# Theory Guide

# QSA Q-Bat Explorer — Theory Guide

### QuickerSim Automotive Ltd

## 1. Software Philosophy

QSA Q-Bat Explorer is an innovative software designed for heat transfer calculations in

battery packs by the means of Finite Element Method (FEM). The software uses Reduced

Order Modelling (ROM) to reduce the computation complexity of the problems and provide

the user with accurate thermal estimations. Q-Bat Explorer can be used to calculate both

individual cells, as well as batteries or whole battery packs coupled hydraulically.

This Theory Guide provides basic theoretical information about the models used in Q-Bat

Explorer and describes the software convention, in order to help the User understand the

Framework workflow.

The software’s UI is developed in a prototype-base convention. At first the User creates a

prototype of each of the unique components in the assembly. Component prototypes are

only virtual and therefore are not automatically added to the simulation domain. At this

stage all of the physical properties of the element are specified and boundary conditions are

applied.

Elements are added to the physical domain by specifying location matrices of their occurrences.

If only one instance is needed in the assembly, specifying the array is still necessary

(it will be an 3×1 array containing the location of the body). All elements can be freely rotated

and translated in the domain. If a similar component with different physical properties

or boundary conditions is needed, a second base component (prototype) is required to be

created

**EXAMPLE**

The User wants to simulate heat transfer in a battery containing 6 cells. All of the cells share

the same geometry and material properties, but 3 of them are discharged by 200 A, while the

other 3 – by twice as much (400 A). In order to create a model of the battery the User has

to create two different cell bases and replicate both of them, specifying two matrices which

describe the location of the cells.

To depict the steps needed to create the cells of the simulated battery a hierarchical tree is

displayed (Figure 1).

Figure 1: Example: hierarchical tree of cells and cell bases

This way the User creates a model consisting of 6 cells, shown in Figure 2. Omission of cell

base replication (creating only the cell bases) would result in an empty model, as cells would

not be added to the simulation domain.

##### Figure 2: Example: cell arrangement

After specifying all of the components of the assembly (their bases and locations), the collective

battery model is created by assigning the component arrays to a higher-level aggregating

component (described in detail in Subsection 2.6).

Thermal contacts (Subsection 2.4) and the initial temperature is specified and applied to specific

components or the whole assembly. The model is reduced and global mass and diffusion

matrices are assembled.

The User defines the parameters of the simulation (length of time step and number of time

steps) and calculations are started. Solutions from each time step can be displayed and

exported afterwards.

To perform a steady-state calculation the User should specify a very large time step (order

of thousands of seconds).

All of the parameters and calculations are in SI units, except for temperature, which is given

in degrees Celsius.

## 2. Mathematical Models of Battery Components

This section focuses on specific battery components and how they are modeled inside the software.

Equations and methods applied to describe the properties and state of the components

are presented in each of the subsections.

Three main types of elements that can be composed in Q-Bat Explorer include:

• Cell

• Cooling plate

• Passive heat component

By using different arrangements of these components the User is able to model any battery.

Methods for creating different elements, which do not match the ones listed above (such as a

thermal pad or battery housing), are described in the following subsections.

A sample geometry used for visualizing specific component models is presented below (Fig.

3).

##### Figure 3: Sample geometry

The geometry comprises three cells (colored red) separated by passive heat components (yellow),

standing on a cooling plate (grey) with a cylindrical pipe. Between the cells and cooling

plate a thermal pad is applied.

The battery housing (shown in transparent grey) surrounds all of the other elements.

6 QuickerSim

### 2.1 Cell

##### Figure 4: Sample geometry – cells

Cells are modeled as prismatic or cylindrical solid bodies.

Heat generation can be set by either specifying the volumetric heat generation (in [ ]) orthe

electrical properties of the cell and applied current load (in [A]), that can vary in time.

To capture the instantaneous capacitive effects, cells are modelled as RC circuits. A model

of the circuit is presented below (Fig. 5).

##### Figure 5: RC circuit

In the presented diagram Uocv is the open circuit voltage and R_{0}, R_{1} and C_{1} are the parameters

of the model (resistance and capacitance).

Two governing equations describing the cell’s instantaneous state are derived from Kirchhoff’s

circuit laws:

(1)

(2)

Where U_{1} is the voltage drop on the RC pair [V], U is the total cell voltage [V] and I is the

cell current [A].

Heat generation is calculated using the Joule heating equation:

(3)

Both heat source and electric current (inputs to the model) can be given as a constant scalar

or linear or non-linear matrix defined in an Excel spreadsheet.

By default the heat is generated as uniform distribution in the cell. This can be modified by

the User by specifying the distribution matrix.

Within the cell heat is transferred by the means of diffusion, following Fourier’s law of thermal

conduction, described in a differential form by:

(4)

Where q [ ] is the local heat flux density, λ [ ] denotes thermal conductivity and ∇ T []

is the local temperature gradient.

Material properties of the cell are defined in an Excel spreadsheet. Properties can be constant

scalars or can be given as a table of variables depending on temperature. Electrical properties

can also depend on the cell’s State of Charge (SOC).

If any of the given properties are temperature-dependent or SOC-dependent variables, Q-Bat

Explorer approximates them linearly between the given points and fixes their values after

exceeding the marginal values of the independent variables.

Q-Bat Explorer supports anisotropic heat conduction – values of λ can be given in an Excel

spreadsheet or entered by the User directly. If the value of λ is given as a 1×1 scalar, the cell

is by default treated as isotropic.

### 2.2 Cooling Plate

##### Figure 6: Sample geometry – cooling plate

A cooling plate is modeled as a solid body containing a solid body pipe. The volume of the

cooling fluid is modeled as a void in the geometry.

Within the cooling plate heat is transferred by the means of diffusion, following Fourier’s law

of thermal conduction, described in a differential form by:

Where is the local heat flux density, denotes thermal conductivity of the

cooling plate material and is the local temperature gradient.

Material properties of the cooling plate are defined in an Excel spreadsheet. Properties can

be constant scalars or can be given as a table variables depending on temperature. When

defining the material properties during cooling plate creation, two different Excel spreadsheets

are needed – one describing the properties of the cooling plate, and the other for the cooling

fluid. The cooling pipe is omitted – the cooling plate and pipe are therefore modeled as one

body.

##### Figure 7: Sample geometry – cooling fluid

properties of the fluid, it is necessary to specify the hydraulic diameter of the pipe (d [m]),

mass flow rate (in ) and the fluid inlet temperature(in ).

Heat between the fluid and cooling pipe is transferred by the means of forced convection,

quantified by Newton’s Law of cooling:

Where is the heat transfer coefficient and is the difference in the cooling

fluid’s temperature between the wall temperature and its surrounding fluid.

The value of α can be given by the User or calculated by Q-Bat Explorer from the definition

of the Nusselt number, where d is the hydraulic diameter of the pipe [m].

The Nusselt number (Nu) depends on the character of the flow and can be obtained from

[2]:

• For Reynold’s number Re < 2000 the Nusselt number is given by Michiejew’s equation

for laminar flows. Grashof number has been omitted, as heat transfer from forced convection

is much greater than that from natural convection, and would require additional

User input:

(8)

• For 2000 < Re < 10000 the Nusselt number is given by Michiejew’s equation for

transitional flows:

(9)

• For 10000 < Re < 5000000 (fully turbulent flow) Sieder and Tate’s equation is used:

(10)

In equations (8)-(10) and denote the Prandtl number and and are the

values of dynamic viscosity in the fluid and on the wall surface respectively. EL is a correction

factor based on the L/d ratio (where L – length of pipe [m]). The set of EL values for different

L/d ratios are given in Table 1.

L/d | 1 | 2 | 5 | 10 | 15 | 20 | 30 | 40 | 50 |

1.9 | 1.7 | 1.44 | 1.28 | 1.18 | 1.13 | 1.05 | 1.02 | 1 |

Table 1: correction factor values for laminar flows

is a correction factor based on the flow’s Reynold’s number. A table of values is

presented below.

2200 | 2300 | 2500 | 3000 | 3500 | 4000 | 5000 | 6000 | 7000 | 8000 | 9000 | 10000 | |

2.2 | 3.6 | 4.9 | 7.5 | 10 | 12.2 | 16.5 | 20 | 24 | 27 | 30 | 33 |

Table 2: correction factor values

The values of and are interpolated linearly between the points specified in the tables.

Higher values of Re are not supported, as they are non-physical in this application.

### 2.3 Passive Heat Component

##### Figure 8: Sample geometry – heat component

Passive heat components are modeled by solid bodies of any geometry.

Within the component heat is transferred by the means of diffusion, following Fourier’s law

of thermal conduction, described in a differential form by:

(11)

Where is the local heat flux density, denotes thermal conductivity of the

cooling plate material and is the local temperature gradient

Material properties of the heat component are defined in an Excel spreadsheet. Properties can

be constant scalars or can be given as a table of variables depending on temperature.

Passive heat components are not meant to generate heat. Although it is possible to assign a

heat source to a passive heat component this practice is not recommended.

### 2.4 Thermal Pads & Other Contacts

##### Figure 9: Sample geometry – thermal pad

By default the conductance of adhering components is equal to zero, meaning heat transfer

between them does not occur. In order to allow heat transfer between the elements it is

necessary to add contacts to the model (and specify their conductance).

Contacts are defined by specifying thermal conductivity and thickness [m] of a

contact. Thermal pads and other significantly thinner components are modeled as thermal

contacts with a specified conductance.

Heat transfer on the contact is described by the one-dimensional form of Fourier’s Law:

(12)

### 2.5 Internal Air & Battery Housing

##### Figure 10: Sample geometry – housing

At this point natural convection is not supported by Q-Bat Explorer. This is to be added in

later editions of the software.

A temporary solution involves creating a domain of air (as a passive heat component) with

artificial deflation of its thermal conductance.

To simulate a battery housing it is recommended to create a passive heat component with the

material properties of air and a corresponding geometry. Thermal contacts between the air

and adhering components are required to be added. The value of the heat transfer coefficient

α can be broadly approximated by the equations for natural convection near a single vertical

wall (described in [3]):

(13)

(14)

Where

is the average Nusselt number, Pr is the Prandtl number for air and Ra is the

Rayleigh number for the assumed air flow. The height of the wall is denoted by l [m],

is the thermal coefficient of air.

The obtained values should range between 5-10 , what is in accordance with typical

values of α in natural convection in air.

While specifying the contact between components and the surrounding air, λ should be equal

to the value of α obtained from eq. 14 and the value of thickness δ should be equal to 1 (the

summary conductance will then be in the range of 5-10 ).

To simulate the heat transfer through the wall of the housing it is recommended to specify

a thermal contact on the outer walls of the air domain. An additional Robbin boundary

condition can be added to consider forced convection around the housing walls.

Internal air should be treated in a similar manner.

### 2.6 Aggregating components

Q-Bat Explorer gives the opportunity to aggregate components, in order to form integral battery parts.

These assemblies can be easily rotated, transformed or duplicated. The supported

aggregating types include cell caskets, battery modules, battery islands and batteries.

These components require the User to append different basic types of components. On this

level it is also possible to add thermal couplings between the objects, as described in subsection

2.4. After creating an aggregated component and assigning contacts between its parts, it can

be aggregated to a higher form, up to a battery.

The supported aggregating types include:

• A cell casket is an object that consists of cells and passive heat components.

• A battery module is an object that consists of cells caskets, passive heat components

and cooling plates.

• A battery island is an object that consists of battery modules, passive heat components

and cooling plates.

• A battery is an object that consists of battery islands and passive heat components.

Batteries are the highest form of aggregating objects, although they can be coupled with one

another hydraulically.

A simple hierarchy of all the components and elements that can be aggregated within them

is shown schematically in the figure below.

##### Figure 11: Aggregating components

*Comment: Each of the components shown above can but does not have to consist of the shown** elements. It can also include multiple objects of the same type. For example a battery module** can consist of three different cell caskets, two identical cooling plates and no passive heat** components.*

## 3. Discretization

As the software uses FEM, discretization of the computational domain is crucial. Q-Bat

Explorer reduces the basis of the solution to simplify the model and therefore substantially

shortens the computational time. This section briefly describes the methods of discretization

used in the software.

### 3.1 Mesh Generation & FEM

Q-Bat Explorer uses Gmsh software [1] to generate Finite Element meshes. Mesh creation

is done automatically, with just the mesh density as the User input. For more complicated

geometries, it is possible to create a custom mesh using the Gmsh software and import it

directly into Q-Bat Explorer. This approach can help the user create a dense mesh only near

the regions of interest, where most of the heat transfer occurs. When importing a msh file all

surfaces must have physical IDs.

The automatic mesh creation takes in a parameter between 0-1, specifying the mesh density.

Values close to 1 correspond to a poor quality mesh, while those closer to 0 correspond to a

denser mesh (what also significantly increases the reduction time).

**EXAMPLE**

Meshes created automatically in Q-Bat Explorer for different mesh densities for a simple

cuboid geometry

##### Figure 12: Mesh-0.1 Figure 13: Mesh – 0.5 Figure 14: Mesh – 0.9

When creating a cooling plate it is also possible to specify the number of points on curvature.

This parameter allows the create a denser mesh only in the proximity of the cooling pipe (if

it is circular).

**EXAMPLE**

Cooling plate meshes created automatically in Q-Bat Explorer for different number of curvature points.

##### Figure 15: Mesh – 3 points on curvature Figure 16: Mesh – 10 points on curvature

On basis of the created meshes, the heat transfer equations are solved by means of Finite

Element Method. The Galerkin approach is utilized for calculating the diffusion matrix.

For each component a local mass and diffusion matrix is created. The matrices are coupled

together depending on occurring contacts. Global matrices for the whole model are assembled

and a linear equation system is solved. The Euler method is used for calculating the

solution.

### 3.2 Model Reduction

For each body, the thermal problem is given by the equation:

(15)

Where M is the mass matrix, D is the diffusion matrix and T is the solution vector. Assuming

a linear problem, the solution can be approximated as:

(16)

Bringing the problem to an eigenvalue equation:

(17)

(18)

Where is the eigenvector.

The obtained eigenvectores are used to create a basis of the solution. A reduction transformation matrix is formed:

(19)

A low-dimensional representation of the original space is created by forming a reduced mass

and diffusion matrix:

(20)

(21)

The software uses the reduced matrices Mr and Kr to find the solution to the problem, which

is later transformed to the full model.

The determined values of eigenvectors are regarded as modes, used to represent the basis of

the solution. A visualization of a few of the modes generated for each object is depicted in

the example below.

**EXAMPLE**

##### Figure 17: Sample mode generation – geometry

Some of the generated modes are displayed below to illustrate the concept of a reduced

solution basis.

Figure 18: Sample mode Figure 19: Sample mode Figure 20: Sample mode

generation – mode 2 generation – mode 3 generation – mode 5

### 3.3 Cooling fluid channel discretization

The cooling fluid is discretized by the User by defining the number of divisions (n) during

cooling plate creation. Control point inside the fluid are created and the fluid is divided into

n − 1 volumes.

The flow in the pipe is calculated using a second order Backward Differentiation Formula

(BDF) in the discretized volume centroids, described by the following equations. The fluid is

coupled with the surrounding cooling pipe in every discretized volume by the wall temperature.

(22)

(23)

Where:

– density in the i-th volume .

– energy in the i-th volume [J].

– i-th volume .

– cross-section area before the i-th volume .

– heat transfer coefficient in the i-th volume .

– side area (pipe area) of the i-th volume .

– wall temperature in the i-th volume [K].

– mass flow rate .

## References

[1] url: https://gmsh.info/.

[2] Wieslaw Gogol. Wymiana ciep la – Tablice i wykresy. s.l. : Wydawnictwa Politechniki

Warszawskiej, 1972.

[3] Tomasz S. Wisniewski Stefan Wisniewski. Wymiana ciep la.