Environmental Engineering Reference
In-Depth Information
where
t ¼
R
i
C
¼
CESR is the short term time constant of the capacitor. Then its
charge efficiency, under constant current charging, is
1
1
þð
2
t=
T
Þ
h
c
¼
ð
10
:
60
Þ
where
T
is the final time at which the capacitor voltage reaches its rated value for
impressed current,
I
. In this context, the charging efficiency is time dependent and
leads to the view of capacitors having a time rating. For example, a 10 or 100 min
capacitor would be the proper nomenclature to identify the component. What this
rating is useful for is capacitor selection when charge duration is known.
Following the same procedure, the discharge efficiency using the set-up of
Figure 10.40(b) can be determined. The procedure is to account for total energy lost
out of the total stored energy that results in a given total available energy. This is
achieved by taking the expression for power lost as given in (10.59) and converting
it to lost energy. Energy lost is power multiplied by discharge time
T
. Available
energy at the terminals is then
W
avail
¼
0
:
5
CU
co
I
2
R
i
T
ð
10
:
61
Þ
The discharge efficiency is then the ratio of available energy divided by total
stored energy. Making this calculation, one obtains
h
d
¼
1
2
t
T
ð
10
:
62
Þ
The discharge efficiency is the reciprocal of the charge efficiency. During
charging, the energy source delivered a total of the stored energy plus the energy
lost to dissipation in the capacitor internal resistance, or its ESR. Similarly, during
discharge the same fraction of energy is lost to dissipation in the capacitor ESR if
the discharge constant current is the same as the charge constant current.
The ultra-capacitor turnaround, or round trip, efficiency under constant current
conditions is therefore the product of (10.60) and (10.62) shown here as (10.63):
½
1
2
ð
=
T
Þ
½
1
þ
2
ðt=
T
Þ
t
h
ta
¼ h
c
h
d
¼
ð
10
:
63
Þ
This section concludes with specifications of available capacitors obtained
from
the
data
sheets
of
Maxwell
Technologies
Inc.
-
a
leading
capacitor
manufacturer.
In Table 10.9 use has been made of (10.55) through (10.62) to compute the
capacitor attributes and metrics. It is noteworthy that, regardless of an individual
cell (first two rows) or modules (second two rows), the capacitor time constants
remain very closely spaced. This is indicative of the technology employed in
manufacturing, electrode construction and electrolyte materials. Second, the large
difference
between
specific
power
at
95%
efficiency
at
power
at
matched
