Texas A&M University Artie McFerrin Department of Chemical
Engineering
Using Mathcad to solve non-linear equations
CHEN 354: Fall 2008 Prof. Perla B. Balbuena
Mathcad is a mathematical software easy to use. Among the capabilities of
Mathcad are: Solving non-linear equations, graphing functions in two or three dimensions,
solving differential equations, performing symbolic calculations, etc.
The objective of this quick guide is to provide you the basic knowledge to use
Mathcad in order to solve systems of non-linear equations. First, we go through the basic
operations in Mathcad, such as defining variables and evaluating functions. Then, we
solve a vapor-liquid equilibrium problem involving a system of non-linear equations.
Mathcad basics
The Mathcad window is shown in the following figure. All the expressions and
objects are inserted in the workspace
T.A. Rafael Callejas Tovar | jrc2267@chemail.tamu.edu Page | 1
Prof. Perla B. Balbuena
The “Math toolbar” is useful to get quick access to the different capabilities of Mathcad.
Defining variables and functions
To define a variable, just place the cursor in the workspace and type its name
followed by the “Definition symbol”, located in the “Evaluation toolbar” or typing the “:”
key. For example, define P equal to 101.33.
If you want to know the numerical value of a previously defined variable, place the
cursor below its definition, type the name of the variable followed by the “Evaluate
numerically symbol” whose shortcut is the “=” key.
Defining
functions is simple.
Type the name of the
function followed by
the name of its independent variables enclosed in parenthesis. Then use the “Definition
symbol” and create the correct expression aided by “Space bar”.
You can evaluate the functions for given values of their independent variables.
T.A. Rafael Callejas-Tovar jrc2267@chemail.tamu.edu
Prof. Perla B. Balbuena
Useful tips
If you want
to
use
Greek
symbols to name
the variables, use
the “Greek symbol
toolbar”. If you know
the equivalent letter
in the Latin alphabet, type the letter then “CTRL+g”. For example, to obtain gamma type
“g” then “CTRL+g”
To insert a textbox just type the “double quote key” and the region will become a
text region. All the characters inside the textbox will be treated as text. When selecting a
math region the cursor is blue, in a text region the cursor is red.
The Given… Find solution block
The “Given… Find block” is the tool that allows us to solve non-linear equations.
The following is an example of its implementation. It is required to provide initial
approximations for the unknowns. The block starts with the word “Given” written as a
math expression; below it we place all the equations we want to solve. These equations
may contain previously defined functions, is important to use the “Boolean equal symbol”
to write the equations (“Ctrl+=”).
T.A. Rafael Callejas-Tovar jrc2267@chemail.tamu.edu
Prof. Perla B.
Balbuena
The “Given… Find block” ends with the “Find” command. Its arguments are the
unknowns that we want to calculate. The result of the “Find” command may be assigned
to a variable or function. In the example shown above the results are stored in the
function “Solution(y1)”. This is useful when we want to solve the system for several values
of y1.
In the example shown above, we have a two-equations/two-unknonws system.
Therefore, the solution porvided by the “Find” command will be a two-rows/one-column
vector. We can split assign each vector element to separate functions as follows.
To make reference to the elements of a vector use the subscript, “[“ key. In
Mathcad the array subscripts begin at zero.
Making plots
To make plots, first define a “range variable”, in general, it will be in the x-axis of
the 2Dplot. A range variable takes an initial value and add it a constant increment until a
final value. To define it, name the variable, use the definition symbol, put the initial value,
type a comma “,”, put the increment and type a semicolon “;”, then type the final value
T.A. Rafael Callejas-Tovar jrc2267@chemail.tamu.edu
Insert a 2D-plot template (“Shift+2”), fill the blanks accordingly to the functions you
want to graph. You can format the plot by selecting it and right click > Format…
Example: Create T-x-y and x-y diagrams
Solve the following problem using Mathcad. For the system methanol(1)/methyl
acetate(2), the following equations provide a reasonable correlation for the activity
coefficients:
γ
)
1
= exp
where
(
2
2
= exp
A =14.25326 −
Ax )γ =
2
2
exp
(
Ax
2
1
2.771 − 0.00523 T
In addition, the following Antoine equations provide vapor pressures:
Sat Sat
P ⎛ 3643.310 ⎞ P ⎛ 2665.54 ⎞
1
= exp 16.59158 −
⎜⎟⎜ ⎟
kPa TK − 33.424 kPa TK − 53.424
⎝⎠⎝ ⎠
Assuming the validity of
yP = x γ P (i = 1,2, …, N )
Sat
i iii
Calculate T and { x }
i
diagrams at this pressure.
, for
P = 101.33 kPa and y =
1
0.40
. Also, prepare T-x-y and x-y
T.A. Rafael Callejas-Tovar jrc2267@chemail.tamu.edu
Using Mathcad to create T-x-y and x-y diagrams
Methanol(1)/Methyl acetate(2)
Input data and expressions
Pressure, kPa
P := 101.33
(T) :=
2.771 −⋅
Temperature dependence of the activity coefficient parameter, T in K
0.00523 T
The activity coefficients depend on temperature and composition, T in K
γ1(x1 , T)
:= exp
⎡⎣
A(T) ⋅ (1 − x1)
2
⎤⎦
( 2)
γ2(x1 , T) := exp A(T) ⋅ x1
I
n
i
t
i
a
l
a
p
p
r
o
x
i
m
a
t
Antoine equations, T in K
⎛⎜⎝
⎞⎟⎠
:= 300
⎞⎟⎠
3643.310
Psat1 (T)
16.59158 −:= exp
T − 33.424 2665.54
T − 53.424
⎛⎜⎝
Psat2 (T)
14.25326 −:= exp
ions for
the
unknown
s
x1 := 0.5
Mathcad
solution
block
Given x1⋅γ1(x1 , T) ⋅ Psat1(T) = y1 P
⋅ (1 − x1) ⋅γ2(x1 , T) ⋅ Psat2(T) = (1 − y1) ⋅ P
Solution(y1) := Find(x1 , T) The solution of the system of equations is written as a function of y1. This is
useful if you want to solve the system for several values of y1.
)
: Solution(y1)0
=
Since we have 2 unknows in the system of equations, the solution
vector is formed by 2 rows and 1column. The order of the
unknows in this vector corresponds to the definition of the "Find"
command.
: Solution(y1)1
=
Suppose that we want to know theIndew-temperature
and
composition
Mathcad the array
origin
subscriptof
is 0 by default.
T(y1)
the first condensed drop for y1 = 0.4. We just have to evaluate the
previously defined functions,T(y1) and x1(y1), at y1 = 0.4 x1(0.4) = 0.46
T(0.4) = 326.697 If we want to create T-x-y and x-y diagrams, we have
to evaluate T(y1) and x1(y1) at several points of y1. Therefore, we
define y1 as a range variable. y1 := 0 , 0.01 .. 1 y1 take values from 0 to
1 using a constant increment of 0.01 First values of y1
y1 =
0
.
0
5
0
.
0Ins
1ert
the
02D.plot
0and
2fill
the
0bla
.nks
0as
3follo
w
0
.
0340
4
0
335
T( y1 ) T( y1 ) 330
325
00.20.40.60.8
y1 , x1 ( y1 )
1
y1
0.5
y1
0
x1 ( y1 ), y1
You can use the previously
defined funtions T(y1) and
x1(y1) to build
other functions,
for example, you can
calculate You can
use the previously defined
funtions T(y1) and
x1(y1) to build other
functions, for
example, you can calculate
ln(γ1) and ln(γ2)
00.20.40.60.8
1
ln(γ1 (
y1 , T( y1 ))) ln(γ2 ( y1 , T( y1 )))
0.5
0 x1 ( y1 )
00.20.40.60.8
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