Calculating temperature change of helium due to Joule-Thomson effect We are trying to fill a rocket helium tank that is 78 liters with helium, final pressure of $\pu{10500 psi}$. We plan to pressurize high pressure tanks to $\pu{20000 psi}$ using a compressing pump system, so that on launch day we don't have to wait for the 78 liter tank to fill slowly by the compressing system. We plan to fill the rocket tanks from the high pressure tanks in a cascade manner (the number and volume of the tanks may vary, depending on the solution of the JT equation(s).

The problem is that helium has a negative JT coefficient. It will heat up during the transfer from the high pressure tanks to the lower pressure tank if I understand it correctly.

So, the concern is how hot will it get. Will the O-rings/seals melt? I have some equations, but it seems to me there may be too many unknowns.

I've seen others make certain assumptions on this site, but its not clear to me how that works. Can anyone advise how you would proceed?
To do this analysis, one needs data up to 20000 psi (1400 bars) on either enthalpy vs temperature and pressure (a graph) or compressibility factor Z vs temperature and pressure.  The only graphs I have found of the former type go up to only 100 bars, which is a factor of 14 too low.  However, I have found data on the compressibility factor Z at temperatures up to 300 K and pressures up to 1000 bars:  https://cds.cern.ch/record/1444601/files/978-1-4419-9979-5_BookBackMatter.pdf
Although this is still a factor of 1.4 too low, it might provide some idea of the temperature rise that might be expected in the valve.  

So here is how the data would be used.

The effect of pressure on enthalpy (per mole) of gas is given by $$\left(\frac{\partial H}{\partial P}\right)_T=V-T\left(\frac{\partial V}{\partial T}\right)_P\tag{1}$$For a real gas, the equation of state in terms of the compressibility factor Z=Z(P,T) is given by$$PV=ZRT\tag{2}$$If we substitute Eqn. 2 into Eqn. 1, we obtain:  $$\left(\frac{\partial H}{\partial P}\right)_T=-\frac{RT^2}{P}\left(\frac{\partial Z}{\partial T}\right)_P\tag{3}$$Integrating Eqn. 3 between P=0 and arbitrary P at constant temperature yields the so-called Residual Enthalpy $H^R$:$$H^R(P,T)=-RT^2\int_0^P{\left(\frac{\partial Z}{\partial T}\right)_{P'}\frac{dP'}{P'}}=-RT^2\frac{\partial}{\partial T}\left(\int_0^P{(Z(T,P')-1)\frac{dP'}{P'}}\right)\tag{4}$$where P' is a dummy variable of integration.  

If the final pressure coming out of the valve is low (so that the gas exiting the valve is in the ideal gas region), we can write:  $$\Delta H=-H^R+C_p\Delta T=0$$where, for a monoatomic gas like Helium, $C_p=\frac{5}{2}R$.  Therefore, $$\Delta T=-\frac{2}{5}T^2\frac{\partial}{\partial T}\left(\int_0^P{(Z(T,P')-1)\frac{dP'}{P'}}\right)\tag{5}$$
This expression would be evaluated using the data presented in the reference above.  

If a reference can be found with data going out to 1400 bar, that would be even better.