1. Computational Method in Chemical Engineering (TKK-2109)
14/15 Semester 5
Computational Method in Chemical Engineering (TKK-2109)
Instructor: Rama Oktavian
Office Hr.: M.13-15, T , W , F

2. Ordinary Differential Equation
An equation relating a dependent variable to one or more independent variables by means of its differential coefficients with respect to the independent variables is called a “differential equation”.
Ordinary differential equation
only one independent variable involved: x
Partial differential equation
more than one independent variable involved: x, y, z, 

3. Ordinary Differential Equation
Ordinary differential equations are classified in terms of order and degree
Order of an ordinary differential equation is the same as the highest order derivative
The degree of a differential equation is the highest power of the highest order differential coefficient that the equation contains after it has been rationalized.

4. Ordinary Differential Equation
Ordinary differential equations are classified in terms of order and degree
3rd order O.D.E.
1st degree O.D.E.
1st order O.D.E.
2nd degree O.D.E.

5. Ordinary Differential Equation
Linear or non-linear
Differential equations are said to be non-linear if any products exist between the dependent variable and its derivatives, or between the derivatives themselves.
Product between two derivatives ---- non-linear
Product between the dependent variable themselves ---- non-linear

6. Ordinary Differential Equation
Swinging pendulum
A second-order
nonlinear ODE.
Falling parachutist problem

7. Ordinary Differential Equation
Ordinary differential equations are classified in terms of order and degree
3rd order O.D.E.
1st degree O.D.E.
1st order O.D.E.
2nd degree O.D.E.

8. Ordinary Differential Equation
ODE in chemical engineering
Illustrative Example: A Blending Process
An unsteady-state mass balance for the blending system:

9. Ordinary Differential Equation
ODE in chemical engineering or
where w1, w2, and w are mass flow rates.
The unsteady-state component balance is:
For constant , Eqs. 2-2 and 2-3 become:

10. Ordinary Differential Equation
ODE in chemical engineering
Equation 2-13 can be simplified by expanding the accumulation term using the “chain rule” for differentiation of a product:
Substitution of (2-14) into (2-13) gives:
Substitution of the mass balance in (2-12) for in (2-15) gives:

11. Ordinary Differential Equation
ODE in chemical engineering
After canceling common terms and rearranging (2-12) and (2-16), a more convenient model form is obtained:

12. Ordinary Differential Equation
ODE in chemical engineering

13. Ordinary Differential Equation
Numerical method for solving ODE
Euler’s method Φ
Step size, h x y
x0,y0
True value
y1, Predicted
value
Slope
Figure 1 Graphical interpretation of the first step of Euler’s method

14. Ordinary Differential Equation
Numerical method for solving ODE
Euler’s method Φ
Step size h
True Value
yi+1, Predicted value yi x y xi
xi+1
Figure 2. General graphical interpretation of Euler’s method

15. Ordinary Differential Equation
Numerical method for solving ODE
Euler’s method
How does one write a first order differential equation in the form of
Example
is rewritten as
In this case

16. Ordinary Differential Equation
Example
Numerical method for solving ODE
Euler’s method
The concentration of salt, in a home made soap maker is given as a function of time by
At the initial time, t = 0, the salt concentration in the tank is 50g/L. Using Euler’s method and a step size of h = 1.5 min, what is the salt concentration after 3 minutes.

17. Ordinary Differential Equation
Example
Numerical method for solving ODE
Euler’s method
Step 1:
is the approximate concentration of salt at

18. Ordinary Differential Equation
Example
Numerical method for solving ODE
Euler’s method
Step 2:
is the approximate concentration of salt at

19. Ordinary Differential Equation
Numerical method for solving ODE
The exact solution of the ordinary differential equation is given by
The solution to this nonlinear equation at t=3 minutes is

20. Ordinary Differential Equation
Numerical method for solving ODE
Figure 3. Comparing exact and Euler’s method

21. Ordinary Differential Equation
Numerical method for solving ODE
Table 1. Concentration of salt at 3 minutes as a function of step size, h 3
1.5
0.75
0.375
0.1875
−362.50
720.31
284.65
10.718
10.714
373.22
−709.60
−273.93 −
3483.0
6622.2
2556.5
Step

22. Ordinary Differential Equation
Numerical method for solving ODE
Figure 4. Comparison of Euler’s method with exact solution for different step sizes

23. Ordinary Differential Equation
Numerical method for solving ODE
Runge Kutta 2nd order method
Taylor’s expansion

24. Ordinary Differential Equation
Numerical method for solving ODE
Runge Kutta 2nd order method
Taylor’s expansion

25. Ordinary Differential Equation
Numerical method for solving ODE
Runge Kutta 2nd order method
However, it is relatively difficult to find second derivative of ODE
For
Runge Kutta 2nd order method is given by
where

26. Ordinary Differential Equation
Numerical method for solving ODE
Heun’s method x y xi
xi+1
yi+1, predicted yi
Here a2=1/2 is chosen
resulting in
where
Figure 1 Runge-Kutta 2nd order method (Heun’s method)

27. Ordinary Differential Equation
Midpoint Method
Numerical method for solving ODE
Here
is chosen, giving
resulting in
where

28. Ordinary Differential Equation
Ralston’s Method
Numerical method for solving ODE
Here
is chosen, giving
resulting in
where

29. Ordinary Differential Equation
Example
The concentration of salt, in a home made soap maker is given as a function of time by
At the initial time, t = 0, the salt concentration in the tank is 50g/L. Using Euler’s method and a step size of h=1.5 min, what is the salt concentration after 3 minutes.

30. Solution Step 1: x1 is the approximate concentration of salt at

31. Solution Step 2: x1 is the approximate concentration of salt at

32. Solution Table 1. Effect of step size for Heun’s method Step size, 3
1.5
0.75
0.375
0.1875
1803.1
3579.6
442.05
11.038
10.718
−1792.4
−3568.9
−431.34 − −
16727
33306
4025.4
3.0079
(exact)

33. Ordinary Differential Equation
Numerical method for solving ODE
Runge Kutta 4th order method
Taylor’s expansion

34. Ordinary Differential Equation
Numerical method for solving ODE
Runge Kutta 4th order method
However, it is relatively difficult to find second and third derivative of ODE
For
Runge Kutta 4th order method is given by

35. Ordinary Differential Equation
Example
The concentration of salt, in a home made soap maker is given as a function of time by
At the initial time, t = 0, the salt concentration in the tank is 50g/L. Using Euler’s method and a step size of h=1.5 min, what is the salt concentration after 3 minutes.

36. Solution
Step 1:

37. Solution is the approximate concentration of salt at

38. Solution
Step 2:

39. Solution is the approximate concentration of salt at

40. Solution
Table 1 Value of concentration of salt at 3 minutes for different step sizes
Step size, 3
1.5
0.75
0.375
0.1875
14120
11455
25.559
10.717
10.715
−14109
−11444
−14.843 − −
131680
106800
138.53
(exact)

41. Thank You !