ENPH 257 Final Presentation (2023)

Here is a walkthrough of my final presentation for ENPH 257 2023S on throttling of a non-ideal gas.

Throttling is a quick process in which pushing gas through a porous plug changes its temperature and pressure downstream of the plug. Also known as the Joule-Thomson (JT) process, it is commonly used in refrigeration cycles to drastically cool down a working substance that can then pull heat out of the refrigerator. Here is how it is defined:

The throttling / Joule Thomson process is always constant enthalpy.

So, the throttling process is constant enthalpy, but if we try to simulate it with an ideal gas, the temperature does not change. So we should try some other gas model if we want some results that represent what happens in real life.

Now that I’ve introduced the Van der Waal model, we can try simulating the throttling process with that instead.

This boxed expression for the Joule Thomson coefficient works for any equation of state: the ideal gas law, the Van der Waal gas law, or any other real gas model. All that needs to be done is to calculate the appropriate $\left(\frac{\partial V}{\partial T}\right)_P$ expression, individual to each model.

To start, we can calculate $\left(\frac{\partial V}{\partial T}\right)_P$ for an ideal gas:

And the JT-coefficient ends up being 0! This is a secondary confirmation that an ideal gas would not change temperature under throttling. Now, we can try it for a Van der Waals gas:

Finally, we end up with the expression for the Joule Thomson Coefficient of a Van der Waals Gas. Notice that it is not a constant, but a function of state variables.

Now, we can take a look at inversion curves, which is the line of which the Joule-Thomson coefficient evaluates to 0.

Overall, inversion curves are individual to each real gas, and outline whether or not the temperature of a real gas would decrease or increase through throttling, given some initial condition of P and T.

The inversion curve shown above is from real experimental data, so now I want to see if we can use our calculate Van der Waals Joule Thomson expression to predict the inversion curve.

At first glance, the experimental and theoretical look pretty similar, but there are some noted discrepancies. However, there are also some noted similarities which show signs of success:

That gives us a nice visual of where we’re at, but now I want to see if we can get some concrete numbers from our expression. Here is an example problem I took from the course textbook (cited at end):

Normally, these problems are done with a specific enthalpy chart, which provides the enthalpy of a given gas at some temperature and pressure:

We end up finding that the gas would cool from 300K → 281.4K using the linear interpolation of the enthalpy chart. Now, let’s use our expression for the Joule Thomson Coefficient and see if we get the same number:

*I should have written out that the reason that $T_f = T_i + \int_{P_i}^{P_f}\phi_\text{VdW} dP$ is because:

Which them motivates us to use a first-order approximation instead of evaluating the integral.

And we end up with a very similar result! This is pretty impressive considering our linear approximation was over 99 bar, or 9,900,000 Pascals.