Water Flow and Energy in Coils

In the United States HVAC industry, fluid flows are almost always reported in gallons per minute (gpm) for design and calculation purposes.  As you are probably well aware, a standard formula to calculate the thermal transfer in a heating or cooling coil is:

Btuh = gpm*500*(To-Ti)

Hmm.

We know that the specific heat of water will relate energy flow to temperature change in the water.  We also know that specific heat refers to a unit mass of water.  We also note that we are looking at Btu-per-hour, while the flow rate is in gallons-per-minute.   So the factor of 500 must accommodate these items, among others.  Let’s investigate a little.  We will be using our old English units for this one.

The density (mass) of water is approximately 1.938 slugs per cubic foot at 40 degF.   Or, multiplying by 32.2, about 62.4 Lbm per cubic foot.

A cubic foot of water is approximately 7.48 gallons.  Which suggest that a gall0n of water weights (62.4 lbm/7.48gal) 8.34 pounds a gallon.  Which is just about right.

The specific heat of water is 1.002 Btu/Lbm-DegR at 40 degF (the Rankine scale has the same incremental values as the Fahrenheit scale fortunately!)

So let’s see where that 500 comes from.  We need to convert gpm to lbs-per-hour.  And this gets multiplied times specific heat:

Gallons/minute * (8.34 Lbs/gallon * 60 minutes/1 hour) = Lbs-per-hour

(8.34 Lbs/gallon * 60 minutes/1 hour) * 1.002 Btu/lbm-DegR = conversion factor.

Which works out to 502.  Close enough for HVAC work!  And indeed, if you tweak the temperatures at which you select the specific heat and density of the water, you can hit the 500 on the nose.

Note, by the way, that we are looking at the coil power, not the energy.  The energy is equal to the coil power multiplied by time!

Lets do an example.  Imaging a coil with a flow of 200 gpm, an entering chilled water temperature of 40 degrees and a leaving chilled water temperature of 52 degrees.

The cooling rate is:  200 gpm * 500 * (52-40) = 120,000 Btuh.

You may recall that cooling power is often reported as tons for large HVAC systems, one ton equivalent to 12,000 Btuh.  If you wish to solve directly for tons, simply divide 500 by 12,000:

The cooling rate is ( 200 gpm * (52-10) ) / 24 = 100 tons

Note that if you wanted to carry the units on the 500 conversion factor, it would be (Btu-min/gallon-hr-DegR)  Which kind of explains why you don’t see it much.

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