# Ch 3 Note
###### tags: `熱傳學`
## 3.6
### A General Conduction Analysis
The general form of theenergy equation for an extended surface ==(Equation 3.66)==

$\dfrac{d}{dx}(A_c\dfrac{dT}{dx})-\dfrac{hP}{k}(T-T_{\infty})=0$

or

$\dfrac{d^2T}{dx^2}+(\dfrac{1}{A_c}\dfrac{dA_c}{dx})\dfrac{dT}{dx}-\dfrac{hP}{kA_c}(T-T_{\infty})=0$
### Fins of Uniform Cross-Sectional Area
For the simplest case of ==straight rectangular== and ==pin fins== of ==uniform cross section==,
with the base surface of temeperature $T(0)=T_b$ and extends into a fluid of temperature $T_{\infty}$

Due to $A_c and P$ are constant, equation 3.66 reduces to

$\dfrac{d^2 T}{dx^2}-\dfrac{hP}{kA_c}(T-T_{\infty})=0$

==Defining excess temperature $\theta$,==$\ \ \ \theta(x)\equiv T(x)-T_{\infty}$

with $T_{\infty}$ is constant, $d\theta/dx=dT/dx$, we get

$\dfrac{d^2\theta}{dx^2}-m^2\theta=0, \ \ \ where\ \ \ m^2\equiv\dfrac{hP}{kA_c}$

The general solution,

$\theta(x)=C_1e^{mx}+C_2e^{-mx}$

The boundary conditions are necessary,there are two condition is the temperature at the ==base of the fin(x=0)==
The second condition is at the ==fin tip(x=L)==, and there are four different physical situations.
- Case A -- Consider convection heat transfer from the fin tip
1. Applying an energy balance to a control surface about this tip

$hA_c[T(L)-T_{\infty}]=-kA_c\dfrac{dT}{dx}\vert_{x=L}$

or

$h\theta(L)=-k\dfrac{d\theta}{dx}\vert_{x=L}$

2. Substituting the general solution into the obove equations, we obtain

$\begin{aligned}
\theta_b&=C_1+C_2\\
h(C_1e^{mL}+C_2e^{-mL})&=km(C_2e^{-mL}-C_1e^{mL})
\end{aligned}$

The temperature distribution is,

$\dfrac{\theta}{\theta_b}=\dfrac{cosh\ m(L-x)+(h/mk)\ sinh\ m(L-x)}{cosh\ mL+(h/mk)\ sinh\ mL}$


3. From the above figure, it is evident that the fin heat tranfer rate can be evaluated in two ways,

- Applying Fourier's law at the fin base

$q_f=q_b=-kA_c\dfrac{dT}{dx}\vert_{x=0}=-kA_c\dfrac{d\theta}{d x}\vert_{x=0}$

- The rate at which heat is transferred by convection from the fin must equal the rate at which it is conducted through the base of the fin,

$\begin{aligned}
q_f&=\int_{A_f}h\ [T(x)-T_{infty}]\ dA_s\\
q_f&=\int_{A_f}h\theta(x)\ dA_s
\end{aligned}$
- Case B -- The convection heat transfer at the tip is negligible

That is, $\quad\dfrac{d\theta}{dx}\vert_{x=L}=0$
- Case C -- The temperature at the fin is prescribed, that is,$\quad\theta(L)=\theta_L$
- Case D -- The very long fin, as $\ \ L\rightarrow\infty,\ \theta_L\rightarrow 0$

### Fin Performance Parameters
- Fin Effectiveness and Fin Resistance