Wade Ortel

# Watt’s a Watt-Hour?

“I lifted 25 degrees Fahrenheit at the gym today.”

The above apocryphal statement is utter nonsense. This conclusion is not reached by contesting whether the speaker did, in fact, lift 25 of something at the gym today, but because it is obvious that they did not lift any quantity of degrees Fahrenheit (°F) today or any day.

From context, it is clear that the individual making the statement is specifying a quantity of weight that they lifted, yet the units, °F, are used to measure temperature. When specifying a quantity, the units that follow the number are critical to the conveyed meaning. While this example may seem trivial, the conflation of units is a remarkably common, though the mistake is rarely so plainly evident.

Let us now consider another example: “I used a lot of energy this month. My power bill says I used 1000 kilowatts.”

At first blush, it may not be obvious why this statement is just as meaningless as the first one. We are accustomed to hearing units like kilowatt (kW) or kilowatt-hour (kWh) when describing quantities of electricity. The above statement seems to be describing a quantity of electrical energy used by an individual on a monthly basis. However, the units provided are a unit of power, not energy.

To better illustrate this point, consider a certain volume of water in a barrel. We might measure that volume in liters (L). Now assume we are filling this barrel from a garden hose, steadily increasing the volume of water stored in the barrel. In order to describe this flow of water, it would be sensible to use a unit of volumetric flow rate, such as liters per second (L/s). That is, in one second, some well-defined number of liters of water flows into the barrel.

Continuing with the water example, it should be apparent that describing the volume of water stored in the barrel using units of liters per second is not sensical. In much the same way, describing the flow from a garden hose in terms of liters alone also makes little sense. To do so would be ignoring the temporal nature of pumping water. This relationship between volume and volumetric flow rate is precisely analogous to the relationship between energy and power; power is the flow rate of energy. Mathematically, we can say that power is the time derivative of energy, or that energy is power integrated over time.

Electrical devices are generally described by how much power they can generate or consume. Therefore, when conversing about electrical devices, it is logical to begin with units of watts, the International System of Units (SI) standard unit of power. However, rather than use the SI unit of energy, the joule, watt-hours are typically used when describing electrical energy. This is done as a matter of convenience; it is easy to see that a 100-watt light bulb allowed to operate for one hour will use 100 watt-hours of electrical energy. It is less obvious that this same quantity can be expressed as 360,000 joules or 360 kilojoules. The common use of watt-hours rather than joules when quantifying electricity can be ascribed to the intuitive advantage that the former units offer. For grid-scale quantities, the SI prefixes ‘kilo’ (103) or ‘mega’ (106) are often used in order to avoid large numbers of trailing zeros.

Now you are equipped to understand the units of measure associated with the distribution and consumption of electricity. Look at your electricity bill and see how many kilowatt-hours (kWh) of electrical energy you are paying for each month. Take a look at a light bulb to see how many watts of electrical power it uses. Realize that these two measures are necessarily presented in different units, but that they share a fundamental relation.