount (Wmin,S) Wmin,S )

∫

Wmin,S )

Wmin,S

0

dw )

∫

xint

0

(

kHV - L

kF (1 - x) -

(

)

1 x KEQ

L - kHV KEQ KEQ + 1 ln 1 x kF KEQ + 1 KEQ int

dx

(42)

)

(43)

Because of the assumption of the straight VLE line, the analytical equation becomes relatively simple. Compared to the analytical equation of eq 23, the theoretical minimum catalyst weight generated by eq 43 has a deviation of less than about 10% if the equilibrium constant is smaller than 1, as shown in Figure 17. Also, eq 16 for a single-phase PFR is also recovered for the case with kH ) 1, L ) 0, and V ) F. By analogy, the simplified analytical equation of the minimum catalyst amount (Wmin,S) for countercurrent cascaded reactors with A(LK)TB(HK) can also be obtained Wmin,S ) -

(

L - kHV KEQ KEQ + 1 1 ln x kF KEQ + 1 KEQ int KEQ

Figure 17. Comparison of the analytical equation between eqs 23 and 43 as KEQ and R vary.

)

where kH is assumed to be equal to (1 - yEQ)/(1 - xEQ).

systems

DMB1TDMB2

DCL3BTDCL2B

EBTOX

method ESTa ACTa OPTa ESTa ACTa OPTa ESTa ACTa OPTa RR Wcat Vs Dc NRb NRXb TAC ∆(%)

13.85 861 187.1 3.06 23.5 2.3 933 -6.4c

14.79 1107 187.9 3.07 24 3 922 1.9d

14.27 1427 181.8 3.02 23 4 905

0.99 1652 25.1 1.17 8.7 30.8 271 14.9c

0.86 1602 19.9 1.15 9 31 221 8.9d

0.97 996 21.2 1.19 11 18 203

39.74 4945 513.3 5.34 62.6 4.4 3556 12.2c

35.39 5686 461.4 5.38 63 5 3156 0.5d

34.90 6719 454.7 5.34 60 6 3140

a EST: Shortcut design. ACT: Actual design with NR,est and NRX,est. OPT: Optimal design. b NR does not include a condenser. NRX includes a reactive reboiler. c ∆RR ) (RREST - RRACT)/RRACT. d ∆TAC ) (TACACT - TACOPT)/TACOPT.

Table 10. Calculation Result of Shortcut Design, Actual Design, and Optimal Design for Three Systems with Reverse Reactions (The Simplified Analytical Equation of the Theoretical Minimum Catalyst Is Used) systems

DMB1TDMB2

DCL3BTDCL2B

EBTOX

method ESTa ACTa OPTa ESTa ACTa OPTa ESTa ACTa OPTa BR Wcat Vs Dc NS b NRXb TAC ∆(%)

1.58 139 19.9 1.00 19.4 3.6 162 -14.4c

1.85 186 23.3 1.09 20 4 140 5.0d

1.67 252 21.0 1.03 21 6 133

3.87 3304 48.7 1.63 10.6 31.7 500 6.7c

3.63 3803 45.7 1.74 11 32 472 0.1d

3.66 3717 46.1 1.75 12 31 472

5.32 746 67.0 1.93 51.9 5.1 550 -6.5c

5.69 1066 71.7 2.13 52 6 573 2.3d

5.26 1478 66.3 2.05 56 9 560

a EST: Shortcut design. ACT: Actual design with NR,est and NRX,est. OPT: Optimal design. b NS includes a reboiler. NRX includes a reactive condenser. c ∆BR ) (BREST - BRACT)/BRACT. d ∆TAC ) (TACACT TACOPT)/TACOPT.

Finally, we apply this simplified analytical equation to generate the shortcut design with r ) 1.2 and w ) 2.4 for the above three systems. The results for the three systems with the bottom reactive section and the top reactive section are summarized in Tables 9 and 10, respectively. Compared to the previous results, the shortcut designs given by the simplified analytical equation are comparable; the average values of ∆RR and ∆TAC are 10.1% and 3.1%, respectively. 6. Extension to Binary Reaction System with Nonequal Molar Reactions

(44)

Besides the isomerization reaction, other types of binary reactions include the decomposition reaction (AT2B) and the dimerization reaction (2ATB). For such reactions, molar flow


in the reactive section changes stage by stage because the reaction is nonequal molar (VT * 0), so eqs 23 and 27 do not apply. However, Lee et al.16 present an alternative method for such reaction systems in which equal mass overflow is assumed if the heat of reaction is not significant. Therefore, the McCabe-Thiele coordinates are expressed in weight fraction. In this framework, a derivation of the expression for the minimum catalyst loading can be conducted for binary systems with nonequal molar reactions. The Fenske and Underwood equations can also be modified accordingly. Consider the decomposition reaction (AT2B) where A is heavy and B is light, so the reaction zone is placed in the bottom of the RD column shown in Figure 9A. The material balance on the jth reactor (see envelope III in Figure 2) when Wcat ) ∆w is Lw(xw,j-1 - xw,j) + Vw(yw,j+1 - yw,j) +

(

MAkF (1 - x) -

)

1 2 x ∆w ) 0 KEQ

(45)

where Lw and Vw are mass flow rates of liquid and vapor, MA is molecular weight, yw is mass fraction of component B, x and xw are liquid mole fraction and liquid mass fraction of component B. The relationship between x and xw is x)

2xw 1 + xw

(46)

Therefore, eq 45 becomes

((

Lw(xw,j-1 - xw,j) + Vw(yw,j+1 - yw,j) + MAkF 1 -

(

2xw,j 1 KEQ 1 + xw,j

)) 2

)

2xw,j 1 + xw,j

∆w ) 0

(47)

According the process of the previous derivation, the expression for the minimum catalyst loading is RVw

Wmin,w )

∫

xw,int

0

- Lw [1 + (R - 1)xw]2 2xw 2xw 1 11 + xw KEQ 1 + xw

((

kF

)

(

)) 2

dxw

(48)

On the basis of the relationship between x and xw (eq 46), the Fenske and Underwood equations can also be employed to estimate the number of stages in the separation section. 7. Conclusion Similar to the conventional distillation column, reflux ratio is also a dominant design variable for design of binary reactive distillation columns. On the basis of the concept of the countercurrent cascaded reactors, an analytical equation for the theoretical minimum catalyst amount can be derived as the number of CCRs approaches infinity. Then, a shortcut method is developed in the form of RRopt = r × RRmin (BRopt = s × BRmin) and Wopt = w × Wmin to generate a near optimal design so that the number of separation stages and the number of reaction stages can be quickly obtained. Heuristically, a suitable range of r (or s) ) 1.2-1.4 and w ) 2.0-3.2 is recommended. For the specific case of r (or s) )1.2 and w ) 2.4, the calculation results of the shortcut design have about 11.3% deviation from that of the shortcut design (followed by rigorous simulation, sizing, and costing). In addition, the total annual cost of the shortcut design is on average 7.25% more than that of the optimal design. Although this shortcut method

11527

cannot guarantee the optimal design, it can be used for a preliminary process design or as the initial case for a rigorous optimization. So far, this shortcut method is limited to a binary system and a small heat of reaction because the derivation of the expression for the theoretical catalyst minimum weight is based on the assumption of equal molar overflow or equal mass overflow and a thermo-neutral reaction. However, the insight obtained from the study of the binary system can be used as a guide to facilitate investigation of more complicated ternary and quaternary systems. Acknowledgment This work was supported by the Ministry of Economic Affairs (98-EC-17-A-09-S1-019) and National Taiwan University. Appendix A: Process Description of a Case with A(HK)TB(LK) An ideal binary reactive distillation system is used to illustrate the effects of R, kF,HK, and KEQ,HK on r and w for short design. Consider the following first order liquid phase reversible reaction

(A1)

A(HK) S B(LK)

The forward and backward specific rates following the Arrhenius law and the rate constants on tray j are kF,j ) aFe-EF/RTj

(A2)

kB,j ) aBe-EB/RTj

(A3)

where aF and aB are the pre-exponential factors, EF and EB are the activation energies, and Tj is the absolute temperature on tray j. The reaction rate on tray j can be expressed in terms of mole fractions of component B (xj) and the catalyst weight (Wcat).

(

Rj,i ) νiWcatkF,j (1 - xj) -

1 x KEQ,j j

)

(A4)

where Rj,i is the reaction rate of component i on the jth tray (mol/s), i is the stoichiometric coefficient that takes a negative value for the reactants, Wcat is the catalyst weight on a reactive tray (kg) and KEQ, j is equal to kF,j/kB,j. Note that catalyst weight comes from the column sizing by considering a HETPcat ) 0.333 m and Fpacking ) 150 kg/cum. Assumptions made in this work include the following: (1) Saturated reactant feed is assumed, and FA ) 12.6 mol/s. (2) Constant relative volatility is assumed. The tray temperature is computed using a bubble point temperature calculation given Antoine vapor pressure coefficients. That is S S P ) xj,APA(T + xj,BPB(T j) j)

ln PSi ) AVP,i -

BVP,i T

(A5) (A6)

where P is total pressure (bar) and PS denotes the vapor pressures. Note that to ensure constant relative volatility, the Antoine coefficients, BVP’s, are the same for both components. (3) Equal molar overflow is assumed because the heat of the reaction of most isomerization reactions is very small. (4) Vapor holdup and pressure drop are neglected. For the process optimization, the TAC is used as the objective function to indentify the economically optimal design and the specifications are 99% purity level for the products. The TAC is defined as

11528


TAC ) operating cost +

capital cost payback time

(A7)

Here, a payback time of 8 years, catalyst price of $3.5/lb, and energy price of $4.7/106 Btu are used. A catalyst life of 3 months is also assumed. The formula for the TAC computation is taken from Kaymak and Luyben.13 Given the product specification (xd ) 0.99), design variables (Figure 9A) include the number of reactive trays (NRX), the numbers of trays in rectifying section (NR), and feed tray location for the reactant (NF). The bottom temperature is the highest, so the feed with the heavy reactant should be introduced into the bottom base to benefit the reaction. In the following optimization procedure, the feed location (NF) is fixed to the bottom base, and the other design variables are changed one at a time to minimize the design objective.

Wn ) Fn )

)

dX

pχEQ

0

kF[(1 - x) -

1 x] KEQ

1 KEQ + 1 1pχEQ KEQ

Fn KEQ 1 ln kF KEQ + 1 1 - p

(

)

)

(B6)

The minimum catalyst amount (Wmin,SF) is the sum of the catalyst amount in all of the reactors: N

)

Subawalla and Fair10 present a method to determine minimum catalyst requirements by simulating a series of isothermal PFRs and ideal separators in series, as shown in Figure 15. In this case, the reaction of ATB is considered. The composition of component B is x, so the composition of component A is 1 x. If the temperature of PFR is specified, an equilibrium conversion (χEQ) at that temperature is

dX RB

(

∑W

n

j)1 N

Appendix B: Derivation of the Subawalla and Fair method

pχEQ

0

Fn KEQ ln ) kF KEQ + 1

Wmin,SF )

χEQ )

Fn

∫ ∫

∑ j)1

)

F1(1 - pχEQ)j-1 KEQ 1 ln kF KEQ + 1 1 - p

(

F1 KEQ 1 ln kF KEQ + 1 1 - p

)

N

( ) ∑ (1 - pχ ) F K 1 1 - (1 - pχ ) ln( k K + 1 1 - p ) 1 - (1 - pχ ) F [1 - (1 - pχ ) ] 1 ln( kp 1 - p) j-1

EQ

j)1

N-1

)

1

F

EQ

EQ

EQ

EQ

N-1

)

1

EQ

F

KEQ KEQ + 1

(B7)

(B1) Substituting eq B5 into B7, we obtain

Assume pure feed (x ) 0) with the flow rate F1 into to the first reactor. Each isothermal PFR contains just enough catalyst to achieve p × 100% equilibrium conversion at that temperature. This means in each PFR the conversion is pXEQ. If the desired overall conversion (χ) is specified, then F1χ ) F1pχEQ + F2pχEQ... + FnpχEQ

(B2)

Fn ) F1(1 - pχEQ)

(B3)

Substituting eq B3 into B2, we obtain F1χ ) F1pχEQ + F1(1 - pχEQ)pχEQ... + F1(1 - pχEQ)n-1pχEQ ) F1pχEQ[1 + (1 - pχEQ)... + (1 - pχEQ)n-1] ) F1pχEQ

(1 - pχEQ)n-1 - 1 (1 - pχEQ) - 1

) F1[1 - (1 - pχEQ)

n-1

]

(B4) Rearranging eq B4, we obtain the required number of reactors (N) N)

ln(1 - χ) +1 ln(1 - pχEQ)

(

(

)

)

Appendix C: Calculation of Number of Separation Stages

The feed flow rate to the feed to the jth PFR is n-1

F1[1 - (1 - pXEQ)ln(1-χ)/ln(1-pχEQ)] 1 ln kFp 1-p F1χ 1 ln ) kFp 1-p (B8)

Wmin,SF )

(B5)

From the design equation of PFR, the catalyst requirement in each reactor can be obtained shown as below

Underwood17 developed an ingenious algebraic procedure to calculate the number of equilibrium separation stages in a staged column. For the case of A(HK)TB(LK) where reaction occurs at the bottom of the column and there is a nonreactive rectifying section, a quantity φ is defined such that RR + 1 )

RHKxHK,D RLKxLK,D + RLK - φ RHK - φ

(C1)

where RLK, RHK, xLK,D and xHK,D are the relative volatility and the distillate composition of the light component and the heavy component. In our case, we set RHK ) 1, RLK ) R, xHK,D ) 1 - xd, xLK,D ) xd. Therefore RR + 1 )

1 - xd Rxd + R-φ 1-φ

(C2)

If the distillate purity is specified and RR is known, φ can be determined from eq C1, where there are two real solutions which satisfy R > φ1 > 1 and 1 > φ2 > 0. Once φ1 and φ2 are obtained, the number of stages in the rectifying section can be calculated by the following equation.

(

)


1 R x + x 1 - φ2 LK,F R - φ2 HK,F NR ) / ln(φ1 /φ2) 1 R xLK,F + xHK,F 1 - φ1 R - φ1

xLK,F )

(C3)

(

xEQ + xint xEQ + xint xHK,F ) 1 2 2

)

(C4)

For the opposite case, where reaction occurs at the top of the column and there is a nonreactive stripping section, a quantity j is defined φ -BR )

RHKxHK,B RLKxLK,B + ¯ RLK - φ RHK - φ¯

(C5)

where RLK, RHK, xLK,B, and xHK,B are the relative volatility and the bottoms composition of the light component and the heavy component. In our case, we set RHK ) 1, RLK ) R, xHK,B ) 1 - xb, xLK,B ) xb. Note that xb is the bottoms composition and xB is the mole fraction of component B. Therefore -BR )

Rxb R - φ¯

+

1 - xb 1 - φ¯

(C6)

j can If the bottom’s purity is specified and RR is known, φ be determined from eq C6, where there are two solutions which j 1 > R and R > φ j 2 > 1. satisfy ∞ > φ j 2 are obtained, the number of stages in the j 1 and φ Once φ stripping section can be calculated by the following equation.

(

)

1 R xLK,F + xHK,F 1 - φ¯ 1 R - φ¯ 1 NS ) / ln(φ¯ 1 /φ¯ 2) 1 R xLK,F + xHK,F 1 - φ¯ 2 R - φ¯ 2 where xi,F is the top composition in stripping section:

)

(C8)

Literature Cited

where xi,F is the composition of component i at the bottom of the rectifying section. Note that the feed is to the reboiler. As mentioned above, the bottom composition in the rectifying section is between xEQ and xint. Note xEQ and xint are very close because RR and RREQ are close. Unlike for the Fenske equation, xint cannot be selected as the bottom composition in the rectifying section, because xint is the composition where the nonreactive operating line intersects the equilibrium curve. Therefore, it is a pinch point and an infinite number of stages would be required. To estimate NR, the bottom composition in rectifying section (xi,F) is modified to xLK,F )

(

xEQ + xint xEQ + xint xHK,F ) 1 2 2

11529

(C7)

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ReceiVed for reView June 16, 2010 ReVised manuscript receiVed August 26, 2010 Accepted September 14, 2010 IE101291F

Determination of Catalyst Loading and Shortcut Design for Binary

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