Efficiency is the useful output of energy. We can mathematically define energy efficiency as the ratio of useful energy output to total energy output. This will give us a value between 0-1. To get this as a percentage, you should multiply by 100%. To calculate efficiency, the following formula can be used:
This simple yet powerful equation allows us to express efficiency as a percentage, where 100% would represent a theoretically perfect device with no energy losses—a condition that, as we will explore, is impossible to achieve in practice due to the fundamental laws of thermodynamics.
NOTE: You can never have energy efficiency over 100%. If you get a value above 1 by using the formula above, make sure you have the USEFUL energy value on the top and TOTAL energy output on the bottom of the equation.
Example 1
An electric motor consumes 100 watts (a joule per second (J/s)) of power to obtain 90 watts of mechanical power. Determine its efficiency.
Solution:
Input to the electric motor is in the form of electrical energy, and the output is mechanical energy.
Using the efficiency equation:
Or efficiency is 90%.
Caution!
This is a simple example because both variables are measured in Watts. If the two variables were measured differently, you would need to convert them to equivalent forms before performing the calculation.
Test Yourself #1
An electric motor consumes 91 watts (a joule per second (J/s)) of power to obtain 83 watts of mechanical power. Determine its efficiency.
- Click to see the solution to Test Yourself #1
Solution:
Useful Energy = 83 Watts
Total Energy = 91 Watts
Efficiency = useful / total
= 83/91
= 0.912
= 91.2%
What if the units are not the same? You need to have the same units in order to determine efficiency.
Example 2
The United States' power plants consumed 39.5 quadrillion Btus of energy and produced 3.675 trillion kWh of electricity. What is the average efficiency of the power plants in the U.S.?
Solution:
Total Energy input = 39.5 x 1015 Btus and the Useful energy output is 3.675 x 1012 kWh. Recall that both units have to be the same. So we need to convert kWh into Btus. Given that 1 kWh = 3412 Btus:
Step 1
Given:
Therefore:
Step 2
Use the formula for efficiency.
Test Yourself #2
The United States power plants consumed 35 quadrillion Btus of energy and produced 3 trillion kWh of electricity. What is the average efficiency of the power plants in the U.S.?
- Click to see the solution for Test Yourself #2
Solution:
The units don’t match, so you need to convert to the same units and we know that 1 kWh = 3,412 Btus.
Useful Energy Output = 3 trillion kWh
= 3 x 1012 kWh x 3,412 Btus / 1 kWh
= 10,236 x 1012 BtusTotal Energy Input = 35 quadrillion Btus
Efficiency = Useful Energy Output / Total Energy Input
= (10,236 x 1012) / (35 x 1015)
= 0.2925
= 29.25%
Energy Efficiencies
Energy efficiencies are not 100%, and sometimes they are pretty low. The table below shows typical efficiencies of some of the devices that are used in day to day life:
| Device | Efficiency |
|---|---|
| Electric Motor | 90 % |
| Home Gas Furnace | 95 % |
| Home Oil Furnace | 80 % |
| Home Coal Stove | 75 % |
| Steam Boiler in a Power Plant | 90 % |
| Overall Power Plant | 36 % |
| Automobile Engine (ICE) | 25 % |
| Electric Bulb: Incandescent | less than 10 % |
| Electric Bulb: Fluorescent | 60 % |
| Electric Bulb: LED | 90 % |
From our discussion on national and global energy usage patterns in Lesson 3, we have seen that:
- about 41% of the US energy is used in power generation;
- about 38% of the US energy is used for transportation.
Yet the energy efficiency of a power plant is about 35%, and the efficiency of automobiles is about 25%. Thus, over 62% of the total primary energy in the U.S. is used in relatively inefficient conversion processes.
Why are power plant and automobile design engineers allowing this? Can they do better?
There are some natural limitations when converting energy from heat to work.