Environmental Engineering Reference
In-Depth Information
have around a 90% capacity factor, which is a large improvement from a 66% capacity factor in 1990.
Nuclear power has had a large amount of funding for R&D in the United States and continues to receive
substantial federal funding. Again, go to the Energy Information Agency for more information.
2.8 MATHEMATICS OF EXPONENTIAL GROWTH
Values of future consumption,
r
, can be calculated from the present rate,
r
0
, and the fractional
growth per time period,
k
:
(2.8)
r
0
r
kt
where
e
is the base of the natural log and
t
is the time.
EXAMPLE 2.5
Present consumption is 100 units/year and growth rate is 7% per year.
r
0
100 units/year,
k
0.07/year
Suppose
t
100 years.
r
100
e
0.07*100
100
e
7
100 * 1,097 1 * 10
5
per year
The consumption per year after 100 years is 1,000 times larger than the present rate of
consumption.
Note
: Exponents never have any units associated with them.
2.8.1 D
OUBLING
T
IME
The doubling time,
T
2 in years, for any growth rate can be calculated from Equation 2.8:
rr
kT
2
r
2
r
0
,
2
or 2
e
kT
2
0
0
Take the natural log ln of both sides of the equation:
ln 2
k
*
T
2,
T
2 0.69/
k
If right-side values are multiplied by 100,
T
2 69/
R
, which is Equation 2.7, where
R
is the percent-
age growth rate per year.
2.8.2 R
ESOURCE
C
ONSUMPTION
The total sum of the resource consumed from any initial time to any time,
T
, can be estimated by
summing up the consumption per year. This can be done by using a spreadsheet on personal com-
puters or calculated. If
r
is known as a function of time, then the total consumption can be found by
integration. For exponential growth, the total consumption is given by
T
¯
¯
C
rdt
r e
kt
dt
0
0
r
k
e
kT
C
0
(
1
)
(2.9)
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