Mann U. Principles of Chemical Reactor Analysis and Design: New Tools for Industrial Chemical Reactor Operations

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ECONOMIC-BASED OPTIMIZATION

In the preceding chapters, we described how to design chemical reactors—how to

determine the reactor size (or operating time) to obtain a speciﬁed production rate,

and how to determine the production rate attainable on an existing reactor.

However, we have not addressed the following question: Is it desirable to achieve

high reaction extents (and use large reactors) or to utilize smaller reactors and

recycle unconverted reactants? When more than one reactant is used, we also

have to ask what is the most economical proportion of the reactants? The answers

to those questions are not straightforward. They depend on the value of the pro-

ducts, the cost of the reactants, the cost of operating the reactor, as well as

the cost of recovering unconverted reactants. The underlying motivation to the

design and operation of chemical reactors is their economic performance. The

dimensionless species operating curves generated from the reactor design relate

the species production rates to the reactor size (through dimensionless operating

or space time, t). The dimensionless energy balance equation ties the utilities

(heating/cooling) needed for the operation to the extents and t. Together, they

provide the means to conduct an economic-based optimization of reactor oper-

ations. In this chapter, we discuss brieﬂy how to apply these relations to optimize

the design and operations of chemical reactors.

Principles of Chemical Reactor Analysis and Design, Second Edition. By Uzi Mann

441

10.1 ECONOMIC-BASED PERFORMANCE OBJECTIVE FUNCTIONS

Optimization of chemical processes is based on the objective of maximizing the

proﬁt of the entire process. For processes involving chemical reactions, we can

write the objective function as

Profit

rate

($=time)

()

Rate of

value of

products

($=time)

;



Rate of

cost of

reactants

($=time)

;



Rate of

operating

expense

($=time)

;

(10:1:1)

For convenience, the operating expenses are divided into different categories:

Rate of

operating

expense

($=time)

;

Rate of

cost of

utilities

($=time)

;

Rate of

cost of

labor

($=time)

;

Rate

cost of

maintenance

($=time)

;

Rate of

amortization of

capital equipment

($=time)

;

(10:1:2)

The revenue portion of Eq. 10.1.1 depends on the composition of the reactor outlet.

A higher purity product is more valuable, and the economics of the process

depends on whether unconverted reactants and undesirable by-products are separ-

ated from the ﬁnal product. When species are separated, the separation cost should

be incorporated into the analysis of the reactor operation. To maximize the proﬁt of

the process, an engineer can adjust the design and several operating parameters:

†

The feed rate per reactor volume (expressed in terms of dimensionless

operating or space time)

†

The proportion of reactants (expressed in terms of y

’s)

†

The hea tin g (or cooling) rate (by adjusting the temperature of the heating ﬂuid, u

)

The proﬁt objective function (Eq. 10.1.1) is then expressed in terms of these (and

other) operating parameters, and we determine the optimal values of the parameters

by solving the following equations:

q{Profit rate}

¼ 0

q{Profit rate}

(0)

¼ 0

q{Profit rate}

¼ 0 (10:1:3)

For most operations with single chemical reactions, the proﬁt function is expressed

in terms of relatively simple functions of these parameters, and Eq. 10.1.3 can be

solved analytically. For operations with multiple chemical reactions, we have to

determine the optimal parameters numerically.

Next, we derive the gross revenue function of the process, expressed in terms of

the extents of the independent reactions. Let Val

denote the value of species j

442 ECONOMIC-BASED OPTIMIZATION

expressed in $/mol. When all the species in the reactor outlet are separated, using

Eq. 2.7.8, the value of the product stream is

out

Val

¼ (F

tot

)

Val

)

(10:1:4)

The value of the feed stream is

Val

¼ (F

tot

)

Val

(10:1:5)

Hence, when the inlet stream is selected as the reference stream, the gross revenue

of the process is

Rate of

gross

revenue

($=time)

;

¼ (F

tot

)

Val

)

(10:1:6)

In many instances, the separation expense is expressed in terms of cost per mole of

product recovered, SC

(in $/mol of j); hence, the separation expense rate is

Rate of

separation

expense

($=time)

;

¼ (F

tot

)

(10:1:7)

Combining Eqs. 10.1.6 and 10.1.7, the gross income rate of the reactor operation

(without accounting for reactor operation expense) is

Rate of

gross

income

($=time)

;

¼ (F

tot

)

(Val

 SC

)



(SC

)

(10:1:8)

When the species in the reactor efﬂuent stream are not separated (unconverted

reactants are discarded with the product), the gross revenue rate is

Gross

revenue

rate

($=time)

;

¼ (F

tot

)

j¼1

Val

)

out

 ( F

tot

)

j¼1

Val

(10:1:9)

Note that the summation in the ﬁrst term on the right of Eq. 10.1.9 is over the

species in the outlet, but the value of products that are not being sold is zero

10.1 ECONOMIC-BASED PERFORMANCE OBJECTIVE FUNCTIONS 443

(i.e., unconverted reactants that are discarded with the product). Also, note that

polluting species in the product stream may have a negative value (the cost of

removing them to meet environmental speciﬁcations). To obtain the net proﬁt of

the operation, we have to substitute into Eq. 10.1.1 the operating expenses of the

reactor, itemized in Eq. 10.1.2.

The example below illustrates an economic-based optimization procedure.

Example 10.1 Product V is produced by reacting reactant A with reactant B

where the following liquid-phase, elementary chemical reaction take place:

Reaction 1: A þ B ! V

Reaction 2: V þ B ! W

where product W is undesirable. A stream of reactant A (C

¼ 16 mol/L,

r ¼ 800 g/L) and a stream of reactant B (C

¼ 20 mol/L, r ¼ 800 g/L) are

available in the plant, and we want to utilize an available 200-L tubular reactor

(plug-ﬂow). The reactor is operated isothermally at 1908C. When each stream is

fed at the same volumetric ﬂow rate, determine:

a. The total feed rate needed to maximize the yield of product V

b. The total feed rate needed to maximize the proﬁt when the outlet stream of

the reactor is not separated

c. The total feed rate when the outlet stream undergoes separation

Data: At 1908C, k

¼ 0:01 L=mol h

1

¼ 0:02 L=mol h

1

Values of the reactants: A ¼ 1$=mol B ¼ 2$=mol

V alues of the products: V(raw) ¼ 30 $=mol V( pure) ¼36 $=mol W ¼3$=mol

Species separ a tion costs: A ¼ 0:02 B ¼ 0:3V¼ 0:3W¼ 0:03 $=mol

Solution The stoichiometric coefﬁcients of the chemical reactions are

¼1 s

¼ 1 s

¼ 0 D

¼1

¼ 0 s

¼1 s

¼ 1 D

¼1

Since each reaction has a species that does not participate in the other, the two

reactions are independent and there is no dependent reaction. We select the inlet

stream to the reactor as a reference stream and denote its ﬂow rate by v

, where

¼ v

þ v

. The concentration of the reference stream is

tot

)

þ v

¼ w

þ (1  w

(a)

444 ECONOMIC-BASED OPTIMIZATION

where w

¼ v

/(v

þ v

) is the volumetric fraction of stream 1 (w

¼ 0.5). The

molar fractions of the species in the reference stream are

¼ w

¼ (1  w

)

(b)

¼ y

¼ 0, and Z

¼ Z

¼ 0. We write Eq. 7.1.1 for each independent

reaction:

¼ r



(c)

¼ r



(d)

Using Eq. 2.7.8, the local species molar ﬂow rates are

¼ (F

tot

)

( y

 Z

) (e)

¼ (F

tot

)

( y

 Z

)(f)

¼ (F

tot

)

 Z

) (g)

¼ (F

tot

)

(h)

For liquid-phase reactions, we use Eq. 7.1.11 to express the species concen-

trations, and the rates are

¼ k

( y

 Z

)( y

 Z

) (i)

¼ k

 Z

)( y

 Z

)(j)

We deﬁne the characteristic reaction time on the basis of reaction 1; hence,

(k)

Substituting (i), ( j), and (k) into (c) and (d), the design equations reduce to

¼ ( y

 Z

)( y

 Z

) (l)



 Z

)( y

 Z

) (m)

a. We solve (l) and (m) numerically subject to the initial condition that at t ¼ 0,

¼ Z

¼ 0. Once we obtain Z

and Z

as functions of t, we use (e) through

(h) to obtain the species operating curves. Figure E10.1.1 shows the reaction

curves, and Figure E10.1.2. shows the species operating curves. From

the curve of product V we determine that highest F

/(F

tot

)

¼ 0.111,

10.1 ECONOMIC-BASED PERFORMANCE OBJECTIVE FUNCTIONS 445

and it is achieved at about t ¼ 1.9. The feed ﬂow rate that provides the high-

est proﬁt rate is

¼ 18:9L=h

b. When the reactor is operated without a separator, product V is contaminated

with unconverted reactants and product W. The proﬁt rate of the operation is

calculated by combining Eqs. 10.1.5 and 10.1.6:

Profit rate

¼ F

Val

þ F

Val

 (F

Val

þ F

Val

) (n)

where F

and F

are given by (g) and (h), respectively. Figure E10.1.3 shows

the proﬁt rate as a function of the dimensionless space time. The curve indi-

cates that the highest proﬁt rate is 60.2 $/h, and it is achieved at t ¼ 1.4. The

feed ﬂow rate that provides the highest proﬁt rate is 25.7 L/h. Note that cal-

culation involves iteration since the value of v

in not known a priori. Once v

is known, we can plot the proﬁt rate as a function of the feed rate, as shown in

Figure E10.1.4.

Figure E10.1.2 Species operating curves.

Figure E10.1.1 Reaction operating curves.

446 ECONOMIC-BASED OPTIMIZATION

c. When the efﬂuent stream of the reactor is fed to a separator, the proﬁt rate of

the combined operation is calculated by Eq. 10.1.9,

Profit rate

¼ F

(36  0:03) þ F

(3  0:03) þ F

(0:1  0:02)

þ F

(0:2  0:03)  [F

(0:1) þ F

(0:2)] (o)

where F

, F

,andF

are given by (e), (f), (g), and (h), respectively.

Figure E.3 shows the pr oﬁt r a te of this operation as a function of the dimension-

less spa ce time. The curve indica tes that the highest proﬁt rate is 162.8 $/h, and

it is achieved at t ¼ 1.15. The feed ﬂo w rate tha t provides the highes t proﬁt rate

is 31.3 L/h. The plot of the proﬁt ra te as a function of the feed rate is shown in

Figure E10.1.4.

Note that the feed ﬂow rate that provides the maximum proﬁt depends on the

mode of opera tion (with or without separation) and is differ ent fr om the feed

ﬂow ra te that pro vides the highest yield of product V. Also note tha t this example

considered a given proportion of the reactants. The procedure can be repeated

for differ ent r eactant proportions to determine the proportion tha t pr o v ides the

highest pr oﬁt.

Figure E10.1.4 Proﬁt vs. feed ﬂow rate.

Figure E10.1.3 Proﬁt curve.

10.1 ECONOMIC-BASED PERFORMANCE OBJECTIVE FUNCTIONS 447

10.2 BATCH AND SEMIBATCH REACTORS

The designs of batch and semibatch reactors, covered in Chapters 6 and 9, addressed

only the operating time needed to obtain a given conversion (or extent). The size

of the batch was not considered. The latter is determined by the required production

rate of the unit. The operation of batch reactors usually consists of four steps:

1. Preparation and ﬁlling, in duration time of t

2. The reaction time (or the operating time), t

3. Discharging time, t

4. Idle time, t

The size of the batch and the duration of the reaction time are determined on the

basis of economic consideration. Usually, the desired production rate of a product,

, is speciﬁed, and the size of the batch, expressed in terms of (N

tot

)

,is

tot

)

þ t

)

(10:2:1)

where (N

tot

)

¼ C

. The extents of the independent reactions are determined by

the design equations for a given operating time. Below, we discuss the determi-

nation of the optimal operating time.

Let us denote the respective operating costs per unit time of these steps by W

, W

, and W

. The total cost of the operation is therefore

Cost

tot

¼ t

þ t

(10:2:2)

When the species in the reactor discharge are not separated, the gross revenue of a

batch is

Gross

revenue

per batch

($)

;

¼ (N

tot

)

j¼1

Val

)

 (N

tot

)

j¼1

Val

(10:2:3)

Combining Eqs. 10.1.1 and 10.1.2, the net proﬁt rate is

Net

profit

rate

($=time)

;

tot

)

þ t

sold

j¼1

Val

)



j¼1

Val



þ t

(10:2:4)

448 ECONOMIC-BASED OPTIMIZATION

Note that the reaction time is expressed in terms of the dimensionless operating

time, t

¼ t

t, and relates to the extents by the design equations and to (N

tot

)

Eq. 10.2.1. To optimize the operation, we determine the values of the parameters

by solving the following equations:

@ profit rate

¼ 0,

@ profit rate

@( y

)

¼ 0 (10:2:5)

For operations with multiple chemical reactions, the optimal parameters are usually

determined numerically. For most operations with single chemical reactions, the

proﬁt function is expressed in terms of relatively simple functions of these par-

ameters, and the optimal operating time can be determined analytically. This is

illustrated in the example below.

Example 10.2 The liquid-phase, ﬁrst-order reaction

A ! B þ C

is carried out in a batch reactor. The value of reactant A is 0.50 $/mol, the value

of product B is 1 $/mol, and the value of product C is 1.50 $/mol. The sum of

the feed, discharge, and idle times is 30 mins. The operating cost of the reactor is

$0.10/min. Determine:

a. The optimal operating (reaction) time.

b. The optimal extent.

c. The reactor volume if the desired production rate of product B is 5 mol/min.

Data: C

(0) ¼ 2 mol=L k ¼ 0:1 min

1

Solution The stoichiometric coefﬁcients are

¼1 s

¼ 1 s

¼ 1

We select the initial state as the reference state; hence, C

¼ C

(0) ¼ 2 mol/L,

and y

¼ 1 and y

¼ y

¼ 0. For a ﬁrst-order reaction, the characteristic reac-

tion time is

¼ 10 min (a)

Using Eq. 2.7.6,

¼ (N

tot

)

(1  Z) (b)

¼ N

¼ (N

tot

)

Z (c)

The dimensionless design equation, in this case, is

¼ 1  Z (d)

10.2 BATCH AND SEMIBATCH REACTORS 449