Chapter 3

Ideal Reactors

LEARNING OBJECTIVES

After completing this chapter, you should be able to

1. explain the differences between the three ideal reactors: batch, continuous stirredtank, and plug flow;

2. explain how the reactant and product concentrations vary spatially in ideal batch,ideal continuous stirred tank, and ideal plug-flow reactors;

3. derive ‘‘design equations’’ for the three ideal reactors, for both homogeneousand heterogeneous catalytic reactions, by performing component material balances;

4. calculate reaction rates using the ‘‘design equation’’ for an ideal continuousstirred-tank reactor;

5. simplify the most general forms of the ‘‘design equations’’ for the case of constantmass density.

The next few chapters will illustrate how the behavior of chemical reactors can bepredicted, and how the size of reactor required for a given ‘‘job,’’ can be determined.These calculations will make use of the principles of reaction stoichiometry and reactionkinetics that were developed in Chapters 1 and 2.

There are many different types of reactor. One of the most important features thatdifferentiates one kind of reactor from another is the nature of mixing in the reactor. Theinfluence of mixing is easiest to understand through the material balance(s) on the reactor.These material balances are the starting point for the discussion of reactor performance.

3.1 GENERALIZED MATERIAL BALANCE

The reaction rate ri is an intensive variable. It describes the rate of formation of species‘‘i’’ at any point in a chemical reactor. However, as we learned in Chapter 2, the rate ofany reaction depends on variables such as temperature and the species concentrations. Ifthese variables change from point to point in the reactor, ri also will change from pointto point.

For the time being, to emphasize that ri depends on temperature and on the variousspecies concentrations, let’s use the nomenclature

ri ! ri"T ; all Ci#

The term ‘‘all Ci’’ reminds us that the reaction rate may be influenced by the concentration ofeach and every species in the system.

Consider an arbitrary volume (V) in which the temperature and the species concen-trations vary from point to point, as shown below.

36

T1, all Ci,1(Point 1)

T2, all Ci,2(Point 2)

T3, all Ci,3(Point 3)

The rate at which ‘‘i’’ is formed in this control volume by a chemical reaction or reactions isdesignated Gi, the generation rate of ‘‘i’’. The units of Gi are moles/time. For a homogeneousreaction, Gi is related to ri by

Generation rate$$homogeneous reaction

Gi !Z Z Z

V

ri dV (3-1)

For a heterogeneous catalytic reaction, where ri has units of moles/time-weight of catalyst,Gi is given by

Generation rate$$heterogenous catalyticreaction

Gi !Z Z Z

W

ri dW !Z Z Z

V

rirB dV (3-2)

Here, rB is the bulk density of the catalyst (weight/volume of reactor). In Eqns. (3-1) and(3-2), ri is the net rate at which ‘‘i’’ is formed by all of the reactions taking place, as given byEqn. (1-17).

Although they are formally correct, Eqns. (3-1) and (3-2) are not very useful in practice.This is because the reaction rate ri is never known as an explicit function of position.Therefore, the indicated integrations cannot be performed directly. The means of resolvingthis apparent dilemma will become evident as we treat some specific cases.

Generalized component material balanceConsider the control volume shown below, with chemical reactions taking place that resultin the formation of species ‘‘i’’ at a rate, Gi. Species ‘‘i’’ flows into the system at a molar flowrate of Fi0 (moles i/time), and flows out of the system at a molar flow rate of Fi.

Fi0

Fi

T1, all Ci,1(Point 1)

T2, all Ci,2(Point 2)

T3, all Ci,3(Point 3)

3.1 Generalized Material Balance 37

The molar material balance for species ‘‘i’’ for this control volume is

rate in$ rate out% rate of generation by chemical reactions ! rate of accumulation

Generalized material balanceon component ‘‘i’’

Fi0 $ Fi % Gi !dNi

dt(3-3)

Here ‘‘t’’ is the time and Ni is the number of moles of ‘‘i’’ in the system at any time.Now let’s consider three special cases that are of practical significance, and allow Eqn.

(3-3) to be simplified to a point that it is useful.

3.2 IDEAL BATCH REACTOR

A batch reactor is defined as a reactor in which there is no flow of mass across the systemboundaries, once the reactants have been charged. The reaction is assumed to begin at someprecise point in time, usually taken as t ! 0. This time may correspond, for example, towhen a catalyst or initiator is added to the batch, or to when the last reactant is added.

As the reaction proceeds, the number of moles of each reactant decreases and thenumber of moles of each product increases. Therefore, the concentrations of the species inthe reactor will change with time. The temperature of the reactor contents may also changewith time. The reaction continues until it reaches chemical equilibrium, or until the limitingreactant is consumed completely, or until some action is taken to stop the reaction, e.g.,cooling, removing the catalyst, adding a chemical inhibitor, etc.

Figure 3-1a Overall view of anominal 7000 gallon batch reactor(in a plant of Syngenta CropProtection, Inc.). This reactor is usedto produce several different products.The reactor has a jacket around it topermit heat to be transferred into orout of the reactor contents via aheating or cooling fluid thatcirculates through the jacket. Thehoists are used for lifting rawmaterials to higher levels of thestructure. (Photo used withpermission of Syngenta CropProtection, Inc.)

38 Chapter 3 Ideal Reactors

Batch reactors are used extensively throughout the chemical and pharmaceuticalindustries to manufacture products on a relatively small scale. Properly equipped, thesereactors are very flexible. A single reactor may be used to produce many different products.

Batch reactors usually are mechanically agitated to ensure that the contents are wellmixed. Agitation also increases the heat-transfer coefficient between the reactor contentsand any heat-transfer surface in the reactor. In multiphase reactors, agitation may also keep asolid catalyst suspended, or may create surface area between two liquid phases or between agas phase and a liquid phase.

Very few reactions are thermally neutral "DHR ! 0#, so it frequently is necessary toeither supply heat or remove heat as the reaction proceeds. The most common means totransfer heat is to circulate a hot or cold fluid, either through a coil that is immersed in thereactor, or through a jacket that is attached to the wall of the reactor, or both.

For a batch reactor, Fi0 ! Fi ! 0. Therefore, for a homogeneous reaction, Eqn. (3-3)becomes

Gi !Z Z Z

V

ri dV ! dNi

dt(3-4)

Ideal batch reactorNow, consider a limiting case of batch-reactor behavior. Suppose that agitation of the reactorcontents is vigorous, i.e., mixing of fluid elements in the reactor is very intense. Then thetemperature and the species concentrations will be the same at every point in the reactor, atevery point in time. A batch reactor that satisfies this condition is called an ideal batchreactor. Many laboratory and commercial reactors can be treated as ideal batch reactors, atleast as a first approximation.

For an ideal batch reactor, ri is not a function of position. Therefore,RRR

Vri dV ! riV

and Eqn. (3-4) becomes

riV !dNi

dt

Figure 3-1b The top of the reactor in Figure 3-1a. The view port in the left front of the picturepermits the contents of the reactor to be observed, and can be opened to permit solids to be charged tothe reactor. A motor that drives an agitator is located in the top center of the picture, and a charging lineand valve actuator connected to a valve are on the left. (Photo used with permission of Syngenta CropProtection, Inc.)

3.2 Ideal Batch Reactor 39

Rearranging

Design equation$$ideal batch reactor$$homogeneous reaction"moles#

1

V

dNi

dt! ri (3-5)

Equation (3-5) is referred to as the design equation for an ideal batch reactor, indifferential form. This equation is valid no matter how many reactions are taking place,provided that Eqn. (1-17) is used to express ri, and provided that all of the reactions arehomogeneous.

The subject of multiple reactions is treated in Chapter 7. Until then, we will be concernedwith the behavior of one, stoichiometrically simple reaction. For that case, ri in Eqn. (3-5) isjust the rate equation for the formation of species ‘‘i’’ in the reaction of concern.

The variable that describes composition in Eqn. (3-5) is Ni, the total moles of species‘‘i’’. It sometimes is more convenient to work problems in terms of either the extent ofreaction j or the fractional conversion of a reactant, usually the limiting reactant. Extent ofreaction is very convenient for problems where more than one reaction takes place.Fractional conversion is convenient for single-reaction problems, but can be a source ofconfusion in problems that involve multiple reactions. The use of all three compositionalvariables, moles (or molar flow rates), fractional conversion, and extent of reaction, will beillustrated in this chapter, and in Chapter 4.

If ‘‘i’’ is a reactant, say A, then the number of moles of A in the reactor at any time can bewritten in terms of the fractional conversion of A.

xA !NA0 $ NA

NA0; NA ! NA0"1$ xA#

In terms of fractional conversion, Eqn. (3-5) is

Design equation$$ideal batch reactor$$homogeneous reaction"fractional conversion#

NA0

V

dxA

dt! $rA (3-6)

If ‘‘A’’ is the limiting reactant, the value of xA that is stoichiometrically attainable will liebetween 0 and 1. However, as discussed in Chapter 1, chemical equilibrium may limit thevalue of xA that can actually be achieved to something less than 1.

Equation (3-6) should not be applied to a product. First, NA will be greater than NA0 if‘‘A’’ is a product. Moreover, if NA0 ! 0; xA is infinite. However, Eqn. (3-5) can be used foreither a product or a reactant.

The design equation can also be written in terms of the extent of reaction. If only onestoichiometrically simple reaction is taking place

j ! DNi

ni! Ni $ Ni0

ni(1-4)

Design equation$$ideal batch reactor$$homogeneous reaction"extent of reaction#

ni

V

dj

dt! ri (3-7)

40 Chapter 3 Ideal Reactors

Equations (3-5)–(3-7) are alternative forms of the design equation for an ideal batchreactor with a homogeneous reaction taking place. Despite the somewhat pretentious name,design equations are nothing more than component material balances, i.e., molar balanceson ‘‘i’’, ‘‘A’’, etc.

The volume V in Eqns. (3-5)–(3-7) is that portion of the overall reactor volume in whichthe reaction actually takes place. This is not necessarily the whole geometrical volume ofthe reactor. For example, consider a reaction that takes place in a liquid that partially fills avessel. If no reaction takes place in the gas-filled ‘‘headspace’’ above the liquid, then V is thevolume of liquid, not the geometrical volume of the vessel, which includes the ‘‘headspace.’’

Equations (3-5)–(3-7) apply to a homogeneous reaction. For a reaction that is catalyzedby a solid, the design equation that is equivalent to Eqn. (3-5) is

Design equation$$ideal batch reactor$$heterogeneous catalytic reaction"moles#

1

W

dNi

dt! ri (3-5a)

EXERCISE 3-1

Derive this equation.

The equivalents to Eqns. (3-6) and (3-7) for heterogeneously catalyzed reactions are given inAppendix 3 at the end of this chapter, and are labeled Eqns. (3-6a) and (3-7a). Be sure thatyou can derive them.

Temperature variation with timeIn developing Eqns. (3-5)–(3-7), we did not assume that the temperature of the reactor contentswas constant, independent of time. Only one assumption was made concerning temperature,i.e., there are no spatial variations in the temperature at any time. An ideal batch reactor is saidto be isothermal when the temperature does not vary with time. The design equations for anideal batch reactor are valid for both isothermal and nonisothermal operation.

Constant volumeIf V is constant, independent of time, Eqn. (3-5) can be written in terms of concentration as

dCi

dt! ri (3-8)

where Ci is the concentration of species i. Similarly, if V is constant, Eqn. (3-6) can bewritten as

CA0dxA

dt! $rA (3-9)

where CA0 is the initial concentration of A. Equations (3-8) and (3-9) are alternative forms ofthe design equations for an ideal, constant-volume batch reactor, in differential form. Thelighter boxes around these equations indicate that they are not as general as Eqns. (3-5) and(3-6) because they contain the assumption of constant volume.

If the volume V is constant, then the mass density of the system, r "mass/volume#, mustalso be constant, since the mass of material in a batch reactor does not change with time. Wecould have specified that the mass density was constant instead of specifying that the reactorvolume was constant. These two statements are equivalent. However, for a batch reactor,

3.2 Ideal Batch Reactor 41

constant volume probably is easier to visualize than constant mass density. For constant-volume (constant mass density) systems, the design equations can be written directly interms of concentrations, which can easily be measured. For systems where the mass densityis not constant, we must work with the most general forms of the design equations, usingmoles, fractional conversion, or extent of reaction.

For a heterogeneous catalytic reaction, derivation of the constant-volume version ofEqn. (3-5a) requires a bit of manipulation.

1

W

dNi

dt! ri (3-5a)

Dividing by the reactor volume V and multiplying by W

1

V

dNi

dt! W

V

! "ri

If V is constant,

Design equation$$ideal batch reactor$$heterogeneous catalytic reaction"constant volume#

dCi

dt! Ccatri (3-8a)

Equation (3-8a) is the design equation for an ideal, constant-volume, batch reactor for areaction that is catalyzed by a solid catalyst. The symbol Ccat represents the massconcentration (mass/volume) of the catalyst. The catalyst concentration does not changewith time if V is constant.

The equivalent of Eqn. (3-9) for a heterogeneous catalytic reaction is given in Appendix3.I at the end of this chapter and is labeled Eqn. (3-9a).

The assumption of constant volume is valid for most industrial batch reactors. The massdensity is approximately constant for a large majority of liquids, even if the temperaturechanges moderately as the reaction proceeds. Therefore, the assumption of constant volumeis reasonable for batch reactions that take place in the liquid phase. Moreover, if a rigidvessel is filled with gas, the gas volume will be constant because the dimensions of the vesselare fixed and do not vary with time.

Variable volumeIf V changes with time, Eqn. (3-5) must be written

1

V

dNi

dt! 1

V

d"Ci V#dt

! dCi

dt% Ci

V

dV

dt! ri

Clearly, this equation is more complex, and harder to work with, than Eqn. (3-8).

EXERCISE 3-2

There are a few batch reactors where the assumption of constantvolume is not appropriate. Can you think of one? Hint: You

probably come within 10 ft of this reactor at least once a week,perhaps even every day.

Integrated forms of the design equationThe design equation must be integrated in order to solve problems in reactor design andanalysis. In order to actually perform the integration, the temperature must be known as afunction of either time or composition. This is because the rate equation ri contains one ormore constants that depend on temperature.

42 Chapter 3 Ideal Reactors

As we shall see in Chapter 8, the energy balance determines how the reactor temper-ature changes as the reaction proceeds. Broadly, there are three possibilities:

1. The energy balance is so complex that the design equation and the energy balance must besolved simultaneously. We shall leave this case for Chapter 8.

2. The reactor can be heated or cooled such that the temperature changes, but is known as afunction of time. An example of this case is treated in Chapter 4.

3. The reactor is adiabatic, or is heated or cooled so that it is isothermal. If the reactor isisothermal, the parameters in the rate equation are constant, i.e., they do not depend oneither time or composition. In the adiabatic case, the temperature can be expressed as afunction of composition. Therefore, the parameters in the rate equation can also bewritten as functions of composition. This will be illustrated in Chapter 8.

For the third case, i.e., an isothermal or adiabatic reactor, ri depends only on concen-tration. If V is constant, or can be expressed as a function of concentration, Eqn. (3-5) can besymbolically integrated from t ! 0; Ni ! Ni0 to t ! t; Ni ! Ni. The result is

ZNi

Ni0

1

V

dNi

ri!Zt

0

dt ! t (3-10)

When the reactor temperature varies with time in a known manner, then ri depends ontime as well as concentration. In such a case, Eqn. (3-5) must be used as a starting pointinstead of Eqn. (3-10). This will be illustrated in the next chapter.

The integrated forms of Eqns. (3-5) through (3-9) for Case 3 above are given inAppendix 3.I, and are labeled as Eqns. (3-10) through (3-14), respectively. Appendix 3.I alsocontains the integrated forms of Eqns. (3-5a) through (3-9a) for Case 3.

Once the integrations of the design equations have been performed, the time requiredto reach a concentration CA, or a fractional conversion xA, or an extent of reaction j can becalculated. Conversely, the value of CA; xA, or j that results for a specified reaction time canalso be calculated. Chapter 4 illustrates the solution of some batch-reactor problems where thereactor is isothermal, or where the temperature is known as a function of time. The simultaneoussolution of the design equation and the energy balance is considered in Chapter 8.

3.3 CONTINUOUS REACTORS

When the demand for a single chemical product reaches a high level, in the region of tens ofmillion pounds per year, there generally will be an economic incentive to manufacture theproduct continuously, using a reactor that is dedicated to that product. The reactor mayoperate at steady state for a year or more, with planned shutdowns only for regularmaintenance, catalyst changes, etc.

Almost all of the reactors in a petroleum refinery operate continuously because of thetremendous annual production rates of the various fuels, lubricants, and chemical inter-mediates that are manufactured in a refinery. Many well-known polymers such as poly-ethylene and polystyrene are also produced in continuous reactors, as are many large-volume chemicals such as styrene, ethylene, ammonia, and methanol.

Figure 3-3 is a simplified flowsheet showing some of the auxiliary equipment that may beassociated with a continuous reactor. In this example, the feed stream is heated to the desiredinlet temperature, first in a feed/product heat exchanger and then in a fired heater. The streamleaving the reactor contains the product(s), the unconverted reactants, and any inert components.

3.3 Continuous Reactors 43

This stream is cooled in the feed/product heat exchanger and then is cooled further to condensesome of its components. The gas and liquid phases are separated. The liquid phase is sent to aseparations section (fractionation unit), where the product is recovered. A purge is taken fromthe gas that leaves the separator, in part to prevent buildup of impurities in the recycle loop. Theremainder of the gas is recycled.

Most of the unconverted reactants from a continuous reactor will be recycled back intothe feed stream, unless the fractional conversion of the reactants is very high. Some of theproduct and/or inert components also may be recycled, to aid in control of the reactortemperature, for example.

Continuous reactors normally operate at steady state. The flowrate and composition ofthe feed stream do not vary with time, and the reactor operating conditions do not vary withtime. We will assume steady state in developing the design equations for the two ideal

Figure 3-2 A continuous reactor,with associated equipment, for thecatalytic isomerization of heavynormal paraffins, containing about 35carbon atoms, to branched paraffins.The catalyst is comprised of platinumon an acidic zeolite that has relativelylarge pores. The reaction produceslubricants that have a high viscosity athigh temperatures, but retain thecharacteristics of a liquid at lowtemperatures. Without theisomerization reaction, the lubricantwould become ‘‘waxy’’ and would notflow at low temperatures. This unit islocated at the ExxonMobil refinery inFawley, UK. (Photo, ExxonMobil2003 Summary Annual Report.)

44 Chapter 3 Ideal Reactors

continuous reactors, the ideal continuous stirred-tank reactor (CSTR), and the ideal plug-flow reactor (PFR).

3.3.1 Ideal Continuous Stirred-Tank Reactor (CSTR)

Like the ideal batch reactor, the ideal CSTR is characterized by intense mixing. Thetemperature and the various concentrations are the same at every point in the reactor. Thefeed stream entering the reactor is mixed instantaneously into the contents of the reactor,immediately destroying the identity of the feed. Since the composition and the temperatureare the same everywhere in the CSTR, it follows that the effluent stream must have exactlythe same composition and temperature as the contents of the reactor.

Feed

Composition andtemperature are thesame everywhere in

the reactor

Composition andtemperature are the

same in the effluent asin the reactor

Effluent

Charge heater

Reactor

Combinedfeed-reactoreffluent exchanger

Effluent condenser

Separator liquidto fractionation

SeparatorNet

separatorgas

Recyclecompressor

Charge

Figure 3-3 A typical flow scheme forthe reactor section of a continuous plant.

1

Several items of heat exchangeequipment, a recycle compressor, and aphase separator are required to supportthe steady-state operation of the reactor.(Figure Copyright 2004 UOP LLC. Allrights reserved. Used with permission.)

1 Stine, M. A., Petroleum Refining, presented at the North Carolina State University AIChE Student ChapterMeeting, November 15, 2002.

3.3 Continuous Reactors 45

On a small scale, e.g., a laboratory reactor, mechanical agitation is usually required toachieve the necessary high intensity of mixing. On a commercial scale, the required mixingsometimes can be obtained by introducing the feed stream into the reactor at a high velocity,such that the resulting turbulence produces intense mixing. A bed of catalyst powder that isfluidized by an incoming gas or liquid stream, i.e., a fluidized-bed reactor, might be treatedas a CSTR, at least as a first approximation. Another reactor configuration that canapproximate a CSTR is a slurry bubble column reactor, in which a gas feed stream issparged through a suspension of catalyst powder in a liquid. Slurry bubble column reactorsare used in some versions of the Fischer–Tropsch process for converting synthesis gas, amixture of H2 and CO, into liquid fuels.

The continuous stirred-tank reactor is also known as a continuous backmix, backmixed,or mixed flow reactor. In addition to the catalytic reactors mentioned in the precedingparagraph, the reactors that are used for certain continuous polymerizations, e.g., thepolymerization of styrene monomer to polystyrene, closely approximate CSTRs.

Because of the intense mixing in a CSTR, temperature and concentration are the sameat every point in the reactor. Therefore, as with the ideal batch reactor, ri does not depend onposition. For a homogeneous reaction, Eqns. (3-1) and (3-3) simplify to

Fi0 $ Fi % riV !dNi

dt(3-15)

Equation (3-15) describes the unsteady-state behavior of a CSTR. This is the equation thatmust be solved to explore strategies for starting up the reactor, or shutting it down, orswitching from one set of operating conditions to another.

At steady state, the concentrations and the temperature of the CSTR do not vary withtime. The exact temperature of operation is determined by the energy balance, as we shallsee in Chapter 8. At steady state, the right-hand side of Eqn. (3-15) is zero.

Fi0 $ Fi % riV ! 0

Design equation$$ideal CSTR$$homogeneous reaction"molar flow rates#

V ! Fi0 $ Fi

$ri(3-16)

Equation (3-16) is the design equation for an ideal CSTR. It can be applied to a reactor wheremore than one reaction is taking place, if Eqn. (1-17) is used to express ri.

For a single reaction, it frequently is convenient to write Eqn. (3-16) in terms of eitherextent of reaction or fractional conversion of a reactant. If ‘‘i’’ is a reactant, say A,

FA ! FA0"1$ xA#

and Eqn. (3-16) becomes

Design equation$$ideal CSTR$$homogeneous reaction"molar flow rates#

V

FA0! xA

$rA(3-17)

Alternatively, if only one, stoichiometrically simple reaction is taking place,

j ! Fi $ Fi0

ni

46 Chapter 3 Ideal Reactors

and Eqn. (3-16) becomes

Design equation$$ideal CSTR$$homogeneous reaction"extent of reaction#

V ! nij

ri(3-18)

Equations (3-16)–(3-18) are equivalent forms of the design equation for an ideal CSTR.In these equations,$rA (or ri) is always evaluated at the exit conditions of the reactor, i.e., atthe temperature and concentrations that exist in the effluent stream, and therefore in thewhole reactor volume. Once again, the design equation is simply a molar componentmaterial balance.

For a heterogeneous catalytic reaction, the equivalent form of Eqn. (3-16) is

Design equation$$ideal CSTR$$heterogeneous catalytic reaction"molar flowrates#

W ! Fi0 $ Fi

$ri(3-16a)

EXERCISE 3-3Derive this equation.

Appendix 3.II gives the forms of the design equation for a heterogeneous catalyticreaction that are equivalent to Eqns. (3-17) and (3-18). These equations are labeled Eqns.(3-17a) and (3-18a).

Space time and space velocityThe molar feed rate, FA0, is the product of the inlet concentration CA0 and the inletvolumetric flow rate y0, i.e.,

FA0 ! y0CA0 (3-19)

For a homogeneous reaction, the space time at inlet conditions t0 is defined as

t0&V=y0 (3-20)

This definition of space time applies to any continuous reactor, whether it is a CSTR or not.For a homogeneous reaction, space time has the dimension of time. It is related to the

average time that the fluid spends in the reactor, although it is not necessarily exactly equalto the average time. However, the space time and the average residence time behave in asimilar manner. If the reactor volume V increases and the volumetric flow rate y0 staysconstant, both the space time and the average residence time increase. Conversely, if thevolumetric flow rate y0 increases and the reactor volume stays constant, both the space timeand the average residence time decrease.

Space time influences reaction behavior in a continuous reactor in the same way thatreal time influences reaction behavior in a batch reactor. In a batch reactor, if the time that thereactants spend in the reactor increases, the fractional conversion and the extent of reactionwill increase, and the concentrations of the reactants will decrease. The same is true forspace time and a continuous reactor. If a continuous reactor is at steady state, the conversionand the extent of reaction will increase, and the reactant concentrations will decrease, whenthe space time is increased.

3.3 Continuous Reactors 47

Using Eqns. (3-19) and (3-20), Eqn. (3-17) can be written as

Design equation$$ideal CSTR$$homogeneous reaction"in terms of space time#

t0 !CA0xA

$rA(3-21)

The concept of space time is also applicable to heterogeneously catalyzed reactions. Inthis case, t0 is defined by

t0&W=y0 (3-22)

Here, the units of t0 are (wt. catalyst-time/volume of fluid). With this definition, Eqn. (3-21)applies to both homogeneous reactions and reactions catalyzed by solids.

The inverse of space time is known as space velocity. Space velocity is designated invarious ways, e.g., SV, GHSV (gas hourly space velocity), and WHSV (weight hourly spacevelocity). ‘‘Space velocity’’ is commonly used in the field of heterogeneous catalysis, andthere can be considerable ambiguity in the definitions that appear in the literature. Forexample, GHSV may be defined as the volumetric flowrate of gas entering the catalystdivided by the weight of catalyst. In this case, the units of space velocity are (volume of fluid/time-wt. catalyst). The volumetric flowrate may correspond to inlet conditions or to STP.However, it is not uncommon to find space velocity defined as the volumetric flowrate of gasdivided by the volume of catalyst bed, or by the volume of catalyst particles. With either ofthese definitions, the units of space time are inverse time, even though the reaction iscatalytic.

When the term ‘‘space velocity’’ is encountered in the literature, it is important topay very careful attention to how this parameter is defined! Analysis of the units mayhelp.

This book will emphasize the use of space time, since it is analogous to real time in abatch reactor. Space velocity can be a bit counterintuitive. Conversion increases as spacetime increases, but conversion decreases as space velocity increases.

Constant fluid densityIf the mass density (mass/volume) of the fluid flowing through the reactor is constant, i.e., ifit is the same in the feed, in the effluent, and at every point in the reactor, then the subscript‘‘0’’ can be dropped from both t and y. In this case Eqn. (3-21) can be written as

t ! CA0xA

$rA(3-23)

When the fluid density is constant, then t"!V=y# is the average residence time that thefluid spends in the reactor. This is true for the CSTR, and for any other continuous reactoroperating at steady state.

If (and only if ) the fluid density is constant,

xA&FA0 $ FA

FA0! yCA0 $ yCA

yCA0! CA0 $ CA

CA0

48 Chapter 3 Ideal Reactors

so that Eqn. (3-23) becomes

t ! CA0 $ CA

$rA(3-24)

Equations (3-23) and (3-24) are design equations for an ideal CSTR with a constant-density fluid. The lighter box around these equations indicates that they are not asgeneral as Eqn. (3-21), which is not restricted to a constant-density fluid. Equations(3-23) and (3-24) apply to both homogeneous and heterogeneously catalyzed reactions,provided that t is calculated from the appropriate equation, either Eqn. (3-20) or Eqn.(3-22).

Calculating the reaction rateThe various forms of the design equation for an ideal CSTR (Eqns. (3-16) through (3-18),(3-21), (3-23), and (3-24)) can be used to calculate a numerical value of the rate of reaction,if all of the other parameters in the equation are known. The following example illustratesthis use of the CSTR design equation.

EXAMPLE 3-1Calculation of Rateof Disappearanceof Thiophene

The catalytic hydrogenolysis of thiophene was carried out in a reactor that behaved as an ideal CSTR.The reactor contained 8.16 g of ‘‘cobalt molybdate’’ catalyst. In one experiment, the feed rate ofthiophene to the reactor was 6:53' 10$5 mol/min. The fractional conversion of thiophene in thereactor effluent was measured and found to be 0.71. Calculate the value of the rate of disappearanceof thiophene for this experiment.

APPROACH Equation (3-16a) is the most fundamental form of the design equation for a heterogeneous catalyticreaction in an ideal CSTR.

W ! Fi0 $ Fi

$ri(3-16a)

Using the subscript ‘‘T’’ for thiophene and rearranging,

$rT !FT0 $ FT

W

From the definition of fractional conversion, FT0 $ FT ! FT0xT. Therefore, all of the param-eters on the right-hand side of the above equation are known and $rT can be calculated.

SOLUTION

$rT !FT0xT

W! 6:53' 10$5"mol/min# ' 0:71

8:16"g# ! 0:57' 10$5"mol/min-g#

3.3.2 Ideal Continuous Plug-Flow Reactor (PFR)

The plug-flow reactor is the third and last of the so-called ‘‘ideal’’ reactors. It is frequentlyrepresented as a tubular reactor, as shown below.

3.3 Continuous Reactors 49

z

Little “plugs” of fluid flow single filethrough reactor: • no mixing in direction of flow, i.e.,one fluid element cannot pass or mixwith another; • no temperature or concentrationvariations normal to flow.

Fi0

Fi

The ideal plug-flow reactor has two defining characteristics:

1. There is no mixing in the direction of flow. Therefore, the concentrations of thereactants decrease in the direction of flow, from the reactor inlet to the reactor outlet. Inaddition, the temperature may vary in the direction of flow, depending on the magnitude ofthe heat of reaction, and on what, if any, heat transfer takes place through the walls of thereactor. Because of the variation of concentration, and possibly temperature, the reactionrate, ri, varies in the direction of flow;

2. There is no variation of temperature or concentration normal to the direction of flow.For a tubular reactor, this means that there is no radial or angular variation of temperature orof any species concentration at a given axial position z. As a consequence, the reaction rate ri

does not vary normal to the direction of flow, at any cross section in the direction of flow.

The plug-flow reactor may be thought of as a series of miniature batch reactors that flowthrough the reactor in single file. Each miniature batch reactor maintains its integrity as itflows from the reactor inlet to the outlet. There is no exchange of mass or energy betweenadjacent ‘‘plugs’’ of fluid.

In order for a real reactor to approximate this ideal condition, the fluid velocity cannotvary normal to the direction of flow. For a tubular reactor, this requires a flat velocity profilein the radial and angular directions, as illustrated below.

50 Chapter 3 Ideal Reactors

For flow through a tube, this flat velocity profile is approached when the flow is highlyturbulent, i.e., at high Reynolds numbers.

Let’s analyze the behavior of an ideal plug-flow reactor. We might be tempted to choosethe whole reactor as a control volume as we did with the ideal batch reactor and the idealCSTR and apply Eqn. (3-3),

Fi0 $ Fi % Gi !dNi

dt(3-3)

Assuming a homogeneous reaction, setting the right-hand side equal to 0 to reflect steadystate, and substituting Eqn. (3-1),

Fi0 $ Fi %Z Z Z

V

ri dV ! 0 (3-25)

For a PFR, the reaction rate varies with position in the direction of flow. Therefore, ri is afunction of V and the above integral cannot be evaluated directly.

We can solve this problem in two ways, the easy way and the hard way.

3.3.2.1 The Easy Way—Choose a Different Control Volume

Let’s choose a different control volume over which to write the component material balance.More specifically, let’s choose the control volume such that ri does not depend on V.

From the above discussion, it should be clear that the new control volume must bedifferential in the direction of flow, since ri varies in this direction. However, the control volumecan span the whole cross section of the reactor, normal to flow, since there are no temperature orconcentration gradients normal to flow. Therefore, ri will be constant over any such cross section.

For a tubular reactor, the control volume is a slice through the reactor perpendicular tothe axis (z direction), with a differential thickness dz, as shown below.

dz

Fi + dFi

Fi

z = z

The inlet face of the control volume is located at an axial position, z. The molar flow rateof ‘‘i’’ into the element is Fi and the molar flow rate of ‘‘i’’ leaving the element is Fi % dFi.For this element, the steady-state material balance on ‘‘i’’ is

Fi $ "Fi % dFi# % ri dV ! 0

Design equation$$ideal PFR$$homogeneous reaction"molar flow rate#

dV ! dFi

ri(3-26)

3.3 Continuous Reactors 51

Equation (3-26) is the design equation for an ideal PFR, in differential form. This equationapplies to a PFR where more than one reaction is taking place, provided that ri is expressedusing Eqn. (1-17).

For a single reaction, it may be convenient to write Eqn. (3-26) in terms of eitherextent of reaction or fractional conversion. If ‘‘i’’ is a reactant, say A, the molar flow ratesmay be written in terms of the fractional conversion xA, i.e., FA ! FA0"1$ xA#, anddFA ! $FA0 dxA. With these transformations, Eqn. (3-26) becomes

Design equation$$ideal PFR$$homogeneous reaction"fractional conversion#

dV

FA0! dxA

$rA(3-27)

If only one, stoichiometrically simple reaction takes place,

j ! Fi $ Fi0

ni; dFi ! nidj

Design equation$$ideal PFR$$homogeneous reaction"extent of reaction#

dV ! nidj

ri(3-28)

Equations (3-26)–(3-28) are various forms of the design equation for a homogeneousreaction in an ideal, plug-flow reactor, in differential form. The equivalents of Eqns. (3-26)–(3-28) for a heterogeneous catalytic reaction are given in Appendix 3.IIIA as Eqns. (3-26a),(3-27a), and (3-28a). Be sure that you can derive them.

Temperature variation with positionIn developing Eqns. (3-26)–(3-28), we assumed that the temperature was constant in anycross section normal to the direction of flow. We did not assume that the temperature wasconstant in the direction of flow. For a PFR, the reactor is said to be isothermal if thetemperature does not vary with position in the direction of flow, e.g., with axial position ina tubular reactor. On the other hand, for nonisothermal operation, the temperature willvary with axial position. Consequently, the rate constant and perhaps other parameters inthe rate equation such as an equilibrium constant will also vary with axial position. Thedesign equations for an ideal PFR are valid for both isothermal and nonisothermaloperation.

Space time and space velocityAs noted in the discussion of the ideal CSTR, the space time at inlet conditions, t0, is defined by

t0&V

y0(3-20)

Using Eqn. (3-20), Eqn. (3-27) can be written in terms of CA0 and t0 as

dt0 ! CA0dxA

"$rA#(3-29)

52 Chapter 3 Ideal Reactors

This equation is also valid for a heterogeneous catalytic reaction, if Eqn. (3-22) is used todefine t0.

The concept of ‘‘space velocity,’’ discussed in connection with the ideal CSTR, alsoapplies to ideal PFRs.

Constant densityIf the mass density of the flowing fluid is the same at every position in the reactor, thesubscript ‘‘0’’ can be dropped from t and y. As noted in the discussion of the ideal CSTR, forthe case of constant density, t"!V=y# is the average residence time of the fluid in the reactor.This is true for any continuous reactor operating at steady state. However, for the ideal PFR, thas an even more exact meaning. Not only is t the average residence time of the fluid in thereactor, it is also the exact residence time that each and every fluid element spends in thereactor. For an ideal PFR, there is no mixing in the direction of flow, i.e., adjacent fluidelements cannot mix with or pass each other. Therefore, every element of fluid must spendexactly the same time in the reactor. That time is t, when the mass density is constant.

For the case of constant density, Eqn. (3-29) becomes

dt ! CA0dxA

"$rA#(3-30)

For constant density, xA ! "CA0 $ CA#=CA0 and dxA ! $dCA=CA0, so that Eqn. (3-30) canbe written as

dt ! $dCA

"$rA#(3-31)

Equations (3-30) and (3-31) are also valid for a PFR with a heterogeneous catalyticreaction taking place, provided that Eqn. (3-22) is used to define t.

Integrated forms of the design equationAs with the batch reactor, the design equations in differential form for the PFR must beintegrated to solve engineering problems. The same three possibilities that were discussedfor the batch reactor also exist here, except that the variable of time for the batch reactor isreplaced by position in the direction of flow for the ideal PFR. For Case 3, where the reactoris either isothermal or adiabatic, Eqns. (3-26) and (3-27) can be integrated symbolically togive

V !ZFi

Fi0

dFi

ri(3-32)

V

FA0!ZxA

0

dxA

$rA(3-33)

The initial conditions for these integrations are V ! 0, Fi ! Fi0, xA ! 0.Appendix 3.III contains the equivalents of these equations for different variables (e.g., j

and t), for heterogeneous catalytic reactions, and for the constant-density case. Thenumbering of the equations in Appendix 3.IIIA continues from Eqn. (3-33) above.

3.3 Continuous Reactors 53

3.3.2.2 The Hard Way—Do the Triple Integration

Let’s return to Eqn. (3-25) and again focus on a tubular reactor, with flow in the axialdirection.

Fi0 $ Fi %Z Z Z

V

ri dV ! 0 (3-25)

The triple integral may be written in terms of three coordinates z (axial position), u (angularposition), and R (radial position).

Fi0 $ Fi %Z2p

0

ZR0

0

ZL

0

ri du R dR dz ! 0 (3-38)

In this equation, R0 is the inside radius of the tube and L is its length. Since there are notemperature or concentration gradients normal to the direction of flow, ri does not depend oneither u or R. Since

R 2p0

R R00 du R dR ! A, where A is the cross-sectional area of the tube

"A ! pR20#, Equation (3-38) may be written as

Fi0 $ Fi % A

ZL

0

ri dz ! 0

Differentiating this equation with respect to z gives dFi=dz ! Ari, which can be rearrangedto

Adz ! dV ! dFi

ri(3-26)

Equation (3-26) has been recaptured. Therefore, all of the equations derived from it can beobtained via the triple integration in Eqn. (3-38).

3.4 GRAPHICAL INTERPRETATION OF THE DESIGN EQUATIONS

Figure 3-4 is a plot of "1=$ rA#, the inverse of the rate of disappearance of Reactant A,versus the fractional conversion of reactant A (xA). The shape of the curve in Figure 3-4 isbased on the assumption that $rA decreases as xA increases. In this case, "1=$rA# willincrease as xA increases. We will refer to this situation as ‘‘normal kinetics.’’

1/(–rA)

Fractional conversion (xA)

Figure 3-4 Inverse of reaction rate (rateof disappearance of reactant A) versusfractional conversion of A.

54 Chapter 3 Ideal Reactors

‘‘Normal kinetics’’ will be observed in a number of situations, e.g., if the reactor isisothermal and the concentration-dependent term in the rate equation obeys Generalization IIIfrom Chapter 2. Recall that Generalization III stated that the concentration-dependent term F (allCi) decreases as the concentrations of the reactants decrease, i.e., as the reactants are consumed.

In the discussion of graphs of#1=$rA

$versus xA, the term ‘‘isothermal’’ will be used to

mean that the temperature does not change as xA changes. This definition is consistent withthe definitions, given previously, for isothermal ideal batch reactors and isothermal idealplug-flow reactors. However, this definition of ‘‘isothermal’’ is more general and can applyto a CSTR or to a series of reactors.

‘‘Normal kinetics’’ will also be observed if Generalization III applies and the reactiontemperature decreases as xA increases. The temperature will decrease as xA increases, forexample, when an endothermic reaction is carried out in an adiabatic reactor.

The shape of the "$1=rA# versus xA curve is not always ‘‘normal.’’ This curve can bevery different if the reaction is exothermic and the reactor is adiabatic, or if the rate equationdoes not obey Generalization III.

Now, let’s reexamine one form of the design equation for an ideal CSTR:

V

FA0! xA

$rA(3-17)

In order to discriminate between the variable xA and the outlet conversion from the CSTR,let’s call the latter xA,e (‘‘e’’ for ‘‘effluent’’), and write the design equation as

V

FA0! xA;e

$rA"xA;e#This equation tells us that "V=FA0# for an ideal CSTR is the product of the fractionalconversion of A in the reactor outlet stream (xA,e) and the inverse of the reaction rate,evaluated at the outlet conditions (1=$rA"xA;e#). This product is shown graphically inFigure 3-5. The length of the shaded area is equal to xA,e, and the height is equal to1=$rA"xA;e#. The area is equal to V=FA0, according to the above equation.

Now, let’s examine the comparable design equation for an ideal PFR:

V

FA0!ZxA;e

0

dxA

$rA(3-33)

Area = V/FA0

xA,e

1/(–rA)

1/[–rA(xA,e)]

Fractional conversion (xA)

Figure 3-5 Graphical representation of the design equation for an ideal CSTR.

3.4 Graphical Interpretation of the Design Equations 55

This equation tells us that (V=FA0) for an ideal PFR is the area under the curve of "1=$rA#versus xA, between the inlet fractional conversion "xA ! 0# and the outlet fractionalconversion (xA,e). This area is shown graphically in Figure 3-6.

Now we can compare the volumes (or weights of catalyst) required to achieve aspecified conversion in each of the two ideal, continuous reactors. Suppose we have an idealCSTR and an ideal PFR. The same reaction is being carried out in both reactors. The PFR isisothermal and operates at the same temperature as the CSTR. The molar feed rate ofReactant A to both reactors is FA0. If the kinetics are ‘‘normal,’’ which reactor will requirethe smaller volume to produce a specified conversion, xA,e, in the effluent stream?

Figure 3-7 shows the graphical answer to this question. For a given FA0, the requiredvolume for an ideal CSTR is proportional to the entire shaded area (both types of shading). Therequired volume for an ideal PFR is proportional to the area under the curve. Clearly, therequired volume for the PFR is substantially less than the required volume for the ideal CSTR.

Area = V/FA0

xA,e

1/(–rA)

Fractional conversion (xA)

Figure 3-6 Graphical representation of the design equation for an ideal PFR.

1/(–rA)

Fractional conversion (xA)

xA,e

1/[–rA(xA,e)]

Figure 3-7 Comparison of the volumes required to achieve a given conversion in an ideal PFRand an ideal CSTR, for a given feed rate, FA0. The required volume for a PFR is proportional to the areaunder the curve. The required volume for a CSTR is proportional to the area of the rectangle (the sumof the two cross-hatched areas).

56 Chapter 3 Ideal Reactors

EXERCISE 3-4

Explain this result qualitatively. What is there about the oper-ation of an ideal CSTR with ‘‘normal kinetics’’ that causes it toneed a larger volume than an ideal PFR to achieve a specifiedoutlet conversion for a fixed FA0?

Hint: Using the above figure, compare the average reactionrate in the PFR with the rate in the CSTR. Why are these ratesdifferent?

The graphical interpretation of the design equations for the two ideal continuousreactors has been illustrated using fractional conversion to measure the progress of thereaction. The analysis could have been carried out using the extent of reaction j with Eqns.(3-18) and (3-34). Moreover, for a constant-density system, the analysis could have beencarried out using the concentration of Reactant A, CA, with Eqns. (3-24) and (3-37).

Plots such as those in Figures 3-5–3-7 are often referred to as ‘‘Levenspiel’’ plots. OctaveLevenspiel, a pioneering figure in the field of chemical reaction engineering, popularized theuse of this type of plot as a pedagogical tool more that 40 years ago.2 ‘‘Levenspiel plots’’ willrecur in Chapter 4, as a means of analyzing the behavior of ‘‘systems’’ of ideal reactors.

SUMMARY OF IMPORTANT CONCEPTS

* Design equations are nothing more than component materialbalances.

* There are no spatial variations of temperature or concentrationin an ideal batch reactor or in an ideal continuous stirred-tankreactor (CSTR).

* There are no spatial variations of temperature or concentrationnormal to flow in an ideal plug-flow reactor (PFR). However,

the concentrations, and perhaps the temperature, do vary in thedirection of flow;

* If (and only if ) the mass density is constant, the designequations can be simplified and written in terms of concen-tration.

PROBLEMS

Problem 3-1 (Level 1) Radial reactors are sometimes used incatalytic processes where pressure drop through the reactor is animportant economic parameter, for example, in ammonia syn-thesis and in naphtha reforming to produce high-octane gasoline.

Top and cross-sectional views of a simplified radial cata-lytic reactor are shown below.

In this configuration, the feed to the reactor, a gas, isintroduced through a pipe into an outer annulus. The gas dis-tributes evenly throughout the annulus, i.e., the total pressure isessentially constant at every position in the annulus. The gas thenflows radially inward through a uniformly packed catalyst bed, inthe shape of a hollow cylinder with outer radius Ro and innerradius Ri. The total pressure is essentially constant along thelength of the central pipe. There is no fluid mixing in the radialdirection. There are no temperature or concentration gradients inthe vertical or angular directions. The catalyst bed contains atotal of W pounds of catalyst.

Reactant A is fed to the reactor at a molar feed rate FA0

(moles A/time), and the average final fractional conversion of Ain the product stream is xA.

Derive the ‘‘design equation,’’ i.e., a relationship among FA0,xA, and W, for a radial reactor operating at steady state.

A more detailed design of a radial fixed-bed reactor isshown below.1

Gas in

Reactorwall

Reactorwall

Centralpipe

Retainingscreen

Gasout

Catalystbed

Inletpipe

Annulus

1 Stine, M. A., Petroleum Refining, presented at the North CarolinaState University AIChE Student Chapter Meeting, November 15,

2002.2 Levenspiel, O., Chemical Reaction Engineering, 1st edition, John

Wiley & Sons, Inc., New York (1962).

Problems 57

Conventional radial flow reactor

Catalyst bed cover plate

Scallop shield

Centerpipe shroud

Scallops (or outer screen)

Catalyst bed (concentrically loadedaround centerpipe)

Centerpipe (punched platewrapped with screen material)

(Figure Copyright 2004 UOP LLC. All rights reserved.Used with permission.)

Problem 3-2 (Level 2) Rate equations of the form

$rA !kCACB

"1% KACA#2

are required to describe the rates of some heterogeneous catalyticreactions. Suppose that the reaction A% B! products occurs inthe liquid phase. Reactant B is present in substantial excess, sothat CB does not change appreciably as reactant A is consumed.

1. The value of$rA goes through a maximum as CA is increased.At what value of CA does this maximum occur?

2. The concentration of A in the feed to a continuous reactor isCA0 ! 1:5=KA. The concentration of A in the effluent is0:50=KA. Make a sketch of "1=$rA# versus CA that coversthis range of concentration. One ideal continuous reactor willbe used to carry out this reaction. Should it be a CSTR orPFR? Explain your answer.

3. Would your answer to Part b be different if the inlet concen-tration was CA0 ! 1:5=KA and the outlet concentration wasCA ! 1:0=KA? Explain your answer.

Problem 3-3 (Level 2) In an ideal, semi-batch reactor, some ofthe reactants are charged initially. The remainder of the reactantsare fed, either continuously or in ‘‘slugs,’’ over time. Thecontents of the reactor are mixed vigorously, so that there areno spatial gradients of temperature or concentration in thereactor at any time.

Consider the single liquid-phase reaction

A% B! products

which takes place in an ideal, semi-batch reactor. The initialvolume of liquid in the reactor is V0 and the initial concentrationsof A and B in this liquid are CA0 and CB0 respectively. Liquidis fed continuously to the reactor at a volumetric flow rate y.The concentrations of A and B in this feed are CAf and CBf,respectively.

1. Derive a design equation for this system by carrying out amaterial balance on ‘‘A’’. Work in terms of CA, not xA or j.

2. Ultimately, we would like to determine CA as a function oftime. Under what conditions is the design equation that youderived sufficient to do this? Assume that the rate equation forthe disappearance of A is known.

Problem 3-4 (Level 1) Plot "1=$rA# versus xA for an iso-thermal, zero-order reaction with a rate constant of k. If thedesired outlet conversion is xA ! 0:50, which of the two idealcontinuous reactors requires the smaller volume, for a fixedvalue of FA0?

What is the outlet conversion from a PFR whenV=FA0 ! 2=k?

Problem 3-5 (Level 1) Develop a graphical interpretation ofthe design equation for an ideal batch reactor.

Problem 3-6 (Level 1) The kinetics of the catalytic reaction3

SO2 % 2H2S! 3S% 2H2O

are being studied in an ideal CSTR. Hydrogen sulfide "H2S# isfed to the reactor at a rate of 1000 mol/h. The rate at which H2Sleaves the reactor is measured and found to be 115 mol/h. Thefeed to the reactor is a mixture of SO2;H2S, and N2 in the molarratio 1/2/7.5. The total pressure and temperature in the reactorare 1.1 atm and 250 8C, respectively. The reactor contains 3.5 kgof catalyst.

What is the rate of disappearance of H2S? What are thecorresponding concentrations of H2S; SO2, S, and H2O?

Problem 3-7 (Level 2) The homogeneous decomposition ofthe free-radical polymerization initiator diethyl peroxydicarbon-ate (DEPDC) has been studied in supercritical carbon dioxideusing an ideal, continuous, stirred-tank reactor (CSTR).4 Theconcentration of DEPDC in the feed to the CSTR was0.30 mmol. Because of this very low concentration, constantfluid density can be assumed. At 70 8C and a space time of10 min, the fractional conversion of DEPDC was 0.21. At 70 8Cand a space time of 30 min, the fractional conversion of DEPDCwas 0.44.

The rate equation for DEPDC decomposition is believed tohave the form

$rDEPDC ! k(DEPDC)n

1. What is the value of n, and what is the value of k, at 70 8C?

2. The activation energy of the decomposition reaction is132 kJ/mol. What is the value of the rate constant k at 85 8C?

3 This reaction is the catalytic portion of the well-known Clausprocess for converting H2S in waste gas streams into elemental sulfur.4Adapted from Charpentier, P. A., DeSimone, J. M., and Roberts,

G. W. Decomposition of polymerization initiators in supercritical

CO2: a novel approach to reaction kinetics in a CSTR, Chem. Eng.Sci., 55, 5341–5349 (2000).

58 Chapter 3 Ideal Reactors

Problem 3-8 (Level 1) An early study of the dehydrogenationof ethylbenzene to styrene5 contains the following commentsconcerning the behavior of a palladium-black catalyst:

‘‘A fair yield (of styrene) was obtained at 400 8C. A 12-gsample of catalyst produced no greater yield than the 8-g sample.By passing air with the ethylbenzene vapor, the dehydrogenationoccurred at even lower temperatures and water was produced.’’

The Pd-black catalyst was contained in a quartz tube and theflow rate of ethylbenzene through the tube was 5 cc (liquid) perhour for all experiments. The measured styrene yields (molesstyrene formed/mole ethylbenzene fed) were about 0.2 with both8 and 12 g of catalyst. For the purposes of this problem, assumethat ‘‘yield’’ of styrene is the same as the conversion of ethyl-benzene, and assume that the dehydrogenation of ethylbenzene isstoichiometrically simple. Also assume that the reaction tookplace at 1 atm total pressure. In answering the questions below,assume that the experimental reactor was an ideal PFR.

1. In view of the design equation for an ideal PFR in which aheterogeneously catalyzed reaction takes place, how wouldyou expect the yield of styrene to have changed when theamount of catalyst was increased from 8 to 12 g?

2. How do you explain the fact that the yield of styrene did notchange when the catalyst weight was increased from 8 to 12 g?

3. How do you explain the behavior of the reaction when air wasadded to the feed?

Problem 3-9 (Level 1) The hydrogenolysis of thiophene"C4H4S# has been studied at 235–265 8C over a cobalt–molyb-date catalyst, using a CSTR containing 8.16 g of catalyst. Thestoichiometry of the system can be represented by

C4H4S% 3H2!C4H8 % H2S Reaction 1

C4H8 % H2!C4H10 Reaction 2

All species are gaseous at reaction conditions.The feed to the CSTR consisted of a mixture of thiophene,

hydrogen, and hydrogen sulfide. The mole fractions of butene"C4H8#, butane "C4H10#, and hydrogen sulfide in the reactoreffluent were measured. The mole fractions of hydrogen andthiophene were not measured.

The data from one particular experimental run are givenbelow:

Total pressure in reactor ! 832 mmHg

Feed rate

Thiophene ! 0:653' 10$4 g+mol/min

Hydrogen ! 4:933' 10$4 g+mol/min

Hydrogen sulfide ! 0

Mole fractions in effluent

H2S ! 0:0719

Butenes "total# ! 0:0178

Butane ! 0:0541

Calculate$rT (the rate of disappearance of thiophene) andthe partial pressures of thiophene, hydrogen, hydrogen sulfide,butenes (total), and butane in the effluent. You may assume thatthe ideal gas laws are valid.

Problem 3-10 (Level 2) The hydrolysis of esters, i.e.,

C

O

OR'

(E) (W) (A)

+ H2O + R'OH

RC

O

OHR

frequently is catalyzed by acids. As the above hydrolysisreaction proceeds, more acid is produced and the concentrationof catalyst increases. This phenomenon, known as ‘‘autocatal-ysis,’’ is captured by the rate equation

$rE ! "k0 % k1(A)#(E)(H2O)

Here k0 is the rate constant in the absence of the organic acid thatis produced by the reaction.

To illustrate the behavior of autocatalytic reactions, let’sarbitrarily assumed the following values:

E0 "ester concentration in feed# ! 1:0 mol/l

W0 "water concentration in feed# ! 1:0 mol/l

A0 "acid concentration in feed# ! 0

The concentration of the alcohol (R0OH) in the feed also iszero. The rate constants are

k0 ! 0:01 l/mol-hk1 ! 0:20 l2/mol2-h

In answering the following questions, assume that thereaction takes place in the liquid phase.

1. Calculate values of $rE at fractional conversions of ester"xE# ! 0; 0:10; 0:20; 0:30; 0:40; 0:50; 0:60; and 0:70:

2. Plot 1=$rE versus xE.

3. What kind of continuous reactor system would you use if afinal ester conversion of 0.60 were desired, and if the reactionwere to take place isothermally. Choose the reactor orcombination of reactors that has the smallest volume. Justifyyour answer.

Problem 3-11 As pointed out in Chapter 2, kinetics are notalways ‘‘normal’’ (e.g., see Figure 2-3). Consider a liquid-phasereaction that obeys the rate equation: $rA ! kC$1

A , over somerange of concentration. Suppose this reaction was to be carried outin a continuous, isothermal reactor, with feed and outlet concen-trations within the range where the rate equation is valid.

1. Make a sketch of "1=$rA# versus CA for the range ofconcentration where the rate equation is valid.

2. What kind of continuous reactor (or system of continuousreactors) would you use for this job, in order to minimize thevolume required? Explain your answer.

5 Taylor, H. S. and McKinney, P. V., Adsorption and activation of carbonmonoxide at palladium surfaces, J. Am. Chem. Soc., 53, 3604 (1931).

Problems 59

I. Ideal batch reactor

A. General—differential form Design equation

VariableHomogeneous

reactionHeterogeneous

catalytic reaction

Moles of species ‘‘i’’, Ni1

V

dNi

dt! ri (3-5)

1

W

dNi

dt! ri (3-5a)

Fractional conversion of reactant A, xANA0

V

dxA

dt! $rA (3-6)

NA0

W

dxA

dt! $rA (3-6a)

Extent of reaction, jni

V

dj

dt! ri (3-7)

ni

W

dj

dt! ri (3-7a)

B. Constant volume—differentialform

Homogeneousreaction

Heterogeneouscatalytic reaction

Concentration of species ‘‘i’’, CidCi

dt! ri (3-8)

dCi

dt! Ccat"ri# (3-8a)

Fractional conversion of reactant A, xA CA0dxA

dt! $rA (3-9) CA0

dxA

dt! Ccat"$rA# (3-9a)

C. General—integrated form (seeNote 1)

Homogeneousreaction

Heterogeneouscatalytic reaction

Moles of species ‘‘i’’, Ni

ZNi

Ni0

1

V

dNi

ri! t (3-10)

1

W

ZNi

Ni0

dNi

ri! t (3-10a)

Fractional conversion of reactant A, xA NA0

Z1

V

dxA

"$rA#! t (3-11)

NA0

W

ZxA

0

dxA

"$rA#! t (3-11a)

Extent of reaction, j ni

Zj

0

1

V

dj

ri! t (3-12)

ni

W

Zj

0

dj

ri! t (3-12a)

D. Constant volume—integrated form(see Note 1)

Homogeneousreaction

Heterogeneouscatalytic reaction

Concentration of species ‘‘i’’, Ci

ZCi

Ci0

dCi

ri! t (3-13)

ZCi

Ci0

dCi

ri! Ccatt (3-13a)

Fractional conversion of reactant A, xA CA0

ZxA

0

dxA

"$rA#! t (3-14) CA0

ZxA

0

dxA

"$rA#! Ccatt (3-14a)

Note 1: For Case 3 (p. 43).

3.5 APPENDIX 3 SUMMARY OF DESIGN EQUATIONS

Warning: Careless use of the equations in this appendix can be damaging to per-fomance. In extreme cases, improper use of these materials can be academically fatal.Always carry out a careful analysis of the problem being solved before using thisappendix.

When in doubt, first carefully choose the type of reactor for which calculations are to beperformed. Then decide whether the reaction is homogeneous or heterogeneous. Finally,begin with the most general form of the appropriate design equation and make anysimplifications that are warranted.

60 Chapter 3 Ideal Reactors

II. Ideal continuous stirred-tank reactor (CSTR)

A. General—in terms of V or W Homogeneousreaction

Heterogeneouscatalytic reaction

Molar flow rate of species ‘‘i’’, Fi V ! Fi0 $ Fi

$ri(3-16) W ! Fi0 $ Fi

$ri(3-16a)

Fractional conversion of reactant A, xAV

FA0! xA

"$rA#(3-17)

W

FA0! xA

"$rA#(3-17a)

Extent of reaction, j V ! nij

ri(3-18) W ! nij

ri(3-18a)

B. General—in terms of t0

(see Note 1)Homogeneous

reactionHeterogeneous

catalytic reaction

Fractional conversion of reactant A, xA t0 !CA0xA

$rA(3-21) t0 !

CA0xA

$rA(3-21)

C. Constant density—in terms of t(see Note 2)

Homogeneousreaction

Heterogeneouscatalytic reaction

Fractional conversion of reactant A, xA t ! CA0xA

$rA(3-23) t ! CA0xA

$rA(3-23)

Concentration of reactant A, CA t ! CA0 $ CA

$rA(3-24) t ! CA0 $ CA

$rA(3-24)

Note 1:

For a homogeneous reaction, t0 ! V=y0 ! VCA0=FA0.

For a heterogeneous reaction, t0 ! W=y0 ! WCA0=FA0.

Note 2:

For a homogeneous reaction, t ! V=y ! VCA0=FA0.

For a heterogeneous reaction, t ! W=y ! WCA0=FA0.

III. Ideal continuous plug-flow reactor (PFR)

A. General—differential form—interms of V or W

Homogeneousreaction

Heterogeneouscatalytic reaction

Molar flow rate of species ‘‘i’’, Fi dV ! dFi

ri(3-26) dW ! dFi

ri(3-26a)

Fractional conversion of reactant A, xAdV

FA0! dxA

$rA(3-27)

dW

FA0! dxA

$rA(3-27a)

Extent of reaction, j dV ! nidj

ri(3-28) dW ! nidj

ri(3-28a)

B. General—differential form–interms of t0 (see Note 1)

Homogeneousreaction

Heterogeneouscatalytic reaction

Fractional conversion of reactant A, xA dt0 ! CA0dxA

"$rA#(3-29) dt0 ! CA0

dxA

"$rA#(3-29)

C. Constant density—differentialform in terms of t (see Note 2)

Homogeneousreaction

Heterogeneous catalyticreaction

Fractional conversion of reactant A, xA dt ! CA0dxA

"$rA#(3-30) dt ! CA0

dxA

"$rA#(3-30)

Concentration of reactant A, CA dt ! $dCA

"$rA#(3-31) dt ! $dCA

"$rA#(3-31)

Appendix 3 Summary of Design Equations 61

D. General—integrated form (seeNote 3)

Homogeneousreaction

Heterogeneouscatalytic reaction

Molar flow rate of species ‘‘i’’, Fi V !ZFi

Fi0

dFi

ri(3-32) W !

ZFi

Fi0

dFi

ri(3-32a)

Fractional conversion of reactant A, xAV

FA0!ZxA

0

dxA

"$rA#(3-33)

W

FA0!ZxA

0

dxA

"$rA#(3-33a)

Extent of reaction, j V !Zj

0

nidj

ri(3-34) W !

Zj

0

nidj

ri(3-34a)

Fractional conversion of reactant A,xA—in terms of t0 (see Note 1)

t0 ! CA0

ZxA

0

dxA

"$rA#(3-35) t0 ! CA0

ZxA

0

dxA

"$rA#(3-35)

E. Integrated form—constantdensity—in terms of t (see Note 2and 3)

Homogeneousreaction

Heterogeneouscatalytic reaction

Fractional conversion of reactant A, xA t ! CA0

ZxA

0

dxA

"$rA#(3-36) t ! CA0

ZxA

0

dxA

"$rA#(3-36)

Concentration of reactant A, CA t ! $ZCA

CA0

dCA

"$rA#(3-37) t ! $

ZCA

CA0

dCA

"$rA#(3-37)

Note 1:

For a homogeneous reaction, t0 ! V=y0 ! VCA0=FA0.

For a heterogeneous reaction, t0 ! W=y0 ! WCA0=FA0.

Note 2:

For a homogeneous reaction, t ! V=y ! VCA0=FA0.

For a heterogeneous reaction, t ! W=y ! WCA0=FA0.

Note 3:

For the PFR equivalent of Case 3 (p. 43). See the discussion on p. 53.

62 Chapter 3 Ideal Reactors