# 热力学系统

## 开放系统

### 流动过程

$\mathrm{d}U=\mathrm{d}U_{in}+\delta Q-\mathrm{d}U_{out}-\delta W\,$

$\delta W=\mathrm{d}(P_{out}V_{out})-\mathrm{d}(P_{in}V_{in})+\delta W_{shaft}\,$

$\mathrm{d}U_{cv}=\mathrm{d}U_{in}+\mathrm{d}(P_{in}V_{in}) - \mathrm{d}U_{out}-\mathrm{d}(P_{out}V_{out})+\delta Q-\delta W_{shaft}\,$

$\mathrm{d}U_{cv}=\mathrm{d}H_{in}-\mathrm{d}H_{out}+\delta Q-\delta W_{shaft}\,$

$\frac{\delta W_{shaft}}{\mathrm{d}t}=\frac{\mathrm{d}H_{in}}{\mathrm{d}t}- \frac{\mathrm{d}H_{out}}{\mathrm{d}t}+\frac{\delta Q}{\mathrm{d}t} \,$

## 封闭系统

$\Delta U=Q-W+m_{in}(h+\frac{1}{2}v^2+gz)_{i}-m_{out}(h+\frac{1}{2}v^2+gz)_{e}$

$\mathrm{d}U={\delta Q}-{\delta W}.$

${\delta W}=P\mathrm{d}V.$

${\delta Q}=T\mathrm{d}S$

${\delta U}=T\mathrm{d}S-P\mathrm{d}V.$

$\sum_{j=1}^m a_{ij}N_j=b_i^0$

## 参考文献

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