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

- Things become apparent when we use appropriate units
- Heat values are often given in different units which are easily converted from one to another.

A look at how to compare the costs of natural gas, gasoline and electricity and how to convert between various heat units including R-values, U-values, BTU/h, W and more.

**Energy Cost Comparison**

**Energy Cost Comparison**

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 June 2010, in $CDN, which today is almost the same as the $US):

**Natural gas**: $15/GJ

**Electricity**: $0.06/kWh

**Gasoline**: $1.15/L

Let's 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) = $17/GJ (remarkably close the natural gas cost).

Second, the energy content of **gasoline** is 32 MJ/L1, so the unit cost in terms of energy is

($1.15/L)(1000/32 L/GJ) = $36/GJ, or about twice the cost of electricity and natural gas. The difference is mostly due to the tax structure.

Now let us consider the environmental consequences, and express the cost in $ per kg CO_{2} and kg CO_{2} per GJ.

**Natural Gas**

This is mostly methane, CH_{4}, which has an energy content of 55 MJ/kg1. Methane has a molecular mass of 16 and that of CO_{2} is 44, so 16 kg of CH_{4} burns to give 44 kg of CO_{2}2. In other words, one tonne of CH_{4} burns to give 2.75 tonnes of CO_{2}. After a little more algebra, we see that $300-worth of natural gas produces one tonne of CO_{2}, and that each GJ of natural gas consumed produces 0.05 tonnes of CO_{2}.

**Electricity**

If your electricity comes from mature hydro-electric dams as does most of the electrical power in BC, then the greenhouse gas production per GJ of energy is tiny compared to other sources. If, however, it comes from coal-fired stations, as does some of the power in BC, then the CO_{2} production is large and easy to estimate. It takes 2.6 Mtonnes of coal to be burnt each year in to produce 1 GW_{e}^{*}, and this produces 8.9 Mtonnes of CO_{2}3. With a mean heat of combustion of 30 MJ/kg1, and a thermal efficiency of 0.4 (i.e. can get 12 MJ electrical from each kg of coal) we can calculate that burning coal produces 0.28 tonnes of CO_{2} per GJe. Note: we are ignoring transmission losses here, which may be substantial.

**Gasoline**

By the same process, we can calculate that burning gasoline produces 0.07 tonnes of CO_{2} per GJ (thermal, multiply by about 3 to obtain the mass per GJ mechanical, as automobile engines are about 25-40% efficient) and costs about $500 per tonne of CO_{2}. Put another way, you make a tonne of CO_{2} for every $500-worth of gasoline you buy.

Parameter |
Natural Gas |
Electricity |
Gasoline |

Cost (Vancouver 2010) |
C$15/GJ | C$0.06/kWh | C$1.15/L |

Heat of combustion (MJ/kg) |
55 | 12 (electrical, from coal) | 46 |

Cost per GJ |
C$15 | C$17 | C$36 |

CO_{2} production (tonnes/GJ) |
0.05 | 0.28 (if from coal) | 0.07 |

Cost of producing a tonne of CO_{2} |
C$300 | C$60 (if from coal) | C$500 |

* When electricity is produced in thermal power plants, we always have to be careful about the distinction between energy and power of the burnt fuel and energy and power in the electricity produced. Where there can be confusion, we use the subscript "th" for thermal power or energy, and "e" for electrical power or energy.

**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 (written usually as just BTU) where one BTU, or British Thermal Unit, is equal to 1055 joules 4. If we buy a large BBQ with a 36,000 BTU/h input, how many watts is that equivalent to?

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**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 material can be found by

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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 5. 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,

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**Quads**

Quads, defined to be 10^{15} BTUs or 1.055 x 10^{18} J 6, are used when discussing large annual energy consumptions. For example, in 2008, the total worldwide energy consumption was

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The USA is responsible for 25% of the world’s energy consumption, or about 115 quads 7.

**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 8.

**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?

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- 1. a. b. c. Useful numbers http://c21.phas.ubc.ca/article/useful-numbers [2]
- 2. Greenhouse gas calculations http://c21.phas.ubc.ca/article/energy-and-greenhouse-gas-ghg-calculations [3]
- 3. Clean coal /article/clean-coal
- 4. Wikipedia.
*British Thermal Unit.*(online). http://en.wikipedia.org/wiki/British_thermal_unit [4] [8 July 2010]. - 5. Wikipedia.
*R-value (insulation)*(online). http://en.wikipedia.org/wiki/R-value_(insulation) [5] [8 July 2010]. - 6. Wikipedia.
*Quad (energ**y**)*(online). http://en.wikipedia.org/wiki/Quad_(energy) [6] [7 July 2010]. - 7. Wikipedia.
*World Energy Resources and Consumption*(online). http://en.wikipedia.org/wiki/World_energy_resources_and_consumption [7] [7 July 2010]. - 8. Wikipedia.
*Barrel (unit)*(online). http://en.wikipedia.org/wiki/Barrel_(unit) [8] [9 August 2010].

© Physics and Astronomy Outreach Program at the University of British Columbia (Chris Waltham & Brittany Tymos 2010-07-13)