A look at formulas relating heat transfer to flow rate.

Last summer I had the privilege of presenting a series of seminars titled "Modern Engineering Concepts for Hydronic Heating Design" in six locations across the United States.

Part of those seminars involved calculations that relate the rate of heat transfer to fluid flow rate and temperature drop.

One very attentive attendee and long-time friend, Ted Lowe, observed that the value of the "fluid factor" in the various formulas changed from one example to another. He went on to suggest it would be nice to know which fluid factor is appropriate for a given situation. I agree. So thanks for the suggestion, Ted; I hope this month's column will clarify those numbers.

## The E=mc2 Of Hydronics

One of the best known and widely used formulas in hydronic heating design is:

Q = 500 x f x DeltaT
Formula 1

Where:
Q = rate of heat transfer (Btu/hr.)

f = flow rate (gallons per minute or gpm)

DeltaT = temperature change (degrees F)

500 = fluid factor based on water as the system fluid

This formula can be used to determine the rate of heat transfer whenever the fluid flow rate and temperature change across a "device" are known.

Let's say, for example, that a flow meter indicates the flow rate of water through a boiler is 10.5 gpm. A thermometer on the inlet of the boiler reads 135 degrees F and another on the boiler outlet reads 154 degrees F. Formula 1 can be used to estimate the rate of heat transfer into the water:

Q=500x10.5x(154-135)=500x10.5x19=99,750 Btu/hr.

It's worth pointing out that the results of this calculation would be the same if the boiler inlet temperature was 95 degrees F and the outlet temperature was 114 degrees F. The DeltaT of 114-95 is still 19 degrees F. In other words, it's always the change in temperature across the device in combination with flow that determines the rate of heat transfer.

Another example would be reading the flow rate of a radiant floor tubing circuit at 1.1 gpm on a manifold flow meter, combined with a known supply temperature of 110 degrees F and a circuit return temperature of 93.5 degrees F. The rate of heat output of the circuit would be estimated as:

Q=500x1.1x(110-93.5)=9,075 Btu/hr.

The bottom line on Formula 1 is this: If you know both flow rate and DeltaT, you can quickly estimate the rate of heat transfer to or from any device that's part of a hydronic system. ## Rule Of Thumb

Many of you know the rule of thumb that 1 gallon per minute of water flow can carry 10,000 Btu/hr. along for the ride. This is based on Formula 1 when a temperature drop of 20 degrees F is assumed.

Q = 500 x f x 20 = 10,000 x f
Formula 2

Where:
Q = rate of heat transfer (Btu/hr.)

f = flow rate of water (gpm)

Remember that Formula 2 only applies when a 20-degree-F temperature drop is assumed or actually measured. It also only applies when the heat transfer fluid is water and, in fact, is still an approximation, as you will see.

## When It's Not 500

The number 500 in Formulas 1 and 2 are based on the specific heat and density of the fluid being circulated, as well as some unit conversion factors. The expanded version of the formula would look like this:

Heat rate=(density)x(specific heat)x(flow rate)x(unit conversion factor)x(temperature change)

The American customary units on these quantities would be as follows:

Btu/hr. = lb./ft. x Btu/lb.xdeg.F x gallon/minute x (1ft3/7.4805 gallon x 60 min./1 hr.) x deg.F/1

If you think back to algebra class, identical quantities (in this case units) that appear in both the top and bottom of fractions cancel each other out. That means this long chain of units can be reduced to:

Bru/hr. = d x c x f x 8.02 x DeltaT
Formula 3

Where:
d = density of the fluid (lb./ft.3)

c = specific heat of fluid (Btu/lb./degrees F)

f = flow rate (gpm)

DT = temperature change (degrees F)

Formula 3 can be used with any fluid provided the units are those stated, and the density and specific heat of the fluid are known.

The density of water at 60 degrees F is 62.355 lb./ft.3, and its specific heat is 0.99987 Btu/lb./degrees F. Putting these numbers into Formula 3 and simplifying yields:

Btu/hr. = 500.02 x f x DeltaT

It's easy to see that the value of 500 comes from rounding off the 500.02. However, this value is based on the density and specific heat of water at 60 degrees F, which is quite low relative to where most hydronic heating systems operate.

To better reflect the true heat-carrying properties of water and other fluids, the density and specific heat values used in Formula 3 should be based on the average temperature of the system fluid under design load conditions. For example, if the boiler supplies 180-degree-F water to the distribution system, and the return temperature under design load is 160 degrees F, then the density and specific heat should be determined at the average water temperature of 170 degrees F.

To give you an idea of how things change, I've evaluated the product of (density x specific heat x 8.02) for water and some antifreeze solutions and posted the results in Table 1 (rounded to the near whole number). For values at other temperatures, use the graph in Figure 1. In both cases, the units on the fluid factor are Btu/hr./gpm/degrees F.

You can see that the changes in the fluid factor (density x specific heat x 8.02) are relatively small for a given fluid over the temperature range given. However, the fluid itself has a major effect on the value. This is primarily due to the pronounced drop in specific heat of glycol solutions at higher concentrations.

Using the correct value of the fluid factor is especially important when designing glycol-based systems such as for snowmelting. ## One Thing Leads To Another

To achieve equivalent heat transport capability with glycol solutions, it's necessary to use higher flow rates. You can estimate how much higher by taking the ratio of the fluid factor (from Table 1 or Figure 1) for the glycol solution and dividing it by the fluid factor for water at the same average fluid temperature.

For example, using a 50 percent glycol solution at 120 degrees F would require 493/451 = 1.093, or 9.3 percent higher flow rate.

The higher flow rates needed for equivalent heat transport ability also increases head loss. So does the higher viscosity of the glycol solution relative to water. The combination of both results in considerably higher pumping energy requirements.

To provide equivalent heat transport ability of a 50 percent propylene glycol solution vs. water at an average fluid temperature of 120 degrees F, the increase in head loss would be approximately:

1.38 x (1.093)1.75 = 1.61 or 61 percent high head loss

The factor of 1.38 comes from comparing the density and viscosity of 50 percent propylene glycol vs. water. The factor of 1.0931.75 is based on the increased flow rate needed for equivalent heat transport ability. The exponent of 1.75 represents the fact that head loss increases with the 1.75 power of flow rate (for smooth tubing circuits). Using glycol solutions instead of water thus produces a "double whammy" effect due to both lower heat capacity and increased viscosity.

In the above situation, a designer seeking equivalent heat transport capability would be selecting a circulator with 9.3 percent higher flow and 17 percent higher head relative to a circulator for the same circuit operating with water. That's quite a difference, and does give you pause to consider high glycol concentrations in systems other than snowmelting.

So, does the number 500 still have a place in hydronic system design? You bet it does. When the system fluid is water, it's easier to make quick mental calculations using 500 instead of 494. Estimating using the 500 factor certainly gets you in the ballpark. It's also likely that other design or installation decisions could affect system performance to a greater extent than using the nontemperature-adjusted value of the fluid factor for a given fluid type. However, if you are using a calculator, spreadsheet or other computational tool, you should use the most accurate data possible. It's part of being a hydronic heating professional.