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MechanicalMan
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I am trying to solve an advanced heat transfer problem and I have a 2nd order ODE. I can solve the homogeneous solution easily, but I am having trouble with the non-homogeneous solution.
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.
Energy balance on small element dx
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?
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?