# [熱力學]第五、六週筆記
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# Ch 4 Energy Analysis of Closed System
For a closed system, first law of thermodynamics
$$\Delta E=Q-w;~\Delta e=q-w$$$$dE=\delta Q-\delta W;~de=\delta q-\delta w$$$$\frac{dE}{dt}=\delta \dot Q-\delta \dot W;~\frac{de}{dt}=\delta \dot q-\delta \dot w$$$$E=U+KE+PE;~e=u+ke+pe$$
- Special cases
1. Stationary System: $$\Delta KE=\Delta PE\Rightarrow \Delta E=\Delta U$$
2. Steady ~: 穩態$$\Rightarrow \Delta E=0$$
3. Undergoing a cycle: $$\Delta E=0$$
4. Adiabatic system: $$Q=0$$
5. Rigid boundary: $$W=0$$

## 4.1 Moving Boundary work/ Boundary work
$$\delta W_b=F\cdot dx=\frac{F}{A}\cdot Adx=PdV$$$$\Rightarrow per~unit~mass:\delta w_b=Pdv$$$$w_b=\int^{2}_{1}\delta w_b=\int^{2}_{1}Pdv$$$$\Rightarrow w_b=area~under~p-v~diagram)$$
### Polytropic process
$P=cV^{-n}~or~PV^n=c~~~~n,~c:~constant$
- Special case
1. n=o --> P=c --> constant pressure$$W_b=\int^{2}_{1}PdV=P(V_2-V_1)$$
2. n=1 --> isothermal process for an ideal gas $$W_b=\int^{2}_{1}PdV=\int^{2}_{1}\frac{c}{V}dV=c\cdot ln(\frac{V_2}{V_1})$$$$\overset{ideal~gas}{\overset{\rightarrow}{PV=c=mRT}}\Bigg\{\begin{align*}W_b=mRTln(\frac{V_2}{V_1})\\w_b=RTln(\frac{V_2}{V_1})~~\end{align*}$$
3. $n\neq 1$ --> $P=cV^{-n}\Rightarrow~$adiabatic process for an calorically perfect gas$$W_b=\int^{2}_{1}PdV=c\int^{2}_{1}V^{-n}dV=\frac{c}{1-n}\cdot V^{1-n}|^{V_2}_{V_1}=\frac{1}{1-n}\cdot [cV^{-n}_2-cV^{-n}_{1}]$$$$\Rightarrow W_b=\frac{P_2V_2-P_1V_1}{1-n}$$$$\overset{ideal~gas}{\rightarrow}\Bigg\{\begin{align*}W_b=\frac{mR}{1-n}(T_2-T_1)\\w_b=\frac{R}{1-n}(T_2-T_1)\end{align*}$$

## 4.2 Specific heat - A way to evaluate $\delta q$
- Specific heat: energy required to raise the temperature of a unit mass of a substance by one degree. unit ($\frac{kJ}{kg\cdot ~^oC}$)

Consider a stationary system with boundary work
$$\big\{ \begin{align*} du&=\delta q-Pdv\\h&=u+Pv \end{align*}$$$$\Rightarrow \big\{\begin{align*}du&=\delta q-Pdv\\dh&=du+d(Pv)=\delta q +vdP\end{align*}$$$$\Rightarrow \big\{ \begin{align*}\delta q|_v^{(fixed~specific~volume)}=du|_v\\\delta q|_P=dh|_P\end{align*}$$
From the postulate: u=u(T,v), h=h(T,P)
by calculus$$\bigg\{ \begin{align*} du&=\big(\frac{\partial u}{\partial T}\big )_vdT+\big(\frac{\partial u}{\partial v}\big )_Tdv\\dh&=\big(\frac{\partial h}{\partial T}\big )_PdT+\big(\frac{\partial h}{\partial P}\big )_TdP\end{align*}$$
$$\Rightarrow \bigg\{ \begin{align*} du|_v&=\big(\frac{\partial u}{\partial T}\big )_vdT\\dh|_P&=\big(\frac{\partial h}{\partial T}\big )_PdT\end{align*}$$
- Define **specific heat at constant volume**$$C_v=\frac{\delta q|_v}{dT}=\frac{du|_v}{dT}=\big(\frac{\partial u}{\partial T}\big )_v$$
- Define **specific heat at constant pressure**$$C_P=\frac{\delta q|_P}{dT}=\frac{dh|_P}{dT}=\big(\frac{\partial h}{\partial T}\big )_P$$
:::info
#### 1
$$\Big\{\begin{align*}C_P&>C_v~~for~a~compressible~ substance\\C_P&=C_v~~for~an~incompressible~ substance\end{align*}$$
> 多一個壓縮能[color=#f4d995]
>> 會在Chapter 12證明 [color=#d60e54]

#### 2
For a real gas $C_v=C_v(T,v)$;$~C_P=C_P(T,P)~$in general
- $If~C_v=C_v(T),~C_P=C_P(T)~for~a~gas\Rightarrow gas~is~thermally~perfect「熱完美氣體」$
- $If~C_v,~C_P=constant~for~a~gas\Rightarrow gas~is~calorically~perfect「恆比熱完美氣體」$

#### 3
For a liquid and solid, since the substance nearly incompressible
$\Rightarrow C_v\approx C_P=C$
:::
## 4.3 Internal energy, enthalpy, and specific heats of ideal gases
:::success
we can prove mathematically in Ch12 that for an ideal gas$~\big(\frac{\partial u}{\partial v }\big)_T=0\Rightarrow u=u(T)~only$
:::
Moreover, since $h= u+ Pv$$$ enthalpy~for~an~ideal~gas,~h=u(T)+RT=h(T)~only$$
Therefore,$$C_v=\big(\frac{\partial u}{\partial T}\big)_v=\frac{du}{dT}$$$$C_P=\big(\frac{\partial h}{\partial T}\big)_P=\frac{dh}{dT}$$Both are function of Temperature$$~C_v=C_v(T),~C_P=C_P(T)$$
Thus, **an ideal gas is thermally perfect**$$\Rightarrow\Big\{\begin{align*}du=C_v\cdot dT\\dh=C_P\cdot dT\end{align*}$$$$\Rightarrow \Bigg\{\begin{align*}\Delta u=\int^{T_2}_{T_1}C_vdT\\\Delta h=\int^{T_2}_{T_1}C_PdT\end{align*}$$

- 實際上我們會用線性近似來求值
$$For~ideal~gas$$$$\Delta u\approx C_v(T_{avg})\cdot (T_2-T_1)$$$$\Delta h\approx C_P(T_{avg})\cdot (T_2-T_1)$$$$T_{avg}=\frac{T_2+T_1}{2}$$
- 另一種方式估計$C_P,~C_v$
- 查表！Table A-2(c)：用多項式函數表示$C_P,~C_v$，查可得知每種氣體對應的係數為何，再將溫度帶入即可得。

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- Specific heat ratio$$k(or~\gamma)\overset{def}{=}\frac{C_P}{C_v}$$==記法：大的（等壓比熱）在上面==
- monatomic air, k=5/3
- **Air, k=1.4**==記起來==

> 大部分氣體都假設為理想氣體來計算

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$$\because h=u+RT,~for~ideal~gas$$$$\Rightarrow \Delta h=\Delta u+R\Delta T$$$$\lim_{\Delta T\rightarrow 0}\frac{\Delta h}{\Delta T}=\lim_{\Delta T\rightarrow 0}\frac{\Delta u}{\Delta T}+R$$$$C_P=C_v+R$$$$\overset{\times M}{\Rightarrow}\overline{C_P}=\overline{C_v}+R_u(universal~gas~constant)$$$$\overline{C_P}:molar~specific…\overline{C_v}:molar~specific…$$

## 4.4 Internal energy, enthalpy, and specific heats of liquids and solids
similar to ideal gas,
:::success
we can prove mathematically in Ch12 $$~\big(\frac{\partial u}{\partial v }\big)_T=0\Rightarrow u=u(T)~only$$$$C_v=\frac{du}{dT}=C_P=\big(\frac{\partial h}{\partial T}\big)_P\overset{def}=C$$
:::
- enthalpy for an incompressible substance$$h\overset{def}=u+Pv$$$$dh=du+d(pv)$$$$dh=du+vdp~(\because incompressible)$$$$\Rightarrow \Delta h=\Delta u+v\Delta P$$$$For~an~incompressible~substance$$$$\Delta u=\int^{T_2}_{T_1}CdT\approx C_v(T_{avg})\cdot (T_2-T_1)$$$$\Delta h=\int^{T_2}_{T_1}C(T)dT+v\Delta P$$

### 1. constant pressure, isobaric
$$\Delta h=\int^{T_2}_{T_1}C(T)dT=\Delta u\approx C(T_{avg}\cdot \Delta T)$$
### 2. constant temperature, isothermal
以液體為例
- $\Delta h=v\Delta P$
- $\Delta u=0$
- $\Delta v=0~(\because incompressible)\Rightarrow v=v_f$
$$h-h_f(T)=v_f(T)\cdot (P-P_{sat}(T))$$$$h=h_f(T)+v_f(T)\cdot (P-P_{sat}(T))$$