Stage Difference Equation and Its Application in Distillation Synthesis

Jun 16, 2009 - This paper introduces the stage difference equation (SDE) for ... stage difference equation (RSDE) is proposed based on reaction and va...
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Ind. Eng. Chem. Res. 2009, 48, 6715–6722

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Stage Difference Equation and Its Application in Distillation Synthesis Wende Tian,*,† Suli Sun,‡ and Qingjie Guo† College of Chemical Engineering and School of Polymer Science and Engineering, Qingdao UniVersity of Science & Technology, Qingdao, P. R. China

This paper introduces the stage difference equation (SDE) for distillation synthesis, which yields a series of liquid composition profiles for a plate tower. SDE is derived from mass balance through one common column section and identifies the relationship between two joined sections by its parameters. As these parameters need to be determined at first before applying SDE, their relations between composition and reflux ratio are then examined. SDE is used in nonreactive distillation and reactive distillation synthesis where the reactive stage difference equation (RSDE) is proposed based on reaction and variable transformations. Application case studies to nonreactive and reactive distillations indicate that these equations can facilitate the determination of operating regime and conveniently produce optimal synthesis solutions. 1. Introduction Exploitation of new products is of increasing importance resulting from growing societal demands. For a given product performance requirement, process synthesis develops the corresponding process flow structure and its inner subsystem function and, then, provides optimal system combination under a given objective.1 As distillation column is one of key unit operations in process industry, its synthesis problem has received more and more attention. Shortcut methods have proven to be highly successful when applied to the distillation of regular mixtures. They introduce equations for estimating the minimum number of theoretical trays and the minimum reflux ratio required for a particular separation. The study of nonideal and azeotropic mixtures has resulted in the development of the residue curve method. For example, Castillo et al.2 employed residue curve and distillation line maps to design single-feed columns for a finite-reflux ratio. With an operating leaf shaped in total and minimum reflux conditions, the procedure is noniterative with respect to the reflux and boil-up ratios. Because residue curve maps enable one to gain insight into infinite-reflux column behavior only, Tapp et al.3,4 used the difference point equation to obtain column profile maps (CPMs). They analyzed the behavior of distillation for ideal thermodynamics and discussed how to create more creative designs using CPMs. Due to growing energy saving concerns, reactive distillation in the chemical and process industries has received more and more attention;5–8 consequently, Barbosa and Doherty9–11 presented phase diagrams for simultaneous chemical reaction and phase equilibrium with ideal and nonideal systems. Ung and Doherty12 then introduced a new set of composition variables to treat phase equilibrium in multicomponent and multireaction systems with or without an inert component. The CPM method includes the following two features. First, because CPM (or residue curve) is focused on batch fractioning initially, it can handle a packed column synthesis problem well but not a plated column synthesis problem yet. It essentially approximates integer programming using a continuous model, so there needs to be several transformation steps when it is applied to a plate column, leading to a low calculation precision * To whom correspondence should be addressed. E-mail: tianwd@ qust.edu.cn. Tel.: +86-532-84022026. † College of Chemical Engineering. ‡ School of Polymer Science and Engineering.

and slow speed consequently. Second, the key synthesis/design concept in these methods is how to define separation boundaries. There remains no clear and exact understanding of separation boundaries and no straightforward way of accurately computing them in practice. Lucia et al.13,14 proposed a geometric methodology to define exact separation boundaries by differential geometry and dynamical systems theory. But formulation is needed to construct a constrained global optimization problem. Moreover, adequate calculations must be performed prior to distillation boundaries’ determination using CPMs, leading to a complex synthesis process. Motivated by the liquid composition change over a distillation column plate through mass balance, this paper proposes a stage difference equation (SDE) which can be developed into a tool and further exploited in the staged distillation synthesis encountered in the chemical industry. Because residue curve and CPM methods which are initially designed to deal with synthesis problem for packed towers, the purpose of SDE is aimed at the staged tower case hence. But with modification about difference operator and variables, SDE can equally be used in a packed tower scene, including reactive distillation, batch and continuous processes, and nonideal systems, etc., as residue curve and CPM are.15–18 But, this is out of scope of this paper. As the composition in SDE is discrete about each stage, the solutions of SDE constitute the composition profile within a staged column. Therefore the operating region for a given product can be readily expressed by SDE, and its boundaries can be easily obtained through certain equation transformation of SDE. The applications of SDE to nonideal columns demonstrate that it can be utilized for distillation synthesis. Since SDE requires no additional data, the high computing load and special programming formulation can be avoided partially. In the present study, a mass balance approach is presented for deriving SDE, with the goal of providing a systematic framework for guiding the choice of suitable operating points in nonreactive and reactive distillation synthesis. This paper is organized in five sections. In the following section, the proposed SDE is derived in more detail first, and its parameters and revision for reactive distillation are discussed thereafter. Section 3 illustrates the distillation synthesis procedure with SDE and the reactive stage difference equation (RSDE), as applied to chloroform-benzene-acetone and methyl tert-butyl ether (MTBE) cases, respectively. The major conclusions reached from the analysis of the application results are presented in section 4.

10.1021/ie8015158 CCC: $40.75  2009 American Chemical Society Published on Web 06/16/2009

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Figure 3. Diagram of material balance between column sections.

Considering the liquid composition change through plate n ∆xi,n+1 ) xi,n+1 - xi,n

Figure 1. Schematic diagram of one column section.

(3)

Equation 2 can be rewritten as ∆xi,n+1 ) xi,n+1 -

V V y + yi,T - xi,T L i,n+1 L

(4)

Neglecting the stage number subscript, given by eq 4 ∆xi )

Vyi,T - Lxi,T V L-V (xi - yi) + xi + L L L

(5)

Let ∆)V-L*0 L ∆ Vyi,T - Lxi,T ) ∆ r∆ )

xi,∆

Figure 2. Diagram of material balance in one column section.

(6) (7) (8)

then the SDE is obtained 2. Derivation of SDE

∆xi )

2.1. Assumptions. (A) The vapor-liquid phase lies in equilibrium on a plate. (B) The molar rate is constant for vapor and liquid flow through a plate. As the frequently used premises in distillation design, these two assumptions are very important to simplify vapor-liquid phase equilibrium calculation.19 2.2. Derivation. For either a simple or complex column, their main column body consists of several sections,20 which have some general characteristics and are separated by feed and/or siding flow (Figure 1). In each section, the general characteristics hereby mean: (i) there exist only interior vapor and liquid flow passing stages; (ii) the constant molar flow assumption implies that the total vapor and liquid molar flow rates remain constant; (iii) heat and mass transfer with phase equilibrium occurs on each stage. SDE is derived from the analysis of these sections. A mass balance ranging from a certain plate to the top of one section, denoted as the rectangle of the dashed and dotted line in Figure 2, can be expressed as follows: Vyi,T + Lxi,n ) Vyi,n+1 + Lxi,T

(1)

So xi,n )

V V y - yi,T + xi,T L i,n+1 L

(2)

(

)

1 1 + 1 (xi - yi) + (xi,∆ - xi) r∆ r∆

(9)

2.3. Discussion of SDE Parameters. (i) ∆ and r∆. It can be seen from their definitions (eqs 6 and 7) that they are merely associated with vapor and liquid flow rate, so they are constant within one section according to the constant molar flow assumption in subsection 2.1. (ii) xi,∆. Equation 1 can also be expressed as Vyi,T - Lxi,T ) Vyi,n+1 - Lxi,n

(10)

Vyi,n+1 - Lxi,n VyTi - LxTi ) ∆ ∆

(11)

and

Equation 11 indicates that, in one section according to the definition of xi,∆ (eq 8), the value of xi,∆ on each plate equals that on the top, thus xi,∆ can be taken as constant within one single section. (iii) Effect of Feed and Side Stream on ∆ and xi,∆. A mass balance of component i between two consecutive column sections, showed as the dashed and dotted rectangle in Figure 3, produces V2yi,T2 - L2xi,T2 + Fxi,F ) V1yi,B1 - L1xi,B1 + Sxi,S (12)

Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

Vyi,T + Lnxi,n ) Vyi,n+1 + LTxi,T + Vi

∑ dε dt

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(18)

where ε denotes the molar extent of reaction ε)

ni - ni0 Vi

(19)

To eliminate the reaction term in eq 18, component k is taken as the reference component Vyk,T + Lnxk,n ) Vyk,n+1 + LTxk,T + Vk

∑ dε dt

(20)

Then eq 18 and 20 are divided by Vi and Vk, respectively, and their difference results in Figure 4. Diagram of material balance in one reactive column section.

V

(

It can bee seen from eqs 1, 6, and 8 that V2yi,T2 - L2xi,T2 ) ∆2xi,∆2

(13)

V1yi,B1 - L1xi,B1 ) V1yi,T1 - L1xi,T1 ) ∆1xi,∆1

(14)

) (

∆1xi,∆1 - Fxi,F + Sxi,S ∆2

Yi,T ) (15) Xi,n )

Similarly, total mass balance around the rectangle in Figure 3 meets ∆2 ) ∆1 - F + S

isobutene + methanol T MTBE

Yi,n+1 )

(16)

Equation 15 and 16 show the effect of feed and side streams on ∆ and xi,∆ of SDE. These two equations indicate that SDE parameters are influenced by the section exterior although this itself originates from section interior deduction. As showed earlier, the SDE can be easily used in one section given ∆, r∆, and xi,∆. For a liquid composition, a profile line can be determined through a ∆xi ) f(xi,yi) relationship of SDE. SDE also depicts the liquid composition change in neighboring plates, which can represent the separating effect of one plate. 2.4. SDE for Reactive Distillation. On the basis of SDE, a reactive stage difference equation (RSDE) will be deduced in this section. For simplicity, chemical reaction, of which reactants and products can exist in the vapor and/or liquid phase, is supposed to take place only in liquid phase. In liquid phase, catalysis is easy to realize and concentration of reactants and products is greater than that in gas phase, so the chemical reaction process becomes more feasible in liquid phase. Accordingly, the liquid phase reaction assumption can be satisfied in many cases. For example, in the nonideal system of MTBE which is an important chemical used as an antiknock agent in gasoline, the following liquid phase reaction occurs9

(

Xi,T )

(

)

( (

) )

yk,T yi,T /(Vk - VTyk,T) Vi Vk

(22a)

xk,n xi,n /(Vk - VTxk,n) Vi Vk

(22b)

)

(22c)

xk,T xi,T /(Vk - VTxk,T) Vi Vk

(22d)

yk,n+1 yi,n+1 /(Vk - VTyk,n+1) Vi Vk

(

)

Equation 21 can be rewritten as V(Vk - VTyk,T)YiT + Ln(Vk - VTxk,n)Xi,n ) V(Vk - VTyk,n+1)Yi,n+1 + LT(Vk - VTxk,T)Xi,T

(23)

or Xi,n )

V(Vk - VTyk,n+1) LT(Vk - VTxk,T) Yi,n+1 + X Ln(Vk - VTxk,n) Ln(Vk - VTxk,n) i,T V(Vk - VTyk,T) Y Ln(Vk - VTxk,n) i,T

(24)

Omitting the stage number subscript, the liquid composition difference between stage n and n + 1 can be presented ∆Xi ) Xi,n+1 - Xi,n LT(Vk - VTxTk) V(Vk - VTyk) Y X + ) Xi Ln(Vk - VTxk) i Ln(Vk - VTxk) Ti V(Vk - VTyTk) Y Ln(Vk - VTxk) Ti V(Vk - VTyk) (X - Yi) + ) Ln(Vk - VTxk) i Ln(Vk - VTxk) - V(Vk - VTyk) Xi + Ln(Vk - VTxk) V(Vk - VTyk,T)Yi,T - LT(Vk - VTxk,T)Xi,T (25) Ln(Vk - VTxk)

(17)

The other assumptions for RSDE are the same as those for SDE. The future work of this paper will investigate the validity of RSDE for both liquid and vapor phase reaction situation simultaneously. The presence of reaction will change the liquid flow rate continuously, so the constant molar flow assumption is not valid any more. Mass balance over a section with reaction (Figure 4) leads to

)

If we now introduce the transformed composition variables11 defined by eqs 22a-22d

So eq 12 can be rewritten as xi,∆2 )

) (

yk,T xk,n yk,n+1 yi,T xi,n yi,n+1 + Ln )V + Vi Vk Vi Vk Vi Vk xk,T xi,T (21) LT Vi Vk

Let

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∆ ) V(Vk - VTyk) - Ln(Vk - VTxk) Ln(Vk - VTxk) ∆

(27)

V(Vk - VTyk,T)Yi,T - LT(Vk - VTxk,T)Xi,T ∆

(28)

r∆ ) X∆ )

(26)

Finally, RSDE is achieved ∆Xi )

(

)

1 1 + 1 (Xi - Yi) + (X∆ - Xi) r∆ r∆

(29)

It can be found that RSDE has the same equation form as SDE despite the different variable and parameter definitions. Consequently, analysis of RSDE parameters can be carried out referring to SDE similarly, and only conclusions are given here: (i) ∆ is constant for one reactive section; (ii) r∆ + 1 )

Vk - VTyk (r + 1) Vk - VTyTk ∆T

Table 1. Parameters in SDE for Simple Column Case SDE parameters section

Where r∆T )

(

)

Vk - VTxTk

1 + 1 (Vk - VTyTk) - (Vk - VTxTk) rext

(iii) X∆ is constant through one reactive section. 3. Distillation Synthesis with SDE This work defines a feasible operating region with triangular coordinates and limits to ternary nonreactive systems or reactive systems with three independent components. Only key components will be considered when analyzing a column with more than four components, e.g. crude oil distillation, with this method. The first step using SDE for distillation synthesis is to determine each section’s operating range (operating leaf). The next step is to present the intersections of all the operating leafs, and to choose the optimal integration solution within them. In this investigation, only a simple column synthesis process is introduced in this section, as illustrated in Figure 5. In this case, parameters in SDE for the rectifying section become ∆)V-L)D r∆ ) xi,∆ )

Figure 5. Diagram of a simple distillation column.

L L ) )R ∆ D

Vxi,D - Lxi,D (V - L)xi,D Vyi,T - Lxi,T ) ) ) xi,D ∆ D D

and those for a stripping section become ∆ ) V' - L' ) -W r∆ ) xi,∆

V′ + W L′ ) ) -S - 1 ∆ -W

-Wxi,W V′yi,T ′ - L′xi,T ′ ) ) xi,W ) ∆ -W

These parameters are listed in Table 1. 3.1. Plot of the Operating Leaf. For distillation synthesis, the operating leaf collects all the possible liquid compositions

∆

rectifying stripping

D -W

r∆

x∆

R -S - 1

xD xW

on a tray under various reflux ratios for a given product. It is a closed region, bounded by a distillation line and pinch point line, and filled by a large number of composition profile lines. The liquid composition profile of a staged column at total reflux is the distillation line that passes through the product. And, the locus of points on composition curves whose tangents pass through the product composition is termed the pinch point line.21 Composition profile lines turn into distillation lines in the case of total reflux, and their end points form the pinch point line under various reflux ratios. Thus, a derivation of SDE when r∆ ) +∞ depicts the distillation line ∆xi ) xi - yi

(30)

In addition, because the points in the pinch point can be reached only when passing infinite number of stages from a product, the driving forces for mass transfer then become infinitesimal and the composition difference through one stage ∆xi approximates zero consequently. SDE is hereby simplified as

(

)

1 1 + 1 (xi - yi) + (x∆ - xi) ) 0 r∆ r∆

(31)

Because yi and xi satisfy the phase equilibrium relationship (eq 32) and eq 31 is a function of xi, xi can be calculated from eq 31 solved. The pinch point line is then formed connecting different xi point under different reflux ratios. p0i γi x yi ) P i

(32)

It can be concluded from the above discussion that liquid composition calculation on a tray is the fundamental step to plot an operating leaf. The calculation model for liquid composition on the nth stage is shown in Figure 6, consisting of two types of variables, liquid composition xn,i and temperature T, and two types of equations, SDE (eq 6) and the normalization equation of xn,i (eq 33).

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c

∑x

n,i

)1

(33)

i)1

In this model, the following variables are known previously: liquid composition on the previous tray xn-1,i, pressure P, R, and xD of SDE (Table 1). Because SDE is a function about the nth stage and the stage number is omitted from this equation for simplicity, liquid compositions are calculated stage by stage. That is, unknown quantity xn-1,i is first solved using the model on stage n - 1, then it turns into one constant to solve xn,i when applying the model at stage n. The vapor composition yn,i is only taken as intermediate variables to reduce variable dimension (variable number), for a large variable dimension can increase computing load and decrease computing robustness. For a chloroform (A)-benzene (B)-acetone (C) system with the required column bottom composition of [0.3 0.6 0.1], Figure 7 demonstrates the operating leaf shape in the stripping section. A right angled triangle coordinate system is used in Figure 7 to express the relation among the three compositions, in which each apex, side, and inner point represent a pure component, binary mixture, and ternary mixture, respectively. The horizontal and vertical axes denote the composition of acetone and chloroform, respectively, the subtractive remainder of which by unity is the composition of benzene. The program is coded in Matlab 6.5 software, and Table 2 provides the process parameters appearing in eqs 9 and 32. The maximum stage number in Table 2 means the calling number of the above model to obtain stage composition points constituting one composition profile line. Because one composition profile line is composed of compositions from stage 1 to +∞, it is impossible to give all the points on it. So an upper limit of point number (maximum stage number) should be given to plot it. The Wilson method is chosen as the activity coefficient model shown in eq 32. Due to the nonlinearity of eq 32 for xn,i and yn,i, the single stage model is solved by a Gauss-Newton algorithm. Such a method differs from the traditional residue curve method. The residue curve method has been widely used in the synthesis and design of distillation column sequences that separate azeotropic mixtures. A residue curve is defined as the locus of the liquid composition during a simple distillation process. It has been shown that residue curves represent operating lines of continuous packed columns operating at total reflux, while distillation lines represent the composition of the liquid phase on each plate for a staged-column operating at total reflux.21 Figure 8 illustrates two different distillation lines for the operating leaf in Figure 7, using SDE and residue curve methods, respectively. A conclusion can be drawn hereby that the residue curve method will introduce some errors and should not be adopted directly in determining the plate type distillation line. On the other hand, the residue curve method is needed to calculate composition profile lines through a large quantity of plates in order to connect their end points to form the pinch point line, unlike SDE which only uses eq 31. The SDE method is characterized by a short calculation time and a high calculation precision, comparatively.

Figure 6. Diagram of the liquid composition calculation model on a tray.

Figure 7. Operating leaf of the stripping section. Table 2. Parameters in Plotting the Operating Leaf parameter

value

maximum stage number molar volume at boiling point, m3/kmol

20 0.080 (A), 0.089 (B), 0.074 (C)

interaction coefficient in Wilson model, kJ/kmol A B C

polynomial coefficient of saturated vapor pressurea A B C pressure, kPa initial temperature, K a

a

A

B

C

0 207.7 486.2

-677.5 0 3387.4

-2122.2 -1436.1 0

b

c

d

e

f

73.7058 -6055.6 0 -8.9189 0.000008 2 169.65 -10314.8 0 -23.5895 0.000021 2 71.3031 -5952.0 0 -8.53128 0.000008 2 101.3 338.83 (average boiling point)

ln(pi0) ) a + b/(c + T) + d ln(T) + eTf where pi0 (kPa), T (K).

Figure 8. Comparison of two methods in determining the operating leaf.

3.2. Feasible Operating District. Each operating leaf includes all the possible liquid compositions for a given product (overhead, tower bottoms, or side runoff), so their overlapping parts represent the feasible operating district of a column. Figure 9 describes an operating regime calculated from the chloroform, benzene, and acetone system, with parameters listed in Table 2. The stripping section leaf is same as Figure 7, and the rectifying section leaf is calculated with the required column overhead composition [0.08 0.02 0.90]. Because the operating

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Figure 9. Feasible district.

leaf of the rectifying section is larger than that of the stripping section, the maximum stage number equals 40 when plotting the latter. Each intersection point in the overlapping district corresponds to a feasible solution (theoretically perfect plate number, reflux ratio, and feed thermal state) for distillation operation. The optimal distillation synthesis program can be chosen from these solutions through an appropriate mathematic programming solver, with the investment cost plus operating cost as an overall objective function. Boundary determination is a key to the feasible district calculation. At present, the residue curve method must plot as many composition profile lines as possible to obtain the operating leaves profile.4 So it has a great calculation workload and cannot exactly get the feasible district boundary. Lucia et al.13,14 give the mathematical proof that distillation boundaries correspond to locally longest residue curves. Perturbation theory and the implicit function theorem explore the relationship between neighboring trajectories while local congruence and a variety of limiting arguments are used to establish the relative distances between neighboring residue curves. Their method is required to formulate a generalized mathematical programming problem and can systematically determine distillation boundaries. In contrast, the SDE method is easier and more convenient since it directly determines the boundary from eqs 28 and 29. 3.3. Plot of the Operating Leaf with RSDE. Although there are some common aspects between distillation synthesis with RSDE and that with SDE, for example, they all make use of distillation line, pinch point line, operating leaf, etc., the former is more complex due to the interaction of reaction and distillation. RSDE is first used to calculate the operating leaves for reactive distillation, during which the reflux/ reboiler ratio step is adaptively adjusted to guarantee composition calculation convergent. In addition, the relative error between two adjacent calculated compositions is monitored to determine the end point of the pinch point line. The next step in synthesis based on RSDE is to obtain feasible operating regime and optimal operating points. Because the feasible regime is enclosed near reactive azeotrope, a small step iteration is needed to produce operating points. Figure 10 illustrates the operating leaves for reactive MTBE system with reaction eq 17 and n-butane as inert. Transformed compositions according to eqs 22 and with MTBE as the reference component are used in Figure 10, where the required head composition of [0.1124 0.0015 0.0086 0.8775] and bottom composition of [0.0001 0.0195 0.9781 0.0023] are transformed to [0.1200 0.0100 0.8700] (XTD) and [0.4945

Figure 10. Operating leaves for reactive distillation. Table 3. Parameters in Plotting Reactive Operating Leaves parameter

value

components

i-butene (A), methanol (B), MTBE (C), n-butane (D) 30 0.094650 (A), 0.040762 (B), 0.11981 (C), 0.099660 (D)

maximum stage number molar volume at boiling point, Vi (m3/kmol)

binary coefficients in Wilson model, Aij (kJ/kmol)a

A

B

C

D

A B C D

0 10822.8 1140.6 0

714.0 0 1706.8 1605.8

-127.0 6229.6 0 0

0 9592.3 0 0

polynomial coefficients of saturated vapor pressureb A B C D pressure, kPa initial temperature, K a

a

b

c

d

e

f

57.8859 59.8373 83.1465 66.9450 800

-4236.31 -6282.89 -6284.79 -4604.09

0 0 0 0

-6.81038 -6.37873 -10.4252 -8.25491

9.39886e-6 4.61746e-6 9.47316e-6 1.15706e-5

2 2 2 2

371.95 (average boiling point)

(

)

Aij Vj exp Vi RT Aij ) 0 f Λij ) 0 implies ideality Λij )

c

ln γi ) 1 - ln(

c

∑xΛ ) - ∑ j

j)1

ij

k)1

xkΛki c

∑xΛ j

b

kj

j)1 f

ln(pi0) ) a + b/(c + T) + d ln(T) + eT where pi0 (kPa), T (K).

0.5043 0.0012] (XTB), respectively. Parameters appearing in such a procedure are listed in Table 3. It can be observed from Figure 10 that bottom product MTBE does not lie in the reactive area, so one nonreactive stripping column is needed besides one reactive rectifying column to obtain pure MTBE. The stripping leaf is bounded by the outer distillation line and bottom horizon pinch point line. The shape of rectifying leaf is somewhat complex, bounded by the left upper pinch point line and right bottom distillation line. Figure 10 indicates that composition changes sharply between XTD and point A [0.18 0.42 0.40], so feasible solutions should be chosen avoiding this composition range.

Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009

Conceptual methods of distillation synthesis frequently use a graphical representation of important design variables, which allows the design engineer to gain insights into and guide the development of promising column designs. It can be seen from Figures 9 and 10 that SDE and RSDE form just one graphical method to assess the feasibility of a proposed column, for both nonreactive and reactive distillation columns, for a given pair of products. That is, points on the composition profiles of the rectifying section and stripping section, limited within the intersection district of two operating leaves, must coincide if the proposed separation is feasible. The McCabe-Thiele method is one classical graphical step-by-step construction utilizing operating and equilibrium lines to compute the number of ideal stages needed to accomplish a definite concentration difference in either the rectifying or the stripping section. In contrast with it, SDE and/or RSDE can handle ternary or quaternary mixtures, include the effect of reaction on liquid molar flow rates, and supply a series of liquid composition trend curves for any column section. In summary, our method is more applicable to multicomponent mixtures with or without reaction. Moreover, transformation of composition variables is of great importance for RSDE because it can remove the reaction term added in the case of reactive distillation and give a similar equation structure as SDE. 4. Conclusions To meet the synthesis requirement of plate type distillation, a stage difference equation is proposed. Its parameters are discussed qualitatively and quantitatively, and then, a reactive distillation difference equation is derived. The case study shows that the synthesis process with these two equations not only finely reflects the composition variation within a column, but it is also more flexible and accurate than the residue curve method. So, a conclusion can be drawn that the equations provide a better theory basis for the whole column design and column sequence synthesis in the future. Acknowledgment The authors gratefully acknowledge that this work is jointly supported by Scientific Research Foundation for the Returned Overseas Chinese Scholars (grant number: 2005-29), Excellent Scholar Research Award Foundation of Shandong Province (grant number: 2006BS05005), and the Taishan Scholar Construction Foundation of Shandong Province (grant number: JS200510036). Nomenclature L ) liquid flow rate with a distillation column [mol/s] Ln ) liquid flow rate leaving plate n [mol/s] LT ) liquid flow rate entering the section top [mol/s] ni ) the number of moles of component i at time t [mol] ni0 ) the initial number of moles of component [mol] xi,n+1 ) liquid composition of component i in plate n + 1 [mol/ mol] yi,T ) composition of component i in the section top vapor flow [mol/mol] Y ) transformed composition of vapor flow [mol/mol] ∆xi,n+1 ) liquid composition change of component i in stage n [mol/ mol] X ) transformed composition of liquid flow [mol/mol] r∆ ) reflux ratio

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r∆T ) reflux ratio on section top rext ) exterior reflux ratio of outlet flow xi,∆ ) composition difference between vapor and liquid phase [mol/ mol] xi,n ) liquid composition of component i in stage n [mol/mol] xi,T ) liquid composition of component i entering section top [mol/ mol] yi,n+1 ) vapor composition of component i in stage n + 1 [mol/ mol] V ) vapor flow rate within a column [mol/s] νi ) stoichiometric coefficient of component i νT ) overall stoichiometric coefficient ε ) reaction depth [mol] ∆ ) flow rate difference between vapor and liquid phase [mol/s] Subscripts B ) section bottom n ) stage number T ) section top i ) component index ∆ ) difference ext ) exterior

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(18) Modise, T. S.; Tapp, M.; Hildebrandt, D.; Glasser, D. Can the Operating Leaves of a Distillation Column Really Be Expanded. Ind. Eng. Chem. Res. 2005, 44 (19), 7511–7519. (19) Stichlmair, J.; Thiel, J. Design of Distillation Columns-state of the art and Future Developments from the Perspective of Universities and Industry. Chem. Eng. Technol. 2003, 75, 1163. (20) Holland, S. T.; Tapp M.; Hildebrandt D. Novel Separation System Design using ‘moving triangles’. Proceedings of the 8th International Symposium on Process Systems Engineering Process, Kunming, China, 2003.

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ReceiVed for reView October 7, 2008 ReVised manuscript receiVed May 13, 2009 Accepted May 27, 2009 IE8015158