= Van der Waals throttling model
{c}
A hypothetical <Van der Waals equation> model illustrates positive and negative <Joule-Thomson coefficient> regions and a finite throttle that first warms, then cools. Use the molar convention already distinguished in <Van der Waals equation>:
$$
p=\frac{RT}{v_{\rm m}-b_{\rm m}}-\frac{a_{\rm m}}{v_{\rm m}^2},
\qquad
u_{\rm m}=C_{V,\rm m}T-\frac{a_{\rm m}}{v_{\rm m}},
\qquad C_{V,\rm m}=\frac32R.
$$
Here $v_{\rm m}$ is <molar volume>, $R$ the <molar gas constant>, $u_{\rm m}$ <internal energy> per <mole>, and $C_{V,\rm m}$ the stipulated constant <molar heat capacity> at fixed <volume (thermodynamics)>. The second equation is an additional caloric assumption: an <equation of state> alone does not specify the <temperature> dependence of <internal energy>. It satisfies the equilibrium identity $(\partial u_{\rm m}/\partial v_{\rm m})_T=T(\partial p/\partial T)_{v_{\rm m}}-p=a_{\rm m}/v_{\rm m}^2$. The arbitrary <energy> reference is set to zero.
\Image[/van-der-waals-throttling-inversion-model.png]
{title=Hypothetical <Van der Waals throttling model>: <Joule-Thomson inversion curve> and finite <temperature> responses}
{id=van-der-waals-throttling-inversion-model}
{description=All curves are original model calculations, not <nitrogen>, <carbon dioxide> or other gas measurements. Colors label the response along constant <molar enthalpy>; arrows point toward lower <pressure>. The displayed states are above the model <Van der Waals critical point> <temperature> and have positive <isothermal compressibility>. The purple path crosses the zero-coefficient curve and has a <temperature> maximum.}
Define dimensionless variables $r=v_{\rm m}/b_{\rm m}$, $\theta=Rb_{\rm m}T/a_{\rm m}$, $\pi=b_{\rm m}^2p/a_{\rm m}$ and $\eta=b_{\rm m}h_{\rm m}/a_{\rm m}$, where $h_{\rm m}$ is <molar enthalpy>. These differ from the critical-value scaling in <Van der Waals law of corresponding states>. The throttle calculation reduces to
$$
\pi=\frac{\theta}{r-1}-\frac1{r^2},\qquad
\eta=\frac32\theta+\pi r-\frac1r,
\qquad
\theta(r;\eta)=\frac{\eta+2/r}{3/2+r/(r-1)}.
$$
Thus a prescribed inlet fixes $\eta$, and solving the first and last equations at the outlet <pressure> gives the final <temperature>. For an inlet $(\theta_1,\pi_1)=(0.8,0.12)$ and outlet $\pi_2=0.025$, the model gives $\theta_2\simeq0.752287$, a decrease of $0.047713$. The inlet $(2.2,0.12)$ instead gives $\theta_2\simeq2.205786$, an increase of $0.005786$. The purple inlet $(1.45,0.30)$ initially warms but ends at $\theta_2\simeq1.437585$ for the same outlet <pressure>. These are reduced toy values, not kelvin readings or experimental evidence. No physical gas parameters were fitted.
For a short inversion derivation, implicit <differentiation> of the <Van der Waals equation> at fixed <pressure> gives
$$
\left(\frac{\partial v_{\rm m}}{\partial T}\right)_p
=\frac{R/(v_{\rm m}-b_{\rm m})}{RT/(v_{\rm m}-b_{\rm m})^2-2a_{\rm m}/v_{\rm m}^3}.
$$
Insert this into $T(\partial v_{\rm m}/\partial T)_p=v_{\rm m}$ and substitute back into the <equation of state>. The <Joule-Thomson inversion curve> is
$$
T_{\rm inv}=\frac{2a_{\rm m}}{Rb_{\rm m}}\left(1-\frac{b_{\rm m}}{v_{\rm m}}\right)^2,
\qquad
\theta_{\rm inv}=2(1-1/r)^2,
\qquad
\pi_{\rm inv}=\frac{2r-3}{r^2}.
$$
Positive inversion <pressure> requires $r>3/2$. Its maximum <pressure> is $\pi=1/3$ at $r=3$, while the upper dilute limit is $\theta_{\rm inv}\to2$. Using <Van der Waals critical point> values, the latter is $T_{\rm inv,max}/T_c=27/4$. The inversion locus depends on the <equation of state>; the chosen <molar heat capacity> additionally determines how much <temperature> changes along each finite throttle.<Thermodynamics and Statistical Mechanics by Richard Fitzpatrick/Joule-Thompson Throttling>[Fitzpatrick, Joule-Thompson Throttling, equations 6.188–6.190 and following paragraph: inversion locus, maximum temperature and limits of the van der Waals prediction]{ref}
The displayed region has $\theta>8/27$, above the model's critical <temperature>, and uses only $r>1$, $p>0$, $(\partial p/\partial v_{\rm m})_T<0$ and positive $C_{p,\rm m}$. Subcritical calculations require attention to <phase coexistence>, <metastability> and the <Maxwell construction>; an unstable homogeneous continuation is not an equilibrium throttle endpoint. Even on a stable branch, this <mean-field theory> with constant $C_{V,\rm m}$ is a qualitative model, not an accurate inversion map for a particular <real gas>. <Phase transitions>, changing composition, appreciable <heat> leakage or bulk <kinetic energy> changes require a different property calculation or <steady-flow energy balance>. At <phase coexistence>, the smooth single-<thermodynamic phase> coefficient formula cannot replace a balance involving the <phase fraction>.
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