Entropy熵

What you'll be able to do本章學習成果

• Understand entropy transfer, entropy production, and the increase-of-entropy principle.
• Evaluate entropy and entropy change, and analyze isentropic processes with property data.
• Represent heat transfer as an area on a T–s diagram.
• Apply entropy balances to closed systems and control volumes, and evaluate isentropic efficiencies of turbines, nozzles, compressors, and pumps.

Key equations重要公式

Defining entropy change

Starting from the Clausius inequality, consider two internally reversible processes linking the same two states. The integral $\int \delta Q/T$ comes out identical for both — it depends only on the end states. A quantity whose change is path-independent is a property. We call it entropy, $S$:

You compute it by imagining any internally reversible path between the states — but because entropy is a property, the result holds for the actual process too, reversible or not. Entropy is extensive; specific entropy $s$ has units kJ/kg·K. For two-phase mixtures, $s = s_f + x\,s_{fg}$, just like $v$, $u$, and $h$.

Entropy & heat transfer熵與熱傳

On a differential basis $dS = (\delta Q/T)_{int\,rev}$. So heat into a system raises its entropy and heat out lowers it — entropy transfer accompanies heat transfer, in the same direction. An internally reversible, adiabatic process has no heat transfer and therefore constant entropy — an isentropic process.

The entropy balance熵的平衡

Entropy is accounted for like mass and energy — but with a production term. For a closed system:

The first term is entropy transfer with heat (evaluated at the boundary); $\sigma$ is entropy production from internal irreversibilities. Its sign is decisive:

• $\sigma = 0$ — no internal irreversibilities
• $\sigma > 0$ — irreversibilities present
• $\sigma < 0$ — impossible

For an adiabatic process the transfer term vanishes, so $\sigma = S_2 - S_1$: a negative entropy change would be impossible. This is the increase-of-entropy principle in action — the test the explorer below applies.

Control-volume entropy balance控制體積的熵平衡

Streams of matter carry entropy across a control surface. At steady state, for one inlet and one exit:

A throttling valve is a clean example: adiabatic with $h_2 = h_1$, so $\dot\sigma_{cv}/\dot m = s_2 - s_1 > 0$ — the entropy production traces directly to the unrestrained expansion to lower pressure.

Calculating entropy change熵變化的計算

The property tables are built from the T dS equations, $T\,ds = du + p\,dv = dh - v\,dp$. Two models give closed forms. For an incompressible substance with constant $c$: $\Delta s = c\,\ln(T_2/T_1)$. For an ideal gas with constant specific heats:

Setting $s_2 = s_1$ recovers the familiar isentropic relation $T_2/T_1 = (p_2/p_1)^{(k-1)/k}$. (Where specific heats vary appreciably, the tabulated $s^\circ$ and relative-pressure $p_r$ data handle the temperature dependence.)

Isentropic efficiency等熵效率

Because adiabatic devices can only reach exit states with $s_2 \ge s_1$, the isentropic (constant-entropy) endpoint is the best case. For a turbine, the most work is the isentropic work; the efficiency is the actual fraction of it:

For a compressor or pump the ideal is the minimum work, so the ratio inverts: $\eta_c = (h_{2s}-h_1)/(h_2-h_1)$. In every case the actual process lands to the right of the isentropic one on a T–s or Mollier diagram — that horizontal shift is the entropy generated.

Isentropic-efficiency lab等熵效率實驗室

Expand or compress air between two pressures at a chosen isentropic efficiency. The ideal path (1→2s) is vertical — constant entropy; the actual path (1→2) leans right by exactly the entropy generated. Watch the work and $\sigma$ respond as you lower the efficiency.

Air as ideal gas, constant $c_p = 1.005$ kJ/kg·K, $k = 1.4$. Adiabatic device, so actual Δs equals the entropy generated.

Entropy change of an ideal gas理想氣體的熵變化