Advanced Heat Transfer - Conduction Cooling in Turbine Blade

[MechanicalMan]

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.

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?

 

[Redbelly98]

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.

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

Shouldn't this be
h[Tg - T(x)] P dx + qx A = [ qx - (dqx/dx) dx ]A
I.e.,

Heat in from ambient
+ Heat in from the +x side
= Heat out the -x side​

You seem to have some of your +/- signs off, plus there was a dx term missing in the "Heat out" expression. Also, partial derivatives don't apply since x is the only dependent variable.

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 agree with this, so it looks like you just had some typos in posting your earlier part of the calculation.

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 don't know if you really meant to write it this way. Where is the negative-root solution? Where is the x in the exponent terms?

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?

Since there are two constants to find (C1 and C2), we need two boundary conditions. An obvious one is the temperature at the base of the fin. Another one would relate to the heat transfer at the fin tip. Find equations to express those conditions, and you should be able to get C1 and C2.

 

[MechanicalMan]

Shouldn't this be
h[Tg - T(x)] P dx + qx A = [ qx - (dqx/dx) dx ]A
I.e.,

Heat in from ambient
+ Heat in from the +x side
= Heat out the -x side​

You seem to have some of your +/- signs off, plus there was a dx term missing in the "Heat out" expression. Also, partial derivatives don't apply since x is the only dependent variable.


I agree with this, so it looks like you just had some typos in posting your earlier part of the calculation.


I don't know if you really meant to write it this way. Where is the negative-root solution? Where is the x in the exponent terms?



Since there are two constants to find (C1 and C2), we need two boundary conditions. An obvious one is the temperature at the base of the fin. Another one would relate to the heat transfer at the fin tip. Find equations to express those conditions, and you should be able to get C1 and C2.

Thanks, I did make some typos in the original statement, which is why it may seem off at some points. I did miss the negative root and I did miss the x in the exponent of my original post. I have the boundary conditions, but my only concern is that if I relate the heat transfer by convection at the tip to the conduction through the fin, I'll have an expression that is dependent on x. I do not know what T(L) is, so if I make that substitution, do I not just complicate the problem even more?

 

[Redbelly98]

I do not know what T(L) is, so if I make that substitution, do I not just complicate the problem even more?

That's right, T(L) is not one of the boundary conditions.

Consider the dx element at the very tip of the fin. What is qx entering into that element (from the x+ direction)?

 

[MechanicalMan]

That's right, T(L) is not one of the boundary conditions.

Consider the dx element at the very tip of the fin. What is qx entering into that element (from the x+ direction)?

I managed to solve the problem, and after pages and pages of algebra, I have the solution in terms of hyperbolic functions. Thanks for the tips.