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Determination of the entropy change profile of a cylindrical lithium-ion battery by heat flux measurements
Journal of Power Sources 330 (2016) 61e69
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
Determination of the entropy change profile of a cylindrical lithium-ion battery by heat flux measurements K.A. Murashko a, *, A.V. Mityakov a, V.Y. Mityakov b, S.Z. Sapozhnikov b, J. Jokiniemi c, € nen a J. Pyrho a b c
Lappeenranta University of Technology LUT, Skinnarilankatu 34, 53850 Lappeenranta, Finland Saint-Petersburg State Polytechnical University, Polytechnicheskaya, 29, 195251 Saint-Petersburg, Russia University of Eastern Finland, Kuopio Campus, P.O. Box 1627, FI-70211 Kuopio, Finland
h i g h l i g h t s DS measurements by using a gradient heat flux sensor are presented. Time required for the DS measurements by the potentiometric method is evaluated. Uncertainty and limitations of the methods under consideration are given. Comparison of the proposed method with the potentiometric method is made.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 June 2016 Received in revised form 9 August 2016 Accepted 30 August 2016
The popularity of lithium-ion (Li-ion) batteries has increased over the recent years. Because of the strong dependence of the Li-ion battery operation characteristics on temperature, heat generation in the battery has to be taken into account. The entropy change of a Li-ion battery has a significant influence on heat generation, especially at a low C-rate current. Therefore, it is necessary to consider the entropy change profile in the estimation of heat generation. In the paper, a method to determine the entropy change (DS) profile by heat flux measurements of a cylindrical Li-ion cell is proposed. The method allows simultaneous measurements of the thermal diffusivity and DS of the cylindrical cell. The thermal diffusivity and DS measurements are carried out by a gradient heat flux sensor (GHFS). The comparison between the DS profile determined by the GHFS method with that obtained using a standard potentiometric method clearly shows that the entropy change measurements could be made by using a GHFS. Even though the uncertainty of the reported method is higher than that of the potentiometric method, a significant decrease in the experiment time compared with the potentiometric method is a major advantage of this method. © 2016 Elsevier B.V. All rights reserved.
Keywords: Li-ion battery Entropy change Thermal diffusivity Heat flux measurements Sensor Thermal model
1. Introduction The increasing popularity of electric and hybrid vehicles over the recent years is caused by a number of factors, including an increase in CO2 emissions and high energy costs. An energy storage device is one of the main parts in electric vehicles (EVs). Nowadays, because of their large number of usable operation cycles and low self-discharge rate, lithium-ion (Li-ion) batteries are considered the most promising technology for the energy storage in EVs. Despite the advantages of the Li-ion batteries, they also have certain disadvantages such as the high sensitivity to overcharge and
* Corresponding author. E-mail address: kirill.murashko@lut.fi (K.A. Murashko). http://dx.doi.org/10.1016/j.jpowsour.2016.08.130 0378-7753/© 2016 Elsevier B.V. All rights reserved.
overdischarge and the risk of a thermal runaway, which can even be a reason of fire. Therefore, a battery management system and a thermal management system are required in the operation of Li-ion batteries. An appropriate thermal management system becomes more important when the surface area per volume ratio of a battery decreases with the increasing battery size, resulting in a lower heat transfer rate per unit rate of heat generation [1]. Therefore, a good thermal management system is required to ensure safe operation of a lithium-ion battery system. The construction of a thermal management system calls for information about heat generated during the operation of the battery system. It was shown in Ref. [2] that the entropy change can account for more than 50% of the total heat generated (in Joules) at the 1 C-rate discharge, which is significant for the battery system operation in EVs, where the typical discharge current is from 0.125 to 0.25 C-rate [3].
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In addition, the entropy change has to be taken into account in plug-in hybrid electrical vehicles, where batteries are typically discharged with a 0.25 C-rate current [3]. Therefore, information about the entropy change is important and should be taken into consideration in the thermal management system. This prevents the system overheating and enhances the reliability and safety of the system. The measurements of DS can be carried out by calorimetric [4,5] or potentiometric [3,6e10] methods. The calorimetric method requires an expensive calorimeter for the measurement of heat generation in a Li-ion battery. Application of the potentiometric method is typically less expensive than the calorimetric method and requires an option to adjust the temperature of the Li-ion battery, which can be done for example by using an environmental chamber. However, the relaxation time of the Li-ion battery strongly affects the measurement results in the potentiometric method. As it was shown in Ref. [8], the entropy profile measurements require 10e30 h per a single state of charge (SoC). The influence of the cell relaxation time can be taken into account as it was done in Ref. [8], where the effective approach to speed up the DS profile measurement was proposed to investigate the Li-ion battery aging and to determine its state of health (SoH). However, also in Ref. [8], 2.5 h was required to measure DS for a single value of the SoC. In this paper, a method for the determination of the entropy change profile is proposed, which is based on heat flux measurements on the surface of a cylindrical Li-ion cell. The cylindrical cell was chosen for two reasons: it is one of the most common cell types, and the heat generation in a cylindrical cell is usually lower than in pouch or prismatic cells. The low heat generation allows to assume that if the heat generation can be estimated with a low uncertainty by heat flux measurements in the cylindrical cell, it could also be estimated in pouch or prismatic cells with an acceptable uncertainty. However, in the case of other cell types than the cylindrical ones, a special thermal model should be established to calculate the heat generation based on the information of temperature and heat dissipation from the cell surface. The proposed method for the cylindrical cell is similar to the calorimetric method; however, it does not require an expensive calorimeter for the measurements of heat generation. The proposed method allows simultaneous measurements of the thermal diffusivity and DS of the cylindrical cell. By using this method, it is possible to reduce the time of the experiment to only 80 min per a single value of the SoC. The quality of the DS measurements was analyzed by comparing the values of DS obtained by the heat flux sensor with the values of DS measured by the potentiometric method. In addition, the limitations of the DS profile determination method and the potentiometric method were analyzed based on the proposed cylindrical cell thermal model.
2. Entropy measurement by the potentiometric method 2.1. Test setup and procedure At a constant temperature T (K) and pressure, the following dependence of the enthalpy H (J mol1) on the Gibbs free energy G (J mol1), the entropy change DS (J mol1 K1), and the temperature can be given [5,6]:
DH ¼ DG þ T$DS;
(1)
DG ¼ n$Uocv $F;
(2)
DS ¼ n$F$
vUocv ; vT
(3)
where F is the Faraday constant (F ¼ 96485 C mol1), n is the
stoichiometric number of electrons participating in the cell reaction (for a Li-ion cell, n ¼ 1), and Uocv is the open-circuit voltage of the cell (V). In this work, DS of a 2.3 A h A123 Systems cylindrical cell (ANR26650M1A) was estimated by the potentiometric method applying Equation (3). The parameters of the cylindrical cell were measured and are given in Table 1. The values of the theoretical heat capacity and thermal conductivity in the radial direction were assumed, as it was done in Refs. [11,12], to determine the limitations of the DS measurements. The cylindrical cell was placed in an environmental chamber (Jeio Thech TH-KE-025), where the temperature and humidity of air can be controlled. The terminal voltage of the cell was measured by a 34420A nanoVolt/micro-Ohm meter. The Potentiostat/Galvanostat Gamry Reference 3000 was used for charging and discharging. A PT100 temperature sensor and a Keithley 2701 bench digital multimeter were used to measure the temperature on the cell surface. The step of the DS measurements was chosen to be 10% of the SoC as a compromise between the number of measured points and the time required to perform the measurements. The DS measurement process consisted of three steps. In the first step, the test preparation was carried out. The available capacity of the cylindrical cell was measured by discharging the cell with a 3.0 A constant current in order to determine the discharge capacity of the cell at the operation current under study. The 3.0 A constant current, that is the maximum operation current of the Potentiostat/ Galvanostat Gamry Reference 3000, was chosen for the cell discharge to reduce the duration of the test. Then, the cell was fully charged according to the manufacturers recommendations [13]. In the second step, the cylindrical cell was discharged with a 3.0 A current for 264 s, and the relaxation period of the cylindrical cell after charging and discharging was 12 h. The internal pressure of the cell was assumed to be constant during charging and discharging. The relaxation time was chosen based on [8], where it was found that the entropy profile measurements require between 10 and 30 h per SoC. The charging, discharging, and relaxation of the cylindrical cell took place at a room temperature of about 24 C. At the end of the second step, the cell was placed in an environmental chamber. Then, the measurements of DS were performed for the considered value of the SoC. The temperature in the chamber was set to 30 C. When the temperature in the cylindrical cell became constant and equal to the air temperature in the chamber, the OCV of the cell was measured. The profile of the temperature changes in the environmental chamber during the DS measurements was similar to that in Ref. [8]. The OCV measurements were made at the temperatures of 30 C, 25 C, 20 C, and 30 C, in this order. In addition, to compensate for the influence of the cell relaxation process on the entropy measurements, a similar method as in Ref. [8] was applied. The change in the OCV, caused by the relaxation process in the cylindrical cell during the DS measurements, was subtracted from the measured data. 2.2. Thermal transient processes in the cylindrical cell The measurements of the OCV after changing the air temperature inside the environmental chamber should be made only after the attenuation of all thermal transient processes in the cell when the temperature in all parts of the cell is equal to the air temperature in the chamber. The time after which the measurements can be carried out was estimated based on the thermal model of the cylindrical cell, which was constructed based on the thermal model of an infinitely long cylinder, presented in Ref. [14]. To simplify the thermal model of the cylindrical cell, ceramic fiber wool was used to provide the thermal isolation of the ends of the cylindrical cell. In this case, the cylindrical cell can be
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Table 1 Parameters of the cylindrical cell [11,12]. Parameter
Value
Parameter
Value
Diameter, (m) Height, (m) Mass, (g)
0.02585 0.06515 70
Theoretical heat capacity, (J g1 K1) Thermal conductivity in radial direction, (W m1 K1) Density, (kg m3)
0.89 0.4 2.0473,106
considered as a cylinder with an infinite length (Fig. 1), and the heat dissipation from the cylinder ends can be neglected. The boundary conditions for the cylindrical cell with thermally isolated cell ends are shown in Fig. 1. The initial temperature of the cell is T0 (K), and the temperature of the surrounding air is Tamb (K). The radius of the cell is r (m), k is the thermal conductivity of the cell (W m1 K1), and h is the convection heat transfer coefficient (W m2 K1). The temperature in the middle of the cell T (K) after a period of time t (s) can be calculated as in Ref. [14]:
T ¼ T0 þ Q$ðTamb T0 Þ;
(4)
where Q is the temperature coefficient, which is calculated as
Q¼1
∞ X
An $J0 ð0Þ$ m2n $Fo ;
(5)
n¼1
where J0 is the zero-order Bassel function, Fo is the thermal Fourier number, and mn and An are auxiliary variables. The thermal Fourier number can be calculated in a similar way as in Refs. [15,16].
Fo ¼
a$t r2
;
(6)
where a is the thermal diffusivity (m2 s1), which is obtained from
a¼
k
r$cth
;
(7)
where cth is the specific heat capacity (J kg1 K1) and r is the cell average density (kg m3). The auxiliary variables are written as
An ¼
2$Bi
m2n þ Bi2 $J0 ðmn Þ
mn ¼
;
Bi$J0 ðmn Þ ; J1 ðmn Þ
(8)
(9)
where J1 is the first-order Bassel function, and Bi is the Biot number, which is calculated as
Bi ¼
h$r : k
(10)
In the calculation of the convection heat transfer coefficient, the airflow, which is produced in the environmental chamber by a 60 W Sirocco fan, was taken into account. A 60 W Sirocco fan produces an airflow of 13 m s1 [17]. The convection heat transfer coefficient h was calculated as in Ref. [18], and it is 16.5 (W m2 K1). The Biot number is calculated for the obtained convection heat transfer coefficient, and it is equal to 0.53. According to Equation (4), the temperature in the middle of the cell equals the temperature in the chamber Tamb when the temperature coefficient is equal to one. The temperature coefficient dependences on the thermal Fourier number and the Biot number are shown in Fig. 2. As it can be seen in Fig. 2, the thermal Fourier number is approximately equal to six for the required temperature coefficient and Biot number. Therefore, the minimum period of time after which the measurements of the OCV of the cylindrical cell can be made in the environmental chamber can be calculated from Equation (6). It is equal to 76 min. 3. Entropy measurements by a heat flux sensor The entropy change of the cell can be calculated based on the
Fig. 1. Schematic representation of the cylindrical cell (a) and its representation as an infinitely long cylinder (b).
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information about heat generation during the operation of the cell. The heat generation Q_ (W) in the Li-ion battery can be calculated based on the energy balance presented by Bernardi et al. in Ref. [19].
vUocv þ Q_ mix þ Q_ pch ; Q_ ¼ I$ðUtab Uocv Þ þ I$T$ vT
(11)
where I is the cell current (A), which is positive during charging and negative during discharging, and Utab is the cell terminal voltage (V), Uocv is the OCV (V), T is the temperature (K), Q_ mix is the heat generation caused by the mixing effect (W), and Q_ pch is the heat generation caused by the phase change effect (W). Equations of the mixing and phase change terms can be found in Ref. [19]. Assuming that the reversible part of the heat generation is unchanged during charging and discharging, it can be obtained as
Q_ Q_ ch : Q_ rev ¼ dis 2
(12)
where Q_ dis is the heat generation during discharging (W), and Q_ ch is the heat generation during charging (W). By neglecting the pressure change during charging and discharging, the entropy change can be estimated as
DS ¼
n$F$Q_ rev : I$T
(13)
where F is the Faraday constant (F ¼ 96485 C mol1), and n is the stoichiometric number of electrons participating in the cell reaction (for a lithium-ion cell, n ¼ 1). The heat generation during charging and discharging that is required for the calculation of DS can be measured in a calorimeter. However, a calorimeter is usually a complex and expensive device. Therefore, in this work, a gradient heat flux sensor (GHFS), made from bismuth single crystal, was chosen as a small and inexpensive sensor by which the heat generation in the Li-ion battery can be measured. Detailed information about the GHFS and its calibration is given in Refs. [20e22]. 3.1. Measurements of heat generation by the GHFS The GHFS and the temperature sensor PT100 were installed on the cylindrical cell surface as it is shown in Fig. 3(a). In addition, the surface of the cylindrical cell was covered with rubber; the thickness of the rubber band is 1 mm. This was done in order to consider the conduction heat transfer from the cylindrical cell surface instead of the convection heat transfer. In addition, the
configuration reduces the influence of the airflow and the influence of changes in the ambient temperature in the laboratory. Ceramic fiber wool was used for thermal isolation of the ends of the cylindrical cell. A Potentiostat/Galvanostat Gamry Reference 3000 was used for the cell charging and discharging. A 34420A nanoVolt/micro-Ohm meter was applied to measure the voltage signal from the GHFS. A PT100 temperature sensor and a Keithley 2701 bench digital multimeter were used to measure the temperature of the cylindrical cell surface. At the beginning of the test, the cylindrical cell was fully charged according to the manufacturers recommendations [13]. Then, it was discharged by 3.0 A current impulses. The duration of a pulse required to discharge the cell by 10% of the SoC is 264 s. The 3.0 A current pulses were used because of the limitation of the Potentiostat/Galvanostat Gamry Reference 3000. After discharging, the cylindrical cell was charged by similar but opposite current pulses with the 3.0 A current as during discharging. After each impulse, the cell relaxation period was 30 min. During the relaxation process, the thermal diffusivity of the cylindrical cell was measured. The requirement for the duration of the relaxation period is based on the limitations of the proposed method, which are given below. The heat generated in the cylindrical cell during charging and discharging is calculated by applying the superposition principle. Two systems are considered. In the first system, we assume that heat is generated; however, all surfaces of the cylindrical cell are fully isolated and the heat dissipation is equal to zero. In this case, the temperature on the cell surface T1 can be given as
T1 ¼
1 $ cth $r$V
Ztend
Q_ ðtÞdt;
(14)
t0
where V is the volume of the cylindrical cell (m3). In the second system, the generated heat is equal to zero, but there is a heat flux from the cell surface. In this case, if the heat flux from the cell surface is constant, the temperature on the cylindrical cell surface can be calculated by the method presented in Ref. [14]. However, in the case under consideration, a variable heat flux is present. Therefore, the measured heat flux curve is divided into N time intervals tn, where a constant value of heat flux was considered. The differences between the values of the heat generated in each interval are obtained from
q_ n ¼ qn qn1 ;
(15)
where n is the number of the intervals n ¼ 1…N and q is the average value of the heat flux from the cylindrical cell surface in the considered interval (W m2). The temperature on the cell surface T2 in the case under study can be determined as
r T2 ¼ T0 $Q S; k
(16)
where T0 is the initial temperature of the cell (K), S is the N1 matrix, which includes the values of q_ n , and Q is the 1N matrix that includes the values of temperature coefficients, which are written as
1 4
Q ¼ 2$Fo þ
∞ X 2 An $J0 mn $exp mn $Fo ;
Fig. 2. Dependence of the temperature coefficient on the thermal Fourier number with the Biot number as a parameter.
(17)
n¼1
where Q is the temperature coefficient and An and mn are auxiliary variables. An is calculated as
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65
Fig. 3. Cylindrical cell with isolation, a GHFS, and a PT100 sensor (a) and its representation as a cylinder of infinite length (b).
An ¼
2
2 ;
(18)
mn $J0 m
1 4
and the auxiliary variable mn is obtained from the equation
Q ¼ 2$Fo þ :
J0 m
¼ 0:
(19)
The thermal Fourier number, which is required in Equation (17), is calculated as
Fo ¼
a$tn r2
;
(20)
where tn is the length of the time interval n ¼ 1…N, into which the heat flux curve is divided. The real value of the temperature on the surface of the cylindrical cell is obtained from
T ¼ T1 þ T2 :
(21)
By substituting Equation (21) into Equations (7), (14) and (15), the following equation can be given
Ztend t0
V Q_ ðtÞdt ¼ $ðk$ðT T0 Þ þ r$Q SÞ:
a
surface is calculated by Equation (16), and the elements of the matrix S by Equation (15). By assuming a linear temperature gradient in the cylindrical cell, Equation (17) can be simplified and the elements of the matrix Q are written as
(22)
The value of the heat generated during charging and discharging in any point in the time interval from t0 to tend can be obtained by numerical integration of Equation (22). When the values of the heat generation during charging and discharging are obtained, DS for each point of the SoC is calculated by Equations (12) and (13). 3.2. Measurements of the thermal diffusivity of the cylindrical cell Thermal diffusivity is measured during the relaxation period after discharging and charging of the cylindrical cell, when the temperature of the cylindrical cell decreases. In this case, the heat generation in the cell is equal to zero, and heat is dissipated by conduction and convection. The temperature of the cylindrical cell
(23)
By substituting Equation (20), and (23) into Equation (16), the thermal diffusivity of the cylindrical cell during cooling can be obtained by 2
r$k$ðT0 T2 Þ r4 $ a¼ 2$t S
PN
n¼1
q_ n
;
(24)
where t is the 1N matrix, which includes the values of the time intervals tn, S is the N 1 matrix, which includes the values of q_ n , and T2 is the temperature on the cylindrical cell surface (K). In order to increase the accuracy of the measurements, the thermal diffusivity for each point of the SoC is calculated as an average value between the values of thermal diffusivity estimated during discharging and charging for the same value of the SoC. The estimated values of thermal diffusivity for each point of the SoC are used in Equation (22) for the estimation of heat generation during charging and discharging in the same point of the SoC. 4. Limitations and uncertainty 4.1. Limitations The following requirements and limitations should be considered before performing the experiments and calculating the required parameters: The ends of the cylindrical lithium-ion cell should be fully isolated in order to stop the heat dissipation from these parts of the cylindrical cell. In this case, the cylindrical cell can be considered as a cylinder with an infinite length, and thus, the methods described in this paper can be applied. The value of thermal conductivity should be known. In this paper, the value of the thermal conductivity was taken to be the same as in Ref. [11].
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The calculation of the thermal diffusivity of the cell should be carried out at a linear temperature gradient in the cylindrical cell. The linear temperature gradient is required to simplify the calculation of the thermal diffusivity of the cylindrical cell. The minimum time of the experiment for the determination of the thermal diffusivity, after which equation (24) can be used, can be calculated by
tmin ¼
Fomin $r 2
a
;
(25)
where Fomin is the minimum thermal Fourier number at which the thermal gradient in the cylindrical cell becomes linear. The assumed parameters of thermal conductivity, heat capacity, and density, which were given in Table 1, were used for the preliminary estimation of the thermal diffusivity. The value of Fomin is obtained by analyzing the temperature gradient in the cylindrical cell during cooling of the cell, when the heat generation in the cylindrical cell is equal to zero and the heat flux from the cell surface, caused by convection and conduction, is known. In this case, the temperate gradient can be written as [14].
PN vT q_ ¼ VG$ n¼1 n ; vr k
(26)
where VG is the dimensionless thermal gradient coefficient. By neglecting the change in the thermal parameter values of the cylindrical cell, which are caused by the temperature gradient, we can assume that the form of the temperature gradient depends only on the temperature gradient coefficient, which can be calculated as in Ref. [14].
VG ¼ h þ
∞ X
2 Cn $J1 mn $h $exp mn $Fo ;
(27)
n¼1
where h is the dimensionless thickness coefficient, and Cn is the auxiliary variable. The auxiliary variable mn is calculated by Equation (19). The dimensionless thickness coefficient and the auxiliary variable Cn are calculated as in Ref. [14].
y r
h¼ ; Cn ¼
(28) 2
mn $J0 mn
:
(29)
The analysis of the dependence of the temperature gradient coefficient on the thermal Fourier number and the dimensionless thickness coefficient is shown in Fig. 4. As it can be seen in Fig. 4, the temperature gradient coefficient has a linear dependence on the dimensionless thickness coefficient if Fo 0.4. Therefore, the minimum value of the thermal Fourier number at which the temperature gradient in the cylindrical cell is linear and at which the thermal diffusivity of the cell can be calculated is equal to 0.4.
4.2. Uncertainty The uncertainty was calculated according to the standard EA-4/ 02 M: 2013 [23]. The standard uncertainty for the single measured value is calculated as
Fig. 4. Dependence of the thermal gradient coefficient on Fo and the dimensionless thickness coefficient.
u2 ðyÞ ¼
N X vf i¼1
vxi
2 $uðxi Þ
;
(30)
where u(y) is the uncertainty of the output estimate y, u(xi) is the uncertainty of the input estimate xi, where i is the number of the input estimates i ¼ 1…N, and f is the function that shows the dependence between the input and the output [16]. The uncertainty of the DS determination and the thermal diffusivity of the cylindrical cell depend on the uncertainty of the temperature sensor, heat flux sensor, and uncertainties in the measurement of time, current, voltage, and dimensions. The uncertainties of the input estimates were calculated as in Ref. [23] for the required parameters or were taken from the equipment documentation provided by the manufacturer. If the manufacturer does not give the uncertainty of the equipment, the uncertainty is calculated as a zero-order uncertainty, which is equal to one-half of the equipment resolution. The values of uncertainty for the applied equipment are shown in Table 2.
5. Results and discussion The evaluation of the time required for the DS measurements by the potentiometric method was carried out by analyzing the OCV change, caused by the relaxation processes in the cell after charging and discharging. In addition, the influence of the ambient temperature on the OCV change during the DS measurements by the potentiometric method was estimated by measuring the OCV changes first at 20 C and then at 30 C. The OCV changes at the values of temperature in the chamber after more than 12 h of the relaxation period are shown in Fig. 5 at each 10% of the SoC. High changes of the OCV are observed close to 0% of the SoC and 100% of the SoC. The negative value of the OCV change at 100% is caused by a sequence of tests where the cylindrical cell was first charged up to 100% of the SoC. The other measured points of the SoC were obtained by discharging of the cell. In addition, as it can be seen in Fig. 5, the OCV change increases when the SoC decreases, which was also noticed in Ref. [8]. This phenomenon can be explained by the diffusivity of the Liþ ions, as it was done in Ref. [8]. The diffusivity of the Liþ ions in graphite increases with the decreasing SoC [24], which may lead to increased OCV changes during the relaxation period. Because of the high OCV changes even after 12 h of the relaxation period, the influence of the cell relaxation, which causes changes in the OCV during the DS measurements by the potentiometric method, should be taken into account
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Table 2 Uncertainty of the input estimates. Input estimates
Uncertainties
Time, (s) Voltage (34420A nanoVolt/micro-Ohm meter), (nV) Temperature (4-Wires RTD Keithley 2701), (+ C) Voltage (Potentiostat/Galvanostat Gamry Reference 3000), (mV) Current (Potentiostat/Galvanostat Gamry Reference 3000), (mA) Dimensions (micrometer), (m) Mass, (g) Heat flux sensor sensitivity, (V W1) Temperature (Chamber Jeio Tech TH-KE), (+ C) Humidity (Chamber Jeio Tech TH-KE), (%)
0.5 30 0.005 0.57 2 0.000005 0.05 6.04,105 [20] 0.05 0.05
Fig. 5. OCV change per minute at 20 C and 30 C of the temperatures in the chamber.
especially for those values of the SoC that are close to 0% and 100%. The dependence of the OCV changes on temperature, which can be seen in Fig. 5, may be explained by considering that the diffusivity of the Liþ ions increases when temperature increases as it was shown in Ref. [24]. Therefore, the increasing temperature during a relaxation period leads to increasing OCV changes during this period. A high temperature decreases the relaxation time. However, the speed of the different side reactions increases with an increasing temperature, which affects the changes of the OCV during the relaxation period and increases the uncertainty of the result. A reduction in the temperature during the test decreases the speed of the kinetic processes in the cell and prolongs the relaxation period required to achieve a constant value of the OCV. Therefore, to decrease the uncertainty of the DS measurements by the potentiometric method, the temperature should be kept close to 25 C. This temperature was suggested in Ref. [25] as the optimal operation temperature of the Li-ion cell. In this case, it may be possible to find a compromise between the reaction speed required to achieve a constant value of the OCV and the speed of unwanted side reactions, which increase the uncertainty of the DS measurements. The calculated time at which the temperature of the cylindrical cell will be equal to the temperature in an environmental chamber was verified by placing the GHFS on the cylindrical cell surface. Heat dissipation from the cell surface was measured after changing the temperature in the environmental camber. The heat flux from the cylindrical cell surface is shown in Fig. 6 for 50% of the SoC and when changing the chamber temperature from 30 C to 25 C. The heat dissipation is 0 W m2 when the temperature of the cylindrical cell is equal to the temperature in the environmental chamber. According to the measurement results, the thermal transient processes in the cylindrical cell under study stop after 76 min, which is in compliance with the previous estimated value
by using Equation (25) Based on the analysis of the OCV changes caused by the relaxation processes in the cell and the time at which the cell temperature is equal to the temperature inside the chamber, the time required for the DS measurements was calculated. The required time to measure the DS values for a single point of the SoC for the cylindrical cell is about 17.1 h if the relaxation period of the cell after charging or discharging is 12 h. This value was obtained assuming that the temperature in the environmental chamber is changed four times (30 C, 25 C, 20 C, and 30 C), and 76 min is required for each value of temperature, while the cell temperature is equal to the temperature inside the chamber. If the relaxation period of 1.5 h is applied, as it was suggested in Ref. [8], it will take about 6.6 h to measure DS for a single value of the SoC. The calculated value of the uncertainty of the DS determination in the environmental chamber Jeio Tech TH-KE at 50% of the SoC is equal to 0.2 J mol1 K1 which is 2% of the measured DS value at the SoC under consideration, which is equal to 11.73 J mol1 K1. By using the GHFS, the time required for the simultaneous determination of DS and thermal diffusivity in a single point of the SoC is 80 min. The time required to measure DS in nine points of the SoC is about 12 h. The time required for the test can be shorter if the heat capacity and thermal conductivity of the cell are known. With the GHFS, the estimated value of the uncertainty of the DS determination at 50% of the SoC was 2.38 J mol1 K1, which is 20% of the measured DS value. The uncertainty of the thermal diffusivity determination at 50% of the SoC is 0.2 mm2 s1, of the measured thermal diffusivity value. The values of the thermal diffusivity, measured by using the GHFS, are given in Table 3 for each point of the SoC. As it can be seen from Table 3, the thermal diffusivity of the cylindrical cell has a small dependence on the SoC. The dependence of the thermal diffusivity on the SoC may be explained by the dependence of the heat capacity and the thermal conductivity on the SoC and temperature, which were shown and explained in
Fig. 6. Heat flux density from the cylindrical cell surface.
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Refs. [26,27]. The measured values of the heat flux from the cylindrical cell surface and the temperatures of the cell surface that were measured during the DS determination by the GHFS are shown in Fig. 7. Based on the proposed method, DS was calculated from the measured data of the cylindrical cell during discharging and charging by 3.0 A current impulses. A comparison between the obtained values of DS by using the potentiometric method and by using the GHFS is shown in Fig. 8. The influence of the entropy change can easily be observed in Fig. 7 even with a more than 1 C-rate operation current. During discharge, the lower heat dissipation and temperature changes can be seen in the middle of the discharge test, which corresponds to the SoC from 35% to 75%. In addition, high heat dissipation and temperature changes can be observed at the end of the discharge test, which corresponds to the SoC from 0% to 10%. An opposite situation can be observed during that part of the test when the cylindrical cell was charged. Such behavior is explained by different heat generation during charging and discharging, which was caused by variation of DS with the SoC shown in Fig. 8. The DS profile obtained by the potentiometric method is similar to these results, which were shown in Ref. [11]. The variation of DS with SoC can be explained by the DS profiles of the separate electrodes under study, which were explained in Ref. [3]. Based on the data presented in Ref. [3], the variation of DS with the SoC of the cell is mostly caused by the variation of DS with the SoC on the negative electrode, which is made of graphite. The variation of DS with the SoC in the positive electrode, which is made of lithium iron phosphate, is significant only at the SoCs close to 0% or 100%. The comparison of the data measured by the potentiometric method and the data obtained by the GHFS shows an opportunity of entropy change measurements by the GHFS. The reason for the deviation of the results obtained by both reported methods may be the internal pressure increases during battery charging and discharging, which was neglected in this article for the simplification of the DS determination. The uncertainty of the proposed method strongly depends on the heat generation in the cylindrical cell during the DS determination. In the case of a small cylindrical Li-ion cell, where the heat generation and the resulting temperature changes are small, the uncertainty of the proposed method is higher than the uncertainty of the potentiometric method. However, even for a cylindrical cell, which was discharged with the 1.3 C-rate current, DS can be measured with an acceptable uncertainty. Taking into account the decrease in the time required for the DS determination, which is up to 5 h per a single point of the SoC, the use of the GHFS to determine DS of a cylindrical cell is recommended. 6. Conclusions In the paper, determination of DS by using a gradient heat flux sensor (GHFS) was considered. The proposed method allows to significantly decrease the time required for the measurement of the DS values. In addition, the method does not require large and expensive devices such as an environmental chamber or a calorimeter. The results of the DS measurements were compared with
Table 3 Thermal diffusivity of the cylindrical cell for different values of the SoC. SoC, %
a, mm2 s1
SoC, %
a, mm2 s1
SoC, %
a, mm2 s1
90 80 70
0.226 ± 0.0022 0.205 ± 0.0021 0.188 ± 0.0019
60 50 40
0.201 ± 0.0020 0.195 ± 0.0020 0.197 ± 0.0020
30 20 10
0.198 ± 0.0020 0.200 ± 0.0020 0.195 ± 0.0020
Fig. 7. Measured values of heat flux and temperatures on the cylindrical cell surface.
Fig. 8. Entropy change measured by the potentiometric method and by using the GHFS.
the values of DS that were measured by the potentiometric method in an environmental chamber. In order to decrease the uncertainty of the DS measurements in the environmental chamber, the limitations on the required relaxation period after charging and discharging of the cell and on the temperature in the environmental chamber were analyzed. It was shown that the OCV changes caused by the relaxation of the cell after charging or discharging should be taken into account even after 12 h of the relaxation period. In addition, the dependence of the OCV changes on the chamber temperature during DS measurements was shown. Moreover, by applying the presented method, an analysis was made of the thermal transient process in a cylindrical cell. It was shown that damping of the thermal transient processes in the cell after a change in the temperature inside the environmental chamber is reached after 76 min, which was verified experimentally. A comparison of the DS values obtained by the potentiometric method in the environmental chamber and by the GHFS shows an opportunity to use the proposed method for the DS determination. Even though the uncertainty of the method is higher than in the potentiometric method, a major advantage of the method is the significant decrease in the experiment time (down to 80 min per a single value of the SoC from the minimum of 6.6 h required for the potentiometric method, where the environmental chamber is used). Therefore, the application of the GHFS is recommended for the DS determination in the case of the cylindrical Li-ion cell.
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References [1] K. Onda, T. Ohshima, M. Nakayama, K. Fukuda, T. Araki, Thermal behavior of small lithium-ion battery during rapid charge and discharge cycles, J. Power Sources 158 (1) (2006) 535e542. [2] J. Hong, H. Maleki, S. Al Hallaj, L. Redey, J.R. Selman, Electrochemical-calorimetric studies of LithiumIon cells, J. Electrochem. Soc. 145 (5) (1998) 1489e1501. [3] V.V. Viswanathan, D. Choi, D. Wang, W. Xu, S. Towne, R.E. Williford, J.G. Zhang, J. Liu, Z. Yang, Effect of entropy change of lithium intercalation in cathodes and anodes on Li-ion battery thermal management, J. Power Sources 195 (11) (2010) 3720e3729. [4] A. Eddahech, O. Briat, J.-M. Vinassa, Thermal characterization of a high-power lithium-ion battery: potentiometric and calorimetric measurement of entropy changes, Energy 61 (2013) 432e439. [5] Q. Huang, M. Yan, Z. Jiang, Thermal study on single electrodes in lithium-ion battery, J. Power Sources 156 (2) (2006) 541e546. [6] K. Jalkanen, T. Aho, K. Vuorilehto, Entropy change effects on the thermal behavior of a LiFePO4/graphite lithium-ion cell at different states of charge, J. Power Sources 243 (2013) 354e360. [7] R.E. Williford, V.V. Viswanathan, J.-G. Zhang, Effects of entropy changes in anodes and cathodes on the thermal behavior of lithium ion batteries, J. Power Sources 189 (1) (2009) 101e107. [8] P.J. Osswald, M. del Rosario, J. Garche, A. Jossen, H.E. Hoster, Fast and accurate measurement of entropy profiles of commercial lithium-ion cells, Electrochimica Acta 177 (2015) 270e276. [9] K. Takano, Y. Saito, K. Kanari, K. Nozaki, K. Kato, A. Negishi, T. Kato, Entropy change in lithium ion cells on charge and discharge, J. Appl. Electrochem. 32 (3) (2002) 251e258. [10] K. Maher, R. Yazami, Effect of overcharge on entropy and enthalpy of lithiumion batteries, Electrochimica Acta 101 (2013) 71e78. [11] M. Rad, D. Danilov, M. Baghalha, M. Kazemeini, P. Notten, Thermal modeling of cylindrical LiFePO4 batteries, J. Mod. Phys. 4 (7b) (2013) 1e7. [12] H. Perez, J. Siegel, X. Lin, A. Stefanopoulou, Y. Ding, M. Castanier, Parameterization and validation of an integrated electro-thermal cylindrical LPF battery model, in: ASME 2012 5th Annual Dynamic Systems and Control Conference Joint with the JSME 2012 11th Motion and Vibration Conference, DSCC 2012MOVIC 3, 2012, pp. 41e50. [13] Akukeskus, A123Systems Lithium-ion Rechargeable ANR26650M1A Cell, 2016
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(accessed 08.08.16), http://www.akukeskus.ee/anr26650m1a0_datasheet_ april_2009.pdf. A. Pekhovich, V. Zhidkikh, Calculation of the Solids Thermal Condition, second ed., Energy, Leningrad, 1976. J.H. Lienhard IV, J.H. Lienhard V, A Heat Transfer Textbook, third ed., Phlogiston Press, Cambridge, MA, U.S.A, 2008. €nen, V. Mityakov, S. Sapozhnikov, Thermal K. Murashko, A. Mityakov, J. Pyrho parameters determination of battery cells by local heat flux measurements, J. Power Sources 271 (2014) 48e54. Brills marketing corporation, http://www.brillsph.com/vector-sirocco-fan. html, 2016 (accessed 09.06.16). F.P. Incropera, D.P. DeWitt, T.L. Bergman, A.S. Lavine, Fundamentals of Heat and Mass Transfer, sixth ed., John Wiley & Sons Inc, New York, 2007. D. Bernardi, E. Pawlikowski, J. Newman, A general energy balance for battery systems, J. Electrochem. Soc. 132 (1) (1985) 5e12, http://dx.doi.org/10.1149/ 1.2113792. A.V. Mityakov, S.Z. Sapozhnikov, V.Y. Mityakov, A.A. Snarskii, M.I. Zhenirovsky, €nen, Gradient heat flux sensors for high temperature environments, J.J. Pyrho Sensors Actuators A Phys. 176 (2012) 1e9. € nen, Local H.K. Jussila, A.V. Mityakov, S.Z. Sapozhnikov, V.Y. Mityakov, J. Pyrho heat flux measurement in a permanent magnet motor at No load, IEEE Trans. Ind. Electron. 60 (11) (2013) 4852e4860. S.Z. Sapozhnikov, V.Y. Mityakov, A.V. Mityakov, A.I. Pokhodun, N.A. Sokolov, M.S. Matveev, The calibration of gradient heat flux sensors, Meas. Tech. 54 (10) (2012) 1155e1159. EA-4/02 M: 2013, http://www.european-accreditation.org/publications, 2016 (accessed 10.06.16). P. Yu, B.N. Popov, J.A. Ritter, R.E. White, Determination of the lithium ion diffusion coefficient in graphite, J. Electrochem. Soc. 146 (1) (1999) 8e14. T. Waldmann, M. Wilka, M. Kasper, M. Fleischhammer, M. Wohlfahrt-Mehrens, Temperature dependent ageing mechanisms in lithium-ion batteries a post-mortem study, J. Power Sources 262 (2014) 129e135. H. Maleki, S.A. Hallaj, J.R. Selman, R.B. Dinwiddie, H. Wang, Thermal properties of lithium-ion battery and components, J. Electrochem. Soc. 146 (3) (1999) 947e954. S.J. Bazinski, X. Wang, Experimental study on the influence of temperature and state-of-charge on the thermophysical properties of an LFP pouch cell, J. Power Sources 293 (2015) 283e291.
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Journal of Power Sources journal homepage: www.elsevier.com/locate/jpowsour
Determination of the entropy change profile of a cylindrical lithium-ion battery by heat flux measurements K.A. Murashko a, *, A.V. Mityakov a, V.Y. Mityakov b, S.Z. Sapozhnikov b, J. Jokiniemi c, € nen a J. Pyrho a b c
Lappeenranta University of Technology LUT, Skinnarilankatu 34, 53850 Lappeenranta, Finland Saint-Petersburg State Polytechnical University, Polytechnicheskaya, 29, 195251 Saint-Petersburg, Russia University of Eastern Finland, Kuopio Campus, P.O. Box 1627, FI-70211 Kuopio, Finland
h i g h l i g h t s DS measurements by using a gradient heat flux sensor are presented. Time required for the DS measurements by the potentiometric method is evaluated. Uncertainty and limitations of the methods under consideration are given. Comparison of the proposed method with the potentiometric method is made.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 June 2016 Received in revised form 9 August 2016 Accepted 30 August 2016
The popularity of lithium-ion (Li-ion) batteries has increased over the recent years. Because of the strong dependence of the Li-ion battery operation characteristics on temperature, heat generation in the battery has to be taken into account. The entropy change of a Li-ion battery has a significant influence on heat generation, especially at a low C-rate current. Therefore, it is necessary to consider the entropy change profile in the estimation of heat generation. In the paper, a method to determine the entropy change (DS) profile by heat flux measurements of a cylindrical Li-ion cell is proposed. The method allows simultaneous measurements of the thermal diffusivity and DS of the cylindrical cell. The thermal diffusivity and DS measurements are carried out by a gradient heat flux sensor (GHFS). The comparison between the DS profile determined by the GHFS method with that obtained using a standard potentiometric method clearly shows that the entropy change measurements could be made by using a GHFS. Even though the uncertainty of the reported method is higher than that of the potentiometric method, a significant decrease in the experiment time compared with the potentiometric method is a major advantage of this method. © 2016 Elsevier B.V. All rights reserved.
Keywords: Li-ion battery Entropy change Thermal diffusivity Heat flux measurements Sensor Thermal model
1. Introduction The increasing popularity of electric and hybrid vehicles over the recent years is caused by a number of factors, including an increase in CO2 emissions and high energy costs. An energy storage device is one of the main parts in electric vehicles (EVs). Nowadays, because of their large number of usable operation cycles and low self-discharge rate, lithium-ion (Li-ion) batteries are considered the most promising technology for the energy storage in EVs. Despite the advantages of the Li-ion batteries, they also have certain disadvantages such as the high sensitivity to overcharge and
* Corresponding author. E-mail address: kirill.murashko@lut.fi (K.A. Murashko). http://dx.doi.org/10.1016/j.jpowsour.2016.08.130 0378-7753/© 2016 Elsevier B.V. All rights reserved.
overdischarge and the risk of a thermal runaway, which can even be a reason of fire. Therefore, a battery management system and a thermal management system are required in the operation of Li-ion batteries. An appropriate thermal management system becomes more important when the surface area per volume ratio of a battery decreases with the increasing battery size, resulting in a lower heat transfer rate per unit rate of heat generation [1]. Therefore, a good thermal management system is required to ensure safe operation of a lithium-ion battery system. The construction of a thermal management system calls for information about heat generated during the operation of the battery system. It was shown in Ref. [2] that the entropy change can account for more than 50% of the total heat generated (in Joules) at the 1 C-rate discharge, which is significant for the battery system operation in EVs, where the typical discharge current is from 0.125 to 0.25 C-rate [3].
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In addition, the entropy change has to be taken into account in plug-in hybrid electrical vehicles, where batteries are typically discharged with a 0.25 C-rate current [3]. Therefore, information about the entropy change is important and should be taken into consideration in the thermal management system. This prevents the system overheating and enhances the reliability and safety of the system. The measurements of DS can be carried out by calorimetric [4,5] or potentiometric [3,6e10] methods. The calorimetric method requires an expensive calorimeter for the measurement of heat generation in a Li-ion battery. Application of the potentiometric method is typically less expensive than the calorimetric method and requires an option to adjust the temperature of the Li-ion battery, which can be done for example by using an environmental chamber. However, the relaxation time of the Li-ion battery strongly affects the measurement results in the potentiometric method. As it was shown in Ref. [8], the entropy profile measurements require 10e30 h per a single state of charge (SoC). The influence of the cell relaxation time can be taken into account as it was done in Ref. [8], where the effective approach to speed up the DS profile measurement was proposed to investigate the Li-ion battery aging and to determine its state of health (SoH). However, also in Ref. [8], 2.5 h was required to measure DS for a single value of the SoC. In this paper, a method for the determination of the entropy change profile is proposed, which is based on heat flux measurements on the surface of a cylindrical Li-ion cell. The cylindrical cell was chosen for two reasons: it is one of the most common cell types, and the heat generation in a cylindrical cell is usually lower than in pouch or prismatic cells. The low heat generation allows to assume that if the heat generation can be estimated with a low uncertainty by heat flux measurements in the cylindrical cell, it could also be estimated in pouch or prismatic cells with an acceptable uncertainty. However, in the case of other cell types than the cylindrical ones, a special thermal model should be established to calculate the heat generation based on the information of temperature and heat dissipation from the cell surface. The proposed method for the cylindrical cell is similar to the calorimetric method; however, it does not require an expensive calorimeter for the measurements of heat generation. The proposed method allows simultaneous measurements of the thermal diffusivity and DS of the cylindrical cell. By using this method, it is possible to reduce the time of the experiment to only 80 min per a single value of the SoC. The quality of the DS measurements was analyzed by comparing the values of DS obtained by the heat flux sensor with the values of DS measured by the potentiometric method. In addition, the limitations of the DS profile determination method and the potentiometric method were analyzed based on the proposed cylindrical cell thermal model.
2. Entropy measurement by the potentiometric method 2.1. Test setup and procedure At a constant temperature T (K) and pressure, the following dependence of the enthalpy H (J mol1) on the Gibbs free energy G (J mol1), the entropy change DS (J mol1 K1), and the temperature can be given [5,6]:
DH ¼ DG þ T$DS;
(1)
DG ¼ n$Uocv $F;
(2)
DS ¼ n$F$
vUocv ; vT
(3)
where F is the Faraday constant (F ¼ 96485 C mol1), n is the
stoichiometric number of electrons participating in the cell reaction (for a Li-ion cell, n ¼ 1), and Uocv is the open-circuit voltage of the cell (V). In this work, DS of a 2.3 A h A123 Systems cylindrical cell (ANR26650M1A) was estimated by the potentiometric method applying Equation (3). The parameters of the cylindrical cell were measured and are given in Table 1. The values of the theoretical heat capacity and thermal conductivity in the radial direction were assumed, as it was done in Refs. [11,12], to determine the limitations of the DS measurements. The cylindrical cell was placed in an environmental chamber (Jeio Thech TH-KE-025), where the temperature and humidity of air can be controlled. The terminal voltage of the cell was measured by a 34420A nanoVolt/micro-Ohm meter. The Potentiostat/Galvanostat Gamry Reference 3000 was used for charging and discharging. A PT100 temperature sensor and a Keithley 2701 bench digital multimeter were used to measure the temperature on the cell surface. The step of the DS measurements was chosen to be 10% of the SoC as a compromise between the number of measured points and the time required to perform the measurements. The DS measurement process consisted of three steps. In the first step, the test preparation was carried out. The available capacity of the cylindrical cell was measured by discharging the cell with a 3.0 A constant current in order to determine the discharge capacity of the cell at the operation current under study. The 3.0 A constant current, that is the maximum operation current of the Potentiostat/ Galvanostat Gamry Reference 3000, was chosen for the cell discharge to reduce the duration of the test. Then, the cell was fully charged according to the manufacturers recommendations [13]. In the second step, the cylindrical cell was discharged with a 3.0 A current for 264 s, and the relaxation period of the cylindrical cell after charging and discharging was 12 h. The internal pressure of the cell was assumed to be constant during charging and discharging. The relaxation time was chosen based on [8], where it was found that the entropy profile measurements require between 10 and 30 h per SoC. The charging, discharging, and relaxation of the cylindrical cell took place at a room temperature of about 24 C. At the end of the second step, the cell was placed in an environmental chamber. Then, the measurements of DS were performed for the considered value of the SoC. The temperature in the chamber was set to 30 C. When the temperature in the cylindrical cell became constant and equal to the air temperature in the chamber, the OCV of the cell was measured. The profile of the temperature changes in the environmental chamber during the DS measurements was similar to that in Ref. [8]. The OCV measurements were made at the temperatures of 30 C, 25 C, 20 C, and 30 C, in this order. In addition, to compensate for the influence of the cell relaxation process on the entropy measurements, a similar method as in Ref. [8] was applied. The change in the OCV, caused by the relaxation process in the cylindrical cell during the DS measurements, was subtracted from the measured data. 2.2. Thermal transient processes in the cylindrical cell The measurements of the OCV after changing the air temperature inside the environmental chamber should be made only after the attenuation of all thermal transient processes in the cell when the temperature in all parts of the cell is equal to the air temperature in the chamber. The time after which the measurements can be carried out was estimated based on the thermal model of the cylindrical cell, which was constructed based on the thermal model of an infinitely long cylinder, presented in Ref. [14]. To simplify the thermal model of the cylindrical cell, ceramic fiber wool was used to provide the thermal isolation of the ends of the cylindrical cell. In this case, the cylindrical cell can be
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63
Table 1 Parameters of the cylindrical cell [11,12]. Parameter
Value
Parameter
Value
Diameter, (m) Height, (m) Mass, (g)
0.02585 0.06515 70
Theoretical heat capacity, (J g1 K1) Thermal conductivity in radial direction, (W m1 K1) Density, (kg m3)
0.89 0.4 2.0473,106
considered as a cylinder with an infinite length (Fig. 1), and the heat dissipation from the cylinder ends can be neglected. The boundary conditions for the cylindrical cell with thermally isolated cell ends are shown in Fig. 1. The initial temperature of the cell is T0 (K), and the temperature of the surrounding air is Tamb (K). The radius of the cell is r (m), k is the thermal conductivity of the cell (W m1 K1), and h is the convection heat transfer coefficient (W m2 K1). The temperature in the middle of the cell T (K) after a period of time t (s) can be calculated as in Ref. [14]:
T ¼ T0 þ Q$ðTamb T0 Þ;
(4)
where Q is the temperature coefficient, which is calculated as
Q¼1
∞ X
An $J0 ð0Þ$ m2n $Fo ;
(5)
n¼1
where J0 is the zero-order Bassel function, Fo is the thermal Fourier number, and mn and An are auxiliary variables. The thermal Fourier number can be calculated in a similar way as in Refs. [15,16].
Fo ¼
a$t r2
;
(6)
where a is the thermal diffusivity (m2 s1), which is obtained from
a¼
k
r$cth
;
(7)
where cth is the specific heat capacity (J kg1 K1) and r is the cell average density (kg m3). The auxiliary variables are written as
An ¼
2$Bi
m2n þ Bi2 $J0 ðmn Þ
mn ¼
;
Bi$J0 ðmn Þ ; J1 ðmn Þ
(8)
(9)
where J1 is the first-order Bassel function, and Bi is the Biot number, which is calculated as
Bi ¼
h$r : k
(10)
In the calculation of the convection heat transfer coefficient, the airflow, which is produced in the environmental chamber by a 60 W Sirocco fan, was taken into account. A 60 W Sirocco fan produces an airflow of 13 m s1 [17]. The convection heat transfer coefficient h was calculated as in Ref. [18], and it is 16.5 (W m2 K1). The Biot number is calculated for the obtained convection heat transfer coefficient, and it is equal to 0.53. According to Equation (4), the temperature in the middle of the cell equals the temperature in the chamber Tamb when the temperature coefficient is equal to one. The temperature coefficient dependences on the thermal Fourier number and the Biot number are shown in Fig. 2. As it can be seen in Fig. 2, the thermal Fourier number is approximately equal to six for the required temperature coefficient and Biot number. Therefore, the minimum period of time after which the measurements of the OCV of the cylindrical cell can be made in the environmental chamber can be calculated from Equation (6). It is equal to 76 min. 3. Entropy measurements by a heat flux sensor The entropy change of the cell can be calculated based on the
Fig. 1. Schematic representation of the cylindrical cell (a) and its representation as an infinitely long cylinder (b).
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K.A. Murashko et al. / Journal of Power Sources 330 (2016) 61e69
information about heat generation during the operation of the cell. The heat generation Q_ (W) in the Li-ion battery can be calculated based on the energy balance presented by Bernardi et al. in Ref. [19].
vUocv þ Q_ mix þ Q_ pch ; Q_ ¼ I$ðUtab Uocv Þ þ I$T$ vT
(11)
where I is the cell current (A), which is positive during charging and negative during discharging, and Utab is the cell terminal voltage (V), Uocv is the OCV (V), T is the temperature (K), Q_ mix is the heat generation caused by the mixing effect (W), and Q_ pch is the heat generation caused by the phase change effect (W). Equations of the mixing and phase change terms can be found in Ref. [19]. Assuming that the reversible part of the heat generation is unchanged during charging and discharging, it can be obtained as
Q_ Q_ ch : Q_ rev ¼ dis 2
(12)
where Q_ dis is the heat generation during discharging (W), and Q_ ch is the heat generation during charging (W). By neglecting the pressure change during charging and discharging, the entropy change can be estimated as
DS ¼
n$F$Q_ rev : I$T
(13)
where F is the Faraday constant (F ¼ 96485 C mol1), and n is the stoichiometric number of electrons participating in the cell reaction (for a lithium-ion cell, n ¼ 1). The heat generation during charging and discharging that is required for the calculation of DS can be measured in a calorimeter. However, a calorimeter is usually a complex and expensive device. Therefore, in this work, a gradient heat flux sensor (GHFS), made from bismuth single crystal, was chosen as a small and inexpensive sensor by which the heat generation in the Li-ion battery can be measured. Detailed information about the GHFS and its calibration is given in Refs. [20e22]. 3.1. Measurements of heat generation by the GHFS The GHFS and the temperature sensor PT100 were installed on the cylindrical cell surface as it is shown in Fig. 3(a). In addition, the surface of the cylindrical cell was covered with rubber; the thickness of the rubber band is 1 mm. This was done in order to consider the conduction heat transfer from the cylindrical cell surface instead of the convection heat transfer. In addition, the
configuration reduces the influence of the airflow and the influence of changes in the ambient temperature in the laboratory. Ceramic fiber wool was used for thermal isolation of the ends of the cylindrical cell. A Potentiostat/Galvanostat Gamry Reference 3000 was used for the cell charging and discharging. A 34420A nanoVolt/micro-Ohm meter was applied to measure the voltage signal from the GHFS. A PT100 temperature sensor and a Keithley 2701 bench digital multimeter were used to measure the temperature of the cylindrical cell surface. At the beginning of the test, the cylindrical cell was fully charged according to the manufacturers recommendations [13]. Then, it was discharged by 3.0 A current impulses. The duration of a pulse required to discharge the cell by 10% of the SoC is 264 s. The 3.0 A current pulses were used because of the limitation of the Potentiostat/Galvanostat Gamry Reference 3000. After discharging, the cylindrical cell was charged by similar but opposite current pulses with the 3.0 A current as during discharging. After each impulse, the cell relaxation period was 30 min. During the relaxation process, the thermal diffusivity of the cylindrical cell was measured. The requirement for the duration of the relaxation period is based on the limitations of the proposed method, which are given below. The heat generated in the cylindrical cell during charging and discharging is calculated by applying the superposition principle. Two systems are considered. In the first system, we assume that heat is generated; however, all surfaces of the cylindrical cell are fully isolated and the heat dissipation is equal to zero. In this case, the temperature on the cell surface T1 can be given as
T1 ¼
1 $ cth $r$V
Ztend
Q_ ðtÞdt;
(14)
t0
where V is the volume of the cylindrical cell (m3). In the second system, the generated heat is equal to zero, but there is a heat flux from the cell surface. In this case, if the heat flux from the cell surface is constant, the temperature on the cylindrical cell surface can be calculated by the method presented in Ref. [14]. However, in the case under consideration, a variable heat flux is present. Therefore, the measured heat flux curve is divided into N time intervals tn, where a constant value of heat flux was considered. The differences between the values of the heat generated in each interval are obtained from
q_ n ¼ qn qn1 ;
(15)
where n is the number of the intervals n ¼ 1…N and q is the average value of the heat flux from the cylindrical cell surface in the considered interval (W m2). The temperature on the cell surface T2 in the case under study can be determined as
r T2 ¼ T0 $Q S; k
(16)
where T0 is the initial temperature of the cell (K), S is the N1 matrix, which includes the values of q_ n , and Q is the 1N matrix that includes the values of temperature coefficients, which are written as
1 4
Q ¼ 2$Fo þ
∞ X 2 An $J0 mn $exp mn $Fo ;
Fig. 2. Dependence of the temperature coefficient on the thermal Fourier number with the Biot number as a parameter.
(17)
n¼1
where Q is the temperature coefficient and An and mn are auxiliary variables. An is calculated as
K.A. Murashko et al. / Journal of Power Sources 330 (2016) 61e69
65
Fig. 3. Cylindrical cell with isolation, a GHFS, and a PT100 sensor (a) and its representation as a cylinder of infinite length (b).
An ¼
2
2 ;
(18)
mn $J0 m
1 4
and the auxiliary variable mn is obtained from the equation
Q ¼ 2$Fo þ :
J0 m
¼ 0:
(19)
The thermal Fourier number, which is required in Equation (17), is calculated as
Fo ¼
a$tn r2
;
(20)
where tn is the length of the time interval n ¼ 1…N, into which the heat flux curve is divided. The real value of the temperature on the surface of the cylindrical cell is obtained from
T ¼ T1 þ T2 :
(21)
By substituting Equation (21) into Equations (7), (14) and (15), the following equation can be given
Ztend t0
V Q_ ðtÞdt ¼ $ðk$ðT T0 Þ þ r$Q SÞ:
a
surface is calculated by Equation (16), and the elements of the matrix S by Equation (15). By assuming a linear temperature gradient in the cylindrical cell, Equation (17) can be simplified and the elements of the matrix Q are written as
(22)
The value of the heat generated during charging and discharging in any point in the time interval from t0 to tend can be obtained by numerical integration of Equation (22). When the values of the heat generation during charging and discharging are obtained, DS for each point of the SoC is calculated by Equations (12) and (13). 3.2. Measurements of the thermal diffusivity of the cylindrical cell Thermal diffusivity is measured during the relaxation period after discharging and charging of the cylindrical cell, when the temperature of the cylindrical cell decreases. In this case, the heat generation in the cell is equal to zero, and heat is dissipated by conduction and convection. The temperature of the cylindrical cell
(23)
By substituting Equation (20), and (23) into Equation (16), the thermal diffusivity of the cylindrical cell during cooling can be obtained by 2
r$k$ðT0 T2 Þ r4 $ a¼ 2$t S
PN
n¼1
q_ n
;
(24)
where t is the 1N matrix, which includes the values of the time intervals tn, S is the N 1 matrix, which includes the values of q_ n , and T2 is the temperature on the cylindrical cell surface (K). In order to increase the accuracy of the measurements, the thermal diffusivity for each point of the SoC is calculated as an average value between the values of thermal diffusivity estimated during discharging and charging for the same value of the SoC. The estimated values of thermal diffusivity for each point of the SoC are used in Equation (22) for the estimation of heat generation during charging and discharging in the same point of the SoC. 4. Limitations and uncertainty 4.1. Limitations The following requirements and limitations should be considered before performing the experiments and calculating the required parameters: The ends of the cylindrical lithium-ion cell should be fully isolated in order to stop the heat dissipation from these parts of the cylindrical cell. In this case, the cylindrical cell can be considered as a cylinder with an infinite length, and thus, the methods described in this paper can be applied. The value of thermal conductivity should be known. In this paper, the value of the thermal conductivity was taken to be the same as in Ref. [11].
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The calculation of the thermal diffusivity of the cell should be carried out at a linear temperature gradient in the cylindrical cell. The linear temperature gradient is required to simplify the calculation of the thermal diffusivity of the cylindrical cell. The minimum time of the experiment for the determination of the thermal diffusivity, after which equation (24) can be used, can be calculated by
tmin ¼
Fomin $r 2
a
;
(25)
where Fomin is the minimum thermal Fourier number at which the thermal gradient in the cylindrical cell becomes linear. The assumed parameters of thermal conductivity, heat capacity, and density, which were given in Table 1, were used for the preliminary estimation of the thermal diffusivity. The value of Fomin is obtained by analyzing the temperature gradient in the cylindrical cell during cooling of the cell, when the heat generation in the cylindrical cell is equal to zero and the heat flux from the cell surface, caused by convection and conduction, is known. In this case, the temperate gradient can be written as [14].
PN vT q_ ¼ VG$ n¼1 n ; vr k
(26)
where VG is the dimensionless thermal gradient coefficient. By neglecting the change in the thermal parameter values of the cylindrical cell, which are caused by the temperature gradient, we can assume that the form of the temperature gradient depends only on the temperature gradient coefficient, which can be calculated as in Ref. [14].
VG ¼ h þ
∞ X
2 Cn $J1 mn $h $exp mn $Fo ;
(27)
n¼1
where h is the dimensionless thickness coefficient, and Cn is the auxiliary variable. The auxiliary variable mn is calculated by Equation (19). The dimensionless thickness coefficient and the auxiliary variable Cn are calculated as in Ref. [14].
y r
h¼ ; Cn ¼
(28) 2
mn $J0 mn
:
(29)
The analysis of the dependence of the temperature gradient coefficient on the thermal Fourier number and the dimensionless thickness coefficient is shown in Fig. 4. As it can be seen in Fig. 4, the temperature gradient coefficient has a linear dependence on the dimensionless thickness coefficient if Fo 0.4. Therefore, the minimum value of the thermal Fourier number at which the temperature gradient in the cylindrical cell is linear and at which the thermal diffusivity of the cell can be calculated is equal to 0.4.
4.2. Uncertainty The uncertainty was calculated according to the standard EA-4/ 02 M: 2013 [23]. The standard uncertainty for the single measured value is calculated as
Fig. 4. Dependence of the thermal gradient coefficient on Fo and the dimensionless thickness coefficient.
u2 ðyÞ ¼
N X vf i¼1
vxi
2 $uðxi Þ
;
(30)
where u(y) is the uncertainty of the output estimate y, u(xi) is the uncertainty of the input estimate xi, where i is the number of the input estimates i ¼ 1…N, and f is the function that shows the dependence between the input and the output [16]. The uncertainty of the DS determination and the thermal diffusivity of the cylindrical cell depend on the uncertainty of the temperature sensor, heat flux sensor, and uncertainties in the measurement of time, current, voltage, and dimensions. The uncertainties of the input estimates were calculated as in Ref. [23] for the required parameters or were taken from the equipment documentation provided by the manufacturer. If the manufacturer does not give the uncertainty of the equipment, the uncertainty is calculated as a zero-order uncertainty, which is equal to one-half of the equipment resolution. The values of uncertainty for the applied equipment are shown in Table 2.
5. Results and discussion The evaluation of the time required for the DS measurements by the potentiometric method was carried out by analyzing the OCV change, caused by the relaxation processes in the cell after charging and discharging. In addition, the influence of the ambient temperature on the OCV change during the DS measurements by the potentiometric method was estimated by measuring the OCV changes first at 20 C and then at 30 C. The OCV changes at the values of temperature in the chamber after more than 12 h of the relaxation period are shown in Fig. 5 at each 10% of the SoC. High changes of the OCV are observed close to 0% of the SoC and 100% of the SoC. The negative value of the OCV change at 100% is caused by a sequence of tests where the cylindrical cell was first charged up to 100% of the SoC. The other measured points of the SoC were obtained by discharging of the cell. In addition, as it can be seen in Fig. 5, the OCV change increases when the SoC decreases, which was also noticed in Ref. [8]. This phenomenon can be explained by the diffusivity of the Liþ ions, as it was done in Ref. [8]. The diffusivity of the Liþ ions in graphite increases with the decreasing SoC [24], which may lead to increased OCV changes during the relaxation period. Because of the high OCV changes even after 12 h of the relaxation period, the influence of the cell relaxation, which causes changes in the OCV during the DS measurements by the potentiometric method, should be taken into account
K.A. Murashko et al. / Journal of Power Sources 330 (2016) 61e69
67
Table 2 Uncertainty of the input estimates. Input estimates
Uncertainties
Time, (s) Voltage (34420A nanoVolt/micro-Ohm meter), (nV) Temperature (4-Wires RTD Keithley 2701), (+ C) Voltage (Potentiostat/Galvanostat Gamry Reference 3000), (mV) Current (Potentiostat/Galvanostat Gamry Reference 3000), (mA) Dimensions (micrometer), (m) Mass, (g) Heat flux sensor sensitivity, (V W1) Temperature (Chamber Jeio Tech TH-KE), (+ C) Humidity (Chamber Jeio Tech TH-KE), (%)
0.5 30 0.005 0.57 2 0.000005 0.05 6.04,105 [20] 0.05 0.05
Fig. 5. OCV change per minute at 20 C and 30 C of the temperatures in the chamber.
especially for those values of the SoC that are close to 0% and 100%. The dependence of the OCV changes on temperature, which can be seen in Fig. 5, may be explained by considering that the diffusivity of the Liþ ions increases when temperature increases as it was shown in Ref. [24]. Therefore, the increasing temperature during a relaxation period leads to increasing OCV changes during this period. A high temperature decreases the relaxation time. However, the speed of the different side reactions increases with an increasing temperature, which affects the changes of the OCV during the relaxation period and increases the uncertainty of the result. A reduction in the temperature during the test decreases the speed of the kinetic processes in the cell and prolongs the relaxation period required to achieve a constant value of the OCV. Therefore, to decrease the uncertainty of the DS measurements by the potentiometric method, the temperature should be kept close to 25 C. This temperature was suggested in Ref. [25] as the optimal operation temperature of the Li-ion cell. In this case, it may be possible to find a compromise between the reaction speed required to achieve a constant value of the OCV and the speed of unwanted side reactions, which increase the uncertainty of the DS measurements. The calculated time at which the temperature of the cylindrical cell will be equal to the temperature in an environmental chamber was verified by placing the GHFS on the cylindrical cell surface. Heat dissipation from the cell surface was measured after changing the temperature in the environmental camber. The heat flux from the cylindrical cell surface is shown in Fig. 6 for 50% of the SoC and when changing the chamber temperature from 30 C to 25 C. The heat dissipation is 0 W m2 when the temperature of the cylindrical cell is equal to the temperature in the environmental chamber. According to the measurement results, the thermal transient processes in the cylindrical cell under study stop after 76 min, which is in compliance with the previous estimated value
by using Equation (25) Based on the analysis of the OCV changes caused by the relaxation processes in the cell and the time at which the cell temperature is equal to the temperature inside the chamber, the time required for the DS measurements was calculated. The required time to measure the DS values for a single point of the SoC for the cylindrical cell is about 17.1 h if the relaxation period of the cell after charging or discharging is 12 h. This value was obtained assuming that the temperature in the environmental chamber is changed four times (30 C, 25 C, 20 C, and 30 C), and 76 min is required for each value of temperature, while the cell temperature is equal to the temperature inside the chamber. If the relaxation period of 1.5 h is applied, as it was suggested in Ref. [8], it will take about 6.6 h to measure DS for a single value of the SoC. The calculated value of the uncertainty of the DS determination in the environmental chamber Jeio Tech TH-KE at 50% of the SoC is equal to 0.2 J mol1 K1 which is 2% of the measured DS value at the SoC under consideration, which is equal to 11.73 J mol1 K1. By using the GHFS, the time required for the simultaneous determination of DS and thermal diffusivity in a single point of the SoC is 80 min. The time required to measure DS in nine points of the SoC is about 12 h. The time required for the test can be shorter if the heat capacity and thermal conductivity of the cell are known. With the GHFS, the estimated value of the uncertainty of the DS determination at 50% of the SoC was 2.38 J mol1 K1, which is 20% of the measured DS value. The uncertainty of the thermal diffusivity determination at 50% of the SoC is 0.2 mm2 s1, of the measured thermal diffusivity value. The values of the thermal diffusivity, measured by using the GHFS, are given in Table 3 for each point of the SoC. As it can be seen from Table 3, the thermal diffusivity of the cylindrical cell has a small dependence on the SoC. The dependence of the thermal diffusivity on the SoC may be explained by the dependence of the heat capacity and the thermal conductivity on the SoC and temperature, which were shown and explained in
Fig. 6. Heat flux density from the cylindrical cell surface.
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K.A. Murashko et al. / Journal of Power Sources 330 (2016) 61e69
Refs. [26,27]. The measured values of the heat flux from the cylindrical cell surface and the temperatures of the cell surface that were measured during the DS determination by the GHFS are shown in Fig. 7. Based on the proposed method, DS was calculated from the measured data of the cylindrical cell during discharging and charging by 3.0 A current impulses. A comparison between the obtained values of DS by using the potentiometric method and by using the GHFS is shown in Fig. 8. The influence of the entropy change can easily be observed in Fig. 7 even with a more than 1 C-rate operation current. During discharge, the lower heat dissipation and temperature changes can be seen in the middle of the discharge test, which corresponds to the SoC from 35% to 75%. In addition, high heat dissipation and temperature changes can be observed at the end of the discharge test, which corresponds to the SoC from 0% to 10%. An opposite situation can be observed during that part of the test when the cylindrical cell was charged. Such behavior is explained by different heat generation during charging and discharging, which was caused by variation of DS with the SoC shown in Fig. 8. The DS profile obtained by the potentiometric method is similar to these results, which were shown in Ref. [11]. The variation of DS with SoC can be explained by the DS profiles of the separate electrodes under study, which were explained in Ref. [3]. Based on the data presented in Ref. [3], the variation of DS with the SoC of the cell is mostly caused by the variation of DS with the SoC on the negative electrode, which is made of graphite. The variation of DS with the SoC in the positive electrode, which is made of lithium iron phosphate, is significant only at the SoCs close to 0% or 100%. The comparison of the data measured by the potentiometric method and the data obtained by the GHFS shows an opportunity of entropy change measurements by the GHFS. The reason for the deviation of the results obtained by both reported methods may be the internal pressure increases during battery charging and discharging, which was neglected in this article for the simplification of the DS determination. The uncertainty of the proposed method strongly depends on the heat generation in the cylindrical cell during the DS determination. In the case of a small cylindrical Li-ion cell, where the heat generation and the resulting temperature changes are small, the uncertainty of the proposed method is higher than the uncertainty of the potentiometric method. However, even for a cylindrical cell, which was discharged with the 1.3 C-rate current, DS can be measured with an acceptable uncertainty. Taking into account the decrease in the time required for the DS determination, which is up to 5 h per a single point of the SoC, the use of the GHFS to determine DS of a cylindrical cell is recommended. 6. Conclusions In the paper, determination of DS by using a gradient heat flux sensor (GHFS) was considered. The proposed method allows to significantly decrease the time required for the measurement of the DS values. In addition, the method does not require large and expensive devices such as an environmental chamber or a calorimeter. The results of the DS measurements were compared with
Table 3 Thermal diffusivity of the cylindrical cell for different values of the SoC. SoC, %
a, mm2 s1
SoC, %
a, mm2 s1
SoC, %
a, mm2 s1
90 80 70
0.226 ± 0.0022 0.205 ± 0.0021 0.188 ± 0.0019
60 50 40
0.201 ± 0.0020 0.195 ± 0.0020 0.197 ± 0.0020
30 20 10
0.198 ± 0.0020 0.200 ± 0.0020 0.195 ± 0.0020
Fig. 7. Measured values of heat flux and temperatures on the cylindrical cell surface.
Fig. 8. Entropy change measured by the potentiometric method and by using the GHFS.
the values of DS that were measured by the potentiometric method in an environmental chamber. In order to decrease the uncertainty of the DS measurements in the environmental chamber, the limitations on the required relaxation period after charging and discharging of the cell and on the temperature in the environmental chamber were analyzed. It was shown that the OCV changes caused by the relaxation of the cell after charging or discharging should be taken into account even after 12 h of the relaxation period. In addition, the dependence of the OCV changes on the chamber temperature during DS measurements was shown. Moreover, by applying the presented method, an analysis was made of the thermal transient process in a cylindrical cell. It was shown that damping of the thermal transient processes in the cell after a change in the temperature inside the environmental chamber is reached after 76 min, which was verified experimentally. A comparison of the DS values obtained by the potentiometric method in the environmental chamber and by the GHFS shows an opportunity to use the proposed method for the DS determination. Even though the uncertainty of the method is higher than in the potentiometric method, a major advantage of the method is the significant decrease in the experiment time (down to 80 min per a single value of the SoC from the minimum of 6.6 h required for the potentiometric method, where the environmental chamber is used). Therefore, the application of the GHFS is recommended for the DS determination in the case of the cylindrical Li-ion cell.
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