Environmental Engineering Reference
In-Depth Information
establish double layer capacitance contributions. However, the spreading resistance
is large through the carbon 'mush' in order to reach down to these fine structures,
sometimes resembling dendrites. At medium frequencies (mHz to low Hz values),
the ions are able to only migrate into meso pores. Here the double layer capacitance
is lower but so is the spreading resistance. The time constants for this band of ion
mobility are shorter, so the discharge/charge time response is much improved.
Finally, at the stage of macro pores the capacitive effects are lower still because the
effective area is not as large but the spreading resistance is lowest since this part of
the carbon electrode is closest to the current collector. For this portion of the total
capacitor, the frequency response is greatest and it represents the high frequency
behaviour of the ultra-capacitor. In addition to the resistance encountered within
the carbon mush, there is also resistance due to kinetics within the electrolyte since
ion mobility is impeded by their passage into meso and micro pores. Various
models of ultra-capacitor behaviour are described in the remainder of this section.
Ultra-capacitors are generally modelled as multi-time constant networks [38].
This model will be referred to as the Toronto model. A realistic, third order system of
vastly different time constants in the Toronto model was shown to very accurately
reflect the performance of ultra-capacitors for time intervals of 30 min or less, which
is sufficient for most ac drive systems used in transportation applications. This is
valid since the majority of vehicle use is for commutes of shorter duration. In fact,
according to the US National Personal Transportation Survey [39] 74% of trips are
30 mi or less. Total trip lengths of 11-20 mi are 60% of the total miles travelled.
L
R
s
R
d
R
L
R
leak
C
so
C
d
C
L
C
s1
(
U
1
)
Figure 10.50 Three time constant model of an ultra-capacitor
The three time constants in the Toronto model (Figure 10.50) are short term,
t
s
,
delayed term,
t
d
, and long term,
t
L
. The voltage dependent capacitor in the short
term branch brings in the non-linear capacitance due to surface effects at the
interfaces. A leakage term is included to model the internal bleed off charge.
Recalling the definition of porosity, the following relations are defined for the
model parameters:
R
s
<
R
d
<
R
L
t
s
< t
d
< t
L
micro
<
meso
<
macro pore
ð
10
:
68
Þ
