[Computer Aided Chemical Engineering] Integrated Design and Simulation of Chemical Processes Volume 35 || Batch Processes

CHAPTER

BATCH PROCESSES 1111.1 INTRODUCTIONBatch processes are widely applied in different branches of chemical process industries, namely in

manufacturing configured-consumer products, such as speciality chemicals, pharmaceuticals, paints,

polymers, food products and cosmetics. Batch processes are preferred for dealing with complex chem-

istry, separation and purification steps and for smaller production scales, in general below 10,000 ton/

year. In these cases, the manufacturing process consists of batches based on ‘recipes’, for which is im-

portant to trace the origin and the operation conditions. However, a similar situation may be valid for

some commodities or industrial products, such as technical polymers, where large manufacturing ca-

pacity is provided by employing batch reactors. Many new products start as a small-scale batch process

and evolve later to a large-scale continuous process, as a result of technology improvement and market

acceptance.

The key advantage of a batch process is flexibility: the flowsheet configuration (connections be-

tween units) and the operation procedure can be adapted to the quality specifications required by

the customer or to the features of raw materials. Typically, a batch process consists of a relatively sim-

ple flowsheet employing standard unit operations (mixing/blending, heating/cooling, size reduction,

batch reaction, distillation, filtering, evaporation, crystallisation, etc.). However, the equipment should

be versatile. Therefore, in term of specific cost, the equipment may be expensive because of smaller

size and more demanding mechanical characteristics.

The main disadvantage of a batch process is lower productivity when compared with an equivalent

continuous operation. The design cannot be a priori optimal for all situations. Thus, flexibility is paid

by oversizing, higher energy demand and more manufacturing time. However, an optimal operation

procedure can be found for fixed equipment configuration and utilities. Only a part of the production

time is spent in reaction and/or in separation sequences. Pre- and post-processing are time consuming,

as the preparation of charges, heating and cooling, etc. Several batch reactors may share the same sep-

aration equipment. For this reason, the scheduling of time and units’ availability is a key activity in

batch processes.

This chapter deals with two major subjects. The first one is the separation of multi-component mix-

tures by batch distillation. It includes the treatment of zeotropic and azeotropic mixtures. Key topics are

the trade-off between purity and recovery, and efficient reflux policy for minimising the energy expen-

diture. Simple computational methods are described for getting useful insights before stepping up to a

sophisticated simulation tool. The second part handles the conceptual issues when employing batch

Computer Aided Chemical Engineering. Volume 35. ISSN 1570-7946. http://dx.doi.org/10.1016/B978-0-444-62700-1.00011-5

© 2014 Elsevier B.V. All rights reserved.449

http://dx.doi.org/10.1016/B978-0-444-62700-1.00011-5

reactors. Emphasise is put on the control of the thermal regime in exothermic reactions, with implica-

tions on safety. A case study for large-scale PVC reactors illustrates the other facet of batch processes,

the manufacturing of large-scale industrial products.

11.2 BATCH DISTILLATION11.2.1 THEORETICAL BACKGROUNDFigure 11.1 shows the three basic arrangements for carrying out a batch distillation operation:

A. bottom vessel with rectifying column (batch rectification),

B. top vessel with stripping column (batch stripping),

C. middle vessel with distillation column with both rectification and stripping sections.

The most employed set-up in industry is batch rectificationwith the bottom vessel linked to a rectifying

column. The pot is loaded with mixture, and distillation is started. If the mixture is zeotropic, the com-

ponents are separated in the top of the rectification column, following the order of increasing boiling

points. Several receivers are provided for collecting the good purity product fractions (cuts), as well as

the slope fractions (off-specs cuts) to be recycled to the next batches.

In batch stripping (inverted configuration), the mixture is fed to the condenser of a distillation col-

umn. The component separates in the reverse order of volatilities in the bottoms of the stripping col-

umn, starting with the heaviest one. Contrary to expectations, this arrangement is the best in term of

operating time when large amount of heavy component has to be removed (Sorensen and

Skogestad, 1996).

D

B

xB

xD

B

D

xD

MxM

B

xB

xB

xD

D

A B C

FIGURE 11.1

Industrial arrangements in batch distillation. (A) Batch rectification; (B) Batch stripping; (C) Middle vessel.

450 CHAPTER 11 BATCH PROCESSES

The set-up withmiddle vessel and stripping/rectification sections is close to the principle of counter-current operation in continuous distillation. This set-up succeeds to overcome the limitations of regular

batch rectification with respect to yield and energy consumption (Meski and Morari, 1994). Applica-

tions in separating both zeotropic and azeotropic mixtures have been presented by Stichlmair and

co-workers (Warter et al., 2004).

The residue curve map (RCM) representation allows highlighting the key problems in separating

multi-component mixtures (Stichlmair and Fair, 1998; Doherty and Malone, 2001). Figure 11.2 illus-

trates the situation with a ternary zeotropic mixture ABC, A being the most volatile, B the intermediate

and C the heaviest. In the case of a simple distillation (Figure 11.2, left), the evolution of the concen-

tration in the vessel is shown by the points x1, x2, x3,. . ., xk, while the concentration of distillate y1, y2,y3,. . ., yk by the tangent at every xk point. It may be seen that at the beginning the distillate purity in A is

high, but it decreases rapidly by incorporating increasing amounts of intermediate B. At a certain point,B becomes dominant, but in short time it begins to contain more and more C. Thus, by simple batch

distillation, the fractions of A and B are of modest purity. Only components with large difference in

volatility might be conveniently separated.

A much more efficient mode is submitting the vapour to separation in a rectification column, by

taking profit from the counter-current mass transfer between vapour and liquid phases. Figure 11.2

(right) explains the process. The distillation starts from the initial composition xF. The separation

of A may take place from the beginning with high purity, depending on the performance of the recti-

fication device: number of stages and reflux. The composition in the still follows a line passing through

xF and the distillate composition xD, located on the edge AB close to the vertex A. As long as the dis-

tillation goes on, the composition approaches the edge BC. Over a certain laps of time, it is possible to

collect a fraction of A of good purity, after which the distillate quality begins to deteriorate rapidly, by

incorporating increasing amounts of component B, as indicated by the point xD0 . After a transient slope

cut, a second fraction of intermediate B may be collected. This intermediate fraction contains at the

beginning A as impurity, but when A is exhausted, the separation takes place only between B and C.Thus, the B fraction contains both A and C as impurities, depending on the decision where to start

A

BC

x1

y1

x2

x3xk

y2

y3

yk

A

BC

xF

X ¢D

X²D

XD

d1

FIGURE 11.2

RCM representation of a batch separation of a zeotropic mixture; left side: simple distillation, right side: batch

rectification.

45111.2 BATCH DISTILLATION

and cut it. In the final stage, the component C may be left in the still, or recovered by vaporisation

from other material as a good purity fraction, with only B as impurity.

Putting in a nutshell, the batch rectification can supply good purity fractions by sequential separa-

tion of components. In between, there are slope mixtures. For given column design (number of stages),

the purity of fractions, and accordingly the necessary energy, depends on the reflux policy. This topic

will be analysed in more details in the subsequent sections.

In the case of azeotropic mixtures, the occurrence of a distillation boundary prevents getting high-

purity components, which however becomes possible by employing a suitable entrainer.

11.2.2 SEPARATION OF ZEOTROPIC MIXTURES BY BATCH RECTIFICATIONTwo operation modes are the most used in batch rectification:

1. Constant reflux ratio and variable distillate composition.

2. Variable reflux ratio and constant distillate composition.

The explanation of separation can be illustrated by means of a McCabe–Thiele diagram, as shown in

Figure 11.3. In the followings, the word reflux will mean reflux ratio. At constant reflux (Figure 11.3,

left), the slope of the operation line is given by the ratio L/V of molar flow rates of reflux and boilup,

respectively. At initial time, the bottoms concentration of the light component is xB,1. Drawing upwardsthe number of stages gives the distillate composition xD,1. After some time, the concentration dimin-

ishes in the pot to xB,2, and accordingly to xD,2 in the distillate. It can be seen that the purity of distillatedepends on the reflux flow rate. During the process, the distillate purity drops significantly. If higher

purity and recovery are desired, the reflux should have a large value from the beginning.

At variable reflux (Figure 11.3, right), the concentration of the distillate xD is aimed to be constant

by increasing continuously the slope of the operation line. If high recovery is desired, the reflux should

grow considerably to the end, raising proportionally the utility consumption.

The following simplified analysis can bring useful insights in the above issues. Let us consider a

binary mixture with close molar vaporisation enthalpies, such that a constant molar distillation rate Dmay be assumed. Another assumption is the negligible hold-up in the column compared with the

x D/(

R+

1)

xD,1

Mole fraction liquid, x

Mol

e fr

actio

n va

pour

, y

x D/(

R+

1)

xD

xB,1

Mole fraction liquid, x

Mol

e fr

actio

n va

pour

, y

xB,2

xD,2

xB,2 xB,1

FIGURE 11.3

Operation modes in batch distillation: constant reflux (left) and variable reflux (right).


amount in the pot, such that the change in the distillate composition follows instantaneously the com-

position variation in the pot. The initial molar amount of the mixture is F with initial concentration xF.The actual molar amount in the pot is B. The differential material balance over an infinitesimal period

of time gives:

�d BxBð Þ¼�BdxB�xBdB¼ xDDdt (11.1)

dB +Ddt¼ 0 (11.2)

Since D dt¼�dB, the following differential equation describes the distillation process:

dB

B¼ dxBxD�xB

(11.3)

The distillate concentration xD is function of the bottoms concentration xB, relative volatility a, thenumber of stages NS and the reflux ratio R.

xD¼ f xB, a,NS, Rð Þ (11.4)

Note that by this approach the evolution of the concentration is decoupled from the time of operation.

Next, the two operation modes will be examined.

11.2.2.1 Operation at constant refluxThe concentrations in the pot and distillate vary continuously. The amount left in the pot can be de-

termined by the integration of the relation (11.3), as follows:

ln B=Fð Þ¼ðxBxF

dxBxD�xB

(11.5)

The integral can be evaluated graphically, by plotting 1/(xD�xB) versus xB for different positions of theoperation line, starting from different values of xB, as explained before. If the integral has the value L,the amount left in the pot is as follows:

B¼Fexp Lð Þ (11.6)

The distillate D is simply the difference to the initial amount:

D¼F�B (11.7)

The average composition of the distillate fraction can be determined from the mass balance:

xDm¼ xF�xBB=F

1�B=F(11.8)

The time of operation is as follows:

t¼F�B

D(11.9)


Of remarkable interest is the solution with the assumption of infinite number of stages (Stichlmair and

Fair, 1998). Two steps in operation can be identified. In the first step, the amount of light component in

the pot is large enough to produce a high-purity vapour with xD¼1. In the second step, the amount of

the light component drops at a level that cannot sustain the production of pure distillate. The interme-

diate pot composition xB0 can be determined as:

x0B¼1

a�1ð ÞR (11.10)

Replacing in Equation (11.5) and integrating gives the amount B0 left in the pot:

B0

F¼ 1�xF1�1= a�1ð ÞRð Þ (11.11)

In the second step, when the pot concentration falls below xB0 , the distillate composition can be de-

scribed by the following relation obtained from the classical McCabe–Thiele analysis:

xD¼ a R + 1ð ÞxB1 + a�1ð ÞxB�RxB (11.12)

The integration of Equation (11.5) gives the expression:

B

B0¼ xB

x0B

� �1= a�1ð Þ1�x0B1�xB

� �a= a�1ð Þ" #1= R + 1ð Þ(11.13)

The mean distillate concentration collected over the second step is as follows:

xDm,2¼B0x0B�BxBF�B

(11.14)

Since the distillation concentration of the first step is xDm,1¼1, the final average concentration is as

follows:

xDm¼F�B0 + B0 �Bð ÞxDm,2F�B

(11.15)

If a, R and xDm are set, the values of xB0 , B0, xB, B and xDm,2 can be calculated by means of the implicit

solution of Equations (11.10)–(11.15). D is calculated from material balance. Accordingly, the recov-ery or recovery of separation can be computed by:

Y¼ D

F

� �xDmxF

� �(11.16)

The energy required for separation can be normalised by reference to the amount of feed (moles) mul-

tiplied by the enthalpy of vaporisation (kJ/mol) (Stichlmair and Fair, 1998). Thus, the number

Q/(FDHv) indicates the minimum energy input in batch rectification with respect to that necessary

for vaporising the feed. This value is named specific heat requirement Qsp in this book. The following

expression holds at constant reflux:


Qsp¼Q

FDHvð Þ¼D

F

� �R+ 1ð Þ (11.17)

Although apparently simple, Equation (11.17) is actually a complex function of purity and recovery.

Qsp should take high values if one wishes simultaneous high purity and recovery, as it will be demon-

strated later in this section.

EXAMPLE 11.1 BATCH RECTIFICATION AT CONSTANT REFLUXExamine the separation of a 100 kmol mixture of methanol/n-propanol by batch rectification. It is aimed to recover at least

95%methanol. The concentration of the methanol is 34.8%. The separation takes place at atmospheric pressure in a column

of three theoretical stages. The reflux ratio is constant at the value 2. Compare the performance of this set-up with the batch

rectification at infinite number of stages.

Solution. Figure 11.4 illustrates the solution.

Fenske equation is used for describing the liquid–vapour equilibrium:

y¼ ax1 + a�1ð Þx (11.18)

The relative volatility is obtained as geometrical average at the normal boiling points of components,

a¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�3:173p ¼ 3:56. The path of distillation can be evaluated by the integration of the relation (11.5). McCabe–Thiele

method is employed for determining the concentration profile in the distillation column. The operation line for the recti-

fication zone can be drawn for an arbitrary selected composition of the vapour distillate. Then, starting from the top, a step-

wise line can be drawn between the equilibrium curve and the operation line, determining the liquid concentration on a

particular stage and the vapour concentration at equilibrium rising from that stage. The solution can be done by using a

spreadsheet (Excel™). The procedure applied here is as follows: starting with y1¼xD, the equilibrium liquid composition x1on the first stage is computed by solving the Fenske equation (11.18) as x¼ f(y). Next, the vapour composition rising from

the stage below y2 is found from the equation of the operation line:

yn + 1¼R

R+ 1xn +

xDR+ 1

(11.19)

Continued

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vap

our

com

posi

tion

Liquid composition

Time 0

Time 3

0

1

2

3

4

5

6

7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

1/(xD–x

B)

xB

ln( / )-

=xB B

xF D B

dxB F

x x

FIGURE 11.4

Batch rectification of the mixture methanol/n-propanol.


Then the liquid concentration x2 is calculated by the Fenske equation, etc. In this way, one gets the composition profile

from the top xD down to the pot xB. The procedure is repeated by selecting different distillate compositions such as to cover the

working domain. The quantity 1/(xD�xB) is represented versus xB, and the integral evaluated by the trapezoidal rule.

Table 11.1 illustrates the computations. The first value xD¼0.912 corresponds to the top distillate delivered by the three-

stage column for the initial methanol composition in the pot xB¼0.348. The last value xD¼0.2 gives a residual concentration

in the pot xB¼0.027. The integral fromEquation (11.5) is 0.6937.1 The amounts of residueB and distillateD can be calculated

by the relations (11.6) and (11.7) as follows: B¼100�exp(0.6937)¼49.974 kmol; D¼100�49.974¼50.026 kmol.

The average concentration of methanol in the distillate given by the relation (11.8) is xDm¼0.669, much lower than the

initial value of 0.912. This means that a significant amount of n-propanol has been carried in the distillate. The recovery is

Y¼DxDm/FxF¼50.026�0.669/(100�0.348)¼96.54%, above the target 95%. Hence, the recovery is good, but the dis-

tillate quality is poor.

Next, we try to see if employing a rectifier with a large number of theoretical stages would improve the process, we keep

the same reflux. The limit of residual composition in the pot is fixed at the limit value 0.027 found before in order to com-

pare the recoveries. Since infinite stages, the distillate composition starts with a purity of 1. This situation lasts until the

methanol concentration in the pot decreases to xB0 ¼0.1948 found with the relation (11.10). The amount of residue B0 in

the pot is 80.98 kmol (relation 11.11), and accordingly an amount D0 ¼100�B0 ¼10.02 kmol of 100% purity is obtained.

In a second step, the methanol molar fraction in the distillate decreases. The ratio B/B0 can be found from (11.13) as 0.707,

from which we get B¼57.26 kmol. The amount of distillate is D¼100�57.26¼42.74 kmol, significantly less than

before, but of higher purity. For the second step, the purity is xDm,2¼0.601 (relation 11.14), and the final purity is

xDm¼0.778 (relation 11.5). The yield is 95.6%, a value close to that got with a three-stage column.

The conclusion of this exercise is that getting simultaneously high purity and high recovery is not possible by batch

rectification in a single run. Further improvement in recovery can be achieved by recycling the slope fractions. Regarding

the efficiency of the process, increasing the number of stages may help substantially, namely for getting high purities.

Increasing the reflux is another possibility, which is examined next.

11.2.2.2 Operation at variable refluxThe principle of operation is explained in Figure 11.3. As before, the number of stages is fixed. A target

composition of the distillate xD is selected for a given reflux rate R. This composition can be kept con-

stant during the process by steadily increasing the reflux. Equation (11.5) gives:

B

F¼ xD�xFxD�xB

(11.20)

The distillate amount and the recovery are given by the relations B¼F�B and Y¼DxD/FxF.

Table 11.1 Batch Rectification at Constant Reflux: Distillate and Still Concentration ofMethanol

xD xB xD�xB 1/(xD�xB) Integral (Equation 11.5)

0.912 0.348 0.564 1.773 –

0.8 0.203 0.598 1.674 0.2509

0.6 0.109 0.491 2.037 0.1733

0.4 0.061 0.339 2.950 0.1198

0.2 0.027 0.173 5.760 0.1498

0.6937

1The integral is over-rated by 2.5% because of few points, but the physical significance is preserved.


Since the reflux is variable, the energy needed for operation is variable too. In the hypothesis of a

constant enthalpy of vaporisation, the incremental amount of energy is as follows:

dQ¼ R+ 1ð ÞDHvdD (11.21)

One may write dD¼�dB. The variation of dB with the composition in the pot can be obtained by the

differentiation of Equation (11.20):

dB¼� xD�xFð Þ R+ 1ð Þ dxB

xD�xBð Þ2 (11.22)

Combining (11.21) and (11.22) leads to the following relation for the specific energy at variable reflux:

Q

FDHv¼ xD�xFð Þ

ðxFxB

R + 1ð ÞdxBxD�xBð Þ2 (11.23)

The assumption of an infinite number of stages offers again an elegant analytical solution for getting

the minimum specific energy (Stichlmair and Fair, 1998). Reflux ratio and pot composition are linked

by the equation of the operation line:

R¼ 1

a�1

xDxB�a

1�xD1�xB

� �(11.24)

After integration, the following expression is obtained:

Q

FDHv¼ xD�xFð Þ

a�1ð ÞxD 1�xDð Þ axD ln1�xF1�xB

� a�1ð ÞxD + 1ð Þ ln xD�xFxD�xB

+ 1�xDð Þ ln xDxF

� �(11.25)

The following example highlights the strategies for carrying out a batch rectification at constant

distillate composition and the differences with the batch rectification at constant reflux.

EXAMPLE 11.2 BATCH RECTIFICATION AT CONSTANT COMPOSITION AND VARIABLEREFLUXConsider the same mixture and a three-stage column as in Example 11.1. Examine the variable reflux policy in two sit-

uations (1) distillate purity of 0.912 and (2) distillate purity of 0.669.

Solution. Themethod ofMcCabe–Thiele (see Figure 11.3) is worked out in Excel™, as previously explained. Table 11.2

summarises the computation for integrating Equation (11.23). Firstly, the reflux ration is set, and then the residual methanol

concentration in the pot is computed at fixed distillate composition. The final results include ratio B/F (residue/feed), D/F

(distillate/feed), recovery Y and specific energy consumption.

In the first situation, when a purity of 0.912 is aimed, D/F is only 21.6% instead of maximum permitted by the initial

charge 34.8/0.912¼38.15%, even if the reflux ratio increases from 2 to 100. The number of stages, only three, is insuf-

ficient for keeping constant the top purity even at infinite reflux. The recovery is poor at 56.65%. In the second case, when

the target is low-purity distillate,D/F is 45.8%, and the recovery at 88.1% (note that previously at constant reflux ratio of 2

the recovery was 96.54%). The explanation of lower recovery is again in the insufficient number of stages.

Thus, operating the batch rectification at variable reflux allows getting higher purity, however, paid by very large reflux

ratios. The variable reflux policy has to be optimised against product specifications and productivity. With respect to the

energy consumption, Table 11.2 seems to indicate that getting higher purity needs more energy than getting higher recov-

ery. This aspect will be analysed in Section 11.2.2.3.

Continued


Table 11.2 Batch Rectification at Constant Distillate Purity: Reflux Policy, Recovery and Energy Consumption

xD¼0.912 xD¼0.669

R xB z¼xD�xB 1/(R+1)/z2 Integral R xB z¼xD�xB 1/(R+1)/z2 Integral

2 0.348 0.565 9.403 0.000 0.04 0.348 0.350 8.473 0.000

4 0.276 0.636 12.367 0.773 0.5 0.244 0.455 7.232 0.825

8 0.235 0.677 19.609 0.666 2 0.132 0.567 9.345 0.921

16 0.211 0.701 34.606 0.635 6 0.078 0.621 18.168 0.745

100 0.192 0.720 195.077 2.143 100 0.051 0.648 240.596 3.518

4.216 6.009

B/F 0.784 B/F 0.542

D/F 0.216 D/F 0.458

Y 56.65% Y 88.10%

Qsp 2.378 Qsp 1.929

11.2.2.3 Energy input for batch rectificationThe energy consumption is a key issue in batch rectification. In this chapter, we present a consistent

comparison of the two main operation modes, constant reflux ratio R versus constant composition xD.The basis is the assumption of an infinite number of stages, in other words the most efficient rectifi-

cation column.

11.2.2.3.1 Constant refluxThe calculation of the specific heatQsp is given by Equation (11.17). For a given R value, the ratio D/Fis determined by the implicit solution of Equations (11.10)–(11.16) by employing the Goal Seek func-

tion in Excel™. The target is the average distillate purity and the manipulated variable is the xB final.

Figure 11.5 presents the results. The initial mixture has xF¼0.5. Two relative volatilities have been

selected for comparison, a¼2 and a¼3, as well as three average distillate purities, 0.99, 0.9 and 0.8. Full

lines show data for a¼3,while dot lines for a¼2. Lines of constant reflux ratioR are drawn for illustration.

Three vertical lines at 0.505, 0.555 and 0.625 represent the limit values of the ratioD/F at 100% recovery,

equal with the ratio xF/xD. The relation between recovery and ratio D/F is given by Equation (11.16).

Firstly, it can be seen that the relative volatility has a major influence upon the energy of distillation.

Significantly higher Qsp is necessary for separating the component with a¼2 than with a¼3. The

second factor is the target purity and recovery. For a given purity approaching high recovery raises

exponentially the energy input and make necessary higher reflux rate. High recovery can be achieved

with smaller D/F but of good purity as with larger D/F of modest purity.

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6

Qsp

D/F

R = 5

R = 4

R = 3

R = 2

R = 2.5

R = 7.5

R = 10

xD= 0.8

xD= 0.9

xD= 0.99

a = 2a = 3

FIGURE 11.5

Batch rectification: energy input at constant reflux ratio and infinite number of stages.


For example, for a¼3 and high recovery of 98.5%, the values D/F are 0.492, 0.546 and 0.614 for

the purities 0.99, 0.9 and 0.8, while Qsp and R are, respectively (5.4; 10), (2.74; 4) and (1.84; 2). If high

purity xD¼0.99 is aimed, selecting D/F of 0.4, 0.45 and 0.49 (recoveries of 79.2%, 89.1% and 97%)

gives from Figure 11.5 the valuesQsp of 1.25, 1.9 and 5.4. Remember thatQspmeans the multiplication

factor for vaporising completely the feed. The explanation of the above behaviour is the considerable

increase in the reflux ratio, from 1.25 to 1.9 and finally to 7.5.

11.2.2.3.2 Variable refluxAt constant distillate composition, the global picture looks similar, but the values are different, as

shown by Figure 11.6. Again, the specific energy input tends to very high values for achieving high

purity or high recovery. For example, at xD¼0.99 and D/F¼0.4 (recovery 79.2%), Qsp is about 1 for

a¼3 but 1.5 for a¼2. As before, for a¼3 and xD¼0.99, considering the ratioD/F of 0.4, 0.45 and 0.49

gives Qsp of 1, 1.2 and 1.6. It can be observed that at the same purity and recovery the operation at

variable reflux ratio needs less energy compared with constant reflux. This is an important practical

result. Note that Figures 11.5 and 11.6 can be seen also as a shortcut tool for assessing the reflux policy

and for estimating the minimum energy input.

11.2.2.3.3 Comparison between energy input at R constant and R variableThe specific energy consumption is plotted in Figure 11.7 for two situations, high purity (xD¼0.99) and

low purity (xD¼0.8) for a¼3. It can be observed that variable reflux needs always less energy. When

high purity is targeted, the two operation modes are equivalent at lower recovery, roughly up to 70%,

0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Qsp

D/F

a = 2a = 3

xD= 0.8

xD= 0.9

xD= 0.99

FIGURE 11.6

Batch rectification: energy input at constant distillate composition and infinite number of stages.


but it starts to diverge considerably at higher recovery. For example, at 99.5% recovery, the constant

reflux policy asks four times more energy. At lower purity, the two operation modes give similar re-

sults. The conclusion is that operating at variable reflux is more advantageous theoretically, but con-

stant reflux is more convenient in practice. A compromise can be found by operating step-wise a

variable reflux policy following a procedure determined by optimisation.

It is interesting to compare the above result with the minimum energy input for continuous oper-

ation and infinite number of trays. This is given by the expression (Stichlmair and Fair, 1998):

Qmin

FDHv

� �¼ 1

a�1

xDxF�a

1�xD1�xF

� �� D

F

� �(11.26)

The energy input in continuous operation depends on the relative volatility and the target purity, but it is

a linear function of D/F, being finite at full recovery. The computation by Equation (11.26) plotted in

Figure 11.7 shows that the minimum energy input is achieved in continuous operation.

11.2.2.4 Optimal reflux and heating policyThe previous sections demonstrated that the separation of zeotropic mixtures by batch distillation

should solve the trade-off of purity versus recovery and energy requirement. Figure 11.8 presents this

relation for 99% purity at infinite number of stages separating the binary methanol/n-propanol by usingthe previous analysis. Right side vertical axis displays the specific energy versus recovery (dashed

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Qsp

D/F

xD= 0.99 xD= 0.8

R ct.xD var.

xD ct.R var.

Continuous

FIGURE 11.7

Energy input at constant reflux ratio versus constant distillate composition (continuous line – variable reflux,

dashed line – constant reflux).


lines) for (1) R variable and xD constant (open square marks) compared with (2) R constant and xDvariable (open circle marks). The left side vertical axis shows the corresponding reflux ratio R, buton logarithmic scale for easier visualisation. It can be observed that for achieving the same recovery

the strategy with variable reflux ratio starts from lower values, but can rise to infinite when approaching

100%. However, the final energy input with variable reflux is always lower than with constant reflux.

At lower recovery, the difference is negligible, in this case up to 70–75%, but it increases significantly

further. For example, at 99% purity and 80% recovery, comparing R variable from 1 to 3 versus R con-

stant at 2.2, gives Qsp 1.0 versus 1.2, respectively. At 95% recovery, R variable from 1 and 12 versus Rconstant at 5, gives Qsp of 1.4 versus 3.0, respectively. For recoveries towards one, the energy input

raises to infinite. The practical manner to deal with variable reflux in industry is by stage-wise policy.

The conclusion is that batch distillation is a good subject for optimisation. The following objective

functions are the most used (Mujtaba, 2004):

1. Minimum time of operation for achieving an amount of distillate of specified purity.

2. Maximum distillate amount of specified purity to be collected in a given time.

3. Maximum profit. This can be defined as the difference between the added value divided by the total

batch time, including additional operations, and the operating costs, including manpower, energy,

equipment utilisation, etc.

There is a vast literature on this subject, for which we recommend specialised monographs (Mujtaba,

2004; Diwekar, 2012). The research started in the early 1960s. Two basic approaches have been

employed: the optimal control theory known as the Pontryagin maximum principle (1962) and the

0

1

2

3

4

5

6

7

8

9

10

1

10

100

0 0.2 0.4 0.6 0.8 1

Spe

cific

ene

rgy

inpu

tRef

lux

ratio

Recovery

xD ct. R var.

R ct xD var.Qsp xD ct.

Qsp R ct.

Ct. reflux at 2.2

Var. reflux 1–3

Ct. reflux at 5.6

Var. reflux 1–12

FIGURE 11.8

Reflux policy in batch distillation.


non-linear programming for dynamic optimisation following the feasible and infeasible path. In the last

years, researches based on genetic algorithms have been developed for integrating the design and op-

erating policies with complex trade-offs among production revenues, utility and capital costs Sorensen

and co-workers (Low and Sorensen, 2003; Barakat et al., 2008).

Hereafter, we present only a qualitative illustration of an optimal reflux policy. Figure 11.9 presents

the separation of a mixture methanol/ethanol/n-propanol of 20/20/20 kmol at atmospheric pressure by

batch rectification with a column of 15 plates. The computation is performedwith the module BatchSep

implemented in Aspen Plus v8.0. The operation is conducted at constant reflux ratio R of 5 and boilup Vof 30 kmol/h. The plate hold-up is 0.2 kmol each and 1 kmol for the reflux drum. The mixture has a

quasi-ideal behaviour, the mean relative volatility of components at the normal boiling points being

a(ethanol/methanol)¼1.721 and a(n-propanol/ethanol)¼2.077. It can be seen that a first methanol-

rich fraction can be obtained, but its purity declines rapidly as the distillation progresses, by incorpo-

rating more and more ethanol. A second ethanol-rich fraction can be separated, but of modest purity.

The third propanol fraction is also of low purity. The fractions of low than required purity, usually

called off-specs, are quite large, with obvious productivity loss.

The second method is a stage-wise programmed reflux policy, as shown in Figure 11.10. For the

first fraction, the following policy is applied: R¼4 for 1 h, then R¼8 for 1.5 h, and R¼12 for 1.5 h;

boilup V¼30 kmol/h. The first fraction has a purity of 99.2%. It follows an off-specs period of 1.5 h by

droppingR to 6 and increasingV to 36. In this way, methanol can be stripped out for getting a second cut

of good purity. The second fraction is drawn off for another 4.5 h at R¼10 and V¼40, the purity being

95% with methanol as main impurity. A short off-specs period of 0.5 h at R¼6 follows for removing

the rest of ethanol. The remaining material in the pot is high-purity n-propanol.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

Mol

ar fr

actio

n

Time (h)

Methanol

Ethanol

n-Propanol

FIGURE 11.9

Separation of the ternary mixture methanol/ethanol/n-propanol by batch rectification at constant reflux policy.


Thus, by variable reflux policy, two high-purity fractions have been obtained, methanol and

n-propanol, the light and the heavy component. The recovery of methanol, ethanol and n-propanolare of 80%, 77% and 98%, giving a total recovery in a single run of 82.8%. The off-specs cuts are

usually recycled to the next batches, or collected and distilled separately as binary mixtures. The price

to pay is higher energy consumption and longer time operation, involving more manpower and

equipment costs.

In the industrial practice, the stage-wise reflux policy is widespread, as it allows the operators adapt-

ing the process control to different tasks and products. Another good practice is limiting the recovery

for getting high purity, but recycling the off-specs mixtures. In this way, the characteristics of a quasi-

continue operation can be obtained (Luyben, 1971, 1988; Luyben and Chien, 2010).

11.2.2.5 Design and simulationThe conceptual design should comply with the first requirement of a batch process, flexibility. Indeed,

this separation device should be capable of handling various components and compositions, and

implicitly various VLE. With respect to vessel design, this should ensure suitable operation in the

pressure/temperature working range, including heat transfer capabilities.

Regarding the column design, this should have a sufficient number of theoretical stages, robust

hydrodynamics and good dynamics for control and operation. Some requirements could be contradictory.

Thus, tall columns deliver better purity and recoveries, but have slower dynamics. On the contrary,

short columns permit faster distillation, but the insufficient separation power gives larger amounts

of slope cuts. A number of 15–20 theoretical stages are often sufficient. Structured packing is

recommended.

0

5

10

15

20

25

30

35

40

45

50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10 11 12

Ref

lux

ratio

and

boi

lup

(km

ol/h

)

Mol

ar fr

actio

n

Time (h)

MethanolEthanoln-propanolReflux ratioBoilup

Cut-1 Cut-2 Cut-3

Offs 1 Offs 2

FIGURE 11.10

Separation of the ternary mixture methanol/ethanol/n-propanol by batch rectification at variable reflux policy.


The hold-up of stages and the dump receiver are important parameters in optimisation. When an

ABC mixture is handled, smaller values are more favourable for the first fraction, while larger values

for the middle fraction. A total hold-up of 10% referred to the initial charge gives good results

(Stichlmair and Fair, 1998).

The computer simulation of batch distillation is today a powerful tool for investigating the feasi-

bility and the operability of various scenarios, saving costly research and development resources. Ded-

icated simulation tools are available, which can be found on the websites of the main suppliers, such as

Aspen Batch Modeller, ChemCAD, SimSci Pro/II, ProSim and gPROMS. The complexity of the

modelling is a distinctive feature. From algorithmic viewpoint, the software deals with large stiff sys-

tems of differential equations, the treatment of events, dynamic optimisation, etc. Obtaining conver-

gence is difficult or impossible without a careful design and/or selection of parameters necessary for

ensuring good conditioning of equations. For example, using molar flows instead of mass flows, molar

boilup instead duties, etc., may help the solver and avoid index problems. The shortcut methods pre-

viously developed may be very helpful to understand the problem.

11.2.3 SEPARATION OF AZEOTROPIC MIXTURESAs explained in Chapter 9, the separation of mixtures forming azeotropes is difficult because of

the existence of separation boundaries. A key topic is the entrainer selection for breaking the

azeotrope. In few cases, there is only one homogeneous distillation region. The entrainer must

be selected such to belong to the same distillation region with the components A and B. This con-dition is simply met when the boiling point of the entrainer lies between those of A and B. In other

situations, the entrainer should form either an intermediate boiling maximum azeotrope with the

lower boiler A or an intermediate minimum boiler azeotrope with the higher boiler B (Stichlmair

and Fair, 1998).

In most cases, the azeotropes are formed by close boiling components. The entrainer selection may

lead to employing homogeneous or heterogeneous azeotropic distillation. This topic was thoroughlyinvestigated by Skogestad and co-workers (Skouras et al., 2005). There is a large analogy between

batch and continuous processes, but also some fundamental differences. Let us consider a binary

AB azeotrope to be separated by an entrainer. In a continuous process, the designer can play on the

sequencing of two or three columns, each with different number of stages and reflux specification.

In a batch process, the configuration is limited to only one column with fixed number of stages. Only

boilup and reflux may be adjustable variables for the distillation part. However, collecting intermediate

fractions and submitting them to further processing may give supplementary means for solving the

problem, as it will be illustrated later in this section.

11.2.3.1 Homogeneous azeotropic distillationThe separation of an azeotropic mixture in components by simple batch distillation is impossible be-

cause at the azeotropic point there is no driving force between liquid and vapour phases. By adding an

entrainer, the phase equilibrium changes dramatically. The occurrence of distillation boundaries adds

difficulties. If the border is straight, this cannot be crossed (Doherty and Malone, 2001). If the border

has a significant curvature, this can be crossed from the concave side. This behaviour gives the oppor-

tunity of separating the components in fractions of good purity, depending on the VLE, the ratio

entrainer/mixture and the operation parameters.


EXAMPLE 11.3 HOMOGENEOUS AZEOTROPIC BATCH DISTILLATIONAcetone (nbp 56.2 �C) and chloroform (61.2 �C) form a homogeneous azeotrope with the composition 34.5 mol% acetone

and nbp 64.7 �C. Example 9.2 demonstrated the separation by continuous distillation using toluene (nbp 110.9 �C). Ex-amine the possibility of separating the mixture by batch distillation. As a numerical example, consider an initial charge of

35/15/50 kmol acetone/chloroform/toluene.

Solution. There is a distillation border linking toluene and azeotrope, which presents a significant curvature, as shown

by Figure 9.4. This can be crossed at finite reflux in continuous distillation, resulting in high purity of both acetone and

chloroform products. The question is what happens when employing batch distillation. The problem is simulated in Aspen

Plus v8.0 bymeans of the module BatchSep. The column has 15 theoretical stages. After some trials, an acceptable solution

was reached by applying the following scenario:

Cut-1: Boilup 40 kmol/h; distillate rate 5 kmol/h; time 5.5 h.

Slope cut-1: Boilup 60 kmol/h; distillate rate 5 kmol/h; time 2.3 h.

Cut-2: Boilup and distillate rate 5 kmol/h; time 1.9 h.

Table 11.3 presents the results. The first cut separates high-purity acetone of 99.66 mol%. The second slope cut gives a

transition mixture of acetone and chloroform in almost equal proportion. The second cut comprises a chloroform-rich frac-

tion of 88.42 mol%. Remarkable, all fractions are toluene free. The difference between the total of cuts and the initial

charge is the hold-up in the column. Note that in this first approach, the overall recovery of acetone is 78.3%mol.

Figure 11.11 shows the evolution of the process in a right triangle RCMwith UNIQUAC as VLE option. The evolution

of the composition in the pot is described by the curve 0–1–2–3, where the points mean initial state, end of cut-1, end of

slope cut and end of cut-2. The composition of vapour received in condenser is plotted on the inclined edge. The perfect

straight line shows that, indeed, there is no toluene entrained in the distillate. Note that the vapour compositions are linked

by tangents with the liquid residue points. The filled-in markers on the condenser curve have the same signification as

before, showing the composition of cuts at the transition points.

Thus, for the cut-1 (zone 0–1), the vapour composition starts at 99.99 dropping to 98.90%. Then, it follows a transition

period (zone 1–2), in which the vapour crosses the azeotropic point. Finally, the third cut is taken between 81.6% and

93.3%. The simulation indicates that it is very difficult to get a high-purity chloroform fraction. Very high ratio en-

trainer/mixture should be employed.

In a second attempt, we can improve the separation by two observations:

Firstly, the chloroform-rich fraction can be concentrated, say to 99%, by simple batch distillation, the final residue

being the azeotrope corresponding to the amount of acetone. In this case, the 100% pure chloroform gives 6.31 kmol with

a recovery of 6.31�100/15¼42%. The rest is dumped into the slope cut, making a total of 14.69 kmol (acetone 7.52,

chloroform 7.17). Secondly, the slope cut can be submitted likewise to simple distillation, as it contains an excess of ac-

etone with respect to azeotrope. In this way, another 3.74 kmol acetone can be recovered, raising the overall molar recovery

to 89%. The final molar recovery of recovering the both components in a single run is 74.9%. The remaining azeotrope

mixture of 12.54 kmol (3.85 acetone, 8.69 chloroform) can be recycled to the next batch.

Table 11.3 Material Balance for Example 11.3

Operation Acetone (kmol) Chloroform (kmol) Total (kmol)

Charge 35 15 50

Cut-1 27.407 0.093 27.5

Slope cut 6.418 5.082 11.5

Cut-2 1.1 8.4 9.5

Total 34.925 13.575 48.5

Hold-up 0.075 1.425 1.5


As it can be observed, the separation by homogeneous azeotropic distillation needs careful inves-

tigation by experimental research and computer simulation. Among important industrial applications

one may cite the separation of hydrochloric acid from aqueous solution by using sulphuric acid

(Stichlmair and Fair, 1998).

A vivid interest was found in the last years by extractive azeotropic distillation. A high boiler en-

trainer can increase radically the volatility of the light component. Typical application is the separation

of ethanol from diluted aqueous solution with ethylene glycol. A suitable arrangement is the middlevessel column (see Figure 11.1c). Initially, the mixture is charged in the middle vessel and the whole

column brought under total reflux. Then the entrainer is fed at the top of the rectification section flow-

ing down in counter-current with the mixture to be separated, issued from the middle vessel. The light

component separates in top, while the mixture heavy/entrainer enters the stripping section, provided

with reboiler, from which the entrainer is removed in bottoms (to be recycled for the next batch), while

the heavy accumulates in the middle vessel.

11.2.3.2 Heterogeneous azeotropic batch distillationLiquid–liquid phase splitting is a powerful method for separating azeotropic mixtures. In this way, the

constraint of the boundary crossing can be overcome. The continuous process for separating ethanol

from water described in Section 9.7.3 has at least two columns. The first one is for removing the water

excess in bottoms with respect to the azeotrope separated in top. The second one is for breaking the

azeotrope with an entrainer: water is carried with the ternary azeotrope in the top, while the ethanol

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Chl

orof

orm

Acetone

1

2

2

3

3

00

1

Cut-1Slope cut-1

Cut-2

Distillation

az.

FIGURE 11.11

Batch distillation of the azeotrope acetone/chloroform with toluene as entrainer.


leaves in the bottoms. In between, there is a decanter for separating the ternary azeotrope in two phases.

The water phase returns to the water column, the organic phase to azeotropic distillation.

A similar principle is applied in batch distillation, shown in Figure 11.12 for the separation of eth-

anol fromwater employing benzene as entrainer. The ternary diagramwas computed by Aspen Plus 8.0

with NRTL as thermodynamic option. In a first step, the initial water solution F is charged together

with recycled aqueous phase w in an inverted rectification set-up (batch stripping). The binary azeo-

trope ethanol–water separates in the top distillate d1, while the excess water goes in the bottoms b1. Thesecond step follows a regular rectification. The charge contains the azeotrope, the make-up entrainer

and the organic phase o resulting from a previous batch. The water is removed with the ternary azeo-

trope, which has the composition benzene/ethanol/water of 0.527/0.281/0.192 (molar fractions) or

0.715/0.225/0.06 (mass fractions) with the nbp 64.2 �C. The ethanol remains in the pot. In this

way, high-purity ethanol over 99% can be obtained. Note the large difference of the boiling points

of the ternary azeotrope compared with the pure ethanol. If slight excess of benzene is introduced, this

can be removed down to ppm level by the next binary azeotrope ethanol/benzene.

11.3 BATCH REACTORS11.3.1 OPERATION MODESBatch reactor is standard equipment in chemical process industries. It consists essentially of an auto-

clave provided with mixing and heat transfer devices, as well as with piping connections for feeding the

reactants and taking-off the products (Figure 11.13). The operation mode is intrinsically dynamic,

Benzene

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

69.35 °C

64.02 °C

78.15 °C

67.71 °C

w

o

f1F

d1

b1

d2

f2

b2

78.3 °C

80.1 °C100 °C

b1

f1

d1

Initial mixture F

Aqueousphase w

Azeotrope Water Azeotrope

EntrainerOrganicphase o Ethanol

Organicphase o

Waterphase w

b2

d2

FIGURE 11.12

Separation of ethanol/water mixture by heterogeneous batch azeotropic distillation.


which means that the properties vary continuously in time, but not in space because of homogenisation

by mixing. For an isothermal operation, there is a perfect analogy between the evolution in time of

concentrations and reaction rate in a batch reactor and the variation over length in an ideal continuous

plug-flow reactor (PFR).

Batch reactors are typically employed in laboratory for investigating the feasibility for manufacturing

chemical products, namely from the viewpoint of chemistry. Another important application is the formu-

lation of the recipe of a future chemical product in term of composition and operation procedure. Among

advantages of batch reactors one can cite flexibility formulti-product processes, and relatively low cost of

equipment andmaintenance. A remarkable feature of batch reactors is the possibility of using directly the

research in laboratory for scaling-up the industrial process. If only the reaction time is examined, a batch

reactor offers the best productivity, the same compared with an ideal PFR. However, with respect to the

total manufacturing time, including auxiliary steps (charging, heating, cooling, discharging and cleaning),

the productivity becomes much lower. Other disadvantages are higher manpower costs, more sophisti-

cated process instrumentation and control, as well as the inherent variability of the product quality.

From operation viewpoint, one can distinguish between two modes:

• Fully batch process: reactants are initially charged and products discharged at the end;

• Semi-continuous process: at least one reactant or one product is removed continuously. This

procedure is employed for manipulating the product distribution for equilibrium-limited reactions,

for playing with kinetic effects for complex reactions and for avoiding runaway effects in the case of

highly exothermic reactions.

11.3.2 ISOTHERMAL BATCH REACTORAn important design parameter for efficient heat transfer in batch reactors is the heat transfer area to

reactor volume ratio. Let us consider the reversible general reaction:

Charge

HX

Thermalagent Discharge

Concentration

Reaction rate

Timedt

dz

FIGURE 11.13

Batch reactor and the comparison of performance with a plug-flow reactor.

46911.3 BATCH REACTORS

aA + bB + � � � ��!Keq

pP + rR + � � � (11.27)

The material balance equation is as follows:

input flowsð Þ- output flowsð Þ+ generation rateð Þ¼ rate of accumulationð Þ (11.28)

Since the input and output flows are zero, the characteristic equation in differential form is as follows:

� 1

V

dNA

dt¼�rA cj, T, pk

� �(11.29)

NA is the number of moles of the reference reactant A, V the reaction volume and �rA the consump-

tion rate of the reactant A. The reaction rate is a function of species concentrations cj, temperature T andof a number of parameters pk expressing chemical kinetics and equilibrium. If the kinetics is known,

then Equation (11.29) can be integrated to get the time for achieving a given conversion XA, f of the

reference reactant. For an isothermal reaction at constant volume, the characteristic equation is as

follows:

tr ¼ cA0

ðXA, f

0

dXA

�rA cj, T, pk� � (11.30)

If the reaction volume is variable, the usual assumption is a linear variation with conversion,

V¼V0(1+ eAXA). The integration of (11.29) leads to:

tr ¼ cA0

ðXA, f

0

dXA

1 + eAXAð Þ �rA cj, T, pk� �� (11.31)

The integration of Equations (11.30) and (11.31) can be done analytically for simple reactions (singlestoichiometric equation and single rate expression) in a number of cases. Some solutions are given in

Table 11.4. The same equations are valid for a PFR, in which the batch time is replaced by the residence

time. When the analytical solution is not available, the integration can be solved using a spreadsheet

and suitable numerical methods, such as trapezoidal or Simpson rules.

In the case of complex reactions, analytical solutions are available for parallel and consecutive

reactions, illustrated in Table 11.5.

In general, for complex reactions, the distribution of the components can be found by numerical

integration of the system of kinetic equations. A suitable variable is the molar extent of reaction xiassociated to each independent reaction i (see Chapter 8). Thus, for a component j participating in

the reaction i, the variation of the number of moles is as given by:

Nj¼Nj0 + njxi (11.32)

The net reaction rate of the component j involved in R reactions is as follows:

1

V

dNj

dt¼XRi¼1

nijdxv, idt¼XRi¼1

nijri (11.33)


in which xv,i is the molar extent of reaction referred to the volume unit. Note that the molar extent of

reaction xm,i referred to the total mass of reaction may be used for when the reaction volume is variable

and when the rate equations are formulated in mass units. The expression (11.33) is equivalent with the

system of equations:

dxv, idt¼ ri xv,1, xv,2, . . . , xv,R

� �for i¼ 1�R (11.34)

If the volume variation is significant, then a supplementary equation should be added, as:

dV

dt¼ eAXA with V¼V0 at XA¼ 0 (11.35)

Applications of time calculation for both simple and complex reactions can be found in the textbooks

for chemical reactions engineering (Fogler, 1999).

Table 11.4 Time–Conversion Relation for Single Reactions in Batch Reactors at Constant

Volume

No. Reaction Rate Expression Reaction Time

1 A!products �rA¼kcA kt¼� ln(1�XA)

2 uA!products �rA¼ kcnA

kt n�1ð Þcn�1A0¼ 1�XAð Þ1�n�1

3 A+B!products �rA¼kcAcB ktcA0(M�1)¼ ln[(M�XA)/M/(1�XA)]

4 A+2B!products �rA¼kcAcB2

ktc2A0 2�Mð Þ2¼ 2 2�Mð ÞXA=M= M�2XAð Þ+ ln M�2XAð Þ=M= 1�XAð Þ½ �

5 A$B �rA¼k1cA�k2cB (k1+k2)t¼� ln(1�XA/XAe)

6 A+B$P+R �rA¼k1cAcB�k2cPcR2k1 1=XAe�1ð ÞcA0t¼� ln

XAe� 2XAe�1ð ÞXA

XAe�XA

for cA0¼cB0 and cP0¼cR0¼0

where XAe is the equilibrium conversion

2A$ 2P �rA¼ k1c2A�k2c

2P

2A$P+R �rA¼k1cA2�k2cPcR

A+B$ 2P �rA¼ k1cAcB�k2c2P

Table 11.5 Time–Concentration Relation for Complex Reactions in Batch Reactor

Reaction Rate Expression Reaction Time Expression

1 A!P, A!R �rA¼k1cA+k2cArP¼k1cArR¼k2cA

cA/cA0¼exp[�(k1+k2)t]cP� cP0¼ k1cA0

k1 + k21�exp � k1 + k2ð Þt½ �ð Þ; cP�cP0

cR�cR0¼ k1k2

2 A!P!R �rA¼k1cArP¼k1cA�k2cPrR¼k2cP

cA/cA0¼exp[�k1t]cP=cA0¼ k1

k2�k1exp �k1tð Þ�exp �k2tð Þ½ �

cP0¼cR0¼0; cA0¼cA+cP+cR

tP,max ¼ ln k1=k2ð Þk1�k2

; cP,max=cA0¼ k1=k2ð Þk2= k1�k2ð Þ


11.3.3 NON-ISOTHERMAL BATCH REACTORThe energy balance can be used to determine the temperature profile. For multiple reactions and con-

stant volume, the equation is as follows:

XCj¼1

mjcp, j

!dT

dt¼XRi¼1

riVð Þ �DHR, ið Þ�Qt (11.36)

Equation (11.36) is solved together with the system (11.34) and (11.35) in order to obtain the profile of

concentrations and temperature. Equations for phase equilibrium and for pressure development should

be considered in a more general approach.

The heat transfer term is described by a general equation of the form:

QT ¼UADTm (11.37)

U is the heat transfer coefficient, A the heat transfer area and DTm the mean temperature difference. For

example, for cooling/heating DTm is the mean logarithmic temperature difference at the inlet and outlet

between reactor and cooling agent.

DTm¼ T�Ta,1ð Þ� T�Ta,2ð Þln T�Ta,1ð Þ= T�Ta,2ð Þ (11.38)

The heat transfer capabilities of a batch reactor are usually:

• Jacket heat transfer

• Coil heat transfer

• External condenser

• External heat exchanger and pumping.

An important issue in the design of a batch reactor is the sensitivity of the thermal regimewith respect tochanges that may occur in operation, introduced voluntarily by the operator or produced accidentally.

This issue is particularly important when dealing with highly exothermic reactions. High activation

energy gives higher sensitivity to temperature. The designer should examine several scenarios in order

to determine the heat transfer capacity in the worst situation. For isothermal operation at any time, the

heat transfer capacity should be larger than the heat generated by reaction. For a single exothermic

reaction A!products, this condition is as follows:

Qt >�rA cj, T� �

V �DHR,Að Þ (11.39)

The equilibrium between the heat generated and the heat transferred is fulfilled dynamically by the

control system. The analysis of this topic may start with a representation in a temperature-conversion

diagram. In this way, the trajectory of the reaction becomes independent of time. Thus, for the reaction

A!products, the characteristic equation is as follows:

dXA

dt¼�rAV

NA0(11.40)


Eliminating the time between Equation (11.36) for single reaction and (11.40) leads to:

dT

dXA¼NA0 �DHR,Að ÞX

mjcp, j� NA0Qt

�rAVð ÞX

mjcp, j(11.41)

Equation (11.41) can be integrated numerically. Let us consider an exothermal first-order reaction, for

which (�rAV)¼k0 exp(�E/RT)NA0(1�XA). In addition, Qt¼UA(T�Tc), in which Tc is the tempera-

ture of the cooling agent, considered constant. The relation (11.41) becomes:

dT

dXA¼DTad� 1

tT

T�Tcð Þk 1�XAð Þ (11.42)

Equation (11.44) highlights two important parameters:

• Adiabatic temperature rise

DTad ¼NA0 �DHR,Að ÞXmjcp, j� � (11.43)

• Time constant

tT ¼ UAXmjcp, j

(11.44)

The integration of Equation (11.42) can be done in Excel by using the Euler method. Figure 11.14 presents

the results of a numerical example. The purpose is examining the possibility of reaching a desired temper-

ature profile bymanipulating theparameters of the heat transfer system, namely theheat transfer coefficient

and the temperature of the thermal agent. In a first attempt, the exercise aims getting an isothermal regime at

373 K for a first-order reaction taking place at constant volume. The reaction is moderate exothermic with

DHR,A¼�30 kcal/mol. The activation energy of about 20 kcal/mol raises by two the reaction rate for

10 K. The cooling agent is available at constant temperature Tc¼303 K. The following data are known:

k¼ 2:0�108exp �9902:2=Tð Þs�1; NA0¼ 10kmol; V¼ 5m3;X

mjcp, j¼ 1400kcal=K

U¼ 600; 570; 560; 550f gkcal= m2hK� �

; A¼ 20m2:

Figure 11.14 highlights the fact that getting a stable temperature profile without process control action is

impossible. WhenU takes the value 600 kcal/m2/h/K, the generated and transferred heat are initially bal-

anced, but the reaction runs to extinction because of the too hard cooling effect. Lower U values of 570

and 560 kcal/m2/h/K may even give a rise in temperature above 373 K for a shorter or longer period. IfUfurther drops to 550 kcal/m2/h/K, the temperature profile jumps and the reaction runs away. Note that the

above variations for the heat transfer coefficient are in the limit of computational errors.


11.3.4 SEMI-CONTINUOUS BATCH REACTORSA batch reactor operates in semi-continuous mode if at least one reactant is charged in the vessel ini-

tially, but the addition of the other reactant(s) or the removal of product(s) takes place continuously. Let

take as example the reaction A+B!P+R. The reactant A is charged initially, while the reactant B is fed

slowly. This is often the case when B is a bubbling gas, as in hydrogenations, oxidation, chlorination orammonolysis reactions. If the reaction is reversible, the removal of a product will shift the equilibrium

to completion. This is typically the case of esterification reactions, where water is removed by distil-

lation. Another interesting application is manipulating the product distribution in complex reactions.

Thus, if the main reaction is A+B!P, and the secondary one P+B!R, the product P is maximised if Bis added in small amounts.

Other situation is of fast exothermic reactions, when a reactant is added in portions, at a rate com-

patible with the heat removal capacity of the reactor.

The component material balance (11.28) applies for determining the reaction time. For multiple Rreactions with C components, the variation of the variable Nj in time is obtained by summing the con-

tribution of each reaction, including input or output flows:

dNj

dt¼Fj, in�Fj,out +

XRi¼1

ni, j �riVð Þ (11.45)

in which Fj,in and Fj,out are inlet and outlet molar flow rates of the component j. Supplementary equa-

tions should be written for Fj,in and Fj,out taking into account the specific operation conditions, for

FIGURE 11.14

Temperature-conversion dependence of a batch reactor.


example, feeding/emptying over a time interval, or periodically, distillation of a product by a given

duty, etc. Assuming constant density, the variable reaction volume is described by:

dV

dt¼Qv, in�Qv,out (11.46)

where Qv,in and Qv,out denote the total volumetric flow rates at inlet and outlet, respectively.

The initial conditions are Nj¼Nj0, V¼V0 at t¼0. The ODE system (11.45)–(11.46) can be inte-

grated by usual numerical methods.

EXAMPLE 11.4 SEMI-BATCH FEED OF REACTANTS IN EXOTHERMIC REACTIONSThe reaction A+B!P+R describes the manufacturing of a specialty product. The reactant A is an organic component,

while the reactant B is NH3. The initial amounts of A and B are 10 and 20 kmol, respectively. NH3 is available as aqueous

solution of 30% weight with the density of 892 kg/m3. The reaction is highly exothermic with DHR¼�240 kJ/mol. Ini-

tially, A is dissolved in 4 m3 solvent. It is planned to carry out the reaction in a batch reactor of 6 m3 isothermally at

400 K. The design of the reactor allows amaximum cooling duty of 750 kW. Examine the possibility of feeding the reactant

B semi-continuously in order to fulfil the constraint. The reaction rate equation is rA¼�2.0�104 exp(�2408/T)cAcB. Thetarget conversion is 99.95%.

Solution. The problem is simulated in Matlab. The required amount of ammonia solution is determined as follows: molar

concentration of ammonia cB0¼892�0.3/17¼15.741 kmol/m3; volumetric feed Qv¼20/15.741¼1.271 m3/h over 1 h.

Thus, the total volume is V¼4+1.271¼5.271 m3. The initial reactant concentration in the batch mode is

cA0¼10/5.271¼1.897 kmol/m3, and cB0¼20/5.271¼3.795 kmol/m3. At 400 K, the reaction rate constant is

k¼2.76�10�4 m3/kmol/s. The reaction time for achieving XA¼0.995 in batch mode calculated with Equation (4) from

Table 11.4 gives 147.5 min.

The initial reaction rate is �rA0¼2.76�10�4�1.897�3.795¼1.99�10�3 kmol/m3/s. Accordingly the heat gener-

ated is Qg¼1.99�10�3�5.271� (�240,000)¼2517.9 kW. Since this is 3.3 times the maximum cooling duty, ammonia

must be added semi-continuously. We charge initially an amount of B such to give a reaction rate compatible with the duty

constraint; a load of 4.5 kmol fulfils this condition. The rest of 15.5 kmol (0.985 m3 solution) is fed during 1 h at constant

flow 985/3600¼0.274 l/s.

Figure 11.15 shows the results obtained with Matlab, dot lines for batch operation and full lines for semi-batch. The

reaction rate is faster in batch mode. In semi-batch, the initial reaction rate is much moderate, since the reactant B is in

Continued

0

1

2

3

4

5

6

0

5

10

15

20

25

0 20 40 60 80 100 120 140 160 180

Vol

ume

(m3 )

Mol

ar a

mou

nt (

kmol

)

Time (min)

Volume

NB

NA

BatchSemi-batch

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120 140 160 180

Dut

y (M

W)

Time (min)

Batch

Semi-batch

FIGURE 11.15

Semi-batch reactors: runaway prevention by semi-continuous feeding of reactants.


smaller amount. The same is valid for the heat generated, whose elimination is spread over a longer period. The reaction

time increases from 147 to 167 min. This example demonstrates that the semi-batch feeding of reactants is a genericmethod

to ensure efficient heat removal for high exothermic reactions and prevent runaway.

11.4 INTEGRATION OF DESIGN AND CONTROL OF BATCH REACTORSThe important conclusion from the previous section is that without the intervention of the control system

is impossible to ensure the temperature profile of a batch reactor, namely isothermal operation. The

reason is that the batch reactor is permanently in unsteady state, unlike continuous reactors running

around an operating point, where the role of the control system is to counteract the effect of distur-

bances. Therefore, the process control of batch reactors is more demanding, particularly in the case

of highly exothermic reactions. Dynamic simulation with control implementation is available today

in process simulators, as well as the modelling tools in Matlab. These can be used for examining dif-

ferent scenarios, and in the first place safety issues. The designer should provide not only sufficient heat

transfer capability for the worst scenario but also security measures in the case of emergency situations.

Preventing the runaway of exothermic reactions in batch reactors is an important issue in industry.

Among numerous papers, we mention the review of Westerterp and Molga (2006). They emphasised

the danger of runaway namely for speciality chemicals, where often the scale-up takes place directly

from laboratory to industrial plant, without kinetic studies and accurate evaluation of the thermo-

chemical effects. The operation of such reactions takes great risks, unfortunately demonstrated by sev-

eral serious accidents (Willey, 1999). On the contrary, the excessive prudent selection of operating

conditions leads to low productivity. In the last years, the Reaction Calorimeter has proved to be a

key experimental tool, particularly for determining the adiabatic temperature rise, previously definedby the relation (11.43). The next two examples illustrate the interrelation between thermal design and

process control of batch reactors.

EXAMPLE 11.5 RUNAWAY SCENARIO FOR EXOTHERMIC REACTIONS IN BATCHREACTORSThe polymerisation reaction in solution A!product has been studied in laboratory. Around 373 K, the reaction is very

sensitive with the temperature, doubling the reaction rate for 10 K. The reaction rate expression is k¼2.0�108 exp

(�10,070/T). The thermal effect is moderately exothermic of 30 kcal/mol. Examine the sizing of an industrial reactor,

as well as the design and process control implementation of a suitable cooling system. The reactor is charged with 10 kmol

A of concentration 2 kmol/m3.

Solution

Sizing. The reaction volume is V¼10/2¼5 m3. The vessel is filled up to 80%; thus, the reactor volume is 6.25 m3. The

reactor geometry (Figure 11.16) can be found from a relation volume–diameter that takes into account the shape factor

s¼Hc/D (see Chapter 20). The reaction vessel is a cylinder provided with a torispherical bottom, for which V¼(0.785s+0.0809)D3. By selecting s of 2.5 gives a vessel diameter of 1.45 m. The height of the cylindrical part is

Hc¼2.505�1450¼3632 mm, while the torispherical bottom is hb¼0.0809�1450¼118 mm, so that the total height

of the space housing the reaction is H¼Hc+hb¼3750 mm.

The mechanical design of the reactor is described in Section 20.1.3. For stainless steel and a temperature up to 300 �C,the assumed value of the stress is s¼100 N/mm2. The vessel is designed to operate up to 10 bar. The wall thickness is

computed by the relation (20.13) d¼PiD/(2s�Pi) as d¼10�0.1/(2�100�1)¼7.25 mm (1 bar¼0.1 N/mm2). Note that

from Table 16.4, the minimum is 7 mm for cylindrical vessels between 1 and 2 m. We select a value of 10 mm.


Thermal design. The cooling system consists of a jacket provided with a spiral baffle such to form a channel around the

vessel. By this construction, high velocity of the cooling agent and higher heat transfer coefficient are achieved. The chan-

nel has a rectangular cross area of 50�140 mm. The cooling jacket is equivalent to 20 spirals. The heat transfer area is

thus 12.5 m2.

Regarding the selection and pre-design of the mixing system, some basic information is given in Section 20.1.3.

More information can be found in specialised works (Taterson, 1994). Since the fluid is a polymer solution of low visco-

sity, we select a four-blade stirrer with the diameter Ds¼0.5�D¼725 mm, the blade width w¼0.2�725¼145 mm and

the speed 100 rotations/min. The heat transfer coefficient can be determined with the relation (20.12) as

Nu¼ 0:36Re0:66Pr0:33. The physical properties at 400 K are: density r¼800 kg/m3, viscosity �¼1 cP; thermal conduc-

tivity l¼0.5 kcal/(m h K) and cp¼0.35 kcal/(kg K). Finally, the heat transfer coefficient bulk reactor/wall has the value

a1¼2600 kcal/(m2 h K).

On the jacket side, we adopt the configuration with heating/cooling water pumped at constant flow rate in a loop. This

arrangement ensures high heat transfer coefficient during the whole process. Only the temperature of the inlet water will

vary, being adapted to the reactor heat balance by injecting hot or cold water (Figure 11.7). The heat transfer coefficient can

be found from the relationNu¼0.027Re0.8Pr0.33 (20.42). The water flow is set at 10 kg/s, which results in water velocity in

the channel of 1.42 m/s. The equivalent diameter is de¼4A/P¼4 � (50 �140)/2 � (50+140)¼74 mm. The physical constants

for water at 303 K are known. The computation leads to a2¼5600 kcal/(m2 h K).

Finally, the thermal resistance of the wall has to be accounted for. Thermal conductivity of stainless steel is chosen as

l¼15 kcal/(m h K). With d¼10 mm, it gives l/d¼1500 kcal/(m2 h K). This value is set at 1000 kcal/(m2 h K) to include

fouling. The overall heat transfer coefficient isU¼ (1/2600+1/5600+1/1000)¼640 kcal/(m2 h K). This value was used in

simulation.

Model equations. The material and heat balance take the form:

dNA tð Þdt¼�r cA, Tð Þ (11.47)

dT tð Þdt¼�rA cA, Tð Þ �DHr,Að Þ�Qt

m�cp (11.48)

Continued

D

Da

Hc

b

hb

Dm

Hm

TC

LC

Circulation pump

Reactor

Primary controller(reactor temperature)

Secondary controller(jacket temperature)

Cold water

SP

SP

MVPV

PVTC

Hot water

FIGURE 11.16

Geometry of the batch reactor and temperature control.

47711.4 INTEGRATION OF DESIGN AND CONTROL OF BATCH REACTORS

The initial conditions are at time t¼0, NA¼10 kmol and T¼373 K.

Assuming that the coolant is in plug-flow around the vessel and neglecting the dynamics of the cooling coil with respect

to the dynamic of the vessel, allow correlating the inlet/outlet jacket temperatures by the heat transferred, as follows:

Tc,out¼ T�exp � UA

Fccp,c

� �T�Tc, inð Þ (11.49)

Qt¼Fccp,c Tc,out�Tc, inð Þ (11.50)

The cooling effect is obtained by manipulating the temperature of the coolant-inlet temperature, Tc,in, which in turn is

modified by using the controller equations:

e tð Þ¼ T tð Þ�TSP (11.51)

Tc, in tð Þ¼Tc, in,bias�Kc� e tð Þ+ 1

Ti

ðt0

e tð Þdt� �

, with the restriction Tc,min < Tc, in <Tc,max (11.52)

Tc,in is implemented by mixing a water from jacket-outlet with (1�a) cold water with Tc,0¼298 K. Therefore,

a¼ (Tc,in�Tc,0)/(Tc,out�Tc,0).

Scenarios. Three scenarios are presented, as illustrated by Figure 11.17. Firstly, we let run the reaction normally, with

the temperature control on. Perfect control of the reactor temperature is achieved. The cooling water starts at 318 K. The

initial difference Tc,out�Tc,in is about 12 K. This diminishes during time, reflecting the decrease in the heat generated by

reaction.

In a second scenario, the pump breaks down after 10 min from the start. This situation, which lasts for 5 min, is sim-

ulated by letting a very small water flow in, and by dropping U from 640 to 100. The reactor runs quasi-adiabatically. The

temperature begins to increase significantly because of high activation energy. The point of no-return is rapidly reached:

the reaction cannot be slowed down even if the full cooling is restored quickly.

In the third scenario, the pump fault occurs later, after 30 min. Now the reaction can recover, and only a peak of about

10 K is recorded. The explanation is that the concentration level dropped enough such that the heat generated is under the

cooling capacity.

In conclusion, even if the thermal design seemed correct, a very dangerous situation developed because of an operation

incident. By dynamic simulation with process control implementation, the designer can examine the appropriate measures

for ensuring a safe operation, particularly when dealing with high exothermic reactions.

EXAMPLE 11.6 LARGE-SCALE BATCH REACTOR FOR s-PVC MANUFACTURINGThis application illustrates the key features of a batch process in the field of polymer technology (s-PVC means by sus-

pensionmethod), namely reactor design and control, setting-up the polymerisation recipe and operating procedures suitable

for different product specifications. The link between operation and product quality is achieved by means of detailed ki-

netic modelling, including molecular weight distribution (MWD). The reactor design is examined from the viewpoint of

heat transfer capability imbedded in a dynamic model with control system implementation, simulated in Matlab. The re-

sults are expressed as profiles in time of temperature, pressure, reaction rate, concentration of monomer, initiator as well as

by MWD. More details can be found in Dimian and Bildea (2008).

The technology of large-scale batch reactors for s-PVCmanufacturing is one of the most spectacular in chemical equip-

ment. As illustrated by Figure 11.18 (Saeki and Emura, 2002), the reactor size increased from few m3 in 1950 to

130–200 m3 today. The technology of large reactors is largely adopted for the new units because significant saving in cap-

ital, instrumentation and manpower.

Reactor design. The polymerisation of vinyl chloride is moderately exothermic (1.025�108 J/kmol or 24.5 kcal/mol at

50 �C), but for large-scale reactors of more than 100 m3 this feature raises a considerable challenge. The design target is


300

320

340

360

380

400

0 60 120 180

Tem

pera

ture

(K

)

Time (min)

T

Tc,out

Tc,in

280300320340360380400420440460480500520540

0 10 20 30

Tem

pera

ture

(K

)

Time (min)

T

Tc,out

Tc,in

300

320

340

360

380

0 60 120 180

Tem

pera

ture

(K

)

Time (min)

T

Tc,outTc,in

FIGURE 11.17

Runaway of a batch reaction: (left) normal operation; (middle) pump failure after 10 min, non-return; (right) pump failure after 30 min, recovery.

Continued

maximising the reactor productivity for a desired product grade, characterised by MWD, by providing an efficient and safe

removal of heat developed. This is achieved by two methods: cooling jacket and external condenser. The jacket takes over

the major part, about 70–80%. Note that water can be added directly for compensating the volume shrinkage, but limited to

5% from the total duty.

The heat transfer area A of the vessel diminishes drastically at larger volume, since A/V�1/V1/3. For example, scaling

up from 30 to 100 m3 reduces the A/V from 1.5 to 1. The effective heat transfer area depends on the shape factorH/D, in turn

determined by the technology of the mixing device. As a rule of thumb, a reactor of 100 m3 has an external heat transfer

surface of about 100 m2.

The overall heat coefficient reactor/jacket U is built-up from several contributions, as follows:

1

U¼ 1

hi+

1

hif+1

ht+

1

hof+

1

ho(11.53)

The partial heat transfer coefficients correspond to inside film transfer (hi), inside fouling (hif), wall conduction (ht),

outside fouling (hof) and outside jacket (ho). The coefficient ho in the jacket can raise over 6500 W/m2K by a coil-type

construction making possible high velocity of the cooling water running in closed loop. Because of mixing the inside heat

transfer coefficient, hi takes high values, about 3500 W/m2K, but possible drop in time might take place because increased

viscosity of the suspension with conversion. The thermal resistance of the wall is of significance. With an average con-

ductivity of 30 W/m2Kmetal and 10 mmwall thickness (8 mmCS and 2 mm stainless steel), the heat transfer is equivalent

FIGURE 11.18

High capacity s-PVC reactors versus traditional batch reactors (Saeki and Emura, 2002).


to 3000 W/m2K. Inserting the above values leads toU¼1(1/6500+1/3500+1/3000)�1300 W/m2K, an exceptionally high

value for such large reactors. At this stage, we have to mention the effect of fouling by polymer deposit (scale). As an order

of magnitude, a thin uniform polymer layer of 200 mm with 0.20 W/mK, gives a conduction of 1000 W/m2K. Hence, wall

fouling by polymer deposit might control the overall heat transfer. Today, this problem is solved by using anti-fouling

agents, electrical polishing and water jet cleaning.

Kinetics. The polymerisation of vinyl chloride by suspension starts in the droplets of monomer dispersed in water,

finally converted to solid particles. The reaction mechanism is similar with the bulk polymerisation. Since the polymer

is not soluble in its own monomer, a polymer-rich phase appears, shortly after the onset of reaction. Consequently,

a two-phase mixture coexists in each droplet, respectively, the monomer-rich phase and the polymer-rich phase. The po-

lymerisation takes place in both phases, the first diminishing and the second growing, up to a critical conversion Xf, where

the monomer-rich phase disappears. The value of Xf depends on temperature, for example, 73% at 50 �C, but it can be

visualised when pressure begins to fall. Beyond this point, the polymerisation resumes only by the monomer captured

in polymer-rich phase, up to the final conversion, of about 85–90%.

The above physical picture, known as the two-phase model, was proposed by Talamini et al. (1998). With different

modifications, this model has proved to be a good support for describing the behaviour of real industrial systems Hamielec

and co-workers (Xie et al., 1991; Dimian et al., 1995; Kiparissides et al., 1997). The main assumption is that the devel-

opment of polymer species takes place independently in each phase, but with different rates. Because of the ‘gel effect’, the

reaction rate in the polymer-rich phase is sensibly higher than in the monomer-rich phase. If Rm is the specific polymer-

isation rate in the monomer-rich phase and Rc in the polymer-rich phase, one can write:

Rc¼QRm (11.54)

where Q is larger than one. The MWD is affected accordingly.

A typical polymerisation mechanism by free radicals includes the following steps: initiation, propagation, chain trans-

fer, formation of chain defect structure, termination and inhibition. Other reactions have effect on the polymer structure,

namely the formation of short- and long-chain branches. A complete list of reactions at s-PVC polymerisationmay be found

in Kiparissides et al. (1997). On the above physical basis, a complex system of kinetic equations may be written. From

overall reactor modelling, two equations are essential for reactor dynamics.

Monomer consumption rate:

d M½ �dt¼ kp M½ �

X1y¼1

Ry

� ¼ kp M½ � rinkt

� �1=2

(11.55)

Initiator consumption rate

rin¼ 2f kd I½ � with I½ � ¼ I0½ �exp �kdtð Þ1� eXð Þ (11.56)

where e is a volume contraction factor due to difference in density between monomer and polymer e¼ (rp�rm)/rp.Figure 11.19 shows the shape of the curves: conversion versus time and reaction rate versus time. By r1 and r2 are

noted the contributions of monomer and polymer-rich phases, respectively, while r is the overall reaction rate. The strong

acceleration effect due to the gel effect may be noted.

By examining the reaction rate evolution (Figure 11.19), it may be observed that a more uniform path is advantageous

both for productivity and temperature control. This effect can be achieved by employing a cocktail of initiators: faster

initiator rapidly exhausted at lower temperature, and slower but active initiator at the reaction temperature, more specif-

ically 0.075 kmol Perkadox and 0.045 kmol TBPD (tert-butyl peroxyneodecanoate).Dynamic simulation. In addition to the equations for mass balance and polymer chain formations, the energy balance is

considered to account for the temperature variation:

mscp,sdTr

dt¼ �DHð Þ

62:5� �dM

dt

� ��Qt (11.57)

Continued

48111.4 INTEGRATION OF DESIGN AND CONTROL OF BATCH REACTORS

The amount of heat removed by jacket is as follows:

Qt¼Fccp, j Tj, in�Tj,out

� �(11.58)

The inlet jacket temperature Tj,0 is adapted continuously to heat balance by injecting hot water of 80�C for heating and

chilled water of 5 �C for cooling. The method for controlling the temperature of the inlet cooling water has been described

in Example 11.5.

Results. Here, we present the results of polymerisation in a 150 m3 reactor operated at 57 �C with the initiation system

mentioned above. Despite the very large reactor size, measures can be taken to ensure a responsive process dynamics for

short polymerisation time (high productivity). Figure 11.20 describes the evolution of conversion, pressure and reaction

rate. The batch starts at a temperature of 20 �C. In a first period, hot water of 80 �C is used for heating. The reaction tem-

perature of 57 �C is reached after 60 min, when the cooling is switched on by injecting a water makeup of 5 �C in the

baffled jacket. The critical conversion of 75% is obtained after a batch time of 300 min. A total batch time of 500 min

is necessary to obtain the final conversion of 85%. It can be seen that the reaction rate profile is different from the situation

with single initiator, in the sense of a more uniform but high polymerisation rate. In this way, superior productivity and safe

operation can be achieved with this gigantic reactor.

Figure 11.21 depicts the evolution of the reactor, jacket-inlet and jacket-outlet temperatures, as well as flow rate of the

cooling water. It can be observed that the control of the temperature is good, on the set-point. The runaway danger is at the

beginning, when switching from heating to cooling, and at the critical conversion when the polymerisation reaction is at

maximum. Despite reaching the maximum cooling capacity, the temperature stays close to the desired value.

0

0.2

0.4

0.6

0.8

0 100 200 300 400

t (min)

Rea

ctio

n ra

te (

% /

min

)

r = r1 + r2

r = r2

r2

r1

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400

t (min)

T = 50 �C T = 50 �C

Criticalconversion

Con

vers

ion

FIGURE 11.19

Conversion of VCM versus time (right) at 50 �C and reaction rate evolution (left).

0

0.2

0.4

0.6

0.8

0 100 200 300 400 500

r (%

/min

)

t (min)t (min)

r1

r2

r

4

5

6

7

8

9

10

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500

p [a

tm]

X [–

]

Xp

FIGURE 11.20

Evolution of conversion, pressure and polymerisation rate in a 150 m3 s-PVC reactor.


Finally, Figure 11.22 presents the evolution of the averageMWD, as numericalMWn and massMWm. At early time, the

molecular weight is higher due to fast initiation and slow termination. As the reaction goes on, the molecular weight drops

rapidly, but it increases slowly to the end. FinalMWm is about 90,000 and polydispersity ratio around two. In the industrial

practice, very detailed kinetic models have been successfully employed for predicting the quality of the polymer from

detailed polymerisation kinetics (Kiparissides et al., 1997; Wieme et al., 2007).

In conclusion, this case study is of significance for product and process design. Typically, a batch reactor has to deliver

products with a variety of specifications. The reactor size and the heat transfer capabilities are imposed. The optimality of

design arises from a better operation procedure, which lead to shorter reaction time and less energy consumption, but ensure

the required product quality. Radical improvements can be achieved by changing the chemistry (catalyst, initiator, etc.), but

the implementation has to consider the environment of the constraints from an engineering viewpoint. Integrating process

design with process control is necessary.

11.5 OTHER BATCH PROCESSESIn addition to batch distillation and batch reactors, there are several additional operations often encoun-

tered in batch processes. This section briefly describes these operations.

0

20

40

60

80

0 100 200 300 400 500

T (

�C)

t (min)

Treactor

Tc,in

Tc,out

0

20

40

60

80

100

0 100 200 300 400 500

Fc

(kg/

s)

t (min)

Fc,cold

Fc,hot

FIGURE 11.21

Evolution of the heating/cooling system in a 150 m3 s-PVC reactor: left, temperature in reactor, inlet and outlet

jacket; right, the flow rate of water for heating and cooling.

0

5

10

15

20

0 100 200 300 400 500

MW

(10

4 )

t (min)

MWm

MWn

FIGURE 11.22

Evolution of average molecular weights in a 150 m3 s-PVC reactor.

48311.5 OTHER BATCH PROCESSES

11.5.1 LIQUID–LIQUID EXTRACTIONLiquid–liquid extraction (LLX) is a method to separate chemicals based on their relative solubilities in

two different immiscible liquids, usually water and organic solvent. Continuous solvent extraction is

used intensively in petrochemistry, as well as in the production of various industrial chemicals. Batch-

wise multi-stage extraction is commonly used on the small scale in chemical labs, in the synthesis of

fine chemicals and pharmaceutical products. In a typical industrial process, an extraction step is used to

transfer the solutes from the aqueous phase to the organic phase – this step being often followed by a

scrubbing and a stripping stage. The organic solvent is then recycled.

While solvent extraction is often carried out on a small scale using a separating funnel, at larger

scale it is performed using adequate equipment as mixer-settlers, spray columns, pulsed columns, cen-

trifugal contactors, thin layer extraction, etc. More details are provided in Perry (2008) and in specia-

lised monographs (Godfrey and Slater, 1995; Treybal, 2008).

11.5.2 CRYSTALLISATION AND PRECIPITATIONCrystallisation is the process of formation of solid crystals from solution, melt or by deposition directly

from a gas phase. Crystallisation is obtained through a variation of the solubility conditions of the sol-

ute in the solvent, by opposition to precipitation that is due to a chemical reaction. This operation is

frequently encountered in the synthesis of pharmaceutical products.

Typically, crystallisation separates a product from a liquid feed stream by cooling the stream or

adding precipitants that lower the solubility of the desired product so that it forms crystals. Well-

formed crystals are usually pure, since each molecule or ion must fit perfectly into the lattice as it leaves

the solution. Impurities that do not fit in the lattice remain in solution. Hence, one can say that molec-

ular recognition is the principle of purification by crystallisation. However, there are instances when

impurities incorporate into the lattice, thus decreasing the purity of the final product. Also, the solvent

may remain into the lattice forming a solvate. In addition, the solvent may be trapped within the crystal,

phenomenon known as inclusion.

The crystallisation process consists of two major events, nucleation and crystal growth:

• Nucleation is the step where the solute molecules dispersed in the solvent start to gather into clusters

that become stable under current operating conditions. These stable clusters constitute the nuclei.

However, when the clusters are not stable, they dissolve. Therefore, the clusters need to reach a

critical size in order to become stable nuclei. Such critical size is dictated by the operating

conditions (e.g. temperature, super-saturation). At the stage of nucleation, the atoms arrange in a

defined/periodic manner that defines the crystal structure (relative arrangement).

• Crystal growth is the subsequent growth of the nuclei. Nucleation and growth continue to occur

simultaneously while the super-saturation exists. Super-saturation is the driving force of

crystallisation. This can be achieved by different methods, as cooling, anti-solvent addition,

evaporation, pH adjustment and chemical reaction. Depending on conditions, either nucleation or

growth may be predominant, and as a result crystals with different sizes and shapes are obtained.

Once the super-saturation exhausted, the solid–liquid system reaches equilibrium, unless the

operating conditions are modified, such as supersaturating the solution again.

Note that many compounds may crystallise as different crystal forms, a phenomenon called polymor-

phism. Each polymorph is in fact a different thermodynamic solid state. Crystal polymorphs exhibit


different physical properties, such as dissolution rate, shape, and melting point. For this reason, poly-

morphism is of major importance in the manufacture of crystalline products.

The tank crystalliser, at atmospheric pressure, under vacuum or under pressure is typically the main

equipment in batch processes, namely in pharmaceutical industry. More information about crystallisa-

tion is available in Perry’s handbook (2008), and in specialised monographs (Mullin, 2001; Myerson,

2001; Tung et al., 2009; Beckmann, 2013).

11.5.3 FILTRATION, LEACHING AND CENTRIFUGATIONFiltration is the operation used for the separation of solids from fluids by interposing a medium through

which only the fluid can pass. Oversize solids in the fluid are retained, but the separation is not com-

plete since the solids are contaminated with some fluid and the filtrate (liquid that passed through)

contains fine particles, depending on the pore size and filter thickness. Filtration is also used in

bio-processes, especially in water treatment and sewage treatment in which the undesired constituents

are removed by absorption into a biological film grown on, or in the filter medium.

Leaching is a technique for removing a soluble fraction, in form of a solution, from an insoluble but

usually permeable solid phase. There are several variations including solid–liquid extraction, percola-

tion, infusion, washing and decantation-settling, as well as corresponding batch or continuous operat-

ing equipment (Perry, 2008).

A more productive alternative to gravitational filtration is the centrifugation. This method is espe-

cially useful for separating solids which do not filter well, such as fine particles or gelatinous suspen-

sions. Note that centrifugation may be also used to separate two immiscible liquids. Filtration or

centrifugation operations are typically found after the chemical reactor in order to separate the main

products from the by-products or from the solvent. Filtration can be also involved in the pre-treatment

of the raw materials before a reaction or separation step. More details about design and equipment are

provided in Perry (2008) and in specialised monographs (Leung, 1998; Sutherland, 2008; Tien, 2013).

11.5.4 DRYINGDrying is amass transfer process consisting of the removal of water (as usual in bio-processes) or solvents

by evaporation from a solid, semi-solid or liquid. This process is often used after filtration or centrifu-

gation, as a final step before selling or packaging the products. Typically, drying involves a source of heat

and a drying agent to remove the vapour produced by the process. In the most cases, a gas stream (air)

applies the heat by convection and carries away the vapour as humidity. Other possibilities are vacuum

drying, where heat is supplied by conduction or radiation (or microwaves), while the vapour produced is

removed by the vacuum system. An indirect technique is drum drying, where a heated surface is used to

provide the energy, and aspirators draw the vapour outside the chamber. More information about drying

technology is given in Perry (2008) and in Tsotsas and Mujumdar (2014). Note that process simulation

offers simple methods capable to replace the classical psychometric charts and tables.

11.6 SUMMARY AND FURTHER READINGBatch processes are widespread in small-scale and multi-product manufacturing processes, as speci-

ality chemicals and consumer-configured products. Key advantages are flexibility and lower capital

investment. However, the batch processes are penalised by lower productivity and higher material

48511.6 SUMMARY AND FURTHER READING

and energy consumptions. These shortcomings are challenges for designer. The unsteady-state nature

of the process raises the need of a fine understanding of its dynamics. In this respect, the simulationmay

help a lot, namely incorporating control and optimisation features.

Batch distillation allows the separation of a variety of multi-component mixtures in low-cost equip-

ment. Possible arrangements are batch rectification, batch stripping and middle vessel. Unlike a con-

tinuous process, obtaining simultaneously high purity and high recovery is not possible in a single run.

The energy consumption is higher too. There are always off-specs fractions to be recycled. The chal-

lenge for designer is getting a trade-off between purity, recovery, energy consumption and time of

operation.

Batch rectification is by far the most employed method. Variable reflux policy needs typically less

energy than constant reflux. Minimum energy consumption can be calculated analytically with

simplified assumptions, such as an infinite number of stages. Providing sufficient number of stages

is advisable. A helpful method for improving the overall component recovery is recycling the off-specs

fractions. Process control and scheduling plays an essential role in designing effective processes.

Heterogeneous azeotropic distillation and extractive distillation can be applied for separating highly

non-ideal mixtures.

The batch reactor is very productive with respect to only chemical reaction period. In a large num-

ber of situations, analytical solutions are available for estimating the time needed for the reaction and

the distribution of components. Continuous feed of some reactants is used to change the product dis-

tribution in complex reactions. Continuous removal of a product helps the equilibrium-limited reac-

tions. An important application of semi-batch operation is mastering the heat balance when dealing

with highly exothermic reactions. Especially the area to volume ratio plays a key role in designing

of batch reactors.

Safety is an essential problem in designing and operating the batch reactors, namely for fine che-

micals, where the kinetics of reaction and the development of exothermic effects are often ignored. The

simulation of scenarios allows the detection of possible catastrophic events at the conceptual stage.

An important problem not treated in this chapter is the scheduling of batch operations. This subjectis common to planning of operations in any manufacturing industry, aiming to maximise the efficiency

and reduce the costs. Scheduling of tasks is important for the operation system of a computer. The al-

gorithmic basis of scheduling is founded by mathematical methods developed in a systems engineering

area known as operational research. A general treatment for a student public can be found in the book

of Pinedo (2002). Specific introductory material for process designers and optimal design of multi-

product batch plants can be found in the book of Biegler et al. (1997). The state of the art has been

reviewed in a landmark paper by Grossmann and co-workers (Mendez et al., 2006).

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[Computer Aided Chemical Engineering] Integrated Design and Simulation of Chemical Processes Volume 35 || Batch Processes