Growth in the world and the U. S. energy consumption as a function of time, follow what is known as exponential function. The exponential increase is characterized as follows. The amount of change (increase in energy consumption) per unit time is proportional to the quantity (or consumption) at that time.

$$\frac{\Delta N}{\Delta t}\propto N$$

or

$$\frac{\Delta N}{\Delta t}=\lambda N$$

Where Greek letter Δ(delta) is the change or increment of the variable and λ (lambda) is the growth rate. After some mathematical methods, it can be shown that the equation changes to the form

$$N={N}_{0}{e}^{\lambda t}$$

where e is a constant = 2.71

We can determine how long it takes for N0 to become 2N0 (twice its original number or double). That time period is called doubling time. After some mathematical steps it can be written as:

Doubling Time = 70 / % Growth Rate per Year

### Illustration

Use of coal is projected to increase at the rate of 1.7% per year in the U.S. How long will it take to double its usage?

$$Doubling\text{\hspace{0.17em}}Time\text{\hspace{0.17em}}(years)=\frac{70}{1.7}=41.17\text{\hspace{0.17em}}years$$

In 41.17 years, the consumption of coal will be twice as much as it is today.