PSOD
Lecture 2
Matrices and vectors in Chemical
& Process Engineering

Appear in calculations when process is
described by the system of equations:
– Piping system
– Cascade of
» Reactors
» Heat exchangers
» Mixers
– System of apparatus and streams in chemical
plant
Matrices in Chemical & Process
Engineering
L, cs
V L, c
1
V L, c
2
V L, c
3
V
L, c4
concentrations give 4-elements vector c
 To find solution we need system of 4
equations
 Equation parameters creates square matrix

V
,
c
s
Matrices in Chemical & Process
Engineering
V
,
c
1
V
,
c
2
V
,
c
3
V
,
c
4
input  output  source(reaction)
Lcs  Lc1  c1kV
Lc1  Lc2  kVc2
Lc2  Lc3  kVc3
Lc3  Lc4  kVc4
c1 L  kV   0c2  0c3  0c4  Lcs
Lc1  L  kV c2  0c3  0c4  0
0c1  Lc2  L  kV c3  0c4  0
0c1  0c2  Lc3  L  kV c4  0
V
,
Matrices in Chemical & Process
Engineering
c
s
V
,
c
1
V
,
c
2
V
,
c
V
,
c
3
4
 L  kV
 L

 0

 0
0
 L  kV 
0
0
L
 L  kV 
0
L


0


0

 L  kV 
0
 c1 
 Lcs 
c 
 0 
 2   
 c3 
 0 
 
 
c
 0 
 4
MathCAD – vectors and matrix
MathCAD – vectors and matrix

Matrix operations
–
–
–
–
–
Multiply by constant
Matrix transpose [ctrl]+[1]
Inverse [^][-][1]
Matrix multiplying
Determinant
MathCAD – vectors and matrix

To read the matrix elements Ar, k: key [[] rrow nr, k – column nr
– e.g. element A1,1 keystrokes: [A][[][1][,][1][=]

To chose matrix column: M<col.nr>
– First column A( A<0>):
keys [A][ctrl]+[6][0]

Default first column&row number is 0,
– (to change : Math/Options/Array Origin)
MathCAD – vectors and matrix

Calculations of dot product and cross
product of vectors
MathCAD – vectors and matrix

Special definition of matrix elements as a
function of row-column number Mi,j=f(i,j)
– E.g. Value of element is equal to product of
column and row number
Constrain: function
arguments have to
be integer

MathCAD 3D graphs
3D graphs of function on the base of matrix :
[ctrl]+[2] [M]
–
M – matrix defined earlier
MathCAD 3D graphs

3D Graphs of function of real type
arguments
–
–
–
Using procedure: CreateMesh(function,
lb_v1, ub_v1, lb_v2, ub_v2, v1grid, v2grid)
Assign result to variable
Plot of the variable is similar to plot of
matrix ([ctrl]+[2])
Boundaries can be the real numbers. (def. –5,5)
Grids have to be integer numbers (def. 20)
MathCAD 3D graphs
MathCAD 3D graphs - formating
MathCAD 3D graphs – formatting: fill options
MathCAD 3D graphs – formatting: fill options
Contours colour
filled
MathCAD 3D graphs – formatting: line options
MathCAD 3D graphs – formatting: Lighting
MathCAD 3D graphs – formatting: Fog and
perspective
MathCAD 3D graphs – formatting: Backplane and
Grids
MathCAD 3D scatter graphs

Data given as three
vectors of each point
coordinates
– Equal vector size
– Button on Graph
toolbar: 3D Scatter
Plot
– In the placeholder
type in brackets the
vectors names
separated by comas
Predefined constants



e = 2,718 – natural logarithm base
g = 9,81 m/s2 – acceleration of gravity
 = 3,142 – circle perimeter/diameter ratio
Solving of algebraic equation
When equation is implicit
 When we don’t want to separate variables

MathCAD equation solvers

Single equation (one unknown value)
1. Given-Find method
»
»
Input start point of variable
Type "Given"
»
»
Type equation with using [=] ([ctrl]+[=])
Type Find(variable)=
MathCAD equation solving

Given-Find – solving methods
–
–
Linear (function of type y=c0x + c1) –starting
point choice do not affects on results.
Nonlinear – according to nonlinear equation.
Obtained result could depend on starting
point. Available methods:
»
»
»
»

Conjugate Gradient
Quasi – Newton
Levenberg-Marquardt
Quadratic
The choice of method is automatic by
default. User can choose method from the
pop-up menu over word Find.
MathCAD equation solving

Single equation (one unknown value)
2. Root procedure:
Root(function, variable, low_limit, up_limit)=
–
Values of function at the bounds must have different signs
or
MathCAD equation solving

Single equation (one unknown value)
2. Root procedure
methods:
1.
2.
Secant method
Mueller method (2nd order polynomial)
y1
x4  x2  y 2
x2
y3
x3
x5
x4
x2  x3
y 2  y3
x1
y2
xi 1  xi 1  yi 1
xi 1  xi
yi 1  yi

MathCAD equation solving
Single equation (one unknown value)
3. Special procedure: polyroots for the
polynomials. Argument of procedure is a
vector of polynomial coefficients (a0, a1...).
The result is a vector too.
Methods:
1. Laguerre's method
2. companion matrix
Laguerre's method
Polynomial p(x) of degree n. Starting from assumed xk.
pxk 
G
p  xk 
pxk 
H G 
p  xk 
2
a
n
G
n  1nH  G 2 
xk 1  xk  a
MathCAD, the system of equations solving

The system of linear equations
–
Solving on the base of matrix toolbar:
» Prepare square matrix of equations coefficients
(A) and vector of free terms (B)
» Do the operation x:=A-1B and show result: x=
Or
» Use the procedure LSOLVE: lsolve(A,B)=
MathCAD, the system of equations solving
MathCAD, the system of equations solving

The system of nonlinear equation
–
Can be solved using given-find method
»
»
Assign starting values to variables
Type Given
»
Type the equations using = sign (bold)
»
Type Find(var1, var2,...)=
MathCAD, the system of equations solving
Differential eq. Solvers in
MathCAD
Ordinary differential equations
solving

Numerical methods:
– Gives only values not function
– Engineer usually needs values
– There is no need to make complicated
transformations (e.g. variables separation)
– Basic method implemented in MathCAD is
Runge-Kutta 4th order method.
Ordinary differential equations
solving

Numerical methods principle
– Calculation involve bounded range of
independent variable only
– Every point is being calculated on the base of
one or few points calculated before or given
starting points.
– Independent variable is calculated using step:
xi+1 = x i + h = xi+Dx
– Dependent value is calculated according to the
method
yi+1 = y i +Dy= y i +Ki
Ordinary differential equations
solving

Runge-Kutta 4th order method principles:
– New point of the integral is calculated on the
base of one point (given/calculated earlier) and
4 intermediate values k1  hF xi , yi 
1
1 

k 2  hF  xi  h, yi  k1 
2
2 

1
1 

k3  hF  xi  h, yi  k 2 
2
2 

k 4  hF  xi  h, yi  k3 
1
k1  2k2  k3   k4 
6
yi 1  yi  K  O h 5
K
 
MathCAD differential equations

Single, first order differential equation
dy
 f ( x, y )
dx
Initial
condition
x  x0 , y x  x0  y0
1. Assign the initial value of dependent variable
(optionally)
2. Define the derivative function
3. Assign to the new variable the integrating function
rkfixed:
R:=rkfixed(init_v, low_bound, up_bound, num_seg, function)
MathCAD, differential equations
4. Result is matrix (table) of two columns: first
contain independent values second dependent ones
 x0

 x1
R   x2

...
x
 N
y1, 0 

y1,1 
y1, 2 

... 

y1, N 
5. To show result as a plot: R<1>@R<0>
MathCAD differential equations
MathCAD differential equations

System of first order differential equations
 dy0



f
x
,
y
,
y
0
1
 dx

 dy1  f  x, y , y 
0
1
 dx
x  x0
y0 xx0  y
0
0
y1xx0  y
0
1
1. Assign the vector of initial conditions of dependent
variables (starting vector)
2. Define the vector function of derivatives (right-hand
sides of equations)
3. Assign to the variable function rkfixed:
R:=rkfixed(init_vect, low_bound, up_bound, num_seg, function)
MathCAD differential equations
4. Result is matrix (table) of three columns:
first contain independent values, 2nd column
contains first dependent variable values,
third second ones :
 x0 y1,0 y2, 0 


 x1 y1,1 y2,1 
R   x2 y1, 2 y2, 2 


... 
... ...
x

 N y1, N y2, N 
5. Results as a plot: R<1>,R<2>@ R<0>
MathCAD differential equations
x  x0
y0 xx0  y00
y1xx0  y10

MathCAD differential equations
Single second order equation
d y
dy 

 f  x, y , 
2
dx
dx 

2
Initial
condition
x  x0 , y x  x0  y0
 dy 
 y0
 
 dx  x  x0
1. Transform the second order equation to the
system of two first order equations:
y  z0 ,
dy dz0

 z1 ,
dx dx
 dz0
 dx  z1

 dz1  f x, z , z 
0 1
 dx
d 2 y dz1

2
dx
dx
x  x0
z0 xx  z00
0
z1xx0  z10
MathCAD differential equations
Example:
Solve the second order differential equation
(calculate: values of function and its first
derivatives) given by equation:

d2y
2

x
 3 y  y
2
dx
While y=10 and y’=-1 for x=0
In the range of x=<0,1>
MathCAD differential equations
System of equations
Starting vector
Vectoral function