I’ve been working on a post that shows how a scatter plot technique can be used to assess the performance of an HVAC airside economizer. It’s based on comparing a plot of outdoor air temperature vs. mixed air temperature for a perfect economizer (left) with that same information taken from data loggers or control system trends for an operating system (right).
Contents
The following links jump you to different topics in the post. At the end of each major topic you will find a link that will bring you back to here.
 Why Worry About the Physics of a Mixed Air Plenum
 Conservation of Mass
 Conservation of Energy
 The First Law of Thermodynamics
 Introducing the Property of Enthalpy
 Simplifying Assumptions vs. Simplifying Substitutions
 Elevation Changes in a Mixed Air Plenum
 Velocity Changes in a Mixed Air Plenum
 Moving from a General Case to a Specific Case
 But Wait, There’s More!
 Specific Heat
 Leveraging the Ideal Gas Relationship for Enthalpy
 Ta Da!
 Other Useful Relationships
 The Psych Chart, A Graphical Approach to the Same Problem
Why Worry About the Physics of a Mixed Air Plenum
To develop the perfect economizer lines, you do math based on the physics of what is going on in a mixed air plenum. The equations you use to do that are also useful for field work, so I wanted to share those. But, I am a believer in understanding where the various mathematical relationships we use in the field came from so you are aware of the constraints, if any, behind them.
For instance, I have done a number of posts on the unit conversion constants that are in familiar equations we use in HVAC, like the 1.08 in the equation for sensible load in an air stream …
… or like the 500 in the water side load equation.
Since the constants include conversions for density and since, for instance, the density of air varies with altitude, you can’t use the sensible heat equation as expressed above for work in Denver, the mile high city, with out correcting the units conversion constant for altitude.
So, to lay the foundation for the economizer scatter plot post, I thought I would develop the relationships behind the equations that it will use. Hopefully, this will be helpful to those who want to understand and/or feel comfortable with the physics behind the math.
And it was a good exercise for me because in the slide set I use for this topic in class, there were a couple of points where I used a technique similar to what the scientist used in the somewhat classic Sidney Harris Then a miracle occurs cartoon (hover over the top thumbnail on the right on the page the link takes you to and it will show up). Now, I have a more complete set of slides for future classes.
Conservation of Mass
From a fundamental physics standpoint, there are two things going on in a mixed air plenum that are of interest to us in the context of our discussion. One is conservation of mass.
In other words, if I bring in 1 pound outdoor air through the intake system and add one pound of return air from inside the building to the mix, there will be two pounds of mixed air leaving the plenum. Stated mathematically, it would look like this.
Not too scary, even if you are mathphobic like I am.
Conservation of Energy
The other fundamental physical principle that is of interest to us in the context of our discussion is conservation of energy.
This is the first law of thermodynamics. There are a number of ways to say it in words, but to quote Herman Stoever in Engineering Thermodynamics …
If any system undergoes a process in which energy is added or removed from it (in the form of work or heat), none of the energy added is destroyed within the system and none of the energy removed is created within the system.
I selected that particular reference because it is one that Bill Coad, one of my mentors used and I always found the book to be very understandable. Some might observe that it was written in 1951 and thus, may not be relevant. But near as I can tell, the fundamental physical principles we apply to HVAC processes are still about the same now as then. And for me, having a reference that I find to be readily understandable is nice.
The First Law of Thermodynamics
The mathematical statement of the first law of thermodynamics is often referred to as the steady flow energy equation, which looks like this.
So, there are a bunch of terms, but mostly only the basics of math; addition, multiplication and division, plus the velocity term is squared. So a bit more intimidating than the conservation of mass equation, but not to bad.
The equation typically is developed by considering an apparatus through which a steady flow process is taking place as shown below. (Apparatus is simply a technical term that I am using for a thing or gizmo to make it sound like I am doing something very knowledgeable and sophisticated.)
From the standpoint of thermodynamics, there are a number of constraints regarding what constitutes a steady flow process but it is reasonable to assume that a mixed air plenum in a steady state condition (the entering and leaving temperatures are steady, the dampers are not moving and the fan is not changing speed so the flow rates are steady) constitutes a steady flow process.
Unfortunately, a mixed air plenum typically involves mixing two air streams at a steady flow rate to create a third air stream, so it is a bit more complex than the apparatus shown above. Specifically, we are concerned with the conservation of mass in addition to the conservation of energy so we have to combine the two equations and account for the second incoming fluid stream somehow if we are going to use the relationships to our benefit by manipulating them mathematically.
That means the mathematics get a bit more complex. If we rewrite the equation to come up with a general case where:

There is a steady flow of mass through the process and potentially,

Work is done on or by the process at a steady rate and/or

Heat is added or removed from the process at a steady rate,
then we end up with something that looks like this.
The bar over the “Q” and “W” terms means that the heat transfer and/or work are being done at some sort of rate, like Btu/hr or ftlb/hr. The dot over the “m” term means a mass flow rate, like pounds per hour.” The “∑” symbol means that the parameters inside the parenthesis are totaled up for all of the fluid streams on each side of the equation.
So now, things are starting to get a little scary. But if you remain calm and continue to breath normally, you will discover that, for our purposes, we can combine some terms and make some simplifying assumptions that will reduce the complexity of the math and still give us meaningful results for our work with HVAC systems.
Introducing the Property of Enthalpy
In thermodynamics, we often talk about properties of a system or substance. In general terms, a property is an observable characteristic of a system, something like temperature or pressure or volume. The p and v terms in the equation above represent the properties of pressure and volume. The u term represents the property of internal energy, which is basically molecular motion. We observe it as temperature.
It turns out that the collection of properties …
.. show up a lot in thermodynamics. As a result, the collection of properties has been given the name enthalpy and the symbol h. Mathematically …
… which means the steady flow energy equation can be rewritten to look like this.
So that is a little bit of progress in making it a bit less intimidating. But there are still a lot of terms to deal with. However, for most HVAC processes, there are two simplifying assumptions that we can make that further reduce the complexity.
Simplifying Assumptions vs. Simplifying Substitutions
Up until now, we have simplified the steady flow energy equation by substituting a new property called enthalpy for a common collection of properties used in the original equation; specifically, the properties of internal energy, pressure, and volume. That is a bit different from what I am about to do, which is eliminate terms from the equation because, in the context of the processes and systems we are talking about, they are insignificant.
Its important to realize the subtle difference; I am about to make assumptions that will be valid for the specific conditions under which I am making them. But change the conditions – for instance, apply the simplified equation to a process where the air flow becomes supersonic – and the results will not be correct. In contrast, the substitution of the property of enthalpy for the collection of terms involving internal energy, pressure and volume will still yield correct results.
Elevation Changes in a Mixed Air Plenum
For virtually any mixed air plenum, including one in a highrise building, even if the intake as on the roof and the outlet was in the basement, the magnitude of the potential energy term …
… will not be significant.
In other words, compared to some of the other terms in the equation, the changes in elevation that occur in an HVAC process like a mixed air plenum can probably be ignored in favor of simplifying the analysis with out impacting the accuracy of the calculation significantly.
For example, lets say that the outdoor air and return air dampers associated with a mixed air plenum were at the top of the plenum entering from the left and right side and that the centerline of the duct leaving the plenum is 10 feet below the centerline of the two ducts entering the plenum. In this case:
That’s a pretty small number, even though the duct centerlines were 10 feet apart. Most of the time, for a mixed air plenum, the difference in elevation between the ducts entering and leaving will only be a foot or two, if any.
In contrast, if the temperature changed 1°F in the process and no moisture was added, the energy content of the air would have changed by about 0.25 Btu per pound. That’s over an order of magnitude more than what was associated with the elevation change.
If we assume that we can ignore elevation changes in a mixed air plenum, then the steady flow energy equation is further simplified:
Velocity Changes in a Mixed Air Plenum
It also turns out that for virtually any mixed air plenum, the changes in the kinetic energy term in the equation …
… will be insignificant.
In other words, the velocity change that we might see in the mixed air plenum of a typical HVAC system, compared to some of the other terms in the equation can probably be ignored in favor of simplifying the analysis with out impacting accuracy.
For example, lets say that the velocity exiting the mixed air plenum is 2,000 feet per minute higher than the velocity entering the mixed air plenum. Since the units for the gravitational constant are in feet per second squared, we need to express 2,000 feet per minute as 33.3333 feet per second.
Once we have done that:
Most mixed air plenums will not have a velocity change of 2,000 feet per minute or more through them. For many, there is virtually no velocity change. And, as demonstrated above, even if the velocity change was as high as 2,000 fpm, the impact on the overall process is an order of magnitude less than the impact of a temperature change of 1°F.
If we simplify the steady flow energy equation by making this assumption, the result now looks like this.
Now we are making progress. If only we could get rid of those scary looking sigma things and maybe some of the parenthesis.
Moving from a General Case to a Specific Case
While the simplifications we have made so far were made in the context of our mixed air plenum discussion, they are also reasonable for the general case in HVAC. But if we now make an adjustment or two for the specific case of a mixed air plenum, we can get rid of some of the mathematical stuff that may be intimidating us.
Specifically, for our mixed air plenum, there will typically be three air streams involved:
 The outdoor air stream, which is an input in terms of mass and energy.
 The return air stream, which is also an input in terms of mass and energy.
 The mixed air stream, which is an output.
Note that some systems may have a separate minimum outdoor air flow duct or fan. But in that case, you could either consider that flow and the maximum outdoor air flow as one. Or you could make similar adjustments to the ones I am about to make and develop the math for four air streams instead of three. That might be helpful if there was a minimum outdoor air fan in the minimum outdoor air flow; more on that in a minute.
Returning to our discussion, the sigma signs are just a fancy way of saying add up all of the occurrences of this term. Meaning, given out three air streams, we could write the equation like this and eliminate them.
Notice that we also were able to eliminate some of the parenthesis since our previous simplifications left only enthalpy to be multiplied by the mass flow rate. So the second curved set of parenthesis were not needed.
But Wait, There’s More!
Unless you have a separate minimum outdoor air duct with a fan in it, there will be no work done on the air going through the mixed air plenum. And, if the mixed air plenum is well insulated, it is reasonable to assume that the heat transfer to or from it is insignificant relative to the remaining terms in the equation, specifically, the enthalpy change that occurs across it.
As a result, we can eliminate the work and heat transfer terms from the equation in most situations, leaving us with the following result.
Definitely an improvement from where we started in terms of complexity. But the equation is in terms of enthalpy, which can be harder to come up with in the field versus a parameter like temperature. And temperature is simply easier to “wrap our head around” since as humans, we are very aware of it.
That’s not so true for enthalpy all though we experience changes in it just as often as we experience changes in temperature because temperature is our gauge for a change in internal energy and enthalpy includes that. We just are not accustomed to thinking of it that way.
Oh, if only there was a way to express the equation above in terms of temperature. That would really make things easier for us field folks.
Specific Heat
It turns out that our wish can come true; there is a way to express enthalpy in terms of temperature and the path to that is the property of specific heat. Specific heat is a measure of the amount of energy it takes to change the temperature of of a unit mass of a substance 1 degree. It is different for different substances and can vary with temperature and pressure, as illustrated below in a chart that documents the constant volume and constant pressure specific heats for air.
That said, for air at very low pressures and temperatures relative to the range shown on the chart, i.e. air in an HVAC system, it is reasonable to assume the specific heats are fairly constant.
The generally accepted value for the constant pressure specific heat (termed c_{p}) for air at or near atmospheric pressure , is 0.241 Btu per pound. Under similar conditions, the constant volume specific heat (termed cv) is generally assumed to be about 0.1712 Btu per pound.
You may recall from some of my previous posts that for HVAC systems, we can treat dry and moist air as if it is an ideal gas. That comes in very handy in the context of our current discussion because it turns out that for an ideal gas, the following relationship applies.
This looks promising since we can fairly easily measure temperature in the field.
Leveraging the Ideal Gas Relationship for Enthalpy
Lets start putting things together to develop a couple of useful relationships. Specifically, let’s combine the simplified steady flow energy equation …
… the conservation of mass equation …
… which can also be written as …
… and the ideal gas relationship for enthalpy.
We will make some substitutions and then do a bit of algebra. First, we will eliminate the m_{ReturnAir} term from the steady flow energy equation by making a substitution as follows.
Since we are interested in the outdoor air percentage, we will divide both sides of the equation by the mass flow of mixed air (the m_{MixedAir} term).
For two of the terms in the resulting equation, the mixed air terms cancel out.
We can further simplify this by rearranging some terms and also recognizing that …
… on a mass basis. But, if the density of the air does not change much in our mixed air plenum, then that percentage also represents the percentage of outdoor air on a volume basis, which is how we tend to think of it for HVAC systems.
The bottom line is that we can write the equation we have developed in the following form and then solve it for %_{OutdoorAir}.
Now we can leverage our assumption about air being an ideal gas and the relationship for constant pressure specific heat that is associated with that. Specifically, because ….
…we can say …
… and.
As a result, we can write the outdoor air percentage equation as follows:
Since the mixing process in a typical HVAC system occurs over a relatively small temperature span in the context of the temperature span that would cause the constant pressure specific heat to vary, it is reasonable to assume that the constant pressure specific heat is constant.
Ta Da!
Assuming the specific heat is constant allows us to cancel out a few more terms in the equation and arrive at a very useful relationship for operations and field work.
Note that all of these calculations assume that the mixed air temperature is the true mixed air temperature. That means that because of the temperature and velocity stratification that exists in a real world mixed air plenum, you need multiple data points to come up with the true mixed air temperature and may even need to take velocity into account.
Or, you need to measure at a point where the air is truly mixed. If there was a fan involved in that process, then you need to correct for the fan heat it added to the air stream. If the motor and/or belts are in the air stream, then you need to correct for those losses too since they will show up as heat.
Also note that the relationship above becomes indeterminate when the outdoor air temperature and return air temperature are the same. That also means that the accuracy of the prediction degrades as those two temperatures approach each other.
Other Useful Relationships
I use the relationship we have been discussing in my perfect economizer spreadsheet as well as for other economizer spreadsheet calculations. For example, the spreadsheet I use for assessing the savings associated with making improvements to an economizer process uses the relationship we derived along with a number of other handy relationship you can derive from it by doing a bit more math and making a few substitutions.
Say for example, that you wanted to determine the outdoor air temperature (t_{OutdoorAir}) that would cause the mixed air temperature to drop to freezing (t_{MixedAir}=32°F), given a return temperature (t_{ReturnAir}) and an outdoor air percentage (%_{OutdoorAir}):
A more general case of the preceding would allow you to solve for a specific mixed air temperature of interest. For instance, you may want to know how cold it would have to get outside given a system′s return temperature and minimum outdoor air percentage before the mixed air temperature would drop below the design mixed air temperature and preheat would be required.
Or, you may want to determine the mixed air temperature (t_{MixedAir}) created by a particular outdoor air percentage (%_{OutdoorAir}), outdoor air temperature (t_{OutdoorAir}), and return air temperature (t_{ReturnAir}).
There are a number of other ways you could manipulate the equations to find out useful information, but those are the ones I use the most. In fact, I have the calculations built into a little spreadsheet that I have posted under the resource links on the blog home page if you are interested in it.
All of the relationships we looked at were developed for dry bulb temperature. But I should note that you can develop similar relationships for the moisture content of the air. In general terms if you simply substitute relative humidity, dew point temperature, wet bulb temperature, etc. for the dry bulb temperature in the preceding relationships, the will give you reasonable results.
The Psych Chart, A Graphical Approach to the Same Problem
If all of that is still a bit to much math for you, you can always solve the problem on a case by case basis using a psych chart. This is not the way to go if you want to assess things for a couple thousand data logger points, thus the advantage of understanding the math. But for a field situation where you are simply trying to understand what is going on, the psych chart may be the way to go. In some ways, it is better because it paints a picture that includes the impact of the moisture in the air.
If you are mixing two air streams on a psych chart, the mix point lies on the line created by the two given points, and the distance between the points on the line is proportional to the temperatures.
For instance, if I wanted to know the outdoor air percentage associated with making freezing mixed air when I had a design condition of 75°F, 40%RH for the return air from a hospital patient room and an extreme outdoor air condition of –3.2°F/50% RH, I would plot those two points on the psych chart and connect them with a line.
Then, I would draw a constant 32°F dry bulb temperature line on the chart. The point where those two lines cross represents the freezing mixed air condition for the state points I plotted.
Next, I measure the total length of the line between the two points.
Note that the dimensions in the screen shots I am using are what I get with the slide projected on my monitor, so you may get different dimensions from the image on the blog as it projects on your monitor. As you will see in a minute, it is not the specific dimensions that matter, it is the ratio of the dimensions.
Next, I measure the distance between the mixed air point and the return air point, which comes up to about 41/2 inches on my monitor.
If I divide the total length of the line (83/16 inches) by the distance from the return air point to the mixed air point (41/2”), I end up with 0.55. In other words, the mixed air point is 55% of the way from the return air point to the outdoor air point, which means the outdoor air percentage that will make a freezing mixed air plenum for this system at the conditions I selected would be 55% outdoor air.
If I use the relationship we developed and do the math in a spreadsheet or with a calculator, I end up with virtually the same number. The minor differences are related to how accurately I plotted my points and read the dimensions off of the psych chart, rounding errors and precision in the calculations, etc.
It is also worth noting that if the building were not humidified and the only source of moisture was the people in the facility, then the indoor condition would likely be more along the lines of 12.5%. If you go through the same exercise, you will discover that while the lengths change, the dimensions still work out and predict the mixed air percentage as 55%, just like the calculation does.
That is the beauty of the psych chart; it can provide an accurate solution to some fairly complex problems using a graphical technique. Kind of attractive if you are math phobic.
That said, now you know about both techniques and have some insight into the physics going on in a mixed air plenum. That information, along with the information in the next post, will set us up to understand the perfect economizer and then use scatter plot techniques to assess how well a real world one stacks up against it.
But the next post will have to wait because I will be tied up tonight going out to dinner with my bride and then watching the Muppet’s new show.
Hopefully you will be able to do the same.
David Sellers
Senior Engineer – Facility Dynamics Engineering