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Procedia Engineering 38 (2012) 2383 - 2397

2012 International Conference on Modeling, Optimisation and Computing

Modeling, Simulation and Control of Middle Vessel Batch

Distillation Column

C. S. Rao* and K. Bank

Department of Chemical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302, INDIA

Abstract

Batch distillation is a very efficient unit operation which allows the fractionation of a multicomponent mixture into its pure constituents in a single column. Due to its flexibility and low capital costs, batch distillation is becoming increasingly important in the fine chemicals and pharmaceutical industries. However, the regular batch distillation processes have some disadvantages, such as high energy demands and high temperatures in the feed vessel. An alternative to the regular batch distillation is the middle vessel batch distillation. With this type of process it is possible to obtain simultaneously light and heavy boiling fractions from the top and the bottom of the column respectively and intermediate boiling fraction accumulates in the middle vessel. This work dealing with the modeling and simulation of middle vessel batch distillation column (MVBDC) neglecting the vapor hold and its influence on the process. Modeling of MVBDC is done based on the combined first principle model and empirical relations. The startup of MVBDC is covered. Here we also deal with the control of distillate composition and condenser hold up and process gain estimation. Predominantly theoretical studies regarding controlled variables are studied.

© 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Noorul Islam Centre for Higher Education

Keywords: Simulation; MVBDC; First principle; Empirical model

1. Introduction

Nomenclature

Hi Hold up on plate i ; Vj Vapor flow rate from plate i ; Li Liquid flow rate from plate i ; Xi Composition of liquid on plate i ; yi Composition of vapor on plate i ; hvap Vapor enthalpy ; hiiq

1877-7058 © 2012 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.06.285

Liquid enthalpy; Qj Heat lossess ; D Distillate flow rate ; QD Condenser heat duty; Qb Reboiler

'Corresponding author. Tel.: +91-9025883372 E-mail address: csankarrao@gmail.com

heat duty ; M Position of middle vessel; R Reflux flow rate ;tc Delay time constant; Xi Integral reset time constant; ri(t) Modeling error parameter ; Kp Process gain ; u Manipulated variable ; Au Change in reflux rate; e(t) Error

Batch distillation is a unit operation, which has a risen increasing interest in industry over the past decade. The reason is a significant change in the market from a quantity-oriented to a quality-oriented demand. It is favorable in the separation of multicomponent mixtures if the amount of feed is small and high purity products are demanded. Nevertheless, there are intrinsic disadvantages associated with the conventional batch distillation process. These are: long batch times, high temperatures in the charge vessel and complex operation. Hence, alternative processes and operation policies, which have the potential to overcome these disadvantages, are being extensively investigated namely, the middle vessel column and multivessel column. The use of such a column for the separation of multicomponent mixtures was first proposed by Robinson and Gilliland [1], but the first analysis of the process was presented by Bortolini and Guarise [2]. Hasebe et al. [3] generalized the principle of complex batch columns and introduced the so-called Multi-Effect Batch Distillation connecting multiple column sections and product vessels. Predominantly theoretical studies regarding the feasibility and optimization of manipulated and controlled variables have been published. Hasebe et al. presented simulation studies, in which process control is realized by keeping the mass holdup in the product vessels constant or by adjusting it according to optimal trajectories.

Most publications dealing with the modeling and simulation of distillation columns neglect the startup period and its influence on the process. Experimental studies by Gruetzmann et al. [4] recently indicated that the deliberate switching of discrete decision variables can lead to a shorter process. In terms of optimization, a focus on the first period, in which hydrodynamic and thermodynamic profiles are formulated, is reasonable. The startup of continuous distillation has been covered by several works. An analysis of the startup period of conventional batch distillation columns with product removal has been considered in only a few cases. For example, Wang et al. [5] and Elgue et al. [6] presented detailed models simulating the startup of batch distillation. However, the total reflux operation of novel batch distillation columns has not been considered. The main objective of this paper is to simulate the novel batch distillation column which is modeled from the combined first principle and empirical relation. Consequently, the feedback controller design has been designed. Apart from this process gain has also been estimated.

2. Mathematical Model

Model development is necessary for the system to simulate, optimize or to control the process. In general, models are developed by first principles method. Empirical methods are also adopted to develop the model. The advantage of this approach is the depth of the insight into the behavior of the system and thus ability to predict the performance. The model will consist of a set of ODES to be obtained from mass and energy balances around each plate of the middle vessel batch distillation column and a set of algebraic equations (for the dynamics of the reboiler and reflux drum) which will be used to predict the physical properties, the plate hydraulics, the mass and heat transfer. The variables that appear on the left hand side of the ODES will be termed the "differential" variables; all other variables except time will be termed as "algebraic" or "procedure" variables. Time will be the independent variable. The following assumptions are made in the development of the proposed model:

• The molar vapor holdup is negligible compared to the molar liquid holdup

• The liquid and vapor leaving the each plate are in thermal equilibrium

The definition of Murphree plate efficiency will apply for each plate and the liquid and vapor are perfectly mixed on each plate. First assumption is quite reasonable since in most systems, the vapor density is considerably smaller than the liquid density. The assumption of perfect mixing and Murphree plate efficiency is necessary to reduce the complexity of the problem.

Tray Modeling. Index i denotes the components and index j denotes the theoretical stages. The numbering is from the bottom to the top. The mathematical model can be described by the following equations.

Material and Energy balance Overall material balance around plate j

-¿^-y^-L, a,

Component balance for component i and around plate j

dHtxt .

= Vj^-V^J + -Lj\J (2) Energy balance around plate j

dHj^kL = Vj+Avap - VjKap+" LA«q ~ Qjms (?)

Heat losses are determined from equation below. The overall heat transfer coefficient k is calculated in advance using common correlations. The surface area is given by the column design.

Qj,ioss=kA{T -Tamblent) (4)

Summation Equations

Vapor-Liquid Equilibrium. If both vapor and liquid streams exist, vapor liquid equilibrium is calculated from

y^i - Kixij (6)

where K denotes the equilibrium constant 2.1. Condenser modeling

= Distillaterate(mol / h) (7)

The condenser reflux ratio is defined as dt

Overall material balance

_ dD dHn

Vl=—(R +1) + —^ (9)

Component balance for component i

dHnx. n dD

—^ = F,y0 -Lqx.d -X.d — (10)

Condenser heat duty, QD is given by

Qd = VAvap-KKin-HDtUq ^ (11)

2.2. Reboiler modeling

= Bottomsrate(mol / h) (12)

The condenser reflux ratio is defined as

Overall material balance

dH, _ j- dB

— -Ln-VB----(14)

dt " B dt

Component balance for component i

dHbxib dB

-^-L^-V^-x^- (15)

Energy balance equation

dH,h, _ , T.. , dB _

-— = Lh , -FA -A,,--O, (16)

^ n rt,hq b b,vap b,liq ^ v '

The plate balances for the bottom sections of the column are same as those found earlier for top except for the plate where middle vessel is located.

2.3. Middle vessel column dynamics

The rigorous model for the middle vessel column is presented below. The model is based on the assumptions of negligible vapor holdup and theoretical trays. A schematic for the middle vessel column is shown in Fig 1.

Fig. 1. Schematic of middle vessle column For the middle vessel the holdup is a function of time; therefore we have the following equations

_ * NT

Here Q is the ratio of vapor boilup rate in the rectifying section to that of the vapor boil up rate in the stripping section.

Overall mass balance around the middle vessel

— V —V +T -T

' JVT+1 NT NT-l ^

Component balance for component i around the middle vessel

~ ^NT+\yi,NT+\ ^NTyi.NT + NT-lXi,NT-\ ^NTXi,NT

Energy balance around the middle vessel

dHmhmlia

m m,liq dt

^NT+\^NT+\,vap ^NT^NT,vap + ^NT-l^NT-lJiq ^ NT ^ NT Mi

3. Simulation

Due to the absence of tray hydraulics data, the initial model is developed by considering a constant L and V on each plate to obtain the Francis Weir relationship. The results are then used to develop a model based on mass, energy and thermodynamic equilibrium data to predict the composition and flow rate of distillate. Here we have assumed a constant relative volatility. The liquid flow rates are calculated using Francis Weir formula and the vapor flow rates are calculated using enthalpy balance. Enthalpy is the function of temperature so it is necessary to have temperature composition correlation. By using the Hildebrand model, bubble point temperatures are estimated. Vapor and liquid enthalpies are determined from the following algebraic equations.

Hv = a + bT + cT2 + dT

H,=HV- (RT2 (B /(C + Tf)

Francis Weir Formula is given below

pwL999[ (

Table 1. Initial conditions

System Ethanol/Propanol/Butanol

Feed (kmol) 1000 kmol

Feed composition 0.4/0.4/0.2

Distillate (D) 0 (start up)

Reboiler duty (Q, kcal/min) 100

Tray holdup (H, kmol) 0.3

Condenser holdup (Hukmol) 1

Middle vessel holdup (kmol) 0.75

Reboiler holdup (kmol) 2

Murphree tray efficiency 0.8

Column diameter (inch) 29.52

Weir length (WL, inch) 19.48

Weir height (Wh, inch) 0.4

Number of trays (NT) 28

Integration time interval (min) 0.005

Pressure (atm) 1

Steps followed to solve modeling equations are:

• Input variables at time (t = 0) are liquid phase compositions (xj), constant parameters including VLE data, Francis weir constants, and murphree tray efficiency.

• Under total reflux condition bottom product rate is zero.

• Compute equilibrium vapor-phase composition (y,) and temperature (T) for each tray based on bubble point and the actual vapor-phase composition employing Murphree relationship.

• Estimate the liquid and vapor phase enthalpies for each tray.

• Calculate the internal liquid flow rates using the Francis- weir formula and the vapor flow rates by solving the energy balance equations.

• Compute the liquid holdup and liquid-phase compositions on all trays for each time step (t+At) solving the total mole balance and component mole balance equations, respectively by using Euler's explicit method.

• To run the process simulator for the next time step, go back to Step 3.

3.1. Simulation results and discussion

Steady state at total reflux to find out the initial conditions once the product is withdrawn. Dynamic

simulation to predict the variation in composition, liquid and distillate flow rate with time. The effect of

reboiler heat duty and number of trays in stripping and rectifying section on top product composition flow

rate and composition has also been studied.

Fig. 2. Composition Profile (a) at distillate (top left); (b) at middle vessel (top right); (c) at reboiler (below) 3.1.1. Composition Profiles

The above Fig 2(a, b, c) shows the variation of composition with time and indicating that the low boiling fraction (ethanol), intermediate boiling fraction (propanol) and heavy boiling fraction (butanol) components comes out as a main product in distillate, middle vessel and reboiler, respectively. The MVBDC is simulated with different number of plates in which the middle vessel position is varied. From the Fig 3 we can suggest that the position of middle vessel will be 13 to 19 because here propanol composition almost constant about to 99%.

Fig. 3. Variation in steady state propanol composition due to change in position of Middle vessel column

The MVBDC is simulated with different heat duty ranging from 50 to 150 kcal/min and the corresponding results are presented in the graphs (Fig 4(a) and 4(b)) which show the variation of ethanol composition and equilibrium time due to change in reboiler heat duty. With increase in reboiler heat duty up to 90 kcal/min equilibrium time increases then fall as increases. The top product composition is also increases as heat duty increases. The column is simulated with different reflux rate ranging from 11 to 16 gmol/min and the corresponding results are presented in the graphs (Fig 5(a) and 5(b)) which show the variation of ethanol composition and equilibrium time due to change in reflux rate. As the reflux rate increases both time taken to reach equilibrium and top product composition falls.

Fig. 4. (a) Composition of ethanol due to change in reboiler heat duty (left); (b) Variation in equilibrium time (right);

The MVBDC is simulated with different number of plates in rectifying and stripping section. By keeping constant number of plates in rectifying section vary the stripping section plates. Similarly, vary the rectifying section plates by keeping number of plates in stripping section constant.

From the simulation results we can predicted as ethanol composition at distillate and butanol composition at reboiler increases with increase in number of trays in rectifying section. Whereas the propanol composition decreases with increase in the rectifying section trays. But, in other case, the propanol composition remains constant as increasing the trays in stripping section. The later one will not affect the composition of propanol because the number of stages in the stripping section are remains constant.

Fig. 5. (a) Variation in composition of ethanol (left); (b) Equilibrium time due to change in reflux rate (right)

4. Robust Feedback Control Design

The MVBDC column is one kind of batch distillation process which is developed by the set of algebraic and differential equations. The mathematical model for the batch distillation process can be represented [7] as

f(x',x,t,u) = 0,t £[0,^] (24)

Where t denotes time, tf is the batch period, x is the vector of state variables (e.g., flows, concentrations, pressures, temperatures, etc.), while u denotes the vector of manipulated variables (e.g., reflux flow).

4.1. Estimation ofKp

In principle, the process gain Kp can be obtained as K =

p du dt du

That is, Kp is the time derivative of the sensitivity function dy/du. An open-loop test is proposed for identifying the process gain through the following procedure (see Figure):

Fig. 6. Change in composition of distillate due to periodic perturbations in reflux rate

Step by step procedure:

The following guidelines are taken from [7]

1. Apply regular relay perturbations u(t) with period P and amplitude Au around the nominal value u. Relay perturbations are commonly carried out in practice to tune linear (e.g., PI and PID) controllers, and its implementation only requires to change periodically valve positions. Since the relay perturbations should reveal the Kp dynamics about an operating region, ±5-10% Au-perturbations and periods P< tf where tf is the batch operation period, are proposed.

2. According to the proposed model td, and since u(t) is discontinuous at td = P/2, P, 3P/2 ;...... , the

output response y(t) is continuous but not differentiable at such time instants. Compute the timederivative y'(t) of the response signal y(t).

3. At the discontinuity times td=P=2, P, 3P=2;...... , compute the estimated high-frequency gain as

^ &x(t+d)-x(t~)

K =- where x is the derivative of y with respect to time at t=td fro left and right.

y(t) = Kp(t)u(t) (26)

y(t) = y(0) + \Kp{(j)u{(j)d(J (27)

Notice that, for u = constant, y(t) is given by the integration of the process gain Kp(t). This is why Kp is called as high-frequency gain.

4.2. Controller design

Based on the proposed model [7], we proceed to construct a robust feedback controller to regulate the distillate composition. In the interest of simplicity in presentation, let us consider the approximate (timeinvariant) model of the MVBDC.

y = K~pu(t) (28)

Where Kp is a constant estimate of the process gain along the operation period tf. For instance, Kp can be taken as the mean of the estimated (time-varying) high-frequency gain Kp(t).

— l'f

Kp=-\Kp{cj)dcj (29)

In a more general case, one can take a time-varying (e.g., time wise constant) estimated high-frequency gain, Kp . In such case, it is expected that the underlying controller will give better stability and performance properties. To compensate for the approximations used to obtain model, consider the following representation of the MVBDC:

y = Kpu(t) + rj(t) (30)

Where Ti(t) is the modeling error that accounts for model/plant mismatches. Based on the MVBDC representation, a robust feedback controller can be developed along the same lines of modeling error compensation approaches. To this end, introduce the tracking error e(t) = y(t) - yr(t) associated to the desired composition trajectory yr(t).

It can be shown that this controller is equivalent to a linear PI compensator with controller gain and integral reset time-constant parameterized by tc, xe and Kp . In fact, by computing the controller transfer function e —> u, it is possible to describe the controller as t

u=udc+ Kce + KjJ e(t)dt (31)

where UdC is a dc-like constant input and the controller Kc and integral KI gains are given by

— L +-]

Kc KP\ Te

Kj KpTcre

Hence, if Kp is time varying, the equivalent PI controller is also of time-varying nature. Probably, the main advantage of the PI controller with parameterization over traditional ones is that the tuning can be carried out quite efficiently. Specifically, for a given closed-loop response time-constant xc, the PI control gains depends only on the estimation time-constant xe, which can be reduced to adjust the modeling error estimation response. That is, the directionality of the control parameters xc and xe is well established in the sense that the control response is enhanced as such time-constants are reduced.

4.3. Tuning of controller parameters

The control design structure described above suggests the following guidelines on tuning of the proposed controller:

(a) In a first step, set the value of the closed-loop time-constant xc. If tf is the batch operation period, values of xc of the order of 0.05tf are suggested. It is a well-known fact that the regulations of the batch distillation operation at the first operating periods are critical to achieve the desired product quality [89]. In this way, the controller response will be at least 20 times faster than the operating batch time.

(b) After the value of the closed-loop time-constant has been set, select a suitable value of the estimation time-constant xe. In principle, the estimation of the modeling error signal t|(t) must be faster than the prescribed closed-loop convergence time xc. However, excessively large values of xe can excite unmodeled (actuator) dynamics and amplify excessively measurement noise. To avoid these undesirable effects, set xe at a value not smaller than the dominant time delay [7].

Keeping the above constraints in consideration numerical simulations are done for different values of time constant and the one giving maximum product of desired purity level is selected.

4.5. Results and discussion

We give a change in reflux rate of ±10% at an interval of 25 minutes as shown in figure 6. The corresponding change in composition is plotted as a function of time. The derivatives are found at each point from right and left and the average of all the derivatives are taken to get the average value of Kp

Table 2. Process gain values with time

Time(min) 5

10.5 22.5 35.5 55

70 80.5 95.05

Process gain, Kp

0.057 0.029 0.0012 0.183 0.034 0.0042 .002 0.221 0.094

From the above values the average value of Kp is found to be 0.057

4.6. Effect of zc, Te and zr

Composition of feed = 0.4, Total number of moles of Ethanol = 300

Table 3. Effect of zc, zr & xe Due to variation in required purity of desired product on percentage recovery

Required purity 0.99

Tr 1c Te Moles recovered Percentage recovery Time (min)

0.5 1 1 207 69 408

0.5 1 0.5 222 74 640

0.5 1 2 174 58 486

0.5 2 2 154.5 51.5 358

0.25 1 1 183 61 575

0.75 1 1 178.5 59.5 518

Required purity 0.98

Tr Xc te Moles recovered Percentage recovery Time(min)

0.5 1 1 249 83 645

0.5 1 0.5 261 87 109

0.5 1 2 237 79 532

0.5 2 2 198 66 482

0.25 1 1 246 82 142

0.75 1 1 241.5 81 123

From the above table we get maximum percentage recovery for xr = 0.5, xc =1 and xe =1. By increasing xr, xc and xe the recovery decrease. This is because it results decrease of Kc and IQ which results in decrease in reflux rate. Other combinations of time constants do give comparable recovery but time taken is large in that case. Thus the values ofKp , IQ and Kc are 0.057, 17.54 and 35.08 respectively. The distillate composition is controlled from the designed feedback controller which is shown in below Fig 7.

Fig. 7. Variation in composition of ethanol with and without control 5. Conclusion

A rigorous mathematical model has been derived to describe the operating behavior of a conventional middle vessel batch distillation column. The work mainly helps to develop a robust model for scheduling and controlling of a middle vessel batch distillation column of mixture ethanol, propanol and butanol. The model takes into account varying molar holdup, varying distillate flow rate, varying reflux flow rate. The effect of reboiler heat duty, reflux flow rate and number of plates on compositions of ethanol, propanol and butanol are studied. There is no impact on the propanol composition at middle vessel by increasing the number of trays in rectifying section because in stripping section trays are remains same. Here we also deal with the control of distillate composition and condenser hold up. And process gain estimation is also studied. Predominantly theoretical studies regarding controlled variables are studied.

References

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