Saturated, Multi-Phase Systems and Proof that a Watched Pot Does Actually Boil – Plotting the Data (and a Bit about Logarithmic Scales)

Just when you thought it was safe to come back with out running into yet another post on multi-phase saturated systems.

I really do think this will be the last one.  But, it occurred to me that it might be interesting to look at the experimental results plotted on the thermodynamic diagrams that I gave examples of towards the end of part 2 in this series.  And, since some of the diagrams use logarithmic scales, I also discuss what they mean and how they are read.  So with out further adieu …

The Data

The table below is the data from the experiment that is plotted on the charts that follow.

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The values that were measured during the experiment were the temperature and pressure.  The remaining values were calculated from these two parameters using the equation of state via REFPROP.

You Only Need Two Points to Use the Charts

I used REFPROP to make the table above because I wanted to have Excel plot lines on my charts so I had nice graphics to present.  I could have done the same thing by looking up the data in something like the Keenan and Keyes steam tables.  But I don’t want that to obscure a very important point.  That is, that given the two data points I had from my experiment (pressure and temperature), I could have plotted the points on any of the charts that follow and read the other values directly from the chart.

Frequently, this is a much faster way to find out things like entropy and enthalpy and density.  And, the procedure allows you to visualize the process, which can be helpful in terms of understanding it.  But the downside is that its usually difficult to read a chart to a bunch of decimal places.  So if you need very precise numbers, then the equation of state and the tables generated using it are the way to go.  But to get quick picture of what is going on and a pretty good feel for the value of the related parameters, plotting things on a thermodynamic chart is hard to beat. 

As many of you know that is exactly what we do when we use a psychrometric chart, typically entering it with a dry bulb temperature and some measure of humidity, like wet bulb temperature (which we can read very readily with a sling psychrometer – the black thing to the right in the picture below) or relative humidity  or dew point (which are parameters that electronic instruments like the one to the left in the picture below typically provide).

Sling and 4-in-1 01

So bottom line, I only needed the first two columns in the table to plot the lines you are about to look at if I was going to do it manually.  Having done that, I could read off the values in the other columns all though not to three decimal places.  And, as you will see, using a particular chart can make some parameters easier to read compared to trying to read them from a different chart, even though they all may be present and plotted on the chart.

Logarithmic Scales

Its important to realize when you are reading some of the charts that follow (or a lot of charts used for HVAC design and analysis for that matter) that one or more axis may be plotted using a logarithmic scale.  For instance, in the P-h diagram that follows, the pressure axis on the left is logarithmic. 

In the “olden days”, we used logarithms a lot because they made it easy to work with big numbers and exponents.  So if you are young, the term may sound unfamiliar or scary (same is true if you are older and haven’t done it for a while) because you typically solve the problem by hitting a button labeled “log” on your calculator.   

That’s really a great advancement, but its probably good to not loose sight of what its about.  So, if you are rusty, it might be worth poking around on the internet a bit to refresh your memory.  (For some, the Jimmy Buffett song “Math Suks” may be coming to mind).

The reason graphs are drawn using logarithmic scales (or “log” scales as way-cool technical people might say) (a.k.a. nerds) (and not to be confused with the instruments the lumber industry uses to weigh lumber)  is  that they can straighten out curved lines, and/or compress the scale of an axis  For instance, in the P-h diagram that follows, if the pressure axis was linear, it would be much, much taller than it is. Or, it would have a much finer grid than is shown given that the minimum value is 0.1 and the maximum value is 10,000.

All that “logarithmic scale” really means is that the distance between one grid line and the next is a function of an exponent in contrast with a  linear scale, where the distance between each grid line is an equal division relative to the others.  So for instance, in the P-h diagram that follows, the distance between 1 and 10 is divided into 9, non-equal intervals with each interval being proportional to a power of 10, as illustrated in this table.

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The pattern repeats for every power of 10; i.e. 1, 10, 100, 1,000, etc., which are often called decades (not to be confused with the popular, St. Louis based, classic rock and roll band).

The Results on a Pressure-Enthalpy (P-h) Diagram

In some ways, this is not a very exciting plot.  That is because, for all intents and purposes, the pressure was virtually constant at 14.7 psia during the entire experiment.  Since the “y” axis on this diagram is pressure, the results show up as a straight line, specifically, the orange line with the yellow square markers.

Experiment P-h Diagram

Incidentally, if you want a larger version of these charts, they are on my Google Drive along with the “clean” versions that are part of the Pacific Energy Center Hot Water and Steam Systems class materials.

The Results on a Temperature-Entropy (T-s) Diagram

This one is a little more interesting because the temperature varies during the experiment.  As was the case in the preceding graph, the experimental data is the orange line with the yellow square markers.

Experiment T-s Diagram

One advantage of this representation may be that it’s nice picture of what happens during the transition from:

  • A sub-cooled liquid to
  • A saturated liquid to
  • A mix of saturated liquid and vapor to
  • A saturated vapor and finally to
  • Superheated vapor.

Things (temperature specifically) rise, level off for a while, and then rise again.  It’s a very similar pattern to the data we plotted from the experiment.  But for the experiment, the x axis is time where as in the T-s diagram, its entropy.

image

With regard to the T-s diagram, following the orange data line from left to right,

  • Generally speaking, our experiment occurred at a constant pressure;  specifically  atmospheric pressure, as I mentioned above.  You can see the how the water in the kettle followed the constant pressure line (which is initially on top of the saturated liquid line) until it reached the saturation temperature associated with the atmospheric pressure that existed at the time, which happened to be very close to the 14.7 psi associated with a  standard sea-level atmosphere.  Up to that point, the liquid is considered a sub-cooled liquid.
  • Once the saturation temperature is reached, both the temperature and the pressure remain constant and the data plots as a horizontal line stretching from the saturated liquid line to the saturated vapor line.
  • Our horizontal line of data represents a significant increase in the energy content of the water/vapor mixture as you follow it from left to right.  This can be seen because it crosses the green enthalpy lines, with the enthalpy value increasing as the orange line representing the data approaches the saturate vapor line.
  • The horizontal line that is our data also represents a significant increase in the volume of the water vapor mixture as you follow it from left to right.  This can be seen because it crosses the violet specific volume lines, with the volume increasing as the orange line representing the data approaches the saturated vapor line.
  • In general terms, the dashed black lines, which are called quality lines and curve from the bottom of the chart towards the critical point, indicate how much of the saturated water/vapor mix is liquid and how much is vapor.  Lower quality means more liquid and less vapor, with the energy content and volume being correspondingly lower.  You will note that the quality increases as you follow the orange line from left to right.
  • For the entire time that the line of data is horizontal, the energy we put into the experimental system via the stove was used to change the liquid water to vapor.  Basically that meant that it went into making the water molecules move faster and farther apart.  Since we were not constraining them, the specific volume increased significantly, as mentioned above.   And the energy increased significantly also, as indicated by the change in enthalpy.
  • Once our data hits the saturated vapor line, it continues to follow the constant atmospheric pressure line.  But the line turns sharply upward because at that point, the energy we were putting was primarily causing the temperature to increase again. The water vapor at this point is considered a superheated vapor because its temperature has been driven up about the saturation temperature for water at atmospheric pressure.

The Results on a Enthalpy-Entropy (h-s) Diagram

This chart is not as exciting in terms of how our data moves around on it as the T-s diagram was.  But to my eye, it might be the prettiest chart because of how the lines curve, converge, and diverge from each other.

Experiment h-s Diagram

Again, our data is the orange line with the yellow markers on it.

A Few Bottom Lines

As you hopefully have seen from this discussion and the related graphics, if you measure or know two parameters for a substance undergoing a process (in our case, the substance was water, the  process was boiling water in a tea kettle, and the parameters we had recoreded for the substance were temperature and pressure), then you can use thermodynamic charts to determine other important parameters, which in are case were enthalpy, entropy, volume (or its reciprocal, density), and quality.

There are a number of different thermodynamic diagrams that can be used to assess a process.  For HVAC work, the psychrometric chart is one of the most useful tehmodynamic diagrams available in terms of understanding what moist air will do. 

For our experiment, we looked at boiling water on three different diagrams, with the T-s diagram providing the best picture of what happened, at least to my eyes.  But the diagram you select to analyze your process is often a function of the type of process you are looking at.   For example a P-h diagram is often the best choice for looking at a refrigeration cycle.  And power cycles are often assessed by using an h-s or a T-s diagram.

So, having probably typed more than you care to know about all of this, I will close and move on to a different topic.

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David Sellers
Senior Engineer – Facility Dynamics Engineering
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