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Heat sources
CHAPTER 4
Heat sources Contents 4.1. Fundamental laws 4.2. Heat of reactions 4.2.1 Reversible or entropic heat of reactions Enthalpy of reaction Entropy of reaction Entropy of electron 4.2.2 Joule heating 4.3. General Joule heating concept 4.4. Heat dissipation 4.4.1 Convection 4.4.2 Conduction 4.4.3 Radiation 4.4.4 Exhausted enthalpy 4.4.5 Equivalent circuit model 4.5. Summary 4.6. Problems
114 118 119 119 120 123 124 125 125 125 126 126 126 127 127 128
Usually a battery, just like any other systems from the thermodynamical point of view, is a system with fixed mass. However, in some cases the battery is considered as a control volume because we have mass transport to or from the battery. Metal–air batteries are of control volume type, whereas lead–acid, lithium-ion, nickel-based batteries, and many other technologies are of system type. Whether they are considered as a system or control volume, they exchange heat with ambient and also act as a heat generator or in some rare cases as sinks. To obtain the battery temperature, we have to deal with thermodynamical laws and relations. Any thermodynamical system is subjected to generation, dissipation, and storage of energy. To predict its temperature distribution, the first law of thermodynamics should be applied on it. In a multiphase multicomponent medium like batteries, special care should be paid to obtain accurate results. In such a system, heat generation and dissipations are due to: 1. Heat generation due to electrochemical reactions. 2. Material phase change. 3. Mixing. 4. Change in heat capacity of the system due to the material change. Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00008-8
Copyright © 2020 Elsevier Inc. All rights reserved.
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5. Electrical work. 6. Heat exchange with surroundings. The traditional methods for prediction of thermal behavior of batteries yield to the estimation of a uniform temperature for the whole cell. This assumption is made because in traditional methods a lumped model is used, in which the battery cell is considered as a uniform system with high thermal conductivity resulting in a uniform temperature. If we like to obtain a temperature gradient inside the battery, we need to introduce a model that estimates both generation and dissipation inside the cell. Such models are based on fundamental governing equations in vector form. The fundamental laws are in the form of convective–diffusive transport equations that include the effect of transient parts, heat convection, heat conduction, and heat sources and sinks. A proper system of governing equations with proper boundary conditions gives an appropriate tool for prediction of temperature profile inside any battery cell. In this chapter, we fully discuss the fundamental thermodynamical equations that govern the energy balance of a battery system. We discuss in detail heat sources and sinks that are the main issue in temperature profile.
4.1 Fundamental laws For a multicomponent multiphase medium in which the energy content is changing and different phases are in contact with each other, the first law of thermodynamics can be written as the equation dHtot = q − IV , dt
(4.1)
in which Htot is the total enthalpy of all the phases: Htot =
j
Vj
ci,j H¯ i,j d∀,
(4.2)
i
where i and j describe the ith species and the jth reaction, respectively. In Eq. (4.1) the rate of heat generation and dissipation is shown by q, and IV represents the electrical work. For simplicity, we can assume a mean composition for each phase; hence we have
dHtot d = ni,j H¯ iavg ,j + dt dt j i
Vj
ci,j H¯ i,j − H¯ iavg d∀ . ,j
(4.3)
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115
The first term on the right-hand side of Eq. (4.3) shows the rate of enthalpy change of the system when all the species are in an equilibrium state and the second term accounts for their deviation from equilibrium. The first term can be divided into three parts:
avg
¯ i ,j ∂H
¯ . =C pi,j avg
∂T
(4.4)
p
The first part in Eq. (4.3) can be calculated as d j
i
dt
¯ i ,j ) = (ni,j H avg
j
ni,j C¯ piavg,j
i
dni,j dT ¯ iavg +H . ,j dt dt
(4.5)
As explained earlier, at each electrode a set of electrochemical reactions occurs, all of which can be defined by the general formula
νi,l Mizi = nl e− .
(4.6)
i
This equation is written so that each species i is in a single phase. Therefore diffusion phenomena from one phase to another phase should be separately considered. Using the balance of chemical species, we can calculate the mole of each species by the equation dni,m νi,l¯il dni,j = − . dt nl F j, j=m dt l
(4.7)
The first term on the right-hand side of the equation represents the rate of generation of consumption of species i due to electrochemical reactions. Moreover, the partial current ¯il is the necessary current that each reaction requires. This current is positive for cathodic reactions and negative for anodic reactions. The second term of Eq. (4.7) indicates the change of mole of species i due to phase change. Integration of Eq. (4.7) results in ni,m = n◦i,m −
(ni,j − n◦i,j ) +
j, j=m
νi ,l l
nl F
◦
t
¯il dt.
(4.8)
The partial molar enthalpy is defined by ◦ 2 H¯ iavg ,m = Hi,m − RT
d avg ln(ai,m ). dT
(4.9)
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On the other hand, the theoretical open-circuit voltage for reaction l in an averaged composition with respect to a reference electrode is ◦ + Ui,avg = Ul◦ − URE
RT RT avg νi,RE ln(aRE νi,l ln(ai,m ). (4.10) i )− nRE F i nl F i
Using the Gibbs–Helmholtz relation, we can calculate the standard enthalpy of any reaction as a function of cell voltage: νi ,l
nl F
i
Hi◦,m
d =T dT
2
Ul◦ . T
(4.11)
According to Eqs. (4.5) and (4.7), we conclude that the share of electrochemical reactions in enthalpy generation can be calculated by the equation m
H¯ iavg ,m
νi,l ¯il
i
=
nl F
l
¯il m
l
nl F
H¯ iavg ,m νi ,l .
(4.12)
i
This value can be expressed as a function of potential of electrode reactions using Eqs. (4.9) and (4.11). The result is ¯il m
l
nl F
H¯ iavg ,m νi ,l
=
i
It
l
d T dT 2
Ul◦ RT 2 d avg avg ln(ai,m )νi,j . − T nl F i dT (4.13)
Since we need to express the potentials with respect to a reference electrode, we rewrite Eq. (4.13) using Eq. (4.10):
¯ iavg H Ul,avg ,m νi ,l 2 d ¯il ¯ = il T . l
m
i
nl F
l
dT
T
(4.14)
The quantity that is multiplied by ¯il in both sides of Eq. (4.14) is called the enthalpy voltage of reaction l. Applying Eqs. (4.7), (4.8), and (4.14) in Eq. (4.5) and using the results in Eq. (4.3) give the general form of enthalpy change of the system as a
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117
function of time, dHtot /dt; using this value, Eq. (4.1) yields: q − IV =
d Ul,avg ¯il T 2 l
Enthalpy of reaction
dT T
d
γ i ,j ∂ ln avg d∀ ∂ T γ i ,j Vj j i
avg γ dn d i , j i , m Hij◦→m − RT 2 − ln avg dT dt γ i ,j j,j=m i t ⎛ ¯il dt dT ⎜ ◦ ¯ avg ◦ ⎜ + ni,j Cpi,j + Cpl dt ⎝ nl F −
dt
j
+
j,j=m
i
ci,j RT 2
l
Enthalpy of mixing Phase change
⎞
¯ avg − C ¯ avg )(ni,j − n◦ )⎠ , (C pi,j pi,m i ,j
Heat capacity
i
(4.15) where Cpl =
¯ avg νi ,l C pi,m
(4.16)
i
and Hij◦→m = Hi◦,m − Hi◦,j .
(4.17)
It should be noted that all the composition properties in Eq. (4.15) are defined with respect to the activity coefficient ai,j = xi,j γi,j . This definition emphasizes the fact that if the activity coefficient of composition is known, then the thermodynamical of that composition is also known. Moreover, in Eq. (4.15) the heat capacity of the battery should be accurately calculated. Since a battery is a multicomponent system, we need to consider the heat capacity of all its elements. Since the heat capacity of the battery changes due to material change, in many practical cases, we can assume a constant mean value for the heat capacity of the battery in all cases. Finally, in Eq. (4.15), we need to consider the effect of heat dissipation to ambient. The majority of heat dissipation is due to convection of heat to ambient, which can be calculated by the equation q = −hA(T − TA ).
(4.18)
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4.2 Heat of reactions Neglecting enthalpy of mixing, phase change, and change of heat capacity, we reduce Eq. (4.15) to
Ul,avg 2 d ¯ il T q = IV + .
dT
l
T
(4.19)
In discharge the chemical energy is directly converted to the electrical work. The maximum electrical work is obtained if the reaction is reversible. In that case the reversible electrical work is IVrev =
¯il Ul,avg .
(4.20)
l
The difference between the real and reversible voltages is called the overpotential: η = V − Vrev .
(4.21)
The overpotential is an indicator of the irreversibility of the system, such as Ohmic drop, charge transfer overpotential, and mass transfer limitations. The product of overpotential and current is called the heat of polarization and consists of Joule heating and energy loss inside electrodes. Beside polarization heat, the enthalpy of reaction includes entropic heat defined as dUl,avg ¯il T (4.22) . qrev = − dT l Eqs. (4.20) and (4.22) are generated by power and heat in a reversible reaction, respectively. The reversible work can be obtained from changes in the Gibbs free energy. Inserting Eqs. (4.20) and (4.22) into Eq. (4.19) results in
q = IV −
l
¯il Ul,avg +
dUl,avg ¯il T . l
dT
(4.23)
This is another expression for Eq. (3.34) and shows that the generated heat of any reaction can be divided into two parts: 1. The generated heat due to chemical bonds of species that are involved in electrochemical reactions and are of a reversible type. The heat changes sign in charge and discharge, meaning that if it is exothermic in charge, then it becomes endothermic in discharge and vice versa.
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2. The irreversible part of heat generation is always positive (both in charge and discharge) and hence always contributes in temperature build–up. The reversible or entropic heat of reactions requires further inspection, whereas in electrochemical reactions, we are dealing with ions and electrons as well as with chemical compounds. The following sections cover this issue.
4.2.1 Reversible or entropic heat of reactions Entropy change in any half-cell reaction gives good information about the distribution of heat generation and consumption inside a battery. The entropy of different chemical compounds is tabulated in thermodynamic tables, which can be used for estimation of reversible heat of the cell for complete reactions. However, concerning the half-reactions, we are dealing with ions other than chemical compounds since any half-cell reaction contains some chemical compound and ions. Therefore we need a way to calculate the entropy of ions. Lampinen and Fomino [34] introduced a proper way as will be discussed here.
Enthalpy of reaction The standard molar enthalpy of a compound material (Hfi◦ ) is equal to enthalpy change of its elements for a unit mass of compound i in standard conditions. Its elements should also be maintained in standard conditions (i.e., standard temperature and pressure). In this case and under standard conditions, the standard enthalpy of formation h◦i is defined as h◦i = Hfi◦ .
(4.24)
Under nonstandard conditions (i.e., in any temperature T and pressure p), the enthalpy of the unit mass of compound i is calculated from the equation hi (T , p, x◦ ) = Hfi◦ (T◦ , p◦ , x◦ ) +
T
Cpi dT +
T◦
p p◦
vi − T
∂ vi dp. (4.25) ∂T
To obtain Eq. (4.25), we used the following thermodynamic relations: ∂ hi = Cpi , ∂T ∂ hi ∂ vi = vi − T , ∂p ∂T
(4.26) (4.27)
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vi =
∂V . ∂ ni
(4.28)
Moreover, the partial mole of species is defined as x = (x1 , . . . , xm ),
(4.29)
ni xi = . nk
(4.30)
where
Since the net electric charge of an electrochemical reaction is constant and does not change during the reaction, we cannot experimentally measure the thermodynamic properties of ions, but we can measure the whole thermodynamic property of a reaction such as 1 Na(s) + H+ (aq) → Na+ (aq) + H2 (g). 2
(4.31)
In this reaction, sodium ions are produced from the main element, and hydrogen gas is released from the solution under standard conditions. In Eq. (4.31) the net electric charge remains unchanged. On the other hand, the standard scale defined for standard enthalpy of formation of ion in water indicates that when the net electric charge is constant in standard conditions, that is, T = 298.15 K and p = 1 bar for ions with activity coefficient of unity that is an indicator of unit molality, we have Hf◦ [Na+ (aq)] = H ◦
[Eq. (4.31)].
(4.32)
According to Eq. (4.32), for hydrogen ion, we obtain Hf◦ [H+ (aq)] = 0.
(4.33)
Consequently, using Eq. (4.25), we can write h◦ [H+ (aq)] = Hf◦ [H+ (aq)] = 0,
h◦ [H2 (g)] = Hf◦ [H+ (aq)] = 0. (4.34)
Entropy of reaction The third law of thermodynamics indicates that the entropy has an absolute value. This scale is called the absolute entropy. By that statement any pure
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element has the absolutely zero entropy at absolute zero. For example, [Na(s), T = 0 K] = 0.
(4.35)
The absolute entropy of elements at higher temperatures above absolute zero is obtained from the integral of heat; as an example,
298.15
sa [Na(s), T = 298.15 K] = 0
dQ = 51.17 J/mol K. T
(4.36)
However, the main problem is finding the absolute entropy of charged particles in solutions, which is not easy to obtain. For example, to calculate sa [Na+ (aq)], we need to use the entropy of reaction. Just for an example, for Eq. (4.31), we can write 1 2
Sf◦ [Na+ (aq)] = s◦a [Na+ (aq)] + s◦a [H2 (g)] − s◦a [Na(s)] − s◦a [H+ (aq)].
(4.37) The absolute entropies s◦a [H2 (g)] and s◦a [Na(s)] are well known, but the value of s◦a [H+ (aq)] should be calculated. Note that s◦a [H+ (aq)] = 0 but Sf◦ [Na(aq)] = 0. Hence the absolute entropy of sodium ions cannot be calculated using Eq. (4.37) unless s◦a [H+ (aq)] is determined in some way. The values that are tabulated as absolute entropies in thermodynamic tables are defined as follows: 1 s◦ [Na+ (aq)] = Sf◦ [Na+ (aq)] − s◦a [H2 (g)] + s◦a [Na(s)]. 2
(4.38)
Here another zero scale is defined for entropy such that s◦ [H+ (aq)] = 0,
(4.39)
where the subscript that indicates the absolute entropy is dropped out, so that it can be distinguished from absolute entropy. In other words, s◦ [Na+ (aq)] = s◦a [Na+ (aq)].
(4.40)
Since s◦a [H+ (aq)] = 0, the new value for entropy can be called the semiabsolute entropy because although it is absolute in some point of view, its zero value differs from the absolute zero.
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Simulation of Battery Systems
Not we have the absolute and semiabsolute entropy values for each element at temperature T and pressure p, which can be obtained from the following relations: ◦
◦
sai (T , p, x◦ ) = sai + s◦i (T , p, x◦ ) = s◦i +
T
T◦ T T◦
p Cpi ∂ vi dT + dp, − T ∂ T p◦ p Cpi ∂ vi dp. − dT + T ∂T p◦
(4.41) (4.42)
In these equations, we used the following thermodynamic relations: cpi ∂ sai ∂ si = = , ∂T ∂T T ∂ sai ∂ si ∂ vi = =− . ∂p ∂p ∂T
(4.43) (4.44)
Now the main problem is that how we can use different entropy scales to calculate the absolute entropy? To answer the question, consider a system with differently charged particles (such as ions and electrons) whose entropy is defined by semiabsolute scale, i = 1, . . . k, and some noncharged particles whose entropy is defined by absolute entropy, i = k + 1, . . . , m. The net entropy of the system then can be calculated as S(T , p, n1 , . . . , nm ) =
k
nisi +
i=1
m
nisai .
(4.45)
i=k+1
However, from the absolute entropy point of view, the absolute entropy of the system is S (T , p, n1 , . . . , nm ) =
m
ni sai .
(4.46)
i=1
Using Eqs. (4.38) and (4.39), we see that for charged particles (zi = 0, i = 1, . . . , k), sai = si + zi s◦ai [H+ (aq)],
(4.47)
and according to Eqs. (4.45) to (4.47), we obtain ◦
+
S − S = sai [H (aq)]
k i=1
ni zi = 0.
(4.48)
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123
This is because according to the balance of electric charge, the net electric charge is zero, that is, q=F
k
ni zi = 0.
(4.49)
i=1
Therefore Eqs. (4.48) and (4.49) show that the simultaneous usage of absolute and semiabsolute entropies for calculation of the absolute entropy will give the same results if 1. all the semiabsolute scales are defined according to a single zero, 2. the entropy of electron is used in calculations. As a result, for calculation of the entropy of electrochemical and chemical reactions, we need to have a value for the entropy of electron. In the succeeding subsection, we discuss this issue.
Entropy of electron According to thermodynamic relations, the entropy of a system can be calculated using the Gibbs free energy that contains electrostatic energy as follows:
∂G . (4.50) S=− ∂ T p,ni ,...,nm According to this equation and Eq. (4.45), the partial entropy of noncharged particles (zi = 0, μ˜ i = μi , where chemical and electrochemical potentials are equal) is sai =
∂S ∂ ∂G ∂ ∂G ∂ μ˜ ∂μ =− =− =− =− , ∂ ni ∂ ni ∂ T ∂ T ∂ ni ∂T ∂T
where
μ˜ =
∂G ∂ ni
(4.51)
.
(4.52)
T ,p,n1 ,...,ni −1,ni +1,...,nm
In the same manner, for charged particles (zi = 0), si =
∂S ∂ μ˜ i =− . ∂ ni ∂T
(4.53)
In an equilibrium state for standard hydrogen, we have μ◦ [H2 (g)] = 2μ˜ ◦ [H+ (aq)] + 2μ˜ ◦ [e− (Pt)].
(4.54)
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Figure 4.1 A resistance model for Joule heating.
Differentiating these equations and using Eqs. (4.51) and (4.53), we obtain s◦a [H2 (g)] = 2s◦ [H+ (aq)] + 2s◦ [e− (Pt)].
(4.55)
Substituting Eq. (4.39) into Eq. (4.55) yields 1 s◦ [e− (Pt)] = s◦a [H2 (g)] = 65.29 J/mol K. 2
(4.56)
Eq. (4.56) gives a tool for calculation of entropy of electron with the same zero scale.
4.2.2 Joule heating In general, the applied current should be conserved throughout a cell. This current enters the cell from an external circuit and enters the solid phase. As can be seen schematically in Fig. 4.1, the current enters the electrolyte through surface reactions with rates determined by the kinetics of the reactions. At the end of the solid phase, all the current enters the electrolyte phase and is carried by ions to the other electrode. On the other electrode surface, this current enters the solid phase again in a reverse manner. This description can be understood better by considering the resistance model, as shown in Fig. 4.1. The electrical resistance of the cell consists of the resistance of the positive and negative electrodes, electrolyte, and separator. In each region the current passes through different phases. Hence Joule heating should be considered in all the phases. Therefore Joule heating can be written as qJoule =
|φk · ik |.
(4.57)
k
In this equation, the absolute value signs should be present since Joule heating is always positive and contributes to a rise in temperature.
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125
The last point about the Joule heating is that although the summation should be carried on all phases, we can neglect the current through the gas phase since it is negligible compared to the current through the solid and liquid phases.
4.3 General Joule heating concept Splitting the heat of reactions into reversible and irreversible parts show that the irreversible part is very similar to Joule heating (i.e., Eq. (4.57)), where the quantity is a product of current and a voltage difference. Consequently, the irreversible part of heat generation has a Joule heating nature and is positive both in charge and discharge in both electrodes. This fact leads to a new concept known as the generalized Joule heating, which is the sum of the classical Joule heating and irreversible part of heat generation due to electrochemical reactions, that is, qGJH = qJoule + qirrev =
|φk · ik | + qirrev .
(4.58)
k
This concept was first introduced by Torabi and Esfahanian [32]. They noted that the generalized Joule heating plays an important role in thermal behavior of a battery, especially in thermal runaway.
4.4 Heat dissipation In addition to heat sources in a battery, there are a variety of procedures ending up by releasing an extra produced heat into the ambient. These mechanisms can dissipate the excess heat in the form of different heat fluxes.
4.4.1 Convection One of the most common transferred heat to the ambient is done by convection mechanism, the equivalent resistance of which is demonstrated in the following equation, where the term h∞ indicates heat transfer coefficient: Rconv =
1 . h∞
The role of convection resistance is illustrated in Fig. 4.2.
(4.59)
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Figure 4.2 Heat sinks.
4.4.2 Conduction Conduction heat transfer occurs in solid parts of battery case leading heat from inside to the outer surface of the battery. Therefore, for calculating the heat dissipation by convection mechanism, heat conduction from cell to cell and to cell case (as shown in Fig. 4.2) should be calculated. The conductive resistance is calculated according to the following equation, where Lcase is the thickness of the case wall, and λcase is its heat conductivity: Rcond =
Lcase λcase
.
(4.60)
4.4.3 Radiation Another mechanism responsible for heat dissipation is radiation, and the amount of transferred heat can be calculated by applying the following equation, where σs is the Stefan–Boltzmann constant: 4 4 − Tamb ). qrad = σs A(Tcase
(4.61)
Since the operational temperature of a ZSOB does not increase too much, the radiation part is negligible.
4.4.4 Exhausted enthalpy In systems with external flows, another heat sink is available and required to be taken into account. This heat can be defined as the exhausted heat
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127
through exhausting gas flow, and its amount can be obtained by using the following equation, where m˙ shows the mass flux of the exhausting gas: qexh = mh ˙
(4.62)
4.4.5 Equivalent circuit model In general, dissipation mechanisms are capable of being simulated based on their equivalent electrical resistance. Most conventional batteries are operating at moderate temperatures (usually less than 100◦ C and are sealed so that no or negligible mass transfer occurs). Consequently, heat dissipation due to radiation and mass transfer can be neglected. In other words, conduction and convection are the only existent heat dissipation mechanisms. In Fig. 4.2, heat sinks are represented for a conventional battery in more detail. This battery consists of three cells in series. There are three important points about this picture [31]. First, there is a big difference between the amount of heat that an internal cell is capable of dissipating to the ambient in comparison with the cells that are located at the sides of the battery. Indeed, a side-cell has more surface area for dissipating heat to the ambient in comparison with an internal cell. Thus the critical cell that has higher temperature can be considered to be an internal cell. Second, the resistance in the upper parts of an internal cell itself is greater than that in the lower parts. As a result, the top areas in an internal cell have the potential of becoming warmer. Finally, there is a symmetric condition in the amount of heat transferred from an internal cell to its neighbors in the form of conduction. According to this symmetry, we can assume that the heat fluxes are required to be zero in the borders in contact with neighbors. As a result, an internal cell can be considered to be isolated from each side, and the only available heat fluxes are those from the upper and lower parts of the cell.
4.5 Summary Heat generation inside a battery is very important because battery performance strongly depends on temperature. Therefore, for a good simulation, the heat sources and sinks should be properly modeled. The general formulation that was given in the present chapter enables us to obtain proper formulation for any simulation. It should be noted that heat sinks are also important in analyzing the thermal behavior of the batteries. The thermal management of batteries
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depends mainly on designing efficient heat sinks so that the battery pack temperature remains within a specific limit.
4.6 Problems 1. Explain the difference between reversible and irreversible heat sources. 2. Is Joule heating reversible? 3. Try to figure out how to model heat sinks in a one-dimensional model.
Heat sources Contents 4.1. Fundamental laws 4.2. Heat of reactions 4.2.1 Reversible or entropic heat of reactions Enthalpy of reaction Entropy of reaction Entropy of electron 4.2.2 Joule heating 4.3. General Joule heating concept 4.4. Heat dissipation 4.4.1 Convection 4.4.2 Conduction 4.4.3 Radiation 4.4.4 Exhausted enthalpy 4.4.5 Equivalent circuit model 4.5. Summary 4.6. Problems
114 118 119 119 120 123 124 125 125 125 126 126 126 127 127 128
Usually a battery, just like any other systems from the thermodynamical point of view, is a system with fixed mass. However, in some cases the battery is considered as a control volume because we have mass transport to or from the battery. Metal–air batteries are of control volume type, whereas lead–acid, lithium-ion, nickel-based batteries, and many other technologies are of system type. Whether they are considered as a system or control volume, they exchange heat with ambient and also act as a heat generator or in some rare cases as sinks. To obtain the battery temperature, we have to deal with thermodynamical laws and relations. Any thermodynamical system is subjected to generation, dissipation, and storage of energy. To predict its temperature distribution, the first law of thermodynamics should be applied on it. In a multiphase multicomponent medium like batteries, special care should be paid to obtain accurate results. In such a system, heat generation and dissipations are due to: 1. Heat generation due to electrochemical reactions. 2. Material phase change. 3. Mixing. 4. Change in heat capacity of the system due to the material change. Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00008-8
Copyright © 2020 Elsevier Inc. All rights reserved.
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5. Electrical work. 6. Heat exchange with surroundings. The traditional methods for prediction of thermal behavior of batteries yield to the estimation of a uniform temperature for the whole cell. This assumption is made because in traditional methods a lumped model is used, in which the battery cell is considered as a uniform system with high thermal conductivity resulting in a uniform temperature. If we like to obtain a temperature gradient inside the battery, we need to introduce a model that estimates both generation and dissipation inside the cell. Such models are based on fundamental governing equations in vector form. The fundamental laws are in the form of convective–diffusive transport equations that include the effect of transient parts, heat convection, heat conduction, and heat sources and sinks. A proper system of governing equations with proper boundary conditions gives an appropriate tool for prediction of temperature profile inside any battery cell. In this chapter, we fully discuss the fundamental thermodynamical equations that govern the energy balance of a battery system. We discuss in detail heat sources and sinks that are the main issue in temperature profile.
4.1 Fundamental laws For a multicomponent multiphase medium in which the energy content is changing and different phases are in contact with each other, the first law of thermodynamics can be written as the equation dHtot = q − IV , dt
(4.1)
in which Htot is the total enthalpy of all the phases: Htot =
j
Vj
ci,j H¯ i,j d∀,
(4.2)
i
where i and j describe the ith species and the jth reaction, respectively. In Eq. (4.1) the rate of heat generation and dissipation is shown by q, and IV represents the electrical work. For simplicity, we can assume a mean composition for each phase; hence we have
dHtot d = ni,j H¯ iavg ,j + dt dt j i
Vj
ci,j H¯ i,j − H¯ iavg d∀ . ,j
(4.3)
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The first term on the right-hand side of Eq. (4.3) shows the rate of enthalpy change of the system when all the species are in an equilibrium state and the second term accounts for their deviation from equilibrium. The first term can be divided into three parts:
avg
¯ i ,j ∂H
¯ . =C pi,j avg
∂T
(4.4)
p
The first part in Eq. (4.3) can be calculated as d j
i
dt
¯ i ,j ) = (ni,j H avg
j
ni,j C¯ piavg,j
i
dni,j dT ¯ iavg +H . ,j dt dt
(4.5)
As explained earlier, at each electrode a set of electrochemical reactions occurs, all of which can be defined by the general formula
νi,l Mizi = nl e− .
(4.6)
i
This equation is written so that each species i is in a single phase. Therefore diffusion phenomena from one phase to another phase should be separately considered. Using the balance of chemical species, we can calculate the mole of each species by the equation dni,m νi,l¯il dni,j = − . dt nl F j, j=m dt l
(4.7)
The first term on the right-hand side of the equation represents the rate of generation of consumption of species i due to electrochemical reactions. Moreover, the partial current ¯il is the necessary current that each reaction requires. This current is positive for cathodic reactions and negative for anodic reactions. The second term of Eq. (4.7) indicates the change of mole of species i due to phase change. Integration of Eq. (4.7) results in ni,m = n◦i,m −
(ni,j − n◦i,j ) +
j, j=m
νi ,l l
nl F
◦
t
¯il dt.
(4.8)
The partial molar enthalpy is defined by ◦ 2 H¯ iavg ,m = Hi,m − RT
d avg ln(ai,m ). dT
(4.9)
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Simulation of Battery Systems
On the other hand, the theoretical open-circuit voltage for reaction l in an averaged composition with respect to a reference electrode is ◦ + Ui,avg = Ul◦ − URE
RT RT avg νi,RE ln(aRE νi,l ln(ai,m ). (4.10) i )− nRE F i nl F i
Using the Gibbs–Helmholtz relation, we can calculate the standard enthalpy of any reaction as a function of cell voltage: νi ,l
nl F
i
Hi◦,m
d =T dT
2
Ul◦ . T
(4.11)
According to Eqs. (4.5) and (4.7), we conclude that the share of electrochemical reactions in enthalpy generation can be calculated by the equation m
H¯ iavg ,m
νi,l ¯il
i
=
nl F
l
¯il m
l
nl F
H¯ iavg ,m νi ,l .
(4.12)
i
This value can be expressed as a function of potential of electrode reactions using Eqs. (4.9) and (4.11). The result is ¯il m
l
nl F
H¯ iavg ,m νi ,l
=
i
It
l
d T dT 2
Ul◦ RT 2 d avg avg ln(ai,m )νi,j . − T nl F i dT (4.13)
Since we need to express the potentials with respect to a reference electrode, we rewrite Eq. (4.13) using Eq. (4.10):
¯ iavg H Ul,avg ,m νi ,l 2 d ¯il ¯ = il T . l
m
i
nl F
l
dT
T
(4.14)
The quantity that is multiplied by ¯il in both sides of Eq. (4.14) is called the enthalpy voltage of reaction l. Applying Eqs. (4.7), (4.8), and (4.14) in Eq. (4.5) and using the results in Eq. (4.3) give the general form of enthalpy change of the system as a
Heat sources
117
function of time, dHtot /dt; using this value, Eq. (4.1) yields: q − IV =
d Ul,avg ¯il T 2 l
Enthalpy of reaction
dT T
d
γ i ,j ∂ ln avg d∀ ∂ T γ i ,j Vj j i
avg γ dn d i , j i , m Hij◦→m − RT 2 − ln avg dT dt γ i ,j j,j=m i t ⎛ ¯il dt dT ⎜ ◦ ¯ avg ◦ ⎜ + ni,j Cpi,j + Cpl dt ⎝ nl F −
dt
j
+
j,j=m
i
ci,j RT 2
l
Enthalpy of mixing Phase change
⎞
¯ avg − C ¯ avg )(ni,j − n◦ )⎠ , (C pi,j pi,m i ,j
Heat capacity
i
(4.15) where Cpl =
¯ avg νi ,l C pi,m
(4.16)
i
and Hij◦→m = Hi◦,m − Hi◦,j .
(4.17)
It should be noted that all the composition properties in Eq. (4.15) are defined with respect to the activity coefficient ai,j = xi,j γi,j . This definition emphasizes the fact that if the activity coefficient of composition is known, then the thermodynamical of that composition is also known. Moreover, in Eq. (4.15) the heat capacity of the battery should be accurately calculated. Since a battery is a multicomponent system, we need to consider the heat capacity of all its elements. Since the heat capacity of the battery changes due to material change, in many practical cases, we can assume a constant mean value for the heat capacity of the battery in all cases. Finally, in Eq. (4.15), we need to consider the effect of heat dissipation to ambient. The majority of heat dissipation is due to convection of heat to ambient, which can be calculated by the equation q = −hA(T − TA ).
(4.18)
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Simulation of Battery Systems
4.2 Heat of reactions Neglecting enthalpy of mixing, phase change, and change of heat capacity, we reduce Eq. (4.15) to
Ul,avg 2 d ¯ il T q = IV + .
dT
l
T
(4.19)
In discharge the chemical energy is directly converted to the electrical work. The maximum electrical work is obtained if the reaction is reversible. In that case the reversible electrical work is IVrev =
¯il Ul,avg .
(4.20)
l
The difference between the real and reversible voltages is called the overpotential: η = V − Vrev .
(4.21)
The overpotential is an indicator of the irreversibility of the system, such as Ohmic drop, charge transfer overpotential, and mass transfer limitations. The product of overpotential and current is called the heat of polarization and consists of Joule heating and energy loss inside electrodes. Beside polarization heat, the enthalpy of reaction includes entropic heat defined as dUl,avg ¯il T (4.22) . qrev = − dT l Eqs. (4.20) and (4.22) are generated by power and heat in a reversible reaction, respectively. The reversible work can be obtained from changes in the Gibbs free energy. Inserting Eqs. (4.20) and (4.22) into Eq. (4.19) results in
q = IV −
l
¯il Ul,avg +
dUl,avg ¯il T . l
dT
(4.23)
This is another expression for Eq. (3.34) and shows that the generated heat of any reaction can be divided into two parts: 1. The generated heat due to chemical bonds of species that are involved in electrochemical reactions and are of a reversible type. The heat changes sign in charge and discharge, meaning that if it is exothermic in charge, then it becomes endothermic in discharge and vice versa.
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Heat sources
2. The irreversible part of heat generation is always positive (both in charge and discharge) and hence always contributes in temperature build–up. The reversible or entropic heat of reactions requires further inspection, whereas in electrochemical reactions, we are dealing with ions and electrons as well as with chemical compounds. The following sections cover this issue.
4.2.1 Reversible or entropic heat of reactions Entropy change in any half-cell reaction gives good information about the distribution of heat generation and consumption inside a battery. The entropy of different chemical compounds is tabulated in thermodynamic tables, which can be used for estimation of reversible heat of the cell for complete reactions. However, concerning the half-reactions, we are dealing with ions other than chemical compounds since any half-cell reaction contains some chemical compound and ions. Therefore we need a way to calculate the entropy of ions. Lampinen and Fomino [34] introduced a proper way as will be discussed here.
Enthalpy of reaction The standard molar enthalpy of a compound material (Hfi◦ ) is equal to enthalpy change of its elements for a unit mass of compound i in standard conditions. Its elements should also be maintained in standard conditions (i.e., standard temperature and pressure). In this case and under standard conditions, the standard enthalpy of formation h◦i is defined as h◦i = Hfi◦ .
(4.24)
Under nonstandard conditions (i.e., in any temperature T and pressure p), the enthalpy of the unit mass of compound i is calculated from the equation hi (T , p, x◦ ) = Hfi◦ (T◦ , p◦ , x◦ ) +
T
Cpi dT +
T◦
p p◦
vi − T
∂ vi dp. (4.25) ∂T
To obtain Eq. (4.25), we used the following thermodynamic relations: ∂ hi = Cpi , ∂T ∂ hi ∂ vi = vi − T , ∂p ∂T
(4.26) (4.27)
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Simulation of Battery Systems
vi =
∂V . ∂ ni
(4.28)
Moreover, the partial mole of species is defined as x = (x1 , . . . , xm ),
(4.29)
ni xi = . nk
(4.30)
where
Since the net electric charge of an electrochemical reaction is constant and does not change during the reaction, we cannot experimentally measure the thermodynamic properties of ions, but we can measure the whole thermodynamic property of a reaction such as 1 Na(s) + H+ (aq) → Na+ (aq) + H2 (g). 2
(4.31)
In this reaction, sodium ions are produced from the main element, and hydrogen gas is released from the solution under standard conditions. In Eq. (4.31) the net electric charge remains unchanged. On the other hand, the standard scale defined for standard enthalpy of formation of ion in water indicates that when the net electric charge is constant in standard conditions, that is, T = 298.15 K and p = 1 bar for ions with activity coefficient of unity that is an indicator of unit molality, we have Hf◦ [Na+ (aq)] = H ◦
[Eq. (4.31)].
(4.32)
According to Eq. (4.32), for hydrogen ion, we obtain Hf◦ [H+ (aq)] = 0.
(4.33)
Consequently, using Eq. (4.25), we can write h◦ [H+ (aq)] = Hf◦ [H+ (aq)] = 0,
h◦ [H2 (g)] = Hf◦ [H+ (aq)] = 0. (4.34)
Entropy of reaction The third law of thermodynamics indicates that the entropy has an absolute value. This scale is called the absolute entropy. By that statement any pure
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Heat sources
element has the absolutely zero entropy at absolute zero. For example, [Na(s), T = 0 K] = 0.
(4.35)
The absolute entropy of elements at higher temperatures above absolute zero is obtained from the integral of heat; as an example,
298.15
sa [Na(s), T = 298.15 K] = 0
dQ = 51.17 J/mol K. T
(4.36)
However, the main problem is finding the absolute entropy of charged particles in solutions, which is not easy to obtain. For example, to calculate sa [Na+ (aq)], we need to use the entropy of reaction. Just for an example, for Eq. (4.31), we can write 1 2
Sf◦ [Na+ (aq)] = s◦a [Na+ (aq)] + s◦a [H2 (g)] − s◦a [Na(s)] − s◦a [H+ (aq)].
(4.37) The absolute entropies s◦a [H2 (g)] and s◦a [Na(s)] are well known, but the value of s◦a [H+ (aq)] should be calculated. Note that s◦a [H+ (aq)] = 0 but Sf◦ [Na(aq)] = 0. Hence the absolute entropy of sodium ions cannot be calculated using Eq. (4.37) unless s◦a [H+ (aq)] is determined in some way. The values that are tabulated as absolute entropies in thermodynamic tables are defined as follows: 1 s◦ [Na+ (aq)] = Sf◦ [Na+ (aq)] − s◦a [H2 (g)] + s◦a [Na(s)]. 2
(4.38)
Here another zero scale is defined for entropy such that s◦ [H+ (aq)] = 0,
(4.39)
where the subscript that indicates the absolute entropy is dropped out, so that it can be distinguished from absolute entropy. In other words, s◦ [Na+ (aq)] = s◦a [Na+ (aq)].
(4.40)
Since s◦a [H+ (aq)] = 0, the new value for entropy can be called the semiabsolute entropy because although it is absolute in some point of view, its zero value differs from the absolute zero.
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Simulation of Battery Systems
Not we have the absolute and semiabsolute entropy values for each element at temperature T and pressure p, which can be obtained from the following relations: ◦
◦
sai (T , p, x◦ ) = sai + s◦i (T , p, x◦ ) = s◦i +
T
T◦ T T◦
p Cpi ∂ vi dT + dp, − T ∂ T p◦ p Cpi ∂ vi dp. − dT + T ∂T p◦
(4.41) (4.42)
In these equations, we used the following thermodynamic relations: cpi ∂ sai ∂ si = = , ∂T ∂T T ∂ sai ∂ si ∂ vi = =− . ∂p ∂p ∂T
(4.43) (4.44)
Now the main problem is that how we can use different entropy scales to calculate the absolute entropy? To answer the question, consider a system with differently charged particles (such as ions and electrons) whose entropy is defined by semiabsolute scale, i = 1, . . . k, and some noncharged particles whose entropy is defined by absolute entropy, i = k + 1, . . . , m. The net entropy of the system then can be calculated as S(T , p, n1 , . . . , nm ) =
k
nisi +
i=1
m
nisai .
(4.45)
i=k+1
However, from the absolute entropy point of view, the absolute entropy of the system is S (T , p, n1 , . . . , nm ) =
m
ni sai .
(4.46)
i=1
Using Eqs. (4.38) and (4.39), we see that for charged particles (zi = 0, i = 1, . . . , k), sai = si + zi s◦ai [H+ (aq)],
(4.47)
and according to Eqs. (4.45) to (4.47), we obtain ◦
+
S − S = sai [H (aq)]
k i=1
ni zi = 0.
(4.48)
Heat sources
123
This is because according to the balance of electric charge, the net electric charge is zero, that is, q=F
k
ni zi = 0.
(4.49)
i=1
Therefore Eqs. (4.48) and (4.49) show that the simultaneous usage of absolute and semiabsolute entropies for calculation of the absolute entropy will give the same results if 1. all the semiabsolute scales are defined according to a single zero, 2. the entropy of electron is used in calculations. As a result, for calculation of the entropy of electrochemical and chemical reactions, we need to have a value for the entropy of electron. In the succeeding subsection, we discuss this issue.
Entropy of electron According to thermodynamic relations, the entropy of a system can be calculated using the Gibbs free energy that contains electrostatic energy as follows:
∂G . (4.50) S=− ∂ T p,ni ,...,nm According to this equation and Eq. (4.45), the partial entropy of noncharged particles (zi = 0, μ˜ i = μi , where chemical and electrochemical potentials are equal) is sai =
∂S ∂ ∂G ∂ ∂G ∂ μ˜ ∂μ =− =− =− =− , ∂ ni ∂ ni ∂ T ∂ T ∂ ni ∂T ∂T
where
μ˜ =
∂G ∂ ni
(4.51)
.
(4.52)
T ,p,n1 ,...,ni −1,ni +1,...,nm
In the same manner, for charged particles (zi = 0), si =
∂S ∂ μ˜ i =− . ∂ ni ∂T
(4.53)
In an equilibrium state for standard hydrogen, we have μ◦ [H2 (g)] = 2μ˜ ◦ [H+ (aq)] + 2μ˜ ◦ [e− (Pt)].
(4.54)
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Simulation of Battery Systems
Figure 4.1 A resistance model for Joule heating.
Differentiating these equations and using Eqs. (4.51) and (4.53), we obtain s◦a [H2 (g)] = 2s◦ [H+ (aq)] + 2s◦ [e− (Pt)].
(4.55)
Substituting Eq. (4.39) into Eq. (4.55) yields 1 s◦ [e− (Pt)] = s◦a [H2 (g)] = 65.29 J/mol K. 2
(4.56)
Eq. (4.56) gives a tool for calculation of entropy of electron with the same zero scale.
4.2.2 Joule heating In general, the applied current should be conserved throughout a cell. This current enters the cell from an external circuit and enters the solid phase. As can be seen schematically in Fig. 4.1, the current enters the electrolyte through surface reactions with rates determined by the kinetics of the reactions. At the end of the solid phase, all the current enters the electrolyte phase and is carried by ions to the other electrode. On the other electrode surface, this current enters the solid phase again in a reverse manner. This description can be understood better by considering the resistance model, as shown in Fig. 4.1. The electrical resistance of the cell consists of the resistance of the positive and negative electrodes, electrolyte, and separator. In each region the current passes through different phases. Hence Joule heating should be considered in all the phases. Therefore Joule heating can be written as qJoule =
|φk · ik |.
(4.57)
k
In this equation, the absolute value signs should be present since Joule heating is always positive and contributes to a rise in temperature.
Heat sources
125
The last point about the Joule heating is that although the summation should be carried on all phases, we can neglect the current through the gas phase since it is negligible compared to the current through the solid and liquid phases.
4.3 General Joule heating concept Splitting the heat of reactions into reversible and irreversible parts show that the irreversible part is very similar to Joule heating (i.e., Eq. (4.57)), where the quantity is a product of current and a voltage difference. Consequently, the irreversible part of heat generation has a Joule heating nature and is positive both in charge and discharge in both electrodes. This fact leads to a new concept known as the generalized Joule heating, which is the sum of the classical Joule heating and irreversible part of heat generation due to electrochemical reactions, that is, qGJH = qJoule + qirrev =
|φk · ik | + qirrev .
(4.58)
k
This concept was first introduced by Torabi and Esfahanian [32]. They noted that the generalized Joule heating plays an important role in thermal behavior of a battery, especially in thermal runaway.
4.4 Heat dissipation In addition to heat sources in a battery, there are a variety of procedures ending up by releasing an extra produced heat into the ambient. These mechanisms can dissipate the excess heat in the form of different heat fluxes.
4.4.1 Convection One of the most common transferred heat to the ambient is done by convection mechanism, the equivalent resistance of which is demonstrated in the following equation, where the term h∞ indicates heat transfer coefficient: Rconv =
1 . h∞
The role of convection resistance is illustrated in Fig. 4.2.
(4.59)
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Simulation of Battery Systems
Figure 4.2 Heat sinks.
4.4.2 Conduction Conduction heat transfer occurs in solid parts of battery case leading heat from inside to the outer surface of the battery. Therefore, for calculating the heat dissipation by convection mechanism, heat conduction from cell to cell and to cell case (as shown in Fig. 4.2) should be calculated. The conductive resistance is calculated according to the following equation, where Lcase is the thickness of the case wall, and λcase is its heat conductivity: Rcond =
Lcase λcase
.
(4.60)
4.4.3 Radiation Another mechanism responsible for heat dissipation is radiation, and the amount of transferred heat can be calculated by applying the following equation, where σs is the Stefan–Boltzmann constant: 4 4 − Tamb ). qrad = σs A(Tcase
(4.61)
Since the operational temperature of a ZSOB does not increase too much, the radiation part is negligible.
4.4.4 Exhausted enthalpy In systems with external flows, another heat sink is available and required to be taken into account. This heat can be defined as the exhausted heat
Heat sources
127
through exhausting gas flow, and its amount can be obtained by using the following equation, where m˙ shows the mass flux of the exhausting gas: qexh = mh ˙
(4.62)
4.4.5 Equivalent circuit model In general, dissipation mechanisms are capable of being simulated based on their equivalent electrical resistance. Most conventional batteries are operating at moderate temperatures (usually less than 100◦ C and are sealed so that no or negligible mass transfer occurs). Consequently, heat dissipation due to radiation and mass transfer can be neglected. In other words, conduction and convection are the only existent heat dissipation mechanisms. In Fig. 4.2, heat sinks are represented for a conventional battery in more detail. This battery consists of three cells in series. There are three important points about this picture [31]. First, there is a big difference between the amount of heat that an internal cell is capable of dissipating to the ambient in comparison with the cells that are located at the sides of the battery. Indeed, a side-cell has more surface area for dissipating heat to the ambient in comparison with an internal cell. Thus the critical cell that has higher temperature can be considered to be an internal cell. Second, the resistance in the upper parts of an internal cell itself is greater than that in the lower parts. As a result, the top areas in an internal cell have the potential of becoming warmer. Finally, there is a symmetric condition in the amount of heat transferred from an internal cell to its neighbors in the form of conduction. According to this symmetry, we can assume that the heat fluxes are required to be zero in the borders in contact with neighbors. As a result, an internal cell can be considered to be isolated from each side, and the only available heat fluxes are those from the upper and lower parts of the cell.
4.5 Summary Heat generation inside a battery is very important because battery performance strongly depends on temperature. Therefore, for a good simulation, the heat sources and sinks should be properly modeled. The general formulation that was given in the present chapter enables us to obtain proper formulation for any simulation. It should be noted that heat sinks are also important in analyzing the thermal behavior of the batteries. The thermal management of batteries
128
Simulation of Battery Systems
depends mainly on designing efficient heat sinks so that the battery pack temperature remains within a specific limit.
4.6 Problems 1. Explain the difference between reversible and irreversible heat sources. 2. Is Joule heating reversible? 3. Try to figure out how to model heat sinks in a one-dimensional model.