= Liquefaction enthalpy budget
{title2=$y=(h_g-h_{\rm in}-q)/(h_g-h_l)$}
Consider an original model of a cold section enclosing a throttle and collecting region, with the precooling <heat exchanger> outside its boundary. The inlet has <mass flow rate> $\dot m>0$ and <specific enthalpy> $h_{\rm in}$. The outlets are equilibrium <liquid> and <vapor> at one fixed <pressure>, with <specific enthalpies> $h_l$ and $h_g$. Let $y$ be the outlet <liquid> <mass fraction>, and let $q=\dot Q/\dot m\geq0$ be incoming unwanted <heat> per unit incoming <mass>. Assume <steady flow>, no shaft <work>, and negligible bulk <kinetic energy> and <gravitational potential energy> changes. The <steady-flow energy balance> gives
$$
h_{\rm in}+q=yh_l+(1-y)h_g,\qquad
L=h_g-h_l>0,\qquad
y=\frac{h_g-h_{\rm in}-q}{L}.
$$
Here $L$ is the <latent heat> per unit <mass> at the chosen outlet <pressure>. The <mass flow rates> of the outlets are $y\dot m$ and $(1-y)\dot m$. The formula describes a flowing cold section; a filling vessel with appreciable <energy> or <mass> accumulation requires storage terms.
For a deliberately dimensionless example, choose
$$
\eta=\frac{h_{\rm in}-h_l}{L}=0.70,\qquad
\ell=\frac qL=0.05,\qquad y=1-\eta-\ell=0.25.
$$
Per unit incoming <mass>, one quarter leaves as <liquid> and three quarters as <vapor>. With zero incoming <heat>, the same inlet would give $y=0.30$. Increasing $q/L$ to $0.20$ reduces $y$ to $0.10$, and $q/L=0.30$ reaches the no-<liquid> boundary. These are chosen model values, not <helium> property data or a historical yield. In normalized <enthalpy> units with $h_l=0$ and $h_g=1$, conservation checks directly: $0.70+0.05=0.25\times0+0.75\times1$.
\Image[/liquefaction-enthalpy-fraction-model.png]
{title=An original normalized <liquefaction enthalpy budget>}
{id=liquefaction-enthalpy-fraction-model}
{description=Each solid segment satisfies the stipulated equilibrium two-outlet model. Dashed negative continuations are algebra only: they do not describe a negative <liquid> <mass fraction>. No <helium> properties or experimental measurements were fitted.}
{source=liquefaction-enthalpy-fraction-model.py}
The admissible interval is $h_l\leq h_{\rm in}+q\leq h_g$, or $0\leq y\leq1$. Outside it, the assumed equilibrium two-phase outlet fails: an <enthalpy> above $h_g$ requires a hotter single-<vapor> outlet under these assumptions, rather than a negative amount of <liquid>. Below $h_l$, a colder <liquid> or a different <thermodynamic phase> must be considered. The balance does not predict its own outlet <pressure>, <temperature> or <helium> property functions.
At fixed outlet <pressure>, hence fixed $L$, the <derivatives> are $\partial y/\partial h_{\rm in}=-1/L$ and $\partial y/\partial q=-1/L$. Lowering inlet <specific enthalpy> by regeneration and reducing incoming <heat> therefore buy the same increase in <liquid> <mass fraction> per unit <energy> per unit <mass> in this model. Adding any constant to all three <specific enthalpies> leaves $y$ unchanged. This independence of the arbitrary <enthalpy> reference makes the balance useful even when property tables choose different zero points.
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