The first law for open systems: enthalpy carries the ledger

Pipes, not pistons

• Real machines are open: mass crosses a control volume 控制体积 boundary and carries energy with it. At steady flow 稳定流动, inflow equals outflow and nothing inside accumulates - the per-port ledger is $q - w = \Delta h + \Delta ke + \Delta pe$.
• The steady-flow energy equation is the first law rewritten for flow: shaft work and heat at the boundary, enthalpy 焓 $h = u + pv$ at the ports, and velocity changes where they matter.

Where the pv goes

• Pushing 1 kg of fluid into the control volume costs $pv$ of flow work 流动功; the same term reappears pushing it out. Absorbing that transport term into the internal energy is exactly what enthalpy is - which is why the open-system ledger runs on $h$, not $u$.

A turbine stage, priced

• Steam enters at $3\ \mathrm{MPa}$, $400^\circ\mathrm{C}$ ($h_1 = 3230.9\ \mathrm{kJ/kg}$, $s_1 = 6.9212\ \mathrm{kJ/(kg\,K)}$) and expands adiabatically to $50\ \mathrm{kPa}$. The ideal leg is isentropic 等熵过程: $s_2 = s_1$ gives $x_{2s} = (6.9212 - 1.0910)/6.5029 = \mathbf{0.89655}$, so $h_{2s} = 340.49 + 0.89655 \times 2305.4 = \mathbf{2407.4}$ and $w_s = 3230.9 - 2407.4 = \mathbf{823.5}\ \mathrm{kJ/kg}$.
• The real leg prices the imperfection with an isentropic efficiency $\eta_t = 0.85$: $w = 0.85 \times 823.5 = \mathbf{700.0}$, back-solving $h_2 = 3230.9 - 700.0 = \mathbf{2530.9}\ \mathrm{kJ/kg}$. Isentropic, then efficiency - the two-step turbine method.

A pump and a nozzle

• A pump lifts liquid water $0.5 \to 3\ \mathrm{MPa}$: liquid $v$ is tiny, so $w_p \approx v\,\Delta p = 0.001005 \times 2500 = \mathbf{2.5}\ \mathrm{kJ/kg}$.
• A nozzle 喷嘴 accelerates air $10 \to 300\ \mathrm{m/s}$ adiabatically with no shaft work: $\Delta h = -\Delta ke = -(300^2 - 10^2)/2 = \mathbf{-44.95}\ \mathrm{kJ/kg}$. Three devices, one equation - and the kinetic term is kept exactly where velocity is the product.

Three port slips

Recap