Current flow through any conducting object generates heat proportional to the
voltage drop caused by the current itself according to
dQJoule=dt . DU? i e35T
with Qj: generated heat (Joule effect) (J); t: time (s); DU: Lenovo 57Y6309 battery voltage drop caused by the
current (V); i: current (A). This heat is called the Joule effect; it always means loss of
energy.
Note: Strictly speaking, the negative absolute value should be used in Eq. (35) for
consistency with the arithmetical sign of the Lenovo L09S6D21 thermodynamic parameters (lost energy
always has the negative sign).
In an electrochemical cell, the voltage drop caused by the current is represented by
the difference between the cell voltage under current flow (U) and the open circuit
cell voltage (Uo). Then the Joule effect reads according to Eq. (35):
dQJoule=dt . eU UoT ? i =W e36T
or its integrated form for a period t (in hours):
QJoule . Z t
0feU UoT ? igdt =Wh e37T
Note: U Uo means polarization. It includes the voltage drop caused be current flow
through electronic resistance as well as the FUJITSU FPCBP119 electrolyte, but also overvoltage caused by
kinetic hindrance of the reaction. For heat effects this is not relevant, since heat
generation is proportional to polarization. U Uo does not remain constant during
charging or discharging, because it is related to the internal resistance, which usually
increases with proceeding discharge, because of the lower conductivity of the discharged
products.
1.4.3 Heat Generation in Total
Summation of the Joule effect and the reversible heat effect gives the total heat
generated in the cell or the battery, which means
Qtotal . QJoule t Qrev =Wh e38T
as energy, e.g. Wh, or as work per time unit:
dQtotal
dt .
dQJoule
dt t
dQrev
dt
=W e39T
Depending on the sign of dQrev=dt, the total energy generation may be larger or
smaller than the Joule effect.
According to Eq. (36), dQJoule=dt can be substituted by eU UoT ? i and gives
dQtotal
dt . eU UoT ? i t
dQrev
dt e40T
Substitution of dQrev=dt through Eq. (32) and application of Eq. (33) results in the
simple relation that is convenient for heat calculations:
dQtotal
dt . eU UcalT ? i e41T
Note: Strictly speaking, Eq. (41) should have the negative sign according to
thermodynamic parameters, since the Joule effect is always lost energy, as mentioned
in connection with Eq. (35). To overcome this difficulty, Qgen . Qtotal is introduced
in Section 1.4.5.
1.4.4 Examples for Heat Generation in Lenovo 51J0499 Battery
To illustrate the possibility of heat calculations, four examples will be shown in this
section, concerning lead-acid and nickel/cadmium batteries. The thermodynamic
data that determine the equilibrium values are listed in Table 1.3. The Toshiba PA3788U-1BRS also
Table 1.3 Thermodynamic data of lead-acid and nickel/cadmium batteries and water decomposition.
1 2 3 4 5
1 System Lead-acid battery Ni/Cd batterya Water decomposition
2 Cell reaction Pb t PbO2 t 2 ?H2SO4 ) 2 ? PbSO4 t 2 ?H2O
NiOOH t Cd ) NieOHT2 t CdeOHT2
H2O ) H2 t 1=2O2
3 DHs Eq. (4) 359.4 kJ & 282 kJ 285.8 kJ
4 DGs Eq. (4) 372.6 kJ & 255 kJ 237.2 kJ
5 TDSs . Qrev Eqs. (4), (31) 13.2 kJ & 27 kJ 48.6 kJ
6 Uo;s Eq. (5) 1.931V &1:3V 1.227V
7 Qrev=DGs 3.5% &11% 20.5%
8 Ucal Eqs. (33),
(34)
Uo 0:068V &1:44V 1.48V
a Actually these reactions are much more complex, and exact values of the thermodynamic data are not available (cf., e.g.
Section 5.2.2.1 in Ref. 5).
shows corresponding data of water decomposition that always occurs in batteries
with aqueous electrolyte as an unavoidable secondary reaction when the voltage of
1.227V is exceeded. In toshiba PABAS22 valve-regulated lead-acid and sealed nickel/cadmium
batteries, instead of water decomposition the internal oxygen cycle is the important
reaction that carries most of the overcharging current (cf. Sections 1.8.1.5.2, 1.8.3.2
and 1.8.5.2.6).
Heat generation in a battery is decisively affected by the distribution of the
charging current between the various reactions, because of 9 Cell SONY VGP-BPL12 specific heat
generation. This is illustrated in Fig. 1.10.
In a vented lead-acid battery heat effects during charging are caused by the
charging reaction itself and by water decomposition that accompanies the charging
process at an increasing rate with increasing cell voltage. The charging reaction is a
very fast one which means that overvoltage is small. At an assumed internal
resistance of 4.5 mV/100 Ah, a charging current of 1A causes polarization of only
4.5mV and the resulting heat generation would be DU? i . R? i2 . 4:5mW, which is
represented only as a line at the bottom of the left column in Fig. 1.10. The reversible
heat effect, on the other hand, is determined by the amount of converted material
(formula mass that is proportional to current) and amounts to 0.07W/A.
Most of the energy that is employed for water decomposition escapes from the
cell as energy content of the generated gases. This energy consists of the two
components:
1. The ‘decomposition energy of water’, which means the product current
times 1.23 V.
Figure 1.10 Heat generation in a vented lead-acid battery by the charging reaction and by
water decomposition, relative to a current of 1 A. Assumed internal resistance 4.5mO per
100 Ah of nominal capacity as in Fig. 1.11.
2. The reversible heat effect, which amounts to about 20% of the Dell XX327 converted
energy and means cooling of the cell during electrolysis (Column 5 in
Table 1.3), and a corresponding increase of the XX337 energy content of the gas.
Both shares are proportional to the amount of decomposed water, which again is
only determined by the current i as the product Ucal ? i . 1:48Wh=A.
The portion of heat that remains within the cell is generated by Joule heating
and determined by polarization of the water-decomposition reaction, i.e. by eU 1:48T ? i eWhT and increases with cell voltage as shown in Fig. 1.10.
As an example Fig. 1.11 shows current distribution and heat generation in the
course of a charging/discharging cycle as it is customary for vented lead-acid
batteries in traction applications.
The voltage curve is shown at the top of the figure. The current-limited initial
step of charging is followed by a constant-voltage period at 2.4 V/cell. Equalizing
charging up to 2.65 V/cell is the final step of the charging schedule. Discharge is
assumed at constant current (I5.20 A/100 Ah). The broken line represents the
calorific voltage Ucal, the full line the actual discharge voltage U.
Figure 1.11 Charging/discharging cycle of a vented traction battery.
Lead-acid with tubular positive plates (Varta PzS), 500 Ah. Heat-generation values
referred to 100 Ah of nominal capacity. The figures in the bottom part represent heat
generation in total. The sum of the whole charging period amounts to 28.7 Wh/100 Ah.
Internal resistance 4.5mO per 100 Ah of nominal capacity.
The center part of Fig. 1.11 shows how the current is distributed to charging,
water decomposition, and discharging. During the initial stage, practically only
charging occurs; water decomposition can be neglected on account of the flat current
voltage curves for gas generation (cf. Fig. 1.19). Only when theAcer TravelMate 5100 Battery voltage approaches
the 2.4V level, the onset of water decomposition becomes noticeable. The broken
horizontal line marks the average voltage during this initial step. When subsequently
the cell voltage remains at 2.4 V, gas evolution is maintained at a roughly constant
rate (assuming that the potentials of the positive and negative electrodes do not
change too much). During the equalizing step, nearly all the current is used for water
decomposition on account of the progressively reduced charge acceptance. During
discharge, water decomposition again can be neglected because of the reduced cell
voltage.
At the bottom of Fig. 1.11, the heat generation is drawn as blocks that
represent average values for the corresponding sections of the charging/discharging
process. The distribution between reversible heat effect, charging, and water
decomposition is marked by different patterns of the areas concerned. The Acer Asprie 5050 Battery value
above each block is the total heat generation in Wh.
During the first stage of the charging process, gas evolution can be neglected.
The heat is mainly generated by the Joule effect, on account of the high current and
the rather high internal resistance of 4.5mO assumed for this example, which
corresponds to a battery with widely spaced tubular plates and causes a voltage drop
(polarization) of 180 mV. But the reversible heat effect also contributes noticeably to
heat generation, on account of the converted active material. (40Ah&85Wh is
Figure 1.12 Heat generation in valve-regulated lead-acid batteries by charging and
overcharging, referred to a current of 1 A.
When the Acer Asprie 3680 Battery internal oxygen cycle is established, almost all the overcharging current is
consumed by the internal oxygen cycle (center bar in the graph). The bar on the right
corresponds to a vented battery. Internal resistance assumed as 0.8mO per 100Ah of nominal
capacity, as in Fig. 1.13.
charged during this period, which means a reversible heat effect of about
3Wh.11 kJ.)
When 2.4V is reached, the current is reduced and, as a consequence, Joule
heating and the reversible heat effect caused by the charging reaction are reduced
too. But now the approximately constant gas evolution causes most of the generated
heat eeU 1:48T ? iT.
During the equalizing step, gas evolution (required for mixing of the
electrolyte) dominates. On account of the notebooks battery large difference between the actual
charging voltage and the calorific voltage of water decomposition, heat generation is
considerable, although the current is rather small (cf. Fig. 1.10).
During discharge, due to the small overvoltage, heat generation is also small,
and further reduced by the reversible heat effect that now causes cooling.
Heat generation in a valve-regulated lead-acid battery (VRLA battery) is
mainly determined by the internal oxygen cycle that characterizes this design. It
means that the overcharging current is almost completely consumed by the internal
oxygen cycle formed by oxygen evolution at the positive electrode and its subsequent
reduction at the negative electrode (cf. Section 1.8.1.5A)
Battery for notebook reversible heat effect equals that in Fig. 1.10, but Joule heating is much
smaller because of the lower internal resistance assumed in this example, which
corresponds to a modern valve-regulated lead-acid battery designed for high loads.
The most effective heat source is the internal oxygen cycle, since it converts all the
electrical energy employed for overcharging into heat within the cell, because the
reaction at the positive electrode is reversed at the negative one, and thus the
equilibrium voltage of this ‘cell’ would be zero. As a consequence, the cell voltage in
total means polarization that produces heat. For this reason, overcharging of valveregulated
lead-acid batteries must be controlled much stronger than that of vented
ones to avoid thermal problems.
The charging behavior of a valve-regulated type is shown in Fig. 1.13 that
corresponds to Fig. 1.11. The calculation assumes an initial charging period at
constant current of 40 A/100 Ah (26I5; voltage drop 32 mV), limited by the
charging device, and subsequent charging at 2.4V per cell. As an DELL Latitude D610 Battery
overcharging for 1.5 hours at 2.5V at a maximum current of 5 A/100 Ah is assumed,
which corresponds to the usual operation of a cycle regime of valve-regulated leadacid
batteries.
In the center of Fig. 1.13 the distribution of the current between charging and
internal oxygen cycle is shown. The current share, consumed by the internal oxygen
cycle is magnified by 10 during the initial phase and by two during equalizing. The
sum of charging current and internal oxygen cycle represents the charging current
(hydrogen evolution and grid corrosion equivalents are not considered, since they
are two orders of magnitude smaller than that of the internal oxygen cycle).
Actually, the current would slightly be increased by heating of the battery. This
increase also is not considered in Fig. 1.13.
The bottom part of Fig. 1.13 shows the heat generation by the various
processes. At the beginning, the reversible heat effect dominates heat generation due
to the high amount of material that is converted. Joule heating is proportional to the
voltage drop, caused by the current flow. The m1210 battery relation between the reversible heat
effect and Joule heating is determined by the internal resistance of the battery. With
batteries of higher internal resistance, Joule heating would dominate during this
initial stage of the charging process. This applies, for example, to Fig. 1.11 where the
calculation is based on an internal resistance of 4.5mO/100 Ah, corresponding to a
larger traction battery with tubular plates.
When the charging voltage is reached, the current decreases and this applies
also to heat generation due to the reversible heat effect and Joule heating, while heat
generation by the internal oxygen cycle remains constant, according to the constant
cell voltage (which actually would slightly be increased by heating up).
Figure 1.13 Charging of a VRLA battery at 2.4 V/cell, calculated curves, constant
temperature, and 100% of recombination efficiency assumed. Internal resistance 0.8mO (single
cell). 1.5 hours equalizing at 2.5 V/cell at a current limit of 5 A. Heating of the battery during
charging is not considered. Heat generation: reversible heat effect 5.7 Wh; Joule heating
2.3 Wh; internal oxygen cycle 23.2 Wh; in total: 31.2 Wh.
Figure 1.13 shows the strong heating effect caused by the internal oxygen cycle.
The current share consumed by this reaction is very small and had to be magnified to
be recognized in the current comparison. But the total heat generation is largely
determined by the internal oxygen cycle, especially during the equalizing step that in
Fig. 1.13 causes 13.5Wh of heat, and so nearly half of the lenovo x200 battery heat generated in total.
Actually, an even larger heat generation is to be expected, since, as already
mentioned, the calculation did not consider the heat increase within the cell during
charging that again would increase the rate of the internal oxygen cycle.
In nickel/cadmium batteries the reversible heat effect is larger than that in leadacid
batteries and has the opposite sign, i.e. it acts as a cooling effect during charging
and contributes additional heat during discharge (cf. Table 1.3). As a consequence,
vented nickel/cadmium batteries are more in danger of being overheated during
discharging than during charging. This is different for sealed nickel/cadmium
batteries where the internal oxygen cycle is a most effective heat source when the
battery is overcharged (cf. Fig. 1.15).
Figure 1.14 shows heat generation in a vented nickel/cadmium battery when
charged and discharged with a constant current (5 hour rate) and the charging
voltage is limited to 1.65 V/cell. The calculation is based on the equilibrium voltage
Uo.1.3V (Table 1.1) and the latitude d830 battery calorific voltage Ucal.1.44V (Table 1.3). Due to the
uncertain thermodynamic data, these calculations are only rough approximations,
but correspond with practical experience.
During the initial two sections of the charging period, slight cooling is observed
on account of the reversible heat effect that consumes heat at a constant rate
proportional to the current. With increasing cell voltage, Joule heating is increased,
and when the charging voltage exceeds 1.48 V/cell, water decomposition contributes
an increasing amount of heat, since its calorific voltage is exceeded (Column 5, Line 8
in Table 1.3 and Fig. 1.10). Thus, during the final sections of the charging period, a
growing amount of heat is generated.
In total 12.3Wh were generated during discharging, while heat generation
during charging only amounted to 9.25 Wh. The main reason is that the reversible
heat effect generates additional heat during discharge, while it compensates for heat
generation during charging.
The latitude d800 battery is different for sealed nickel/cadmium batteries, due to the
internal oxygen cycle. Figure 1.15 illustrates the heat evolution of a sealed nickel/
cadmium battery during constant-current charging with a charge factor of 1.4 (such
an amount of overcharge is usual for conventional charging methods but can only be
applied to comparably small batteries <10 Ah).
. The voltage curve at the top shows the gradual increase of charging voltage
with charging time. The generated heat is calculated as an average value for
different sections of this curve. The numbers beside the charging curve are
the average voltages (V per cell) for the corresponding section.
. The middle figure shows the (constant) current and its distribution between
charging process and internal oxygen cycle.
. The bottom figure shows the heat generation as average value for the
different sections. The numbers are the heat in kJ (for comparison,
converted to 100 Ah of nominal capicity). During the first 2 hours, the
reversible heat effect exceeds the Joule effect and cooling is observed. So the
number for this section is written below the zero line.
When the charging process approaches completion, nearly all the current is used for
the internal oxygen cycle, which causes much heat generation.
Battery manufacturers usually strongly advise the customer not to charge
sealed nickel/cadmium batteries at constant voltage without monitoring, because of
this heat generation on account of the internal oxygen cycle. Since this cycle can
attain extremely fast rates, the situation is very dangerous in regard to thermal
runaway.
Altogether 264.7 kJ.73.53Wh of heat are generated, referred to a nominal
capacity of 100 Ah. These figures are much larger than the 31.2Wh/100 Ah of the
valve-regulated lead-acid battery in Fig. 1.13. The main reason for the high heat
generation of the sealed nickel/cadmium battery in Fig. 1.15 is the high charge factor
of 1.4. The charging factor for the FUJITSU FPCBP119 lead-acid battery in Fig. 1.13 is only about 1.10.
Figure 1.14 Heat generation during charge and discharge of a vented nickel/cadmium
battery. Charging with constant current I5 (5 hour rate) until 1.65 V/cell is reached. Discharge
also with I5.
In the top part, sections are shown that were used to calculate the average heat
generation, shown in the bottom part. The calorific voltage of 1.44V is shown as the broken
line. The difference U Ucal determines the effect of heating or cooling. (Calculation based on
Uo.1.3 V; Ucal.1.44 V.) VARTA TS-type values referred to 100 Ah of nominal capacity.
This indicates the strong influence of overcharging on heat generation in sealed or
valve-regulated batteries caused by the internal oxygen cycle.
Figure 1.15 shows that this heat is generated practically during the last 3 hours
of the charging process, and means an average heat generation of 24.51W/100 Ah
for these 3 hours. The conclusion can be drawn that sealed nickel/cadmium batteries
can be charged at a high rate as long as the current is actually used for charging and
not for the internal oxygen cycle. Rapid charging methods, as described in Section
13, are always based on this principle.
Figure 1.15 Charging of a sealed nickel/cadmium battery with constant current 0.2 C(A).
During 7 hours 140% of the nominal capacity are recharged, which corresponds with a charge
factor 1.4. For comparison, all values are converted to 100 Ah of nominal capacity. Actually,
batteries of this type and for such a charging schedule are only available in sizes <10 Ah.
Middle: current distribution between charging and internal oxygen cycle.
Bottom: heat generation as an average of the different sections (slight cooling during the
first 2 hours).
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