LINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

The general form of the equation: where P, Q, R, and G are given functions Samples of 2nd order ODE: Legendre’s equation Bessel’s equation Hypergeometric equation

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 a<x<b then for any twice differentiable function f on a<x<b, the linear differential operator L is defined as: L[f] = f” + pf’ +qf L[y] = y” + p(x)y’ + q(x)y = 0

Solutions of Homogeneous Equation Theorem If y = y1(x) and y = y2(x) are solutions of the differential equation L[y] = 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.

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?

r + Dr r a R

Solution

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:

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.

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.

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

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.

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 k=220 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

Pipe Flange r + Dr r ½ in 2 in