AOFW CHG433_Part I_wip_Nov2023.pdf presentation

System analysis

1. University of Lagos
Almoruf O. F. Williams
Chemical Engineering Analysis
(CHG 433), CH= 3,0

2. University of Lagos
Course Outline
•COURSE OUTLINE -HIGH LEVEL
•Use of mathematical tools for the analysis of chemical engineering
operations.
•Process modeling and dynamic analysis.
•Statistical tests, regression analysis.
•Design of Experiments.

3. University of Lagos
Course Outline
COURSE OUTLINE
A First Things First.
•Incentives for mathematical modeling
•Systems and Its Classifications
•Classification of Models
•Building steps for a mathematical model
•Conservation Laws (Total mass balance, Component balance, Momentum balance, Energy
•balance)
•Thermodynamics relations
•Chemical kinetics
•Degrees of Freedom Analyses
B Development of Common Models (Lumped & Distributed Parameter) & Solutions using Mathematical
Tols
•Derivation of Models of Common Chemical Engineering Systems
•Analytical Solutions –including Laplace Transform Method
•Graphical Plots/Presentations using Software

4. University of Lagos
Course Outline
COURSE OUTLINE
B Development of Common Models (Lumped & Distributed Parameter) & Solutions using
Mathematical Tools
•Numerical Solutions of Models (Numerical Techniques)
➢Numerical derivatives (Forward difference formula, Backward difference formula, Central
➢difference formula),
➢Solution of non-linear equations (Bisection method, Newton-Raphson method, Secant
method),
➢Linearization of non-linear equations),
➢Solution of Linear equations (Cramer method, Gauss Elimination method, Matrix Inversion
➢method, Gauss-Seidel method),
➢Curve Fitting (Least square method-Nonlinear optimization) (Linear regression, Polynomial
➢regression,
➢Interpolation (Newton’s Interpolating polynomials, Lagrange Interpolating polynomials),
➢Numerical Integration (Trapezoidal rule, Simpson 1/3 rule, Simpson 3/8 rule),
➢Solution of differential equations (Euler method, Runge-Kutta method, Adams
➢method))

5. University of Lagos
Course Outline
COURSE OUTLINE
Specific goals for the course:
•Understand how to develop models of common chemical engineering processes and
systems;,
•Solve numerical equations (e.g., linear and non-linear algebraic equations, polynomial
equations and differential equations) related to many chemical engineering problems,
• Perform numerical calculus including integration, differentiation, and matrix, vector,
• Simulate unit operations related to fundamental chemical engineering processes,
•Carry-out Design of Experiments, Statistical Testing, Plot and fit experimental data,
Perform error analysis and data fitting and perform data analysis
•Use Modern Computer Software and
•Learn how technical reports etc.
Students will learn (i) the importance of computer-based numerical analysis for solving
chemical engineering problems, (ii) recent modern process simulations formchemical
engineering processes, (iii) fundamental and application of statistics, (iv) how to present
results and prepare written reports for non-technical, and (v) technical audiences through
team projects.

6. University of Lagos
Course Outline contd.
COURSE TEXTBOOKS AND REFERENCE MATERIALS/SOFTWARE TOOLS
TEXTBOOKS/REFERENCE MATERIALS:
•B. A. Ogunnaike,. (1985). Principles of Mathematical Modeling and Analysis in Chemical
Engineering , Done Publishers, Lagos, Nigeria.
•Rasmuson, A, et al., Mathematical Modeling in Chemical Engineering, Cambridge, US, 2014
•Bruce Finlayson, Introduction to Chemical Engineering Computing, Wiley, 2 Ed., 2012
•Mark E. Davies, Numerical Methods for Chemical Engineers, Dover , New York, US, 2013
Reprint of 1984 ed by Wiley
•Mariano Martin Martin (Ed). Introduction to Software for Chemical Engineers, CRC Press,
2015
•W. L. Luyben – Process Modeling, Simulation and Control for Chemical Engineers, 2
nd
. Ed.,
McGaw-Hill, 1990
•D. Seborg, T. Edgar, D. Mellicamp - Process Dynamics and Control, J. Wiley, 3
rd
. Ed., 2011
•Edwin Zondervan, A Numerical Primer for the Chemical Engineer, CRC Press, 2015
•Kamal I. M. Al-Malah MATLAB numerical methods with chemical engineering applications,
McGraw-Hill Education, 2014

7. University of Lagos
Course Outline contd.
COURSE TEXTBOOKS AND REFERENCE MATERIALS/SOFTWARE TOOLS
SOFTWARE TOOLS:
•Computer Numerical/Graphics Software: Matlab/Simulink (Commercial) ,Octave (Open
Source), SciLab (Open Source)
•Computer Algebra Systems: Mathematica, Maple, MATHCAD (Commercial but cheap for
students), and Maxima (Open Source – completely free)
•MS-Excel (Microsoft Office Productivity tool)- commercial
•Process Simulators: CHEMCAD (Commercial), ASPEN Plus /HYSYS(Commercial), DWSIM
(Open Source), COCO (Open Source)
•LaTex e.g. MikTex (Mathematical typesetting software) – Open Source
•Dia Diagram Editor (Open Source drawing software – completely free), Microsoft Visio
(Commercial). These software help you to draw professional diagrams
•Others: CFD – Fluent (Commercial), COMSOL for microscale modelling & simulation
(Commercial), Open Modelica (Open source), OpenFoam (Open Source)
•Programming Languages: Traditional – FORTRAN; Modern – C, C++, Python, Java etc.

14. Mathematical Modeling of
Chemical Processes
Mathematical Model (Eykhoff, 1974)
“a representation of the essential aspects of an existing system (or a system to
be constructed) which represents knowledge of that system in a usable form”
Everything should be made as simple as possible, but no simpler.

20. Table Classification of models according to scale

27. General Modeling Principles
•The model equations are at best an approximation to the real
process.
•Adage: “All models are wrong, but some are useful.”
•Modeling inherently involves a compromise between model
accuracy and complexity on one hand, and the cost and effort
required to develop the model, on the other hand.
•Process modeling is both an art and a science. Creativity is
required to make simplifying assumptions that result in an
appropriate model.
•Dynamic models of chemical processes consist of ordinary
differential equations (ODE) and/or partial differential equations
(PDE), plus related algebraic equations.

28. Table 2.1. A Systematic Approach for
Developing Dynamic Models
1.State the modeling objectives and the end use of the model.
They determine the required levels of model detail and model
accuracy.
2.Draw a schematic diagram of the process and label all process
variables.
3.List all of the assumptions that are involved in developing the
model. Try for parsimony; the model should be no more
complicated than necessary to meet the modeling objectives.
4.Determine whether spatial variations of process variables are
important. If so, a partial differential equation model will be
required.
5.Write appropriate conservation equations (mass, component,
energy, and so forth).

29. 6.Introduce equilibrium relations and other algebraic
equations (from thermodynamics, transport phenomena,
chemical kinetics, equipment geometry, etc.).
7.Perform a degrees of freedom analysis (Section 2.3) to
ensure that the model equations can be solved.
8.Simplify the model. It is often possible to arrange the
equations so that the dependent variables (outputs) appear
on the left side and the independent variables (inputs)
appear on the right side. This model form is convenient
for computer simulation and subsequent analysis.
9.Classify inputs as disturbance variables or as manipulated
variables.
Table 2.1. (continued)

30. Modeling Approaches
➢ Physical/chemical (fundamental, global)
•Model structure by theoretical analysis
▪Material/energy balances
▪Heat, mass, and momentum transfer
▪Thermodynamics, chemical kinetics
▪Physical property relationships
•Model complexity must be determined
(assumptions)
• Can be computationally expensive (not real-
time)
• May be expensive/time-consuming to obtain
• Good for extrapolation, scale-up
• Does not require experimental data to obtain
(data required for validation and fitting)

31. •Conservation Laws
Theoretical models of chemical processes are based on conservation laws.
Conservation of Massrateof mass rateof mass rateof mass
(2-6)
accumulation in out
     
=−     
     
Conservation of Component irateof componenti rateof componenti
accumulation in
rateof componenti rateof componenti
(2-7)
out produced
   
=   
   
   
−+   
   

32. Conservation of Energy
The general law of energy conservation is also called the First Law of Thermodynamics.
It can be expressed as:     
=−     
     


++


rateof energy rateof energyin rateof energy out
accumulation by convection by convection
netrateof heataddition netrateof work
to the system from performed on the sys
the surroundings





tem (2-8)
by the surroundings
The total energy of a thermodynamic system, U
tot, is the sum of its internal energy,
kinetic energy, and potential energy:int
(2-9)
tot KE PE
U U U U= + +

33. ➢ Black box (empirical)
•Large number of unknown parameters
•Can be obtained quickly (e.g., linear regression)
•Model structure is subjective
•Dangerous to extrapolate
➢ Semi-empirical
•Compromise of first two approaches
•Model structure may be simpler
•Typically 2 to 10 physical parameters estimated
(nonlinear regression)
•Good versatility, can be extrapolated
•Can be run in real-time

34. •linear regression
•nonlinear regression
•number of parameters affects accuracy of model,
but confidence limits on the parameters fitted must
be evaluated
•objective function for data fitting – minimize sum of
squares of errors between data points and model
predictions (use optimization code to fit
parameters)
•nonlinear models such as neural nets are
becoming popular (automatic modeling) 2
210 xcxccy ++= ( )
/
1
t
eKy
−
−=

35. Number of sightings of storks
Number of
births (West
Germany) Uses of Mathematical Modeling
•to improve understanding of the process
•to optimize process design/operating conditions
•to design a control strategy for the process
•to train operating personnel

38. Table 2.2. Degrees of Freedom Analysis
1.List all quantities in the model that are known constants (or
parameters that can be specified) on the basis of equipment
dimensions, known physical properties, etc.
2.Determine the number of equations N
E and the number of
process variables, N
V. Note that time t is not considered to be a
process variable because it is neither a process input nor a
process output.
3.Calculate the number of degrees of freedom, N
F = N
V - N
E.
4.Identify the N
E output variables that will be obtained by solving
the process model.
5.Identify the N
F input variables that must be specified as either
disturbance variables or manipulated variables, in order to
utilize the N
F degrees of freedom.

45. Grouping of models into opposite pairs
Linear versus nonlinear
➢Linear models exhibit the important property of superposition; nonlinear ones do not.
➢Equations (and thus models) are linear if the dependent variables or their derivatives
appear only to the first power; otherwise they are nonlinear.
➢ In practice, the ability to use a linear model for a process is of great significance. General
analytical methods for equation solving are all based on linearity.
➢Only special classes of nonlinear models can be attacked with mathematical methods. For
the general case, where a numerical method is required, the amount of computation is also
much less for linear models, and in addition error estimates and convergence criteria are
usually derived under linear assumptions.

46. Grouping of models into opposite pairs
Steady state versus transient
➢Other synonyms for steady state are time invariant, static, or stationary. These terms
➢refer to a process in which the point values of the dependent variables remain constant over
time, as at steady state and at equilibrium.
➢Non-steady-state processes are also called unsteady state, transient, or dynamic, and
represent a situation in which the process dependent variables change with respect to time.
➢A typical example of an non-steady-state process is the startup of a distillation column which
would eventually reach a pseudo steady-state set of operating conditions.
➢ Inherently transient processes include fixed-bed adsorption, batch distillation, and reactors,
drying, and filtration/ sedimentation.

47. Grouping of models into opposite pairs
Lumped parameter versus distributed parameter
➢A lumped-parameter representation means that spatial variations are ignored, and the
various properties and the state of a system can be considered homogeneous throughout the
entire volume.
➢A distributed-parameter representation, in contrast, takes into account detailed variations in
behavior from point to point throughout the system.
➢All real systems are, of course, distributed in that there some variations occur throughout
them. As the variations are often relatively small, they may be ignored, and the system may
then be “lumped.”
➢The answer to the question whether or not lumping is valid for a process model is far from
simple. A good rule of thumb is that if the response of the process is “instantaneous”
throughout the process, then the process can be lumped. If the response shows instantaneous
differences throughout the process (or vessel), then it should not be lumped. Note that the
purpose of the model affects its validity. Had the purpose been, for example, to study
mixing in a stirred tank reactor, a lumped model would be completely unsuitable because it
has assumed from the first that the mixing is perfect and the concentration a single variable.
➢Because the mathematical procedures for solving lumped-parameter models are simpler
than those for solving distributed-parameter models, we often approximate the latter using
an equivalent lumped-parameter system. Whilst lumping is often possible, we must be
careful to avoid masking the salient features of a distributed element and subsequently
building an inadequate model by lumping.

48. Grouping of models into opposite pairs
Lumped parameter versus distributed parameter
➢As an example of the use of lumped versus distributed mathematical models, consider the
equilibrium stage concept of distillation, extraction, and similar processes.
➢As shown in the figure below. we usually assume that the entire stage acts as a whole, and
we do not consider variations in temperature, composition, or pressure in various parts of
the stage. All of these variables are “lumped” together into some overall average. The errors
introduced are compensated for by the stage efficiency factor.
Figure Lumped-parameter and distributed-parameter visualization of a distillation tray.
(a) Actual plate with complex flow patterns and resulting variations in properties from point to
point. (b) Idealized equilibrium stage ignoring all internal variations.

49. Grouping of models into opposite pairs
Continuous versus discrete variables
➢Continuous means that the variables can assume any values within an interval; discrete means
that a variable can take on only distinct values within an interval.
➢For example, concentrations in a countercurrent packed bed are usually modeled in terms of
continuous variables, whereas plate absorbers are modeled in terms of staged multi-
compartment models in which a concentration is uniform at each stage but differs from stage
to stage in discrete jumps.
➢Continuous models are described by differential equations and discrete models by difference
equations.
➢Figure in the next slide illustrates the two configurations.
➢The left-hand figure shows a packed column modeled as a continuous system, whereas the
right-hand figure represents the column as a sequence of discrete (staged) units.
➢The concentrations in the left-hand column would be continuous variables; those in the right
hand column would involve discontinuous jumps. The tic marks in the left-hand column
represent hypothetical stages for analysis.
➢It is, of course, possible to model the packed column in terms of imaginary segregated stages
and to treat the plate column in terms of partial differential equations in which the
concentrations are continuous variables.

50. Grouping of models into opposite pairs
Figure Continuous versus discrete modeling of a packed column absorber.
Continuous versus discrete variables

51. Grouping of models into opposite pairs
Deterministic versus stochastic
➢Deterministic models or elements of models are those in which each variable and parameter
can be assigned a definite fixed number, or a series of numbers, for any given set of
conditions, i.e. the model has no components that are inherently uncertain.
➢ In contrast, the principle of uncertainty is introduced in stochastic or probabilistic models. The
variables or parameters used to describe the input–output relationships and the structure of the
elements (and the constraints) are not precisely known.
➢A stochastic model involves parameters characterized by probability distributions. Due to this
the stochastic model will produce different results in each realization.
➢Stochastic models play an important role in understanding chaotic phenomena such as
Brownian motion and turbulence.
➢They are also used to describe highly heterogeneous systems, e.g. transport in fractured media.
Stochastic models are used in control theory to account for the irregular nature of disturbances.
In the present context, we will focus upon deterministic models.

52. Grouping of models into opposite pairs
Interpolation versus extrapolation
➢A model based on interpolation implies that the model is fitted to experimentally determined
values at different points and that the model is used to interpolate between these points.
➢A model used for extrapolation, in comparison, goes beyond the range of experimental data.
➢Typically, thermodynamic models are used for interpolation as well as correlations
➢in complicated transport phenomena applications.
➢Extrapolation requires, in general, a detailed mechanistic understanding of the system. The
procedure requires great care to avoid misleading conclusions.
➢The figure in the next slide illustrates an exaggerated case of extrapolation by means of a
linear model into a region beyond the range of experimental data for a chemical reaction that
reaches a maximum yield in time.
➢In the safety analysis of nuclear waste repositories models are used to predict the fate of
leaking radionuclides into the surrounding rock formation over geological time scales.
Naturally, it is of the utmost importance that these models are physically/chemically sound
and based on well-understood mechanistic principles.

53. Grouping of models into opposite pairs
Figure Danger of extrapolation. Yield of a chemical reactor versus time.

54. Grouping of models into opposite pairs
Mechanistic versus empirical
➢Mechanistic means that models are based on the underlying physics and chemistry governing
the behavior of a process;
➢Empirical means that models are based on correlated experimental data.
➢Empirical modeling depends on the availability of process data, whereas mechanistic
modeling does not; however, a fundamental understanding of the physics and chemistry of
the process is required.
➢Mechanistic models are preferably used in process design, whereas empirical models can be
used when only trends are needed, such as in process control. Semi-empirical models cover
the range in between.

55. Grouping of models into opposite pairs
Coupled versus not coupled
➢When a model consists of two or more interacting relations, we have a coupled model.
➢The coupling may be weak or strong.
➢ If the interaction only works in one direction, we speak of weak coupling (one-way
coupling); if it operates in both directions we speak of strong (two-way) coupling.
➢Forced convection is an example of one-way coupling, and free convection is an
example of two-way coupling.
➢In forced convection, the flow field is independent of the transport of energy and can
be solved first and then introduced into the energy equation.
➢ In free convection, flow and energy transport are intimately coupled since the flow is
generated by density differences originating from temperature differences.
➢A model of a pneumatic conveying dryer involves a high degree of coupling.

56. Classification based on mathematical complexity
Table Classification of mathematical problems and their ease of solution using
analytical methods

57. Classification based on mathematical complexity

58. Lumped Parameter Models Equation Forms
➢Lumped parameter models, i.e. models developed by ignoring spatial variation
of a physical quantity of interest, are very often used for describing steady state
or dynamic behavior of systems encountered in process industry.
➢For example, in a tank in which two or more fluids of different concentrations
and compositions are mixed, assuming that the contents of the tank are well
mixed and there is no spatial variation in concentrations or temperature inside
the tank considerably simplifies the model development.
➢Let us examine some typical systems encountered in the process industry, which
are modelled as lumped parameter systems, and write the modeling equation
forms.

59. Lumped Parameter Models Equation Forms
Linear Algebraic Equations
➢Plant wide or section wide mass balances are carried out at design stage or later
during operation for keeping material audit.
➢These models are typical examples of systems of simultaneous linear algebraic
equations..
Example 1
➢ Recovery of acetone from air -acetone mixture is achieved using an absorber
and a flash separator (see Figure on next slide).
➢A model for this system is developed under following conditions
•All acetone is absorbed in water
•Air entering the absorber contains no water vapor
• Air leaving the absorber contains 3 mass % water vapor

60. Lumped Parameter Models Equation Forms
Linear Algebraic Equations
Figure: Acetone Recovery using absorber and flash separator

61. Lumped Parameter Models
Linear Algebraic Equations

62. Lumped Parameter Models
Linear Algebraic Equations

63. Lumped Parameter Models
Linear Algebraic Equations

64. Lumped Parameter Models
Non-Linear Algebraic Equations
Figure: Flash Drum Unit
above

65. Lumped Parameter Models
Non-Linear Algebraic Equations
➢Consider a stream of two components A and B at a high pressure Pf and
temperature Tf as shown in the Figure above.
➢ If the Pf is greater than the bubble point pressure at Tf , no vapor will be
present. The liquid stream passes through a restriction (valve) and is flashed in
the drum, i.e. pressure is reduced from Pf to P. This abrupt expansion takes
place under constant enthalpy
➢If the pressure P in the flash drum is less than the bubble point pressureof the
liquid feed at Tf , the liquid will partially vaporize and two phases at the
equilibrium with each other will be present in the flash drum. The equilibrium
relationships are
•Temperature of the liquid phase = temperature of the vapor phase.
• Pressure of the liquid phase = pressure of the vapor phase.
•Chemical potential of the i0th component in the liquid phase = Chemical
potential of the i0th component in the vapor phase

66. Lumped Parameter Models
Non-Linear Algebraic Equations
Example: Flash Vaporization
Table: K values and compositions for flash vaporization example

67. Lumped Parameter Models
Non-Linear Algebraic Equations
Example: Flash Vaporization
➢Consider flash vaporization unit shown in Figure above.
➢A hydrocarbon mixture containing 25 mole % of n butane, 45 mole % of n-hexane
is to be separated in a simple flash vaporization process operated at 10 atm. and
2700F.
➢The equilibrium k- values at this composition are reported in the Table on the
previous slide.
➢Let xi represent mole fraction of the component I in liquid phase and yi represent
mole fraction of the component i in vapor phase.
➢Model equations for the flash vaporizer are

68. Lumped Parameter Models
Non-Linear Algebraic Equations
Example: Flash Vaporization

69. Lumped Parameter Models
Non-Linear Algebraic Equations
Example: Flash Vaporization

70. Lumped Parameter Models
Non-Linear Algebraic Equations
Example: Flash Vaporization

71. Lumped Parameter Models
Optimization-based Formulations
A Variety of modeling and design problems in chemical engineering are
formulated as optimization problems.
Consider a simple reaction
•carried out in a batch reactor (see Figure below). It is desired to find the kinetic
parameters ko;E and n from the experimental data.
•The data reported in Table below is collected from batch experiments in a reactor
at different temperatures
Example

72. Lumped Parameter Models
Optimization-based Formulations
Figure: Batch reactor and typical reactant concentration profiles
Table : Reaction Rates at Di¤erent Temperaturtes and Concentrations in a batch experiment

73. Lumped Parameter Models
Optimization-based Formulations

74. Lumped Parameter Models
Optimization-based Formulations
Example
➢Cooling water is to be allocated to three distillation columns. Up to 8 million liters
per day are available, and any amount up to this limit may be used.
➢The costs of supplying water to each equipment are

75. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
➢For most of the processing systems of interest to the chemical engineer, there are
three fundamental quantities :mass, energy and momentum.
➢These quantities can be characterized by variables such as density, concentration,
temperature, pressure and flow rate.
➢These characterizing variables are called as state of the processing system.
➢The equations that relate the state variables (dependent variables) to the
independent variables are derived from application of conservation principle on
the fundamental quantities and are called the state equations

76. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Figure: General System

77. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
➢Figure in the previous slide shows schematic diagram of a general system and its
interaction with external world.
➢Typical dynamic model equations are as follows:

78. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Variables and parameters appearing in these equations are described in Table
below

79. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
➢By convention, a quantity is considered positive if it fows in and negative if it fows
out.
➢The state equations with the associated variables constitute a .lumped parameter
mathematical model.of a process, which yields the dynamic or static behavior of
the process.
➢The application of the conservation principle stated above will yield a set of
differential equations with the fundamental quantities as the dependent variables
and time as independent variable. The solution of the differential equations will
determine how the state variables change with time i.e., it will determine the
dynamic behavior of the process.
➢The process is said to be at the steady state if the state variables do not change
with time.
➢ In this case, the rate of accumulation of the fundamental quantity S is zero and
the resulting balance yields a set of algebraic equations

80. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Example Stirred Tank Heater System
Total momentum of the system remains constant and will not be considered.
Total mass balance: Total mass in the tank at

81. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Example Stirred Tank Heater System

82. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Example Stirred Tank Heater System

83. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Example Stirred Tank Heater System

84. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Example Stirred Tank Heater System

85. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
Example
Consider three isothermal CSTRs in series in which a .rst order liquid phase reaction
of the form
Three isothermal CSTRs in series

86. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
ExampleThree isothermal CSTRs in series

87. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)
➢The model we considered above did not contain variation of the variables
with respect to space.
➢Such models are called as .Lumped parameter models.and are described by
ordinary differential equations of the form

88. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)

89. Lumped Parameter Models
Ordinary Differential Equations - Initial Value Problem(ODE-IVP)

90. Distributed Parameter Models –Equation Forms
➢Most of the systems encountered in chemical engineering are distributed parameter
systems i.e. DPS
➢Even though behavior of some of these systems can be adequately represented by
lumped parameter models (LPS), such simplifying assumptions may fail to provide
accurate picture of system behavior in many situations and variations of variables
along time and space have to be considered while modeling.
➢This typically results in a set of partial differential equations (PDEs) or ordinary
differential equations with boundary conditions speci.ed (ODE-Boundary value
Problems or ODE-BVP).
➢We shall look at some simple examples here.

91. Distributed Parameter Models –Equation Forms
➢Consider the double pipe heat exchanger in which a liquid fowing in the inner
tube is heated by steam fowing counter-currently around the tube (see figure
below).
➢The temperature in the pipe changes not only with time but also along the axial
direction z.
➢While developing the model, it is assumed that the temperature does not change
along the radius of the pipe.
Figure: Steam-heated double pipe heat exchanger
Steam-heated Heat Exchanger

92. Distributed Parameter Models –Equation Forms

93. Distributed Parameter Models –Equation Forms

94. Distributed Parameter Models –Equation Forms
Double-pipe Heat Exchanger
➢Consider the situation where some hot liquid is used on the shell side to heat the
tube side fluid (see Figure below).
➢The model equations for this case can be stated as
Figure: Double-pipe heat exchanger

95. Distributed Parameter Models –Equation Forms
Double-pipe Heat Exchanger

96. Distributed Parameter Models –Equation Forms
Double-pipe Heat Exchanger
➢Equations (91-92) represent coupled ordinary differential equations.
➢The need to compute steady state profiles for the counter-current double pipe heat
exchanger results in a boundary value problem (ODE-BVP) as one variable is
specified at z = 0 while the other is specified at z = 1.

97. Distributed Parameter Models –Equation Forms
Review of Terminologies Associated with PDEs
➢Definition - Order of PDE: Order of a PDE is highest order of derivative occurring in
PDE.
➢Definition Degree of PDE: Power to which highest order derivative is raised.
Consider the
PDE:
(95)
Here the Order = 2 and Degree = n

98. Distributed Parameter Models –Equation Forms
Review of Terminologies Associated with PDEs
➢Solutions of PDEs are sought such that it is satisfied in the domain and on the
boundaries.
➢A problem is said to be well posed when the solution is uniquely determined and
it is sufficiently smooth and differentiable function of the
➢independent variables.
➢The boundary conditions have to be consistent with one another in order for a
problem to be well posed.
➢This implies that at the points common to boundaries, the conditions should not
violate or contradict each other.

99. Distributed Parameter Models –Equation Forms
Review of Terminologies Associated with PDEs
Classification of PDEs

100. Distributed Parameter Models –Equation Forms
Review of Terminologies Associated with PDEs
Classification of PDEs
➢Similarly, the boundary conditions can be homogeneous or non homogeneous
depending on whether they contain terms independent of dependent variables.
➢The PDEs typically encountered in engineering applications are 2
nd
order PDEs
(reaction-diffusion systems, heat transfer, fluid-flow etc.)

101. Distributed Parameter Models –Equation Forms
Review of Terminologies Associated with PDEs
Classification of 2
nd
Order PDEs

102. Distributed Parameter Models –Equation Forms
Review of Terminologies Associated with PDEs
Classification of 2
nd
Order PDEs

103. Distributed Parameter Models –Equation Forms
Review of Terminologies Associated with PDEs
Classification of 2
nd
Order PDEs
➢Elliptic Problems typically arise while studying steady-state behavior of diffusive
systems.
➢Parabolic or hyperbolic problems typically arise when studying transient behavior
of diffusive systems.