# Heat Transfer: Nature Versus Math

## Formulas and Assumptions Don’t Always Predict what Happens

The following formula has been around the North American hydronics industry for a long time:

**Btuh = 500 x gpm x delta T**

It can be used to estimate the rate of heat transfer into or out of a device that has a stream of water flowing through it at a known flow rate, and with a measured temperature change between the inlet and outlet of that water stream.

For example: If a heat emitter is operating at a flow rate of 8 gpm and if the temperature drop from its water inlet to its water outlet was measured to be 20°F, then that stream is delivering heat to the heat emitter at a rate of:

**Btuh = 500 x 8 x 20 = 80,000 Btuh**

You also could apply this formula to calculate the rate of heat transfer delivered by a stream of water flowing through a heat emitter at 16 gpm and undergoing a temperature drop of 10°.

**Btuh = 500 x 16 x 10 = 80,000 Btuh**

If you think about the math involved, you probably realize an infinite number of combinations of water flow rate and temperature change exist that would give the same result of 80,000 Btuh. Here are just a few …

**Btuh = 500 x 32 x 5 = 80,000 Btuh**

**Btuh = 500 x 160 x 1 = 80,000 Btuh**

**Btuh = 500 x 80 x 2 = 80,000 Btuh**

However, don’t interpret this math to imply that all these combinations are achievable operating conditions. For example, if you assumed a heat emitter could operate with a 800° temperature drop and a flow rate of 0.2 gpm, the math would work out as:

**Btuh = 500 x 0.2 x 800 = 80,000 Btuh**

But do you really think you could pass water through a hydronic heat emitter at 0.2 gpm and have that water undergo an 800° temperature drop? If so, I suggest you run some tests. If it works, immediately file a patent application on the process, because you’re going to become a billionaire.

##
**Math vs. Reality**

There is a distinct difference between what the formula (Btuh = 500 x gpm x delta T) calculates as the rate heat transfer from measured flow rate and temperature drop values, and how this formula should be used to predict the heat transfer rate from assumed flow rate and temperature drop numbers.

Suppose you designed a distribution system that would operate at a flow rate of 8 gpm and a 20° water temperature drop under design-load conditions. Then you pondered how the system should operate under half-load conditions (e.g., a heat output of 40,000 Btuh). You might assume that by slowing the flow rate from 8 gpm to 4 gpm and maintaining a 20° temperature drop, the system would release half its design-load output. After all, the math works out as follows:

**Btuh = 500 x 4 x 20 = 40,000 Btuh**

Unfortunately, nature doesn’t agree. To see why, take a look at Figure 1, which shows the heat output versus flow rate relationship for a 12-foot-long piece of typical residential fin-tube baseboard being supplied with 160° water. The baseboard this graph is based on is rated to release 600 Btuh per foot of element at a 200° water temperature and 1 gpm flow rate.

Figure 1 is based on a detailed engineering model of fin-tube baseboard that accounts for continually decreasing water temperature along the element. The heat output of a 1-inch-long segment of element, at some location along the baseboard, is slightly lower than the output from a 1-inch segment just upstream of that location and slightly higher than a 1-inch segment just downstream of that location.

The model used to construct Figure 1 also is based on information in a baseboard rating standard published by the former Institute of Boiler and Radiator Manufacturers (IBR). Specifically, that heat output varies with the 0.04 power of flow rate. If the heat output from a baseboard at a flow rate of 1 gpm is 250 Btuh per foot, then the heat output from that baseboard at a flow rate of 4 gpm (and the same water temperature) would be:

**250 x (4)0.04 = 250 x 1.057 = 264 Btuh**

If you check heat output tables from baseboard manufacturers, you’ll find this factor of 1.057 is often the ratio between the output number listed for a flow rate of 4 gpm and the output listed for a flow rate of 1 gpm.

##
**Reality vs. Assumption**

Let’s use Figure 1, starting with a water flow rate of 4 gpm, entering the baseboard at 160°. Room air flows into the fin-tube element at 68°. Under these conditions, the baseboard’s heat output is about 4,350 Btuh.

Now look what happens if you maintain the same inlet water temperature and same inlet air temperature, but reduce the flow rate by 50 percent (e.g., from 4 gpm to 2 gpm). The heat output drops from 4,350 Btuh to about 4,200 Btuh — a drop of only about 3.5 percent.

If the flow is cut in half again, down to 1 gpm, the heat output drops to about 3,900 Btuh, a drop of about 10.3 percent from the heat output at a 4 gpm flow rate.

The curve in Figure 1 doesn’t go below flow rates of 0.4 gpm. That’s because at flow rates in the range of 0.3 gpm, the water passing through the 3/4-inch copper fin-tube transitions from turbulent to laminar flow. This will cause a significant drop in heat output. But at exactly what flow rate below 0.4 gpm this transition occurs is hard to predict, so this portion of the graph can’t be shown with reasonable certainty.

The take away is that even after reducing the flow rate from 4 gpm to 0.4 gpm, the heat output of the baseboard is still about 3,500 Btu, or roughly 20 percent lower than at 4 gpm.

##
**Twenty is Not a Magic Number**

Figure 2 shows what happens with the temperature drop across the same 12-foot baseboard as the flow varies across the same range, while the supply water temperature remains at 160°.

At 4 gpm, the temperature drop across this baseboard is only about 2.2°. At 2 gpm, it’s about 4.2° and at 1 gpm, the delta T is about 8°. If the flow gets down to 0.4 gpm, the delta T is just under 18°. The take away here is that the temperature drop along the baseboard naturally changes as the flow rate through the baseboard changes. The notion that a system wants to or even can remain at a fixed temperature drop as the flow rate changes is not supported by these results.

This behavior is not limited to fin-tube baseboard. It also applies to other heat emitters such as fan-coils and radiant panel circuits. What happens with upward heat output and temperature drop for a 300-foot-long circuit of 1/2-inch PEX tubing buried in a bare 4-inch-thick concrete slab at tube spacing of 12 inches? The circuit is supplied with water at 110°. The room temperature above the floor is 70°.

Reducing the flow rate from 2 gpm to 1 gpm only reduces heat output from about 9,300 Btuh to about 8,200 Btuh, a drop of about 12 percent. The corresponding temperature drop along the circuit changes from about 9.5° at 2 gpm to about 17° at 1 gpm.

Don’t assume that just because a mathematical formula is balanced on both sides of the equals sign that nature will mold itself to make sure it agrees with the math.

In a future Hydronics Zone column, we’ll look at the number 500 in the following formula:

**Btuh = 500 x gpm x delta T**

You’ll see where it comes from and that it, too, is subject to change, depending on the fluid used and the temperature at which that fluid operates.

Publication date: 1/27/2014

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