Effectiveness of a Helix Tube to Water Cool a Battery Module

Abstract

This chapter presents an investigation of the effectiveness of water cooling a battery module using a heat-sink prototype in the form of a thin copper helix tube within an aluminium block. A thermal model for the module containing six single cells is developed and numerically solved by coupling the heat energy transport equation with the fluid governing equations. The rate of generation of heat from the cells is calculated using a 2D model of a single cell with the resulting heat flux used as a Neumann boundary condition for the energy equation within a computational fluid dynamics code. Particular attention is given to the battery module operating in extreme ambient temperature conditions. The cooling strategy used is shown to satisfy two of the main concerns when managing the thermal performance of a battery module, that is, a suitable operating temperature range is maintained, and there is reasonable uniformity of temperature across the battery module. This should increase the battery cell life cycle together with enhancement of the charge and discharge performances. Variation of parameters such as the velocity of water within the tube and the number of turns used for the helix were investigated.

1. Introduction

Due to their outstanding properties regarding high energy density, power density, high service life, low self-discharge and rare pollution, the use of lithium ion batteries for battery electric vehicles (BEVs) and hybrid electric vehicles (HEVs) shows excellent promise [1]. Battery charging and discharging are strongly coupled to the operating temperature with a higher temperature increasing the number of undesired reactions, that is, the decomposition of the electrolyte, thickening of the solid electrolyte interface (SEI) film, and so on, which have the effect of degrading the battery capacity [2, 3, 4]. Lithium-ion battery cells are also highly sensitive to low temperatures. For example, at below −10°C, it has been shown that batteries’ performance deteriorates considerably [5, 6]. Safety is also an issue in that at high temperatures, lithium-ion batteries are prone to uncontrollable temperature build-up (thermal runaway) [7]. In addition, a battery pack is usually composed of up to hundreds of single cells connected in series and/or in parallel to produce output voltage and power/energy capacity to achieve the BEVs’ or HEVs’ operating needs. This contributes to the accumulation of heat and uneven temperature distributions across the battery pack so degrading performance. To counteract these serious problems, it is necessary to cool a battery pack effectively and uniformly.

Important to the development of a realistic thermal model of a battery module is good knowledge of heat generation either from the battery module as a whole or from individual cells within the module. The battery heat generation rate can be estimated by direct measurement [8], by indirect measurement, for example by measuring equilibrium voltage [9], measuring internal electrical resistance [10] or using the electrochemical mechanism [11, 12]. For direct measurement, the results apply to only a few operating points, and for indirect measurements, accuracy is a problem as the heat generation rate is a nonlinear function of the state of charge (SOC), current and operating temperature [13]. Work on single cell thermal-electrochemical modelling has already been done [13, 14, 15, 16], and it is the view here that an essential component for the modelling of battery modules and eventually battery packs is to start with accurate calculations of heat generation within a given single cell.

There are a number of ways of cooling battery modules with the major division being between air cooling and liquid cooling. There are advantages and disadvantages associated with each choice. For liquid cooling systems, a disadvantage is that they have the potential to leak, which could cause an electrical short. Also when liquid is used, maintenance and repair can be more costly and more complicated, and generally liquid cooled systems are more heavy and require more components. However, a design can be used, as in the present application where the battery module and liquid can be separated by aluminium. Air cooling systems are generally less effective at maintaining a uniform temperature within and between cells in a battery module, and they cannot carry as much heat away from the battery as quickly as a liquid-based system. Air convection (natural or forced) quite often is insufficient for effective heat dissipation from batteries under abuse conditions leading often to nonuniform temperature distributions within battery packs [17, 18]. Although air cooling systems are lighter, have a lower cost and generally a longer life, water cooling is favoured for this work due to its much superior cooling effect. There are many investigations of cooling systems reported in the literature with the majority of the work focusing on lumped parameter models [19]. The effectiveness of passive air cooling using phase change materials (PCMs) combined with forced cooling has been investigated [17], as has different single cell spacing arrangements [11, 20, 21]. It has been stated and demonstrated in the literature that indirect liquid cooling of battery packs (both passive and active) can prove an efficient method for dissipation or addition of heat [22, 23, 24, 25].

This chapter presents an investigation of the thermal performance of indirect water cooling, using the proposed design prototype of a thin copper helix tube contained within an aluminium block, for a lithium ion battery module. A thermal model is developed for the battery module where electrochemical equations are solved to provide boundary conditions for a computational fluid dynamics code in which the energy and fluid flow conservation equations are solved. Variation of parameters such as the velocity of water within the tube and the number of turns used for the helix were investigated.

2. Mathematical model

2.1. Modelling within a single cell

The mathematical model for a single lithium-ion battery developed here is based on the work of Doyle et al. [26]. The battery cells used are cylindrical with a central mandrel, with thin layers of anode, cathode, current collector and separator rolling up on the mandrel and with protection provided by a battery can. The anode is made of graphite derivatives and the cathode material is a metallic oxide such as LiFePO₄ and LiM₂O₄. A schematic of a lithium ion cell is shown in Figure 1.

Figure 1.

Schematic of a lithium ion battery.

Generally, a lithium ion battery consists of the current collector, the positive electrode, the separator and the negative electrode. A lithiated organic solution fills the porous components and serves as the electrolyte. Several assumptions are needed, that is, the active electrode material is composed of spherical particles with uniform radius and the winding zone of the battery is a lumped model with homogeneous electrochemical properties. The material balance for the Li ions in an active solid material particle is governed by Fick’s second law, here expressed in spherical coordinates

$$\frac{\partial c_{s,i}}{\partial t}=D_{s,i}\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial c_{s,i}}{\partial r}\right),$$E1

where $i=p,s$ and $i=n$ for the positive and negative electrodes, respectively. At the centre of the particle, there is no flux, and on the surface of the particle, the flux is equal to the consuming/producing rate of Li ions due to the chemical reaction occurring at the solid/liquid surface giving the boundary conditions

$$-D_{s,i}{\left.\frac{\partial c_{s,i}}{\partial r}\right|}_{r=0}=0,-D_{s,i}{\left.\frac{\partial c_{s,i}}{\partial r}\right|}_{r=r_s}=J,$$E2

where $J$ is the flux of lithium ions away from the surface of the spherical particles. The mass conservation of Li in the electrode solution is given by the concentration solution theory and can be expressed as

$$\epsilon {\prime }_i\frac{\partial c_i}{\partial t}=D_{\mathit{eff},i}\frac{{\partial }^2{c}_i}{\partial x^2}+\left(1-t_{+}^0\right)a_i{J}_i,$$E3

where $i=p,s$ and $n$ and $a_i$ are the electrode surface area per unit volume of the electrode. In the separator, the pore wall flux $J_s$ is equal to zero, and at the two ends of the cell in the $x$-direction, there is no mass flux

$$-D_{\mathit{eff},p}{\left.\frac{\partial c_p}{\partial x}\right|}_{x=0}=0,-D_{\mathit{eff},n}{\left.\frac{\partial c_n}{\partial x}\right|}_{x=L_p+L_s+L_n}=0$$E4

At the interfaces between the positive electrode/separator and separator/negative electrode, the concentration of the binary electrolyte and its flux is continuous

$${\left.c_p\right|}_{x=L_p^{-}}={\left.c_s\right|}_{x=L_p^{+},}{\left.c_s\right|}_{x={\left(L_p+L_s\right)}^{-}}={\left.c_n\right|}_{x={\left(L_p+L_s\right)}^{+},}$$E5

$$-D_{\mathit{eff},p}{\left.\frac{\partial c_p}{\partial x}\right|}_{x=L_p^{-}}=-D_{\mathit{eff},s}{\left.\frac{\partial c_s}{\partial x}\right|}_{x=L_p^{+}}$$E6

$$-D_{\mathit{eff},s}{\left.\frac{\partial c_s}{\partial x}\right|}_{x=L_p^{-}}=-D_{\mathit{eff},n}{\left.\frac{\partial c_n}{\partial x}\right|}_{x={\left(L_p+L_s\right)}^{+}}$$E7

The effective diffusion coefficient, $D_{\mathit{eff}}$ of Li in the electrode can be represented as $D_{\mathit{eff},i}=D_{ϵ,i}{\epsilon }_i^{{\mathit{\text{brugg}}}_i}$. The specific surface area for the electrode particles, $a$, is given by $a=3{\epsilon }_s/r_s.$

The charge balance in the solid phase is governed by Ohm’s law

$${\sigma }_{\mathit{eff},i}\frac{{\partial }^2{\varphi }_{s,i}}{\partial x^2}=a_{i}F{J}_i$$E8

where $i=p$ and $n$. Here ${\sigma }_{\mathit{eff}}$ is the effective electric conductivity and is given by ${\sigma }_{\mathit{eff}}=\sigma \bullet {\epsilon }_s$.

The boundary conditions are expressed as

$$-{\sigma }_{\mathit{eff},p}{\left.\frac{\partial {\varphi }_{s,p}}{\partial x}\right|}_{x=0}=I_{\mathit{app}},-{\sigma }_{\mathit{eff},p}{\left.\frac{\partial {\varphi }_{s,p}}{\partial x}\right|}_{x=L_p}=0,-{\sigma }_{\mathit{eff},n}{\left.\frac{\partial {\varphi }_{s,n}}{\partial x}\right|}_{x=L_p+L_s}=0$$E9

The potential of the solid phase at the right end of the cell (Figure 1) is set to zero, ${\left.{\varphi }_{s,n}\right|}_{x=L_p+L_s+L_n}$= 0 and the potential of the solid phase at $x=0,{\left.{\varphi }_{1,p}\right|}_{x=0}$ is equal to $E_{\mathit{\text{cell}}}$. The charge balance in the liquid phase is based on Ohm’s law, and it is given by

$$-\frac{\partial }{\partial x}\left({\kappa }_{\mathit{eff},i}\frac{\partial {\varphi }_{2,i}}{\partial x}\right)+\frac{2\mathit{RT}(1-t^{+})}{F_d}\frac{\partial }{\partial x}\left({\kappa }_{\mathit{eff},i}\frac{\partial \left(ln,c_i\right)}{\partial x}\right)=\frac{F_d}{F_i}{J}_i$$E10

where $i=p,s$ and $n$, and, the specific conductivity of the electrolyte is a function of the concentration of the electrolyte in the liquid phase [27]

$${\kappa }_{\mathit{eff},i}={\kappa }_i{\epsilon }_i^{{\mathit{\text{brugg}}}_i}$$E11

At the two ends of the cell, there is no charge flux in the liquid phase

$$-{\kappa }_{\mathit{eff},p}{\left.\frac{\partial {\varphi }_{2,p}}{\partial x}\right|}_{x=0}=0,-{\kappa }_{\mathit{eff},n}{\left.\frac{\partial {\varphi }_{2,n}}{\partial x}\right|}_{x=L_p+L_s+L_n}=0$$E12

In the abovementioned equations, the pore wall flux, $J_i$ is determined by the Butler-Volmer equation

$$J_i=k_i{\left(c_{s,i,\mathit{maz}}-{\left.c_s\right|}_{r=r_s}\right)}^{{\alpha }_a}\bullet {\left.c_s\right|}_{r=r_s}^{{\alpha }_c}\bullet c^{{\alpha }_a}\left\{exp\left(\frac{{\alpha }_{a}F{\eta }_i}{\mathit{RT}}\right)-exp\left(-\frac{{\alpha }_{c}F{\eta }_i}{\mathit{RT}}\right)\right\}$$E13

where ${\eta }_i$ is the over-potential of battery electrodes and is given by

$${\eta }_i={\varphi }_{s,i}-{\varphi }_{2,i}-U_i$$E14

The open circuit voltage of the electrode materials $U_i$ is determined by cell temperature and Li concentrations at the surface of the spherical particle. The energy balance is given by [15]

$$\rho C_p\frac{\partial T}{\partial t}=\nabla \bullet \nabla \left(k_{T}T\right)+Q_{\mathit{rea}}+Q_{\mathit{rev}}+Q_{\mathit{ohm}}$$E15

with the boundary conditions determined by Newton’s cooling law

$$-\lambda {\left.\frac{\partial T}{\partial x}\right|}_{x=0}=h\left(T_{\infty }-T\right),{\left.-\lambda \frac{\partial T}{\partial x}\right|}_{x=L_p+L_s+L_n}=h\left(T-T_{\infty }\right)$$E16

where $h$ is the heat transfer coefficient, $T_{\infty }$ is the ambient temperature, ${\dot{Q}}_{\mathit{rea}}$ is the total reaction heat generation rate, ${\dot{Q}}_{\mathit{rev}}$ is the total reversible heat generation rate, ${\dot{Q}}_{\mathit{ohm}}$ is the total ohmic heat generation rate. The heat fluxes are defined by

$${\dot{Q}}_{\mathit{rea}}=\mathit{aF}{J}_i{\eta }_i$$E17

$${\dot{Q}}_{\mathit{rev}}=\mathit{aF}{J}_{i}T\bullet \frac{d{U}_i}{\mathit{dT}}$$E18

$${\dot{Q}}_{\mathit{ohm}}={\sigma }_{\mathit{eff}}{\left(\frac{\partial {\varphi }_s}{\partial x}\right)}^2+{\kappa }_{\mathit{eff}}{\left(\frac{\partial {\varphi }_2}{\partial x}\right)}^2+\frac{2k_{\mathit{eff}}\mathit{RT}\left(1-t_{+}^0\right)}{F}\left(1+\frac{dln{f}_{\pm }}{dlnc}\right)\frac{\partial lnc}{\partial x}\frac{\partial {\varphi }_2}{\partial x}+\mathit{aF}{J}_i∆{\varphi }_{\mathit{SEI}}$$E19

2.2. Modelling within the aluminium block

The battery module cooling system used here is a heat sink approach, where the lithium-ion battery cells are placed in an aluminium block and also surrounded by a copper helix coil through which water is pumped. The method employed is fundamentally to surround the cells with a conducting material, that is, a form of heat sink, and to remove or add heat using fluid. The cooling design is shown in Figure 2. The model solves in 3D, with fluid pumped through a central vertical tube and returned through the copper helix tube just within the aluminium block for efficient heat transfer and protection against damage.

Figure 2.

Copper helix coil within the aluminium block a. front and b. plan.

The fluid is conditioned using a heater/refrigerator unit placed on the top surface of the plenum chamber as shown in Figure 3. The aim of the overall thermal management system is to maintain a battery module at an optimum average temperature, as dictated by life and performance trade-off. Important is that an even temperature, perhaps with small variations, is maintained between the cells and within the module. However, when designing such a system, regard must also be paid to the fact that the battery module should be compact, lightweight, have low cost manufacture and maintenance, and have easy access for maintenance. The management system should also have low parasitic power, allow the module to operate under a wide range of climatic conditions and provide ventilation if the battery generates potentially hazardous gases.

Figure 3.

Schematic of thermal management system.

The calculation domain has two subdomains, that is, a fluid region and a solid region.

2.2.1. Fluid section

For non-steady flow, the equations of continuity, momentum and energy can be expressed in the general conservation form

$$\frac{\partial }{\partial t}\left(\mathit{\rho \phi }\right)+\frac{\partial }{\partial x_i}\left(\rho u_i\phi \right)=\frac{\partial }{\partial x_i}\left({\mathrm{\Gamma }}_{\phi }\frac{\partial \phi }{\partial x_i}\right)+S_{\phi }$$E20

where $\rho$ is the liquid density, $u_i$ is the velocity vector components, ${\mathrm{\Gamma }}_{\phi }$ is the effective exchange coefficient of $\phi$ and $S_{\phi }$ is the source rate per unit volume. The source rate and the effective exchange coefficient corresponding to each variable $\phi$ solved in this study are given in Table 1. Included in Table 1 are the transport equations for the standard k-$\epsilon$ turbulence model.

Equation	$\phi$	$S_{\phi }$	${\mathrm{\Gamma }}_{\phi }$
Continuity	1	0	0
Momentum	$U_i$	$\frac{\partial p}{\partial x_i}+{\rho }_{\mathit{ref}}{g}_i$	$\rho \left(\mu +{\mu }_t\right)$
Enthalpy	h	$\frac{\mathit{Dp}}{\mathit{Dt}}+\text{heat sources}\left(\text{sinks}\right)$	$\rho \left(\frac{{\mu }_t}{{\sigma }_t}+\frac{\mu }{\mathit{Pr}}\right)$
Kinetic energy of turbulence	k	$G_k-\mathit{\rho \epsilon }$	$\frac{\mu }{{\sigma }_k}$
Eddy dissipation rate	$\epsilon$	$C_1\left(\frac{\epsilon }{k}\right)G-C_2\rho \left(\frac{{\epsilon }^2}{k}\right)$	$\frac{\mu }{{\sigma }_{\epsilon }}$

Table 1.

Source rate and effective exchange coefficient for each $\mathit{\phi }$.

In Table 1, $\mu$ is the viscosity, $\sigma$ is the Prandtl number for $\epsilon$ and $G_k=\mu \left(\partial u_i/\partial x_j+\partial u_j/\partial x_i\right)\left(\partial u_i/\partial x_j\right)$ the turbulence production rate. The values of the constants $C_1$ and $C_2$ are 1.44 and 1.92 respectively and for ${\sigma }_k$ and ${\sigma }_{\epsilon }$, 1.0 and 1.3 respectively [28]. The eddy viscosity term is

$${\mu }_t=\rho C_{\mu }\frac{k^2}{\epsilon }$$E21

where $C_{\mu }=0.09$.

In addition to the standard k-$\epsilon$ turbulence model, the realisable k-$\epsilon$ model with standard wall functions, and, non-equilibrium wall functions for the near-wall treatments were used to model turbulent flow as they show good performance in modelling flow structures [29, 30]. Although low-Reynolds number modelling (LRNM) may give more accurate simulation results, this requires very fine cells close to the wall to resolve the near wall region, which increases the difficulties of grid generation and computing time cost. Also the grid used by LRNM is unsuitable for the high-Reynolds number turbulence models used here because the very fine cells close to the wall cannot satisfy the first node near the wall located out of the viscous sub-layer [31]. The equations for the realisable k-$\epsilon$ turbulence model are

$$\frac{\partial }{\partial t}\left(\mathit{\rho k}\right)+\frac{\partial }{{\partial x}_j}\left(\mathit{\rho k}{u}_j\right)=\frac{\partial }{\partial x_j}\left(\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right)+G_k-\mathit{\rho \epsilon }$$E22

$$\frac{\partial }{\partial t}\left(\mathit{\rho \epsilon }\right)+\frac{\partial }{\partial x_j}\left(\mathit{\rho \epsilon }{u}_j\right)=\frac{\partial }{\partial x_j}\left(\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right)+C_1\rho S_{\epsilon }-C_2\rho \frac{{\epsilon }^2}{k+\sqrt{\mathit{\mu \epsilon }}}$$E23

$${\mu }_t=\rho C_{\mu }\frac{k^2}{\epsilon }=\frac{1}{4.04+A_{s}k{u}^{\ast }/\epsilon }$$E24

$$u^{\ast }=\sqrt{S_{\mathit{ij}}{S}_{\mathit{ij}}{\tilde{\mathrm{\Omega }}}_{\mathit{ij}}{\tilde{\mathrm{\Omega }}}_{\mathit{ij}}},{\tilde{\mathrm{\Omega }}}_{\mathit{ij}}={\mathrm{\Omega }}_{\mathit{ij}}-2{\epsilon }_{\mathit{ijk}}{\omega }_k$$E25

$$A_s=\sqrt{6}cos{\phi }^{\prime },{\phi }^{\prime }=\frac{1}{3}{cos}^{-1}\left(\sqrt{6}W\right),W=\frac{S_{\mathit{ij}}{S}_{\mathit{ik}}{S}_{\mathit{jk}}}{{\tilde{S}}^3}$$E26

$$\tilde{S}=\sqrt{S_{\mathit{ij}}{S}_{\mathit{ij}}},C_1=max\left[0.43,\frac{\tilde{\mu }}{\tilde{\mu }+5}\right]$$E27

where ${\sigma }_k=1.0,{\sigma }_{\epsilon }=1.2$ and $C_2=1.9$. The standard wall functions used here are based on the work of Launder and Spalding [32] and have been found to be suitable for a broad range of wall-boundary flows. The law-of-the-wall for mean velocity gives

$$U^{\ast }=\left\{\begin{array}{c}{y}^{\ast }(y^{\ast }>11.225\\ \frac{1}{\kappa }ln\left(E{y}^{\ast }\right)(y^{\ast }<11.225\end{array}\right.$$E28

where

$$U^{\ast }=\frac{U_p{C}_{\mu }{k}_p^{1/2}}{{\tau }_w/\rho }$$E29

is the dimensionless velocity, and

$$y^{\ast }=\frac{\rho C_{\mu }^{1/4}{k}_p^{1/2}{y}_p}{\mu }$$E30

is the dimensionless distance from the wall and $\kappa$ is the von Karman constant (=0.4187), $E$ is the empirical constant (=0.9793), $U_p$ is the mean velocity of the fluid at the near-wall node P, $k_p$ is the turbulence kinetic energy at the near-wall node P, $y_p$ is the distance from the point P to the wall, and $\mu$ is the dynamic viscosity of the fluid. The temperature wall functions include the contribution from the viscous heating, and for incompressible flow calculations, the law-of-the-wall for the temperature field has the following composite form

$$T^{\ast }=\frac{\left(T_w-T_p\right)\left(C_{\mu }^{1/4}{k}_p^{1/2}\right)}{\left(q_w/\rho C_p\right)}=\left\{\begin{array}{c}\mathit{Pr}{y}^{\ast }(y^{\ast }<y_T^{\ast }\\ {\mathit{Pr}}_t\left[\frac{1}{\kappa }ln\left(E{y}^{\ast }\right)+P\right](y^{\ast }>y_T^{\ast }\end{array}\right.$$E31

where $P$ is given by Jayatilleke [33]

$$P=9.24\left(\frac{\mathit{Pr}}{{\mathit{Pr}}_t}-1\right){\left(\frac{\mathit{Pr}}{{\mathit{Pr}}_t}\right)}^{-1/4}$$E32

where $y_T^{\ast }$ is the dimensionless thermal sublayer thickness, $C_p$ is the specific heat of the fluid, $q_w$ the wall heat flux, $T_p$ is the temperature at the first near-wall node P, $T_w$ is the temperature of the wall, $\mathit{Pr}$ is the molecular Prandtl number ($=\mu C_p/a^{\prime }$), $a^{\prime }$ is the coefficient of heat diffusion, and ${\mathit{Pr}}_t$ is the turbulent Prandtl number (−0.85 at the wall).

The standard wall functions tend to become less reliable when the flow situations depart from the ideal conditions and are subjected to severe pressure gradients and strong non-equilibrium. The non-equilibrium wall functions are introduced and can potentially improve the results in the above mentioned situations [34]. The law-of-the wall for mean temperature remains the same as in the standard wall functions already described and the log-law for mean velocity sensitised to the pressure gradient is

$$\frac{\tilde{U}{C}_{\mu }^{1/4}{k}^{1/2}}{{\tau }_w/\rho }=\frac{1}{\kappa }\left(E\frac{\rho C_{\mu }^{1/4}{k}^{1/2}y}{\mu }\right)$$E33

where

$$\tilde{U}=U-\frac{1}{2}\frac{\mathit{dp}}{\mathit{dx}}\left[\frac{y_{\nu }}{\mathit{\rho \kappa }\sqrt{k}}ln\left(\frac{y}{y_{\nu }}\right)+\frac{y-y_{\nu }}{\mathit{\rho \kappa }\sqrt{k}}+\frac{y_{\nu }^2}{\mu }\right]$$E34

and $y_{\nu }$ is the physical viscous sublayer thickness, and computed from

$$y_{\nu }=\frac{\mu y_{\nu }^{\ast }}{\rho C_{\mu }^{1/4}{k}_p^{1/2}}$$E35

where $y_{\nu }^{\ast }=11.225$.

In this study, the Boussinesq model was used to treat the variable water density in which the water density is taken as a constant in all terms of the solved equations, except for the buoyancy term in the momentum equation

$$\left(\rho -{\rho }_0\right)g=-{\rho }_0\beta \left(T-T_0\right)g$$E36

where ${\rho }_0$ is the reference density of the water flow (kg/m³); $T_0$ is the reference temperature (K); and, Eq. (37) is obtained by the Boussinesq approximation $\rho ={\rho }_0\left(1-\beta ∆T\right)$ to replace the buoyancy terms. This approximation is acceptable so long as changes in actual density are small. Specifically, it is valid when $\beta \left(T-T_0\right)\ll 1$, and should not be used if the temperature difference in the domain is large.

2.2.2. Solid section

The solid section in this work consists of three components, namely, the aluminium block, the wall of the helix coil and the battery module. When the velocity is set to zero in Eq. (20), the equation governing pure conductive heat transfer by diffusion is obtained, that is,

$$\frac{\partial }{\partial t}\left(\mathit{\rho \phi }\right)=\frac{\partial }{\partial x_i}\left({\mathrm{\Gamma }}_{\phi }\frac{\partial \phi }{\partial x_i}\right)+S_{\phi }$$E37

Conjugate heat transfer was used between the solid domain and fluid domain.

2.3. Boundary conditions and settings

The cooling fluid is modelled using the material properties of water calculated using the inlet temperature. The settings and boundary conditions are set out in Tables 2 and 3.

Battery module outside dimensions; Cooling pipe radius ($r_{\mathit{\text{inner}}})$; Coolant; Final mesh size

Radius = 130 mm, height = 360 mm; 5–15 mm; H2O; 10^7

Table 2.

Various settings used during the calculations.

Cylinder outer wall; Coolant/solid interface; Cooling pipe inlet; Cooling pipe outlet; Cell initial temperatures

293.15 K; Conjugate heat transfer; $u_{\mathit{\text{in}}}=0.005-0.1$m/s; $T_{\mathit{\text{in}}}=293.15$K; 101,325 Pa; $T_{\mathit{\text{init}}}=313.15$ K; $T_{\mathit{\text{init}}}=349.15$K

Table 3.

Boundary conditions used during the calculations.

2.4. Grid dependence and computer storage

The requirements set out in two recent guidelines referring to good CFD practice [35, 36] were followed in the present work. Non-uniform structured Cartesian grids were used throughout this work with extensive tests for independence of grid size checked by increasing the grid numbers until further refinement was shown not to be of significance. Close to solid surfaces, the grid was refined using geometric progression with an expansion ratio from the solid surfaces of less than 1.1 and with the ${\mathrm{Y}}^{+}$ values adjacent to a solid boundary held around a value of 20, so complying with recommendations given in the literature that ${\mathrm{Y}}^{+}$ should be between 11.5 and 300 to ensure accuracy when using a high-Reynolds turbulence model [37]. Tests for grid independent solutions were carried out using 2.0 $\times {10}^5$, 2.0$\times {10}^6$ and 1.0$\times$ 10⁷. The overall change in the residual for each variable between the last two numbers of elements was less than 0.1%, indicating grid independence had been achieved. The calculations were performed on a Dell T5500 workstation with 32 nm six-core Intel Xeon 5600 series processor and main memory of 24 GB. A typical CPU time for a transient run with a grid having 10⁶ nodes was just over 24 h.

3. Results

3.1. Heat generation within a single cell

The thermal characteristics of a Li-ion battery cell are first investigated using Eqs. (1)–(19), which form thermal-electrochemical coupled model. The cell used in these calculations has an electrolyte consisting of zinc and lithium salts dissolved in water. When the battery is fully charged, the anode consists of nonporous zinc and the cathode of porous Mn₂O₄. It is important to note that some of the electrochemical calculations are strongly dependent on coefficients, which are in turn strongly dependent of experimental results. For example, for the electrolyte just described, the specific conductivity Eq. (11) of the electrolyte is a function of temperature and the concentration of the electrolyte in the liquid phase, and so the ionic conductivity, ${\mathit{\kappa }}_{\mathit{i}}$ had to be determined by experiment, the results of which are summarised in Figure 4.

Figure 4.

Ionic conductivity of electrolyte consisting of a ZnCl₂ and LiCl aqueous solution.

The effect of using different current rates during discharge of the battery cell is illustrated in Figure 5. Here temperature on the cell surface is calculated against the depth of discharge (DOD), which indicates the state of discharge of the battery cell starting at 100% fully charged. In these calculations, DOD was calculated as $\text{time}\ast \frac{C\text{rate}}{3600}$ and the heat transfer coefficient $h$ was set at 1.0 Wm⁻² K⁻¹. As would be expected, the cell surface gets hotter as the discharge current rate increases. There is a ‘kink’ in the curve at lower current rates, which is thought to be due to an interaction between the ohmic and reversible heat in the energy balance equation.

Figure 5.

Cell surface temperature during discharge at different current rates.

Also calculated were the profiles of cell surface temperatures over a long time of discharge. Two limiting cases, that is, adiabatic and isothermal were used as the boundaries for this study with different heat transfer coefficients used for the intervening calculations, as shown in Figure 6. It can be seen that the heat transfer coefficient 1.0 W⁻² K⁻¹ gives a reasonable result and keeps the battery cell well within the desired operating range, while the 0.1 W⁻² K⁻¹ setting allows the battery cell wall temperature to reach the upper region of the desired range.

Figure 6.

Cell surface temperature for different cooling conditions.

3.2. Cooling the battery module

In this part of the study, the temperature history of the battery module was modelled with ambient conditions (${\mathit{T}}_{\mathit{\infty }})$ set at 293.15 K, and each of the cells sets initially at ${\mathit{T}}_{\mathit{\text{init}}}\mathit{=}313.15$K and then at ${\mathit{T}}_{\mathit{init}}\mathit{=}349.15$ K. This part of the calculations is important to the design process in that, in addition to testing, if the chosen geometry parameters are suitable, it also gives an indication concerning the selection of a suitable pump and heater/refrigeration unit. Typical velocity contours for the liquid coolant are shown in Figure 7. The important part here is that heat can be removed from the coolant in the plenum chamber efficiently. From Figure 7, it can be seen that there is slow moving water adjacent to the heating/refrigeration unit, and hence there is sufficient time for dissipation of heat.

Figure 7.

Velocity contours, $u_{\mathit{\text{in}}}=0.01$ m/s, $T_{\mathit{\text{in}}}=293.15$ K, $r_{\mathit{\text{inner}}}=10$ mm, $T_{\infty }=293.15$ K.

Several tests were conducted, where the battery module was cooled to find appropriate values for the parameters, $u_{\mathit{\text{in}}}$ and the optimum number of helix coil turns. Figure 8 shows temperature values obtained at the centre of a cell versus time during cooling. The figure shows results for two initial cell temperatures, that is, 313.15 and 349.15 K, and the temperature profiles were obtained for different inlet velocities to the helix tube ranging from 0.005 to 0.1 m/s. It was found that when using an inlet velocity of 0.005 m/s, the cell temperature did not reach an acceptable temperature over what was regarded as a reasonable time. When the inlet velocity was increased to 0.01 m/s, acceptable temperatures were calculated after 80 and 20 s for the higher and lower initial temperatures, respectively. With the much higher inlet velocity of coolant to the helix pipe, that is, 0.1 m/s, there was a definite faster reduction in temperature. However, this faster velocity has design implications in that more powerful pumps together with a greater danger of coolant leakage make use of this inlet velocity value less attractive. Therefore, it was decided to continue the study with $u_{\mathit{\text{in}}}=0.01$ m/s. It is also noticeable from Figure 8 that as the initial temperature of the cells was reduced, increasing the velocity of the coolant through the pipe had much less effect on the cooling rate. This could mean that, although the coolant at higher velocity had more capacity to carry heat energy away from the battery pack, the temperature gradient between solid helix pipe and water was no longer sufficient to drive heat energy from solid to liquid effectively. This could possibly be due to the complex nature of the flow within the helix tube. Inside the tube, the flow is stretched from the inner wall, where most of the heat energy enters the liquid towards the outer wall due to centrifugal forces. Secondary flow also results due to the centrifugal forces. This aspect of the design needs further research.

Figure 8.

Temperature cooling profiles versus time at the centre of a cell for different initial cell temperatures and fluid velocities in the helix tube. Here $r_{\mathit{\text{inner}}}=10$ mm, $T_{\infty }=293.15$ K and $T_{\mathit{\text{in}}}=293.15$ K.

It is important to know the optimum number of turns the helix coil makes for a number of reasons. One is that if too many coils are used, then more expense occurs in the manufacturing stage and also the integrity of the structure may suffer. If the number of coils is too small, then cooling of the battery to its ideal operating temperature range may become unacceptable. As can be seen from Figure 9, there is a big advantage to the cooling system when increasing the number of turns for 5 to say 15, but after that, the cooling effect of increasing the number of turns is greatly diminished. Increasing the number of turns is equal to lengthening the heat transfer path. According to Figure 9, as the number of turns increases, the amount of the heat transfer coefficient decreases significantly and after, say, 15 turns, it remains almost constant. The Prandtl number for water is larger than the one which would make the thermal entrance greater than the hydraulic entrance length. This means that after about 15 turns the thermal entrance length has been passed. An optimum number of turns appear to be around 10. The rest of this study continues with the number of helix turns in the aluminium block to be 10.

Figure 9.

Temperature values at the centre of a cell for different number of helix coil turn after 50 s. Here, $T_{\mathit{\text{init}}}=313.15$K, $u_{\mathit{\text{in}}}=0.01$ m/s, $r_{\mathit{\text{inner}}}=10$ mm, $T_{\infty }=293.15$ K, $T_{\mathit{\text{in}}}=293.15$ K.

It is important for the lengthening of battery cell life and the enhancement of charging and discharging performance that uniformity of temperature is achieved throughout each cell, in addition to uniformity of cell temperatures across the cells within the battery module. To confirm that uniformity of temperature could be achieved across a single cell lodged within the battery module, temperature profiles were calculated in the radial direction through the battery module. As can be seen from Figure 10, where the radius at 0 is the battery module centre and radius at 130 mm is the outer wall of the module, the temperature profiles gradually move from an initial profile distorted by the hot cell to an acceptable final uniform distribution after about 15 s.

Figure 10.

Temperature profiles in the radial direction for $T_{\mathit{\text{init}}}=313.15$K, $u_{\mathit{\text{in}}}=0.01$ m/s, $r_{\mathit{\text{inner}}}=10$ mm, $T_{\infty }=293.15$ K, $T_{\mathit{\text{in}}}=293.15$ K.

Another series of tests were conducted on the battery module which had, in addition to initial temperatures of 313.15 and 349.15 K, an internal heat source for each cell of either 0.25, 0.5 or 1 W. Results for the module with each of the cells having internal heat sources of 1 W are shown in Figures 11 and 12. As with the previous tests, what was important was the control of temperature between acceptable limits and a good uniformity of temperature across each cell.

Figure 11.

Front view of temperature profiles for $T_{\mathit{\text{init}}}$ =313.15 K, $u_{\mathit{\text{in}}}=0.01\mathrm{m}/\mathrm{s},r_{\mathit{\text{inner}}}=10\mathrm{mm},T_{\mathit{\text{in}}}=293.15\mathrm{K},\text{and heat source within each cell of}1\mathrm{W}.$

Figure 12.

Plan view of temperature profiles for $T_{\mathit{\text{init}}}$ =313.15 K, $u_{\mathit{\text{in}}}=0.01\mathrm{m}/\mathrm{s},r_{\mathit{\text{inner}}}=10\mathrm{mm},T_{\mathit{\text{in}}}=293.15\mathrm{K},\text{and heat}\text{source within each cell of}1\mathrm{W}.$

It can be seen from Figures 11 and 12 that in the early stages of cooling, non-uniformity was found, but after, say, 1 min, uniformity was acceptable throughout each cell, and after 2 min, each cell was within the desired operating temperature limits (Note: each sub-figure has its own temperature scale). At times greater than 2 min, the heat loss to the atmosphere was slightly higher than heat production within the cells, even at 1 W, so reducing the need for further cooling. In the propose prototype, the temperature would be monitored using a thermostat, and further cooling would ensue intermittently as necessary.

4. Conclusion(s)

Preliminary results useful to the final design of a prototype battery module have been produced. The values found for the important parameters help in confirming the chosen geometry, and give indications of necessary pump and heating/refrigeration specifications needed when assembling the prototype. The calculations were made first inside a given battery cell to indicate what sort of temperature differences may be expected. It was found that these calculations depended heavily on experimental work to find appropriate coefficients for the coupled equations. After getting the range of temperatures arising from single battery cell calculations, a method was developed to find the temperature characteristics of the battery module, with stress being put on uniformity of temperature both within an individual cell and across the complete battery module.

Conflict of interest

There is no conflict of interest associated with this chapter.

Nomenclature

a	specific surface area, m−2
c	electrolyte concentration in the solution phase, mol−3
cs	Li concentration in electrode particles, mol m−3
C1,C2	k -ε turbulence model constants
Cp	specific heat
Cμ	turbulence model coefficient
Deff	effective diffusion coefficient of Li in the electrode
Ds	diffusion coefficient of Li in electrode particles, m2 s−1
E	potential
F	Faraday’s constant
h	heat transfer coefficient
I	current
J	surface reaction flux, mol m−2 s−1
Ji	pore wall flux
k	turbulence kinetic energy
kp	turbulence kinetic energy at near wall node
kT	lumped thermal conductivity of cell
Ln	anode thickness, m
Lp	cathode thickness, m
Ls	separator thickness
Pr	Prandtl number
qw	wall heat flux
Q˙rea	heat generation due to electrochemical reaction
Q˙rev	reversible heat
Q˙ohm	ohmic heat
r	radius/radial
R	universal gas constant
t	time
t+0	transference number
T	temperature
T∞	ambient temperature
T0	reference temperature
Tw	wall temperature
U	mean velocity
Ui	open circuit voltage
xi	spatial coordinates
yv	viscous sublayer thickness
y+	dimensionless velocity
β	constant, Boussinesq model
ε	dissipation
ε′	volume fraction of electrolyte
ηi	overpotential
κ	von Karman constant
λ	coefficient, Newton’s cooling law
μ	absolute viscosity
ρ	density
ρ0	reference density
σeff	effective electric conductivity
σk	constant in turbulence model
σε	constant in turbulence model
τw	wall shear stress
ϕ	potential
Γφ	effective exchange coefficient