NUMERICAL METHODS IN CHEMICAL ENGINEERING

Course Description
Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed. The examples will use MATLAB®.

EXPRESSING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS AS: Ax=b

We wish to solve systems of simultaneous linear algebraic equations of the general form:


a11x1 + a12x2 + … + a1nxn = b1

a21x1 + a22x2 + … + a2nxn = b2

::

::

an1x1 +an2x2 + … + annxn = bn (1.1.1-1)

Where we have N equations for the N unknowns x1, x2, …, xn.

As a particular example, consider the following set of these three equations (N=3) for the three unknowns x1, x2, x3:

x1 + x2 + x3 = 4

2x1 + x2 + 3x3 = 7

3x1 + x2 +6x3 = 2 (1.1.1-2)

aij = constant coefficient (usually real) multiplying unknown xj in equation #i.

Bi = constant “right-hand-side” coefficient for equation #i.

For the system (1.1.1-2) above,

a11 = 1 a12 = 1 a13 = 1 b1 = 4

a21 = 2 a22 = 1 a23 = 3 b2 = 7

a31 = 3 a32 = 1 a33 = 6 b3 = 2

It is common to write linear systems in matrix/vector for as:


Ax=b (1.1.1-3)

Where the vector of unknowns x is written as:

x = (1.1.1-4) n::21xxx

The vector of right-hand-side coefficients b is written:

b = (1.1.1-5) n::21bbb

The matrix of coefficients A is written in a form with N rows and N columns,

A = (1.1.1-6) nna ...aaa: : : :: : : :a ...aaaa ...aaa n3 n2 n12n 23 22 211n 13 12 11

We see that row ‘i’ contains the values ai1, ai2, …, aiN that are the coefficients multiplying each unknown x1, x2, …, xN in equation #i.

Rows �� equations Columns �� coefficients multiplying a specific unknown in each equation.

aij = element of A in ith row and jth column

= coefficient multiplying xj in equation #i.
 
After we will write the coefficients in matrix form explicitly, so that we may


write Ax=bas:

nna ...aaa: : : :: : : :a ...aaaa ...aaa n3 n2 n12n 23 22 211n 13 12 11N21x::]xx = (1.1.1-7) n::21bbb

For the example system (1.1.1-2):

x1 + x2 + x3 = 4

2x1 + x2 + 3x3 = 7

3x1 + x2 +6x3 = 2 (1.1.1-2, repeated)

We have:

A = 6 1 33 1 21 1 1b = (1.1.1-8) 



274

As we will represent our linear systems as matrices “acting on” vectors, some review of basic vector notation is required.
For full lecture on this course and lecture notes, email us at millenniumtutors@yahoo.com

Second Module
http://clickserve.cc-dt.com/link/tplclick?lid=41000000011066022&pubid=21000000000269318

1.2.3 Pivoting Techniques in Gaussian Elimination


Let us consider again the 1st row operation in Gaussian Elimination, where we start with the original augmented matrix of the system. to get the full articles, email us at millenniumtutors@yahoo.com