Polymath Tutorial
CBE 40445
Monday, 29th August 2011
Polymath Capabilities
•
•
•
•
Linear Equations Solver
Nonlinear Equations Solver
Differential Equations Solver
Regression
– Linear & Polynomial
– Data Table
– Multiple Linear or Multiple Nonlinear Regression
• Additional Capabilities
– Export to Excel
– Calculator and unit conversion tools
– Polymath Export to Matlab (see help Menu for more
information)
Today’s Class
• Provide information on Polymath
– How to open and use the software
• Learn how to input and solve nonlinear and
differential equations
– Provide step-by-step instructions so you can practice
and compare results
• Address the question: Why learn another
software program?
Starting Polymath
1. Open Polymath Fogler from the start menu.
Navigating the Menu
1 2 3 4
5 6 7 8
9 10 11 12 13 14 15 16 17 18 19
1. New program
13. Solve system of ODEs
2. Open file
14. Regress and analyze data
3. Open recent file
15. Calculator
4. Save file
16. Unit converter
5. Cut
17. Scientific constants
6. Copy
18. Setup preferences
7. Paste
19. Help
8. Delete
9. Find
10. Find and replace
11. Solve system of linear equations
12. Solve system of nonlinear equations
For more information: Help Menu
Help Menu
Example 1: Nonlinear Equation Solver
Consider the following set of equations:
k·CA12 = v·(CA0-CA1)/V
k·CA22 = v·(CA1-CA2)/V
where k = 0.075, v = 30,
CA0 = 1.6, and CA2 = 0.2 CA0.
Therefore, we have two remaining variables: variables
CA1 and V.
We will use initial estimates: CA1=1 and V=300.
How do we do this in Polymath?
Nonlinear Equation Solver
Rearrange:
0 = k·CA12 – v(CA0 - CA1)/V
0 = k·CA22 – v(CA1-CA2)/V
In Polymath, the appropriate forms for these equations
are:
f(CA1) = k·CA12 – v(CA0 - CA1)/V = 0
f(V) = k·CA22 – v(CA1-CA2)/V = 0
Can f(CA1) and f(V) be switched?
1. Open a new sheet by clicking on the blank sheet in the upper
corner.
2. Open Nonlinear Equations
3. In the upper left corner, click the f(x)+ button. This will allow
you to enter one of the two equations.
4. A new window will open.
5. Type in the first equation as shown before.
3. In the upper left corner, click the f(x)+ button. This will allow
you to enter one of the two equations.
4. A new window will open.
5. Type in the first equation as shown before.
6. Press Done.
7. You should see the new equation in the worksheet, as well as
the comments.
8. Enter the second equation by repeating the same steps.
9. You will see the two equations and two initial value guesses in the
worksheet.
10. The red X displays the undefined variables.
11. Enter the undefined variables by either 1) clicking on the x(=)+
button in the menu or 2) entering directly into the worksheet.
9. You will see the two equations and two initial value guesses in the
worksheet.
10. The red X displays the undefined variables.
11. Enter the undefined variables by either 1) clicking on the x(=)+
button in the menu or 2) entering directly into the worksheet.
12. Now you should see a blue check mark indicating this system of
equations can be solved.
13. To solve, click the purple arrow.
Results:
CA1 initial guess: 1
V initial guess: 300
CA1 = 0.602
V= 1102
Example 2: Solving a System of
Ordinary Differential Equations (ODEs)
This system may contain two types of equations: 1) first order
ordinary differential equations and 2) explicit algebraic
equations.
The differential equations must be entered in the following form:
d(x)/d(t)= ……….
Here t is the independent variable and x is a dependent variable.
Auxiliary algebraic equations must be entered in the form:
x = …….
For differential equations, an initial value is needed for all
variables. A final value is needed for the independent variable.
1. Open a new program (blank sheet)
2. Select Differential Equations
3. Select the d(x)+ button to add a new differential equation.
What should we solve?
Series Reaction:
k1
k2
ABC
Using Polymath, we can monitor the conversion
of A, the formation (and disappearance) of B,
and the formation of C.
Species Balances:
d ( A)
 k1* A
d (t )
d ( B)
 k1* A  k 2 * B
d (t )
d (C )
 k2* B
d (t )
Let’s solve this series of differential equations.
4. Select the d(x)+ button to add a new differential equation.
5. Add the three equations with the following initial
parameters: A(0)=1, B(0)=0, C(0)=0
6. We still have two undefined variables: k2 and k1
7. Enter k1 as 1 and k2 as 2.
5. Add the three equations with the following initial
parameters: A(0)=1, B(0)=0, C(0)=0
6. We still have two undefined variables: k2 and k1
7. Enter k1 as 1 and k2 as 2.
We have an error. We need to enter the initial and final values for
our independent variable (t).
8. To add the final values for t, select Edit followed by
Define Initial and Final values…
9. Then enter the initial and final values for t: 0 and 3.
8. To add the final values for t, select Edit followed by Define
Initial and Final values…
9. Then enter the initial and final values for t: 0 and 3.
10. Select Problem and Arrange Equations to sort the list.
8. To add the final values for t, select Edit followed by Define
Initial and Final values…
9. Then enter the initial and final values for t: 0 and 3.
10. Select Problem and Arrange Equations to sort the list.
Before we solve, let’s add a title.
11. Select Edit and Enter Problem Title…
12. Click the Graph box to add graphing options.
Graph
13. To run the program, click the purple arrow.
Run Program
Final values:
A = 0.050
B = 0.047
C = 0.903
Concentration Profiles
Graph generated to show disappearance of reactant A.
Concentration Profiles
Graph generated to show disappearance of reactant A.
Click
Graph
to
modify
Concentration Profiles
Click
brush to
edit
How can B maximized?
k1
k2
• ABC
• If k1>>k2, what happens?
• Change k2 to 0.5 and k1 to 2
With k1 = 1 and k2 =2,
Bmax = 0.25
With k1 = 2 and k2 =0.5,
Bmax = 0.63
To order Polymath
• http://www.polymath-software.com/
• 5 day free trial
– Use the trial to see if you would like to purchase
the software
• $20/4 months