- #1

- 25

- 0

## Homework Statement

Given a turbine blade and asked to model as a one-dimensional fin, subject to the following constraints:

Troot = 900 deg F

Lfin = 3.6 in

A = 0.506 in^2

Tg (external gas flowing over the fin/blade) = 1500 deg F

Pfin = 2.1 in

k = 8.1 BTU/hr ft deg F

h = 36.6 BTU/hr ft^2 deg F

There is a small element dx with the following thermal energy in/out of it: qx coming out in the negative x-direction, qg into it from the ambient and q(x+dx) coming into the element. At the root, there is a value q0 transferring into the disk through conduction. The temperature along the length of the blade is a function of x.

## Homework Equations

Energy balance on small element dx

## The Attempt at a Solution

This is what I have so far:

qin = qout

h[T(x) - Tg]Pdx + [(qx - ∂qx/∂x)A] = - qxA

h[T(x) - Tg]P + ∂qx/∂x*A = 0

h[T(x) - Tg]P + ∂/∂x{-k∂T(x)/∂x}A = 0

h[T(x) - Tg]P - k (∂2T/∂x2)A = 0

d2T/dx2 - (hP/Ak)*T(x) = -(hP/Ak)*Tg

I know that the solution to this non-homogeneous ODE is a combination of the complimentary solution and the particular solution. The roots of the homogeneous ODE yield:

r = +/- (hP/Ak)^1/2

Therefore the complementary portion of the solution is

T(x) = C1e^(hP/Ak)^0.5 + C2e^(hP/Ak)^0.5

I'm stuck now trying to find the particular solution. I am not sure which method to use. I tried variation of parameters but I can't seem to get something that makes sense. I know I should end up with a hyperbolic function, but I'm stuck. Any ideas?