Juan C. Ramirez, PhD, P.E.

When can we use the Approximate Sensible Heat Equation?​

In HVAC applications, we commonly use the formula q = (1.1)(CFM)(ΔT) as a “quick” approximation to characterize a sensible heating process. This equation is very convenient because we typically work with volumetric flows in cubic feet per minute (cfm) and this formula provides the heating rate in BTU per hour. Whether you are familiar with it or not, it pays to revisit its origins and when it is appropriate to use it.

Consider air flowing at a mass flow rate m. The heat q required to impart a temperature change ΔT is given by q = m Δh where h is the specific enthalpy of air. Using a constant specific heat c to calculate the enthalpy change, we have q = m c ΔT. In HVAC applications, it is more common to work with volume flow Q instead of mass flow. Therefore, we can rewrite the energy balance as

q =  (ρ Q) c ΔT

where ρ is the air density. Defining a new variable CFM=Q/[60min/hour] allows us to use flow in cfm in the energy balance above and still obtain q in BTU/hr:

q = ρ CFM [60 min/hour] c ΔT

Moist air is a mixture of two ideal gases: dry air and water vapor. Therefore, the specific heat of humid air is the weighted average of the specific heat of dry air and water vapor: c = (0.24+0.45⋅W) BTU/(lbm⋅°F) where W is the humidity ratio in pounds of water vapor per pound of dry air. Because in many conventional air conditioning applications the typical value of W is in the order of 0.01, we can make our first “sweeping” approximation and say that regardless of W, we will use c = 0.24 + (0.45)(0.01) = 0.2445 BTU/(lbm⋅°F). Now, our energy balance looks like this:

q = ρ CFM [60 min/hour] [0.2445 BTU/(lbm⋅°F)] ΔT

Finally, we will use a “standard” air density of 0.075 lb/ft3 in the equation above to obtain:

q = CFM [0.075 lb/ft3] [60 min/hour] [0.2445 BTU/(lbm⋅°F)] ΔT

​which you can verify simplifies to:

q = (1.1) CFM ΔT

and the units of that factor 1.1 are (BTU/hour)/(°F)/cfm. Technically speaking, these “cfm” are standard cfm (scfm) because its the volume flow corresponding to a given mass flow at “standard” conditions. So, now you know all the assumptions that are “baked into” this formula. But, let's put it to the test with an example, yes?

Calculate the sensible heating requirement (BTU/hour) to raise the temperature of 4,500 pounds per hour of:

• A) Dry air from 77°F to 100°F.
• B) Saturated air from 77°F to 100°F.​

Let’s start with the formal energy balance calculation using enthalpy changes. We can use the psychrometric chart as shown below to calculate Δh for process A (red lines) and for process B (blue lines). 

A detailed breakdown of the assumptions, derivation, and real-world accuracy limits of the Approximate Sensible Heat Equation, including comparison with psychrometric chart calculations.

• For process A, we find that Δh = (23.96 – 18.44) = 5.52 BTU/lb, therefore q = 24,810 BTU/hour
• For process B, we find that Δh = (46.19 – 40.47) = 5.72 BTU/lb, therefore q = 25,730 BTU/hour

So those are the “exact” values (I actually used software to get those enthalpy values with four significant figures). What do we get with our approximate heat gain equation? Whatever we get, it will be the same for both scenarios, because our approximation only depends on ΔT. The given mass flow rate corresponds to (4,500 lb/hour)/(0.075 lb/ft3)(1 hour/60 minutes) = 1,000 scfm. Hence:

q = (1.1)(1,000)(100 – 77) = 25,300 BTU/hour

For scenario A, our approximation is within 2% of the exact result, and for scenario B, we are within less than 1.7%.

Note that our approximation is based on using a constant value of 0.01 for W, but here we have successfully used it to model a process with zero humidity and a process with twice the humidity used in the derivation of the formula! So, clearly W = 0.02 is not an upper bound for the region of validity for our trusty approximation.
The question naturally follows: what is this upper bound? When do we have “too much” moisture so that we should not use the approximation? That depends on our tolerance for error. Let’s say we never want to be off by more than 5%. In this case, the exact result for the heat required to raise the temperature of 4,500 lb/hour of saturated air by 23°F would have to be (1.05)(25,300 BTU/hour) = 26,565 BTU/hour for the error to be unacceptable.

Using psychrometric software, I found that the heat required to raise by 23°F the temperature of 4,500 lb/hour of saturated air with W = 0.038 is 26,570 BTU/hour. This value of humidity ratio is quite literally “off the chart” for the ASHRAE Psychrometric Chart No. 1 included in the NCEES reference handbook.

The take away message from all this rambling is that you are “safe” using q = (1.1)(CFM)(ΔT) anywhere within the visible space of ASHRAE Psychrometric Chart No. 1, and only when moisture levels start reaching 0.04 or more pounds of water per pound of dry air should you worry about loss of accuracy.

So, here is a challenge for you.
​I mentioned I used software to calculate the heat required to raise by 23°F the temperature of a stream of 4,500 lb/hour of saturated air with a humidity ratio of 0.038. You don’t really need software for this.

Can you use the resources available in the NCEES reference handbook to solve that problem?​