Joule-Thomson Throttle: Exact van der Waals Solver

This is the 2D companion to the 3D Joule-Thomson throttle simulator. Where the 3D version colors its particle stream using the standard linear approximation μ_JT ≈ (1/Cp)(2a/RT − b), this simulator solves the van der Waals equation of state exactly for the molar volume at both the inlet and outlet conditions, then finds the outlet temperature that exactly conserves molar enthalpy across the valve via a nonlinear bisection solve — no shortcuts. The live chart plots the real curved isenthalp this exact solver traces from the high-side pressure down to atmospheric alongside the straight-line linear approximation, so you can see directly where the textbook formula holds up and where — especially for helium near its very low inversion temperature, or for any gas across a large industrial pressure drop — it quietly gives the wrong answer. Switch gases, adjust inlet temperature and high-side pressure, and watch both curves move together at small drops and peel apart at large ones.