Both the energy and power density are important characteristics of capacitors, and are considered in this section. The change in energy associated with a change in capacitor voltage is
(11.36)
The total energy stored in the capacitor can be obtained by integration
(11.37)
where we have assumed that C is constant. Recall that capacitance has units of farads, which are equivalent to J V−2 or C V−1. Using Equation 11.37, we can estimate the amount of energy stored in different types of capacitors, as demonstrated in the following illustration.
ILLUSTRATION 11.8
Calculate the energy stored in the three capacitors described in Illustration 11.2. Take the separator area as 10 cm2, and assume that the voltage for each is as follows:
Electrostatic, 20 V
Electrolytic, 100 V
EDLC, 3 V
For a 1 F capacitor, a surface area 5.65 × 104 m2 was required. This corresponds to 1.77 × 10−5 F·m−2.
Note that the capacitance per unit area used for the EDLC calculation is half the value from Illustration 11.2 in order to account for the two capacitors in series that are inherent in the EDLC. The voltage is the total voltage across the EDLC.
The power is energy per unit time, and is perhaps most easily expressed as
(11.38)
The power will be a maximum for the initial discharge of a capacitor. The potential used in Equation 11.38 is the terminal potential, which is lower because of the resistive losses in the ESR. For an EDLC capacitor, one can show (Problem 11.19) that the maximum usable power that can be obtained is
(11.39)
where ESReff is the total effective series resistance defined by Equation 11.35, V is the terminal voltage across the EDLC, and Ohms law has been used to relate the current to the voltage. Equations 11.37 and 11.39 can be used to illustrate the trade-off between energy and power. We will explore this through Illustration 11.9.
ILLUSTRATION 11.9
For the capacitor from Illustration 11.7, calculate the maximum specific energy and specific power from this device assuming that the maximum potential is 1 V. All of the parameters are the same as those in Illustration 11.7. In addition,
To determine the specific energy and power, we need the mass of the device. We will include the mass of all the EDLC components including the separator, two carbon electrodes, and the electrolyte. The mass of the casing is not included in this calculation.
which when converted to W·h is 1.68 Wh·kg−1.
The maximum specific power is
where ESReff was determined as part of Illustration 11.7.
The same procedure shown in Illustration 11.9 is used to generate Figure 11.16, which shows the specific energy and the maximum specific power for several different electrode thicknesses ranging from 1 μm to 1 mm. We note from Figure 11.16 that for thick electrodes, the rate capability is much lower and the specific energy approaches a constant value. The constant value of the specific energy is reached as the electrodes become sufficiently thick that the mass of the other cell components is no longer significant; we are therefore left with the specific energy of the electrodes themselves, which does not change with electrode thickness, even though the absolute energy stored in the electrodes does increase with thickness. Similarly, for thin electrodes, the maximum power approaches a constant as the resistance of the electrodes becomes less important as they get thinner, and the maximum power is limited only by ohmic losses in the separator and contact resistance. The specific energy, however, continues to drop as the electrodes are made thinner because the energy stored is directly proportional to the electrode thickness.
Figure 11.16 Comparison of maximum specific power and specific energy. The parameters are from Illustration 11.6 and the thickness of electrode varied.
Finally, we note that heat is generated during both charging and discharging of an EDLC. As long as the device is not limited by the supply of ions in the electrolyte (see below), the power lost as heat can be estimated as
(11.40)
The round-trip efficiency of an EDLC is defined as the ratio of the energy delivered by a capacitor to the energy that was supplied to it during a specific discharge/charge cycle, provided that the beginning and ending SOC are the same. For a constant current charge and discharge below the cutoff frequency,
(11.41)
Equation 11.41 assumes that the capacitor can be discharged only to half of Vmax. Using the parameters in Illustration 11.7, the efficiency (right ordinate) and average power (left ordinate) are shown in Figure 11.17. Low currents correspond to long discharge times. The losses are resistive, and low currents therefore translate to high efficiency. For discharge times on the order of the RΩC or less, the efficiency falls off sharply because of the high resistive losses. In extreme cases, it is possible to damage the device by excessive heat generation.
Figure 11.17 Effect of rate of charge and discharge on the efficiency of a capacitor modeled with a simplified series-RC circuit.
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