Built Under Pressure: Part 3 — Thermodynamics in Action

Photo by Bozhin Karaivanov on Unsplash

All models are wrong, but some are useful.

During the summer before my junior year in undergrad, I was in need of a bit of scratch and I saw and opportunity for leading the thermodynamics workshop managed by the PROMES program at the University of Houston. Basically, it’s a peer-led program where upper-classmen facilitate under-classmen learning a particular topic.

I ended up being a facilitator for that workshop for 5 semesters. If you had asked me at the time if I would stay that long, I’d have said no. Turns out, I loved teaching.

What I didn’t expect was how much reinforcing this knowledge would benefit me later on.

Putting it bluntly, if you don’t understand the thermodynamics, then you’ll have a rough time analyzing these machines.

Textbook Flashbacks: Thermodynamics at Play

At the core of compressor thermodynamics lies the concept of compression ratio, defined as the ratio of the discharge pressure to the suction pressure. This ratio governs effectively everything about these machines.

I said before in the last piece that one of my favorite questions to ask junior engineers learning this material for the first time is:

“aside from increase in pressure, what can you be sure always happens when you compress a gas?”

The answer was that the gas got hotter, but let’s see if we can use first principles to predict by how much .

Let’s focus on ideal situations. If you compress something quickly enough with minimal friction, we call this a reversible process. If the environment is also perfectly insulated, we call it adiabatic. If a compression is reversible and adiabatic, then it is now an isentropic process. Knowing this definition is crucial.

Let’s now applying this. Take an example compressor drawn below and let’s look at the 1st Law of Thermodynamics — the energy balance.

If the gas is ideal and no heat is exchanged, then we can simplify the expression to:

For ideal gases, the heat capacities are constant and then the math is even easier:

…okay, so now what?

Ideality and Reality

Well, then we have to take a philosophical approach to this math. There’s reality, and then there’s ideality. Remember that for compressors, ideality means isentropic. As an engineer, we’ll have to merge the two and make sense of the results.

In ideality, the isentropic case applies and we can now relate the outlet temperature and the inlet temperatures together.

with some shuffling, you get:

Great, but how does this relate to reality? Well, you can’t develop an easily usable expression for reality. However, what we can do instead is generate an expression for efficiency that you can easily use to determine the health of a machine.

Efficiency of anything is simply a ratio of the ideal to the real. So let’s do this with math:

Tying this back to first law and assuming constant heat capacities:

or, even more simply:

Finally, after all this theory, we arrive to the expression that carries weight and matters.

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Let’s look at this expressions above carefully:

• If the compression ratio goes up, the discharge temperature goes up.
• If the suction temperature goes up, the discharge temperature goes up.
• The smaller difference in measured temperatures between discharge and suction, the higher the efficiency.

All of this makes sense, right?

Let’s also recap what we assumed to get here:

1. Heat capacities are constant
2. The ideal case for the compressor is an isentropic process

These are common expression for modeling performance of a compression stage. For centrifugal compressors, it’s more common in industry to use polytropic efficiency, but it is not as easy to use without efficiency curves provided by the manufacturer. Unless I have those curves, I use isentropic efficiencies to model behavior regardless of compressor type.

When Reality Bites: Real Gases and Compressors

Up to this point, we’ve assumed the gas behaves ideally — meaning it follows the ideal gas law and plays nicely with our math. But let’s be honest: reality rarely cooperates.

Real gases don’t always behave ideally, especially under high pressure or low temperature — just the kinds of conditions we might see in multi-stage compression, refrigeration, or gas injection. Deviations from ideal gas behavior become significant, and our neat equations start to wobble.

So what changes?

PV ≠ nRT anymore.
The ideal gas law breaks down. Instead, we start using real gas equations of state — like Van der Waals, Redlich-Kwong, or Peng-Robinson. These introduce corrections for intermolecular forces and molecular volumes. They’re more complex, but more accurate.

Isentropic Relationships Need Adjustment
The tidy isentropic relationships between pressure and temperature we derived for ideal gases? They assume constant Cp and Cv and ideal equations of state. For real gases, you have to use real property data — often tabulated in charts, or integrated numerically from thermodynamic libraries.

Efficiency Definitions Still Hold
Here’s the good news: efficiency still makes sense. You just need to redefine your “ideal” baseline using real-gas isentropic properties — values you might get from software or detailed property tables.

You’ll need enthalpy values at actual and isentropic endpoints, all based on real-gas behavior.

This seems like a lot, how do I use this practically?

In the field, you’ll have to rely on whatever you have at your disposal. In my days, I actually didn’t have immediate access to ASPEN or other modeling tools (mostly because of cost reasons). I have always found the following refrain very reassuring, so much so that I made it the subtitle of this article:

“All models are wrong, but some are useful”

-British Statistician, George E.P. Box

To be honest, I’ve used the ideal gas, isentropic assumption when modeling compressors very successfully, regardless of compression ranges and molecular complexity. Even though the models will only get better by using more complicated equations of state, the added mathematical complexity is almost never worth it when doing field troubleshooting. In designing these machines however, the story may be different.

Using Efficiency Models — How can the ideal gas assumption work so well?

We spent most of this article mainly reviewing textbook fundamentals to get to expressions that we can use. Now that we have, let’s state a particularly critical point:

The exact efficiency of a single stage at any given moment is less critical than how that efficiency changes over time. It’s the trend — whether performance is stable, improving, or degrading — that provides meaningful insight into machine health and operational reliability.

I’ll explain.

Everyone obviously wishes for the highest efficiency possible. Max efficiency means max gas throughput, the least amount of wasted power, etcetera. However, the true value in using the efficiency expression above, is knowing when maintenance needs to be done on the compressor.

This is where the value of a mathematically derived expression — built on simplifying assumptions — becomes clear. Even though reality is messy, the idealized model gives us a reference baseline. Once you’ve defined an expected efficiency under stable operating conditions, deviations from that baseline become a diagnostic tool.

When a compressor begins to fail, it doesn’t just lose efficiency in a vacuum. You’ll typically observe a simultaneous shift in compression ratio and a rise in actual discharge temperature relative to its predicted ideal discharge temperature. This is a strong indication that something is wrong.

This makes sense intuitively: if the compressor is no longer moving fresh gas efficiently, it’s likely recompressing part of the same volume repeatedly. In positive displacement machines, this might be due to damaged or leaking piston rings, worn-out valves, or improper valve seating. In centrifugal stages, it could reflect distorted flow profiles due to fouling, worn impellers, or internal recirculation.

The result? You expend more energy per unit mass of gas moved, but see less pressure gain than expected — hence a lower observed compression ratio and a higher discharge temperature than the isentropic prediction. That temperature excess isn’t just a number — it’s wasted work and mechanical strain.

By monitoring both efficiency and compression ratio trends over time, and comparing actual vs. ideal discharge temperatures, you can pinpoint which stage is underperforming and infer the likely cause. For example:

• Step drop in efficiency with stable compression ratio? Possibly instrumentation or measurement error.
• Drop in efficiency accompanied by lower compression ratio and higher discharge T? Likely internal leakage or recompression.
• Inconsistent trends across stages? Suggests a localized issue rather than a process-wide disturbance.

With this kind of quantitative monitoring, you’re not just reacting to failures — you’re forecasting them. And better yet, you’re doing it with the help of first principles, applied in a practical context.

References

“Polytropic Efficiency.” ScienceDirect, www.sciencedirect.com/topics/engineering/polytropic-efficiency. Accessed 17 Apr. 2025.