Numerical Methods for Chemical Engineers

CHE F242

Ajaya Kumar Pani

BITS Pilani Department of Chemical Engineering
B.I.T.S-Pilani, Pilani Campus
Pilani Campus
 Lecture-1
17-01-2017
BITS Pilani
Pilani Campus
 Introduction
BITS Pilani
Pilani Campus
CHE F242 Numerical Methods for Chemical Engineers

Chemical Engineering
II-1 III-1
1. Chemical Process Calculations
1. Chemical Engineering Lab I
2. Chemical Engineering
Thermodynamics 2. Separation Process II
3. Fluid Mechanics 3. Kinetics and Reactor Design
4. Engineering Chemistry 4. Process Design Principles I

II-2
III-2
5. Heat Transfer
6. Numerical Methods for Chemical
5. Chemical Engineering Lab II
Engineers 6. Process Dynamics and
7. Material Science and Engineering Control
8. Separation Process I 7. Process Design Principles II
Why numerical methods in Chemical Engg???
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Why NMCE
A polymer blend is to be formed from three compounds whose
compositions and approximate formulas are listed in the table.
Determine the percentages of each compound A, B, C to be
introduced into the mixture to achieve the desired composition
Compound (%)
Composition A B C Desired
Mixture
(CH4)x 25 35 55 30
(C2H6)x 35 20 40 30
(C3H8)x 40 45 5 40

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Why NMCE

An equation for the heat capacity of carbon was given as:

Cp = 1.2+0.005T-0000021T2
The calculated value of enthalpy for carbon at 1000 F is
1510 Btu/lb. what is the reference temperature?

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Why NMCE

Experimental values of the heat capacity Cp have been

determined as follows. Fit a second order polynomial in
temperature to the data (Cp = a+bT+cT2)
T Cp
100 40.54
200 43.81
300 46.99
400 49.33
500 51.25

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Why NMCE

Determine the equilibrium composition of a mixture of

gases at constant
T = 1000 K and P = 1 atm in the steam reforming of
methane

Solution procedure to this type of problem involves

minimizing the total Gibbs free energy of the system by
changing the number of moles of the species in the
system, subject to the conservation of atoms.

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Why NMCE
F0, h and F all vary with time

For disturbance in F0, to get h Vs time,

requires integration

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CHE F242 Numerical Methods for Chemical Engineers

Why numerical methods

• Problem solving tools, can handle large systems of

equations, non-linearities which are common in
engineering practice and often impossible to solve
analytically
• Commercially available software tools have programs
that involve numerical methods
• Many problems cannot be approached using canned
programs. Knowledge of numerical methods and
computer programming will help in designing own
program
“Motivated students who enjoy numerical methods, computers
and mathematics will in the end make better engineers”
Chapra & Canale
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Why NMCE

Fluid mechanics

Transport
Heat Transfer
Phenomena

Reaction Thermodynamics
Engineering

Numerical
Techniques

Environmental Chemical Process

Engg Calculations

Process Control Mass Transfer

Process Design

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CHE F242 Numerical Methods for Chemical Engineers

Numerical Methods

• Mathematical problems are formulated so that they can

be solved with arithmetic and logical operations
• Involve large number of tedious calculations
• Pre-computer era
• Analytical solution (only for limited class of problems)
• Graphical solution (not very precise result, tedious, limited to fewer dimensions)
• Calculators and slide rule (use of numerical techniques manually)
• Rigorous computation requirement limited their
application
• With computers, more implementation of numerical
techniques
• Numerical methods  Computer mathematics

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CHE F242 Numerical Methods for Chemical Engineers

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CHE F242 Numerical Methods for Chemical Engineers

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CHE F242 Numerical Methods for Chemical Engineers

Hand out
Course No. : CHE F242
Course Title : Numerical Methods for Chemical Engineers
Instructor-in-Charge : AJAYA KUMAR PANI
Instructors : Ajaya Kumar Pani & Srinivas Appari
1. Course Description
Introduction to mathematical modeling and engineering problem solving, Use
of software packages and programming, Errors and approximations
including error propagation and numerical error, Roots of equations: Linear
algebraic equations, 1-D and multi-dimensional unconstrained optimization
including gradient methods, Linear programming, Non-linear constrained
optimization, Optimization with packages, Least Squares Regression
including quantification of error, Polynomial regression, Lagrange, inverse
and spline interpolation and Fourier approximation, Engineering
applications, Numerical differentiation and integration, Ordinary differential
equations, Partial differential equations, Engineering applications.

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CHE F242 Numerical Methods for Chemical Engineers

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2. Scope and Objective
Several chemical engineering problems involve finding roots in a higher order equation,
solving simultaneous set of algebraic equations, solving differential equations etc. Quite
often, these equations are not amenable to analytical solutions. In such cases, use of
numerical methods is necessary which then provides a way for the engineer to translate
the language of mathematics and physics into information that may use to make
engineering decisions. This course will provide students with an exposure to numerical
techniques which can be used to solve algebraic and differential equations of varying
complexity. Numerical methods for differentiation, integration and curve fitting
techniques will also be covered. Strong emphasis will be placed on problem solving
based on case studies in engineering. Specific focus in case studies will be application
of numerical techniques and scientific computing to the practice of chemical
engineering. The role of computers and softwares along with identification, quantification
and minimization of errors involved in numerical analysis will also be highlighted.
3. Text Book
Chapra, S. C. and R. P. Canale, Numerical Methods for Engineers, 7th Edition, McGraw Hill
Education (India) Pvt. Ltd., New Delhi, 2015.
Reference Books
Kenneth J Beers, Numerical Methods for Chemical Engineering, Cambridge University Press, 2007.
Chapra, S. C., Applied Numerical Methods with MATLAB for Engineers and Scientists, 3rd Edition,
McGraw Hill, 2012.
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CHE F242 Numerical Methods for Chemical Engineers

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Lecture No. Learning Objectives Topics to be covered Text Book Chap.

Introduction to the course, significance in chemical

1 Introduction -
engineering, Hand out

Concept of simple mathematical model and conservation

Modeling, software, error
2-6 laws, Role of programming and software, Introduction to Ch. 1-4
analysis
MATLAB

Roots of equations and Engineering practice, introduction

Roots of equations (Bracketing to graphical method for finding root, Bisection method &
7 –8 Ch. 5
Methods) False
Position methods, Incremental searches and initial guess

Single point Iteration, Newton Raphson method, Secant

Roots of equations (Open
9 –10 method, Brent’s method, Multiple roots and system of Ch. 6
Methods)
nonlinear equations

Polynomials in Engineering, Computing with polynomials,

Roots of Polynomials, related Muller and Bairstow’s methods, Case studies in
11-14 Ch. 7, Ch 8
MATLAB functions Engineering, MATLAB application for finding roots of
equations

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Linear algebraic equations and

Engineering practice, Gauss Elimination,
Linear Algebraic
15-16 Naïve Gauss elimination, pitfalls, Ch. 9
equations
Techniques for
improving solutions, Gauss Jordan method

Linear Algebraic LU Decomposition and Matrix Inversion

17-18 Ch. 10
equations methods

Special Matrices, Gauss Seidel method,

Linear Algebraic
19-21 Case studies, MATLAB application for Ch 12
Equations, MATLAB
solving system of equations

Optimization (One Golden section, Newton's method, Case

22-23 Ch 16
dimensional) Studies, MATLAB programming

Curve fitting and Engineering Practice,

24 Curve fitting Least square fit of straight line, Ch.17
Linearization of non-linear relationships

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Polynomial regression, Multiple linear regression, Non-linear
Regression, Divided difference Interpolation formula, Ch. 17, Ch 18 and
25-28 Curve fitting
Lagrange’s interpolation, Spline interpolation, Case studies, Ch 20
Solving regression problems by MATLAB

Newton Cotes formula, Trapezoidal rule,

29-30 Numerical Integration Simpson’s 1/3 and 3/8 rule, Unequal segment Integration, Ch. 21
Multiple integrals,

High accuracy differentiation formulas, Case studies in

Numerical
31 – 32 Engineering, Numerical differentiation and integration using Ch. 23 and Ch 24
Differentiation
MATLAB

ODE’s and Engineering Practice, Euler’s method and error

Ordinary Differential
33-34 analysis, Runge Kutta methods (2nd and Higher order), Ch. 25
equations (ODE)
System of ODE’s, Adaptive Runge Kutta method

Concept of stiffness, Multistep methods (Non-starting Heun’s

method), Methods for Boundary value problems, Eigen value
Ordinary Differential
35-38 problems, Ch. 26-28
equations (ODE)
Case studies in Engineering, Use of MATLAB for solving
ODEs
PDE’s and Engineering Practice, Elliptic PDE’s, Laplace
equation and solution technique, Introduction to control
Partial Differential
39-40 volume approach, Parabolic equation, Heat conduction Ch. 29-30
equations (PDE)
equation, Explicit and
Implicit methods

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Evaluation Scheme

EC Evaluation Weightage Date, Time & Nature of

Duration
No. Component % Venue Component
CB and/or
1. Mid Semester Test 90 min 30
OB
During
CB and/or
2. Tutorial Tests* Tutorial 15
OB
Classes
During
CB and/or
3. Surprise Tests** Lecture 8
OB
Classes
Take
4. Assignment 7 OB
Home
Comprehensive CB
5. 180 min. 40
Exam. and/or

. * Best 5 tut marks out of total 6 will be considered. Each tut test will carry a weightage of 3%.
** There will be total 4 surprise quizzes each carrying a weightage of 2%.

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THANK YOU

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Mathematical Modeling and

Engineering Problem solving
• Requires understanding of engineering systems
– By observation and experiment
– Theoretical analysis and generalization
• Computers are great tools, however, without
fundamental understanding of engineering problems,
they will be useless

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Fig. 1.1

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Acid Rain
• A mathematical model is represented as a functional relationship of
the form

Dependent independent forcing

Variable = f variables, parameters, functions

• Dependent variable: Characteristic that usually reflects the state of

the system
• Independent variables: Dimensions such as time ans space along
which the systems behavior is being determined
• Parameters: reflect the system’s properties or composition
• Forcing functions: external influences acting upon the system

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Newton’s 2nd law of Motion

States that “the time rate change of momentum of a body is equal

to the resulting force acting on it.”
The model is formulated as
F = m a (1.2)
F=net force acting on the body (N)
m=mass of the object (kg)
a=its acceleration (m/s2)

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CHE F242 Numerical Methods for Chemical Engineers

Some mathematical models of physical phenomena may

be much more complex.

Complex models may not be solved exactly or require

more sophisticated mathematical techniques than
simple algebra for their solution

– Example, modeling of a falling parachutist:

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CHE F242 Numerical Methods for Chemical Engineers

dv F

dt m

F FD  FU
FD mg
FU  cv
dv mg  cv

dt m

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CHE F242 Numerical Methods for Chemical Engineers

dv c
g  v
dt m
This is a differential equation and is written in terms
of the differential rate of change dv/dt of the
variable that we are interested in predicting.
If the parachutist is initially at rest (v=0 at t=0),
using calculus Independent variable

gm
v(t ) 
c

1 e  (c / m )t

Dependent variable
Forcing function Parameters
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Analytical solution
c = 12.5 kg/s, m = 68.1 kg
v = 53.39(1-e-0.18355t)

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Numerical solution
Reformulation of the problem to solve by arithmatic operations
dv v vt i 1  vt i  v(ti+1) = v(ti)+[g-(c/m)v(ti)](ti+1-ti))
 
dt t t i 1  t i

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Conservation Laws and

Engineering
• Conservation laws are the most important and
fundamental laws that are used in engineering.
Change = increases – decreases (1.13)
• Change implies changes with time (transient).
If the change is nonexistent (steady-state), Eq.
1.13 becomes
Increases =Decreases

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Fig 1.6

For steady-state incompressible fluid flow in pipes:

Flow in = Flow out
or
100 + 80 = 120 + Flow4
Flow4 = 60
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