Conceptualisation, Investigation, Methodology, Roles/Writing - original draft
[orcid=0000-0002-3054-8638] Funding acquisition, Conceptualisation,
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# Multifaceted thermal regulation in electrochemical batteries using cooling
channels and foam-embedded phase change materials

Mehdi V. Bozorg  Juan F. Torres  ANU HEAT Lab, School of Engineering, The
Australian National University, Canberra, Australia

###### Abstract

Lithium-ion batteries are widely used in electric vehicles and grid energy
storage systems. Compared to cylindrical batteries, prismatic cells are the
primary choice because of their advantage for dense packing. However, thermal
runaway and temperature inhomogeneities are the main thermal regulation
problems that affect their reliability, safety, and useful life. Here, we
propose and assess a multifaceted cooling system composed of water channels
(active cooling) and metallic foam embedded with two types of phase-change
materials or PCMs (passive cooling) with different melting points. We show
that a multifaceted thermal regulation strategy can improve both cooling
effectiveness and temperature homogeneity through a representative prismatic
battery module. Our numerical results indicate that for a battery pack cooled
with a water channel ($3\text{\,}\mathrm{C}$ discharge rate), a dual-PCM
arrangement can reduce the maximum temperature by
$1.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$2.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ compared to a mono-PCM arrangement
and a battery pack without PCM, respectively. The maximum temperature
difference within the cell is also
$1.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Therefore, multi-PCM thermal
management systems show better performance than their mono-PCM predecessors in
terms of lowering the maximum battery temperature and improving thermal
homogeneity. This work motivates the development of multifaceted thermal
management systems with active and passive cooling to improve the long-term
performance of electrochemical battery cells.

###### keywords:

lithium–ion battery dual PCMs metal foam forced liquid cooling computational
fluid dynamics thermal management system

##  1 Introduction

Global warming is the main threat to Earth and comes from excessive carbon
emissions from fossil fuel consumption. Much research has targeted
decarbonisation strategies to reduce greenhouse gas emissions by developing
energy conversion technologies based on renewable resources that reduce our
dependence on hydrocarbon fuels. Moreover, more than one-fifth of global
carbon emissions are related to the transportation sector [16]. Therefore,
significant efforts have been made by engineers, scientists, entrepreneurs,
and governments to produce sustainable electric vehicles (EVs) [11]. Today,
hybrid electric vehicles (HEVs) and EVs are significantly promoting the
development of private and public transportation systems to globalise green
travel. The lithium ion (Li-ion) battery has become the industry standard for
EVs/HEVs to power their electric motors [6]. In addition, the electricity
generated by renewable resources could be stored in large-scale Li-ion
batteries to feed electricity into the grid [8]. For example, the surplus of
electricity produced from off-grid photovoltaic systems during peak solar
hours could be transferred to Li-ion battery packs and then used to charge
EVs. The advantages of Li-ion batteries include a low self-discharge rate,
relatively high energy density rates ($>$
$180\text{\,}\mathrm{W}\text{\,}\mathrm{h}\text{\,}{\mathrm{kg}}^{-1}$) [23],
large power-per-weight ($1.5\text{\,}\mathrm{kW}\text{\,}{\mathrm{kg}}^{-1}$)
[29], high efficiency, reliability, and a relatively long life cycle compared
to other electrochemical batteries. However, large-scale lithium batteries
still face numerous challenges, such as degradation [17], safety issues due to
instability at high temperatures (runaway) [10], production with
environmentally friendly materials [18], and recyclability [14].

The performance of Li-ion batteries is closely related to the maximum
temperature of the module and the homogeneity of the temperature
$(\Delta\textit{T}=\textit{T}_{\mathrm{max}}-\textit{T}_{\mathrm{min}})$
within the battery module. These temperatures are transient during the charge
and discharge phases. Consequently, exceeding a specific temperature range
deteriorates the reliability and safety of lithium ion batteries. Exothermic
reactions that trigger fire and explosion [24] and reduced lifetime due to
cell degradation (loss of battery capacity during charge / discharge cycles)
are the result of perturbations in electrochemical reactions whose magnitude
is affected by the temperature of the cell [26]. One of the most serious
challenges of Li-ion batteries is thermal runaway (TR), which is a major
safety concern. TR causes rapid battery heating [33], first at a local level
in a few battery cells within a battery module, and then the overheating
expands to the module array within a battery pack. Battery packs are installed
in EVs; while connected in large numbers, they can form a grid energy storage
system. Li-ion battery packs commonly used in EVs have been reported to
exhibit high-temperature variations [27]. Fire accidents due to TR in their
Li-ion batteries have been reported for many types of EVs from manufacturers
including Tesla (Model S and X), BYD Auto (model e6) [9], and even buses [30].
Figure 1 illustrates the concept of using battery packs, their thermal
regulation problems, the pros and cons of cooling strategies, and the thermal
management system of the battery proposed in this study. Note that in addition
to the TR problem, unbalanced charging occurs in Li-ion batteries subject to a
high temperature inhomogeneity due to variations in the temperature-dependent
thermophysical properties [12]. This unbalanced charging reduces battery
efficiency and reliability. Therefore, both maximum temperature
$(\textit{T}_{\mathrm{max}})$ and temperature uniformity $(\Delta\textit{T})$
are essential factors that need to be addressed when designing thermal
management systems for Li-ion batteries.

Battery thermal management systems (BTMS) generally have four cooling methods:
air-based, liquid-based, PCM-based, and heat pipes. As an active method, air-
based BTMS is generally limited to small electric vehicles [7]. However, the
low heat capacity of air can easily result in the inability of air-based BTMS
to prevent TR propagation [7] and large temperature inhomogeneities [38]. In
contrast, heat pipe BTMS have a high cooling capacity that ensures moderate
maximum temperatures, but their temperature uniformity is a problem because it
can lead to TR [35]. PCM-based BTMS are widely used passive methods that are
suitable for absorbing large amounts of heat generated from rapid
charge–discharge cycles. PCMs are excellent local heat absorbers as a result
of their large latent heat capacity. However, heat transfer in battery modules
with PCM-based cooling is poor because PCMs have low thermal conductivity.
After melting, the low heat capacity and poor thermal conductivity of PCMs
also lead to large temperature increases [24]. Therefore, due to safety
concerns caused by poor heat conduction, PCM-based cooling systems are
generally not recommended for BTMS [32]. Instead, liquid-based BTMS have
become the main choice in EVs due to their high heat transfer performance
[37], albeit there is no latent heat storage that could help mitigate hot
spots. To address the issues encountered by single thermal control strategies,
hybrid BTMSs have emerged [28].

The combination of liquid cooling and mono-PCM has been found to improve the
cooling performance of BTMS. Bai et al. [5] reported that for a Li-ion battery
module with water channels and PCM in discharge rate of
$2\text{\,}\mathrm{C}$, water cooling decreased the maximum temperature, while
the PCM contributed to the temperature uniformity of the module. Consequently,
Xin et al. [34] analysed the thermal performance of a combined liquid cooling-
PCM BTMS at high operating temperatures. Their results showed that high mass
flow rates of the coolant guaranteed safe temperature ranges during fast
discharging in ambient temperatures as high as
$40\text{\,}\mathrm{\SIUnitSymbolCelsius}$. However, a lower coolant mass flow
rate is desired to achieve an efficient BTMS because it reduces pumping power.
Although effective, high coolant mass flow rates can increase temperature
inhomogeneity within battery modules.

Integration of foam-embedded PCMs with liquid cooling BTMS significantly
enhances thermal conductivity, resulting in improvements in cooling
performance, particularly for lower coolant velocities. This improvement
ensures not only a reduction in maximum temperatures but also better
temperature uniformity. Zhao et al. [40] showed that the use of a copper
foam/paraffin-based BTMS in cylindrical Li-ion batteries at a very low
discharge rate of $0.5\text{\,}\mathrm{C}$ decreased the maximum temperature
at high ambient temperatures of up to
$45\text{\,}\mathrm{\SIUnitSymbolCelsius}$. However, a high mass flow rate of
the coolant increased the inhomogeneity of the temperature field. Another
drawback of their proposed battery pack is that only one type of PCM was
considered. In contrast, integration of the BTMS with dual PCMs could ensure
high cooling efficiencies in a wider range of ambient temperatures. In
addition, because of the lower space occupation of the battery pack, prismatic
battery cells are commonly used in EVs. Compared to cylindrical batteries,
prismatic cells exhibit weak active cooling performance with nonuniformity at
high temperature because of a smaller contact surface area with the coolant
channels.

Here, we propose and analyse the performance of a multifaceted BTMS comprised
of metal foam-embedded dual PCMs with coolant channels. The foam is made of
aluminium; the PCMs are 28HC and 35HC [21]; the coolant is water. This system
is designed for high-density rectangular battery cells and is investigated
using a computational fluid dynamics (CFD) model, which is verified and
validated in this work. To improve the PCM-based cooling systems introduced in
the literature, we integrated metal foam into the PCM to increase the thermal
conductivity of the PCM elements. In addition, using a dual-PCM configuration,
we increased the temperature flexibility of the PCM components. Our proposed
foam-embedded dual-PCM BTMS keeps the maximum temperature of the batteries in
a safe range. Because the BTMS operates efficiently at low coolant velocities,
high-temperature homogeneity is achieved. In fact, the second PCM with a
higher melting temperature decreases the battery temperatures during the end
of discharge, where the batteries generate excessive heat. In addition, a dual
PCM BTMS maintains a lower battery temperature when discharging at high
ambient temperatures. In general, the proposed multifaceted BTMS demonstrates
good performance in mitigating thermal regulation problems and improving
safety.

Figure 1: Concept of multifaceted thermal management strategies for
electrochemical batteries. (a) Applications of electrochemical batteries,
their stringent conditions, thermal regulation problems including thermal
runaway and inhomogeneity, as well as associated safety risks. (b)
Multifaceted cooling strategy for battery packs including advantages and
disadvantages of each element. The proposed battery thermal management system
is comprised of an active method using water channels, and a passive methods
using dual foam-embedded PCMs. Our hypothesis is an improved cooling
performance when implementing a multifaceted thermal management strategy.

##  2 Proposed multifaceted thermal management

In the battery pack shown in Fig. 2a, each battery module is composed of five
rectangular battery cells. The configuration of BTMS with PCM embedded in
metal foam and water channels is illustrated in Fig. 2b. The upper and lower
faces of the battery cells are perfectly insulated. The thermal and physical
properties of the battery cell, the aluminium frame, and the coolant are
listed in Table 1. To achieve an effective multifaceted BTMS, the performance
of the proposed active/passive cooling system is analysed for different PCMs
whose thermophysical properties are shown in Table 1. The candidate materials
for PCMs are selected based on the following criteria: (1) a melting
temperature near the operating conditions of the battery, (2) a high amount of
latent heat to minimise the volume of the required PCM and create a more
compact battery pack, (3) a high value of specific heat to maximise the amount
of sensible heat, and (4) cost-effectiveness. In our case, the thermal
conductivity of the PCMs is not important since metallic porous fillers are
integrated with the PCMs. To analyse the performance of the battery module
BTMS, the governing equations for mass, momentum, and energy are solved using
a CFD model based on a finite-volume method [25]. The boundary conditions
imposed on the model are shown in Fig. 2 c. Moreover, for discharge rates of
$1\text{\,}\mathrm{C}$, $2\text{\,}\mathrm{C}$ and $3\text{\,}\mathrm{C}$, the
heat generation of each battery cell during discharge is adopted from [20],
with Fig. 2d indicating the corresponding transients in volumetric heat
generation for each discharge rate. As shown in Fig. 2e, two BTMS
configurations are analysed: a single chamber with two porous PCM containers
near each water channel, and double chambers with an extra porous PCM column.
The first BTMS model has a water channel with dimensions of $3.75\times
180\times$1.5\text{\,}\mathrm{mm}$$ and a PCM container of dimensions of
$3.75\times 180\times$1.5\text{\,}\mathrm{mm}$$. The second BTMS model has a
water channel with dimensions of $3.75\times
180\times$1.5\text{\,}\mathrm{mm}$$ and two PCM containers of dimensions of
$3.75\times 180\times$1.5\text{\,}\mathrm{mm}$$ and $3.25\times
180\times$4\text{\,}\mathrm{mm}$$. These sizes were chosen to fit the
different components in the battery module and are depicted in Fig. 2b. In
addition, the double-chamber BTMS contains two types of PCMs with higher and
lower melting points to reduce temperature inhomogeneities and decrease the
maximum temperature in the cell.

Figure 2: Schematics of the proposed battery packs and integrated battery thermal management system (BTMS). The dimensions and different sections are indicated. (a) Battery pack. (b) Top view of a battery module section. (c) Boundary conditions imposed to the computational fluid dynamics (CFD) model of a battery module. (d) Imposed transient heat generation densities for discharge rates of $1\text{\,}\mathrm{C}$, $2\text{\,}\mathrm{C}$ and $3\text{\,}\mathrm{C}$. (e) Two configurations are proposed for the BTMS: single and double chamber are embodied in Model 1 and Model 2, respectively. Table 1: Thermophysical properties of water, PCMs, battery and aluminium. The reported properties for the material ‘battery’ are the mass average of representative cell materials. The thermophysical properties include the density $\rho$, the heat capacity $C_{\mathrm{p}}$, thermal conductivity $\lambda$, dynamic viscosity $\mu$, thermal expansion coefficient $B$, solidus temperature $T_{\mathrm{s}}$, liquidus temperature $T_{\mathrm{l}}$ and enthalpy $h$. Component | Ref. | $\rho$ | $C_{\mathrm{p}}$ | $\lambda$ | $\mu$ | $B$ | $T_{\mathrm{s}}$ | $T_{\mathrm{l}}$ | $h$
---|---|---|---|---|---|---|---|---|---
|  | ($\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$) | ($\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$) | ($\mathrm{W}\text{\,}{\mathrm{m}}^{-1}\text{\,}{\mathrm{K}}^{-1}$) | ($\mathrm{kg}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | (${\mathrm{K}}^{-1}$) | ($\mathrm{\SIUnitSymbolCelsius}$) | ($\mathrm{\SIUnitSymbolCelsius}$) | ($\mathrm{kJ}\text{\,}{\mathrm{kg}}^{-1}$)
Water | [13] | 998.7 | 4179 | 0.613 | 0.0007 | 0.001 | — | — | —
28HC | [19, 22] | 880 (solid) | 2000 | 0.2 | 0.0269 | 0.00011 | 27 | 29 | 250
|  | 770 (liquid) |  |  |  |  |  |  |
35HC | [19, 22] | 880 (solid) | 2000 | 0.2 | 0.0269 | 0.00011 | 34 | 36 | 240
|  | 770 (liquid) |  |  |  |  |  |  |
44HC | [19, 22] | 800 (solid) | 2000 | 0.2 | 0.0269 | 0.00011 | 41 | 44 | 250
|  | 700 (liquid) |  |  |  |  |  |  |
n-octadecane | [15, 1] | 814 (solid) | 2150 (solid) | 0.36 (solid) | 0.0032 | 0.00085 | 28 | 30 | 244
|  | 724 (liquid) | 2180 (liquid) | 0.15 (liquid) |  |  |  |  |
Battery | [4] | 3000 | 1005.91 |  $\lambda$x = $\lambda$y = 0.302 | — | — | — | — | —
|  |  |  |  $\lambda$z = 22.48 |  |  |  |  |
Aluminium | [2] | 2719 | 871 | 202.4 | — | — | — | — | —

##  3 Methods

###  3.1 Governing equations

Porous PCM enclosure. The PCM and aluminium foam are assumed in local thermal
non-equilibrium. The metal foam is considered homogeneous and isotropic. For
modelling the melting process of the PCM inside the frame filled with
aluminium foam, the following mass, momentum and energy conservation equations
were solved

$\frac{\mathrm{\partial}\rho}{\mathrm{\partial}t}+\frac{\partial\mathrm{(}\rho
u_{i}\mathrm{)}}{\partial x_{i}}\mathrm{=0},$ (1)

$\begin{split}\frac{1}{\emptyset}\frac{\partial(\rho u_{i})}{\partial
t}+\frac{1}{\emptyset^{2}}\frac{\partial}{\partial x_{j}}\left(\rho
u_{i}u_{j}\right)&=-\frac{\partial p}{\partial x_{i}}\\\ &\begin{aligned}
&+\frac{1}{\emptyset}\frac{\partial}{\partial
x_{j}}\left[\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial
u_{j}}{\partial x_{i}}\right)\right]\\\
&+\rho\beta\left(T-T_{0}\right)\overrightarrow{g}-\frac{\mu\overrightarrow{v}}{K}\\\
&-\frac{\rho
C_{f}}{\sqrt{K}}\overrightarrow{v}\left|\overrightarrow{v}\right|+S,\end{aligned}\end{split}$
(2)

$\emptyset\frac{\mathrm{\partial}(\rho
H)}{\mathrm{\partial}t}+\emptyset\frac{\partial}{\partial x_{j}}\left(\rho
C_{\mathrm{p}}u_{j}T\right)\mathrm{=}\frac{\partial}{\partial
x_{j}}\left({\lambda}_{\mathrm{eff}}\frac{\partial\mathrm{T}}{\partial
x_{j}}\right)+h_{\mathrm{sf}}{\alpha}_{\mathrm{sf}}(T_{\mathrm{m}}-T),$ (3)

where $\emptyset$, $T$, $T_{\mathrm{m}}$, $h_{\mathrm{sf}}$ and
$\alpha_{\mathrm{sf}}$ represent the metal foam porosity, PCM temperature,
foam temperature, heat transfer coefficient and specific surface area,
respectively. Moreover, $K$ and $C_{\mathrm{f}}$ denote the Al foam
permeability and inertial factor and are given as

$K=0.00073(1-\emptyset)^{-0.224}d^{-1.11}_{\mathrm{f}}d^{3.11}_{\mathrm{p}},$
(4)

$C_{f}=0.0012(1-\emptyset)^{-0.132}\left(\frac{d_{\mathrm{f}}}{d_{\mathrm{p}}}\right)^{-1.63},$
(5)

where $d_{\mathrm{p}}$ and $d_{\mathrm{f}}$ represent the pore and ligament
diameters of the metal foam and are taken as $1.93\text{\,}\mathrm{mm}$ and
$0.406\text{\,}\mathrm{mm}$, respectively [3]. In addition, $\emptyset$ and
PPI of the Al foam are taken as 0.794 and 10, respectively [3]. For the metal
foam, the energy equation is given as

$\left(1-\emptyset\right)\frac{\mathrm{\partial}\left({\rho}_{\mathrm{m}}C_{\mathrm{p,m}}T_{\mathrm{m}}\right)}{\mathrm{\partial}t}\mathrm{=}\frac{\partial}{\partial
x_{j}}\left({\lambda}_{\mathrm{m},\text{eff}}\frac{\partial
T_{\mathrm{m}}}{\partial
x_{j}}\right)-h_{\mathrm{sf}}{\alpha}_{\mathrm{sf}}(T_{\mathrm{m}}-T).$ (6)

The specific surface area is written as [31]

${\alpha}_{\mathrm{sf}}=\frac{3\pi d_{\mathrm{f}}}{d_{\mathrm{p}}^{2}}.$ (7)

The heat transfer coefficient is calculated based on the following correlation
[39]

$h_{\mathrm{sf}}=\frac{\lambda}{d_{\mathrm{f}}}\left(0.36+\frac{0.518{Ra}^{0.25}}{{\left(1+{\left(\frac{0.559}{Pr}\right)}^{{9}/{16}}\right)}^{{4}/{9}}}\right),$
(8)

where Pr and Ra are the PCM Prandtl and Rayleigh numbers that are obtained as

$Pr=\frac{C_{\mathrm{p}}\mu}{\lambda},$ (9)
$Ra=\frac{C_{\mathrm{p}}{\rho}^{2}g\beta\pi{d_{\mathrm{f}}}^{3}\Delta
T}{\lambda\mu}.$ (10)

To model the phase-change process of PCMs, we used an enthalpy-porosity
method. Consequently, the terms $S$ and $H$ that appeared in Eqs. (2) and (3)
represent the momentum sink term and the total enthalpy, respectively. $S$ is
given as

$S=\frac{{(1-\gamma)^{2}}}{{(\gamma^{3}+\epsilon)}}A_{\mathrm{mush}}u,$ (11)

where $A_{\mathrm{mush}}$ and $\gamma$ are mushy zone constant and liquid
faction, respectively. The details of enthalpy-porosity method can be find in
[36].

Energy equation in batteries. The heat equation for the battery region is
based on Fourier’s law and is as follows.

$\begin{split}{\rho}_{\mathrm{b}}C_{\mathrm{P,b}}\frac{\mathrm{\partial}T}{\mathrm{\partial}t}&=\frac{\mathrm{\partial}}{\mathrm{\partial}x}\left({\lambda}_{\mathrm{b,x}}\frac{\mathrm{\partial}T}{\mathrm{\partial}x}\right)\\\
&\quad+\frac{\mathrm{\partial}}{\mathrm{\partial}y}\left({\lambda}_{\mathrm{b,y}}\frac{\mathrm{\partial}T}{\mathrm{\partial}y}\right)\\\
&\quad+\frac{\mathrm{\partial}}{\mathrm{\partial}z}\left({\lambda}_{\mathrm{b,z}}\frac{\mathrm{\partial}T}{\mathrm{\partial}z}\right)+Q_{\mathrm{b}},\end{split}$
(12)

where $\lambda_{\mathrm{b},x}$, $\lambda_{\mathrm{b},y}$ and
$\lambda_{\mathrm{b},z}$ stand for the battery thermal conductivity in x, y
and z directions. In addition, $Q_{\mathrm{b}}$ is the density of battery heat
generation that is adopted from [20] (see Fig. 2d).

Numerical solution procedure. The governing equations are solved using a
finite volume method in which the velocity-pressure fields are coupled using
the SIMPLE algorithm [25]. Moreover, an upwind second-order scheme is utilised
to discretise the velocity spatial derivatives. In addition, to discretise the
spatial derivative of pressure, a standard scheme is used. To satisfy the
convergence, the residuals corresponding to the continuity, momentum and
energy equations are chosen $10^{-4}$, $10^{-6}$ and $10^{-7}$, respectively.
Here, Ansys Fluent software is used to simulate the thermal performance of
battery modules.

###  3.2 Verification and validation of CFD model

Mesh resolution. To guarantee mesh-independent results from our computational
fluid dynamics (CFD) model, a mesh refinement study is undertaken with a
water-cooled battery thermal management system (BTMS) having a double-chamber
battery module (see Fig. 2e), integrated with dual PCMs of 28HC and 35HC, as
benchmark metric. A water channel mass flow rate of
$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ ensures active cooling
during the discharge process at a discharge rate of $3\text{\,}\mathrm{C}$.
The ambient temperature is taken as
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The results of the grid
independence test are summarised in Table 2. Time-averaged values of the
maximum temperature of the batteries, liquid fraction of PCM 1 (first
chamber), and liquid fraction of PCM 2 (second chamber), both embedded in
aluminium foam, are compared for five different mesh densities. The hexahedral
structural grids shown in Fig. 3 are generated using Gambit, a commercial
meshing software. According to Table 2, a meshed geometry composed of ca.
$\sim$600,000 cells (Mesh No. 3) yields precise results with an optimal
computational cost.

Table 2: Mesh independence test of our CFD model. The effect of grid size on time-averaged maximum temperature of batteries and time-averaged liquid fractions of PCM 1 (28 HC) and PCM 2 (35 HC) for a double-chamber battery module at a discharge rate of 3C, initial temperature of $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and water channel mass flow rate of $\dot{m}_{\mathrm{cha}}=$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$$. Mesh No. | Cell No. |  $\bar{T}_{\text{max,b}}$ ($\mathrm{\SIUnitSymbolCelsius}$) | Difference (%) | $\bar{\gamma}_{\text{PCM 1}}$ | Difference (%) | $\bar{\gamma}_{\text{PCM 2}}$ | Difference (%)
---|---|---|---|---|---|---|---
Mesh 1 | 298,902 | 35.8347 | 1.78 | 0.6412 | 1.25 | 0.0508 | 1.74
Mesh 2 | 378,048 | 35.5591 | 1.00 | 0.6448 | 0.69 | 0.0513 | 0.77
Mesh 3 | 597,200 | 35.2418 | 0.09 | 0.6488 | 0.08 | 0.0517 | 0.00
Mesh 4 | 1,200,544 | 35.2103 | 0.00 | 0.6491 | 0.03 | 0.0517 | 0.00
Mesh 5 | 2,348,260 | 35.2095 | – | 0.6493 | – | 0.0517 | –
Figure 3: Structured mesh in our computational fluid dynamics (CFD) model. (a)
Grids on the modelled section of a battery module.(b) A magnified view of grid
cells through the BTMS.

Time independence study. Time step independence tests for our CFD model are
carried out for the same metric of a water-cooled, double-chamber battery pack
integrated with 28HC and 35HC. The water channel mass flow rate, initial
temperature of the battery module and discharge rate are
$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$,
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and $3\text{\,}\mathrm{C}$,
respectively. According to Fig. 4, a time step of $0.1\text{\,}\mathrm{s}$
yields a precise prediction of the transient maximum temperature in the
battery with a maximum error of 0.1%. This time step used in our CFD
modelling.

Figure 4: Time independence study for the CFD model. The effect of time step
value (indicated in the legend) on the computed transient maximum temperature
in the batteries is presented for a double-chamber BTMS integrated with dual
Al foam-embedded PCMs (28HC and 35HC). Simulation conditions include a
discharge rate of $3\text{\,}\mathrm{C}$, initial temperature of
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and water channel mass flow rate of
$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$.

Comparison with theoretical study [20]. To prove the reliability of the
present CFD model, the cooling performance of a rectangular battery pack
integrated with water channels is compared with that modelled by Liu et al.
[20]. The dimensions of the modelled section of the battery pack are
$$6.75\text{\,}\mathrm{mm}$\times$180\text{\,}\mathrm{mm}$\times$35\text{\,}\mathrm{mm}$$.
Moreover, the dimensions of the water channel are
$$4\text{\,}\mathrm{mm}$\times$180\text{\,}\mathrm{mm}$\times$1.5\text{\,}\mathrm{mm}$$.
Figure 5a illustrates the temporal profile of the temperature difference
through the battery for the coolant mass flow rate of
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ and a discharge rate of
$3\text{\,}\mathrm{C}$. The analysis indicates a maximum error of 3.4% between
our CFD results and those reported by Liu et al. [20].

Comparison with experimental study [31]. To validate our numerical non-
equilibrium model applied for heat transfer simulation through PCM–metal foam
composites, melting of RT58 within a rectangular enclosure is modelled for
comparison with experimental results reported in the literature. The
dimensions of the foam-embedded enclosure are
$$200\text{\,}\mathrm{mm}$\times$120\text{\,}\mathrm{mm}$\times$25\text{\,}\mathrm{mm}$$.
The porosity of the aluminium foam is 0.85. In addition, the bottom surface of
the porous enclosure is under a constant heat flux. The temperature profile of
the PCM bulk in $y=$8\text{\,}\mathrm{mm}$$, where $y$ is the vertical
distance from the heating plate, is calculated and compared with the
experimental data reported by Tian and Zhao [31]. According to Fig. 5b, there
is a good agreement between our results and the experimental data reported by
Tian and Zhao [31]. The maximum error is 0.9% and it was encountered in the
initial stages ($\lesssim$1500\text{\,}\mathrm{s}$$) of the process, with a
better agreement for times exceeding $2000\text{\,}\mathrm{s}$.

Figure 5:  Validation of our CFD model. (a) Comparison of the temperature
inhomogeneity of batteries obtained by the present model with that presented
by Liu et al. [20] for a discharge rate of $3\text{\,}\mathrm{C}$ and water
mass flow rate of $2.25\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ for a
BTMS with water channels. (b) Melting through an aluminum foam-RT58 PCM
composite resulted from the present non-equilibrium model and its comparison
with Tian and Zhao’s experimental data [31].

##  4 Results

###  4.1 Selection of BTMS configuration and PCM materials

We compare various configurations of the BTMS in a battery module, all
containing the same water channel. Specifically, three different BTMS
configurations are considered: without phase change material (PCM), with a
single chamber containing one type of PCM, and a double chamber configuration
containing two types of PCM (Fig. 2e). The size of the water channel is the
same. Additionally, the BTMS of the double chamber configuration is integrated
with either one or two types of PCM. We compare different candidate PCMs,
namely RT-28HC, RT-35HC, RT-44HC, and n-octadecane. The thermophysical
properties of these PCMs are detailed in Table 1 with corresponding source in
the literature. Furthermore, the water channel mass flow rate and initial
battery temperature are set to
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ and
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. The water mass flow
rate per channel is set $2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ on
the basis of achieving a Reynolds number of 611 whereby the flow regime is
well within the laminar regime. For example, the critical Reynolds number is
about 2000 but our maximum Reynolds number for
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ was 611.

To demonstrate the advantages of inserting a foam-embedded PCM chamber, Fig.
6a displays the maximum temperature profiles of batteries for three BTMS
configurations, i.e. water channel without PCM, with metal-foam-free 28HC PCM,
and 28HC PCM/Al foam at a discharge rate of $3\text{\,}\mathrm{C}$. The
results show that a battery module integrated with a foam-embedded PCM
exhibits a significantly better cooling performance. For instance, at
$t=$400\text{\,}\mathrm{s}$$, the maximum temperature in the battery module
for a case with 28HC PCM/Al foam is
$2.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than that of a battery
module without PCM and $1.0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than
that corresponding to a non-porous chamber filled only with 28HC PCM. The
temperature distribution contours through the batteries at
$t=$400\text{\,}\mathrm{s}$$ are depicted in the insets of Fig. 6a. However,
for operating times higher than approximately $1000\text{\,}\mathrm{s}$, the
maximum battery temperature becomes approximately the same for all three BTMS
cases. The reason is that the entire 28HC PCM melts so, after the latent heat
stops playing a role, sensible heat starts accumulating and elevating the
temperature of all batteries to similar levels. For example at
$1200\text{\,}\mathrm{s}$, the $T_{\mathrm{max}}$ corresponding to Al foam-
embedded 28HC PCM is only $0.5\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$0.8\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than the cases with non-
porous 28HC PCM and without PCM, respectively.

We then explore the possibility of introducing 35HC PCM, whose melting point
is $7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ higher than that of 28HC (see
Table 1) and investigate the possibility of increasing the PCM thermal mass
with a double-chamber configuration. Figure 6b compares the thermal
performance of three BTMS configurations having a single foam-embedded 28HC
PCM, single foam-embedded 35HC PCM, and double-chamber foam-embedded 28HC PCM.
These results show that, due to the higher heat storage capacity of the double
chamber configuration, the battery temperature is considerably lower than the
single chamber case. Additionally, a comparison of single chamber
configurations of 28HC and 35HC PCMs reveals that, after
$t=$800\text{\,}\mathrm{s}$$, 38HC PCM starts to melt and slightly improves
the battery cooling performance (i.e. a lower $T_{\mathrm{max}}$) compared to
the case with 28HC. During melting, however, the PCM with lower melting
temperature (28HC) performs better. For example, at
$t=$600\text{\,}\mathrm{s}$$, the $T_{\mathrm{max}}$ of a battery module with
double chamber Al foam/28HC PCM is $2.9\text{\,}\mathrm{\SIUnitSymbolCelsius}$
and $4.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ colder than the case with
single chambers of 28HC and 35HC, respectively. At
$t=$1200\text{\,}\mathrm{s}$$, these temperature reductions change to
$0.6\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1.0\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. The insets in Fig.
6b show the temperature contours on the three battery modules at
$t=$600\text{\,}\mathrm{s}$$. A battery configuration with double-chamber
foam-embedded 28HC PCM clearly presents and improvement in cooling performance
over the single chamber configurations.

We then consider the simultaneous implementation of two types of PCM in a
double chamber configuration as means of further improving the effective
cooling performance throughout the entire discharging time. Figure 6c displays
$T_{\mathrm{max}}$ within battery modules with dual PCMs. For
$t\lesssim$1000\text{\,}\mathrm{s}$$, double porous chambers of 28HC and
n-octadecane PCMs maintain the battery at significantly lower temperatures.
For example at $t=$600\text{\,}\mathrm{s}$$, $T_{\mathrm{max}}$ of a battery
module integrated with foam-embedded 28HC and n-octadecane PCM chambers is
$2.0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than cases with either
28HC/35HC PCMs or 28HC/44HC. Temperature distributions on battery surfaces for
the three different double chamber BTMS configurations are depicted in the
insets of Fig. 6c. Despite its better cooling performance during melting, the
configuration of the dual 28HC/n-octadecane PCMs is unable to efficiently
reduce the maximum temperatures of the batteries for operating times of
$t\lesssim$1000\text{\,}\mathrm{s}$$. To guarantee a reliable cooling process
for long-hour discharging conditions, a multifaceted BTMS configuration with
28HC/35HC foam-embedded PCMs is recommended. For example, Fig. 6c shows that
at $t=$1200\text{\,}\mathrm{s}$$ the $T_{\mathrm{max}}$ of a module
configuration with dual 28HC and 35HC foam-embedded PCMs is
$1.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ colder than cases with 28HC/44HC
and 28HC/n-octadecane PCMs, respectively.

Figure 6: Cooling performance for different BTMS configurations. Discharge
rate, channel mass flow rate and ambient temperature are
$3\text{\,}\mathrm{C}$, $2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$
and $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. (a) Cases
without PCM, with non-porous PCM of 28HC and alumnium foam-28HC PCM columns.
The insets depict temperature contours on batteries with different BTMS
configuration at $400\text{\,}\mathrm{s}$. (b) Comparisons between single
chamber porous PCM of 28HC, single chamber porous PCM of 35HC and double
chambers porous PCM of 28HC in addition to temperature contours of each
configuration at $t=$600\text{\,}\mathrm{s}$$ depicted in the insets. (c)
Comparison of different dual-PCM configurations in dual porous chambers and
the corresponding temperature contours at $600\text{\,}\mathrm{s}$ depicted in
the insets. All inset labels are indicated in the corresponding figure legend.

In addition to reducing the maximum temperature within the battery
($T_{\mathrm{max}}$), BTMS should also reduce the temperature inhomogeneity
($\Delta T=T_{\mathrm{max}}-T_{\mathrm{min}}$). Figure 7a shows $\Delta T$ for
water cooled BTMS configurations without PCM, foam-free 28HC PCM chamber, and
28HC PCM/Al foam chamber at a discharge rate of $3\text{\,}\mathrm{C}$. The
results indicate that a foam-embedded PCM configuration provides better
temperature homogeneity than cases without PCM and foam-free PCM. This is an
added benefit to that of reducing $T_{\mathrm{max}}$ in Fig. 6a. For example
at $t=$400\text{\,}\mathrm{s}$$, a battery module equipped with Al foam/28HC
PCM is $2.0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ more homogeneous (lower $\Delta
T$) than those corresponding to cases without PCM and Al foam-free PCM,
respectively. Temperature contours for these cases are illustrated at
$t=$600\text{\,}\mathrm{s}$$ in the insets of Fig. 7a. Further investigation
into the double chamber configuration reveals that these are more homogeneous
than single chamber configurations during melting, as shown in Fig. 7b. At
$t\lesssim$950\text{\,}\mathrm{s}$$, due to its higher heat removal capacity
(higher mass of PCM), a double chamber configuration with 28HC PCM provides
reduces temperature inhomogeneity. For example at
$t=$600\text{\,}\mathrm{s}$$, the $\Delta\textit{T}$ resulting from a double
chamber 28HC PCM module is $2.1\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$3.1\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than the single chamber
configurations with 28HC and 35HC PCMs, respectively. Additionally, as shown
in Fig. 7c, a double chamber configuration with 28HC and 35HC PCMs maintains
temperature homogeneity throughout the entire operating time. Comparison of
the data in Figs. 7b,c reveals that at $t=$600\text{\,}\mathrm{s}$$ and
$1200\text{\,}\mathrm{s}$ the $\Delta T$ corresponding to a double chamber
configuration with 28HC and 35HC PCMs are
$1.8\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than those for a battery
module with a double chamber 28HC PCM.

Figure 7: Temperature inhomogeneity of battery modules with different foam-
embedded PCM configurations. The results are compared for an operating
condition where the discharge rate, ambient temperature and water channel mass
flow rate are $3\text{\,}\mathrm{C}$,
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, respectively. (a) The
effect of introducing aluminium foam and 28HC PCM. The insets illustrate the
temperature contours at $600\text{\,}\mathrm{s}$. (b) Cooling performance of
cases with 28HC-Al foam single chamber, 35HC-Al foam single chamber and double
chambers porous PCM of 28HC. The temperature distributions on the battery
surfaces at $800\text{\,}\mathrm{s}$ are shown in the insets. (c) The effect
of introducing dual PCMs in a double-chamber BTMS. Temperature contours at t =
$600\text{\,}\mathrm{s}$ are depicted in the insets. All inset labels are
indicated in the corresponding figure legend.

###  4.2 Performance of BTMS with foam-embedded PCMs

We compared the cooling performance of actively cooled battery modules without
PCM (water channels only) with concurrent active / passive cooling approaches
that have PCMs for discharge rates of $1\text{\,}\mathrm{C}$,
$2\text{\,}\mathrm{C}$, and $3\text{\,}\mathrm{C}$. Four different water-
cooled BTMS configurations were considered: (1) baseline case without PCM, (2)
single chamber with Al foam/28HC PCM, (3) double chambers with Al foam/28HC
PCM, and (4) double chambers with Al foam / dual PCM of 28HC and 35HC PCM. The
water mass flow rate and initial temperature were set to
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ and
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. Figure 8 illustrates
the breakdown of the heat generated and removed for the discharge rates of
$1\text{\,}\mathrm{C}$, $2\text{\,}\mathrm{C}$, and $3\text{\,}\mathrm{C}$.
According to Fig. 8a for a discharge rate of $1\text{\,}\mathrm{C}$, most of
the total heat generated by the batteries is actively removed by the water
channels for cases without PCM (88.9% of the total heat generated), single
chamber 28HC (86. 7%), double chambers 28HC (85. 2%) and double chambers
28HC/35HC (85. 3%). Therefore, the integration of metal foam-embedded PCMs
with a battery module at a discharge rate of $1\text{\,}\mathrm{C}$ may not be
economically viable. However, for discharge rates of $2\text{\,}\mathrm{C}$
and $3\text{\,}\mathrm{C}$, the role of foam-embedded PCMs in the cooling
process becomes more important. For example, for discharge rates of
$2\text{\,}\mathrm{C}$ and $3\text{\,}\mathrm{C}$ in Fig. 8b and Fig. 8c,
respectively, 37.5% and 35.0% of the battery heat is removed by the PCMs in
the fourth multifaceted configuration of a double-chamber foam-embedded
28HC/35HC. For a double chamber mono-pcm BTMS, these values are 42.6% and
39.9% for discharge rates of $2\text{\,}\mathrm{C}$ and
$3\text{\,}\mathrm{C}$, respectively. Although the amount of heat removed by
PCMs in the double chamber 28HC PCM configuration is slightly higher than that
dissipated by PCMs in the 28HC/35HC configuration, the latter provides lower
$T_{\mathrm{max}}$ and $\Delta T$ for long-period operations at discharge
rates $2\text{\,}\mathrm{C}$ and $3\text{\,}\mathrm{C}$ discharge rates
(recall Figs. 6c and 7c).

Figure 8: Heat removal breakdown for different battery module configurations.
The portions of active cooling (water channel), passive cooling (porous PCM)
and residual heat within the batteries for a section of the battery module
shown in (Fig. 2e) are shown in an ambient temperature of 25∘C. The discharge
rate is (a) $1\text{\,}\mathrm{C}$, (b) $2\text{\,}\mathrm{C}$, and (c)
$3\text{\,}\mathrm{C}$.

According to the results shown in Fig. 9a for a discharge rate of
$1\text{\,}\mathrm{C}$, the integration of porous PCMs in the BTMS does not
improve its cooling performance for discharge times of
$t<$3400\text{\,}\mathrm{s}$$. However, in harsh ambient temperatures, a
25HC/35HC double chamber configuration can keep the batteries at lower
temperatures (discussed later in Sec. 4.4). Regarding Fig. 9b, in an operating
condition where the discharge rate is $2\text{\,}\mathrm{C}$,the cooling
performance of a battery module with double chambers filled with 28HC/35HC PCM
becomes better than the double chamber 28HC PCM configuration at
$t\gtrsim$1900\text{\,}\mathrm{s}$$. For example at
$t=$2000\text{\,}\mathrm{s}$$, $T_{\mathrm{max}}$ and $\Delta T$ of the
battery module of double chamber 28HC/35HC are
$1.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$2.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than those of double chamber
28HC (see Appendix A1 for corresponding $\Delta T$ and PCM liquid fraction
$\gamma$). As shown in Fig. 9c, when the battery operates at a discharge rate
of $3\text{\,}\mathrm{C}$, for $t\lesssim$1000\text{\,}\mathrm{s}$$, the
double chamber Al foam / 28HC PCM system provides the best cooling performance
(and temperature homogeneity) in the batteries. For
$t\gtrsim$1000\text{\,}\mathrm{s}$$, a configuration of double chambers with
Al foam/dual 28HC and 35HC PCMs guarantees the minimum $\Delta T$ and thermal
stress (Fig. S?). For example at $t=$1200\text{\,}\mathrm{s}$$, for 28HC/35HC
PCMs with double chamber foam embedded in discharge rate of
$3\text{\,}\mathrm{C}$, $T_{\mathrm{max}}$ and $\Delta T$ are
$1.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than the case of 28HC PCMs
with double chamber foam embedded.

Figure 9: Cooling performance of four battery modules at different discharge
rates. The cooling performance is presented in terms of the maximum
temperature in the cell, $T_{\mathrm{max}}$ for four battery modules: without
PCM, single-chamber (porous PCM) and double-chamber configurations with 28HC
PCM and 28HC/35HC PCM. The ambient temperature and the coolant mass flow rate
per channel are taken as $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, respectively. (a)
Discharge rate of $1\text{\,}\mathrm{C}$; insets depict the temperature
contours of the batteries at $t=$3000\text{\,}\mathrm{s}$$. (b) Discharge rate
of $2\text{\,}\mathrm{C}$; insets show the temperature contours at
$t=$1400\text{\,}\mathrm{s}$$. (c) Discharge rate of $3\text{\,}\mathrm{C}$;
temperature contours at $t=$600\text{\,}\mathrm{s}$$ are shown in the insets.
All inset labels are indicated in the corresponding figure legend.

Figures 10a-c show the inhomogeneity of the battery temperature of four
different water-cooled BTMS configurations: (1) baseline case without PCM, (2)
double chambers with Al foam/28HC PCM, (3) double chambers with Al foam/28HC
PCM, and (4) double chambers with Al foam / dual PCM of 28HC and 35HC PCM
discharge rates of $1\text{\,}\mathrm{C}$, $2\text{\,}\mathrm{C}$, and
$3\text{\,}\mathrm{C}$. The water channel mass flow rate and initial
temperature were set to $2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$
and $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. According to
Fig. 10c, at a discharge rate of $3\text{\,}\mathrm{C}$, a dual-PCM BTMS
reduces $\Delta T$ up to $2.0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ than the
case without PCM and $1.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ compared to a
double chamber mono-PCM BTMS. The transient fraction of the PCM liquid at
different discharge rates is shown in Figs. 10d-f. As an important point, it
should be mentioned that installing foam-PCM columns in between batteries, in
addition to the cooling plate, significantly improves the cooling performance
of the battery module (see the Appendix). However, when PCM components are
used between the batteries, the latent heat released during the off-use cycle,
i.e. standby, could be directly absorbed by the adjacent batteries. This could
lead to thermal runaway problems. Therefore, in the present work, the foam PCM
elements are integrated only with the cooling plate, not between the
batteries.

Figure 10: Cooling performance of four battery modules at different discharge
rates.Cooling performance is presented in terms of the temperature
inhomogeneity in the cell, $\Delta T=T_{\mathrm{max}}-T_{\mathrm{min}}$ for
four battery modules: without PCM (a.1, b.1 and c.1), single chamber porous
PCM (a.2, b.2 and c.2) and double chamber configurations with 28HC PCM (a.3,
b.3 and c.3) and 28HC/35HC PCMs (a.4, b.4 and c.4). The insets show the
temperature profile on the cell surfaces. The ambient temperature and the
coolant mass flow rate per channel were
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, respectively. (a)
Discharge rate of $1\text{\,}\mathrm{C}$; insets depict the temperature
contours of the batteries at $t=$3600\text{\,}\mathrm{s}$$. (b) Discharge rate
of $2\text{\,}\mathrm{C}$; insets show the temperature contours at
$t=$2000\text{\,}\mathrm{s}$$. (c) Discharge rate of $3\text{\,}\mathrm{C}$;
temperature contours at $t=$1200\text{\,}\mathrm{s}$$ are shown in the insets.
(d)–(f) Corresponding PCM liquid fraction for each discharge rate analysed in
this study.

###  4.3 Effect of coolant flow rate on BTMS performance

We evaluated the cooling performance of double-chamber battery modules for the
cases with only 28HC PCM and dual-PCM 28HC/35HC with water channel mass flow
rates of
$\dot{m}_{\mathrm{cha}}=$0.56\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$$,
$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ for ambient temperature of
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$. This mass flow rate
$\dot{m}_{\mathrm{cha}}$ is for a single channel, noting that the proposed
BTMS module has 14 channels. Figure 11a shows that, for a discharge rate of
$1\text{\,}\mathrm{C}$, the cooling performance of the 28HC double chamber PCM
becomes slightly more efficient after $1200\text{\,}\mathrm{s}$,
$3150\text{\,}\mathrm{s}$, and $3400\text{\,}\mathrm{s}$ for water channel
mass flow rates of $0.56\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$,
$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, respectively. Before the
aforementioned critical times, the cooling performance of the two
configurations is the same. For example, at a water channel mass flow rate of
$0.56\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ at
$t=$3600\text{\,}\mathrm{s}$$, $T_{\mathrm{max}}$ and $\Delta T$ corresponding
to the double chamber configuration of the 28HC PCM are
$0.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than the case of 28HC/35HC
PCMs.

For the discharge rate of $2\text{\,}\mathrm{C}$, the cooling performance of
double chamber battery modules with 28HC PCM and 28HC/35HC PCMs is depicted in
Fig. 11b. As can be inferred from the figure, for all mass flow rates and
operating times $t<$1760\text{\,}\mathrm{s}$$, the single 28HC PCM results in
lower $T_{\mathrm{max}}$ and $\Delta T$. In contrast, as the discharging
progresses further, the dual-28HC/35HC-PCM configuration shows a better
thermal performance as evidenced by its lower $T_{\mathrm{max}}$. For example,
for $2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ at
$t=$2100\text{\,}\mathrm{s}$$, the reductions (from the case of 28HC) in
$T_{\mathrm{max}}$ and $\Delta\textit{T}$ for the configuration of 28HC/35HC
PCM are $1.9\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$0.6\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for a mass flow rate of the water
channel of $0.56\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, while these
reductions become $0.5\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$0.4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for a mass flow rate of
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$.

For a higher discharge rate of $3\text{\,}\mathrm{C}$, Fig. 11c depicts the
cooling performance of the proposed BTMS. The time at which the ‘best’
configuration (with lowest $T_{\mathrm{max}}$) changes from single (28HC) to
dual-PCM (28HC/35HC) for mass flow rates of
$0.56\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$,
$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ is
$536\text{\,}\mathrm{s}$, $765\text{\,}\mathrm{s}$ and
$990\text{\,}\mathrm{s}$, respectively. This suggests that BTMS with dual-PCM
are less effective short term, and for low channel mass flow rates where
convective heat transfer is less dominant. In contrast, for long operating
times and high mass flow rates, the cooling performance of the dual 28HC/35HC
configuration is superior. For example, at $t=$1200\text{\,}\mathrm{s}$$, the
reductions in $T_{\mathrm{max}}$ and $\Delta T$ for a battery module with
28HC/35HC PCMs compared to a double chamber 28HC configuration are
$0.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$0.1\text{\,}\mathrm{\SIUnitSymbolCelsius}$,respectively, for a channel mass
flow rate of $0.56\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$. In contrast,
for a mass flow rate of $2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$,
these reductions in $T_{\mathrm{max}}$ and $\Delta T$ become
$1.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$, respectively. As a result, the
usage of multi PCMs of 28HC and 35HC is an effective passive method to reduce
the maximum temperature of the battery and increase temperature homogeneity.
Furthermore, as can be seen from Fig. 11c, after
$t=$1200\text{\,}\mathrm{s}$$, 7.5% and 26.9% of the initial volume of 35HC
PCM is not melted at flow rates of
$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, respectively. Hence, if
the discharging process continues further, chamber 2 (35HC) will play a role
in the temperature regulation of the battery pack. In the mono-PCM (28HC)
configuration, all PCM in both chambers are melted before
$t=$1200\text{\,}\mathrm{s}$$ at the three water flow rates.

Figure 11: Effect of channel mass flow rate on the cooling performance of
BTMS. Water-cooled BTMS include double-chamber configurations with single- and
dual-PCMs. The ambient temperature is
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for all cases. The peak temperature
in the battery ($T_{\mathrm{max}}$), the inhomogeneity of the temperature
within the battery ($T_{\mathrm{max}}-T_{\mathrm{min}}$), and liquid fractions
($\gamma$) of the PCM inside the first and second chambers are shown as a
function of time. Analyses were carried out for discharge rates of (a)
$1\text{\,}\mathrm{C}$, (b) $2\text{\,}\mathrm{C}$, and (c)
$3\text{\,}\mathrm{C}$. All plots share the same legend, indicated on the top
left. Three water channel mass flow rates $\dot{m}_{\mathrm{cha}}$ were
analysed for a double-chamber configuration containing only 28HC and both 28HC
and 35HC.

###  4.4 Effect of ambient temperatures on BTMS performance

To gain insight into the cooling performance of double chamber battery module
configurations with either 28HC or 28HC/35HC under high ambient temperatures,
simulations are conducted for ambient temperatures $T_{\mathrm{am}}$
representative of hot summer days. The ambient temperatures analysed were
$T_{\mathrm{am}}=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ (‘comfortable
ambient temperature), $29\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$33\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (hot summer day) with a channel
mass flow rate of $1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$. The
results for discharge rates of $1\text{\,}\mathrm{C}$, $2\text{\,}\mathrm{C}$
and $3\text{\,}\mathrm{C}$ are shown in Fig. 12. We found that the 28HC/35HC
dual-PCM configuration significantly improves the safety of the battery module
at an ambient temperature of $33\text{\,}\mathrm{\SIUnitSymbolCelsius}$. For
example, for a discharge rate of $1\text{\,}\mathrm{C}$ at
$t=$3600\text{\,}\mathrm{s}$$, $T_{\mathrm{max}}$ is
$0.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower in 28HC/35HC compared to
28HC cases. Regarding the discharge rate of $2\text{\,}\mathrm{C}$, the values
of $T_{\mathrm{max}}$ for the dual-28HC/35HC-PCM configuration are lower at
$t>$2000\text{\,}\mathrm{s}$$ for
$T_{\mathrm{am}}=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. Furthermore,
dual-28HC/35HC-PCM yields a lower temperature for
$t>$1000\text{\,}\mathrm{s}$$ and
$T_{\mathrm{am}}=$29\text{\,}\mathrm{\SIUnitSymbolCelsius}$$, and at
$t>$500\text{\,}\mathrm{s}$$ and
$T_{\mathrm{am}}=$33\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. When the battery
is at a discharge rate $3\text{\,}\mathrm{C}$, the threshold times at which
the maximum battery temperature of the 28HC/35HC configuration is lower than
the case of the double chamber with 28HC PCM are approximately
$900\text{\,}\mathrm{s}$ for
$T_{\mathrm{am}}=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ and approximately
$300\text{\,}\mathrm{s}$ for
$T_{\mathrm{am}}=$29\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ and
$33\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

Figure 12: Effect of ambient temperature on the cooling performance of BTMS.
The analyses were performed for a water-cooled BTMS with double-chamber
configuration. 28HC fills both chambers (blue bars) or 28HC and 35HC fill
chambers 1 and 2, respectively (orange bars). The water channel mass flow rate
was set to $1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$. The maximum
battery temperature $T_{\mathrm{max}}$ is reported for ambient temperatures of
(a) $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, (b)
$29\text{\,}\mathrm{\SIUnitSymbolCelsius}$, and (c)
$33\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

.

##  5 Conclusions

The maximum temperature and inhomogeneity of the temperature within the
lithium ion battery packs determine their degradation rates and the risk of
failure. When batteries operate at high ambient temperatures, cell
temperatures could potentially reach critical levels, particularly at
discharge rates exceeding $1\text{\,}\mathrm{C}$. Thermal regulation problems
lead to safety concerns and motivate further research. Here, we proposed a
multifaceted battery thermal management system (BTMS) design that integrates
active liquid cooling (through water channels) with latent heat thermal
removal (with foam-embedded phase-change materials, or PCMs). The effects of
implementing two types of PCM in the passive section of the BTMS were
investigated, with one PCM having a low melting temperature (i.e. 28HC at ca.
$28\text{\,}\mathrm{\SIUnitSymbolCelsius}$) and another one a high melting
temperature (i.e. 35HC at ca. $35\text{\,}\mathrm{\SIUnitSymbolCelsius}$). We
found that in the proposed multifaceted dual PCM BTMS, 28HC effectively
absorbed the heat generated during the initial stages of the battery
discharging process, while 35HC reduced the maximum temperature caused by
excessive heat generation toward the end of the discharging phase. We analysed
the performance of the proposed cooling system under different operating
conditions to determine cases where the cooling effectiveness of the dual-PCM
28HC/35HC configuration exceeds that of the double chamber 28HC BTMS. We
demonstrated that a multifaceted passive/active BTMS with dual-PCM
configuration is an effective means for mitigating thermal runaway or other
degradation issues brought about by thermal inhomogeneities.

We numerically investigated three materials as secondary PCM with a higher
melting temperature: 35HC, 44HC, and n-octadecane, while 28HC was used as the
primary PCM. We found that the dual-PCM 28HC/35HC BTMS provides temperatures
up to $1.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1.2\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than those cases with
28HC/44HC and 28HC/n-octadecane PCMs, respectively, for an ambient temperature
of $T_{\mathrm{am}}=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ and a water
channel mass flow rate of
$\dot{m}_{\mathrm{cha}}=$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$$.
For the same operating conditions and the case in which both chambers are
filled with Al foam/28HC PCM, 42.6% and 39.9% of the heat is removed by the
foam-embedded PCM for discharge rates of $2\text{\,}\mathrm{C}$ and
$3\text{\,}\mathrm{C}$, respectively. These values become 37. 5% and 35. 0%
for the double chamber configuration 28HC / 35HC. Our findings suggest that
dual-PCM designs provide considerably lower maximum temperature while reducing
temperature inhomogeneity. For example, the values of $T_{\mathrm{max}}$ and
$\Delta T$ with a 28HC/35HC BTMS are
$1.8\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$1.3\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than those of the double
chamber 28HC for discharge rates of $2\text{\,}\mathrm{C}$ and
$3\text{\,}\mathrm{C}$, respectively.

For a discharge rate of $1\text{\,}\mathrm{C}$ and a channel mass flow rate of
$\dot{m}_{\mathrm{cha}}=$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$$,
the cooling performances of the double-chamber and single-chamber
configurations are nearly identical for an ambient temperature of
$T_{\mathrm{am}}=$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. Generally, for
battery operations at high ambient temperatures, the 28HC / 35HC double
chamber results in lower maximum battery temperatures. For example, in the
case where
$\dot{m}_{\mathrm{cha}}=$1.12\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$$
and $T_{\mathrm{am}}=$33\text{\,}\mathrm{\SIUnitSymbolCelsius}$$, the maximum
temperature corresponding to a double chamber 28HC / 35HC configuration is up
to $2.0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than those of a double
chamber packed only with 28HC PCM. However, a lower coolant mass flow rate
could be applied for a double chamber 28HC configuration. This may have
benefits in reducing the pumping power, albeit the low flow rates in our BTMS
suggest that pumping power is not a major concern. The experimental
implementation and energy efficiency analysis of the proposed BTMS could be
the focus of future studies.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this article.

## Appendix. Metal foam-embedded PCM between battery cells

As mentioned in Section 4.1 of the manuscript, the integration of foam-
embedded PCM between the cooling plate and each battery cell improves the
cooling process during the discharge phase. However, when a metal foam-PCM
layer is introduced between battery cells, due to the release of latent heat
to the battery cells during standby, the temperature of the batteries will
increase. In other words, solidification of the PCM injects a large portion of
heat into the adjacent battery cells. Consequently, thermal runaway could
occur in the battery pack. Therefore, the battery module configuration with
the metal foam-PCM layer between the battery cells configuration was not
discussed in the Results section. To gain insight into the cooling performance
of battery modules with PCM foam components in the cooling plate and in
between battery cells, this configuration is modelled for the discharge rate
of $3\text{\,}\mathrm{C}$. The conceptual figure of this battery module
configuration is shown in Fig. 1a. The ambient temperature and the coolant
mass flow rate per channel are taken as
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$, respectively. As shown in
Fig. 1b, as 28 HC PCM foam is introduced between the batteries, the maximum
temperature of the battery module decreases significantly. As an example, at
$t=$1200\text{\,}\mathrm{s}$$, $T_{\mathrm{max}}$ through a battery module
with 28HC PCM foam in the cooling plate and between batteries is
$7.7\text{\,}\mathrm{\SIUnitSymbolCelsius}$ lower than the case with dual
28HC/35HC PCMs used only in the cooling plate (not between batteries).
According to Fig. 1c, significant improvement in the temperature homogeneity
through the battery module is resulted when 28HC PCM is used in between
batteries and the cooling plate. As a result, installing foam-embedded PCM in
between battery cells in conjunction with the cooling plate significantly
improves the cooling process. However, since the PCM component is adjacent to
the batteries, the release of a high amount of latent heat during standby
might trigger thermal runaway. This hinders the practical use of foam-PCM
layers between batteries. Therefore, in the present work, foam-embedded PCMs
are used only in the cooling plate.

Fig. A1 Cooling performance of four battery modules: single chamber (28 HC-
foam), double chamber with 28HC PCM, double chamber with 28HC/35HC PCMs, and a
combination of single chamber and foam-PCM in between batteries (28HC). The
ambient temperature, coolant mass flow rate per channel and discharge rate are
taken as $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
$2.25\text{\,}\mathrm{g}\text{\,}{\mathrm{s}}^{-1}$ and
$3\text{\,}\mathrm{C}$, respectively. (a) Schematic of a battery pack with
mono PCM-metal foam used in the cooling plate and in between batteries. (b)
Maximum temperature versus time through the modules. The inset illustrates the
temperature contour at $600\text{\,}\mathrm{s}$ for the case where foam
embedded PCM layers are installed between the batteries. (c) Temperature
inhomogeneity in the battery modules. The inset shows the temperature
distribution at $1200\text{\,}\mathrm{s}$ for the module configuration in
which foam-PCM is used between the batteries.

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# On some results on Practical Numbers

Sai Teja Somu, Ting Hon Stanford Li and Andrzej Kukla

(Date: August 28, 2024)

###### Abstract.

A positive integer $n$ is said to be a practical number if every integer in
$[1,n]$ can be represented as the sum of distinct divisors of $n$. In this
article, we consider practical numbers of a given polynomial form. We give a
necessary and sufficient condition on coefficients $a$ and $b$ for there to be
infinitely many practical numbers of the form $an+b$. We also give a necessary
and sufficient for a quadratic polynomial to contain infinitely many practical
numbers, using which we solve first part of a conjecture mentioned in [9]. In
the final section, we prove that every number of $8k+1$ form can be expressed
as a sum of a practical number and a square, and for every
$j\in\\{0,\ldots,7\\}\setminus\\{1\\}$ there are infinitely many natural
numbers of $8k+j$ form which cannot be written as sum of a square and a
practical number.

###### Key words and phrases:

Practical Numbers, polynomial representations.

###### 2020 Mathematics Subject Classification:

Primary 11B83; Secondary 11D85

##  1\. Introduction

A positive integer $n$ is said to be a practical number if every integer in
$[1,n]$ can be expressed as the sum of distinct divisors of $n$. Practical
numbers were introduced by Srinivasan in [6]. In [7], Stewart showed that
$n=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}$, where
$p_{1}<p_{2}<\cdots<p_{k}$ are primes and integers $\alpha_{i}\geq 1$ is
practical if either $n=1$ or $p_{1}=2$ and for all $2\leq i\leq k$ we have
$p_{i}\leq\sigma(p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots
p_{i-1}^{\alpha_{i-1}})+1$ (here $\sigma$ denotes the sum of divisors
function). This implies that if $m$ is a practical number then for all natural
numbers $n\leq\sigma(m)+1$, $mn$ is a practical number. We will be using this
property several times in this article.

There are many properties of practical numbers that are similar to that of
properties of prime numbers. All practical numbers except for $1$ are even
while all primes except for $2$ are odd. Let $\pi(x)$ and $P(x)$ denote number
of primes less than or equal to $x$ and number of practical numbers less than
or equal to $x$ respectively. From prime number theorem, we have
$\pi(x)\sim\frac{x}{\log x}$ and Weingartner in [8] showed that there exists a
positive constant $c$ such that $P(x)\sim\frac{cx}{\log x}$.

Margenstern in [3] made a Goldbach-type conjecture that every even positive
integer can be expressed as sum of two practical numbers. In the same paper,
he conjectured that there are infinitely many positive integers $m$ such that
$m-2,m,m+2$ are all practical numbers. Melfi proved both the conjectures in
[4].

In Section 2, we consider practical numbers of a specific form. We prove that
there exists no non-constant polynomial $P(x)$ with integer coefficients such
that $P(n)$ is a practical number for all natural numbers $n$. We give a
necessary and sufficient condition for any linear polynomial for containing
$0$ or $1$ or infinitely many practical numbers. Later we consider quadratic
representations of practical numbers. We could prove first part of Conjecture
1.1 of [9] using our results.

In Section 3, we prove that every number of the form $8k+1$ can be represented
as a sum of a square and a practical number. We also show that for all
$j\in\\{0,\cdots,7\\}$ and $j\neq 1$ there exist infinitely many natural
numbers $k$ such that $8k+j$ is not a practical number.

##  2\. Practical Numbers of a specific form

Goldbach showed that no non-constant polynomial with integer coefficients can
be a prime for all integer values (see [2] and [5]). We show an analogous
result for practical numbers.

######  Proposition 2.1.

For every non-constant polynomial $P$ with integer coefficients and positive
leading coefficient there exists a natural number $n$ such that $P(n)$ is not
a practical number.

###### Proof.

Let $d$ be the greatest common divisor of $\\{P(1)$, $P(2),\cdots\\}$. Let
prime factorization of $d$ be $p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}$ with
$p_{1}<p_{2}<\cdots<p_{k}$. Let $V=\\{p_{1},\cdots,p_{k}\\}$ and let $T$ be
set of all primes less than or equal to $\sigma(d)+1$ which are not in the set
$V$. Since highest power of $p_{i}$ dividing $d$ is $p_{i}^{a_{i}}$, there
exists a congruence $x\mod p_{i}^{a_{i}+1}$ such that $P(x)$ is not a multiple
of $p_{i}^{a_{i}+1}$. Since $d$ is not divisible by any primes in $T$, for all
primes $t\in T$, there exists a congruence $x_{t}\mod t$ such that $P(x_{t})$
is not a multiple of $t$. From Chinese remainder theorem, there exists a
congruence $n\mod d\prod_{t\in T}t$ such that $P(n)$ is not divisible by any
prime in $T$ and highest power of $p_{i}$ dividing $P(n)$ is $a_{i}$.

Let $m$ be a sufficiently large natural number such that $m\equiv n\mod
d\prod_{t\in T}t$ and $P(m)=dq$ where $q>1$ is not divisible by any prime
$\leq\sigma(d)+1$. From [7, Theorem 1], $P(m)$ is not a practical number. ∎

###  2.1. Practical numbers of the form $an+b$

Let $a$ and $b$ be two positive integers. In the following theorem we give
necessary and condition for $an+b$ to contain infinitely many practical
numbers.

######  Theorem 2.2.

Let $a$ and $b$ be two positive integers and let $d$ be equal to largest
practical number divisor of greatest common divisor of $a,b$. We have:

  * (a)

If there exists a prime number $p\leq\sigma(d)+1$ such that
$p\nmid\frac{a}{d}$ then there exist infinitely many practical numbers of the
form $an+b$.

  * (b)

If $b$ is a practical number and all the primes $p\leq\sigma(d)+1$ divide
$\frac{a}{d}$ then there is exactly one practical number of the form $an+b$.

  * (c)

If $b$ is not a practical number and all the primes $p\leq\sigma(d)+1$ divide
$\frac{a}{d}$ then there are no practical numbers of the form $an+b$.

###### Proof.

Let $a_{1}=\frac{a}{d}$ and $b_{1}=\frac{b}{d}$. Let us first prove (a). If
there exists a prime $p\leq\sigma(d)+1$ such that $p\nmid a_{1}$ then we claim
that there are infinitely many practical numbers of the form $an+b$. Suppose
for a contradiction there are only finitely many practical numbers of the form
$an+b$, then there exists an $A$ such that for all $an+b\geq A$, $an+b$ is not
a practical number.

Choose a natural number $k$ such that $p^{k}\geq A$, $p^{k}\geq b_{1}$ and
$\sigma(dp^{k})+1\geq a_{1}+1$. As $p\nmid a_{1}$, there exists a solution to
$a_{1}x\equiv-b_{1}\mod p^{k}$ and therefore there exists a natural number
$1\leq n\leq p^{k}$ such that $p^{k}|(a_{1}n+b_{1})$. As $d$ is a practical
number and $p\leq\sigma(d)+1$, from [7, Theorem 1] we have $dp^{k}$ is a
practical number. Now, $an+b=(dp^{k})\frac{(a_{1}n+b_{1})}{p^{k}}$ is a
practical number as $dp^{k}$ is a practical number and

$\frac{a_{1}n+b_{1}}{p^{k}}\leq\frac{a_{1}p^{k}+b_{1}}{p^{k}}\leq
a_{1}+1\leq\sigma(dp^{k})+1.$

As $p^{k}|an+b$, we have $an+b\geq p^{k}\geq A$. This contradicts the
assumption that there are no practical numbers of the form $an+b$ which are at
least $A$. Hence there are infinitely many practical numbers of the form
$an+b$. This proves (a).

Now let us prove (b) and (c). If for all primes $p\leq\sigma(d)+1$ we have
$p|a_{1}$ then observe that for all primes $p\leq\sigma(d)+1$, we have $p\nmid
b_{1}$ (otherwise $dp$ is a practical number larger than $d$ and is a divisor
of $\text{gcd}(a,b)$). Hence $a_{1}n+b_{1}$ does not contain any prime factors
less than or equal to $\sigma(d)+1$. We will now prove that for $n\geq 1$,
$an+b$ is not a practical number. Let $P$ be the smallest prime factor of
$a_{1}n+b_{1}$. Let the prime factorizations of $d$ and $a_{1}n+b_{1}$ be
$p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}$ and $P^{b_{1}}P_{2}^{b_{2}}\cdots
P_{r}^{b_{r}}$, respectively, where $p_{1}<p_{2}<\cdots<P<P_{2}<\cdots<P_{r}$.
Now, as $P>\sigma(d)+1=\sigma(p_{1}^{a_{1}}\cdots p_{k}^{a_{k}})+1$, from [7,
Theorem 1] we can conclude that $an+b$ is not a practical number. Therefore,
if all primes $p\leq\sigma(d)+1$ are divisors of $\frac{a}{d}$ then $an+b$ is
not a practical number for $n\geq 1$.

Therefore, if $b$ is a practical number and if all primes $p\leq\sigma(d)+1$
are divisors of $a_{1}$ then there is only one practical number of the form
$an+b$, namely $b$. This proves (b). If $b$ is not a practical number and if
all primes $p\leq\sigma(d)+1$ are divisors of $a_{1}$ then there are no
practical numbers of the form $an+b$. This proves (c). ∎

We have a corollary of Theorem 2.2.

######  Corollary 2.3.

Let $a$ be an odd positive integer and $b$ be a positive integer. There exist
infinitely many practical numbers of the form of $an+b$.

###### Proof.

Let $d$ be the largest practical number divisor of $\text{gcd}(a,b)$. Since
there exists a prime $p=2\leq\sigma(d)+1$ which does not divide $\frac{a}{d}$,
from Theorem 2.2 we can conclude that there are infinitely many practical
numbers of the form $an+b$. ∎

###  2.2. Practical numbers of the form $an^{2}+bn+c$

Let $q(n)=an^{2}+bn+c$ be any quadratic polynomial having integer coefficients
and a positive leading coefficient. For every prime $p$, let
$m_{q}(p):=\sup\\{k\in\mathbb{N}\cup\\{0\\}:q(n)\equiv 0\mod p^{k}\text{ has a
solution}\\}$ ($m_{q}(p)=\infty$ is possible). From Hensel’s lemma, it follows
that there exists a prime number $p$ such that $m_{q}(p)=\infty$. Let us now
work on giving a necessary and sufficient condition for a quadratic polynomial
$q(n)$ to represent infinitely many practical numbers. We require a lemma in
order to prove Theorem 2.5.

######  Lemma 2.4.

Let $q(n)=an^{2}+bn+c$ be any quadratic polynomial with integer coefficients.
Let $Q=\\{p:\text{ $p$ prime and there exists $n\in\mathbb{N}$ such that
$p|q(n)$}\\}$ then $\prod_{p\in Q}\big{(}1+\frac{1}{p}\big{)}$ diverges.

###### Proof.

Let $f$ be any irreducible divisor of $q$ and let $\rho_{f}(p)$ denote the
number of solutions modulo $p$ of the congruence $f(x)\equiv 0\mod p$. From
prime ideal theorem, it follows that $\sum_{p}\frac{\rho_{f}(p)}{p}$ diverges
(see Corollary 3.2.2 of [1]). As $\rho_{f}(p)\leq 2$, it follows that
$\sum_{p\in Q}\frac{1}{p}$ diverges and $\prod_{p\in Q}(1+\frac{1}{p})$
diverges. ∎

######  Theorem 2.5.

Let $q$ be a quadratic polynomial with integer coefficients and with positive
leading coefficient. Let $p_{n}$ denote $n$th prime number and let $r$ be the
least positive integer such that $m_{q}(p_{r})=\infty$. There are infinitely
many practical numbers of the form $q(n)$ if and only if
$p_{1}^{m_{q}(p_{1})}p_{2}^{m_{q}(p_{2})}\cdots p_{r-1}^{m_{q}(p_{r-1})}p_{r}$
is a practical number.

###### Proof.

Suppose $p_{1}^{m_{q}(p_{1})}p_{2}^{m_{q}(p_{2})}\cdots
p_{r-1}^{m_{q}(p_{r-1})}p_{r}$ is not a practical number. Then either
$m_{q}(p_{1})=m_{q}(2)=0$ or there exists an $2\leq i\leq r$ such that
$m_{q}(p_{i})>0$ and
$p_{i}>\sigma\left(p_{1}^{m_{q}(p_{1})}p_{2}^{m_{q}(p_{2})}\cdots
p_{i-1}^{m_{q}(p_{i-1})}\right)+1$. If $m_{q}(2)=0$ then all $q(n)$ are odd
and all $q(n)>1$ are not practical numbers. Hence there are only finitely many
practical numbers of the form $q(n)$.

If there exists an $2\leq i\leq r$ such that $m_{q}(p_{i})>0$ and
$p_{i}>\sigma(p_{1}^{m_{q}(p_{1})}\cdots p_{i-1}^{m_{q}(p_{i-1})})+1$, then we
claim that for all natural numbers $n$ such that
$q(n)>p_{1}^{m_{q}(p_{1})}\cdots p_{r-1}^{m_{q}(p_{r-1})}p_{r}$, $q(n)$ is not
a practical number. Observe that $q(n)$ should have at least one prime factor
$p\geq p_{i}$ (otherwise from the definition of $m_{q}(p_{i})$, we must have
$q(n)\leq p_{1}^{m_{q}(p_{1})}\cdots p_{i-1}^{m_{q}(p_{i-1})}$). Let $p$ be
the smallest prime factor of $q(n)$ satisfying $p\geq p_{i}$. Now,
$q(n)=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{i-1}^{a_{i-1}}p^{k}Q$ where $Q$ is
either $1$ or the least prime factor of $Q$ is greater than $p_{i}$. Now as
$p\geq p_{i}>\sigma(p_{1}^{a_{1}}\cdots
p_{i-1}^{a_{i-1}})+1\geq\sigma(p_{1}^{m_{q}(p_{1})}p_{2}^{m_{q}(p_{2})}\cdots
p_{i-1}^{m_{q}(p_{i-1})})+1$, from [7, Theorem 1] we can conclude that $q(n)$
is not a practical number. Hence there are only finitely many $q(n)$ such that
$q(n)$ is a practical number. This proves one part of the theorem.

If $p_{1}^{m_{q}(p_{1})}\cdots p_{r-1}^{m_{q}(p_{r-1})}p_{r}$ is a practical
number. We claim that there are infinitely many practical numbers of the form
$q(n)$. For the sake of contradiction, assume that there are only finitely
many practical numbers of the form $q(n)$. Then there exists a number $A$ such
that there are no practical numbers $q(n)$ such that $q(n)\geq A$. Let
$Q=\\{p:\text{ $p$ prime and there exists $n\in\mathbb{N}$ such that
$p|q(n)$}\\}.$ From previous lemma, it follows that $\prod_{p\in
Q}(1+\frac{1}{p})$ diverges. Hence there exist primes $t_{1},\ldots,t_{s}\in
Q$ that are greater than $p_{r}$ such that
$\prod_{i=1}^{s}(1+\frac{1}{t_{i}})>a+b+c$. Let $k$ be a natural number such
that $p_{r}^{k}>A$ and $p_{1}^{m_{q}(p_{1})}\cdots
p_{r-1}^{m_{q}(p_{r-1})}p_{r}^{k}\geq t_{1}\cdots t_{s}$. Since
$m_{q}(p_{r})=\infty$ there exists a solution $x_{r}\mod p_{r}^{k}$ such that
$q(x_{r})\equiv 0\mod p_{r}^{k}$. For $1\leq i\leq r-1$, there exists a
solution $x_{i}\mod p_{i}^{m_{q}(p_{i})}$ such that $f(x_{i})\equiv 0\mod
p_{i}^{m_{q}(p_{i})}$ and for $1\leq j\leq s$ there exists $y_{j}\mod t_{j}$
such that $q(y_{j})\equiv 0\mod t_{j}$. Hence there exists an $1\leq n\leq D$
such that $q(n)$ is divisible by $D$ where $D=p_{1}^{m_{q}(p_{1})}\cdots
p_{r-1}^{m_{q}(p_{r-1})}p_{r}^{k}t_{1}t_{2}\cdots t_{s}$. Observe that as
$p_{1}^{m_{q}(p_{1})}\cdots p_{r-1}^{m_{q}(p_{r-1})}p_{r}^{k}$ is a practical
number and since $t_{1}t_{2}\cdots t_{r}\leq
p_{1}^{m_{q}(p_{1})}p_{2}^{m_{q}(p_{2})}\cdots
p_{r-1}^{m_{q}(p_{r-1})}p_{r}^{k}$ we have
$D=p_{1}^{m_{q}(p_{1})}p_{2}^{m_{q}(p_{2})}\cdots
p_{r-1}^{m_{q}(p_{r-1})}p_{r}^{k}t_{1}t_{2}\cdots t_{s}$ is a practical
number.

As $q(n)=D(\frac{q(n)}{D})$ and

$\sigma(D)=D\left(\frac{\sigma(D)}{D}\right)\geq
n\prod_{j=1}^{s}\left(1+\frac{1}{t_{j}}\right)>n(a+b+c)\geq\frac{an^{2}+bn+c}{n}\geq\frac{q(n)}{n}\geq\frac{q(n)}{D},$

we have $q(n)$ is a practical number greater than $A$, contradicting our
assumption. Hence there are infinitely many practical numbers of the form
$q(n)$. This proves the other part of the theorem. ∎

Now before proving a corollary of Theorem 2.5, we prove a lemma.

######  Lemma 2.6.

Let $a,b,c$ be positive integers with $2\nmid ab$ and $2|c$. Then for all
positive integers $k$ there exists a natural number $n$ such that
$an^{2}+bn+c\equiv 0\mod 2^{k}$.

###### Proof.

We prove using induction on $k$. For $k=1$, as $a.1^{2}+b.1+c\equiv 0\mod 2$,
the statement is true for $k=1$. Suppose the statement is true for $k=l$, then
there exists an $n$ such that $an^{2}+bn+c\equiv 0\mod 2^{l}$. Now consider
$m=2^{l}x+n$ then as $a(2^{l}x+n)^{2}+b(2^{l}x+n)+c\equiv
an^{2}+2^{l}bx+bn+c\mod 2^{l+1}$, if $x\equiv\frac{an^{2}+bn+c}{2^{l}}\mod 2$
then $a(2^{l}x+n)^{2}+b(2^{l}x+n)+c\equiv 0\mod 2^{l+1}$. Hence the statement
is true for $l+1$ and the lemma follows from the principle of mathematical
induction. ∎

######  Corollary 2.7.

Let $a,b,c$ be positive integers with $2\nmid ab$ and $2|c$. Then there are
infinitely many practical numbers of the form $an^{2}+bn+c$.

###### Proof.

Let $q(n)=an^{2}+bn+c$ then previous lemma implies that $m_{q}(2)=\infty$. As
$2$ is a practical number, from Theorem 2.5 we can conclude that there are
infinitely many practical numbers of the form $an^{2}+bn+c$.

The above corollary solves the first part of [9, Conjecture 1.1]. ∎

###  2.3. Palindromic practical numbers

It is not known whether there are infinitely many palindromic primes but for
practical numbers it is easy to show that there are infinitely many
palindromic practical numbers.

######  Proposition 2.8.

There are infinitely many palindromic practical numbers.

###### Proof.

Let $A_{n}=\frac{8(10^{2^{n}}-1)}{9}$. Observe that $A_{n}$ is palindrome for
all natural numbers $n$. We claim that $A_{n}$ is a practical number for all
positive integers $n$. We prove this by using induction on $n$. When $n=1$,
$A_{1}=88$ which is a practical number. Suppose $A_{n}$ is a practical number,
then as $A_{n+1}=A_{n}(10^{2^{n}}+1)$ and $\sigma(A_{n})\geq
2A_{n}-1>10^{2^{n}}+1$ we can conclude that $A_{n+1}$ is also a practical
number. Hence $A_{n}$ is practical for all positive integers $n$. Hence there
are infinitely many palindromic practical numbers. ∎

##  3\. On Natural Numbers which can be represented as sum of a square and a
practical number

We now prove that every natural number of $8k+1$ can expressed as a sum of
practical number and a square. We will prove this using some lemmas.

######  Lemma 3.1.

If $n=2^{k}m$ with $k\geq 1$ and $m\leq 2^{k+1}$, then $n$ is a practical
number.

###### Proof.

As $2^{k}$ is a practical number and $m\leq 2^{k+1}=\sigma(2^{k})+1$ we can
conclude that $2^{k}m$ is a practical number. ∎

######  Lemma 3.2.

Let $m$ be a natural number such that $m\equiv 1\mod 8$ and let $k$ be a
positive integer. There exists a natural number $1\leq x\leq 2^{k}-1$
satisfying $x^{2}\equiv m\ {\text{(mod }}2^{k+2})$.

###### Proof.

Let $m$ be any natural number such that $m\equiv 1\mod 8$. We prove that for
all positive integers $k$ there exists a natural number $1\leq x\leq 2^{k}-1$
such that $x^{2}\equiv m\mod 2^{k+2}$ using mathematical induction. The
statement is true for $k=1$ as $1^{2}\equiv m\mod 2^{3}$. As $m\equiv 1\mod 8$
implies $m\equiv 1^{2}\mod 2^{4}$ or $m\equiv 3^{2}\mod 2^{4}$, the statement
is true for $k=2$.

Assume that the statement is true for $k=s\geq 2$ then there exists a natural
number $1\leq x_{s}\leq 2^{s}-1$ such that $x_{s}^{2}\equiv m\mod 2^{s+2}$.
Now, $x_{s}^{2}\equiv m\mod 2^{s+3}$ or $x_{s}^{2}\equiv m+2^{s+2}\mod
2^{s+3}$. Let $x_{s+1}=x_{s}$ if $x_{s}^{2}\equiv m\mod 2^{s+3}$ and
$x_{s+1}=2^{s+1}-x_{s}$ if $x_{s}^{2}\equiv m+2^{s+2}\mod 2^{s+3}$. Note that
$1\leq x_{s+1}\leq 2^{s+1}-1$ and we claim that $x_{s+1}^{2}\equiv m\mod
2^{s+3}$. If $x_{s}^{2}\equiv m\mod 2^{s+3}$ then as $x_{s+1}=x_{s}$ we have
$x_{s+1}^{2}\equiv m\mod 2^{s+3}$. If $x_{s}^{2}\equiv m+2^{s+2}\mod 2^{s+3}$
then

$\displaystyle x_{s+1}^{2}$ $\displaystyle\equiv(2^{s+1}-x_{s})^{2}\mod
2^{s+3}$ $\displaystyle\equiv 2^{2s+2}+x_{s}^{2}-2^{s+2}x_{s}\mod 2^{s+3}$
$\displaystyle\equiv m+2^{s+2}-2^{s+2}x_{s}\mod 2^{s+3}$ $\displaystyle\equiv
m+2^{s+2}(1-x_{s})\mod 2^{s+3}$ $\displaystyle\equiv m\mod 2^{s+3}\text{ (as
$(1-x_{s})$ is even, $2^{s+2}(1-x_{s})$ is divisible by $2^{s+3}$)}.$

Hence the statement is true for $k=s+1$ and the lemma follows from the
principle of mathematical induction. ∎

Now we are ready to prove the main result of this section.

######  Theorem 3.3.

Every natural number of the form $8k+1$ can be expressed as sum of a square
and a practical number and for every $j\in\\{0,\cdots,7\\}\setminus\\{1\\}$
there exist infinitely many $k$ such that $8k+j$ cannot be written as sum of a
practical number and a square.

###### Proof.

Let $n$ be a natural number $>1$ of the form $8k+1$. Let
$m=\left\lfloor\log_{2}\sqrt{8k+1}\right\rfloor$ (this implies $2^{2m}\leq
8k+1<2^{2m+2}$), then as $m\geq 1$, from Lemma 3.2 there exists $1\leq x\leq
2^{m}-1$ such that $x^{2}\equiv 8k+1\ {\text{(mod }}2^{m+2})$. Therefore,
$8k+1-x^{2}=2^{m+2}s$ for some positive integer $s$ (observe that, $s$ is a
positive integer as $x^{2}<2^{2m}\leq 8k+1$). As $2^{m+2}s\leq 8k+1\leq
2^{2m+2}$ we have $s\leq 2^{m}$. Hence from Lemma 3.1, $2^{m+2}s$ is a
practical number and $8k+1=x^{2}+2^{m+2}s$ is sum of a square and a practical
number. This proves the first part of the theorem.

Let us now prove the second part by considering each
$j\in\\{0,\cdots,7\\}\setminus\\{1\\}$ separately.

  1. ($j=0$):

Consider numbers $m$ such that $m\equiv 24\ {\text{(mod }}32)$, $m\equiv 2\
{\text{(mod }}3)$, $m\equiv 2\ {\text{(mod }}5)$, $m\equiv-1\ {\text{(mod
}}7)$, $m\equiv-1\ {\text{(mod }}11)$ and $m\equiv 2\ {\text{(mod }}13)$. Then
$m$ is of the form $8k$ and $m\equiv x^{2}\ {\text{(mod }}16)$, $m\equiv
x^{2}\mod 3$, $m\equiv x^{2}\ {\text{(mod }}5)$, $m\equiv x^{2}\ {\text{(mod
}}7)$, $m\equiv x^{2}\ {\text{(mod }}11)$ and $m\equiv x^{2}\ {\text{(mod
}}13)$ have no solutions. Hence if $m=n^{2}+P$ then $16\nmid P$, $3\nmid P$,
$5\nmid P$, $7\nmid P$, $11\nmid P$ and $13\nmid P$. Highest power of $2$
dividing $P$ is less than or equal to $8$, and $P$ is not divisible by any
prime less than or equal to $\sigma(8)+1=16$. Hence $P$ cannot be a practical
number and $m$ cannot be written as sum of a square and a practical number.
Hence there are infinitely many numbers of the form $8k$ which cannot be
written as sum of a square and a practical number.

  2. ($j=2$):

Consider numbers $m$ of the form $24n+2$ such that $24n+1$ is not a perfect
square. Then $m\equiv 2\ {\text{(mod }}8)$, $m\equiv x^{2}\ {\text{(mod }}4)$
and $m\equiv x^{2}\ {\text{(mod }}3)$ have no solutions. Hence highest power
of $2$ dividing $m-x^{2}$ for any $x$ is less than or equal to $2$ and since
$m-x^{2}$ is not divisible by any prime less than or equal to $\sigma(2)+1=4$,
we can conclude that $m-x^{2}$ is not a practical number. Hence there are
infinitely many numbers of the form $8k+2$ which cannot be written as sum of a
square and a practical number.

  3. ($j=3$):

Consider numbers $m$ of the form $24n+11$. Then $m\equiv 3\ {\text{(mod }}8)$
and $m\equiv x^{2}\ {\text{(mod }}4)$, $m\equiv x^{2}\ {\text{(mod }}3)$ have
no solutions. Hence highest power of $2$ dividing $m-x^{2}$ for any $x$ is
less than or equal to $2$ and since $m-x^{2}$ is not divisible by any prime
less than or equal to $\sigma(2)+1=4$, $m-x^{2}$ is not a practical number.
Hence there are infinitely many numbers of the form $8k+3$ which cannot be
written as sum of a square and a practical number.

  4. ($j=4$):

Consider numbers $m$ such that $m\equiv 12\ {\text{(mod }}16)$, $m\equiv 2\
{\text{(mod }}3)$, $m\equiv 2\ {\text{(mod }}5)$, $m\equiv-1\ {\text{(mod
}}7)$, $m\equiv-1\ {\text{(mod }}11)$, and $m\equiv 2\ {\text{(mod }}13)$.
Then $m$ is of the form $8k+4$ and $m\equiv x^{2}\ {\text{(mod }}16)$,
$m\equiv x^{2}\ {\text{(mod }}3)$, $m\equiv x^{2}\ {\text{(mod }}5)$, $m\equiv
x^{2}\ {\text{(mod }}7)$, $m\equiv x^{2}\ {\text{(mod }}11)$, and $m\equiv
x^{2}\ {\text{(mod }}13)$ have no solutions. Hence if $m=n^{2}+P$ then
$16\nmid P$, $3\nmid P$, $5\nmid P$, $7\nmid P$, $11\nmid P$ and $13\nmid P$.
Highest power of $2$ dividing $P$ is less than or equal to $8$, and $P$ is not
divisible by any prime less than or equal to $\sigma(8)+1=16$. Hence $P$
cannot be a practical number and $m$ cannot be written as sum of a square and
a practical number. Hence there are infinitely many numbers of the form $8k+4$
which cannot be written as sum of a square and a practical number.

  5. ($j=5$):

Consider numbers $m$ such that $m\equiv 5\ {\text{(mod }}8)$, $m\equiv 2\
{\text{(mod }}3)$, $m\equiv 2\ {\text{(mod }}5)$, and $m\equiv-1\ {\text{(mod
}}7)$. Then $m$ is of the form $8k+5$ and $m\equiv x^{2}\ {\text{(mod }}8)$,
$m\equiv x^{2}\ {\text{(mod }}3)$, $m\equiv x^{2}\ {\text{(mod }}5)$, and
$m\equiv x^{2}\ {\text{(mod }}7)$ have no solutions. Hence if $m=n^{2}+P$ then
$8\nmid P$, $3\nmid P$, $5\nmid P$ and $7\nmid P$. Highest power of $2$
dividing $P$ is less than or equal to $4$, and $P$ is not divisible by any
prime less than or equal to $\sigma(4)+1=8$. Hence $P$ cannot be a practical
number and $m$ cannot be written as sum of a square and a practical number.
Hence there are infinitely many numbers of the form $8k+5$ which cannot be
written as sum of a square and a practical number.

  6. ($j=6$):

Consider numbers $m$ of the form $24n+14$. Then $m\equiv 6\ {\text{(mod }}8)$,
$m\equiv x^{2}\ {\text{(mod }}4)$ and $m\equiv x^{2}\ {\text{(mod }}3)$ have
no solutions. Hence highest power of $2$ dividing $m-x^{2}$ for any $x$ is
less than or equal to $2$ and since $m-x^{2}$ is not divisible by any prime
less than or equal to $\sigma(2)+1=4$, we have $m-x^{2}$ is not a practical
number. Hence there are infinitely many numbers of the form $8k+6$ which
cannot be written as sum of a square and a practical number.

  7. ($j=7$):

Consider numbers $m$ of the form $24n+23$. Then $m\equiv 7\ {\text{(mod }}8)$,
and $m\equiv x^{2}\ {\text{(mod }}4)$ and $m\equiv x^{2}\ {\text{(mod }}3)$
have no solutions. Hence highest power of $2$ dividing $m-x^{2}$ for any $x$
is less than or equal to $2$ and since $m-x^{2}$ is not divisible by any prime
less than or equal to $\sigma(2)+1=4$, $m-x^{2}$ is not a practical number.
Hence there are infinitely many numbers of the form $8k+7$ which cannot be
written as sum of a square and a practical number.

∎

##  4\. Future prospects

In this paper, we have given necessary and sufficient conditions for linear
and quadratic polynomials to represent infinitely many practical numbers
(Theorem 2.2 and Theorem 2.5). It would be worthwhile to explore such
necessary and sufficient conditions for cubic and biquadratic polynomials with
integer coefficients.

##  5\. Disclosure Statement

No potential conflict of interest was reported by the authors.

##  6\. Data availability statement

There is no data associated with this article.

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