Reactor Design 1

CHE3044F :

Reactor Design 1

Klaus MöllerDepartment of Chemical Engineering

University of Cape Town

March 8, 2012

Contents

1 Mole (or mass) balances 51.1 Chemical Reaction Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Generic flow diagram of a chemical process . . . . . . . . . . . . . . . 6

1.1.2 What do we need to know . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.3 Industrial processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.4 The Chemical Feed stocks . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Rate of a reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Defining when a chemical reaction takes place . . . . . . . . . . . . . 12

1.2.2 Defining the rate of reaction . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 The volume in the reaction rate and its relationship to the mass of thereacting system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.4 The rate equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 The general mole balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Mole balance for batch reactors . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Mole balance for continuous stirred tank reactor (CSTR) . . . . . . . . . . . . 18

1.6 Mole balance for a homogeneous plug flow reactor (PFR) . . . . . . . . . . . 19

1.7 Mole balance for a heterogeneous fixed bed reactor (FBR) . . . . . . . . . . . 22

1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Reactor sizing for single reactions using conversion 24

3 Properties and definitions of the reaction rate equation (reaction rate law) (Foglerchapter 3) 47

4 Reaction stoichiometry and the reaction rate law (Fogler chapter 3) 60

5 Isothermal reactor design for single reactions (Fogler chapter 4) 78

6 Isothermal reactor design for multiple reactions (Fogler chapter 6) 79

7 Collection and data analysis (Fogler chapter 5) 807.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1.1 Limitations on data collected . . . . . . . . . . . . . . . . . . . . . . . 81

7.1.2 Postulate a rate law . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.1.2.1 Homogeneous reactions : . . . . . . . . . . . . . . . . . . . 82

1

CONTENTS CONTENTS

7.1.2.2 Heterogeneous reactions (those with a catalyst): . . . . . . . 82

7.1.3 Choose the same reactor type that matches the data collected . . . . 82

7.1.4 Write the rate equation in terms of the data that has been collected . 83

7.1.5 Make simplifications based on good chemical reaction engineering principles 83

7.1.6 Differential analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.1.7 Integral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.1.8 Goodness of fit and variance in model parameters . . . . . . . . . . . . 85

7.2 batch reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.2.1 Method of excess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.2.2 Example : Differential method . . . . . . . . . . . . . . . . . . . . . . 86

7.2.3 Example : Integral method . . . . . . . . . . . . . . . . . . . . . . . . 91

7.2.3.1 The standard (linearised) analyses . . . . . . . . . . . . . . . 91

7.2.3.2 Example 5.2 (Fogler) . . . . . . . . . . . . . . . . . . . . . 93

7.2.4 Example : Non-linear regression . . . . . . . . . . . . . . . . . . . . . 94

7.3 Method of initial rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3.1 Example (Fogler 5-4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.4 Method of half lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.5 Differential reactors (including CSTR and recycle reactors) . . . . . . . . . . . 99

7.5.1 Mole balances : CSTR (and recycle reactor with rapid recycle) . . . . 100

7.5.2 Mole balances : Tubular and packed bed systems . . . . . . . . . . . . 101

7.5.3 Example : Fogler 5-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8 Developing rate laws from reaction mechanisms and reaction pathways 107

9 Bio-reactor engineering mechanisms 108

10 Bio-reactor design 109

11 Analysis of reactor flow patterns on reactor performance (Fogler chapter 13) 110

2

List of Figures

1.1 Generic flow diagram of a process. (adapted from Schmidt, 2005) . . . . . . 6

1.2 Phthalic anhydride process. (Fogler, 2006) . . . . . . . . . . . . . . . . . . . 6

1.3 The face of chemical reaction engineering. (Fogler, 2006) . . . . . . . . . . . 7

1.4 The multidisciplinary nature of chemical reaction engineering. (Schmidt, 2005) 8

1.5 The reaction rate in a reactor. . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 General mole balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7 Batch reactor. (Fogler, 2006) . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 Change of moles with time in a batch reactor. . . . . . . . . . . . . . . . . . 17

1.9 Continuous stirred tank reactor, abbreviated as CSTR. . . . . . . . . . . . . . 18

1.10 Diagrams of typical plug flow reactors commonly known as PFR’s. . . . . . . 19

1.11 The PFR mole balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.12 Arbitrary shaped PFR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.13 Molar flow rate profiles across the PFR. . . . . . . . . . . . . . . . . . . . . 21

1.14 Tubular fixed bed reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.1 Cubic and polynomial fit to the data. . . . . . . . . . . . . . . . . . . . . . . 89

7.2 Comparing the linear data fit with the non-linear data fit. The green line repre-sents the non-linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.3 Batch reactor integral analysis linear plots for zero, first and second order kineticsusing concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.4 Standard regression plot for the second order reaction. . . . . . . . . . . . . 94

7.5 Predicted and experimental data when using the non-linear leasts squares methodapplied to concentration data. . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.6 Data for example (Fogler5-4). . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.7 Linearised initial rate plot for the dissolution process. . . . . . . . . . . . . . 98

7.8 Half-life analysis plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.9 Differential reactor operation : concepts. . . . . . . . . . . . . . . . . . . . . 100

7.10 Regression plot for the −rCO dependence on PCO. . . . . . . . . . . . . . . . 104

7.11 Plotting the −rCO dependence on PH2. . . . . . . . . . . . . . . . . . . . . . . 104

7.12 The prediction of the reaction rate when regressing for all the rate law constants.106

7.13 The prediction of the reaction rate when regressing for k, KH2 in the rate law. . 106

3

List of Tables

1.1 Summary of reactor mole balances. . . . . . . . . . . . . . . . . . . . . . . . 23

7.1 Reaction rate as a function of concentration using cubic spline and polynomialfitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2 Experimental reaction data for the conversion of CO to CH4. . . . . . . . . . 102

7.3 Processed reaction data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4

Chapter 1

Mole (or mass) balances

5

1.1. CHEMICAL REACTION ENGINEERINGCHAPTER 1. MOLE (OR MASS) BALANCES

1.1 Chemical Reaction Engineering

1.1.1 Generic flow diagram of a chemical process

Figure 1.1: Generic flow diagram of a process. (adapted from Schmidt, 2005)

Figure 1.2: Phthalic anhydride process. (Fogler, 2006)

• Reactors are the key unit operation on plant. A plant is typically designed around a reactor.The operation of the reactor determines the operability and performance of the plant. SeeFig’s 1.1 and 1.2.

• Almost all the worlds commodities and essential items are made using a reactive process.

• The reactors can be as simple as a long hollow tube or a complex reaction vessel many10’s of meters tall with many reactions occurring in multiple phases and with the aid of acatalyst.

• It is their knowledge of chemical reaction engineering that makes chemical engineers unique

Reactors come in all shapes and sizes and arise in many interesting places, see Fig 1.3

6


Figure 1.3: The face of chemical reaction engineering. (Fogler, 2006)

7


1.1.2 What do we need to know

Some facts about practicing chemical engineers

• Single reactions never happen. Reactors are complex unit operations with many simultane-ous chemical reactions occuring in multiple phases. The engineer needs to know enoughabout the reactions, the flow patterns in the vessel, the interaction of heat and masstransfer in order to assemble the basic understanding of the reactor operation.

• The kinetic rate equation almost never exists. Rate equations must be estimated, approx-imated from reaction engineering fundamental knowledge and from experimental or plantdata. The rate equation indicates how fast a reaction proceeds and thus determines boththe size, productivity and selectivity of the reacctor.

• Industrial processes are often severely limited by heat and mass transfer. The engineermust know how to integrate these into the reactor design to yield its actual performance.

Reactor design typically proceeds according to a well defined path

Bench Scale reactor (batch, continuous) → pilot plant → operating plant

Very few engineers will have the honour of following this process. Practicing engineers will mostlikely encounter and old reactor that has been modified many times which the engineer mustnow operate in such a way to double the through-put, improve selectivity and reduce the waste.Alternatively, reactor has failed and the engineer must understand the reasons of failure andre-commission the reactor to operate better without failing.

The typical tasks are

• Maintain and operate a process,

• Fix a problem,

• Increase the capacity or selectivity at minimum cost.

The task must be solved as quickly as possible,for there are many more tasks that need doing.Often, gaining a deeper understanding is often not possible due to the presssure of other morepressing tasks. Besides being competent at “back of the envelope” calculations and having agood intuitive feel for reaction engineering, the engineer also needs to be able to build quickpractical models of the reaction process by integrating all the other engineering disciplines, seeFig 1.4.

Figure 1.4: The multidisciplinary nature of chemical reaction engineering. (Schmidt, 2005)

8


This course is the first course of two in reaction engineering and will aim to develop goodreactor engineering skills for optimal homogeneous, isothermal reactor design by combining theknowledge of

• kinetic models and their structure,

• elementary reactor design equations,

• various type of reactors and configuration of multiple reactor sequences,

• multiple reactions and nature of the reactions,

• experimental data to determine the rate equation.

There will essentially be two types of problems

1. The ones that can be solved in 3 lines often without a calculator. These will build goodreaction engineering intuition;

2. The ones that require more rigorous calculation and in many cases a computer. Theseare the design type problems that will exercise the procedural skills and study the “whatif” scenario’s.

1.1.3 Industrial processes

The following table shows the interesting contrasts between the large volume producers andhigh value chemicals produced by the pharmaceutical companies.

(Schmidt, 2005)

9


1.1.4 The Chemical Feed stocks

The following tables shows the usage of chemicals in the USA in 1994. This ranking of thesechemicals isn constantly changing. It provides an interesting incetive for the development ofalternative processes.

(Schmidt, 2005)

This is in contrast to the prices of chemicals shown in the following table. Note the low valueof the crude oil in contrast to solvents and more so pharmaceuticals (see insulin).

10


11

1.2. RATE OF A REACTION CHAPTER 1. MOLE (OR MASS) BALANCES

(Schmidt, 2005)

1.2 Rate of a reaction

1.2.1 Defining when a chemical reaction takes place

Chemical species are defined by the type, the number and arrangement of the atoms. Forexample isomers differ only by the arrangement of the chemical species

These, isomers, although they have the same type and number of atoms, the different arrange-ment of the atoms results in these molecules having different physical properties and are thuschemically distinct from each other.

A chemical reaction has taken place when one of more of the chemical species in the systemof study are transformed by an infinitesimal amount into (an)other chemical species.

This transformation can be one of the following

• Change the arrangement of the atoms :double bond isomerisation :

skeletal isomerisation :

• Decomposition :

12


• disproportionation :

• Addition :

• and many more other types of reactions (see organic chemistry hand book!)

Often it is simpler to work with letters such as A, B, C, D for chemical species. Thus thereaction of methanol to dimethyl-ether and water written as

2CH3OH → CH3OCH3 +H2O

can be written as2A→ D +W

orA→ 1

2D + 1

2W (1.1)

1.2.2 Defining the rate of reaction

Consider that A is reacting in the vessel in Fig 1.5. −rA represents the consumption of A atsome position in the reactor vessel and it is defined as follows.

Figure 1.5: The reaction rate in a reactor.

Definition of reaction rate : The number of moles of A that are being reacted (consumed orproduced) per unit volume per unit time [mol.m−3.s−1].

13


Definition of rA : rA represents the rate of formation of species A. Thus −rA represents therate of consumption of A. Thus rA is positive for a product and negative for a reactant.

Reaction stoichiometry : In reaction 1.1, the consumption of one mole of A will yield 0.5 molesof D and 0.5 moles of W. For example if the rate of consumption of A is -5 mol.m−3.s−1

then rA = −5 or −rA = 5 and rD = 2.5 and rW = 2.5. Similarly the rate equations for A,D and W are related by the reaction stoichiometry: −rA = 2rD = 2rW =

rD1/2=rW1/2

.

It is good practice to decide on a convention and stick to it. This course will work with thedefinition of rA as given above.

1.2.3 The volume in the reaction rate and its relationship to the mass ofthe reacting system.

Homogeneous reactors well mixed : When the vessel is well mixed such that the concentra-tion of A is equal trhough the reacting vessel, the volume represented by the m3 termrefers to the vessel volume. In gas phase systems the actual volume of the vessel is used.In liquid phase systems, if the vessel is not completely filled, the liquid volume is used.

Homogeneous reactors with concentration gradient : When the species concentrations varythroughout the reactor volume, the volume refered to in the rate equation is that of adifferential element in which it can be assumed that the concentration is uniform for allspecies (i.e. well mixed).

Heterogeneous reactors : In these reactors the reaction takes place on a catalyst surface (ora solid surface or within the catalyst pores). The volume refered to in the rate equation isthen the volume of the catalyst. Since the density of a catalyst is constant, it is easy toconvert the rate equation into a mass basis using the catalyst density and then the rateequation is defined as moles of A reacting per mass of catalyst per time [mol.kg−1.s−1] andis denoted as r ′A. (Note that the activity per unit surface area can similarly be convertedinto a mass basis by using the catalyst properties which define the surface area to massratio.

Multi-phase reactions : If the reaction contains a vapour and a liquid phase, then there will bea rate equation for each phase, r vapA and r l iqA and the volume refered to will be the vapourphase volume and the liquid phase volume respectively. Similarly, for more phase includingsolids and catalysts.

1.2.4 The rate equation

The rate equationrA = f unction(Ci , T, P, Ccatalyst)

is a function of the concentration (activities) of all or some of the chemical species in the reactingsystem, or the temperature and pressure of the system and the properties of the catalyst. Therate equation is an algebraic expression defining how the reaction rate varies with the chemicaland physical properties of the reacting system.

NOTE : The rate equation is independent of the reactor in which the reaction is beingcarried out in.

14

1.3. THE GENERAL MOLE BALANCE CHAPTER 1. MOLE (OR MASS) BALANCES

Examples of possible rate equations for the reaction

A→ P roducts

• Linear variation of reaction rate with concentration : rA = −kCA

• Quadratic variation of the reaction rate with concentration : rA = −kC2A

• Saturation of the reaction rate : rA =−k1CA

(1 + k2CA)2

NOTE : Although rA =dCAdt

is true from the definition of the units of the reaction rate, thisequation is actually the design equation for a constant volume batch reactor, which will bedeveloped later in this course. It is not the definition of the rate equation.

1.3 The general mole balance

Consider a mole balance of species j in a arbritary shaped reaction vessel (or control volume) inFig 1.6. , For example in the reaction

A→ products

A would represent j . It is not necessary the know what the other products are at this stage.Also species j can be a reactant of a product (i.e. sometimes products are fed to the reactoras a result of recycle)

Figure 1.6: General mole balance.

The mole balance then can be written as

IN −OUT + Generation = Accumulationrate of flowof j INTOthe system[mol.s−1]

−

rate of flowof jOUT ofthe system[mol.s−1]

+

rate of generationof jby chemical

reaction in the system[mol.s−1]

=

rate of accumulationof jwithinthe system[mol.s−1]

15

1.4. MOLE BALANCE FOR BATCH REACTORSCHAPTER 1. MOLE (OR MASS) BALANCES

and in symbols this is

Fj,0 − Fj + Gj =dNjdt

(1.2)

where Nj is the total number of moles of j in the system. Gj represents the total transformationof j that has occurred when passing through this reaction vessel. Gj can be related to thereaction rate for two special cases

1. When the concentration of all species is constant throughout the entire system volume: Then the reaction rate rj is the same throught the whole reaction vessel and Gj = rjVwhere V is the volume of the vessel. The general mole balance, equa 1.2 then becomes

Fj,0 − Fj + rjV =dNjdt

(1.3)

2. When the concentration of all species varies throughout the reaction volume : Then thereaction rate will be different at all locations. Dividing the system volume into differentialelements (see Fig 1.6), ∆Vi and assuming that the concentration of all species is constantin each differential element, means that the reaction rate is constant on each differentialelement. Thus the rate of generation of species j in each element is given by rj,i∆Vi . Therate of generation of species j in the system is thus the sum of all the differential elements

: Gj = rj,1∆V1 + rj,2∆V2 + rj,3∆V3 · · · =M∑i=1

rj,i∆Vj . In the differential limit, as ∆V → 0 and

M →∞ then Gj =∫ VrjdV . The general mole balance in this case becomes

Fj,0 − Fj +V∫rjdV =

dNjdt

(1.4)

Equation 1.4 is the GENERAL DESIGN EQUATION for chemical reaction engineering. Whenused on its own, it is applicable to all ISOTHERMAL reaction systems. For NON-ISOTHERMALreaction systems an ENERGY BALANCE will also be needed in order to solve equation 1.4.

1.4 Mole balance for batch reactors

Figure 1.7: Batch reactor. (Fogler, 2006)

16

1.4. MOLE BALANCE FOR BATCH REACTORSCHAPTER 1. MOLE (OR MASS) BALANCES

Fig 1.7(A) shows a schematic diagram of a industrial batch reactor. Its a vessel with a lid sothat it can be opened for cleaning. It has ports for adding and removing chemicals. It almostalways has a stirrer (agitator) and they are almost always used for liquid phase operations. Thereare coils or a jacket around the outside of the vessel for heating and cooling. Some vessels havecoils internally, others use evaporation cooling or direct steam heating. The vessel is alwaysraised off the ground.

Fig 1.7(B) shows the text book schematic of a batch reactor as a vessel filled with liquid and astirrer to provide perfect mixing, such that the concentration off all species in the reactor arethe same everywhere in the vessel. The reactor volume is that of the reactive mixture, in thiscase the liquid volume and NOT the volume of the vessel.

A batch reactor is operated by filling the vessel with the required reactants. The reactor isthen heated to the desired reaction temperature and the reaction is allowed to proceed for apredetermined amount of time, called the reaction time. During this time there is no flow intoor out of the reactor. An the end of the the reaction time the mixture is removed (pumped...)out of the vessel. The reactor is then cleaned and the cycle starts again with fresh reactants.

Since there is no flow into or out of the vessel, the general mole balance on species j becomes

V∫rjdV =

dNjdt

Assuming that the system is well mixed, the reaction rate equation will be the same throughthe whole vessel, thus the design equation is

dNjdt= rjV (1.5)

Notice that the general equation, equa 1.5 is based on the number of moles in the vessel andnot the concentration. More on this later.

For a reaction A→ B the number of moles of A decrease with time while the number of molesof B increase with time, as shown in Fig 1.8.

Figure 1.8: Change of moles with time in a batch reactor.

17

1.5. MOLE BALANCE FOR CONTINUOUS STIRRED TANK REACTOR (CSTR)CHAPTER 1. MOLE (OR MASS) BALANCES

The time t1 that is required to consume (NA0−NA) moles can be obtained by integrating equa1.5:

t1 =

∫ NA1NA0

dNArAV

(1.6)

where NA0 are the number of moles at the start of the reaction, t = 0 and NA1 are the numberfo moles of A that are remaining after a reaction time of t1.

1.5 Mole balance for continuous stirred tank reactor (CSTR)

(A) (B)

Figure 1.9: Continuous stirred tank reactor, abbreviated as CSTR.

Fig 1.9(A) shows a schematic diagram of a continuous stirred tank reactor (CSTR). Thesereactors always have a stirrer or some form of vigorous agitation to ensure that the compositionof all species is uniform throughout the reaction system. They also have a heating/cooling jacketof coils to control the temperature in the vessel. They can be closed vessels or open vessels.They can be filled to capacity or have partially filled with or without two phase operation.

These reactors are most often used for liquid applications involving poorly miscible streamsand/or solids. They are also used for systems which require the additiion of another phase e.g.like bubbling a reactive gas through a liquid reactant.

Fig 1.9(B) shows the diagramatic way in which a CSTR is represented for calculation purposes.The mole balance for species j across this vessel is


dNjdt

If the reactor is operated at steady state, the there is no variation with time of all species on

the reactor, thus,dNjdt= 0 and the mole balance becomes;

Fj,0 − Fj +V∫rjdV = 0

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1.6. MOLE BALANCE FOR A HOMOGENEOUS PLUG FLOW REACTOR (PFR)CHAPTER 1. MOLE (OR MASS) BALANCES

In addition, since there are no spatial variations in the concentrations as the contents of thevessel is well mixed, the reaction rate, rj is constant throughout the whole vessel, which meansthat

∫ VrjdV = rjV and the mole balance then becomes;

Fj,0 − Fj + rjV = 0 (1.7)

V =Fj0 − Fjrj

This is the final design equation for the CSTR. It is important to note that the concentrationof the species in the reactor have the same concentration as the species leaving the reactorin the exit flow.

1.6 Mole balance for a homogeneous plug flow reactor (PFR)

Figure 1.10: Diagrams of typical plug flow reactors commonly known as PFR’s.

A plug flow reactor, Fig 1.10 is basically an empty tube through which a reactive fluid flows. Thereactive streams are mixed at the reactor inlet. The reactor can be heated or cooled dependingon the reaction requirements by using a jacket, by making steam on the outside of the tubesor coils in the vessel. The important aspect of these reactors is that they are extremely longin relation to their diameter, as it is best to have turbulent flow. This is to promote plug flow.Plug flow is defined as flow in which there is no axial mixing (no mixing along the length) bygood radial mixing (mixing perpendicular to the flow). This means that all the species that enterthe reactor spend exactly the same amount of time in the vessel. Species interact only radiallyand not axially. Plug flow reactor can be mounted inside furnaces for very high temperatureoperation.

19


Figure 1.11: The PFR mole balance.

In developing the general mole balance for a PFR, consider the schematic in Fig 1.11. Thegeneral mole balance can be applied to the whole reaction vessel, thus for species j it is


dNjdt

Since the species concentrations (or molar flow rates) are not constant, it is not possible tosimplify the integral expression. Furthermore, when more than one reaction is taking place,the reaction rate term, rj does not only depend on species j but can also depend on the otherspecies in the reaction vessel, preventing the integration from being carried out. Thus in thiscase the general mole balance in this form is not particularly useful. Assuming steady state

makesdNjdt= 0. Then differentiating with respect to the reactor volume yields

dFj,0dV

−dFjdV+ rj = 0

and since the feed does not depend on the reactor volume, thusdFj,0dV

= 0 and thus

dFjdV= rj (1.8)

which is the general mole balance for a PFR. Note that this is a differential equation.

The reactor mole balance can be obtained in a more intuitive way by considering a differentialelement, in which it can be assumed that the species concentrations are constant and thus thereaction rate is also constant. The differential mole balance over the differential element is then

IN −OUT + Generation = Accumulationmolar flow rate

of species jIN at V[mol.s−1]

−

molar flow rateof species j

OUT at V + ∆V[mol.s−1]

+

Molar rate ofgeneration ofspecies j in ∆V[mol.s−1]

=

rate of accumulationof species jwithin ∆V[mol.s−1]

which in symbols, for a reactor at steady state becomes

Fj |V − Fj |V+∆V + rj∆V = 0

Dividing through by ∆V and re-arranging yields

Fj |V+∆V − Fj |V∆V

= rj

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Since the flow rate us a function of V in the sense that F = f (V ) the termFj |V+∆V − Fj |V

∆Vrepresents an approximation to the dereivative of F with respect to V . In the limit as ∆V →∞the PFR mole balance becomes

dFjdV= rj

as before.

IMPORTANT NOTE : for plug flow conditions, the shape of the reactor does not matterand is arbitrary (in the derivation volume was used with no mention of the shape of thevessel). Thus Fig 1.12 could also have been used for the derivation of the mole balance. Thisis no longer true once other factors are accounted for, on in particular is pressure drop, whichdepends on the velocity, which would vary with cross-setional area.

Figure 1.12: Arbitrary shaped PFR.

Consider again the reaction A→ B but this time in a PFR. The molar flow rate profiles acrossthe reactor as shown in Fig 1.13. These are similar to the batch reactor, except that volumehas replaced time.

Figure 1.13: Molar flow rate profiles across the PFR.

The design equation for this reaction is

dFAdV= rA (1.9)

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1.7. MOLE BALANCE FOR A HETEROGENEOUS FIXED BED REACTOR (FBR)CHAPTER 1. MOLE (OR MASS) BALANCES

To determine the volume of reactor required to achieve the exit flow rates FA1 and FB1 in Fig1.13 can be obtained by integrating equation 1.9 from the beginning of the reactor where V = 0and FA − FA0 until V1 where FA = FA1;∫ V1

0

dV = V1 =

∫ FA1FA0

dFArA

1.7 Mole balance for a heterogeneous fixed bed reactor (FBR)

(a) (b)

Figure 1.14: Tubular fixed bed reactor.

Fig 1.14(a) shows a fixed bed reactor with the catalyst packed in the tubes and coolant flowingaround it (sometimes steam is made in this way). This configuration is used if the reaction is veryexothermic (produces a lot of heat) and/or the reactants or products are heat sensitive and/orthe catalyst deactivates at higher temperatures and/or the catalyst selectivity deteriorates andit is crucial to keep the reactor as close to isothermal operation as possible. For other reactionswhere the exothermicity can be tolerated or the reaction is near thermo-neutral, the vessel cansimply be packed with catalyst particles as a uniform bed.

The particle size and the bed diameter are key design parameters. The isothermal vessels aretypically long and thin with a relatively small vessel : particle diameter ratios (20-40). Adiabaticvessels are typically short and squat with large vessel : particle diameter ratios (>‌>100). Massand heat transfer in the catalyst particle to/from the active site becomes more difficult as theparticle size increases. In contrast, the pressure drop across the bed increases as the particlesize decreases. These opposing effects need to be optimised during the reactor design phase.The interplay of heat and mass transfer in catalytic reactions is a leads to exciting reactionsystems. More of this will be the topic of reactor design 2.

In this first level mole balance, it will be assumed that heat and mass can be neglected and thatpressure drop can be ignored and that the flow in the vessel is perfect plug flow. The reactionrate is thus based on the catalyst mass (or catalyst volume), since it is the catalyst that isresponsible for the reaction that is taking place. (Vessel volume and catalyst volume can easilybe used or converted to, through the density of the catalyst particles). Thus the reaction rate

22

1.8. SUMMARY CHAPTER 1. MOLE (OR MASS) BALANCES

is defined asr ′A = [(mol A reacted).s

−1.g−1catalyst ]

Fig 1.14(b) shows the differential element used in the mole balance;

IN −OUT + Generation = Accumulation

Fj |W − Fj |W+∆W + r ′j∆W = 0

Dividing through by ∆V and re-arranging yields

Fj |W+∆W − Fj |W∆W

= r ′j

Since the flow rate us a function of W in the sense that F = f (W ) the termFj |W+∆W − Fj |W

∆Wrepresents an approximation to the dereivative of F with respect to W . In the limit as ∆W →∞the PFR mole balance becomes

dFjdW

= r ′j (1.10)

and the catalyst weight required to achieve a exit floe rate of FA is given by

W =

∫ FAFA0

dFAr ′A

1.8 Summary

Table shows summary of all the mole balances carried out in this chapter.

Table 1.1: Summary of reactor mole balances.reactor Comment Mole balance

differential algebraic integral

Batch no spatial variationdNAdt= rAV t =

∫ NANA0

dNArAV

CSTR no spatial variation, steady state - V =FA0 − FA−rA

PFR steady statedFAdV= rA V =

∫ FAFA0

dFArA

FBR steady statedFAdW

= r ′A W =

∫ FAFA0

dFArA

(Fogler, 2006)

These represent the definite equations of the reactor design 1 course and for that matter allreaction engineering design problems can be solved using these equations. The rest of thiscourse is on the application of these equations to specific problems.

23

Chapter 2

Reactor sizing for single reactions usingconversion

24

Chapter 3

Properties and definitions of the reactionrate equation (reaction rate law) (Foglerchapter 3)

47

Chapter 4

Reaction stoichiometry and the reactionrate law (Fogler chapter 3)

60

Chapter 5

Isothermal reactor design for singlereactions (Fogler chapter 4)

78

Chapter 6

Isothermal reactor design for multiplereactions (Fogler chapter 6)

79

Chapter 7

Collection and data analysis (Foglerchapter 5)

80

7.1. ALGORITHMCHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)

This chapter develops some of the techniques that are used to obtain rate laws for chemical re-actions, which are then ultimately used in the design of a pilot plant and ultimately a commercialunit. THE KNOWLEDGE AND TOOLS GAINED HERE ARE EXTREMELY IMPORTANT.The purpose of this chapter is to develop the following knowledge

1. To be able to estimate the reaction rate constant, the reaction order and other constantsin the rate law from appropriately collected experimental data.

2. To be able to study experimental data and decide how it should be analysed

3. To be able to develop the appropriate reactor design equation and rate law that willdescribe the experimental data

4. To be able to formulate the design equation in the appropriate form such that it is suitablefor regression with experimental data to determine the rate law parameters

5. To be able to formulate the objective function and use appropriate computer code to carryout the non-linear regression

6. To be able to select the best model based on good statistical methods.

7.1 Algorithm

The following steps are general and apply to all the examples and reactor types that follow.

7.1.1 Limitations on data collected

The following are guidelines when collecting data or looking at data to be analysed

1. Make every effort to ensure that the data is collected under isothermal and isobaric con-ditions

2. Make sure that the data are collected at steady state and that the MASS BALANCE ISOBEYED.

3. Make every effort to ensure that the data collected is free of transport limitations such asfilm resistance, pore diffusion. (see later chapters)

4. Make every effort to ensure that the idealisations of the assumed reactor type are obeyed.For example, if a plug flow reactor is assumed, the reactor configuration choosen mustbehave like a plug flow reactor or else the estimated rate constant will depend on theshape of the reactor and CANNOT BE USED FOR SCALE-UP. For example in a twophase batch reactor (gas bubbles in the liquid), it is necessary that the stirring is fastenough such that the bubbles are small enough to ensure that there are no mass transferlimitations.

5. Make every effort to ensure that there are no unexpected concentration and temperaturegradients. For example, in a flow reaction bomb (CSTR) in which a combustion reactionis taking place, gfeat care needs to be taken to ensure that the mixing (stirring) is fasenough to ensure that there are NO temoperature and concentration gradients in thevessel. The same applies to fast liquid phase reactions in a batch reactor, the stirringmust be fast enough to ensure complete mixing at all times during the reaction.

81


6. There are ways to analyse non-isothermal and non-isobaric data, but it is considerablymore difficult to get estimates of the kinetic constants that are reliable. It is also beyondthre scope of this course. Similarly, non-ideal reactor behaviour can be accounted for, butthis also leads to less accurate estimations of the rate constants.

7.1.2 Postulate a rate law

The design and development of rate laws is the topic of the next section (Fogler!!!!). In manycases the rate law needs to be developed in conjunction with the data analysis and experimentalobservations.

Good places to start

7.1.2.1 Homogeneous reactions :

mono-molecular : −rA = kCnA = k ′P αA (7.1)

bi-molecular : −rA = kCαACβB = k

′P αA PβB (7.2)

reversible : −rA = k(CACB −

CCCDKe

)(7.3)

In particular for reversible reactions it is important that the rate law is able to describe thechemical equilibrium of the reaction(s) i.e. for reaction 7.3 the equilibrium limit is

Ke =CCeCDeCAeCBe

where the Cie are the equilibrium concentrations when rA = 0, and obtained from thermody-namics.

7.1.2.2 Heterogeneous reactions (those with a catalyst):

Langmiur-Hinshilwood (LH) : −rA =kPA

1 +KAPA(7.4)

LH-reversible : −rA =k

(PA −

PBKe

)1 +KAPA +KBPB

(7.5)

Also here the rate law must obey the thermodynamic limitations of the reaction thermodynamics.

7.1.3 Choose the same reactor type that matches the data collected

Batch reactor :

−rA =1

V

d(NA)

dt=dCAdt=dCAdt=1

RT

dPAdt

(7.6)

PFR :

−rA =dFAdV= −FA0

dXAdV

(7.7)

CSTR :

−rA =FA0 − FAV

=FA0VXA (7.8)

82


7.1.4 Write the rate equation in terms of the data that has been collected

e.g. PA vs t, NA vs t, XA vs W etc.

7.1.5 Make simplifications based on good chemical reaction engineeringprinciples

Some typical examples are;

1. Reactants in excess : This means that for example that when the concentration of Bexceeds the concentration of A by more that a factor of 100, then the second orderreaction −rA = kCαAC

βB can be approximated by −rA = k ′CαA where k ′ = kCβB. NOTE that

in this csae the rate constant k ′ is a function of the concentration of B (CB). NOTE alsothat this rate law cannot be used for conditions in which the concentration of B is of thesame order of magnitude as A. For situations like this new experimental data will have tobe collected.

2. Reactants are dilute in a reaction mixture : If for example the mole fractions of A,B in areacting mixture are less than 1%, then any volume expansion, pressure change, flow ratechange, due to the change the number of moles, can be neglected (i.e. ε = 0). Similarlyfor liquid phase reactions.

3. Fast reactions can be assumed to be at chemical equilibrium, which simplifies the rateequations.

7.1.6 Differential analysis

This method of analysis takes the approach that the data will be in the form

REACTION RATE vs RATE LAW

−rA =[−dCAdt,dFAdV,−dPAdt,FA0VXA, etc

]vs kCαAC

βB (7.9)

In a CSTR, the data automatically comes in this form i.e. −rA =FA0VXA . However, for

all other reactors this provides an approximation to the data collected. All other reactors willcollect primarily C vs t or F vs V data which has to be converted into rate data by numericaldifferentiation. Numerical differentiation increases the scatter (error) in the data and leadsto poor data analysis and this method is not recommended, although its use will be quicklydemonsrated as is done in Fogler.

Often, data is collected in what is known as a differential reactor, in which the feed entering isalready partially converted (i.e. Xin > 0) and products leaving only increase the conversion bya few percent (i.e. Xout = Xin+ ≤ 0.05), then the reactor can be treated similarly to a CSTR,

viz. −rA = −FA0Xin −Xout∆V

.

In all cases the data analysis normally follows the following approach:

−rA = kCαACβB (7.10)

83


taking logs on both sides:

ln(−rA) = ln(−dCAdt

)= ln(k) + αln(CA) + βln(CB) (7.11)

which is a linear equation in (ln(k), α, β) which can then be found by linear regression. This isa good approach when the reaction order are unknown and are needed to be estimated, sincethe equation is linear in the reaction orders. NOTE : this methods uses log weighting of thedata, and this will have slightly different answers than the direct non-linear regression, whichcan be formulated as :

min [f (k, α, β)] =

n∑i=1

[−rA,i − kCαA,iC

βB,i

]2 ≈ 0where−rA,i represents the measured data data at each experimental value i of the concentrationsCA,i , CB,i in the reaction system when a total number of n experiments have been carried out.The objective function f (k, α, β) must be minimised by adjusting k, α, β until the least squarederror is as close to zero as possible (using excel’s solver or scilab’s lsqrsolve).

7.1.7 Integral analysis

This analysis is more applicable to standard reactors which generate data in the form:

CA, FA, PA, XA vs t, V, W, τ,W

FA0(7.12)

In all these cases the design equation must be integrated. Note that integration is a smoothingprocess and reduces the errors in the estimated rate constants as opposed to the differentialmethod. For the simple αth order reaction −rA = kCαA in a liquid phase batch reactor this yields:

CA = CA0exp(−kt) α = 1 (7.13)

C(1−α)A = −k(1− α)t + C(1−α)A0 (7.14)

For the first equation (α = 1) the linear plot ln(CA) vs t will yield rate constant from the slope.However, for the second equation, the linear plot of C(1−α)A vs t will require an estimate of α.Thus the linear regression requires repeated guesses of α, which is silly. Thus this type of datacan only be analysed using non-linear analysis except in some special cases. (typically when thereaction order is given, linearisation is often possible).

Non-linear regression in this case yields:

min [f (k, α)] =

n∑i=1

[C(1−α)A,i + k(1− α)ti − C(1−α)A0

]2≈ 0 (7.15)

or in terms of the measured variable, CA it is

min [f (k, α)] =

n∑i=1

CA,i − (−k(1− α)ti − C(1−α)A0

) 1

1− α

2 ≈ 0 (7.16)

There are many situations where integration is not possible. For example, consider the secondorder non-equi-molar reaction with the rate law : −rA = kCαAC

βB. Writing in terms of conversions

84


for a constant flow rate system yields, the PFR design equation is:

FA0dXAdV

= kCα+βA0 (1−XA)α

(CB0CA0−XA

)β(7.17)

V

FA0kCα+βA0 =

XA∫0

1

(1−XA)α(CB0CA0−XA

)β dXA (7.18)

which can be formulated as an objective function

min [f (k, α, β)] =

n∑i=1

VFA0,i kCα+βA0 −XA,i∫0

1

(1−XA)α(CB0CA0−XA

)β dXA2

≈ 0 (7.19)

where the different conversion are obtained by varying the flow rate (this is the most commonway, changing V is the other way). In this formulation the integral must be evaluated each timefor each value of XA,i . All this is not a problem on a computer. (NOTE, that if CA0 = CB0then it is no longer possible to distinguish α from β. The advantage of this approach is thatit is general, requires no guessing of α, β and has no limitations of these values either. Thedisadvantage is that it needs some computer code to be written!

It is possible to have multiple reactions with multiple rate constants, where integration using anODE solver is needed each time is also easily done on a computer.

7.1.8 Goodness of fit and variance in model parameters

The goodness of fit can be obtained in two ways

(i) A correlation coefficient can be obtained between the experimental and the predicted datae.g plotting (−rA)exp versus (−rA)model and estimating the mean and variance.

(ii) looking at the least squares error :

√√√√√n∑i=1

[(−rA)exp − (−rA)model

]2n −m where n is he number

of data points and m is the number of estimated parameters (3 in the case shown here)

The variance in the model parameters can be obtained by studying the Jacobian matrix of the

objective function : Ji ,j =(∂f (p)j∂pi

)where p is the vector p = [k, α, β] and f (p) is evaluated

at each data point j , and thus J is a m × n matrix. Then the variance in the parameters p isgiven by

pi = pi ± σi ,i ; σi ,i =

√√√√ f (p)× [(JTJ)−1]i ,i

n −m (7.20)

where[(JTJ

)−1]i ,i

refers to the diagonals of the matrix. When σi ,i � 0.2pi then the parameter

is a poor estimated and the model is most likely over parameterised and that means that thereare most likely too many model parameters or there is not sufficient data.

85

7.2. BATCH REACTORSCHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)

7.2 batch reactors

7.2.1 Method of excess

The idea of this method is to reduce the dependence of the rate law on multiple concentrationsinto one concentration. This method is particularlygood for getting first estimates of the reactionorder with respect to the key component. For example, the reaction :

A+ B → products

with the rate law−rA = kCαAC

βB (7.21)

is difficult to analyse when both α, β are unknown. The idea then is to carry out the reation intwo ways (i) Using excess B such that CB = CB0 and does not change significantly during thereaction. Then

−rA = kCαACβB = kC

βB0C

αA = k

′CαA (7.22)

which can then be used to find α.

(ii) Using excess A such that CA = CA0 and remains effectively constant during the reaction,similarly

−rA = kCαACβB = kC

αA0C

βB = k

′′CβB (7.23)

which can be used to find β.

(iii) Once α, β have been found, then normal concentration data can be analysed for k .

7.2.2 Example : Differential method

(Fogler, ex 5.1, p260)

The reaction of triphenyl methyl chloride (A) with methanol(B) viz

(C6H5)3 CCl + CH3OH → (C6H5)3 COCl +HCl

A+ B → C +D

is carried out in a batch reactor in a solution of benzene and pyridine at 25°C. HCl reacts withthe pyridine, which then precipitates making the reaction irreversible. The concentration timedata is collected

time (min) 0 50 100 150 200 250 300Concentration of A, CA[mmol.L−1] 50 38 30.6 25.6 22.2 19.5 17.4

The initial concentration of methanol was 500 mmol.L−1.

1. determine the reaction order with respect to triphenyl-methyl-chloride

2. the reaction order of methanol was given as 1, determine the reaction rate constant

SOLUTIONPART 1Looking at the data

the time vector is given by : [t] = [t1 t2 t3 ...tn]

the concentration vector is given by : [CA] = [CA,1 CA,2 CA,3 ...CA,n]

86


where n is the number of data points. This also means that you cannot estimate more than nmodel parameters.Proposed rate law : −rA = kCαAC

βB

now, CB0 = 10CA0 thus the it may be assumed that CB = CB0 ≈ constant over the range ofthe data collected (well, not really that constant, by the change is less than 5%)Then −rA = k ′CαA where k ′ = kCβB0

Mole balance :dNAdt= rAV

stoichiometry : NA = NB = −NC = −NDSince this is dilute liquid phase system, the density (volume) of the system is constant, thus thedesign equation becomes;

−dCAdt= k ′CαA (7.24)

differential analysis then requires;

ln

(−dCAdt

)i

= ln(k ′) + αln(CA,i) (7.25)

where the i values run over all data points. Thus it is necessary to find the reaction rate as afunction of time.The best way to do this is to fit a cubic-spline to the data

//fit a cubic-spline to the datad = splin(tt,Ca);function Cat=spln(t)

Cat=interp(t,tt,Ca,d)

endfunctionCas=interp(t,tt,Ca,d) //spline interpolated values

and then to differentiate the cubic-spline//get the gradientttt=tt;ttt(n)=ttt(n)-0.1; //takingcare of the last pointdCa_dt_s=diag(numdiff(spln,ttt));disp(’-dCa/dt=’+string(-dCa_dt_s))

. This yields the following data in table 7.1.An alternative is to fit a polynomial to the data. This is done as follows;The error between each data point and the polynomial is given by

εi = CA,i − (a1 + a2t + a3t2 + a4t3 + a5t4...) (7.26)

which can be vectorised as follows since it is a linear operation with basis sets

εi = CA,i − [p]i [a]T (7.27)

[p]i = [1 ti t2i t3i t4i ...] [a] = [a1 a2 a3 a4 a5] (7.28)

Here the vectot [p] represents the basis sets. Note that these can be any set of independentfunctions e.g. [p] = [1 t sin(t) exp(t) t3]. This can be expanded for every data point n :ε1ε2ε3...εn

=cA,1CA,2CA,3

...CA,n

−1 1 1 1 1

t1 t2 · · · · · · · · ·t21 t22

. . . ...

t31. . . ...

tn1 · · · · · · · · · tnn

a1a2a3...am

= [ε] = [CA]− [p][a]T ≈ 0 (7.29)

87


Note that m is the number of parameters [a] which in this case is 5. Note also that m ≤ nfor a meaningful solution. When m = n then the solution is unique, one parameter for everyequation. However, as with most data, n > m and the solution will never be exactly zero, ratherthe objective function needs to be minimized, thus

minΨ = [ε]T [ε] (7.30)

Matlab’s and Scilab’s backslash operator (\) does exactly that (i.e. the overdetermined system,Ax = b yields the least squares solution x = A\b). Applying this concept here yields theparameters of the polynomial that fits the CA vs t data;

[a]T = [p]\[CA] = [50 − 0.2978 1.34E − 03 − 3.485E − 06 3.67E − 09]

The code that does this is and the data is given in the table.

//polynomial fit to the data using a non-linear leasts squaressolver (the long way)

function f=polfit(a,n)

tx=[ones(1:n); tt; tt^2; tt^3; tt^4]’; //define thevector of polynomial basis sets

f=Ca’-tx*a’ //calculate the error at each point

endfunctiona0=[1 1 1 1 1]; //initial guessa=lsqrsolve(a0,polfit,n) //solve the non-linear problemdisp(’a_cubic=[’+string(a)+’]’) //displaytx=[ones(1:nt); t; t^2; t^3; t^4]’;Cap=tx*a’; //calculate the polynomial fitted valuesfunction f=CaPol(t) //define a function in terms of t

tx=[ones(1:n)’ t t^2 t^3 t^4]; //changes since t is acolumn vector

f=tx*a’ //the value of Ca

endfunction//polynomial regression using linear leasts squares (the short

but abstract way)txx=[ones(1:n); tt; tt^2; tt^3; tt^4];aa=txx’\Ca’disp(’a_poly=[’+string(a)+’]’)

Table 7.1: Reaction rate as a function of concentration using cubic spline and polynomial fitting.

CA 50 38 30.6 25.6 22.2 19.5 17.4

−(dCAdt

)spl ine

0.303 0.185 0.119 0.0814 0.0590 0.0485 0.0351

−(dCAdt

)poly

0.298 0.188 0.119 0.0801 0.0603 0.0485 .0334

The fitted data is shown in Fig 7.1.

88


15

20

25

30

35

40

45

50

0 50 100 150 200 250 300

time; t [s]

concentration;CA[mmol:L¡1]

Figure 7.1: Cubic and polynomial fit to the data.

The linear regression using equa 7.25 is

//linear regression for the rate constant and reaction order//using cubic spline dataaa_s=regress(log(Ca’),log(-dCa_dt_s)); //such thatlog(-dCa_dt)=aa(1)+aa(2)*log(Ca)kr=exp(aa_s(1))alpha=aa_s(2)disp(’k=’+string(kr)+’ alpha=’+string(alpha));//using polynomial dataaa_p=regress(log(Ca’),log(-dCa_dt_p)); //such thatlog(-dCa_dt)=aa(1)+aa(2)*log(Ca)kr=exp(aa_p(1))alpha=aa_p(2)disp(’k=’+string(kr)+’ alpha=’+string(alpha));

Linear regression of the spline fit data this data yields :

ln(k) = −9.116; k ′ = 0.0001099 L.mmol−1.min−1 and α = 2.035 and l sqrErr ror = 0.0136

Linear regression of the poly fit data this data yields :

ln(k) = −9.116; k ′ = 0.0001046 L.mmol−1.min−1 and α = 2.048 and l sqrErr ror = 0.0199

Direct non-linear regression of the cubic spline data uses this equation

min f (k ′, α) =

n∑i=1

(−(dCAdt

)i

− k ′CαA,i)

and the following code

89


//direct non-linear regressionfunction f=EE(b,n)

k=b(1);alpha=b(2)f=-dCa_dt_s-k*(Ca^alpha)’

endfunctionb0=[1 1];b=lsqrsolve(b0,EE,n)disp(’k=’+string(b(1))+’ alpha=’+string(b(2)));

this data yields :k ′ = 0.0001467 L.mmol−1.min−1 and α = 1.954 and l sqrErr ror = 0.00902So why are these values different :Figure 7.2 shows the two different plots which effectively represent the two different methods.

10

10

10

-2

-1

0

10 101 2

CA [mmol:L¡1]

¡dCA

dt[mmol:L¡1:min¡1]

(a) linear-fit

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

15 20 25 30 35 40 45 50

CA [mmol:L¡1]

¡dCA

dt[mmol:L¡1:min¡1]

(b) non-linear-fit

Figure 7.2: Comparing the linear data fit with the non-linear data fit. The green line representsthe non-linear fit.

The linear regression transforms the data into a log-log scale. This means that the error forthe small values of the concentrations and small rates are weighted much more heavily than thevalues at the large concentrations. However, typically, small concentrations are more difficultto measure and thus also have the largest experimental error. This error is magnified by thelog-log scaling. The non-linear fitting, on the other hand, leaves the data in its original for andweights all errors on this basis.Figure 7.2(a) shows the log-log plot is not able to distinguish between the different regressions.On the other hand Fig 7.2(b) shows that the non-linear regression (green) shows a superior fit tothe rate data at high concentrations. The non-linear fitting thus provides a more representativeset of rate constants. This deviation is more pronounced in the rate constant.PART 2Since β = 1 then the true rate constant k can eb calculated:

k =k ′

CB0= 2.92E − 7 [L.mol−1.min−1]2

Thus the rate law is given by

−rA = 0.292C2ACB [mol.L−1.s−1]

90


It is possible to fit for all 3 parameters k αβ if the concentration of B can be found from theconcentration of A. (normally B should also be measured). This would give an idea of the errorsmade in assuming that the concentration of B is constant. From the reaction stoichiometry

CA = CA0(1−X); X = 1−CACA0

CB = CB0 − CA0X = CB0 − CA0 + CAThen the rate law becomes

−rA = kCαA(CB0 − CA0 + CA)β

which can be used for dirrect non-linear regression or linear regression using the log-log approach

non-linear

min f (k, α, β) =

n∑i=1

((−rA,i)− kCαA,i(CB0 − CA0 + CA,i)β

)2which can be solved by a non-linear leasts squares, Scilab’s “lsqrsolve”

[k, α, β] = [180, 2.13, −2.37] with l sqrErr ror = 0.00712.

This has the smallest error of them all. Interesting, WHY

Linear log-log:

ln(−rA)i = ln(k) + αln(CA,i) + βln(CB0 − CA0 + CA,i)ln(−rA) = p ∗ a

p =

1 ln(CA) ln(CB0 − CA0 + CA,i)1

......

......

...

1...

...

a =

ln(k)

α

β

which can be solved by the “backslash” (a = p\ln(−rA)).[k, α, β] = [9.44E13, 2.47, −6.93] with l sqrErr ror = 0.067.

This solution has failed. This is due to the linear dependence of the two concentration columns.

7.2.3 Example : Integral method

Contrary to what Fogler says, this is not a trial and error method. When the reaction orders areknown, then there are many standard rate laws for which the design equations can be integratedand a appropriate regression can be carried out. If not, then they simply become partr of theregression procedure.

7.2.3.1 The standard (linearised) analyses

For the reaction A → products it is always useful to test the data with the standard cases,or make a standard plot. Assuming a constant volume batch reactor, the following analysescan be used for zero, first, second and nth order reactions. Fig 7.3 shows these plots on aconcentration basis for a batch reactor.

91


Zero order reaction

CA = CA0¡ ktslope = ¡ k

WARNING

There is no natural limit

on the smallest value of CA

it can go negative

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

time; t [s]

con

centr

atio

n;CA

[mol:m

¡3]

¯rst order reaction

log

µCACA0

¶= ¡kt

slope = ¡ k

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

time; t [s]

logµCA

CA0¶

orlog(1¡X)

2nd order reaction; kC2A

1

CA= kt+

1

CA0

slope = k

Rxn order <

2

Rxn order >

2

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

time; t [s]

1 CA

2nd order reaction; kCACB

log

µCA

CB0¡ CA0 + CA

¶= ¡k(CB0¡ CA0)t+ log

µCA0CB0

¶

slope = ¡ k(CA0¡ CB0)

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

time; t [s]

logµ

CA

CB0¡CA0+CA

¶

Figure 7.3: Batch reactor integral analysis linear plots for zero, first and second order kineticsusing concentrations.

Zero order reaction :dCAdt= −k ; CA = CA0 − kt; X =

k

CA0t (7.31)

So plotting CA vs t is a linear plot with an intercept CA0 and slope −k , as shown in Fig

7.3. (or plotting X vs t has a slope ofk

CA0. Note that it is not necessary to know the

initial concentration CA0 when making the CA vs t plot, since this given my the intercept.

First order reaction :

dCAdt= −kCA;

CACA0= exp(−kt); 1−X = exp(−kt) (7.32)

linearisation yields

ln

(CACA0

)= −kt; ln(1−X) = −kt (7.33)

So when plotting ln(CACA0

)vs t the slope of the line is −k , as shown in Fig 7.3 (similarly

for plotting of ln(1−X) vs t) Note that re-arrangement to

ln(CA) = ln(CA0)− kt (7.34)

92


and plotting ln(CA) vs t also does not require knowledge of the intial concentration.Furthermore, the reaction rate does not depend on the concentration, and thus it is notimportant to keep the initial concentration constant during experimentation.

Second order reaction :dCAdt

= −kC2A;dX

dt= kCA0(1−X)2 Also for CA0 = CB0 (7.35)

dCAdt

= −kCACB = −kCA(CB0 − CA0 + CA);dX

dt= kCA0(1−X)

(CB0CA0−X

)(7.36)

and their integrated forms re-arranged into linear equations for CA0 = CB0:

1

CA= kt +

1

CA0plot :

1

CAvs t slope : k intercept :

1

CA0(7.37)

X

1−X = kCA0t plot :X

1−X vs t slope: kCA0 (7.38)

And for CA0 6= CB0 :

ln

(CA

CB0 − CA0 + CA

)= −k(CB0 − CA0)t + ln

(CA0CB0

)(7.39)

plot : ln(

CACB0 − CA0 + CA

)vs t slope : − k(CB0 − CA0) intercept : ln

(CA0CB0

)(7.40)

ln

CA0CB0X − 1

X − 1

= k(CB0 − CA0)t (7.41)

plot : ln

CA0CB0X − 1

X − 1

vs t slope : k(CB0 − CA0) (7.42)

Some of these standard plots are made in Fig 7.3. In all cases, except for1

CA= kt +

1

CA0it is necessary to know the concentrations of A and B at the start of the reaction.

qth order reaction :

dCAdt= −kCqA;

dX

dt= kCq−1A0 (1−X)

q (7.43)

1

Cq−1A

= −k(1− q)t +1

Cq−1A0

;1− (1−X)q−1

(1− x)q−1 = k(1− q)t (7.44)

In these linearised equations it is necessary to know the reaction order q. (see Fig 7.3)

7.2.3.2 Example 5.2 (Fogler)

Continuation of example 5.1 : The purpose now is to regress the concentration data directlywithout finding the reaction rate. Thus, assuming that teh reaction is second order;

−dCAdt= k ′C2A;

1

CA= k ′t +

1

CA0(7.45)

93


Regression can be done by noting that the integrated equation is related to y = a + bx when

y =1

CAand x = t. The the solution of this equation by for example Scilab’s “regress” routine,

yields b = k ′ and a =1

CA0. The value obtained for a provides a check on the validity of the

model, since CA0 is already known. So k ′ = 0.0001248 and ”CA0” = 49.7. So the model fitsthe data well as shown in Fig 7.4..

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.050

0.055

0.060

0 50 100 150 200 250 300 350

time; t [min]

1 CA[L:mol¡1]

Figure 7.4: Standard regression plot for the second order reaction.

7.2.4 Example : Non-linear regression

Non-linear regression requires the development of a objective function which is then used in anon-linear leasts squares solver (Scilab’s “lsqrsolve” is a particularly good one, EBE-matlab doesnot have such routines and the nextbest bet is EXCEL!!!)

Continuation of Foglers example 5-1:

Noting that k ′ and α are to be found and that the data is given in the form CexpA,i vs ti , ittherefore makes sense to integrate a general α-order rate equation and its integrated form.;

dCAdt= −k ′CαA ; CmodelA =

[C1−αA0 − (1− α)kt

] 1

1− α

(7.46)

This is the batch reactor design equation and will from now on be called the model. Now thepurpose of this method is to minimise the error between the experimental concentrations andthe model predicted concentrations, thus this will form the objective function;

min f (k ′, α) =

n∑i=1

εiεi = εεT (7.47)

94

7.3. METHOD OF INITIAL RATESCHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)

where n is the number of experimental data points and εi is the error between the experimentaland model concentrations;

εi = CexpA,i − C

modelA,i (7.48)

εi = CexpA,i −

[C1−αA0 − (1− α)kti

] 1

1− α

(7.49)

where ε is a vector of values,one for each data point collected

ε =[ε1 ε2 . . . . . . εn

](7.50)

The solution proceeds by developing a function that calculates the εi values at each data pointand then call the scilab routine lsqrsolve to carry out the least squares minimisation. The scilabfunction is

function f=model1(x,n)

kr=x(1);alpha=x(2); //x is a vector of two parametersf=Ca-(Ca(1)^(1-alpha)-(1-alpha)*kr*tt)^(1/(1-alpha))

endfunctionThe calling of the least squares solver then has hte code

//call the least squares solverx0=[0.0001 2]; //initial guessx=lsqrsolve(x0,model1,n)kr=x(1);alpha=x(2);disp(’kr=’+string(kr)+’ alpha=’+string(alpha))Cam=(Ca(1)^(1-alpha)-(1-alpha)*kr*t)^(1/(1-alpha)) //model predicted val-

uesThe model parameters obtained are :

k ′ = 0.000111 α = 2.037

And the quality of the fata fit can be seen in Fig 7.5.It is possible to use the same code to regress with α = 2. When this is done then k ′ = 0.000126similar to those obtained before. The values of k ′ are summarised here:

method k ′ α

differential : Linear regression of the spline fit 0.1109 2.035differential : Linear regression of the poly fit 0.1046 2.048differential : non-linear regression 0.1467 1.95Integral 0.1248 2.000non-linear regression CA 0.111 2.037non-linear regression 0.126 2.000

7.3 Method of initial rates

The idea here is to study the reaction at very short reactions times, or more precisely, at smallconversions (say < 5% change in conversion over the reaction period). This avoids complicationsfrom any possible reversible reaction steps, deactivation as a result of by-products forming,etc. These experiments are carried out a various inlet concentrations (CA0) and/or variousconversions at the inlet. Then the reaction rate at these conditions is estimated from datacollected over a very small conversion range. Possible sequences might be

95


15

20

25

30

35

40

45

50

55

0 50 100 150 200 250 300

time; t [min]

CA[mol:L¡1]

Figure 7.5: Predicted and experimental data when using the non-linear leasts squares methodapplied to concentration data.

CA0 20 10 5 2 1CA 19 9.5 4.75 1.9 0.95t 10 15 12 10 8

Which yields the reaction rate∆CA∆t

=CA0 − CAt

at the concentrationCA0 + CA2

from whichthe law parameters can be obtained. Or possibly

Xin 0.1 0.2 0.3 0.4 0.5Xout 0.11 0.21 0.31 0.41 0.51FA0 10 15 20 25 30

From which the reaction rate can be obtained from a differential reactor balance (see later)Limitations of this method

1. Real reactions undergo deactivation. If deactivation is fast (FCC) then this method failssince it becomes impossible to esrimate the initial reaction rate without the influence ofthe deactivation.

2. Real reactors operate at very high conversions, contrary to the very low conversions usedhere. High conversions almost always produce side products which can influence thereaction kinetics, especially at high conversions, which are the conditions used under normaloperation. These methods fail to analyse such reaction systems

7.3.1 Example (Fogler 5-4)

The dissolution kinetics Calcium-Magnesium-carbonate within HCl has been measured and isshown in the graphs below. The data in the tables have been extracted from the graphs. Thereaction is

4HCl + CaMg(CO3)2 → Mg2+ + Ca2+4Cl− + 2CO2 + 2H2O (7.51)

96


Determine the reaction order with respect to HCl?DATA:Run1, 4N HClCHCl 4.0000 3.9993 3.9986 3.9980 3.9968time [min] 0 2 4 6 8Run2, 1N HClCHCl 1 0.9996 0.9991 0.9986 0.9980time [min] 0 2 4 6 8

Figure 7.6: Data for example (Fogler5-4).

SOLUTIONAssume a rate law with reaction order q :

−rHCl = kCqHCl (7.52)

This is a batch reactor, so the design equation for a constant volume system is

−dCHCldt

= kCqHCl (7.53)

Linearising:

ln

(−dCHCldt

)= ln(k) + qln(CHCl) (7.54)

Thus need to estimate the reaction rate at t = 0 in each case. This can be done with a forwarddifference formula since the data is equally spaced;

dCCCldt

=−3Ct=0HCl + 4Ct=2HCl − Ct=4HCl

2∆t; ∆t = 2 (7.55)

which yields

CHCl,0[mol.L−1] 1 4 2 0.1 0.5

−dCHCldt

[mol.L−1.min−1] 1.74E-4 3.50E-4 2.49E-4 0.66E-4 1.36E-4

Regressing these to get the reaction order: q = 0.45 and Fig 7.7 shows that the model fits thedata well.

97

7.4. METHOD OF HALF LIVESCHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)

10

10

10

-5

-4

-3

10 10 10-1 0 1

µ ¡dCA

dt

¶ 0

[mol:L¡1:min¡1]

Figure 7.7: Linearised initial rate plot for the dissolution process.

7.4 Method of half lives

The half life of a reaction is the time it takes to reduce the conversion (concentration) by halfthe initial value. Thus a irreversible reaction, A → products with a qth order rate law in abatch reactor the design equation is;

−rA = −kCqA −dCAdt= kCqA (7.56)

Integration yields the reaction time to reach the concentration CA;

t =1

k(q − 1)

(1

Cq−1A

−1

Cq−1A0

)=

1

kCq−1A0 (q − 1)

[(CA0CA

)q−1− 1

](7.57)

Using the half-life concept : t = t1/2 then CA =1

2CA0, thus

t1/2 =2q−1 − 1k(q − 1)

(1

Cq−1A0

)(7.58)

or using a general concept, the nthlife time, such that t = t1/n when CA =1

nCA0 yields

t1/n =nq−1 − 1k(q − 1)

(1

Cq−1A0

)(7.59)

Linearising this equation

ln(t1/2) = ln

(2q−1 − 1k(q − 1)

)+ (1− q)ln(CA)

The slope of the plot is (1− q) from which the reaction order can be obtained as shown in Fig7.8.

98

7.5. DIFFERENTIAL REACTORS (INCLUDING CSTR AND RECYCLE REACTORS)CHAPTER 7. COLLECTION AND DATA ANALYSIS (FOGLER CHAPTER 5)

Figure 7.8: Half-life analysis plot.

7.5 Differential reactors (including CSTR and recycle reac-tors)

The idea of these kinds of reactors is to directly calculate the reaction rate and not have todifferentiate the concentration-time data. This reduces the error. These reactors also operatewith no pressure drop and isothermally as a result of the reaction zone and conversion beingsmall. There are essentially 2 types of differential reactors :

Fixed bed (plug flow) : In these reactors the conversion in the reactive zone is small. Thefeed enters the reactor at various degrees of conversion which is achieved by dilution withproduct species. In this way the reaction rate can be estimated at different conversions.The flow rate of reactants to the reactor remains largely unchanged.

CSTR’s and recycle reactors : The CSTR delivers the reaction rate at any concentration di-rectly from the measured variables. The conversion is changed by changing the feed rateand not the feed conversion as is needed for the fixed bed reactor. A recycle reactor (noseparation of the products must take place) can be assumed to have the same analysis asa CSTR when the recycle ratio is greater than 20, which translates into a conversion perpass of <5%.

These type of experiments are not generally carried out in Batch type reactors, since it is difficultto precisely define the conditions at t = 0. These ideas can be viewed in Fig 7.9.

99


Figure 7.9: Differential reactor operation : concepts.

7.5.1 Mole balances : CSTR (and recycle reactor with rapid recycle)

−rA[mol.L−1.s−1] =FA0 − FAV

=FA0VX =

ν0CA0VX =

CA0τX (7.60)

−rA[mol.kg−1.s−1] =FA0 − FAW

=FA0WX = (WHSV )CA0 (7.61)

So CSTR’s yield the reaction rate directly at any concentration or conversion of the reactantsor products (even for multiple reactions). This is the ideal reactor configuration to use for thedevelopment of rate laws. However, it is very difficult to achieve good CSTR operation, andgreat care must be taken to ensure that the mixing is adequate such that the CSTR assumptionsare valid. The variation in reaction rate is obtained by changing the volume (V,W ), the flowrate (FA0) or more precisely the τ,WHSV .

Fig 7.9 shows the concepts of how a recycle system relates to a real integral reactor. Essentiallythe catalytic zone (reactive zone) represents a slice of the normal reactor. When this reactorsection of the reactor is placed within a recycle loop in which the recycle flow rate exceeds 20times the feed flow rate, the reactor feed will no longer be pure feed, by will be mixed withproducts of the reaction and will thus enter partially converted. The high flow rates through thereactor section ensure that (i) the conversion is low <5% (ii) the film mass transfer limitationsare negligible. It thus provides a ideal reactor for studying the rate law of reactions.

100


7.5.2 Mole balances : Tubular and packed bed systems

Fig 7.9 also shows that the differential reactor is also a slice of the full scale reactor. However,in this reactor the feed must be pre-mixed with products to yield feeds that reproduce theconversion profile of the full-scale reactor. For complex reactions this proves to be impossible.In practice for complex reaction systems, the experimental system operates with two reactorsin series, the first one large enough to generate the conversion desired to be used in the seconddifferential reactor. These systems are not popular for the study of complex chemical reactionsystem kinetics.

The mole balances are best viewed through the first order approximation to the PBR designequation, thus;

FA|Z − FA|Z+∆Z − rA∆W = 0;dFAdW

= ν0dCAdW

= −rA; FA0dX

dW= rA (7.62)

Now making a first order approximation of the differential to yield the average reaction ratewithin the differential interval ∆W is

−rA =FA,out − FA,in∆W

= ν0CA,out − CA,in

∆W= FA0

Xout −Xin∆W

(7.63)

Here the conditions FA0 and CA0 are reserved for the flow rate and concentration of the un-converted feed and correspond to X = 0. The reaction rate so evaluated represents, to a

first approximation the reaction rate at the middle of the ∆W interval, i.e. atFA,out + FA,in

2;

CA,out + CA,in2

,Xout +Xin

2. It is not the reaction rate at the inlet to the differential reactor as

proposed by Fogler, i.e. draw a parallel with the trapazoidal integration where the integrationuses the average value between the end points of the interval if the integration step.

7.5.3 Example : Fogler 5-5

The formation of methane from carbon monoxide(A) and hydrogen(B)

CO + 3H2 → CH4 +H2O

A+ 3B → C + 2D

is being studied at 260°C in a differential reactor where the concentration of methane leavingthe reactor is measured.

(a) From the data supplied determine the conversion for each experiment and the partial pres-sures of CO and H2 at the inlet and outlet of the reactor. Determine also the averagereaction rate of CO across the reactor.

(b) The reaction rate equation is proportional to the partial pressure of CO with the functionf (CO) and proportional to the partial pressure of H2 with the function g(H2);

−rCO = f (CO) · g(H2)

Determine the reaction order with respect to PCOby assuming the rate equation;

−rCO = k1P aCO

101


(c) Show that the data suggests a rate law for the hydrogen dependence of

−rCO =k2P

b1H2

1 +KH2Pb2H2

(d) By combining the two rate laws, yields the overall rate equation

−rCO =kPCOP

b1H2

1 +KH2Pb2H2

Regress this equation for the constants k, b1, b2, KH2.

DATA:

Table 7.2: Experimental reaction data for the conversion of CO to CH4.Run PCO[atm] PH2[atm] CCH4[mol.L−1]×1041 1 1 1.732 1.18 1 4.403 4.08 1 10.04 1 0.1 1.655 1 0.5 2.476 1 4.0 1.75

The exit volumetric flow rate was maintained at 300 [L.min−1]. The catalyst bed contained10 g of catalyst. PCO and PH2were measured at the reactor inlet and CCH4 was measured at thereactor exit.

SOLUTION(a)

Looking at the data

(i) There are 2 sets of data, runs 1-3 are at constant PH2 and runs 1,4-6 are at constant PCO.Thus the CO and H2 dependence can be analysed separately

(ii) The data required will require a stoichiometric table to relate all the quantities to each other.Make the usual assumptions : ideal gases, isothermal reactor, ...

Design equation for the reaction rate of CO is

−rCO = FA0Xout∆W

∝ f (PCO) · g(PH2)

Thus it will be necessary to determine the conversion,X, the molar flow rate of CO, FA0 andthe average partial pressures of CO and H2 across the reactor.

Stoichiometric table gives

A B C Din FA0 FB0 0 0

out FA0(1−X) FA0

(FB0FA0− 3X

)FA0X 2FA0X

total F0 = FA0 (1 + α) FT = FA0 (1 + α−X) α =FB0FA0=PB0PA0

102


From the data, the methane concentration gives the reaction rate, and since the exit volumeflow rate (ν) is given, the molar flow rate of methane can be calculated from

FC = νCCH4 = FA0X

The conversion can be obtained by looking at the methane concentration

CCH4 =FCFT

P

RT=

X

1 + α−XCT

Re-arrangement gives the conversion

X =CCH4(1 + α)

CCH4 + CT

Note that the total presssure P and thus the total concentration, CT is not constant since,P = PA0+PB0 and varies from run to run. Thus FA0 can be obtained from the above equation.The partial pressures of CO and H2 at the reactor exit are given by

PA =1−X1 + α−X (PA0 + PB0); PB =

α− 3X1 + α−X (PA0 − PB0)

and the reaction rate of CO can be obtained from the above equation. Making a new table 7.3.

Table 7.3: Processed reaction data.Run PA,in PB,in α P CT X FA0 PA,out PB,out −rCO PA,ave PB,ave1 1.00 1.00 1.000 2.00 0.0457 0.0075 6.88 0.996 0.981 0.0052 0.998 0.9902 1.18 1.00 0.847 2.18 0.0498 0.0161 8.16 1.171 0.951 0.0132 1.175 0.9753 4.08 1.00 0.245 5.08 0.116 0.0106 28.2 4.071 0.877 0.0300 4.075 0.9384 1.00 0.10 0.100 1.10 0.0251 0.0071 6.90 0.999 0.079 0.0049 0.999 0.0895 1.00 0.50 0.500 1.50 0.0342 0.0107 6.90 0.996 0.471 0.0074 0.998 0.4856 1.00 4.00 4.000 5.00 0.114 0.0076 6.86 0.993 3.983 0.0052 0.996 3.99

PA is relatively constant, however the biggest variation (still not much) is in PB across thereactor even when PB0 is a constant, since 3 moles of B required for every mole of A that reactsaway.

(b)

Regression for k1, a :ln (−rCO) = ln(k1) + a · ln(PCO)

thus k1 = 0.00745 and a = 1.03. Fig 7.10 shows that the data scatter is large and too fewpoints!!

103


10

10

10

-3

-2

-1

10 10 10-1 0 1

PCO [atm]

ln(¡r CO)[mol:L¡1:min¡1]

Figure 7.10: Regression plot for the −rCO dependence on PCO.

(c)Looking at the second table and the rate law

−rCO =k2P

b1H2

1 +KH2Pb2H2

and making a plot of the data, Fig 7.11.

0.0045

0.0050

0.0055

0.0060

0.0065

0.0070

0.0075

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

PH2 [atm]

ln(¡r CO)[mol:L¡1:min¡1]

Figure 7.11: Plotting the −rCO dependence on PH2.

(i) plotting the data shows that at low P the rate is low,it then increase as PH2 increases, goesthrough a maximum and then decreases again.

104


(ii) At low PH2: then 1 � KH2Pb2H2

and the rate law reduces to : −rCO = k2P b1H2. Thus asPH2 increases, so does the reaction rate.

(ii) At high PH2: then 1� KH2Pb2H2

and the rate law reduces to −rCO =k2KH2P b1−b2H2

. If b1−b2 < 0then increasing the pressure will decrease the reaction rate.

The data is rather few, but the trend of the rate law obeys the rate data at least conceptually.

(d)

Regressing all the data,use a non-linear regression for all the constants

min obj(k, b1, b2, KH2) =

6∑i=1

(−rCO,i −

kPCO,iPb1H2,i

1 +KH2Pb2H2,i

)In Scilab this looks like this

//least squares regression for all the parametersfunction f=model(p,n)

k=p(1);b1=p(2);b2=p(3);Kh2=p(4);//b1=0.5;b2=1;f=rate-k*Paa.*Pbb^b1./(1+Kh2*Pbb^b2)

endfunctionAnd calling the least squares solver with an initial guess

p0=[0.02 0.5 1 1];[p v info]=lsqrsolve(p0,model,n)disp(info,’info’);disp(norm(v),’lsqrE’);disp(p,’p’)k=p(1);b1=p(2);b2=p(3);Kh2=p(4);rate_m=k*Paa.*Pbb^b1./(1+Kh2*Pbb^b2) //predict the reaction rate from

parameters

The parameters are p = [k2, b1, b2, KH2] = [0.01135, 0.33, 1.115, 0.510] with a LSQRerror of0.0050 and a variance of σ = [0.79, 26, 46, 106] i.e p = p ± σ. The fit of the model to thereaction rate is shown inFig 7.12. It can be seen that the reaction rate at constant PH2 is notwas well predicted as the variation with PH2. This is already noted in the scatter when fittingfor the reaction order of CO. Furthermore the variance in the model parameters is very high,in fact the model parameters are useless, since the variance exeeds the parameter value.

The rate law then is

−rCO =0.01135PCOP

0.33H2

1 + 0.510P 1.115H2

Also the reaction orders are non-integer values, which from mechanistic considerations is notrealistic. Typically reaction orders are [0, 0.5, 1, 1.5, ...]

With this in mind and the fact that there are too many parameters for such few data, roundingthe orders of the reactions off to the nearest accepable value, yields

−rCO =kPCOP

0.5H2

1 +KH2PH2

Now regressing again yields k = 0.0287 and KH2 = 2.90 with a LSQRerror of 0.0050 and avariance σ = [0.0484, 3.79] which is still very high, but at least better then before. Fig 7.13shows the fit of the reaction rate.

105


0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

PH2 [atm]

¡r CO[mol:L¡1:min¡1]

Figure 7.12: The prediction of the reaction rate when regressing for all the rate law constants.

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

PH2 [atm]

¡r CO[mol:L¡1:min¡1]

Figure 7.13: The prediction of the reaction rate when regressing for k, KH2 in the rate law.

This yields the final rate law

−rCO =0.0189PCOP

0.5H2

1 + 1.48PH2

The rate law is OK, just there are too few data points ot make these values reliable. Note, thatthe LSQRerror does not change much between the last 2 rate laws. There are not enough datapoints to accurately evaluate the reaction orders.

The Scilab code “FoglerEx5-5.sce” does all the calculations.

106

Chapter 8

Developing rate laws from reactionmechanisms and reaction pathways

107

Chapter 9

Bio-reactor engineering mechanisms

108

Chapter 10

Bio-reactor design

109

Chapter 11

Analysis of reactor flow patterns onreactor performance (Fogler chapter 13)

110

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