Homework Statement
I'm unsure of what exactly is changing the heat transfer direction in the triangular fin.
Homework Equations
$$q_{x} = -kA(x)\frac{dT(x)}{dx} (1)$$
$$q_{x+dx} = -kA(x)\frac{dT(x)}{dx} - k\frac{d}{dx}[A(x)\frac{dT(x)}{dx}] (2)$$
$$dq_{conv} = h(x)dS(x)P[T(x) - T_{∞}] (3)$$...
Homework Statement
Solve for
xy'' + y' +αy + βxy = 0
α and β are constants
The Attempt at a Solution
What I initially had in mind was:
xy'' + y' +αy + βxy = x²y'' + xy' +αxy + βx²y = 0
y = \sum_{n=0}^\infty a_n x^{n}
xy = \sum_{n=0}^\infty a_n x^{n+1} = \sum_{n=1}^\infty a_{n-1} x^{n} = a_0x...
Wouldn't it have to be
x²θ′′+xθ′−mθ=0
for it to be an Euler differential equation?
I tried to do power series but I got stuck in the end:
y=\sum_{n=0}^\infty a_k x^k
θ'=\sum_{n=1}^\infty ka_k x^{k-1}
θ''=\sum_{n=2}^\infty k(k-1)a_k x^{k-2}
\sum_{n=2}^\infty k(k-1)a_k x^{k-1} +...
I don't think I expressed myself properly before but anyway, finally managed to get it in the way I wanted:
http://i.share.pho.to/0a247f78_o.jpeg
http://i.share.pho.to/57739467_o.jpeg
T(x) - T_e = θ(x)
m = \frac{h_e(L/e)(1+(e/L)²)^{1/2}}{k}
xθ'' + θ' - mθ = 0
Just need to solve that now.
I'm trying to deduce the differential equation for temperature for a triangular fin:
I know that for a rectangular fin, such as:
I can do:
Energy entering the left:
q_x= -kA\frac{dT(x)}{dx}
Energy leaving the right:
q_{x+dx} = -kA\frac{dT(x)}{dx} - kA\frac{d² T(x)}{dx²}dx
Energy lost by...