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Ind. Eng. Chem. Res. 2008, 47, 3930–3936
Estimation of Still Trajectory for Batch Reactive Distillation Systems James Chin and Jae W. Lee* Department of Chemical Engineering, The City College of New York, New York, New York 10031
This work addresses a new method for estimating the liquid still composition trajectory of batch reactive distillation systems based on material balance and reaction equilibrium data. The still trajectory information is essential to determining whether pure products are reachable from this still pot trajectory when several distillation boundaries are present, initial feed compositions can vary, and the number of components exceeds our visualization capability. For a given feed to product flow ratio with constant feed charge and product compositions, the still pot composition trajectory is mathematically confined to the intersection between the reaction equilibrium manifold and a “material balance plane” that is the union of stoichiometric lines and material balance rays connecting still and product compositions. Starting from this estimated still pot trajectory, we can easily extend feasibility studies of various batch reactive distillation configurations even for multireaction systems. Introduction Reactive distillation is the combination of chemical reaction and V-L (or V-L-L) phase separation in the same unit. Conducting reaction and distillation simultaneously allows each to overcome the limitations of the other, thus allowing for potential reductions in the capital costs, operating costs, and energy usage of chemical processes.1–3 It can greatly simplify processes that would otherwise consist of many single-operation units and can sometimes avoid the use of separate extractive agents. However, the most important question that should be answered before we integrate reaction and distillation is whether or not we can obtain pure products by combining these two tasks. Although reaction conversion and selectivity can be improved with simultaneous separation, we still have to put in additional separation units unless the products from a reactive distillation unit satisfy purity requirements. Our previous work4–10 has determined the feasibility criteria for producing pure products in various reactive distillation column structures in terms of residue curve maps and the upper and lower bounds of internal reflux ratios. The feasibility criteria were applied to any stoichiometry of ternary or quaternary single reaction. In cases where a reactive mixture meets a feasibility criterion, it is easy to evaluate the feasibility of reactive distillation. For example, if there are two distillation regions but the light product is the only unstable node, then a batch reactive rectifier can always produce pure products regardless of where the reaction equilibrium manifold is located. However, if there are multiple distillation regions and only some of those regions meet the feasibility criterion, then feasibility depends upon where the reaction equilibrium manifold is located, what the initial feed charge is, and where the still composition profiles go. Thus, we need to develop a method to estimate the composition trajectory of a given feed composition and to determine its feasibility by considering the relative positions of feed compositions, distillation boundaries, and reaction equilibrium manifolds in multireaction systems. This work will present a new method of estimating the composition trajectories of batch reactive distillation systems for given feed and product specifications and reaction/phase * To whom correspondence should be addressed. E-mail:
[email protected].
equilibrium information without performing dynamic simulations. We will mathematically derive a material balance plane that confines the liquid still pot trajectory to the union between stoichiometric lines and material balance operating lines. We will then intersect this material balance plane with the reaction equilibrium manifolds to estimate the liquid composition trajectory. Finally, we will demonstrate the utility of the composition trajectory estimation in promoting the selectivity of a particular reaction when two reactions occur in series. Trajectory Estimation in Reactive Batch Columns Liquid trajectory estimation in batch reactive distillation systems assumes (1) phase and reaction equilibrium in still pot, (2) pseudosteady state for nonreactive column trays (zero holdup on trays), (3) constant feed and product compositions, (4) a constant ratio between the feed and production rates, and (5) no reaction multiplicity. Still Trajectory for Batch Reactive Rectifier with a Single Reaction. The composition of the still pot changes due to the removal of product in the distillate and due to reaction. These changes are described by the following equation: Hs dxs ) (xs - xD) + (ν - νTxs)r (1) dξ D where xs is the composition of the still pot, xD is the composition of the distillate product, ν is the stoichiometric coefficient vector, VT is the sum of the stoichiometric coefficients, ξ is a nondimensionalized time (dξ ) (D/Hs) dt), Hs is the holdup in the still pot, D is the molar flow rate of distillate, and r is the net reaction rate between forward and reverse reactions per mole of mixture in the still pot. As the reactive distillation progresses, xs will move through composition space in a linear combination of the vectors (xs xD) and (ν - νTxs) in eq 1. For a nonisomolar reaction, (ν νTxs) will be equal to νT(δR - xs) because the reaction difference point is defined as δR ) v/νT.11 The direction of either vector will change as xs moves along the other vector as shown in Figure 1. However, the change in either vector is always dependent on the other vector. Thus, these two vectors define a plane that we shall call the “material balance plane”; the still pot composition is confined to this plane. The material balance plane does not change so long as the product composition does not change. Its mathematical proof is given as follows:
10.1021/ie0713947 CCC: $40.75 2008 American Chemical Society Published on Web 05/10/2008
Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3931
Figure 1. Material balance plane under the assumption of constant product composition.
Lemma. Given only the withdrawal of a constant composition xD and a single nonisomolar reaction along stoichiometric lines that pass through a reaction difference point, δR, the still trajectory (xs) in eq 1 is confined to the two-dimensional linear variation that contains xso, δR, and xD. Proof. Rewriting eq 1 gives
(
)
νTHs νTHsr dxs ) 1δ xs - xD + dξ D D R The total molar balance in the reactive rectifier is
(2)
dHs ) -D + νTHsr (3) dt Replacing dξ by (D/Hs) dt and combining eqs 2 and 3 yields dxs
(D ⁄ Hs) dt
νTHsr 1 dHs x - xD + δ D dt s D R
)-
(4)
Rearranging eq 4 and taking integrals gives
( ) [∫ ]
xs - xso ) ln
Hso x Hs s
t
0
[ ∫ r dt]δ
D dt xD + νT Hs
t
0
R
(5)
Finally eq 5 can be written as xs ) λ1(t)xso + λ2(t)xD + λ3(t)δR where
[ ( )] [ ∫ ][ ( )] [ ∫ ][ ( )]
λ1(t) ) 1 - ln λ2(t) ) -
λ3(t) ) νT
t
0
Hso Hs
-1
Hso D dt 1 - ln Hs Hs
t
0
r dt 1 - ln
(6)
Hso Hs
(7) -1
(8)
-1
(9)
λ1, λ2, and λ3are time-varying coefficients, and the sum of these three coefficients is equal to 1, which can be easily proved using eq 3. Therefore, eq 6 confirms that the still pot composition (xs) will be confined to the 2D linear variation (material balance plane) defined by xso, xD, and δR. The same argument is valid for an isomolar reaction case. The only difference is that the sum of stoichiometric coefficients (νT) is equal to 0 in eqs 1 and 3 for an isomolar reaction. The detailed proof is available in Appendix 1. Figure 1 shows the construction of the material balance plane. To geometrically determine the material balance plane, we start by drawing the geometric ray starting at xs and in the direction of the vector (xs - xD) (this is headed away from xD). Let us call it the “material balance ray”. We then consider the set of all stoichiometric lines that pass through the material balance ray. This union is the “material balance plane” as shown in Figure 1.
Figure 2. Quick estimation of the reactive still trajectory with RCM 43012,13 and a reaction of I T L + H. (a) Still trajectory on the reaction equilibrium curve and (b) residue curves starting from the still trajectory. The numbers in parentheses represent the boiling point order from lowest to highest.
The next step is to find the intersection between the material balance plane and the reaction equilibrium manifold to determine the still trajectory. To do this, we take an arbitrary set of points on the material balance ray and draw stoichiometric lines from each point to the reaction equilibrium manifold. This gives us a set of points that lie on both the material balance plane and the reaction equilibrium manifold simultaneously. This is our estimate of the still pot trajectory in Figure 2a. Our last step is to verify the constant distillate assumption. The distillate composition at any given moment in time depends upon the still pot composition, and the two are connected by a column composition profile. The feasibility of the nonreactive column part is evaluated by residue curves since, in the limit of infinite reflux, the feasible column profiles follow the residue curves and they connect the still pot composition to the distillate composition. So, we then draw a residue curve from each point on our estimated still pot trajectory and follow the residue curves to the most volatile composition. If the residue curves all lead to the product composition, then the assumption of a constant product composition is correct and the column is feasible for the given feed charge. This is shown in Figure 2b: all of the residue curves end at the assumed distillate composition. It should be noted that in the three-component system, the intersection between the material balance plane and the reaction equilibrium curve is the same reaction equilibrium curve, but in systems with more than three components, the intersection between the material balance plane and the reaction equilibrium surface is a curve, a subset of the surface as shown for the four-component case in Figure 3. Thus, for any system having more than three components, estimating still trajectories does not return the entire reaction equilibrium manifold as a result.
3932 Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008
λ2(t) ) -
[ ( )] ][ ( )] ][ ( )]
[∫ HD dt] 1 - ln t
0
s
[ ∫ r dt
λ3i(t) ) νTi
t
i
0
[ ∫ r dt
λ4k(t) ) νTPk
t
0
k
1 - ln
1 - ln
Hso Hs
-1
Hso Hs
-1
Hso Hs
(16) (17) -1
λ5k(t) ) -λ4k(t)
Figure 3. Quick estimation of the reactive still trajectory with I + K T L + H. Solid grid curves are reaction equilibrium curves.
Still Trajectory for Batch Reactive Rectifier with Multiple Reactions. In the case of multiple reactions (n reactions), the second right-hand term in eq 1 is repeated for each additional reaction, and the equation becomes: Hs n dxs ) (xs - xD) + (ν - νTixs)ri dξ D i)1 i
∑
(10)
The vectors still change only in proportion to each other, so the material balance plane becomes a material balance hyperplane with (n + 1) dimensions because there are n reactions and is always a linear variation as proven below. Lemma. Given the withdrawal of a constant composition xD, P nonisomolar reactions along stoichiometric lines that each pass through a unique δRi, and Q isomolar reactions along stoichiometric lines that each have a unique direction νk, where n ) P + Q, xs is confined to the (1 + P + Q)-dimensional linear variation that contains xso and xD, all of the points δRi, and all of the lines of directions νk that pass through them. Proof. As in the previous sections, we can rewrite eq 10
(
)
dxs Hs P Hs P Hs Q ) 1νTi xs - xD + νTiriδRi + rν dξ D i)1 D i)1 D k)1 i i
∑
∑
∑
(11) The total molar balance in the reactive rectifier is P dHs ) -D + Hs νTiri dt i)1
∑
(12)
Replacing dξ by (D/Hs) dt, combining eqs 11 and 12, and taking integrals yields
( )
xs - xso ) ln
Hso x Hs s
[∫ HD dt]x + ∑ (ν ∫ r dt)δ P
t
t Ti 0 i
D
0
s
i)1
Q
∑ (∫ r dt)ν k)1
t
0
Ri +
TPk(cPk - cRk)
k
(13)
Finally eq 13 can be written as P
xs ) λ1(t)xso + λ2(t)xD +
∑λ
Q
(t)δRi +
3i
i)1
∑ (λ
(t)cpk + λ5k(t)cRk)
4k
k)1
(14) where
[ ( )]
λ1(t) ) 1 - ln
Hso Hs
-1
(15)
(18) (19)
The sum of these time-varying coefficients is equal to 1, and xs is expressed as a linear combination of xso and xD, all of the points δR,i, and all of the lines of directions νk. The construction procedure for still trajectories with n reactions can be summarized as follows: 1. Fix a product composition (i.e., xD for the rectifier) and an initial feed charge composition (xso). 2. React the initial feed so that it is in simultaneous equilibrium with all reactions. This is done by taking the stoichiometric linear variation through the feed and finding the intersection (xs) between the stoichiometric linear variation and the reaction equilibrium manifold. 3. Draw a material balance ray connecting the product composition and the intersection point obtained in the previous step. 4. Extend linear stoichiometric variations from the material balance ray within the composition space simplex until they reach the points on the simultaneous reaction equilibrium manifold. These points will form an estimated still trajectory. Still Trajectories for Stripper and Middle Vessel Column. For batch reactive strippers, the analysis for batch reactive rectifiers applies identically. The still pot composition only changes due to two influences: the removal of bottoms product and chemical reaction. Removal of a product stream from the column will cause the still pot composition to move away from the composition of the product stream. Reaction will cause the still pot composition to move back or forth along a stoichiometric line. These two influences are material balance constraints and they only allow the still pot composition to change within a geometric plane (again, this is a two-dimensional linear variation for a single-reaction case or an (n + 1)-dimensional linear variation for an n-reaction case regardless of how many chemical components are involved). The equation is Hs n dxs ) (xs - xB) + (ν - νTixs)ri dξ B i)1 i
∑
(20)
where xB is the composition of the bottoms and is assumed to be constant. To determine the material balance plane, we start by drawing the material balance ray starting at xs and in the direction of the vector (xs - xB) (this is headed away from xB). We then consider the set of all stoichiometric lines that pass through the material balance ray to find the material balance plane that the still pot composition is constrained to. The still pot trajectory must lie on the material balance plane as long as the constant-bottoms-composition assumption is true. The analysis for a batch-reactive middle Vessel column (MVC) is similar to the analysis for rectifiers and strippers, though xs can now change because of three influences: removal of distillate, removal of bottom product, and chemical reaction. If the relative flow rates of distillate and bottoms are not constrained, then xs may vary along a three-dimensional material balance hyperplane rather than a material balance plane. One can conceivably find the intersection between such a material balance hyperplane and the reaction equilibrium manifold, but
Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3933
this will increase the calculation burden dramatically. We instead fix the compositions and relative flow rates of distillate and bottom product. dxs Hs n ) (xs - xP) + (ν - νTixs)ri dξ P i)1 i
∑
(21)
where P is the total product molar flow rate (P ) D + B) and xP is the overall product composition (xP ) (DxD + BxB)/P) which was also defined as a pseudofeed.14 Figure 4a shows that if the material balance line starting at xs is coincident with the stoichiometric line passing through xs, then the composition change due to the removal of product exactly cancels the composition change due to reaction. The result is that the still pot composition is invariant with respect to time, and it is only necessary to draw the residue curves from the still pot composition alone to verify feasibility. This will occur only when the initial feed composition lies on the same stoichiometric line as the overall product composition. Otherwise, the still trajectory (xs) will move along the reaction equilibrium curve in the direction going away from the product that is being withdrawn at the faster flow rate. In Figure 4b, the bottom product flow rate (B) is higher than the distillate rate (D), so the still trajectory move away from the bottom product composition as distillation proceeds. Still Trajectories for Side-Feed Batch Reactive Distillation Columns. Consider an initial feed charge to the reactive still pot of reactive extractive distillation columns that have a side-feed stream in the rectifier, stripper, and middle vessel column as shown in Figure 5. Side-feed rectifier and side-feed middle vessel column are also called batch reactive extractive distillation (BRED) columns.6,10 Assume zero holdup on the column trays, constant distillate composition, constant extractive feed composition, and that reactions occur only in the still pot fast enough that the still pot composition is always in reaction equilibrium. The side-feed batch columns introduce a feed stream (F), and the derivative of xs becomes dxs Hs n F ) (xs - xD) - (xs - xF) + (ν - νTixs)ri dξ P P i)1 i
∑
(22)
where xF is the side-feed composition and F is the side-feed molar flow rate (fixed by design). P is total product molar flow rate. P is equal to D, B, or (D + B) for side-feed rectifier, sidefeed stripper, or side-feed MVC, respectively. Because we fix the compositions of xD and xF and the ratio of F/P, the first two terms on the right side of eq 22 represent a single basis vector. Thus, the still liquid trajectory (xs) is also confined to the (n + 1)-dimensional material balance hyperplane. For the given compositions and F/P ratio, we can estimate the still pot trajectory in the same way as in the previous sections. Figure 6 shows a hypothetical L, I, H system with the reaction 2I T L + H and a heavy inert entrainer (E). Suppose that L, I, and H, by themselves, have a phase equilibrium behavior of 313-S as used for our previous work.10 Thus, there are three binary node azeotropes and a ternary saddle azeotrope. If a heavy inert entrainer is introduced to this ternary system, then two distillation regions are formed by one distillation boundary.10 The molar flow ratio between entrainer and distillate (or the F/D ratio) is 20.0. We take an initial pot composition of pure reactant I and react it to equilibrium. Then, given the F/D ratio, we draw the material balance plane by taking the union of stoichiometric lines and material balance rays as shown in Figure 6a. We then find where the stoichiometric lines intersect the
Figure 4. Still trajectory of MVC with RCM 010-S12,13 and a reaction of I T L + H. (a) Stationary still trajectory and (b) moving still trajectory.
Figure 5. Side-feed batch reactive distillation columns. (a) Side-feed rectifier, (b) side-feed stripper, and (c) side-feed MVC.
reaction equilibrium surface; this intersection is our trajectory estimate. The estimated trajectory in Figure 6a almost overlaps the trajectory from the previous dynamic simulation result in Figure 6b.10 Thus, to determine the feasibility of batch reactive distillation, quick and reliable estimation of still trajectories is essential because dynamic simulations are time-consuming; our dynamic simulation runs have required anywhere between several hours and several days of CPU time on a single Pentium-3 PC. To the best of our knowledge, this work is the first to estimate dynamic still trajectories without dynamic simulations within a few minutes of computation time. The trajectory estimation is easily incorporated into the feasible region10 of side-feed batch reactive distillation where the upper bound of reflux ratio is greater than the lower bound of reflux ratio as shown in Figure 6. Feasible extractive column profiles always lie within the feasible region and so should the still trajectories. Trajectory Estimation of a Multireaction System To Increase Reaction Selectivity In this section, the utility of still trajectory estimation is applied for a system with more than one reaction. As we already showed in eq 22, we can use various batch column structures and can estimate dynamic still trajectories to investigate their feasibility. Consider the diethyl carbonate (DEC) production system.15 Dimethyl carbonate (DMC) and ethanol are reactants, DEC and
3934 Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008
Figure 6. Still trajectory of side-rectifier with RCM 313-S12,13 and a reaction of 2I T L + H with F/D ) 20.0. (a) Estimation and (b) dynamic simulation. Component E is an inert entrainer. Dotted areas are feasible regions.
methanol are products, and methyl-ethyl carbonate (MEC) is an intermediate. The two unstable nodes are the DMC-methanol and MEC-ethanol azeotropes. DEC is the only stable node, and MEC is a saddle. Thus, two unstable nodes and one stable node make two distillation regions with a common stable node in both distillation regions because there is only one pair of stable node and unstable node in a single distillation region.16 There are two reactions in this DEC production system: H3C-OOCOCH3(DMC) + C2H5OH(EtOH) T H3C-OOCOC2H5(MEC) + CH3OH(MeOH) H3C-OOCOC2H5(MEC) + C2H5OH(EtOH) T H5C2-OOCOC2H5(DEC) + CH3OH(MeOH)
(23)
If DEC is the desired final product, then both reactions should be encouraged. Since DEC is the only stable node in the residue curve map and reachable from both distillation regions, the residue curves lead to the stable node DEC, and a batch reactive stripper can produce pure DEC. Figure 7a shows an estimated still trajectory. Figure 7b presents a plot of a rigorous column simulation with kinetic data available in the literature.15 The still pot trajectory has an initial feed charge of [0.33, 0.67, 0, 0, 0] (compositions in order of DMC, ethanol, MEC, ethanol, and DEC) for both cases. This feed charge has the exact stoichiometric ratio between DMC and ethanol because 2 mol of ethanol are involved in the two reactions while 1 mol of DMC participates in the first reaction of eq 23. The estimated still trajectory in Figure 7a provides an excellent approximation of the rigorously simulated still trajectory shown in Figure 7b. From any point in the still trajectory, the residue curves reach the stable node product DEC. Thus, we can produce pure DEC using a reactive batch stripper. For the dynamic simulation, we use 50 nonreactive trays with a top
Figure 7. Still trajectory in MEC-projected composition space: (a) estimation and (b) dynamic simulation. DMC to EtOH ratio in the feed charge is stoichiometric.
reactive still. We also use a reboil ratio of 4000 because the production rate (B) is so small. Our production policy is that we withdraw DEC at the bottom in proportion to the holdup of DEC in the still pot. The final yield of DEC is around 90%. However, if MEC is the desired product, then the first reaction should be encouraged while the second reaction is discouraged. This can be accomplished by using an excess of DMC, which only promotes the first reaction in eq 23 and consumes almost all of the ethanol. Because no large amount of DEC forms, the phase equilibrium behavior of DMC, MEC, methanol, and ethanol dominates, and among those four components, MEC is the only stable node. Starting with an initial feed charge that contains a large excess of DMC, a stripping section can be used to isolate MEC from the reacting mixture, and a small amount of fresh ethanol can be continually fed into the still pot to continue the reaction while maintaining excess DMC in the still pot. Figure 8a shows an estimated trajectory by setting P ) B in eq 22. The initial feed charge is [0.95, 0.05, 0, 0, 0] for DMC, ethanol, MEC, ethanol, and DEC, respectively, and the feed stream (F) contains pure EtOH with a fixed F/B ) 1, which means that EtOH is provided in the same molar amount as MEC is withdrawn at the bottom. The estimated trajectory does not contain a significant DEC fraction (less than 10-4), and it approaches the MeOH-EtOH edge because MEC is removed from the still and DMC is consumed in the still pot as shown in Figure 8a. The dynamic simulation result is shown in Figure 8b with 20 nonreactive trays and a reboil ratio of 200. The total yield for MEC is 87.0% with respect to the limiting reactant of EtOH. The feed stream (F) is pure ethanol with a fixed ratio of F/B ) 1, which means that ethanol is fed to the pot in the same amount that MEC is withdrawn at the bottom. Here the estimated trajectory in Figure 8a is approximately identical to the rigorously simulated trajectory in Figure 8b.
Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 3935
Appendix 1: Linear Variation of Still Trajectories in the Material Balance Plane for Isomolar Reaction Lemma. Given only the withdrawal of a constant composition xD and a single isomolar reaction along stoichiometric lines in the direction of ν, xs is confined to the two-dimensional linear variation that contains xD, xso, and lines of direction ν that pass through both of them. Proof. For an isomolar reaction, eqs 1 and 3 will be Hsr dxs ) (xs - xD) + ν (A1) dξ D dHs ) -D (A2) dt Replacing dξ by (D/Hs) dt and combining eqs A1 and A2 yields Hsr dxs ) xs - xD + v D ⁄ H dt D ( s)
(A3)
Rearranging eq A3 and combining it with eq A2 gives
( ) ( )
dxs ) -
dHs dHs x + x + (r dt)v Hs s Hs D
(A4)
Integrating eq A4 and decomposing ν to the two vectors of normalized product stoichiometric vector and normalized stoichiometric reactant vector14 will yield xs ) λ1(t)xso + λ2(t)xD + λ3(t)cP + λ4(t)cR Figure 8. Still trajectory in DEC-projected composition space: (a) estimation and (b) dynamic simulation. DMC to EtOH ratio in the feed charge ) 19 to 1.
where
λ2(t) ) - ln λ3(t) ) νPT
Conclusions We have presented a quick method of estimating the still pot trajectory of various batch reactive distillation columns. If we assume constant product and feed charge compositions with constant production and feed ratios, then the still pot trajectory lies in the union of material balance rays and stoichiometric lines. The union defines a material balance plane and is a linear variation regardless of the number of components. Then the still pot trajectory is estimated as the intersection between the material balance plane and a reaction equilibrium manifold. This simple estimation of still trajectory is very important in evaluating the feasibility of the batch reactive distillation especially when many distillation boundaries and multiple reactions are involved in a reacting mixture. From the estimated trajectory, we can easily investigate the possibility of producing a desired product with high purity and selectivity in various reactive distillation systems. Acknowledgment We are grateful for the support of American Chemical Society, Petroleum Research Fund (ACS-PRF) and STX Co. Ltd. We really appreciate the first reviewer who gave valuable comments on the revision of the article. We also wish to thank Prof. Wolfgang Marquardt at RWTH Aachen for the continued use of his phase equilibrium calculation code, material properties calculation code, and quaternary system visualization program code.
[ ( )] [ ( )][ ( )] [∫ ][ ( )]
λ1(t) ) 1 - ln
Typically, the still trajectory estimation takes several minutes compared to 7 days of dynamic simulations in a single Pentium-3 PC. Furthermore, the still trajectory estimation is extremely useful to determine whether a desired product in multireaction can be selectively produced in spite of visualization limitations.
t
-1
(A6)
Hso Hs
-1
1 - ln
Hso Hs
-1
r dt 1 - ln
Hso Hs
0
Hso Hs
λ4(t) ) -λ3(t)
(A5)
(A7) (A8) (A9)
One has to note that the normalized product and the normalized reactant vectors (cP and cR) lie in the composition space and νTP is the sum of product stoichiometric coefficients that is equal to the absolute sum of reactant stoichiometric coefficients (νTR). The sum of λ1, λ2, λ3, and λ4 is equal to 1. Thus, eq A5 tells us that xs is confined to the linear variation defined by the vectors xso, xD, and the lines of direction ν () νTP(cP - cR)). Notation B ) molar flow rate of the bottom product stream (mol/time) D ) molar flow rate of the distillate product stream (mol/time) F ) molar flow rate of side-feed to batch column (mol/time) Hs ) total molar holdup of the still pot (mol) at time t Hso ) total molar holdup of the still pot (mol) at the initial charge. P ) number of nonisomolar reactions in eqs 11–14 P ) sum of molar flow rates of products (mol/time) in eqs 21 and 22 Q ) Number of isomolar reactions in eqs 11–14 n ) number of reactions xB ) molar composition of the bottom product stream xD ) molar composition of the distillate product stream xF ) molar composition of side feed to the column xP ) weighted average of the molar compositions of all product streams xs ) molar composition of the still pot
3936 Ind. Eng. Chem. Res., Vol. 47, No. 11, 2008 xso ) initial molar composition of the still pot ri ) net reaction rate between the forward and the reverse directions of reaction i (mol/time); subscript omitted if only one reaction Greek Letters δRi ) reaction difference point for reaction i ξ ) dimensionless time νi ) stoichiometric coefficients for reaction i; subscript omitted if only one reaction νTi ) sum of the stoichiometric coefficients of reaction i; subscript omitted if only one reaction νTPk ) sum of the product stoichiometric coefficients of reaction k νTRk ) absolute sum of the reactant stoichiometric coefficients of reaction k
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ReceiVed for reView October 16, 2007 ReVised manuscript receiVed March 1, 2008 Accepted March 3, 2008 IE0713947
