Unit Conversion Constants; Not Always Constant

Recently, Bill Acker gave me a call discuss the economizer damper test results I had been talking about in the blog over the first several posts.  Bill pointed out that many engineering calculations (including the built-in calculations in some measuring equipment) use engineering constants that are based on the properties of the measured variable at a fixed condition.

For instance, a differential pressure based air flow meter might base its flow calculations on air at standard sea level conditions. As a result, measurements taken and assessed at other conditions may need to be corrected. Bill postulated that perhaps one reason for some of the unexpected results I was seeing was just such an issue. As it turned out, the meter I was using does in fact compensate for variations in barometric pressure and temperature. But, I enjoyed our discussion and the opportunity to get to know someone else who is passionate about what they do and is in involved in our industry.

In addition, Bill’s point was well taken; its very easy to loose sight of the fact that the units conversion constants we recite from memory and use for some of our calculations are based on some very specific conditions. Take the equation we all use for calculating the sensible load that is picked up by an air stream as it goes through a temperature change. Some of you will recall it as being:

image

Others might say:

image

So is the difference simply due to rounding things off, or is there something else at play?

It turns out that the answer could be “both”.  It’s easy to see how 1.08 could be rounded to 1.1. To understand what other factors could come into play, we need to take a look at where the units conversion constant in the equation comes from.

At its core, the equation says that flowing air going through a temperature change equates to an energy flow rate, all other things being equal. But, the energy flow rate in the equation is expressed in terms of btus per hour on the left side while the air flow rate and temperature change are expressed in terms of cubic feet per minute and degrees F respectively on the right.

The problem is that btus per hour are not equal to cubic feet degrees F per minute.
Cubic feet per minute – a volumetric flow rate – needs to be converted to pounds per hour, and the specific heat of air needs to be brought into the picture to convert the temperature change of the resulting mass flow rate to an energy flow rate. That’s where the units conversion constant comes into play. Its really not a single unit-less number; rather, its an amalgamation of several dimensioned numbers that make the terms on both sides of the equation consistent.

This is illustrated below for dry, sea level air at 70°F.

108-cancel 2

Notice how the units all cancel each other out and the numbers multiply together to create the familiar 1.08, which could easily be rounded off to 1.1. But, if we change the density from that associated with dry air at 70°F to that associated with dry air at 60°F we also arrive at a factor of 1.1, but not because we rounded off 1.08.

11 cfm t 2

The point is that the units conversion constant is an amalgamation of numbers that are specific to certain conditions; change the conditions and the numbers and associated units
conversion constant change.

For our example, the density of air will vary as a function of temperature, moisture content, and pressure, sometimes significantly, as can be seen from the figure below. Specific heat is also impacted, but not to the extent that density is.

Density and specific heat graph

Using 1.08 or 1.1 for sensible energy gain calculations for air flowing in typical commercial HVAC systems will usually yield satisfactory results in most situations. But, if we are working on a system that is high up in the mountains, or if the air is coming off a process that is very hot and humid, then we may need to re-assess the units conversion constant we are using if we want a reasonable answer.

The table embedded in the graph above shows how much the constant in our equation can vary with different temperature and humidity levels. Similar effects occur if you vary barometric pressure.

The bottom line is that as technical professionals, we need to be aware of the limitations of the equations and rules of thumb we apply in our practice. And, if we encounter an out of limits condition, we need to know how to adjust the rules and equations to yield the desired level of accuracy.

Hopefully, this post has given you some insight into both the need and the technique. If I’ve made you curious and you want to try your hand at a common constant, see if you can derive the 500 in:

image

I’ll post the answer in a week or so.


David Sellers
Senior Engineer – Facility Dynamics Engineering

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