LINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

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LINEAR SECOND ORDER ORDINARY DIFFERENTIAL
EQUATIONS
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• The general form of the equation
• where P, Q, R, and G are given functions
• Samples of 2nd order ODE
• Legendres equation
• Bessels equation
• Hypergeometric equation

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• When function g(x) is set to zero
• This is the homogeneous form of 2nd order ODE
• Suppose p and q in eqn above are continuous on
  altxltb then for any twice differentiable function
  f on altxltb, the linear differential operator L is
  defined as
• Lf f pf qf
• Ly y p(x)y q(x)y 0

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Solutions of Homogeneous Equation

• Theorem
• If y y1(x) and y y2(x) are solutions of the
  differential equation
• Ly y p(x)y q(x)y 0
• then the linear combination of y c1y1 (x)
  c2y2 (x), with c1 and c2 being arbitrary
  constants is also a solution.

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Application of 2nd Order ODE

• Two concentric cylindrical metallic shells are
  separated by a solid material. If the two metal
  surfaces are maintained at different constant
  temperatures, what is the steady state
  temperature distribution within the separating
  material?

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Solution
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Sample of Transport Model (1)

• Consider the axial flow of an incompressible
  fluid in a circular tube of radius R. By
  considering long tube and assuming q-component
  and r-component of velocities are negligible, one
  can reduce the z-component for constant r and m
• Equation of continuity reduces to

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Sample of Transport Model (2)

• Derive the temperature profile T, in a solid
  cylinder with heat generation if the governing
  differential equation is
• where the coordinate system indicates the
  independent variables r is the mass density and
  Cp the specific heat.

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Example 1

• Consider a long solid tube, insulated at the
  outer radius ro and cooled at the inner radius ri
  with uniform heat generation q within the solid.
• Determine the general solution for the
  temperature profile in the tube
• Suppose the maximum permissible temperature at
  the insulated surface ro is To. Identify
  appropriate boundary conditions that could be
  used to determine the arbitrary constants
  appearing in the general solution and find the
  temperature distribution.
• What is the heat of removal rate per unit length
  of tube?
• If the coolant is available at a temperature T,
  obtain an expression for the convection
  coefficient that would have to be maintained at
  the inner surface to allow for operation at the
  prescribed values of To and q.

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• Assumptions
• Steady state conditions
• One dimensional radial conduction
• Physical properties are constant
• Volumetric heat generation is constant
• Outer surface is adiabatic

ri
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Example 2

• A tubular reactor of length L and
  cross-sectional area 1.0 m2 is used to carry out
  a first order chemical reaction of the type
• A ? B
• The rate coefficient is k (sec-1). In a given
  feed rate of u m3/sec, the initial feed
  concentration of species A is Co and the
  diffusivity of A is D m2/sec. What is the
  concentration of A as a function of the reactor
  length? It may be assumed that during the
  reaction the volume remains constant and that
  steady-state conditions are established. Also
  there is no concentration variation in the
  section following the reactor.

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Example 3

• Two thin wall metal pieces of 1 outside diameter
  are connected by ½ thick and 4 diameter flanges
  that are carrying steam at 250oF. Determine the
  rate of heat loss from the pipe and the
  proportion that leaves the rim of the flange.
• Thermal conductivity of the flange metal is
  k220 Btu/h ft2oF ft-1
• The exposed surfaces of the flanges lose heat to
  the surroundings at T1 60oF according to heat
  transfer coefficient h 2 Btu/h ft2oF

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Pipe Flange
r
½ in
2 in