Ordinary Differential Equations Chapter 2 Ordinary Differential Equations

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

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

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.

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

Homogeneous Equations

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

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

VI VII Divide VII by VI

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

Equation Solved by Integrating Factors

Example

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

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

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

Thermal equilibrium of Liquid Input rate Output rate Accumulation rate

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

Dependent variable does not occur explicitly

Independent variable does not occur explicitly

Homogeneous Second sub

Second sub

Independent variable does not occur explicitly

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

Heat Balance Sectional area (A) = 0.0177 m2 Input Output Accumulation = 0

By Integrating factor Back Substitution Can’t Integrate

Assume constant heat loos

Using negative because As x increase T decrease

But

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

Complementary Function Auxiliary Eqn Assume Solution Unequal roots

Equal roots Example

Example

Particular Integral Undetermined Coefficients Inverse Operators

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

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

Example

(iv) f(x)

(iv) Modified f(x) Example

Example

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

Remove by reaction of A Input - Output

Simultaneous : Elimination t1 80 oC t2 t3 20 oC 0.96 kg/s Cp 1500 J/kgoC I II T2 45 oC T1 88 oC T0 174 oC 1.25 kg/s h1=1150 W/m2K h2=750 W/m2K V = 4500 kg A1 6.28 m2 A2 8.65 m2 Water stop for 1 hr: what are temperature 2. Then supply water at 1.25 kg/s for 1hr

Water Failed In - out = Acc Tank 1 Tank 2

Using Integrating Factor At 1 hr

Water Restore T1 T1 t2 t1

| | | | + + | | | | | | |

At 1 hr oC