1. Ordinary Differential Equations
Chapter 2
Ordinary Differential Equations

2. 2.1 Introduction Ordinary Differential Equation
(2.1)
Partial Differential Equation (Chapter 8)
(2.2)

3. 2.2 Order and Degree
Order = 3, Degree =1, Non Linear

4. Equations in which the variables can be separated Homogeneous
2.3 First Order
Exact equations
Equations in which the variables can be separated
Homogeneous
Equation solvable by an integrating factor.

5. Equations in which the variables can be separated
Exact equations
Equations in which the variables can be separated

6. Homogeneous Equations

7. Sparging chlorine gas into benzene
Example Batch Reactor
Sparging chlorine gas into benzene
How much chroline must be added to give maximum yield of C6H6Cl
Assume isothermal

8. p = moles of chlorine present q = moles of benzene present
Basis 1.0 mole of Chlorine, at time q, V=const
p = moles of chlorine present
q = moles of benzene present
r = moles of monochlorobenzene present
s = moles of dichlorobenzene present
t = moles of trichlorobenzene present
q+r+s+t =1
Amount of chlorine consumed
y=r+2s+3t

9. VI
VII
Divide VII by VI

12. MatLab Divide VIII by VI gives
After eliminate r … using integrating factor
The result is :
MatLab
To determine t using : q+r+s+t =1

14. Equation Solved by Integrating Factors

15. Example

17. Example3 Horizontal tank 1m diameter 2 m long
Insulate with asbestos l = 4 cm.
Charge 95 oC liquid
Hold for 5 days

18. k =0.2 W/m K
h1=150 W/m2K
r =1000 kg/m3
Cp = 2500 J/kg K
h2=10 W/m2K
Plot Liquid temperature with time

19. Surface Area
Rate of heat loss by liquid
Rate of heat loss through lagging
Rate of heat loss to surrounding
All Rates are equal , eliminate Tw

20. Thermal equilibrium of Liquid
Input rate
Output rate
Accumulation rate

22. Second Order Differential Eqs
Non-linear
Dependent Variables does not occur explicitly
Independent Variables does not occur explicitly
Homogeneous Equation
Linear
The coefficients in the equation are constant
The coefficients are functions of independent varible

23. Dependent variable does not occur explicitly

24. Independent variable does not occur explicitly

25. Homogeneous
Second sub

26. Second sub

27. Independent variable does not occur explicitly

28. Example Estimate the duty of water cooler h = 1.7 W/m2 oC 20 oC
Insulator
Water Cooler
50 oC
1500 oC
Furnace
f = 15 cm
150 oC
30 cm
Estimate the duty of water cooler

29. Heat Balance
Sectional area (A) = m2
Input
Output
Accumulation = 0

31. By Integrating factor
Back Substitution
Can’t Integrate

32. Assume constant heat loos

34. Using negative because
As x increase T decrease

35. But

37. Linear Differential Equation
Second Order and
Complementary Function 2 constants
Particular Integral

38. Complementary Function
Auxiliary Eqn
Assume Solution
Unequal roots

39. Equal roots
Example

40. Example

41. Particular Integral
Undetermined Coefficients
Inverse Operators

42. Undetermined Coefficients
(i)f(x) constant ,C

43. (ii) f(x) polynomial
(iii) f(x) Terx

44. Example

45. (iv) f(x)

46. (iv) Modified f(x)
Example

47. Example

48. Example A B y,c =Conc. A Bulk Flow of A Diffusion of A k L=1 m u m3/sdx x
y,c =Conc. A
Bulk Flow of A
Diffusion of A

49. Remove by reaction of A
Input - Output

51. Simultaneous : Elimination
t1 80 oC t2
t3 20 oC
0.96 kg/s
Cp J/kgoC I II
T2 45 oC
T1 88 oC
T oC
1.25 kg/s
h1=1150 W/m2K
h2=750 W/m2K
V = 4500 kg
A m2
A m2
Water stop for 1 hr: what are temperature
2. Then supply water at 1.25 kg/s for 1hr

52. Water Failed
In - out = Acc
Tank 1
Tank 2

53. Using Integrating Factor
At 1 hr

54. Water Restore T1 T1 t2 t1

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57. At 1 hr oC