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Ind. Eng. Chem. Res. 2011, 50, 1666–1673
Nonlinear Control of a Batch Reactive Rectifier Anirban Patra, Darshankumar M. Dave, and Amiya K. Jana* Department of Chemical Engineering, Indian Institute of Technology-Kharagpur, West Bengal-721 302, India
This article considers the issue of designing a nonlinear control strategy for a batch reactive rectifier. The hybrid control strategy consists of a state estimator, namely Luenberger-like nonlinear estimator (LNE) and the feedback linearizing controller (FLC). The LNE observation scheme aims to estimate the imprecisely known parameter (augmented state) along with the unmeasured state using the available state information. The state predictor includes only two ordinary differential equations and is formulated based on the component continuity equation around the condenser-reflux drum envelope. It implies that the predictor model is not a perfect representation of the actual process. For effective handling of this structural discrepancy, the corrector model is coupled with the predictor in the LNE estimator. The next part of this article synthesizes the FLC controller for the example system. A comparison is made in terms of closed-loop performance between the FLC-LNE scheme and gain-scheduled proportional integral (GSPI) controller. Extensive simulation results reveal that the proposed structure appears to be a promising approach to the control of the batch distillation column. 1. Introduction Batch processing is becoming more important due to the rapid expansion of the fine chemical, pharmaceutical, and food industries. Batch operations are widely applied in these industries because of the low amount and the frequent change of the products. These days the product purity requirements are usually strict, and a significant economic penalty exists when the specification requirements are not met. Therefore, it is necessary to develop a correct methodology for tight composition control. Batch reactive rectifier is a nonlinear, complex, and highorder separation process that offers limited flexibility because of the interactive effect of reaction on separation. It is intrinsically dynamic, making the control a more challenging task. Various techniques for designing nonlinear controllers are reported in the literature. Most of the nonlinear controllers require the feedback of state variables. In practice, however, it is infeasible in most applications to obtain complete state information through direct measurement. It is, therefore, required that a state estimator/observer is used. We can define the state estimator as a tool responsible for gathering valuable measurements to infer the desired information. So far, much effort has been concentrated on the design of a nonlinear state estimator and control scheme for nonreactive batch distillation systems.1-3 Relatively very few articles address the issue of estimator-based nonlinear control of batch reactive rectifiers.4-6 It is with this intention that the present work has been undertaken. It is well-recognized that the extended Kalman filter (EKF) is one of the most widely diffused estimators among other nonlinear estimation schemes based on linearization approaches. In an EKF scheme,7 a Riccati equation should be solved to obtain the observer gain. This scheme assumes the knowledge of the noise model in order to obtain the optimum estimated value. However, that model is generally unknown and the assumed noise model could lead to biased estimates or even diverge. Most of the state observation algorithms estimate the full state vector involving complexity and lengthy formulation. However, information on all the states is not always required, especially * To whom correspondence should be addressed. E-mail: akjana@ che.iitkgp.ernet.in. Tel.: +91-3222-283918. Fax: +91-3222-282250.
when the objective is to track only a few unmeasured variables. For this type of problems, the reduced-order estimators8 have been proposed in the literature. The order of such estimation techniques typically corresponds to the difference between the system order and the number of measurements.9 Owing to the small number of measured variables, the reduction in the number of variables to estimate is usually insignificant. Recently, the preferential estimation technique was introduced,9 and it provides a significant reduction in the order of the estimator. The idea of the proposed method is that the most important characteristics of the dynamics of a complex system can be described by the dominant subspace. Thus, the system can be described by a dynamic model whose dimension is equal to that of the dominant subspace. In this contribution, a nonlinear estimation scheme has been formulated to compute a limited number of states (true and augmented). The selection of state variables depends on the requirement for computing the control actions. The Luenbergerlike nonlinear estimator (LNE)7 has been used for this purpose, and the developed LNE scheme provides state information required exclusively for the nonlinear controller simulation. In this work, a batch reactive rectifier has been modeled and simulated for nonlinear state estimation and control. For computing the state variables as per the controller demands, the predictor model is formulated on the basis of only the component mole balance equation around the condenser-reflux drum system and an extra state equation having no dynamics. Since the predictor model is not a perfect representation of the actual process, the corrector model is coupled with the predictor in the LNE estimator for effective handling of the structural mismatch. In the next, the feedback linearizing control (FLC) law, consisting of a transformer and an external linear controller, has been derived for the representative batch column. In the proposed FLC-LNE strategy, the transformer that linearizes the nonlinear process globally receives information from the linear controller, state estimator, and the process. To the best of our knowledge, no work has reported the formulation of the LNE-based FLC control strategy and its application.
10.1021/ie101315b 2011 American Chemical Society Published on Web 12/28/2010
Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011
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Figure 1. The basic FLC structure.
2. Nonlinear Control Algorithm Figure 1 shows the basic FLC control structure that combines a transformer (I/O linearizing state feedback) and an external linear controller. The transformer has the ability to linearize the nonlinear process globally and it is, therefore, appropriate to use a linear controller around the linearized υ-y system. The details of these two control blocks for a single-input/singleoutput (SISO) system are presented in generalized form in the following. 2.1. Feedback Linearizing Control Scheme. The discretetime state-space model for a SISO system has the following form: x(k + 1) ) Φ[x(k), u(k)] y(k) ) h[x(k)]
(1)
For a process of the form of eq 1 with a finite relative order r, the above equation implies that the algebraic equation hr-1[Φ(x, u)] ) y
(7)
is locally solvable in u (via the implicit function theorem). The corresponding implicit function will be abbreviated by u ) Ψ0(x, y)
(8)
and will be supposed to be well-defined and unique on X × h(X). The state feedback transformation that can be calculated from the definition and properties of the relative order has the following form: r
Here, x represents the vector of state variables, u is the manipulated input, and y is an output, all in the form of deviation variables. We suppose that x ∈ X ⊂ Rn, and u ∈ U ⊂ R, where X and U are open-connected sets that include the origin (i.e., the nominal equilibrium point). Φ(x, u) is an analytic vector function on X × U, and h(x) is an analytic scalar function on X. The objective of input-output linearization is to obtain a correlation8 u(k) ) Ψ[x(k), υ(k)]
(2)
such that the resulting scheme, shown in Figure 1, consists of a linear transfer between υ and y:
hr-1{Φ[x(k), u(k)]} ) β0υ(k) -
∑βh
r-l
l
[x(k)]
(9)
l)1
It transforms the input-output system (eq 1) into r
y(k + r) +
∑ β y(k + r - l) ) β υ(k) l
0
(10)
l)1
Recalling the definition of the function Ψ0 (eqs 7 and 8), the transformer of the form of eq 9 gets the following form: r
u(k) ) ψ0{x(k), β0υ(k) -
∑βh
r-l
l
[x(k)]}
(11)
l)1
β0 y(z) ) r r-1 υ(z) z + β1z + ... + βr-1z + βr
(3)
where r is the relative order of the nonlinear system, and βl, l ) 0, ...,r are constant scalars (tuning parameters) with β0 * 0. For a system of the form of eq 1, the relative order of the output y with respect to the manipulated variable u is the smallest integer r for which ∂Φ(x, u) ∂Φ(x, u) *0 [ ∂h(x) ∂x ][ ∂x ] [ ∂u ] r-1
(4)
In the subsequent discussion, the following notation has been used:
{
∆
h0(x))h(x)
(5)
∆
hl(x))hl-1[Φ(x, u)],
l ) 1, .....r - 1
Accordingly, ∂ r-1 ∂h(x) ∂Φ(x, u) h [Φ(x, u)] ) ∂u ∂x ∂x
[
][
u) *0 ] [ ∂Φ(x, ∂u ] r-1
(6)
For the purpose of online implementation of the external controller, a minimal-order state-space realization8 can be represented as follows: ξ1(k + 1) ) -γ1ξ1(k) - ... - γr-1ξr-1(k) + (1 + γ1 + ... + γr-1)ξr(k) + (1 + γ1 + ... + γr)e(k) ξ2(k + 1) ) ξ1(k) l ξr(k + 1) ) ξr-1(k) υ(k) ) (((β1 - γ1)ξ1(k) + ... + (βr-1 - γr-1)ξr-1(k) + (1 + γ1 + ... + γr-1 + βr)ξr(k))/β0) + ((1 + γ1 + ...γr-1 + γr)e(k)/β0) (12) 2.2. Luenberger-like Nonlinear Estimation Scheme. Figure 1 clearly indicates the availability of complete state information. But this is not the case in practice. It motivates the use of a state estimator within the FLC structure. The resulting control scheme shown in Figure 2 is called here a nonlinear control algorithm. In this section, we present the details of LNE scheme. For a nonlinear discrete system of the form of eq 1, the LNE estimator7 has the following form:
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Figure 2. The proposed nonlinear control structure.
xˆ(k + 1) ) xˆ(k) + ∆t[f[xˆ(k)] + g[xˆ(k)]u(k) + O-1[xˆ(k)]KLNE{y(k) - h[xˆ(k)]}]
(13)
Like other closed-loop estimators, the LNE scheme consists of the predictor and corrector parts. The corrector part contains the nonlinear estimator gain represented by Ο-1KLNE. Here, KLNE is a matrix of constants to be developed and Ο represents the Jacobian of the vector φ(x). In this variable gain observer, the vector φ(x) constitutes a nonlinear change of coordinates. Aiming to make easier the estimator design, it is required to transform the original process representation to get a transformed one. The transformed process model includes known parameters, say A and C, which are inserted into the Lyapunov equation below for designing the gain KLNE: (A - KLNEC)TP + P(A - KLNEC) ) -Q
(14)
Here P and Q designate positive definite matrix that must satisfy eq 14. In addition, the following constraint should be satisfied: -qmin + 2pmax(Lγ + LωU) < 0
(15)
where pmax and qmin are the maximum and minimum eigenvalues of P and Q, respectively. Lγ and Lω are Lipschitz constants of the process. Therefore, the dynamics of the estimation error e(k) [) x(k) - xˆ(k)] will be stable. Provided that Ο is invertible, that U is a bound for the input u and known the initial condition xˆ(0), the following property behaves for R > 0: |e(t)| e λ exp(-Rt)|e(0)|
(16)
with λ > 0. As a result, the norm of the estimation error goes to zero as t f ∞. Then, the convergence of the estimation scheme is guaranteed. 3. The Process and the Model To illustrate the proposed FLC-LNE scheme, a batch reactive distillation column4,5 shown in Figure 3 is modeled for simulation. Avoiding the specific complexities that can exist in some real chemical systems, simple vapor-liquid equilibrium, reaction kinetics and physical properties have been considered. The reaction in ethyl acetate (EtAc) production is the esterification of ethanol (EtOH) with acetic acid (HAc): component number boiling point (K)
acetic acid + 1 391.1
ethanol a ethyl acetate + 2 3 351.5 350.3
water 4 373.2
The reaction takes place in the liquid phase and is slightly endothermic. The kinetic data presented by Mujtaba and Macchietto10 and the operating conditions are listed in Table 1.
Figure 3. Schematic representation of the batch reactive rectifier. Table 1. Operating Conditions and Reaction Kinetic Data system
acetic acid/ethanol/ ethyl acetate/water
fresh feed charged, kmol liquid holdup (startup) in each tray, kmol liquid holdup in reflux drum, kmol reboiler duty, kJ/min distillate flow rate (production phase), mol/min
30.0 0.075 0.6 3200 4
component
feed composition
distillate composition
bottoms composition
HAc EtOH EtAc H2O
0.45 0.45 0.0 0.1
5.1932 × 10-5 0.064823 0.934515 6.0712 × 10-4
0.25572 0.25603 0.20096 0.28729
kinetic data rate of reaction (kmol/(L min)): r ) k1c1c2 - k2c3c4 rate constants: k1 ) 4.76 × 10-4; k2 ) 1.63 × 10-4 where ci ) concentration (kmol/L) for the ith component
The sample batch column consists of eight trays, excluding the reboiler and total condenser. The tray numbering has been started from bottom; bottom tray is first tray and top tray is eighth tray. The modeling equations of a sample tray are presented in the Appendix. It can be seen that among the components, ethyl acetate is the lightest one. The continuous withdrawal of ethyl acetate as distillate shifts the chemical
Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011
equilibrium further to the right and consequently, the reactant conversion is improved. 4. Formulation of Control Algorithm for the Example Process The control objective is to maintain the composition of ethyl acetate in the distillate (xD,3) by the manipulation of the reflux rate (R). As shown in Figure 2, the estimates from the state observer serve as input to the FLC controller and the decision based on such feedback information is then implemented on the process. The three control elements, namely a transformer, a PI controller, and the LNE observation scheme, are discussed in the following. 4.1. Nonlinear Control. 4.1.1. Transformer-cum External PI Controller. The control law is synthesized by following the steps of the FLC method: 1. The dynamic component holdup in the condenser-reflux drum system is expressed as m ˙ Dx˙D,3 ) V8y8,3 - (D + R)xD,3
(17)
considered as a measured variable, the online estimation of this state is required to run the observer model for calculating the residual. In this nonlinear control scheme, the two observed states are xD,3 and V8y8,3. In deriving the LNE observation approach, we consider V8y8,3 as an extra state (augmented state) having no dynamics. The estimator dynamics is governed by the following predictor model:
[
] [
∆t xD,3(k + 1) ) xD,3(k) + [V y (k)-(D(k) + mD 8 8,3 R(k))xD,3(k)] (18) 3. To determine the relative order, we calculate,
[
[ ] ] [
∂xD,3 ∂h(x) ) ) [1], ∂x ∂xD,3
]
u) [ ∂Φ(x, ∂u
) -
∆txD,3 mD
[
∆t
mD
0
Let,
]
Vˆ8yˆ8,3(k) - D(k)xˆD,3(k) f[xˆ(k)] ) mD 0
(19)
where, y(k) ) xD,3(k). 5. The digital form of a PI controller for a SISO system with r ) 1 is obtained from eq 12 as ξ(k + 1) ) ξ(k) + (1 + γ1)e(k) 1 + β1 1 + γ1 ξ(k) + e(k) β0 β0
y(k) ) h[x(k)] ) xD,3(k)
(21)
(22)
(23)
The observer gain matrix, Ο-1KLNE corrects the predicted outputs using the error between the values of measured variable, xD,3 and model output, h(xˆ). As mentioned previously, KLNE is a matrix of constants and Ο represents the Jacobian of the vector φ(x) that is defined as
[ ]
(24)
∂h(x) f(x) ∂x
(25)
∂Lfi-1h(x) f(x) ∂x
(26)
where Lfh(x) is the Lie derivative of h(x) in the direction of f(x). Accordingly,
Assuming r ) 1, the condition defined in eq 4 is satisfied and we select the relative order equal to unity. 4. The linearized υ-y system (eq 10) gets the following form:
υ(k) )
] [ ] [ ]
-xˆD,3(k) mD + ∆t [R(k)] 0
h(x) Lfh(x) φ(x) ) l Lfn-1h(x)
and
y(k + 1) + β1y(k) ) β0υ(k)
]
xˆD,3(k + 1) xˆD,3(k) ) ˆ + Vˆ8yˆ8,3(k + 1) V8yˆ8,3(k) Vˆ8yˆ8,3(k) - D(k)xˆD,3(k)
and obviously, 2. A level controller (P-only) has been implemented to take care on the restricted holdup (mD) variations. Accordingly, eq 17 yields the following discrete form:
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(20)
where, ξ(0) ) y(0) ) xD,3(0). 6. It is evident from the FLC formulation that the two tuning parameters, namely β1 and γ1 are involved. The values of β1 and γ1 are obtained using the tuning guidelines11 as -0.8999 and -0.98, respectively. 4.1.2. LNE Estimator. We must note that both mD and xD,3 are considered as measured variables, and their values are used to calculate D and R, respectively. It is now clear that for simulating the transformer model presented above, additionally we need the information on component vapor flow rate leaving top tray (V8y8,3). Although, the top product composition is
Lfh(x) ) Lfi h(x) )
According to the guidelines given earlier, the gain KLNE is set to KLNE )
[ ] 34 320
4.2. Gain-Scheduled PI Controller. In a batch rectifier, a low-plant-gain composition space (i.e., the steady state space) changes to a high-plant-gain composition space. This implies that the control gain should be increased during the batch operation. For this purpose, a gain-scheduled PI law12 has been formulated for the representative process. This nonlinear controller is represented by
(
R ) RS + KC(xD,i) e +
1 τi
∫ e dt) t
0
(27)
where, xD,i denotes the scheduling variable. The controller gain, KC is varied for keeping KCKP constant, which then keeps the stability margin constant. If the process gain is characterized as a function of the scheduling variable, KP(xD, i), the controller gain can be scheduled as
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KC(xD,i) )
KC(xD0,i)KP(xD0,i) KP(xD,i)
(28)
The gain of the GSPI controller can be written as A. When xD,3 > xD0,3, KC(xD,3) ) KC0
1 - xD0,3 1 - xD,3
(29)
where, KP(xD,3) ) 1 - xD,3 and KC(xD0,3) ) KC0. B. When xD,3 < xD0,3 KC(xD,3) ) KC0
(30)
It should be noted that this is a one-way approach. Since the process gain increases with lower purity, maintaining a constant controller gain speeds up the response when the distillate is less pure. The GSPI controller is tuned using the integral square error (ISE) performance criteria and the values of the control parameters are obtained as: KC0 ) 6.1 and τi ) 1.22 min. 5. Simulation Experiments and Results The performance of the state estimator and the nonlinear control scheme is tested by applying them to the simulated batch reactive rectifier. Just after starting the batch operation, the column is brought to the steady state by considering the total reflux startup procedure. Then both the controller and estimator are switched on in the production phase, and the controller as well as estimator parameters are tuned. It is worth mentioning that as we start the product withdrawal, the estimator output may be inaccurate enough and this, in turn, may lead to the production of aggressive control action. It happens because of the shifting of total reflux operation to partial reflux. At the start of a batch operation, the results are produced considering total reflux flow with no esterification reaction. In this situation, the batch distillation, originally a nonreactive ternary (acetic acid/ethanol/water) batch process, reaches at steady state within about 10 min. In the next stage, the start-up phase runs under complete reflux condition but with an esterification reaction. At the end of start-up period, the reactive batch column attains another steady state with the ethyl acetate composition of 0.9345. This value imposes a limit in the achievable product composition under batch operation. Figure 4 presents the start-up dynamics of the uncontrolled distillate composition, with no reaction in the first part followed by esterification reaction in the last part. In the following, first the tracking performance of the LNE estimator is inspected carrying out simulation experiments. The subsequent part covers the performance of the proposed nonlinear controller in comparison with the gain-scheduled PI law. 5.1. Open-Loop Estimator Performance. Rejection of Initialization Error. Figure 5 shows the result of sensitivity test on the estimator performance under +8.9% initialization error in V8y8,3. This testing is carried out because it is usual in practice that the model parameters are not exactly known. It is evident that the closed-loop observer shows excellent convergence of estimation error toward zero. Disturbance in Heat Input to the Reboiler. In this simulation experiment, we are interested in assessing the effect of disturbance in reboiler duty. Accordingly, the heat load has been changed twice (step increase: 3200f 3520 kJ/min at time )2000 min, and step decrease: 3520 f 3200 kJ/min at time )
Figure 4. Open-loop process dynamics at the start-up phase under complete reflux condition with a reaction (no reaction only in the first 10 min).
Figure 5. Comparison of the estimated outputs and process outputs with +8.9% initialization error in V8y8,3 (0.8176 f 0.89 kmol/min).
3000 min) in Figure 6. The open-loop simulation results show satisfactory tracking performance of the LNE estimator.
Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011
Figure 6. Comparison of the estimated outputs and process outputs for ethyl acetate with two consecutive step changes in heat input to the reboiler.
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Figure 8. Comparative control performance for maintaining constant composition. Table 2. ISE Values ISE value Figure
test
8
constant composition (0.9345) control servo test: step changes in set point of xD,3 regulatory test: step changes in QR regulatory test: step changes in tray efficiency
9 10 11
Figure 7. Comparison of the estimated outputs and process outputs for ethyl acetate with two consecutive step changes in tray efficiency.
Uncertain Tray Efficiency. The effects of uncertain tray efficiency have been demonstrated in Figure 7. To provide a realistic test scenario for the proposed estimator, two consecutive step changes (step increase: 0.75 f 0.825 at time ) 2000 min, and step decrease: 0.825 f 0.75 at time ) 3000 min) have been
nonlinear
GSPI
7.243 × 10-6 1.466 × 10-5 0.0003
0.00047
8.66 × 10-6 8.31 × 10-5 1.049 × 10-5 1.57 × 10-4
introduced. It is observed from the results that an excellent agreement has been achieved between the estimated and true process values in the presence of structural discrepancy. 5.2. Comparative Closed-Loop Performance. In this section, a comparative closed-loop performance study is presented between the proposed FLC-LNE and GSPI controller. Several simulation experiments have been conducted in terms of constant composition control, and set point tracking, and disturbance rejection performance. Constant Composition Control. Figure 8 presents a comparative performance between the nonlinear control law and the GSPI scheme for maintaining the top product composition of the concerned batch reactive rectifier at the value of 0.9345 (reference/steady state composition). The results show that at the beginning of production phase, the controller responses are aggressive enough due to the reason stated earlier. Comparatively, the control action provided by the GSPI is more aggressive than that of the nonlinear control system. Table 2 evaluates the comparative performance in terms of the ISE values. Servo Test. A comparative closed-loop performance is investigated in terms of servo performance in Figure 9. In this simulation experiment, a pulse change is introduced in distillate composition of ethyl acetate (step decrease: 0.9345 f 0.9 at time
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Figure 9. Comparative servo performance with a pulse change in set point of xD,3.
Figure 10. Comparative regulatory performance with a pulse change in reboiler duty.
) 2000 min, and step increase: 0.9 f 0.9345 at time ) 3000 min). Despite the discrepancy in process/predictor model structure, the proposed nonlinear controller outperforms the GSPI scheme. To confirm the superiority of the proposed strategy, Table 2 presents the ISE values. Regulatory Test. In the second closed-loop testing, Figure 10 displays the disturbance rejection performance of the nonlinear scheme and GSPI. Two consecutive step changes (step increase: 3200 f 3520 kJ/min at time ) 2000 min, and step decrease: 3520 f 3200 kJ/min at time ) 3000 min) are considered in reboiler heat duty. This situation is more efficiently handled by the nonlinear control algorithm over the GSPI law. A comparison is made in Figure 11 under a pulse change in tray efficiency (step increase: 0.75 f 0.825 at time ) 2000 min, and step decrease: 0.825 f 0.75 at time ) 3000 min). Rejecting the effect of uncertain parameter, the nonlinear control law provides better performance than the GSPI controller. Table 2 supports this observation. 6. Conclusions A nonlinear control system is proposed for a batch reactive distillation column. The strategy comprises a transformer, an external controller, and a closed-loop Luenberger-like nonlinear state estimator. Structural and parametric mismatches were considered between the actual process and its model in order to provide a realistic test scenario for the proposed estimator. It is shown that the developed LNE observer showed promising error convergence ability and the proposed control law outperformed the GSPI controller. Simple structure, ease of design,
Figure 11. Comparative regulatory performance with a pulse change in tray efficiency.
and promising performance make the nonlinear controller attractive for online use.
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Appendix
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C
For developing the model of a batch reactive rectifier, the following assumptions have been made. • negligible tray vapor holdups • variable liquid holdup in each tray • perfect mixing and equilibrium on all trays • no chemical reactions in the vapor phase • reactions occurred on all the trays, except the condenser • fast energy dynamics • constant operating pressure (atmospheric) and tray efficiencies (Murphree vapor-phase efficiency ) 75%) • Raoult’s law with Antoine’s vapor pressure correlation for vapor-liquid equilibrium (VLE) description • nonlinear Francis weir formula13 for liquid hydraulics calculations • constant liquid holdup in the reflux drum (perfectly controlled by a conventional proportional (P-only) controller with a proportional gain of -10.00) The scheme of a typical nth plate is shown in Figure 12. The plate is fed with a liquid feed mixture. Side streams are withdrawn in the state of both liquid and vapor. The model of a typical nth plate consisting of MESH (material balance, vaporliquid equilibrium, mole fraction summation, and heat balance) equations is presented below.
Total mole balance m ˙ n ) Ln+1 + Vn-1 + Fn - (Ln + SLn ) - (Vn + SVn ) +
∑y
n,i
)1
(A6)
i)1
Here, Q denotes the heat loss (kJ/min), x is the liquid-phase mole fraction, y is the vapor-phase mole fraction, z is the feed composition, L is the liquid flow rate (kmol/min), V is the vapor flow rate (kmol/min), F is the feed flow rate (kmol/min), m is the tray holdup (kmol), H is the enthalpy (kJ/kmol), S is the flow of side stream (kmol/min), K is the vapor-liquid equilibrium coefficient, ε is the volume of the liquid holdup (m3), and γ is the stoichiometric coefficient. The subscript/superscript n represents the tray, i is the component, L is the liquid, V is the vapor, and C is the total number of components. The dot symbol (.) is used to represent the time derivative. The time derivative of the multiplication of two variables, say m and x, is denoted here by m ˙ x˙ () d(mx)/dt). Heat of formation is considered while computing the enthalpies of streams by means of which the heat of reaction term can be removed from the energy balance equation. In the simulation, an algebraic form of equations14 has been used to compute the vapor and liquid enthalpies. It is worth mentioning that, for the batch distillation, values of feed flow rate (F) and bottom flow rate (B) are taken as zero. Moreover, no side draws are considered here, that is, SL ) SV ) 0. Literature Cited
C
∑γr ε ) i n n
(A1)
i)1
Component mole balance m ˙ nx˙n,i ) Ln+1xn+1,i + Vn-1yn-1,i + Fnzn,i - (Ln + SLn )xn,i (Vn + SVn )yn,i + γirnεn (A2) Energy balance L V ˙ Ln ) Ln+1Hn+1 + Vn-1Hn-1 + FnHFn - (Ln + SLn )HLn m ˙ nH (Vn + SVn )HVn + rnεnHrn - Qn (A3) Equilibrium yn,i ) Kn,ixn,i
(A4)
Summation C
∑x
n,i
)1
i)1
(A5)
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ReceiVed for reView June 19, 2010 ReVised manuscript receiVed November 28, 2010 Accepted December 2, 2010 Figure 12. Quantities associated with a typical nth tray.
IE101315B
