CHP 445 Notes ver1.pdf

7/23/2019 CHP 445 Notes ver1.pdf

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Moi University

School of Engineering

Department of Chemical & ProcessEngineering

Osembo S Otieno

Office: T 76

Ext.: 496

CHP 445:

PROCESS MODELLING &

SIMULATION


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CHP 445: Process Modelling & Simulation

Course Content

Introduction: definition of a process model, model of a typical system; strategy for modeldevelopment; classes of models; procedure for model building. Physico-chemical (functional)

models.

Numerical solution techniques for system of algebraic equations, ordinary differential equationsand partial differential equations.

Numerical simulation of a process system using one of the programming languages (e.gPASCAL, FORTRAN, C++).

Computational simulation of chemical processes using the softwares (e.g. ASPEN PLUS,FLUENT, PROSIM).

Course Plan

Week 1 Registration

Week 2 Introduction, classes of models, model of a typical system

Week 3 Strategy for model development, procedure for model building

Week 4 Physico-chemical models

Week 5 CAT I

Week 6 Numerical solution techniques for algebraic equations and polynomials

Week 7 Numerical solution techniques for ODE and PDE

Week 8 CAT II

Week 9 Computational simulation packages review

Week 10 Numerical simulation of a process system using programming languageWeek 11 Presentation of assignment

Week 12 Presentation of assignment

Week 13 Revision

Assessment:

S/No Item Mark, %

1 Semester Exam 402 CAT I & II 203 Assignments 204 Class participation 20

List of Useful Books

1. Process Modeling, Simulation & Control for Chemical EngineersW.L. LuybenInternational Student Edition

McGraw-Hill, London


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2. Process Dynamics: Modelling, Analysis & SimulationB. Wayne Bequette

Prentice Hall PTR, New Jersey 07458, 1998ISBN 0-13-206889-3

3. Process Modeling

Morton M DennLongman, 1986

ISBN 0-582-00556-6

4. Process Modelling & SimulationR.W. Gaikwad, Dr. DhirendraCentral Techno Publications, Nagpur, 2003

ISBN 81-87316-71-3

5. Chemical Process Modelling and Computer SimulationAmiya K. Jana

Prentice-Hall of India Private Ltd, New Delhi, 2008

ISBN 978-81-203-3196-9

6. Problem Solving in Chemical Engineering with Numerical MethodsMichael B. Cutlip, Mordechai ShachamPrentice Hall PTR, London, 2000

ISBN 0-13-862566-27. Numerical Methods for Engineers 3rdEd.

Steven C. Chapra, Raymond P. Canale

McGraw-Hill, Boston, 1998ISBN 0-07-010938-9


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Chapter One

Introduction

Engineers, particularly process engineers, are symbolic analysts1. Process engineers use fundamental

scientific principles as a basis for mathematical models that characterise the behaviour of a chemicalprocess. Symbols are used to represent physical variables, such as pressure temperature or concentration

Input information is specified and numerical algorithms are used to solve the models (simulating a

physical system). Process engineers analyse the results of these simulations to make decisions or

recommendations regarding the design or operation of a process.

Process engineers are responsible for technical troubleshooting in the day-to-day operations of a

particular chemical process. Some are responsible for designing feed back control systems so thatprocess variables (such as temperature or pressure) can be maintained at desired values. Others may be

responsible for redesigning a chemical process to provide more profitability. All these require anunderstanding of time-dependent (dynamic) behaviour of a chemical process.

Working definition: Process Model

A Process model is a set of equations (including necessary input data to solve the equations) that allows one topredict the behaviour of a chemical process system.

It is presumed that each variable appearing in the equations of the model can be identified with an entityassociated with the process; each entity must be measurable, at least in principle. A quantity that can

never be measured in principle has no physical meaning.

1.1 Classification of Models

There are three identifiable methodologies used to obtain the equations for a mathematical model. These

can be categorised as follows:

1. Fundamental: Use offundamentalor first principles models, based on known physical-chemicalrelationships. This includes the conservation of mass, conservation of energy, reaction kinetics,

transport phenomena, and thermodynamic relationships.2. Empirical: Use direct observations to develop equations that describe the experiments. An

empirical model is simply an equation that records the relationship between system inputs and

outputs. An empirical model might be used if the process is too complex for fundamental model.

3. Analogy: Use the equations describing a system believed to be analogous, with variablesidentified by analogy on one-to-one basis. The essence of modelling by analogy is identifying a

1 Symbolic analysts solve, identify and broker problems by manipulating symbols. They simplify reality into abstract

images that can be rearranged, juggled, experimented with, communicated to other specialists, and then, eventually,

transformed back into reality. The manipulations are done with analytic tools, shaped by experience. The tools may be

mathematical algorithms, legal arguments, financial gimmicks, scientific principles, psychological insights about how to

persuade or amuse, systems of induction or deduction, or any other set of techniques for doing conceptual puzzles. (italics

added for emphasis)


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well-understood process that seems to have the essential features of the process of interest. This

form of modelling is too specialised and intuitive.

Generally, the preferred models are those based on fundamental knowledge of chemical-physical

relationships. Fundamental models will generally be accurate over a much larger range of conditions

than empirical models. Empirical models, also known as black box models, may be useful forinterpolation but are generally not used for extrapolation; i.e., an empirical model will only be used

over the range of conditions used for the fit of the data.

As a result focus will be on fundamental models (also known as theoretical models), with particularattention to the logical structure of model development and simplification (system analysis). However, it

is important to note that elements of empiricism and analogy, in even the most fundamental models, are

found. This presence is a major factor in the process of validation.

In addition to the above classification, models can generally be grouped according to;

Linear/non-linear

Steady state/unsteady state Lumped parameter/distributed parameter Continuous/discrete variables

Linear vs nonlinear models. If the output,y, of a subsystem is completely determined by the input, x

the parameters of the subsystem and the initial and boundary conditions, in general sense can berepresent the subsystem symbolically by

( )xHy= 1.1

whereHrepresents any form of conversion ofxintoy.

Suppose that two separate inputs are applied simultaneously to the subsystem so that

( ) ( ) ( ) 212121 yyxHxHxxHu +=+=+= 1.2

His then, by definition a linear operator. Operations involving inverse, square, exponential and natural

logarithm are plotted in Fig. 1.1. It can be seen that all of them are nonlinear operators, especially forsmall values of independent variable. Therefore, equations are linear if the independent variables or their

derivatives appear only to the first power otherwise they are nonlinear. A system is termed linear if its

operator H is linear and the model of a linear system, which is represented by linear equations and

boundary conditions, is called linear model. Otherwise the model is nonlinear.


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Fig. 1.1. Plot of some nonlinear operations (inverse, square, exponential and logarithm)

Steady state vs unsteady state. Steady state means that the accumulation terms in the various balances

of interest are zero. In each balance, if the boundary conditions are time independent, the dependent

variables within the system can gradually reach constant values with respect to time at a given point.

Standard Chemical Engineering design techniques for unit operation, reaction kinetics and so on have

been dealt entirely with steady state operations. When process control began to be extensivelyconsidered, it was found that non steady state operations were of significance. To design a plant on the

basis of steady state information and then to add controls afterwards is now felt to be inadequate; both

the units and control system should be designed together.

A typical example of unsteady state process might be the start-up of a distillation column, which wouldeventually reach a steady state set of operating conditions. In fact, when examined in detail, the column

always will prove to be operating in the unsteady state with minor fluctuations in temperature,composition, etc, taking place all the time, but possibly ranging about average steady state values.

Distributed vs lumped parameter. A lumped parameter representation means that spatial variations areignored and the various properties and the state (dependent variables) of the system can be considered

homogeneous throughout the entire system. A distributed parameter on the other hand, takes into

account detailed variations in behaviour from point to point throughout the system. All the real systems

are of course, distributed in that there are some variations throughout them. Many times, however, thevariations are relatively small, so they may be ignored and the system may then be lumped.

y = x-1

R2= 1

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0 5 10 15 20 25

y = x 2

R2= 1

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

y = e x

R2= 1

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8

y = Ln(x)

R2= 1

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25


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It is difficult to decide whether to lump a parameter, but the rule of the thumb is that if the response of

the element is for all practical purposes instantaneous throughout the element, then the element

parameter can be lumped. If the response shows instantaneous differences along the element, then itshould not be. By response is meant the velocity of propagation of the input through the element.

Continuous vs discrete variables. Continuous means that a variable can assume any value within aninterval, discrete means the variable can take only distinct values in the interval.

Modelling can be valuable because it is an abstraction and help avoid repetitive experimentations and

observations. However, the potential cost and time savings must be weighed against the fact that themodel only imitates reality and does not incorporate all features of the real system being modelled.

1.2 How Models are used

Given a set of input data, a model is used to predict the output response. A model is used to solve the

following types of problems:

Synthesis, what process can be used to manufacture a product? Design, what type and size of equipment is required to produce a product? Operation, what operating conditions will maximise the yield of a product? Control, how can a process input be manipulated to maintain a measured process output at a

desired value?

Safety, if an equipment failure occurs, what will be the impact on the operating personnel andother process equipment?

Environment, how long will it take to biodegrade hazardous waste? Allocation, if there are several sources of raw materials, and several manufacturing plants, how

can the raw materials be distributed among the plants, and what products can each plant

produce? Marketing, if the price of a product is increased, how much will the demand decrease?

Many of the models cited above are based on a steady-state analysis. Previously, chemical process

design was based solely on steady-state analysis. However, it is important to consider the dynamic

operability characteristics of a process during the design phase. Also, batch processes that are commonlyused in the pharmaceutical or specialty chemical industries are inherently dynamic and cannot be

simulated with steady-state models.

Mathematical models consist of the following types of equations (including combinations)

Algebraic equations Ordinary differential equations (ODEs) Partial differential equations (PDEs)

The ODEs generally result from macroscopic balances around processes, with assumption of a perfectly

mixed system. To find the steady-state solution of a set of ODEs, then a set of algebraic equations has to

be solved. PDE models result from microscopic balances.


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Chapter Two

Constructing a Model

2.1 Model Building

Model building can be divided into four phases

Problem definition and formulation Preliminary and detailed analysis Evaluation phase Application

Fig. 2.1. Steps in model building

Problem definition and formulation phase. In this phase the problem to solved must be defined and

important elements that pertain to the problem and its solution identified. The degree of accuracy needed


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in the model and the potential uses of the model must be determined. One must also evaluate the

structure and complexity of the model and ascertain

Number of independent variables to be included in the model Number of independent equations required to describe the system Number of unknown parameters in the model

The fundamental physical and chemical laws are used in their general form with time derivatives

included in order to study dynamics of the system. Reasonable assumptions are made to simplify themodel without which the model could be too complex that would take a long time to develop and might

be impossible to solve. The assumptions that are made should be carefully considered and listed. They

pose limitations on the model that should always be kept in mind when evaluating its predicted results.

It is usually a good idea to make sure that the number of variables equals the number of equations. This

means that degree of freedom of the system must be zero in order to obtain a solution. Checking to see

that the units of all terms in the equations are consistent is essential particularly the time units ofparameters in dynamic models. A sketch of a logical flow diagram for modelling is shown in Fig. 2.2.

Before carrying out actual modelling work, it is important to evaluate the economical justification for

the effort of modelling and the capacity of the supporting staff for carrying out such a project. The

available solution techniques and tools must be kept in mind as a mathematical model is developed.

Design phase. This phase involves specification of the information content, general description of the

programming logic and algorithms necessary to develop and employ a useful model, formulation of the

mathematical description of such model and simulation of the model.

First define input and output variables and determine the system. Also select the specific mathematical

representation to be used in the model, as well as the assumptions and limitations of the model resultingfrom its translation into actual computer code. Specify computer input/output media, develop program

logic and flow-sheets and define program modules and their relationships. Use of existing subroutines

and databases saves a lot of time.

Evaluation phase. This phase is intended as a final check of the model. Testing of individual models

elements should be conducted during the earlier phases. Evaluation of model is carried out according to

the evaluation criteria and test plan established in the problem definition phase. Next carry outsensitivity testing of model inputs and parameters and determine if the apparent relationships are

physically meaningful. Use actual data in the model when possible. This step is also referred to as

diagnostic checking, and may entail statistical analysis of the fitted parameters.

Model validation consists of three parts

Validation of logic Validation of model assumptions Validation of model behaviour


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variables (time and threes spatial) coordinates and model inputs. The model inputs play a different role

in the analysis of the model response.

Modelling of physical systems will always require the application of one or more of the fundamental

conservation principles: conservation of mass, momentum and energy. These quantities are known as

thefundamentaldependent variables.

Apart from mass, the fundamental variables are not measurable, even in principle. Energy is measured

in terms of temperature, pressure, composition, velocity etc. Similarly, the momentum is computed

from measured velocities and masses. It is this larger collection of measurable variables thatcharacterises the fundamental quantities, and the model equations are written in terms of thesecharacterising dependent variables. It is necessary to select the minimum set of characterising

variables2that uniquely defines the fundamental variables. This set defines the stateof the system.

2.3Constitutive Equations

In modelling the fundamental variables do not provide enough equations in the model to solve for all thestate variables. There are other relationships required so as to make the model completely defined, i.e

one that has "as many equations as unknowns"3. These required relationships are known as constitutive

equations.

Constitutive equations are those additional relationships between state variables that are required for acomplete mathematical description. Constitutive equations are usually associated with molecular

phenomena.

Constitutive equations come in most cases from experiment, usually guided by some theory and perhapsdimensional analysis or other invariance arguments. Many constitutive equations are available in the

form of dimension less engineering correlation e.g. ( )Pr,GrRe,NuNu= . Several examples ofconstitutive equation are shown in this section.

2.3.1 Gas Law

Process systems containing a gas will normally need a gas law expression in the model. The ideal gaslaw is commonly used to relate molar volume, pressure and temperature:

RTPv= 2.1

The van der Waals PvT relationship contains two parameters (a and b) that are system specific:

2There is no generally accepted terminology called characterising variables. They are often called state variablesin control

and systems engineering literature. However, it is important to note that the term state variable has entirely different

meaning in the thermodynamics literature.3This is convenient and common shorthand. It is not a rigorous equivalent, as the counter example of finding real solutions

ofxand yto the single equation 0yx 22 =+ illustrates.


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( ) RTbvv

aP

2 =

+ 2.2

For other gas law, see a thermodynamics text.

2.3.2Chemical Reactions

The rate of reaction per unit volume is usually a function of the concentration of the reacting species

Consider the reactionA+2B C+3D. If the rate of the reaction ofAis first order in bothAandB, thefollowing expression is used:

BAA ckcr = 2.3

The reaction rates are normally expressed in terms of generation of a species. As a result we have

BAAD

BAAC

BAAB

ckc3r3r

ckcrr

ckc2r2r

==

==

==

Usually, the reaction rate coefficient, k, is a function of temperature. The most commonly usedrepresentation is the Arrhenius rate law

( ) RTEAeTk = 2.4

The frequency factor (pre-experimental factor) A, and activation energy, E can be estimated from the

date of the reaction constant as a function of reaction temperature.

2.3.3 Equilibrium Relationships

The relationship between the liquid and vapour compositions of component i, when the phases are in

equilibrium, can be represented by:

iii xKy = 2.5

The equilibrium constant, Ki, is a function of composition and temperature.

To simplify the vapour/liquid equilibrium models, a constant relative volatility assumption is oftenmade. In a binary system, the relationship between the vapour and liquid phases for the light componen

often used is

( )x11x

y+

=

2.6


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generally described by PDEs while lumped systems are usually described by ODEs (or algebraic

equations if changes in time are not of interest).


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Chapter Three

Process Modelling

3.1 Balance Equations

The dynamic balances material and energy balances are in the form

=

systema

leavingenergy

ormassofrate

systema

enteringenergy

ormassofrate

systemain

onaccumulatienergy

ormassofrate

3.1

The rate of mass accumulation in a system has the form dm/dtwhere mis the total mass in a system

Similarly the rate of energy accumulation has the form dE/dtwhereEis the total energy in a system. Ifni is used to represent the moles of component i in a system, then dni/dt represents the rate of

accumulation of component iin the system.

When developing a dynamic model, one of two general viewpoints can be taken. One viewpoint is

based on integral balance while the other is based on instantaneous balance. Integral balances are

particularly useful when developing models for distributed parameter systems, which result in PDEs

Another viewpoint is the instantaneous balance where the time rate change is written directly.

3.1.1 Integral Balances

An integral balance is developed by viewing a system at two different snapshots in time. Consider afinite interval,t, and perform the balance over that time interval.

( ) ( ) ( )

+

+

=

+ tttotfrom

systemleaving

energyormass

tttotfrom

systementering

energyormass

tat

systeminside

energyormass

ttat

systeminside

energyormass

3.2

The mean-value theorems of integral and differential calculus are then used to reduce the equations todifferential equations.

ExampleConsider a tabular reactor where a chemical reaction changes the concentration of the fluid as it moves

down the tube. A volume element V and a time element t is used. The total moles of species A

contained in theVis (V)cA. The amount of speciesAentering the volume is VAFc and the amount

of species leaving the volume is VVAFc + . The rate of A leaving by reaction (assuming a 1st order

reaction) is ( ) VkcA .


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The material balance is then

( ) ( ) ( )[ ]dtVkcFcFccVcVtt

t AVVAVAtAttA

+

++ =

3.3

The R.H.S. of (3.3) can be written using the mean value theorem of integral calculus, as

( )[ ] ( )( ) tVkcFcFcdtVkcFcFc ttAVVAVAtt

t AVVAVA

++

+

+ = 3.4

where 10


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=

ktanofoutwater

offlowratemass

ktanointwater

offlowratemass

ktaninwaterofmass

ofchangeofrate 3.16

The total mass of water in the tank is V , the rate of change is ( ) dtVd , and the density of the outletstream is equal to the tank contents:

( ) FFVdt

dff = 3.17

Assuming that the density is constant:

( ) FFVdt

df = 3.18

State variable is V, Ffand Fare input variables. If the density is retained, then it is the parameter of the

system in order to solve this problem the inputs Ff(t) and F(t) and the initial condition V(0)must bespecified.

3.2 Material and Energy Balances

Many chemical processes have important thermal effects so it is necessary to develop material and

energy balance models. One key is that a basis must always be selected when evaluating an intensive

property such as enthalpy.

Proper application of conservation of energy requires the use of some basic thermodynamic concepts

There is no way that the proper use of thermodynamics can be avoided when dealing with the energy of

a system. There are attempts to bypass rigour and substitute intuition with a result of incorrecequation.

3.2.1 Thermodynamic Variables

Total energy,E, of a system is the sum of its kinetic, potential and internal energies abbreviated as K.E.P.E. and Urespectively. Thus:

UPEKEE ++= 3.19

Energy per unit mass will be denoted with an underbar:

UPEKEE ++= 3.20

Quantities per unit mole will be denoted with a double underbar:

UPEKEE ++= 3.21


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In flowing systems, it is often more convenient to work in terms of enthalpy,H, defined as

PVUH += 3.22

where Vis the volume and Pthe mean pressure. Enthalpy per unit mass is then

PUH += 3.23

and enthaly per unit mole is

PMUH W+= 3.24

where WM is the average molecular weight. For an ideal gas

RTUH += 3.25

Thermodynamic state is specified by the composition and two of the three characterising variables

(pressure-temperature-volume, PVT). Internal energy is usually defined in terms of the volume andtemperature, while enthalpy is usually defined in terms of pressure and temperature.

The heat capacities at constant pressure are defined to be

ncompositio,P

p

ncompositio,P

p

T

Hc

T

Hc

=

=

3.26

The heat capacities at constant volume are defined as

ncompositio,

v

ncompositio,

v

T

Uc

T

Uc

=

=

3.27

For an ideal gas, U is a function of only T. in this case

Rcc vp += 3.28


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For liquids at moderate pressures and temperatures, cpand cvare nearly equal.

There is one more thermodynamic quantity required. Let ni be the number of moles of speciescontained in volume V. Partial molar enthalpyis defined as

ttanconsn,P,Ti ijnHH

== 3.29

From the Gibbs-Duhen equation, it can be deduced that

=i

iiHnH 3.30

or, equivalently,

=i

iiHc1H

3.31

where ciis the molar concentration of species i.

In an ideal solution, molecules of species interact with molecules of all other species in the same way as

with their own, thusi

i HH = . In a non-ideal solutioni

i HH and there will be enthalpy changes

associated with the mixing of different species.

3.2.2 Conservation of Mass and Energy

The reaction

++++ NMBA 3.32

takes place in a well-stirred tank. (There is no loss of generality in taking the stoichiometric coefficient

of A equal to unity). Because of the well-mixed assumption, the entire tank is taken as the controvolume.

Mass is characterized by the density, ; concentrations (in molar units) NMBA c,c,c,c etc ofA,B, M

N, respectively; and liquid volume, V. The volumetric flow-rate is F.

The principle of conservation of mass as applied to total mass in the system is unchanged by the fact of

chemical reaction and is identical to Eq (3.17).

( ) FFVdt

dff = 3.17


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Taking rito be the net molar rate of formation of species iby chemical reaction per unit volume. The

equation of conservation of mass for speciesAis therefore

( ) iififi VrFccFVcdt

d+= 3.33

If then ris defined as the net rate of disappearance of Aby reaction per unit volume, then Eq.(3.33) canbe written as

( )

( )

( )

( ) VrFccFVcdtd

dtdn

VrFccFVcdt

d

dt

dn

VrFccFVcdt

d

dt

dn

VrFccFVcdt

d

dt

dn

NfNfN

N

MfMfM

M

BfBfB

A

AfAfAA

+==

+==

==

==

3.34

r is often referred to as the instrinsic reaction rate.

The principle of conservation of energy applied to this control volume is

( ) ( ) ( ) Teeefffff WQPEKEUFPEKEUFPEKEUdt

d++++++=++ 3.35

The first two terms on R.H.S. are the rates of convective flow of energy in and out respectively. The

subscript "e" denotes the effluent stream despite perfect mixing the energy will be different from that of

the tank. Qis the rate of heat addition through the boundaries typically from a heating or cooling coil or

jacket, WTis the rate at which work is done on the system (i.e. power input).

Work is done on the system when fluid is forced in and is done by the system to expel the effluent

stream; the rate of the former is FfPfwhile the rate of the latter is FPewhere Pfand Peare the pressuresjust prior to the entrance and exit, respectively. It is convenient to separate out these work terms and

refer to the remaining work term as Wsfor rate of shaft work. Thus Eq (3.35) is rewritten as

s

e

e

f

f

fff WQP

UFP

UF

dt

dU++

+

+=

3.36

KEand PEterms have been dropped since they are usually unimportant if temperature changes of even

a few degrees can occur. From the definitions of enthalpy, Eq (3.23) Eq (3.36) can be written as

sefff WQHFHFdt

dU++= 3.37


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Writing the Uin terms of Hand dropping the subscript "e" on eH since the enthalpy per unit mass is

the same everywhere in the tank and will approximately equal the enthalpy of the effluent;

( ) sfff WQHFHFPVdt

d

dt

dH++= 3.38

The term ( ) dtPVd is neglected since it is negligible in liquid systems and it is zero if volume andpressure are constant; thus for liquid systems,

( ) ( ) sffff WQTHFTHFdt

dH++= 3.39

Eq. (3.39) is often quoted as the starting point in modelling and referred to as the "enthalpy balance"4.

Eq (3.39) must now be expressed in terms of state variables, the first step is to refer all enthalpies to the

same temperature, which is most conveniently taken as the tank temperature. From the definition of cp(Eq. 3.26).

+= fT

T pffff dTc)T(H)T(H 3.40

Approximating cpto be constant and writing the integral as )TT(c fpf , Eq 3.39 is now written as

( ) ( ) ( ) sffffpfff WQTHFTHFTTcFdt

dH+++= 3.41

H is a function of T, P and the number of moles of all component species { }in , and thus an implicitfunction of time. This it can be written

+

+

=

i

i

i dt

dn

n

H

dt

dP

P

H

dt

dT

T

H

dt

dH 3.42

The term PH can be shown to be negligible in most cases for liquid systems 5, and it is zero for ideal

gases. TH is simply pVc , from Eq.(3.26), while inH is the definition of iH . Thus,

4There is no such thing as "enthalpy balance" since Eq 3.39 is totally incorrect for gaseous systems. The "enthalpy balance"

is worsen by including a term to account for the "rate of enthalpy-or energy-generation because of chemical reaction".

Systems containing more than one phase cause particular problems for believers in "enthalpy balances".5It is shown in thermodynamics textbooks that

ii n,Pn,T T

VTV

P

H

=


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+=i

iipdt

dnH

dt

dTVc

dt

dH 3.43

With Eq. (3.34), the sum in Eq. (3.43) can be written as

[ ]BAMNi

ii

i

ifif

i

ii HHHHVrHcFHcFdt

dnH ++= 3.44

Finally, using Eq. (3.31), the term HFHcFi

ii = , while fffi

ififf HFHcF = ; combination of

Eq. (3.41) through (3.44) then becomes6

( ) [ ] ( ) ++++=i

iififfsBAMNfpfffp HHcFWQHHHHVrTTcFdt

dTVc 3.45

The last term in R.H.S. of Eq. (3.45) is neglected as it is small in relative to enthalpy term multiplyingVrand is zero for ideal gases.

The enthalpy term ++ BAMN HHHH is the enthalpy change of reaction, often called the

heat of reaction, and denotedR

H ;R

H is negative for exothermic reaction and positive for

endothermic reaction. Enthalpies of reaction can be calculated from tabulated "heats of formation" and

"heats of combustion". The enthalpies of reaction can be measured in a calorimeter experiment. Thefinal form of energy equation is therefore;

( ) ( ) sRfpfffp WQVrHTTcFdt

dT

Vc +++=

3.46

This equation contains a large number of approximations none of which should be serious for

liquid systems.

6 i

ififf

i

iiff

HcFHcF

[ ]

[ ][ ] ( )

[ ] ( )( ) ( ) ( )

( ) [ ] ( )

++++=

+++=

++++

+++=

+++=

+++=

i

iififfs

BAMNfpfffp

sffffpfff

i

ifiiff

BAMNfffp

i

ifiiffBAMNfff

i

ififf

i

iiffBAMN

i

ii

i

ififf

i

ififf

i

fiiff

BAMN

i

ii

i

iiff

i

ii

HHcFWQHHHHVrTTcFdt

dTVc

WQTHFTHFTTcF

HHcFHHHHVrHFHFdt

dTVc

HHcFHHHHVrHFHF

HcFHcFHHHHVrHcFHcF

HcFHcFHHHHVrHcFHcFdt

dnH


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The rate of heat transfer, Q depends on the configuration used for heating or cooling. The simples

configuration to assume is that the reactor is jacketed and that the jacket fluid is well mixed liquid

Using Eq (3.46)7and assuming that the jacket fluid is not reactive, then

( ) jjjfpjfjfjf

j

pjjj QTTcF

dt

dTcV += 3.47

If it is assumed that there is no loss of heat to the surrounding, then QQj = . Heat transfer rates vary

linearly with heat transfer area and with the temperature differences so it can be written that;

( )TThAQQ jj == 3.48

Ais the area available for heat transfer, and his the heat transfer coefficient. Many correlations exist fo

heat transfer coefficients.

Taking that the liquid volume in the jacket will not change and that cpjf is independent of temperatureand that the density is constant, then Eq (3.47) can be written as

( ) ( )jjjfpjfjfjf

j

pjjj TThATTcF

dt

dTcV += 3.49

3.3 Batch and Tubular Reactors

The batch reactor is a well-stirred reactor for which 0FFf == . Assuming a constant density, implies

that 0dtdV = and for a single reaction Eq (3.34) becomes

rdt

dc,r

dt

dc

rdt

dc,r

dt

dc

NM

BA

==

== 3.50

Tubular reactors behave like moving batch reactors if axial mixing is not taken into consideration. If the

fluid is marked over a small spatial regionzwith a tracer, that fluid element will retain its integrity as it

7Eq. (3.46) is commonly written incorrectly as

1. ( ) ( ) s

Rpfpfffp

WQVrHTFcTcFVTdt

dc +++=

2. ( ) ( ) ( ) sRfpfffp

WQVrHTTcFTVcdt

d+++=


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25

passes through the reactor. Since batch reactor equations do not depend on the size of the contro

volume,zhere is arbitrary and can be as small as possible.

This Lagrangian description can be converted to Eulerian description or a fixed laboratory coordinate

system by noting that the time required to travel distance dzis vdt, where vis the mean linear velocity

of the reactor, so vdzdt= and Eq. (3.50) becomes

== ,rdz

dcv,r

dz

dcv BA 3.51

This assumes that the system is at steady state when viewed from a fixed laboratory frame.

The energy equation for a batch reactor is obtained directly from Eq (3.46) by setting 0Ff = :

( ) sRp WQVrHdtdT

Vc ++= 3.52

For tubular reactors, the heat transfer term is first put in an appropriate form. Let va be the area

available for heat transfer per unit volume of reactor. Thus VaA v= and Eq (3.52) can be written as;

( ) ( )TTVhaVrHdt

dTVc jvRp += 3.53

The shaft work is rarely relevant in a tubular reactor. If dtis replaced by vdz , then the equation for a

tubular reactor becomes;

( ) ( )TTd

hrH

dz

dTvc jRp +=

4 3.54


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d4av = for a tube of diameter d.

The area per unit volume in the jacket of outer diameter djis ( )22j ddd4 , assuming concentricity andthin walls. The corresponding equation for the jacket is then

( )( )j22

j

j

pjjj TTdd

hd4

dt

dTcv

= 3.55

The flow is taken as counter-current if vand vjhave opposite algebraic signs, otherwise it is concurrent.

It should be noted that this derivation of the tubular reactor equations is valid only for steady state, and it

assumes that radial mixing is so rapid that there are no radial concentration or temperature gradients.

3.4 Density of Liquids

The relationship between density and concentration of a liquid system is governed by intermolecularforces. The density of a liquid mixture ofNspecies is

N21 ccc +++= 3.56

where { }ic are the concentrations of all species in mass units (e.g. kg/m3). In thermodynamics, the

Gibbs-Duhen equationestablishes that the density of a liquid mixture at constant temperature is unique

function of 1N concentrations.

Consider a large volume V of a liquid, containing nA moles of A and nB moles of B, V is uniquely

determined by nAand nB. At a constant temperature and pressure a differential amount ofAis added and

the differential volume change measured; this experiment defines thepartial molar volume, AV as


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A

B

n,P,TB

B

n,P,TA

A

nVV

n

VV

=

=

3.57

AV and BV are intensive properties and depend only on the molar ratio nB/nA. It is a consequence o

the mathematics of exact differential forms or an equivalent physical argument, that

BBAA VnVnV +=

3.58

It would be convenient to define thepartial densities

B

WB

B

A

WA

A

V

M,

V

M== 3.59

The partial densities will be functions of the mass concentration ratio cB/cA. Now from Eq (3.56)

+

+=+=+=

A

B

BA

A

B

WBBWAA

BA

cc11

cc

1

V

Mn

V

Mncc

3.60

Eq. (3.60) can be rearranged as

B

B

AA c1

+=

3.61


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28

Since A and B are functions only of cB/cAor equivalently Bc , Eq. (3.61) establishes a unique but

implicit relationship between and cB.

3.5 Dimensionless Models

Most models contain a large number of parameters and variables that may differ in value by severalorders of magnitude. It is often desirable, at least for analysis purposes, to develop models composed odimensionless parameters and variables.

Just for illustration, consider a constant volume, isothermal CSTR modelled by a simple first-orderreaction

( ) AAAf

A kcccV

F

dt

dc=

Let 0AfA ccx= ; 0Afc is nominal ( steady-state) feed concentration ofA. Thus

xkVFx

VF

dtdx f +=

Now taking = tt , where t is a scaling parameter, then dtdt = ; this implies that

xkV

Fx

V

F

dt

dxf

+=

Natural choice for t appears to be FV , the residence time, so;


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xF

kV1x

d

dxf

+=

FkV is a dimensionless number known as a Damkholer number in reaction engineering.

Assuming that the feed concentration is constant; 1x f = and letting FkV= , the

xx1d

dx

=

Therefore a single parameter can be used to characterise the behaviour of all first order, isothermalchemical reactions.


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4.Numerical Methods

4.1 Introduction

Process modelling leads to a system of complex model equations. It is necessary to solve the equations

in order to investigate the process characteristics. There are two ways of finding solutions, namely,analytical method and numerical method. Where possible an analytical method produces exact solutions

usually in the form of general mathematical expressions. On the other hand numerical methods produce

approximate solutions in the form of discrete values or numbers.

In order to develop a dynamic process simulator after the mathematical model development it is

imperative to have a good knowledge of numerical methods. Some of these methods will be covered in

this chapter.

4.2 Iterative Convergence Methods

4.2.1 Bisection Method (Interval Halving)

In order to solve =() = 0 4.1the following steps according to the Bisection method can be followed

Step 1: Find two guess values of (say 1and 2at the 1stiteration), so that one where() < 0and another where() > 0.Step 2: Find the midpoint and then evaluate

(

)at that midpoint.

Step 3: Among the two guess values of , one should be replaced by the value of at midpointReplace the bracket limit that has the same sign as the function value at the midpoint,with the midpoint value.Step 4: Check for convergence. if not converged, go back to Step 2.

If the interval shrinks below a tolerance level, an approximate value of the root has been found. The

Bisection method is also known as interval halvingmethod since it can be halve the size of the interval

in each iteration.

The Bisection method actually locates a root repeatedly narrowing the distance between the two guesses

When an interval contains a root, this simple numerical method never fails. However, the maindrawback of the Bisection technique is the slow convergence rate. Also, it is not easily extended to

multivariable systems.

4.2.2 Secant Method

Although this approach is similar to the Bisection method, it is required to construct a secant line and

find its -intercept as the next root estimate.


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To solve Eq. 4.1, 1and are assumed as the two approximations to the root. Constructing a straightline (secantor chord) through points [1 ,(1)]and [ ,()]gives the slope as

=() (1)

1 4.2

To compute the next approximation +1, a straight line equation is formed, thus(+1) () =(+1 ) =() (1) 1 (+1 ) 4.3For finding its -intercept at(+1) = 0. Simplifying gives

+1 = 1() (1)() 4.4Further simplifying results

+1 =1() (1)() (1) 4.5The iteration is done until the guess is sufficiently close to the root. If the approximations are such that()(1) < 0, then the approach, as represented by Eq. 4.4 or Eq. 4.5, is known as the FalsePosition, or Regula Falsi method. Note that the main difference between these two convergence

techniques is that the Secant method retains the most recent two estimates, while the False Position

method keeps the most recent estimate and the next recent one which has an opposite sign in thefunction value.

4.2.3 Newton-Raphson Method

The Newton-Raphson method is the most common and popular method for solving nonlinear algebraic

equations. It is derived from Taylor series of():( +) =() + ()1! + 2()2 ()22! + 3()3 ()33! + = 0 4.6

Neglecting all terms of order two and higher, Eq. 4.6 yields:

( +) =() + () = 0 4.7That means

= ()() 4.8


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Calculating the guess for at iteration + 1as a function of the value at iteration by defining+1 =+1 4.9

thus Eq. 4.8 becomes

+1 = ()() 4.10From Eqs. 4.9 and 4.10, the following equation is obtained

+1 =()() 4.11Eq. 4.11 represents the Newton-Raphson convergence method for a single-variable problem. The

extension of the Newton-Raphson algorithm to multivariable systems is fairly simple andstraightforward. Considering a multivariable system represented by:

() = 0 4.12This equation consists of a set of by variables (1, 2, , )as:1(1, 2, ,)2(1, 2, ,)

(

1,

2,

,

)

=000

4.13The Taylor series gives for eachafter neglecting the second and higher-order derivative terms as:

( +) =() +()=1 = 0 4.14

The above equation yields the following matrix form:

(

) +

= 0 4.15

where, the Jacobian matrix

=111

121 2


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From Eq. 4.15,

=1() 4.16Therefore, Newton-Raphson method for multivariable is+1 = 1() 4.17The Newton-Raphson method is very efficient iterative convergence technique compared to many other

simple methods. However, at each step, this method requires the calculation of the derivative of a

function at the reference point, which is not always easy. It also may sometimes lead to stability

problems particularly if the function is strongly nonlinear and if the initial guess is very poor.

4.2.4 Muller Method

This is an iterative convergence method based on quadratic equation. Consider a polynomial of seconddegree:

() =02 +1 +2 = 0 4.18where 0( 0), 1and 2are three arbitrary parameters. In this convergence approach, three values ofthe unknown variable are guessed. Let 2, 1and are three approximations to the actual root of() = 0. To obtain 0, 1and 2the following conditions are used:

2 =2 =022 +12 + 2 4.191

=

1=

012 +

11+

2 4.19

= =02 +1 +2 4.19and then substituting 0, 1and 2in Eq. 4.18 gives() = ( 1)( )

(2 1)(2 )2 + ( 2)( )(1 2)(1 )1 +

( 2)( 1)( 2)( 1) = 0 4.20

Eq. 4.20 can be converted to:

( + )1(1 +)2( + + 1)1 1 + ( +)( + +1)( +1) = 0 4.21where, = , = 1 and 1 =1 2. Further assuming = , =1 and = 1 +, Eq. 4.21 gets the form:

2 + + = 0 4.22


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=(, ) 4.28where(, )is, in general, a nonlinear function and the initial condition for is as: (0) =0at time

= 0. Eq. 4.28 can be solved by employing the Euler method by two different ways, namely, explicit

approach and implicit approach.

Explicit Euler approach

A forward difference approximation of Eq. 4.28 yields

= +1 =(, ) 4.29The time increment is known as the step sizeor integration interval. Rearranging Eq. 4.29 gives:

+1=

+

(

,

) 4.30

That is,

+1 = + (,) 4.31Eq. 4.30 or 4.31 represents the Explicit Eulermethod. If sufficiently small integration step size istaken, then estimate +1 will be very close to the correct value. The Euler integration approach isextremely simple to implement for solving even highly nonlinear multivariable complex systems having

a large number of ODEs.

Implicit Euler approachThis approach uses a backward difference approximation and accordingly, Eq. 4.28 gives:+1 =(+1, +1) 4.32Rearranging gives:

+1 = +(+1, +1) 4.33That is,

+1 = + ( ,) 4.34Eq. 4.33 or 4.34 represents the Implicit Euler method. This method in Eq. 4.34 indicates that the

derivative needs to be evaluated at the next step in time +1. It is simple for a linear system but fornonlinear system, a resulting algebraic expression is solved using one of nonlinear algebraic solution

technique such as Newton-Raphson method. The implicit method approach is stable for almost any


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value of and will not oscillate. The Explicit Euler method, on the other hand, may have instabilityproblem with oscillating outputs for a large step size.

4.3.2 Runge-Kutta Methods

The Runge-Kutta method is commonly chosen as a better explicit integration algorithm than thestandard Explicit Euler method due to:

truncation error per step associated with the Euler method is higher Explicit Euler technique is prone to numerical instabilities

Note that the Euler integration technique is sometimes called the 1st-order Runge-Kutta method.

2nd

-order Runge-Kutta approach

This is also known as theMidpoint Eulermethod. In this approach, first the Euler technique is employed

to predict at the midpoint of the integration interval (step size = 2 ). The value of at the end ofthe step (step size =

) is estimated as:

+1 = +2 4.35where 2 = +

21, +

21 =(, )

This integration technique provides better accuracy than the Explicit Euler method but the Euler

approach runs almost twice faster.

4

th

-order Runge-Kutta approachThe 4th

-order RK method is given by

+1 = + 6

[1 + 22 + 23 +4] 4.36where 4 =( +3, +)3 = +

22, +

2

2=

+

21,

+

2

1 =(, )Comparing the 2

nd-order and 4

th-order RK integration approaches, it is easy to observe that the

complexity as well as computational time increases with the increase of the order. To obtain greater

accuracy in estimation, the 4th

-order RK method is preferred over the Euler and 2nd

-order RK

approaches.


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4.3.3 Runge-Kutta-Fehlberg (RKF45) Method

Another efficient and popular technique for solving ODEs is the Runge-Kutta-Fehlberg 4th

-5th

-order(RKF45) method. This ODE integrator can exert some adaptive control over its own performance by

making frequent changes in its step size. At each step, the RKF45 produces two estimates of a state

variable (+1and +1). Here, a numerical estimate of the error, ((+1 +1) , is required to becomputed at each time step. If the estimated error is less than the tolerance level (), the step size +1, to be used in the next step to generate +2is increased to speed up the computations and viceversa. If the value of error is nearly equal to , the two estimates are in close agreement and the value ofstep size is accepted without any correction.

In order to solve Eq. 4.28, each RKF step requires the use of the following six values:

1 =(, ) ()

2 =

+ 1

4,

+

4

(

)

3 = + 3321 + 932 2, + 38 ()4 = + 19322197

1 72002197

2 + 72962197

3, + 1213 ()

5 = + 439216

1 82 + 36805133 845

41044, + ()6 = 8

271 + 22 3544

25653 + 1859

41044 11

405, +

2 ()

The two estimates can be obtained using the following two equations:

+1 = + 25

216 1 +1408

2565 3 +2197

4104 41

5 5 4.37+1 = + 16135

1 + 665612825

3 + 2856156430

4 9505 + 2

556 4.38

Note that +1and +1are obtained using RK method of 4thand 5thorder respectively.The optimal step size ,can be determined using:

,=

2|+1 +1|1 4

4.39

Even though the calculations involved in this approach are tedious and time consuming, this method

gives more accurate results. The generalization of this method to deal with systems of coupled 1st-order

ODEs is fairly obvious.

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