Where Do The Constants Come From?
John Siegenthaler, PE
Over the years you’ve seen plenty of formulas in my columns. As you’ve probably figured out, I’m a big believer in “running the numbers” as a part of making informed design choices. Without doing so, design becomes a matter of guestimating, and oversizing often runs rampant. It needlessly drives up cost and lowers performance.
A few months ago, a very accomplished hydronic heating professional was looking over some material I had just written for the second edition of the RPA Radiant Precision training manual. He asked me a question that really made me reflect back on all the formulas I’ve put into columns and other publicatons over the years. Formulas that I’ve stressed were important for proper system design. His question was simply: “Where do the constants in the formulas come from?”
In case you don’t have fond memories of high school math jargon, a constant is simply a number. Its value never changes and that’s why it’s called a constant.
Here’s an example of a formula with a constant. This formula determines the number of cubic yards of concrete in a 1 1/2-inch thick concrete thin slab having a known area in square feet:
V = (0.00463) x A
V= volume of concrete required (cubic yards)
A = area to be covered by 1.5-inch thick slab (ft2)
0.00463 = constant
So why does 0.00463 belong in this formula?
The answer depends on basic geometry and the units of measurement that we want to use, in this case square feet for area and cubic yards for the calculated volume of concrete.
Let’s start with the geometry. The volume of any rectangular solid equals its length times its width times its depth. Stated as a formula this would be:
volume = length x width x depth
If we use symbols instead of words, the formula reads:
V = l x w x d
V = volume
l = length
w = width
d = depth
As it stands now, we could use any units of measurement for length, width and depth. Suppose, for example, we measured length in feet, width in yards and depth in inches, and then multiplied these together. Would we get a volume? The answer is yes, but the units for that volume would be ft•yd•in. The dots between the units mean multiplication. You would pronounce this unit as “foot yard inch.”
I defy anyone to get a practical feel for a volume expressed in such units. It’s like trying to get a practical feel for speed of a car measured in furlongs per millisecond. Although such units are valid, they are certainly not common. Imagine calling a concrete batch plant and telling them you need 1,250 “foot yard inches” of concrete.
So, following established practice, we express length, width and depth in the same units. Since we’re in the United States, let’s use feet. The volume we calculate would have units of ft•ft•ft or simplified ft3 (pronounced as “cubic feet,” or if you like “feet cubed”).
Here’s an example: Determine the volume of a concrete slab having a length of 50 feet, a width of 20 feet and a depth of 1/2 foot.
V = l x w x d = 50 ft x 20 ft x 1/2 ft = 500 ft3
Now back to Formula 1. If we want the volume to come out in cubic yards (the common unit by which concrete is ordered), we have to be sure the units on the right side of the equal sign also reduce down to cubic yards. Here’s how we do that:
V = depth x area = 1 1/2 inch x (A) ft2
A = area of slab (ft2)
What we need to do is multiply by one or more unit conversion fractions to make the units of inch•ft2 change to cubic yards. Here’s how that’s done:
Notice that the formula has been multiplied by two fractions. The first one is (1 ft/12 inch), and the second is (1 yrd3/27 ft3). Although the units in the top and bottom part of these fractions are different, the physical quantity represented in the top and bottom of each fraction are identical (e.g., 1 ft is exactly the same length as 12 inches, and 1 cubic yard is exactly the same volume as 27 cubic feet).
When the top and bottom of any fraction are equivalent, as is the case with these fractions, the value of the fraction is 1, and multiplying anything by 1 doesn’t change that thing. Thus, these fractions are unit conversions but don’t change the size of the quantity that the units apply to.
Notice that the units in the formula are color-coded. Whenever the same unit appears as a multiplying factor in the top and bottom of a formula, these units cancel each other out, (e.g., they simply go away). Thus the unit inch after 1.5 cancels out the unit inch after 12. The same holds true for ft2 x ft, which equals ft3, in the top of the formula and the ft3 in the bottom. They cancel each other out.
V= volume of concrete required (cubic yards or yrd3)
A = area to be covered by 1.5-inch thick slab (ft2)
0.00463 = constant
The constant of 0.00463 is thus the combined result of geometry and making the left side of the formula come out in the desired units of cubic yards.
A Very Important FormulaSo how does this apply in hydronics? Let’s take at look.
One of the often-used formulas in hydronic heating design calculates the rate of heat transport represented by a water flowing around a piping circuit as a known flow rate and known temperature drop. I’m sure you’ll recognize this formula:
Q = 500 x f x ∆T
Q = rate of heat flow (Btu/hr)
f = flow rate (gpm)
∆T = temperature change of the water (degrees F)
500 = a constant
Although you’ve probably used this formula many times, have you ever wondered why it contains the number 500? The reason is again a combination of the basic physics and the desired units in the final formula (e.g., we want to use gpm for flow rate and degrees F for temperature change).
Start with the basic physics: heat flow rate = (mass flow rate) x (specific heat) x (temperature change).
You can find this equation in just about any textbook on thermodynamics or heat transfer, both in North America and other parts of the world. It represents basic physics, which are independent of the units used. In other words, the pure physics relationship between flow rate, temperature drop and heat transport rate doesn’t change, regardless of what units of measurement we use.
Expressed in symbols, this formula becomes:
Q = m dot x c x (∆T)
The symbol m with the dot over it is pronounced “m dot.” It represents the rate at which fluid mass is passing through the circuit. The m stands for mass, and the dot is mathematical shorthand for rate. The letter c represents the specific heat of the fluid, and ∆T represents the temperature change of the fluid as it flows from the beginning to the end of the circuit.
Mass flow rate can be stated in terms of the volume flow rate and the density of the fluid that’s flowing: mass flow rate = (volume flow rate) x (density of fluid).
In symbols this is:
m dot = f x D
m dot = mass flow rate
f = volume flow rate
D = density of the fluid
Substituting this into Formula 2 yields:
Q = f x D x c x (∆T)
The next step is to start multiplying the right side by unit conversion factors to make the resulting units on the right side of the equal sign come out to Btu/hr (the units on the left side of the equal sign).
Remember from the previous discussion that the top and bottom of these fractions always represent the same physical quantity; thus, the fraction as a whole always represents 1. Multiplying anything by one does not change its numerical value. Thus, you could multiply by any number of unit conversion fraction factors - thousands if necessary - without actually changing the physical quanity you started with. Only the units it’s expressed in.
The unit conversion factors are arranged so that the “undesired units” cancel out and leave behind the “desired units.” For example, the units of minutes in the bottom of the first fraction on right side of the equal sign cancels with the unit of minutes in the top of the last unit conversion fraction.
The units that cancel out are all shown in different matching colors below:
This simplifies to:
Q = (500.5) x f x ∆T
The constant of 500.5 then gets rounded off to 500, leaving:
Q ≈ 500 x f x ∆T
For this formula to be valid, the units of flow rate (f) must be gpm, and the units of temperature drop must be degrees F. The rate of heat transport (Q) will then come out in Btu/hr.
In Europe where the same physics would be expressed in metric units, the formula would be:
Q = (4.1092) x f x ∆T
Q = rate of heat flow (kilowatt)
f = flow rate (liters per second)
∆T = temperature change of the water (degrees C)
4.1092 = a constant
The value of the constant changes significantly based on use of metric units, and yet the formula represents the exact same physics.
There are probably millions of formulas that decribe the physical world around us. Even within the field of hydronic heating there are hundreds of formulas containing constants. I hope this discussion helps you understand where constants in formulas come from, and why it’s critically important to use data expressed in the necessary units when “running the numbers” with any formula. As the saying goes: Garbage in equals garbage out. The last thing we are trying to design are “garbage-grade” hydronic systems.