ORDINARY DIFFERENTIAL EQUATIONS (ODE)

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ORDINARY DIFFERENTIAL EQUATIONS(ODE)
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Differential Equations

• Heat transfer
• Mass transfer
• Conservation of momentum, thermal energy or mass

(4.1)
(4.2)
(4.3)
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ODE

• Definition
• Example
• A 3rd order differential equation for r r(t)
• Solution

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Important Issues

1. Existence of a solution
2. Uniqueness of the solution
3. How to determine a solution

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Linear Equation (1)

1. Rewrite 4.9
2. Determine

where m(t) is called an integrating factor
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Linear Equation (2)

• Multiply both sides of equation 4.10 by m(t)
• Observe that the left-hand side of eqn 4.12 can
  be written as
• or

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Linear Equation (3)

• Equation (4.12) can be rephrase as
• Integrate both sides of Equation (4.14) with
  respect to the independent variable

(4.14)
where c is the constant of integration
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Example 1

• Water containing 0.5 kg of salt per liter is
  poured into a tank at a rate of 2 l/min, and the
  well-stirred mixture leaves at the same rate.
  After 10 minutes, the process is stopped and
  fresh water is poured into the tank at a rate of
  2 l/min, with the new mixture leaving at 2 l/min.
  Determine the amount (kg) of salt in the tank at
  the end of 20 minutes if there were 100 liters of
  pure water initially in the tank.

2 l/min
½ kg salt/l
CA
2 l/min, CA (l/min)
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Solution
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Example 2

• Consider a tank with a 500 l capacity that
  initially contains 200 l of water with 100 kg of
  salt in solution. Water containing 1 kg of salt/l
  is entering at a rate of 3 l/min, and the mixture
  is allowed to flow out of the tank at a rate of
  2 l/min. Determine the amount (kg) of salt in the
  tank at any time prior to the instant when the
  solution begins to overflow. Determine the
  concentration (kg/l) of salt in the tank when it
  is at the point of overflowing. Compare this
  concentration with the theoretical limiting
  concentration if the tank had infinite capacity.

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Solution
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THEOREM

• If the functions p and g are continuous on an
  open interval a lt x lt b containing the point x
  x0, then there exists a unique function y f(x)
  that satisfies the differential equation
• y p(x)y g(x)
• for a lt x lt b , and that also satisfies the
  initial condition
• y(x0) y0
• where y0 is an arbitrary prescribed initial value.

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Higher ODE Reduces to 1st Order
In general, it is sufficient to solve first-order
ordinary differential equations of the form
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• Nonlinear equations can be reduced to linear ones
  by a substitution. Example
• y p(x)y q(x)yn
• and if n ¹ 0,1 then
• n(x) y1-n(x)
• reduces the above equation to a linear equation.

(4.16)
(4.17)
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Example 3

• Suppose that in a certain autocatalytic chemical
  reaction a compound A reacts to form a compound
  B. Further, suppose that the initial
  concentration of A is CA0 and that CB(t) is the
  concentration of B at time t. Then CA0 CB (t)
  is the concentration of A at time t. Determine
  CB(t) if CB(0) CB0.

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Solution
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NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS
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NONLINEAR EQUATIONS

• Rewrite as

If M is a function of t only, and N is a function
of r only, then
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NONLINEAR EQUATIONS

• Consider

subject to
Then, it is separable and results in
(4.16)
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• Simplifying left-hand side 1st consider the
  fraction

(4.17)
where a and g are constants to be determined.
Then
If we put
then
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If we put
then

• Rewrite equation 4.17

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• And equation 4.16 becomes
• which integrates to

where m1 is an arbitrary constant to be
determined with the given initial condition. _at_ t
0, CB CB0, then
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Example of Problem Setup

• Consider the continuous extraction of benzoic
  acid from a mixture of benzoic acid and toluene,
  using water as the extracting solvent. Both
  streams are fed into a tank where they are
  stirred efficiently and the mixture is then
  pumped into a second tank where it is allowed to
  settle into two layers. The upper organic phase
  and the lower aqueous phase are removed
  separately, and the problem is to determine what
  proportion of the acid has passed into the
  solvent phase.

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Example (cont)

• List of assumptions
• Combine the two tanks into a single stage
• Express stream-flow rates on solute-free basis
• Steady flowrate for each phase
• Toluene and water are immiscible
• Feed concentration is constant
• Mixing is efficient, the two streams leaving the
  stage are in equilibrium with each other given by
  y mx
• Composition stream leaving is the same with the
  composition in the stage
• The stage initially contains V1 liter toluene, V2
  liter water and no benzoic acid

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Problem 1

• Consider an engine that generates heat at a rate
  of 8,530 Btu/min. Suppose this engine is cooled
  with air, and the air in the engine housing is
  circulated rapidly enough so that the air
  temperature can be assumed uniform and is the
  same as that of the outlet air. The air is fed to
  the housing at 6lb-mole/min and 65oF. Also, an
  average of 0.20 lb-mole of air is contained
  within the engine housing and its temperature
  variation can be neglected. If heat is lost from
  the housing to its surroundings at a rate of
  Ql(Btu/min) 33.0(T-65oF) and the engine is
  started with the inside air temperature equal to
  65oF.
• Derive a differential equation for the variation
  of the outlet temperature with time.
• Calculate the steady state air temperature if the
  engine runs continuously for indefinite period of
  time, using Cv 5.00 Btu/lb-mole oF.

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Problem 2

• A liquid-phase chemical reaction with
  stoichiometry A ? B takes place in a semi-batch
  reactor. The rate of consumption of A per unit
  volume of the reactor is given by the first order
  rate expression
• rA (mol/liter.s) kCA
• where CA (mol/liter) is the reactant
  concentration. The tank is initially empty. At
  time t0, a solution containing A at a
  concentration CA0(mol/liter) is fed to the tank
  at a steady rate f(liters/s). Develop
  differential balances on the total volume of the
  tank contents, V, and on the moles of A in the
  tank, nA .

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Solving ODEs using Numerical Methods

• Initial Value Problem (IVP)
• y -yx
• y(0) 2, y(0) 1
• Boundary Value Problem (BVP)
• y -yx
• y(0) 2, y(1) 1

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General Procedure

• Re-write the dy and dx terms as ?y and ?x and
  multiply by ?x
• Literally doing this is Eulers method

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Tank mixing problem
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Mixing tank
Dt Error Et at t600
300 1.4
150 0.61
100 0.39
50 0.19
30 0.11
15 0.055
10 0.036
5 0.018
3 0.011
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Matlab output
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Error analysis

• We saw that the error depended on the time step
  size
• Why?
• Extrapolating the curve using a linear function

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Improvements to Eulers Method

• Euler
• Heuns method (predictor-corrector)
• Procedure
• calc yi1 with Euler (predictor)
• calc slope at yi1
• calc average slope
• use this slope to calc new yi1 (corrector)

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Heun example
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Midpoint Method

• Use Euler to calculate a midpoint location
• evaluate slope y at the midpoint
• use that to calculate full step location

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Runge-Kutta
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R-K General form
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R-K 1st Order Form
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R-K 2nd Order Form
y(x)
xi xi1 x
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RK2 Options
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RK2 Options
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R-K 2nd Order Form
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RK 3rd Order Form
y(x)
xi xi1 x
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RK 4th Order
y(x)
xi xi1 x
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Example y?xy, y(0)0
x yo k1fi k2f(xh/2,yh/2k1) k3f(xh/2,yh/2k2) k4f(xh,yhk3) ynyo1/6(k12k22k3k4)h
0 0 0 0.1 0.11 0.222 0.02140
0.2 0.0214 0.221 0.344 0.356 0.493 0.0918
0.4 0.092 0.492 0.641 0.656 0.823 0.2221
0.6 0.222 0.822 1.004 1.023 1.227 0.4255
0.8 0.426 1.226 1.448 1.470 1.720 0.718
1 0.718 1.718 1.990 2.017 2.322 1.120
1.2 1.120 2.320 2.652 2.685 3.057 1.655
1.4 1.655 3.055 3.461 3.501 3.955 2.353
1.6 2.353 3.953 4.448 4.498 5.052 3.250
1.8 3.250 5.050 5.654 5.715 6.393 4.389