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.

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

Figure 1. Schematic of a lithium ion battery.

$$\frac{\partial \mathbf{c}\_{s,i}}{\partial t} = D\_{s,i} \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \mathbf{c}\_{s,i}}{\partial r} \right) , \tag{1}$$

The boundary conditions are expressed as

∂ϕs, <sup>p</sup> ∂x x¼0

� ∂ ∂x

κeff ,i

concentration of the electrolyte in the liquid phase [27]

�κeff ,p

Ji ¼ ki cs,i,maz � csj

�λ ∂T ∂x x¼0

heat generation rate. The heat fluxes are defined by

heat generation rate, Q\_

∂ϕ2,i ∂x 

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

∂ϕ2, <sup>p</sup> ∂x x¼0

r¼rs

where η<sup>i</sup> is the over-potential of battery electrodes and is given by

with the boundary conditions determined by Newton's cooling law

where h is the heat transfer coefficient, T<sup>∞</sup> is the ambient temperature, Q\_

<sup>¼</sup> h Tð Þ <sup>∞</sup> � <sup>T</sup> , �<sup>λ</sup> <sup>∂</sup><sup>T</sup>

∙csj αc r¼rs ∙c <sup>α</sup><sup>a</sup> exp

<sup>α</sup><sup>a</sup>

rCp ∂T

¼ Iapp, � σeff , <sup>p</sup>

∂ϕs,p ∂x x¼Lp

The potential of the solid phase at the right end of the cell (Figure 1) is set to zero,

where i ¼ p, s and n, and, the specific conductivity of the electrolyte is a function of the

κeff ,i ¼ κiε

¼ 0, � κeff ,n

In the abovementioned equations, the pore wall flux, Ji is determined by the Butler-Volmer

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

> ∂x

rev is the total reversible heat generation rate, Q\_

x¼LpþLsþLn

∂ ∂x

bruggi

∂ϕ2,n ∂x 

αaFη<sup>i</sup>

κeff ,i

∂ð Þ ln ci ∂x 

x¼LpþLsþLn

RT � exp � <sup>α</sup>cFη<sup>i</sup>

η<sup>i</sup> ¼ ϕs,i � ϕ2,i � Ui (14)

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∇</sup>∙∇ð Þþ kTT Qrea <sup>þ</sup> Qrev <sup>þ</sup> Qohm (15)

RT (13)

= 0 and the potential of the solid phase at x ¼ 0, ϕ1,p

2RT 1 � t <sup>þ</sup> ð Þ Fd

charge balance in the liquid phase is based on Ohm's law, and it is given by

þ

¼ 0, � σeff ,n

∂ϕs,n ∂x x¼LpþLs

Effectiveness of a Helix Tube to Water Cool a Battery Module

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> ¼ Fd Fi

<sup>i</sup> (11)

¼ 0 (9)

131

<sup>x</sup>¼<sup>0</sup> is equal to Ecell. The

¼ 0 (12)

¼ h Tð Þ � T<sup>∞</sup> (16)

rea is the total reaction

ohm is the total ohmic

Ji (10)

�σeff ,p

ϕs,n 

equation

x¼LpþLsþLn

where i ¼ p, s and i ¼ 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

$$\left. -D\_{s,i} \frac{\partial c\_{s,i}}{\partial r} \right|\_{r=0} = 0, \quad -D\_{s,i} \frac{\partial c\_{s,i}}{\partial r} \Big|\_{r=r\_s} = J\_{\prime} \tag{2}$$

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

$$\varepsilon'\_{i}\frac{\partial c\_{i}}{\partial t} = D\_{\text{eff},i}\frac{\partial^{2}c\_{i}}{\partial x^{2}} + (1 - t\_{+}^{0})a\_{i}l\_{i\prime} \tag{3}$$

where i ¼ p, s and n and ai are the electrode surface area per unit volume of the electrode. In the separator, the pore wall flux Js is equal to zero, and at the two ends of the cell in the x-direction, there is no mass flux

$$\left. -D\_{\epsilon \mathcal{Y}, p} \frac{\partial c\_p}{\partial \mathbf{x}} \right|\_{\mathbf{x}=0} = \mathbf{0}, \quad -D\_{\epsilon \mathcal{Y}, n} \frac{\partial c\_n}{\partial \mathbf{x}} \Big|\_{\mathbf{x}=L\_p + L\_o + L\_n} = \mathbf{0} \tag{4}$$

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

$$\left.c\_{\mathcal{P}}\right|\_{\mathbf{x}=L\_{p}^{-}} = \left.c\_{s}\right|\_{\mathbf{x}=L\_{p}^{+}} \qquad \left.\left.c\_{s}\right|\_{\mathbf{x}=\left(L\_{p}+L\_{s}\right)^{-}} = \left.c\_{n}\right|\_{\mathbf{x}=\left(L\_{p}+L\_{s}\right)^{+}} \tag{5}$$

$$-D\_{\epsilon\mathcal{Y},p} \frac{\partial \mathbf{c}\_p}{\partial \mathbf{x}}\bigg|\_{\mathbf{x}=L\_p^-} = -D\_{\epsilon\mathcal{Y},s} \frac{\partial \mathbf{c}\_s}{\partial \mathbf{x}}\bigg|\_{\mathbf{x}=L\_p^+} \tag{6}$$

$$-D\_{\mathfrak{eff},s} \frac{\mathfrak{dc}\_s}{\mathfrak{d}\mathfrak{x}}\Big|\_{\mathfrak{x}=L\_p^-} = -D\_{\mathfrak{eff},n} \frac{\mathfrak{dc}\_n}{\mathfrak{d}\mathfrak{x}}\Big|\_{\mathfrak{x}=\left(L\_p+L\_q\right)^+} \tag{7}$$

The effective diffusion coefficient, Deff of Li in the electrode can be represented as Deff ,i ¼ DE,iε bruggi <sup>i</sup> . The specific surface area for the electrode particles, a, is given by a ¼ 3εs=rs: The charge balance in the solid phase is governed by Ohm's law

$$
\sigma\_{\it eff,i} \frac{\partial^2 \phi\_{s,i}}{\partial x^2} = a\_i F l\_i \tag{8}
$$

where i ¼ p and n. Here σeff is the effective electric conductivity and is given by σeff ¼ σ∙εs.

The boundary conditions are expressed as

∂cs,i <sup>∂</sup><sup>t</sup> <sup>¼</sup> Ds,i

130 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

�Ds,i

ε0 i ∂ci <sup>∂</sup><sup>t</sup> <sup>¼</sup> Deff ,i

∂cp ∂x x¼0

the concentration of the binary electrolyte and its flux is continuous

�Deff ,p

∂cs ∂x x¼L� p

�Deff ,s

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

�Deff , <sup>p</sup>

cp x¼L� p ¼ csj x¼L<sup>þ</sup>

∂cs,i ∂r r¼0

surface giving the boundary conditions

can be expressed as

Deff ,i ¼ DE,iε

bruggi

x-direction, there is no mass flux

1 r2 ∂ ∂r r <sup>2</sup> ∂cs,i ∂r 

where i ¼ p, s and i ¼ 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

¼ 0, � Ds,i

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

> ∂2 ci <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>1</sup> � <sup>t</sup>

where i ¼ p, s and n and ai are the electrode surface area per unit volume of the electrode. In the separator, the pore wall flux Js is equal to zero, and at the two ends of the cell in the

¼ 0, � Deff ,n

At the interfaces between the positive electrode/separator and separator/negative electrode,

<sup>x</sup>¼ð Þ LpþLs

¼ �Deff ,s

¼ �Deff ,n

The effective diffusion coefficient, Deff of Li in the electrode can be represented as

<sup>p</sup> , csj

∂cp ∂x x¼L� p

σeff ,i ∂2 ϕs,i

where i ¼ p and n. Here σeff is the effective electric conductivity and is given by σeff ¼ σ∙εs.

∂cs,i ∂r r¼rs

> 0 þ

∂cn ∂x 

x¼LpþLsþLn

� ¼ cnj

∂cs ∂x x¼L<sup>þ</sup> p

∂cn ∂x <sup>x</sup>¼ð Þ LpþLs þ

<sup>i</sup> . The specific surface area for the electrode particles, a, is given by a ¼ 3εs=rs:

<sup>x</sup>¼ð Þ LpþLs þ

<sup>∂</sup>x<sup>2</sup> <sup>¼</sup> aiFJi (8)

, (1)

¼ J, (2)

¼ 0 (4)

, (5)

(6)

(7)

aiJi, (3)

$$\left. -\sigma\_{\text{eff},p} \frac{\partial \phi\_{s,p}}{\partial \mathbf{x}} \right|\_{\mathbf{x}=0} = I\_{\text{app}} \quad -\sigma\_{\text{eff},p} \frac{\partial \phi\_{s,p}}{\partial \mathbf{x}} \bigg|\_{\mathbf{x}=L\_p} = 0, \quad -\sigma\_{\text{eff},n} \frac{\partial \phi\_{s,n}}{\partial \mathbf{x}} \bigg|\_{\mathbf{x}=L\_p+L\_s} = 0 \tag{9}$$

The potential of the solid phase at the right end of the cell (Figure 1) is set to zero, ϕs,n x¼LpþLsþLn = 0 and the potential of the solid phase at x ¼ 0, ϕ1,p <sup>x</sup>¼<sup>0</sup> is equal to Ecell. The charge balance in the liquid phase is based on Ohm's law, and it is given by

$$-\frac{\partial}{\partial \mathbf{x}} \left( \kappa\_{\varepsilon \| \boldsymbol{f}, i} \frac{\partial \phi\_{2,i}}{\partial \mathbf{x}} \right) + \frac{2RT(1 - t^{+})}{F\_{d}} \frac{\partial}{\partial \mathbf{x}} \left( \kappa\_{\varepsilon \| \boldsymbol{f}, i} \frac{\partial (\ln c\_{i})}{\partial \mathbf{x}} \right) = \frac{F\_{d}}{F\_{i}} I\_{i} \tag{10}$$

where i ¼ 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]

$$\kappa\_{\mathfrak{e}\mathfrak{f},i} = \kappa\_i \varepsilon\_i^{brugg\_i} \tag{11}$$

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

$$\left. -\kappa\_{\text{eff},p} \frac{\partial \phi\_{2,p}}{\partial x} \right|\_{x=0} = 0, \quad -\kappa\_{\text{eff},n} \frac{\partial \phi\_{2,n}}{\partial x} \Big|\_{x=L\_p + L\_q + L\_a} = 0 \tag{12}$$

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

$$J\_i = k\_i \left( \left. c\_{s,i,max} - \left. c\_s \right|\_{r=r\_s} \right)^{a\_s} \cdot c\_s \right|\_{r=r\_s}^{a\_c} \cdot c^{a\_s} \left\{ \exp \left( \frac{a\_d F \eta\_i}{RT} \right) - \exp \left( - \frac{a\_c F \eta\_i}{RT} \right) \right\} \tag{13}$$

where η<sup>i</sup> is the over-potential of battery electrodes and is given by

$$
\eta\_i = \phi\_{s,i} - \phi\_{\mathfrak{Z},i} - \mathcal{U}\_i \tag{14}
$$

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

$$
\rho \mathbf{C}\_p \frac{\partial T}{\partial t} = \nabla \cdot \nabla (k\_T T) + Q\_{\text{ra}} + Q\_{\text{rev}} + Q\_{\text{ohm}} \tag{15}
$$

with the boundary conditions determined by Newton's cooling law

$$-\lambda \frac{\partial T}{\partial \mathbf{x}}\Big|\_{\mathbf{x}=0} = h(T\_{\infty} - T), \qquad -\lambda \frac{\partial T}{\partial \mathbf{x}}\Big|\_{\mathbf{x}=L\_{p}+L\_{q}+L\_{\mathbf{u}}} = h(T - T\_{\infty})\tag{16}$$

where h is the heat transfer coefficient, T<sup>∞</sup> is the ambient temperature, Q\_ rea is the total reaction heat generation rate, Q\_ rev is the total reversible heat generation rate, Q\_ ohm is the total ohmic heat generation rate. The heat fluxes are defined by

$$
\dot{Q}\_{rea} = a \text{F} \mathfrak{J}\_i \mathfrak{\eta}\_i \tag{17}
$$

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

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

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

where r is the liquid density, ui is the velocity vector components, Γ<sup>φ</sup> is the effective exchange coefficient of φ and S<sup>φ</sup> is the source rate per unit volume. The source rate and the effective exchange coefficient corresponding to each variable φ solved in this study are given in Table 1.

In Table 1, μ is the viscosity, σ is the Prandtl number for ε and Gk ¼ μ ∂ui=∂xj þ ∂uj=∂xi

 the turbulence production rate. The values of the constants C<sup>1</sup> and C<sup>2</sup> are 1.44 and 1.92 respectively and for σ<sup>k</sup> and σε, 1.0 and 1.3 respectively [28]. The eddy viscosity term is

μ<sup>t</sup> ¼ rC<sup>μ</sup>

Equation φ S<sup>φ</sup> Γ<sup>φ</sup> Continuity 1 0 0

Kinetic energy of turbulence <sup>k</sup> Gk � <sup>r</sup><sup>ε</sup> <sup>μ</sup>

k2

∂xi

k <sup>G</sup> � <sup>C</sup>2<sup>r</sup> <sup>ε</sup><sup>2</sup>

∂ ∂xi

Γφ ∂φ ∂xi 

þ S<sup>φ</sup> (20)

Effectiveness of a Helix Tube to Water Cool a Battery Module

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133

<sup>ε</sup> (21)

<sup>þ</sup> <sup>r</sup>ref gi <sup>r</sup> <sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup>

<sup>μ</sup>

Dt <sup>þ</sup> heat sources sinks ð Þ <sup>r</sup> <sup>μ</sup><sup>t</sup>

k

<sup>σ</sup><sup>t</sup> <sup>þ</sup> <sup>μ</sup> Pr

σk

σε

ð Þ¼ ruiφ

Included in Table 1 are the transport equations for the standard k-ε turbulence model.

potentially hazardous gases.

the general conservation form

∂ ∂t

Momentum Ui <sup>∂</sup><sup>p</sup>

Enthalpy h Dp

Eddy dissipation rate <sup>ε</sup> <sup>C</sup><sup>1</sup> <sup>ε</sup>

Table 1. Source rate and effective exchange coefficient for each φ.

ð Þþ rφ

∂ ∂xi

2.2.1. Fluid section

∂ui=∂xj

where C<sup>μ</sup> ¼ 0:09.

$$
\dot{Q}\_{rev} = aFf\_i T \cdot \frac{dI I\_i}{dT} \tag{18}
$$

$$\dot{Q}\_{\text{ohm}} = \sigma\_{\text{eff}} \left( \frac{\partial \phi\_s}{\partial \mathbf{x}} \right)^2 + \kappa\_{\text{eff}} \left( \frac{\partial \phi\_2}{\partial \mathbf{x}} \right)^2 + \frac{2k\_{\text{eff}}RT \left( 1 - l\_+^0 \right)}{F} \left( 1 + \frac{d \ln f\_\pm}{d \ln c} \right) \frac{\partial \ln c}{\partial \mathbf{x}} \frac{\partial \phi\_2}{\partial \mathbf{x}} + aF\_l \Delta \phi\_{\text{SEI}} \tag{19}$$

#### 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.

Figure 3. Schematic of thermal management system.

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.

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

#### 2.2.1. Fluid section

Q\_

rev ¼ aFJiT∙

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

2keffRT 1 � t

dUi

0 þ <sup>F</sup> <sup>1</sup> <sup>þ</sup>

Q\_

þ

132 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

<sup>Q</sup>\_ ohm <sup>¼</sup> <sup>σ</sup>eff

∂ϕ<sup>s</sup> ∂x <sup>2</sup>

þ κeff

2.2. Modelling within the aluminium block

∂ϕ<sup>2</sup> ∂x <sup>2</sup>

block for efficient heat transfer and protection against damage.

Figure 2. Copper helix coil within the aluminium block a. front and b. plan.

Figure 3. Schematic of thermal management system.

rea ¼ aFJiη<sup>i</sup> (17)

∂x

<sup>d</sup> ln <sup>f</sup> � d ln c ∂ ln c

dT (18)

∂ϕ<sup>2</sup> ∂x

þ aFJiΔϕSEI (19)

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

$$\frac{\partial}{\partial t}(\rho \rho) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho u\_i \rho) = \frac{\partial}{\partial \mathbf{x}\_i} \left(\Gamma\_\phi \frac{\partial \rho}{\partial \mathbf{x}\_i}\right) + \mathcal{S}\_\phi\tag{20}$$

where r is the liquid density, ui is the velocity vector components, Γ<sup>φ</sup> is the effective exchange coefficient of φ and S<sup>φ</sup> is the source rate per unit volume. The source rate and the effective exchange coefficient corresponding to each variable φ solved in this study are given in Table 1. Included in Table 1 are the transport equations for the standard k-ε turbulence model.

In Table 1, μ is the viscosity, σ is the Prandtl number for ε and Gk ¼ μ ∂ui=∂xj þ ∂uj=∂xi ∂ui=∂xj the turbulence production rate. The values of the constants C<sup>1</sup> and C<sup>2</sup> are 1.44 and 1.92 respectively and for σ<sup>k</sup> and σε, 1.0 and 1.3 respectively [28]. The eddy viscosity term is

$$
\mu\_t = \rho \mathbf{C}\_{\mu} \frac{k^2}{\varepsilon} \tag{21}
$$

where C<sup>μ</sup> ¼ 0:09.


Table 1. Source rate and effective exchange coefficient for each φ.

In addition to the standard k-ε turbulence model, the realisable k-ε 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-ε turbulence model are

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial \mathbf{x}\_j}(\rho k u\_j) = \frac{\partial}{\partial \mathbf{x}\_j} \left( \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right) + \mathbf{G}\_k - \rho \varepsilon \tag{22}$$

$$\frac{\partial}{\partial t}(\rho \varepsilon) + \frac{\partial}{\partial \mathbf{x}\_j}(\rho \varepsilon u\_j) = \frac{\partial}{\partial \mathbf{x}\_j} \left( \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right) + \mathbf{C}\_1 \rho \mathbf{S}\_\varepsilon - \mathbf{C}\_2 \rho \frac{\varepsilon^2}{k + \sqrt{\mu \varepsilon}} \tag{23}$$

$$
\mu\_t = \rho \mathbb{C}\_{\mu} \frac{k^2}{\varepsilon} = \frac{1}{4.04 + A\_s k u^\*/\varepsilon} \tag{24}
$$

the turbulence kinetic energy at the near-wall node P, yp is the distance from the point P to the wall, and μ 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-

> 8 < :

Prt 1 κ

> Prt � ��1=<sup>4</sup>

<sup>T</sup> is the dimensionless thermal sublayer thickness, Cp is the specific heat of the fluid, qw

the wall heat flux, Tp is the temperature at the first near-wall node P, Tw is the temperature of

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 nonequilibrium. 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

> κ E rC<sup>1</sup>=<sup>4</sup> <sup>μ</sup> k 1=2 y

<sup>y</sup><sup>ν</sup> <sup>¼</sup> <sup>μ</sup>y<sup>∗</sup>

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

ν rC<sup>1</sup>=<sup>4</sup> <sup>μ</sup> <sup>k</sup><sup>1</sup>=<sup>2</sup> p

yν rκ ffiffi k <sup>p</sup> ln <sup>y</sup> yν � �

μ !

<sup>þ</sup> <sup>y</sup> � <sup>y</sup><sup>ν</sup> rκ ffiffi k <sup>p</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> ν μ � � (34)

r � r<sup>0</sup> ð Þg ¼ �r0βð Þ T � T<sup>0</sup> g (36)

Pry<sup>∗</sup> <sup>ð</sup>y<sup>∗</sup> <sup>&</sup>lt; <sup>y</sup><sup>∗</sup>

ln Ey<sup>∗</sup> ð Þþ <sup>P</sup> � � T

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(31)

135

(32)

(33)

(35)

<sup>ð</sup>y<sup>∗</sup> <sup>&</sup>gt; <sup>y</sup><sup>∗</sup> T

Effectiveness of a Helix Tube to Water Cool a Battery Module

), a<sup>0</sup> is the coefficient of heat diffusion,

the-wall for the temperature field has the following composite form

qw=rCp � � ¼

<sup>μ</sup> k 1=2 p � �

<sup>P</sup> <sup>¼</sup> <sup>9</sup>:<sup>24</sup> Pr

Prt � 1 � � Pr

Tw � Tp � � C<sup>1</sup>=<sup>4</sup>

the wall, Pr is the molecular Prandtl number (¼ μCp=a<sup>0</sup>

velocity sensitised to the pressure gradient is

and Prt is the turbulent Prandtl number (�0.85 at the wall).

UC~ <sup>1</sup>=<sup>4</sup> <sup>μ</sup> k 1=2 <sup>τ</sup>w=<sup>r</sup> <sup>¼</sup> <sup>1</sup>

> 2 dp dx

and y<sup>ν</sup> is the physical viscous sublayer thickness, and computed from

<sup>U</sup><sup>~</sup> <sup>¼</sup> <sup>U</sup> � <sup>1</sup>

<sup>T</sup><sup>∗</sup> <sup>¼</sup>

where P is given by Jayatilleke [33]

where y<sup>∗</sup>

where

where y<sup>∗</sup>

<sup>ν</sup> ¼ 11:225.

buoyancy term in the momentum equation

$$
\mu^\* = \sqrt{\mathcal{S}\_{\vec{\eta}} \mathcal{S}\_{\vec{\eta}} \widetilde{\mathcal{Q}}\_{\vec{\eta}} \widetilde{\mathcal{Q}}\_{\vec{\eta}}} \quad \tilde{\Omega}\_{\vec{\eta}} = \Omega\_{\vec{\eta}} - 2\varepsilon\_{\vec{\eta}k} \omega\_k \tag{25}
$$

$$A\_s = \sqrt{6}\cos q', \quad \mathbf{q'} = \frac{1}{3}\cos^{-1}\left(\sqrt{6}\mathcal{W}\right), \quad \mathbf{W} = \frac{S\_{\hat{\mathbb{W}}}S\_{\hat{\mathbb{K}}}S\_{\hat{\mathbb{K}}}}{\hat{\mathbb{S}}^3} \tag{26}$$

$$\tilde{S} = \sqrt{S\_{\tilde{\eta}} S\_{\tilde{\eta}\prime}} \quad \mathcal{C}\_1 = \max\left[0.43, \frac{\tilde{\mu}}{\tilde{\mu} + \mathfrak{F}}\right] \tag{27}$$

where σ<sup>k</sup> ¼ 1:0, σε ¼ 1:2 and C<sup>2</sup> ¼ 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

$$\mathcal{U}^\* = \begin{cases} \mathcal{Y}^\* & (\mathcal{Y}^\* > 11.225\\ 1 \frac{1}{\mathcal{X}} \ln \left( E \mathcal{Y}^\* \right) & (\mathcal{Y}^\* < 11.225 \end{cases} \tag{28}$$

where

$$\mathcal{U}^\* = \frac{\mathcal{U}\_p \mathbb{C}\_{\mu} k\_p^{1/2}}{\tau\_w / \rho} \tag{29}$$

is the dimensionless velocity, and

$$y^\* = \frac{\rho \mathbb{C}\_{\mu}^{1/4} k\_p^{1/2} y\_p}{\mu} \tag{30}$$

is the dimensionless distance from the wall and κ is the von Karman constant (=0.4187), E is the empirical constant (=0.9793), Up is the mean velocity of the fluid at the near-wall node P, kp is the turbulence kinetic energy at the near-wall node P, yp is the distance from the point P to the wall, and μ 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-ofthe-wall for the temperature field has the following composite form

$$T^\* = \frac{\left(T\_w - T\_p\right)\left(\mathbb{C}\_{\mu}^{1/4}k\_p^{1/2}\right)}{\left(q\_w/\rho\mathbb{C}\_p\right)} = \begin{cases} Pry^\* & \left(y^\* < y\_T^\*\right) \\ Pr\_t\left[\frac{1}{\mathbb{K}}\ln\left(Ey^\*\right) + P\right] & \left(y^\* > y\_T^\*\right) \end{cases} \tag{31}$$

where P is given by Jayatilleke [33]

$$P = 9.24 \left( \frac{Pr}{Pr\_t} - 1 \right) \left( \frac{Pr}{Pr\_t} \right)^{-1/4} \tag{32}$$

where y<sup>∗</sup> <sup>T</sup> is the dimensionless thermal sublayer thickness, Cp is the specific heat of the fluid, qw the wall heat flux, Tp is the temperature at the first near-wall node P, Tw is the temperature of the wall, Pr is the molecular Prandtl number (¼ μCp=a<sup>0</sup> ), a<sup>0</sup> is the coefficient of heat diffusion, and Prt 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 nonequilibrium. 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

$$\frac{\tilde{\mathcal{U}}\mathcal{C}\_{\mu}^{1/4}k^{1/2}}{\tau\_w/\rho} = \frac{1}{\kappa} \left( E \frac{\rho \mathcal{C}\_{\mu}^{1/4}k^{1/2}y}{\mu} \right) \tag{33}$$

where

In addition to the standard k-ε turbulence model, the realisable k-ε 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-ε turbulence model are

∂xj

k2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SijSijΩ~ ijΩ~ ij

, <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffi SijSij q

> 1 κ

8 < :

<sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σε � � ∂ε

� �

<sup>ε</sup> <sup>¼</sup> <sup>1</sup>

<sup>3</sup> cos �<sup>1</sup> ffiffiffi

where σ<sup>k</sup> ¼ 1:0, σε ¼ 1:2 and C<sup>2</sup> ¼ 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

<sup>U</sup><sup>∗</sup> <sup>¼</sup> UpCμk<sup>1</sup>=<sup>2</sup>

<sup>y</sup><sup>∗</sup> <sup>¼</sup> <sup>r</sup>C<sup>1</sup>=<sup>4</sup>

is the dimensionless distance from the wall and κ is the von Karman constant (=0.4187), E is the empirical constant (=0.9793), Up is the mean velocity of the fluid at the near-wall node P, kp is

, C<sup>1</sup> <sup>¼</sup> max 0:43; <sup>μ</sup><sup>~</sup>

<sup>y</sup><sup>∗</sup> <sup>ð</sup>y<sup>∗</sup> <sup>&</sup>gt; <sup>11</sup>:<sup>225</sup>

ln Ey<sup>∗</sup> ð Þðy<sup>∗</sup> <sup>&</sup>lt; <sup>11</sup>:<sup>225</sup>

p τw=r

<sup>μ</sup> <sup>k</sup><sup>1</sup>=<sup>2</sup> <sup>p</sup> yp

6 <sup>p</sup> <sup>W</sup> � �

�

<sup>μ</sup> <sup>þ</sup> <sup>μ</sup><sup>t</sup> σk � � ∂k

∂xj

� �

∂xj

þ C1rS<sup>ε</sup> � C2r

þ Gk � rε (22)

με <sup>p</sup> (23)

<sup>S</sup>~<sup>3</sup> (26)

(27)

(28)

(29)

ε2 k þ ffiffiffiffiffi

<sup>4</sup>:<sup>04</sup> <sup>þ</sup> Asku<sup>∗</sup>=<sup>ε</sup> (24)

, <sup>Ω</sup><sup>~</sup> ij <sup>¼</sup> <sup>Ω</sup>ij � <sup>2</sup>εijkω<sup>k</sup> (25)

<sup>μ</sup> (30)

, W <sup>¼</sup> SijSikSjk

μ~ þ 5 �

rkuj � � <sup>¼</sup> <sup>∂</sup>

134 Heat and Mass Transfer - Advances in Modelling and Experimental Study for Industrial Applications

∂xj

μ<sup>t</sup> ¼ rC<sup>μ</sup>

q

∂ ∂t

∂ ∂t ð Þþ rε

where

is the dimensionless velocity, and

ð Þþ <sup>r</sup><sup>k</sup> <sup>∂</sup> ∂xj

> rεuj � � <sup>¼</sup> <sup>∂</sup>

<sup>u</sup><sup>∗</sup> <sup>¼</sup>

<sup>S</sup><sup>~</sup> <sup>¼</sup>

wall-boundary flows. The law-of-the-wall for mean velocity gives

<sup>U</sup><sup>∗</sup> <sup>¼</sup>

∂ ∂xj

As <sup>¼</sup> ffiffiffi 6 <sup>p</sup> cosφ<sup>0</sup>

$$\tilde{\mathcal{U}} = \mathcal{U} - \frac{1}{2} \frac{dp}{d\mathbf{x}} \left[ \frac{y\_v}{\rho \kappa \sqrt{k}} \ln \left( \frac{y}{y\_v} \right) + \frac{y - y\_v}{\rho \kappa \sqrt{k}} + \frac{y\_v^2}{\mu} \right] \tag{34}$$

and y<sup>ν</sup> is the physical viscous sublayer thickness, and computed from

$$y\_v = \frac{\mu y\_v^\*}{\rho \mathbb{C}\_{\mu}^{1/4} k\_p^{1/2}} \tag{35}$$

where y<sup>∗</sup> <sup>ν</sup> ¼ 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

$$(\rho - \rho\_0)\mathbf{g} = -\rho\_0\mathcal{J}(T - T\_0)\mathbf{g} \tag{36}$$

where <sup>r</sup><sup>0</sup> is the reference density of the water flow (kg/m<sup>3</sup> ); T<sup>0</sup> is the reference temperature (K); and, Eq. (37) is obtained by the Boussinesq approximation <sup>r</sup> <sup>¼</sup> <sup>r</sup><sup>0</sup> <sup>1</sup> � <sup>β</sup>Δ<sup>T</sup> to replace the buoyancy terms. This approximation is acceptable so long as changes in actual density are small. Specifically, it is valid when βð Þ T � T<sup>0</sup> ≪ 1, and should not be used if the temperature difference in the domain is large.
