How do I compare heat and energy values given in different units?

Where I live, in Vancouver BC, energy to power my life comes in three major forms: natural gas to heat my home, electricity to light it and run appliances (mostly from hydro-electric dams), and oil in the form of the gasoline that I put in our car. Natural gas is sold by the GJ, electricity by the kWh, and gasoline by the litre. The unit costs I pay are, approximately (as of September 2019, in $\$$CDN)[note] Price of Natural Gas, /article/price-of-natural-gas/[/note][note] Price of Electricity, /article/price-of-electricity/[/note]:

**Natural gas**: $\$$12/GJ

**Electricity**: $\$$0.15/kWh

**Gasoline**: $\$$1.50/L

However, these are averaged costs for a single-family dwelling. Now we need to consider incremental costs – the cost of one extra GJ, kWh or L, regardless of the circumstances:

**Natural gas**: $\$$10/GJ

**Electricity**: $\$$0.11/kWh

**Gasoline**: $\$$1.50/L

Note: the number for gasoline is the same, as there is no fixed infrastructure charge.

Now to compare apples with apples and compare the costs in terms of $\$$/J and $\$$/tonne CO_{2}.

First, 1 kWh is 3.6 MJ (3600 s/h times 1000 W), the **electricity** costs ($\$$0.06 /kWh)(1000/3.6 kWh/GJ) = $\$$31/GJ.

Second, the enthalpy of combustion of **gasoline** is 34 MJ/L [note] Fossil Fuels for Transport, /article/fossil-fuels-for-transport/[/note], so the unit cost in terms of energy is ($\$$1.50/L)(1000/34 L/GJ) = $\$$44/GJ.

To summarize the incremental cost of a gigajoule of energy:

**Natural gas**: $\$$10/GJ

**Electricity**: $\$31$/GJ

**Gasoline**: $\$44$/GJ

Ten years ago, when we did this analysis, the numbers were much closer together. In real terms, natural gas is much cheaper, while electricity and gasoline are somewhat more expensive.

**Converting Heat Units**

**Converting Heat Units**

**BTU/h**

Power is the rate of doing work or the rate at which energy is converted. Watts, the SI units for power, are defined to be joules of energy per second. Most of us are familiar with Watts, whether we have had to buy a light bulb, a power supply for our computers or an amp for our guitars. On the other hand, when buying barbeques, air conditioners or furnaces, the power is given instead in BTU/h (usually written as just BTU) where one BTU, or British Thermal Unit, is equal to 1055 joules [note]Wikipedia. *British Thermal Unit. *(online). http://en.wikipedia.org/wiki/British_thermal_unit [2019-09-30]. [/note]. If we buy a large BBQ with a 36,000 BTU/h input, how many watts is that equivalent to?

$\dfrac{36,000 \textnormal{ BTU}}{\textnormal{ h}} \times \dfrac{1055 \textnormal{ J}}{\textnormal{ BTU}} \times \dfrac{1 \textnormal{ h}}{3600 \textnormal{ s}} = 10,550 \textnormal{ W} \tag{1}$

**Thermal conductivity, U-values & R-values**

Thermal conductivity, *k*, is the ability of a material to conduct heat and is measured in W/mK. The power loss through the material can be found by

$P = \dfrac{kA \Delta T}{x} \tag{2}$

where *k* is the thermal conductivity of a material, *A* is the area of the material, Δ*T* is the temperature difference, and *x* is the thickness of the material.

The rate of heat transfer through a building can be calculated by dividing the thermal conductivity by the thickness of the material. This is called the U-value, or the overall heat transfer coefficient, and has units W/m^{2}K.

The inverse of the U-value, the R-value, describes the thermal resistance and is used frequently in construction. The R-value can be calculated by dividing the thickness of the material by its thermal conductivity, so by increasing the thickness of the insulated layer, the thermal resistance is increased [note]Wikipedia. *R-value (insulation)* (online). http://en.wikipedia.org/wiki/R-value_(insulation) [2019-09-30].[/note]. The SI units of the R-value are m^{2}k/W, but in the US it is given in ft^{2}•^{o}F•h/BTU (though usually the values are given without units, i.e. R-10). Let’s take a look at how to convert ft^{2}•^{o}F•h/BTU to m^{2}k/W.

We know:

1 ft = 0.3048 m

1 ^{o}F = 5/9 K

1BTU = 1055 J

So,

$(\dfrac{1 \textnormal{ft}^{2o}\textnormal{F}}{\textnormal{BTU}})(\dfrac{5 \textnormal{ K}}{9 \textnormal{ F}})(\dfrac{0.3048 \textnormal{ m}}{1 \textnormal{ ft}})^2(\dfrac{1 \textnormal{ BTU}}{1055 \textnormal{ J}})(\dfrac{60 \textnormal{ min}}{1 \textnormal{ h}})(\dfrac{60 \textnormal{ s}}{1 \textnormal{ min}}) = 0.176 \dfrac{\textnormal{ m}^2 \textnormal{K}}{\textnormal{W}} \tag{3}$

**Quads**

Quads, defined to be 10^{15} BTUs or 1.055 x 10^{18} J [note]Wikipedia. *Quad (energ**y**)* (online). http://en.wikipedia.org/wiki/Quad_(energy) [2019-09-30] [/note], are used when discussing large annual energy consumptions. The USA is responsible for 25% of the world’s energy consumption, or about 100 quads [note] Wikipedia. *World Energy Resources and Consumption* (online). https://www.eia.gov/energyexplained/us-energy-facts/[2019-09-30]. [/note].

**Oil Barrel**

Oil has not actually been shipped in barrels since the mid-1800s, but it is still used as a unit for measuring and pricing oil. One barrel of oil is equal to ~159 L [note]Wikipedia.*Barrel (unit)* (online). http://en.wikipedia.org/wiki/Barrel_(unit) [2019-09-30]. [/note].

**Calories per Day**

Also, the power obtained through food can also be converted into watts. In one day, the average person consumes 2200 food calories. One food calorie (1 Cal) is equivalent to 1kcal which is equivalent to 4200 J. How many watts is 2200 food calories?

$ 2200 \textnormal{ Cal} \times \dfrac{1 \textnormal{ kcal}}{1 \textnormal{ Cal}} \times \dfrac{4200 \textnormal{ J}}{1 \textnormal{ kcal}} \times \dfrac{ 1 \textnormal{ day}}{86400 \textnormal{ s}} = 107 \textnormal{ W} \tag{4}$

**Updated by CEW 2019-09-30**