# Humph!

I read a quite a lot of stuff on the Internet about various forms of alternative energy. A pretty sensitive warning sign of confusion is abuse of kilowatts and kilowatt-hours, at least when it's not obviously just a typo. This page is intended as a reference location to send people to when untangling such confusion seems like it might be helpful. (Some cases are sufficiently hopeless that it's just not worth getting involved.)

The typical errors are:

- To use units of power as units of energy, for example to write that a solar panel produces 5 kW per day, and
- To use units of energy as units of power, e.g., saying that the output of a power station is 2 GWh without specifing a time period.

# Symbols and Names

First, let's get the really trivial bits out of the way. I think I've seen pretty much every possible capitalization of the symbol "kWh". All but one of them are wrong and some could potentially be confusing; in particular, an upper-case "K" could be taken for the symbol for the kelvin which is a temperature unit and therefore likely to be used in related contexts.

A good place to look for information on the use of the SI system of units are these pages published by the US Government NIST: Essentials of the SI. Wikipedia is pretty handy, too. Somewhat more definitive is the BIPM's SI brochure. The basic rules for the case of symbols and prefixes in SI are simple enough though:

- Symbols for units not named after people (e.g., the hour in the kWh) are written in lower case.
- Symbols for units named after people (e.g., the "W" which is the watt, named after James Watt of steam engine fame) are written in initial capitals.
- Prefixes for multipliers of 1 000 and smaller (i.e., "k" for kilo-, "h" for hecto-, "da" for deca- and all the prefixes for multipliers smaller than one) are written in lower case.
- Prefixes for multipliers of 1 000 000 and greater (e.g., "M" for mega-, "G" for giga-) are written in initial capitals.

Note, though the symbols for units named after people are written in initial capitals (e.g., "W") the actual names of the units are written in lower case ("watt"). Also, no plural "s" is appended to the symbol when there is more than one of whatever: it's "5 kW", not 5 "kWs" though "five kilowatts" is correct.

# Basic Energy Units

The SI unit of energy is the
joule
(symbol "J") named after
James Prescott Joule.
It's the amount of energy involved in a force of one
newton
(symbol "N") moving through one metre. A newton, in turn, is
the force required to accelerate an object of mass 1 kg
at 1 m·s^{-2}.

The acceleration due to
gravity
varies across the surface of the Earth but is about 9.8 m·s^{-2}.
In such a gravitational field an object with a mass of 1 kg weighs
9.8 N. Expending 1 J will lift this mass by (1 / 9.8) m, about 102 mm.
Another way of looking at it is that the joule is the amount of energy
involved in lifting 1/9.8 kg through one metre: perhaps lifting an
apple on to a table.
Either way, the joule is a pretty small unit so the use of the
kilojoule (kJ) can be helpful.

The specific heat capacity of water is about 4.2 kJ·kg^{-1}·K^{-1}.
That is, it takes 4 200 joules to raise the temperature of one
kilogram of water by one kelvin (one degree Celcius). This also
shows that the joule is a little on the small side for many
day-to-day uses.

There are various slightly different definitions of the BTU but they're all a bit more than 1 kJ.

# Power Units

Power is the rate at which energy is flowing: being collected, expended or just moved around. It is measured in joules per second, which are called watts (symbol "W").

A 60 W light bulb, for example, converts electrical energy into light, heat, buzzing noises or whatever at the rate of 60 joules per second.

The joule is, as shown above, a smallish unit but then a second is not a long period of time so the watt is a fairly practical unit for low power applications. Where somewhat larger amounts of power are used, for bulk heating of small quantities of water, mechanical work, and so on, the watt is a bit small and the kilowatt (kW) is commonly used.

The horsepower is about 745 W or roughly three quarters of a kilowatt.

Quite often the BTU is used as a unit of power. This is pretty sloppy; as noted above the BTU is a unit of energy. When it is used for power (typically in building heating and cooling applications) it is usually implicity meant as BTU/h. One BTU/h is about a third of a watt.

# Further Units of Energy

Because the joule is so small, larger units of energy are commonly used. The obvious ones are the kilojoule (kJ) and megajoule (MJ). In practice, though, units based on the watt and periods of time are more widely used.

As a watt is a joule per second, a joule is a watt-second. I.e., a joule is a power of one watt flowing for one second. The watt-second is a valid but rather redundant unit.

A flow of one watt for an hour, on the other hand, is more interesting. As there are 3 600 (60 × 60) seconds in an hour a Wh is 3 600 J or 3.6 kJ: the amount of heat needed to raise the temperature of 1 kg water by about 0.85 °C. Of course, technically, the hour is not an SI unit but it is a non-SI unit accepted for use with the International System of Units.

Even a watt-hour is a fairly small unit of energy; it won't boil you a cup of tea, for example. Therefore the most common unit of energy is the kilowatt-hour (kWh). 1 kWh is 3 600 000 J or 3.6 MJ.

A Therm (100 000 BTUs) is about 105 MJ or 29.3 kWh.

# Further Units of Power

In principle, the W, kW, MW, GW, TW, etc., would make a perfectly sensible and useful set of units to measure increasingly large amounts of power. However, once we start using kilowatt-hours to measure energy it's often handy to follow that when measuring power, particularly over longer periods of time.

When the flow of energy varies over a day or longer
it often makes sense to use that period in the
measurement of power. For example, a solar panel
with a peak output of 1.5 kW at a time of year which has
an average of effectively two hours of sunshine
each day
(taking into account effects like cloud and the sun
not shining straight on to the panel early and late
in the day) will produce
an average of 3 kWh of energy. It therefore makes sense
to say that its output is 3 kWh per day or 3 kWh·d^{-1}.

Since a day is 86 400 (60 × 60 × 24) seconds long this is the same as (3 000 W) × (3 600 seconds) / (86 400 seconds) so this is just an easier to understand way of saying 125 W.

On the other hand, it does seem a bit daft to talk about, say, 3 kWh per hour, rather than just saying 3 kW, but people do it thereby showing they haven't really got a grip on what they're talking about.