= Steady-flow energy balance
{title2=$\dot Q-\dot W_s=\dot m\,\Delta(h+w^2/2+gz)$}
For <steady flow> through a fixed <control volume> with one inlet and one outlet, the <first law of thermodynamics> and conservation of <mass> give
$$
\dot Q-\dot W_s=\dot m\left[h_2-h_1+\frac{w_2^2-w_1^2}{2}+g(z_2-z_1)\right].
$$
Here $\dot m$ is the <mass flow rate>, $\dot Q$ incoming <heat> per unit time, $\dot W_s$ outgoing non-flow <work> per unit time, $w$ the flow speed, $z$ elevation and $g$ <gravitational acceleration>. The <specific enthalpy> $h=u+p/\rho$ includes both the transported <internal energy> per unit <mass> $u$ and the <pressure> <work> needed to push <volume (thermodynamics)> through the boundary; $\rho$ is <mass density>. The inlet supplies that <pressure> <work>, while the outlet consumes it to displace the downstream <fluid>.<Thermodynamics and Propulsion>[§2.5.1–2.5.2, equation 2.11: mass conservation, transported enthalpy and the steady-flow energy equation]{ref}
For an insulated stationary throttle, require negligible <heat> transfer, no shaft or other non-flow <work>, and negligible changes in bulk <kinetic energy> and gravitational <potential energy>. Then $h_2=h_1$. With unchanged composition this is also equality of <molar enthalpy>. The upstream and downstream states must admit equilibrium <pressure>, <temperature> and property values, although the constriction itself is irreversible and can be far from <thermodynamic equilibrium>. Start-up storage of <energy> in the walls and <fluid>, heat leakage, or a large outlet speed invalidates the reduced balance. A nozzle can convert <specific enthalpy> into bulk <kinetic energy> even when it is insulated.
The same accounting for a fixed transported amount is $U_2-U_1=p_1V_1-p_2V_2$, hence $U_1+p_1V_1=U_2+p_2V_2$. This explains why dropping the two <pressure> <work> terms would give the wrong conserved quantity.<Thermodynamics and Statistical Mechanics by Richard Fitzpatrick/Joule-Thompson Throttling>[Fitzpatrick, Joule-Thompson Throttling, equations 6.167–6.174: inlet/outlet pressure work and enthalpy conservation]{ref}
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