Saturated, Multi-Phase Systems and Proof that a Watched Pot Does Actually Boil – Part 2

Working with REFPROP to develop my Mollier diagrams for water and R134a reminded me of how much we rely on the work of those who have gone before us to do what we do every day.  That made me want to know more about where the numbers in the thermodynamic charts came from.

To some extent, this was covered by the content of my Thermodynamics class back in my college days.  But at the time, I was so focused on trying to pass the course, that I would have to say that I didn’t really appreciate what it all really meant on personal terms and the timeline of it all.  Thus, my little history lesson which follows.

A Brief History of the Work Behind the Numbers

As I understand it, most of the thermodynamic charts have there roots in equations of state for the particular fluid that is the focus of the chart.   The original equation of state is the ideal gas law, which evolved out of work in the mid 1600s through the early 1800’s by Boyle, Dalton, Charles, Gay-Lussac and others (the link takes you to some cool little animations that illustrate the various laws).


For the work we do in HVAC systems, air tends to follow the ideal gas equation.  But, because the ideal gas equation ignores the size of molecules and their attraction to each other, it does not really predict what will happen under all circumstances.  Water boiling and becoming steam is a good example because during the transition from liquid to vapor, the pressure and temperature remain constant while the volume changes by a factor of about 1,600.

In the late 1800’s Johannes van der Waals made a huge breakthrough by proposing a modified form of the ideal gas equation that included factors for both molecular size and inter-molecular attraction.  Appropriately enough, this is called the Van der Waals equation and I believe he received a Nobel Prize for the work.


As computers evolved, our ability to perform complex calculations improved, leading to polynomial based equations that further refined the predicted behavior of substances.  These equations are called virial equations of state. 

For instance, if you look at the DuPont Technical Information document T-134a Thermodynamic Properties of HFC-134a, you will discover that the equation of state looks something like this.


There are 15 polynomials that define the temperature dependence of the  “a” coefficient.   And, when you look at them, you will discover that those polynomials contain 32 additional “b” coefficients.  So, if you are math-phobic (which I am to some extent),  that is just a bit intimidating.

A Picture is Worth a Thousand Words

In contrast to the equation above, a picture can truly be worth a thousand words, or more specifically, a thousand numbers.  By using REFPROP, I was able to have the computer do the math on the latest and greatest equation of state and then take the data tables that were generated and make some pictures including a pressure-enthalpy diagram (p-h diagram) …

Water - P-h Diagram

… a temperature-entropy diagram (t-s diagram) …

Water - T-s Diagram

… and a enthalpy-entropy diagram (h-s diagram).

Water - h-s Diagram

If you want to see larger copies, they are included with the content from the recent PEC class on steam and hot water systems.  I have links to the most recent class materials on the right hand side of the opening page of the blog under 03 – Materials from Classes and Presentations.

A Few More Resources

The Penn State College of Earth and Mineral Sciences has a very good narrative discussion of the history of Equation Of State (EOS) on the e-learning site for the Department of Energy and Mineral Engineering.   It is part of a course on phase relations targeted at Natural Gas and Petroleum Engineers developed by Dr. Michale Adewumi, Professor, Petroleum & Natural Gas Engineering.  But the opening modules are general in nature and use water as an example for developing the concepts.  I found it to be very well written and very understandable.  So, if you are new to the concept of phase behavior and thermodynamics, you may find the information on the Penn State site to be a great introduction to the topic.

If you want more detailed explanations of thermodynamic properties like entropy and enthalpy in layman’s terms, then I have found the opening chapters of Principles of Refrigeration to be very valuable.  The most information can be found in the current, 5th edition, which is co-authored by Thomas Horan.  Since the 5th edition is the current edition, you would have to purchase a new or used copy.  But its well worth the price if you are working in this business (or taking thermo and struggling like I did).  Having said that, the first edition is also very good, just not as detailed.  But it can be downloaded at no cost from a non-profit called Internet Archive.

Thus ends the history lesson.  Next I will talk about why all this should matter to you if you are working with HVAC systems.


David Sellers
Senior Engineer – Facility Dynamics Engineering

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