Chapter 7：Entropy 熵

熵

\[ d S=\left(\frac{d Q}{T}\right)_{\text {int,rev }} \]

熵的定义：熵被定义为热量交换与温度之比，在一个内部可逆过程中。它是一个用于描述系统混乱度的物理量。

For the special case of internally reversible, Isthermal proves（内部可逆等温过程）

\[ \Delta S = \frac Q {T_0} \]

Increase of entropy principle:：熵增原理

\(S_{gen}\ge0\)： entropy genered during the process
\(S_{gen} = 0\)： si reversible

entropy change：熵变

Pure substances 纯物质

Any： \(\Delta s=s_{2}-s_1\).
Insentropic 等熵：\(s_{2}=s_1\)

Incompressible substances 不可压

\(Q=\operatorname{Cavg}\left(T_{2}-T_{1}\right)\)
\(S_{2}-S_{1}=\text { Cavg } \cdot \ln \frac{T_{2}}{T_{1}}\)

Ideal gaze 理想气体

\(Q=U-W=C_{v \text { arg }} \cdot d T+\frac R{dv} \cdot d V = C_{v\text { avg }} \cdot d T+\frac{R T}{d V} \cdot d V\)
\(d S=\left(\frac{Q}{T}\right)_{\text {int, rev }}=\frac{C_{v\text {avg }}}{T} \cdot d T+\frac{R}{d V} \cdot d V\)
得到理想气体状态下熵变：

\[ \begin{array}{r}S_2-S_1=C_{V \text { avg } }\ln \frac{T_2}{T_1}+R \ln \frac{V_2}{V_1} \\S_2-S_1=C_{P \text { avg }} \ln \frac{T_2}{T_1}-R \ln \frac{P_2}{P_1} \\\quad Q=H+V \cdot d P \end{array} \]

isentropic process 等熵过程

\[ \begin{aligned}& \left(\frac{T_2}{T_1}\right)_{S=\text { cosst }}=\left(\frac{V_1}{V_2}\right)^{K-1} \\& \left(\frac{T_2}{T_1}\right)_{S=\text { const }}=\left(\frac{P_1}{P_2}\right)^{K-1 / K} \\& \left(\frac{P_2}{P_1}\right)_{S=\text { const }}=\left(\frac{V_1}{V_2}\right)^K\end{aligned} \]

Steady-flow work for a reversible process 可逆过程的稳态流功

\[ W_{\text {rev }}=-\int_{1}^{v} v \cdot d_{p}-\Delta K_{e}-\Delta{p_ e} \]

for incompressible substances 不可压过程

\[ W_{\text {rev }}=-V\left(p_{2}-p_{1}\right)-\Delta K_{e}-\Delta p_{e} \]

在 incompressible substances 中, 压缩时应使 \(V\) 尽司能小, 使功输入最小化; 扩展时, V应尽量大, 使功输出最大化

Isentropic 等熵过程

\[ P V^{k}=constant \]

\[ W_{\text {comp. in }}=\frac{k}{k-1} \cdot R\left(T_{2}-T_{1}\right)=\frac{k}{k-1} R T_{1} \cdot\left(\left(\frac{P_{2}}{P_{1}}\right)^{1-1 / k}-1\right) \]

Polytropic 多边形过程

\[ P V^{n}=constant \]

\[ W_{\text {comp. } i n}=\frac{n R}{n-1}\left(T_{2}-T_{1}\right)=\frac{n}{n-1} R T_{1}\left(\left(\frac{P_{2}}{P_{1}}\right)^{1-1 / n}-1\right) \]

Isothermal 等温过程

\[ P V= constant \]

\[ W_{\text {comp. in }}=R T \ln \left(\frac{T_{2}}{T_1}\right) \]

Isentropic Device et Isentropic/Adiabatic efficiently 等熵设备和绝热效率

Turbine

\[ \eta_{T}=\frac{W_{\text {actual }}}{W_{S}} \cong \frac{h_{1}-h_{2 a}}{h_{1}-h_{2 s}} \]

Compressors

\[ \eta_c = \frac {w_s}{w_a}\cong\frac{h_{2s}-h_1}{h_{2a}-h_1} \]

Nozzles 喷嘴

\[ \eta_{N}=\frac{V_{2 a}^{2}}{V_{2 s}^{2}} \cong \frac{h_{1}-h_{2 a}}{h_{1}-h_{2 s}} \]

Entropy balance 熵平衡

\[\begin{aligned}& \left\{\begin{array}{l}S_{\text {in }}-S_{\text {ont }}+S_{\text {gen }}=\Delta S_{\text {system }} \\\dot{S}_{\text {in }}-\dot{S}_{\text {ont }}+\dot{S}_{\text {gen }}=d S_{\text {System }} / d t\end{array}\right. \\& \dot{S}_{\text {gen }}=\sum \dot{m}_{e} S_{e}-\sum \dot{m}_{i} \dot{S}_{i}-\sum \frac{\dot{Q}_{k}}{T_{k}}\end{aligned}\]