Heat Tranfer Project Report
-
Upload
pramod-srinivasan -
Category
Documents
-
view
225 -
download
0
Transcript of Heat Tranfer Project Report
-
8/22/2019 Heat Tranfer Project Report
1/97
Tata Technologies Limited
A Report on
1D and 2D heat transfer project(including a general overview of company)
Aerospace DepartmentTATA Technologies Limited, Hinjewadi
By:
Vishesh Gupta 2011A4PS273P
At
TATA Technologies Limited, Hinjewadi, Pune
A Practice School I station of
Birla Institute of Technology and Science,
Pilani, Rajasthan
July, 2013
-
8/22/2019 Heat Tranfer Project Report
2/97
Tata Technologies Limited
A Report on
1D and 2D heat transfer project
(including a general overview of company)
Aerospace Department
TATA Technologies Limited, Hinjewadi
By:
Vishesh Gupta 2011A4PS273P
Prepared in partial fulfillment of the
Practice school I Course
At
TATA Technologies Limited, Hinjewadi, Pune
A Practice School I station of
Birla Institute of Technology and Science,
Pilani, Rajasthan
July, 2013
-
8/22/2019 Heat Tranfer Project Report
3/97
Tata Technologies Limited
Acknowledgement
I am grateful to Tata Technologies Limited for giving me this
opportunity to gain experience at the organization and forgiving me invaluable guidance right at the start.
I would also like to thank Mr Sumeet Dhar (Project Manager),
Mrs Prachi Dhadheech (HR) for their valuable support
throughout our tenure. I would like to make a special mention
of Vinay Shekhar (employee at the aerospace department) for
providing me deep insight at crucial junctures of my project
and small helps in debugging the code. I would also like toextend my gratitude towards Mr Manoj (aerospace project
manager) for giving me the freedom to work through my
project and patiently listening to my queries and explanations
also giving worthy advice. I would further like to express my
gratitude towards Dr Biswadip Shome for giving me worthy
suggestions and sufficient time to extend my project to 2D
analysis in the short time I worked under him.
I would finally like to thank Dr Ranjit Patil( PS Instructor
from BITS Pilani) for being a constant source of motivation for
me throughout my internship at Tata Technologies.
-
8/22/2019 Heat Tranfer Project Report
4/97
Tata Technologies Limited
Birla Institute of Technology and Science
Pilani (Rajasthan)
Practice School Division
Station: Tata Technologies Ltd. Centre: Hinjewadi,
Pune
Duration: 53 Days Date of Start: 22/05/2013Date of Submission: 07/07/2013
Title of Project: Simulation tool for a floor fire test
and the behavior of the material in the test by modeling
of a transient heat transfer process through a series of
slabs.
Name of Student I.D.No. Discipline
Vishesh Gupta 2011A4PS273P B.E. Mechanical
Name and designation of experts:
Dr Biswadip Shome (Head Aerospace Design &
Validation)
Manoj Radle (Project Manager)Name of PS Faculty: Dr. Ranjit Patil
Key words: heat transfer, finite difference analysis
Project Areas: transient heat transfer, partial
differential equations, fully implicit FDM
-
8/22/2019 Heat Tranfer Project Report
5/97
Tata Technologies Limited
Abstract
This project report has been primarily divided into four
chapters. Chapter 1 deals with company information. Chapters
2, 3 and 4 give detailed analysis of my project related work.
Chapter 1 aims at giving a general overview of the company,
mainly information pertaining to what was observed during
the orientation sessions.
Chapter 2 deals with 1 dimensional heat transfer solved by the
analytical method using techniques like transformation of non-
homogenous boundary conditions to homogenous boundary
conditions by superposition principle, separation of variables
and Greens function.
Chapter 3 concerns with the numerical approach adopted to
solving the 1D heat transfer problem. The fully implicit finite
difference scheme has been utilized here which is
unconditionally stable and convergent. The matrix formed has
been solved both directly using in-built MATLAB methods as
well as Gauss-Seidel iterative procedures.
Finally, Chapter 4 showcases the 2D heat transfer problem
solved by a numerical approach which uses a slight variant of
the fully implicit finite difference scheme called the alternating
direction implicit method (ADI).
Signature (Student) Signature (Project Guide) Signature (PS Faculty)
Date: Date: Date:
-
8/22/2019 Heat Tranfer Project Report
6/97
Tata Technologies Limited
Table of Contents
S No. Topic Page No.
1 CHAPTER I
Introduction 5
Orientation Related Information 11
ERM 20
Software Partners 24
2 CHAPTER II
1D Analytical approach 34
Improvements & Conclusion 46
3 CHAPTER III
1D Numerical Approach 47
Improvements & Conclusion 58
4 CHAPTER IV
2D Numerical Approach 59
Improvements and Conclusion 70
5 Appendix A 71
6 Appendix B 777 Appendix C 87
8 References 92
-
8/22/2019 Heat Tranfer Project Report
7/97
Tata Technologies Limited
List Of Illustrations
Figures
Fig. 1.1 Tata overview
Fig. 1.2 Presence of Tata Technologies over the
globe
Fig. 1.3 Product Cycle at Tata TechnologiesFig. 1.4 Historical PerformanceFig. 1.5 Global initiatives to sustain and grow
employee engagement
Fig. 1.6 Risk Management ProcessFig. 1.7 Some clients of Tata TechnologiesFig. 1.8 Aerospace Design ProcedureFig. 1.9 Airplane structure components
Fig. 2.1 Time-Temperature Curve
Fig. 2.2 Values of source temperature and time
Fig. 2.3 Transient heat transfer through a
series of blocks
Fig. 3.1 1 dimensional heat transfer through a
series of blocks
Fig. 4.1 2 dimensional heat transfer through a
series of blocks
-
8/22/2019 Heat Tranfer Project Report
8/97
Tata Technologies Limited
Graphs
Graph 2.1 Variation of source and interface 1
temperature with time
Graph 2.2 Variation of source and interface 2temperature with time
Graph 2.3 Variation of source and all interfacetemperature with time
Graph 3.1 Variation of source and interfacetemperature with time by direct
solving of matrix
Graph 3.2 Variation of source and interfacetemperature with time (Gauss-Seidel
iteration)Graph 3.3 Variation of source and interface
temperature with time (both direct by
direct matrix and Gauss-Seidel
iteration)
Graph 3.4 Convergence of temperature at
particular time by Gauss-Seidel
iterations
Graph 4.1 Variation of node 1 temperatures of
all interfaces with time
Graph 4.2 Variation of temperature of all nodes
at top surface with timeGraph 4.3 Temperature of all nodes at 30 min at
top surface
-
8/22/2019 Heat Tranfer Project Report
9/97
Tata Technologies Limited
CHAPTER I
Introduction
The origins of Tata Technologies lie with disparate
organizations started in India, Europe and the USA over 20
years ago. Gradually the European and North American
companies came together until in 1989 a single entity emerged
INCAT. Following growth and acquisition, that company
firmly established itself in North America, Europe and Japan.
And in 2004, INCAT was launched on the London StockExchange. Tata Technologies started in Singapore in 1994 as a
provider of specialized IT enabled consulting, services and
products to leading manufacturers. Establishing itself in the
USA, UK and India, the companys first major outsourcing
contract was with Tata Motors. Then, in 2005, Tata
Technologies acquired INCAT.
Today, the enlarged company serves the major automotive andaerospace OEMs and their suppliers. Tata Technologies is
active in North America, Europe, the Middle East and the
Asia-Pacific region, and currently services all of the top ten
aerospace original equipment manufacturers (OEMs) and all of
the top ten automotive OEMs.
-
8/22/2019 Heat Tranfer Project Report
10/97
Tata Technologies Limited1
The Tata group
Tata Technologies is part of the Tata group, one of Indias
oldest, largest and most respected businesses. Founded by
Jamsetji Tata in 1868, the group pioneered industries of
national importance to India: steel, power, hospitality and
airlines. With revenues over US $100 billion, the group today
has 3.2 million shareholders, more than 100 operating
companies in six continents and employs more than 450,000
people. It focuses on steel production, power generation,
commercial and passenger vehicles, chemicals, hotels, textiles,
consumer goods, consultancy, information technology and
telecommunications. Earning the trust and respect of millions
of stakeholders around the world, the group and its enterprises
have adhered to a rigorous set of business ethics and a strong
commitment to corporate social responsibility for over 140
years.
Fig. 1.1: Tata overview
-
8/22/2019 Heat Tranfer Project Report
11/97
Tata Technologies Limited1
Fig. 1.2: Presence of Tata Technologies over the globe
-
8/22/2019 Heat Tranfer Project Report
12/97
Tata Technologies Limited1
Fig. 1.3: Product Cycle at Tata Technologies (1st half)
-
8/22/2019 Heat Tranfer Project Report
13/97
Tata Technologies Limited1
Fig. 1.3: Product Cycle at Tata Technologies (2nd half)
-
8/22/2019 Heat Tranfer Project Report
14/97
Tata Technologies Limited1
Fig. 1.4: Historical Performance
-
8/22/2019 Heat Tranfer Project Report
15/97
Tata Technologies Limited1
Orientation Related Information
Tata Technologies enables manufacturing companies to design
and build better products through engineering services and the
application of information technology to product development
and manufacturing enterprise processes. With over 6,000
engineers, representing 17 nationalities, Tata Technologies
covers every aspect of the value chain from conceptualization,
manufacturing and aftermarket/MRO support. The three
major industry verticals that it serves include aerospace,
automotive and machinery manufacturing. Tata Technologies
supports clients through comprehensive engineering services
and IT processes and tools to manage product development
and the complete manufacturing ecosystem. It serves clients in
25 countries, with a delivery model specifically designed for
engineering and IT engagements that offers a unique blend of
deep, local expertise integrated with our nine global delivery
centers based in Detroit (USA), Coventry (UK), Pune and
Bangalore (India), Brasov, Craiova & Iasi (Romania), Stuttgart
(Germany), and Bangkok (Thailand).
-
8/22/2019 Heat Tranfer Project Report
16/97
Tata Technologies Limited1
VPD Vehicle Programs & Development (VPD)
The VPD provides complete outsourced program management,
concept development, detail design, validation and
manufacturing planning services. Projects of this scale and
complexity are achieved through a combination of automotive
experts in the US and Europe, coupled with Indias most
experienced automotive engineers. Programs include electric
vehicles, or EV variants that help achieve sustainability
targets, while providing mobility at an affordable price point.
The eMO-C electric delivery vehicle concept, designed by Tata
Technologies, follows the eMO showcased at the 2012
International Auto Show in Detroit, was a first for any India-
based engineering services firm.
E&D Outsourced Engineering & Design
The services of this group include concept development,
VA/VE, CAE, detailed engineering, embedded software
development, product verification, and manufacturing process
design, tool design and validation, applied to major product
subsystems and components. Offerings include the provision of
providing services from offshore engineering centers in India,
Romania and Thailand.
PLM Product Lifecycle Management Solutions
The PLM optimizes product development processes,
implementing collaborative PLM tools, a major contributor to
ER&D investment efficiency, especially for global engineering
teams with extensive supply chains. Tata Technologies is the
worlds largest independent reseller of PLM technology. It and
-
8/22/2019 Heat Tranfer Project Report
17/97
Tata Technologies Limited1
its customers are among the worlds top users of PLM
technology. The companys engineers use these products to
efficiently deliver services to our clients worldwide. It packages
the insights and best practices we have developed through ourengineering heritage into service offering templates to improve
the efficiency of engineering teams at Tata Technologies and
its clients.
ESG Enterprise Solutions Group
The ESG provides consulting and IT solutions that help
manufacturing customers in optimizing critical enterpriseprocesses through the application and data analytics of
Enterprise Resource Planning, (ERP), Manufacturing
Execution Systems, (MES), and Customer Relationship
Management, (CRM), including the use of social media and
improving manufacturing planning and performance. It also
has extensive experience in rapidly integrating the processes,
systems and data of companies acquired by manufacturers.
-
8/22/2019 Heat Tranfer Project Report
18/97
Tata Technologies Limited1
Overview and Outlook
1. Manufacturing
The sectors that it serves aerospace, automotive andmachinery manufacturing, have emerged from the global
recession and engineering R&D (ER&D) spends have grown
consistently. Demand is being driven by an increasingly
competitive market and changing demographics in the west,
where availability of a skilled engineering workforce is on the
decline in many countries, adds NASSCOM. Tata Technologies
helps its partners in the manufacturing industry tacklechallenges relating to the shortage of engineering resources
and skills. It partners with the worlds top automotive OEMs
and tier one auto suppliers, top aerospace companies and
leading machinery manufacturing OEMs to supply high-end
capability and variable capacity. It can scale up rapidly,
flexing resources and specific skills to meet the project
demands of our clients, helping them move from fixed tovariable costs, yet retaining access to top talent, who
understand their products and processes.
2. Aerospace
Global air travel is poised to grow significantly over the next
few years and expand the addressable market for airlines and
their partners. While innovation in business and operatingmodels evolve at a rapid pace, product innovation needs to
gather exponential momentum for airlines to win the battle
against high operating costs. The aerospace sector is projected
to be worth over $4 trillion by 2030 according to Airbus and
-
8/22/2019 Heat Tranfer Project Report
19/97
Tata Technologies Limited1
Boeing with 86,000 new aircraft expected to be delivered over
the next 20 years.
The backlog of orders for large commercial fixed-wing aircraft
stands at 8,000 and is growing, driven mainly by the
introduction of new aircraft such as Airbus A320neo and
A350, and Boeings 737 MAX and 787. The depressed market
for business jets is starting to recover and new product
development is on the rise with expectations that the market
will grow five percent annually over the next few years. Even
the global commercial helicopter market is recovering sooner
than expected. Booz & Co identifies four major challenges
confronting the civilian aerospace sector: increasing production
rates; unsustainable development cost and value distribution;
growing demand for more efficient aircraft; and digitization of
the industry. The spurt in production of aircraft has led to a
slip-up in the supply chain, as not all suppliers are geared to
meet the increased demand. The existing tooling,
infrastructure and capabilities will be pushed to the limit
across the upstream supply chain because of increased
outsourcing and the complexity of sub-systems and assemblies.
-
8/22/2019 Heat Tranfer Project Report
20/97
Tata Technologies Limited2
3. Automotive
Rarely has an industry confronted the magnitude of
multidimensional change that the automotive industry faces
today. Meanwhile, 2012 sales in Europe reached the lowest
level in 17 years, due to a weak economy and low consumer
confidence. In the United States, strong pent-up demand and a
falling unemployment rate helped auto sales reach a five-year
high in 2012. The regional shift in vehicle sales opportunities
translates into a need to develop products that resonate with
new groups of consumers. The automotive industry is being
shaped by a number of forces, according to KPMG's 2013
Global Automotive Executive Survey. Among them are
environmental pressures leading to more efficient engines.
Electric mobility
To capitalize on these industry trends, Tata Technologies
formed a Vehicle Programs Development group in2011. The
group has demonstrated its capabilities to develop a complete
vehicle. In 2012, Tata Technologies' eMO (electric mobility)
vehicle study was introduced to the industry at the North
American International Auto Show in Detroit. The worlds first
complete vehicle study developed by an India-based
engineering services company, the eMO showcases innovation
in automotive packaging and design, manufacturing processes,
as well as electric vehicle engineering benchmarks, and was
developed to go to market at the disruptive price point of
$20,000.
-
8/22/2019 Heat Tranfer Project Report
21/97
Tata Technologies Limited2
In 2013, the group introduced the eMO-C, a commercial
vehicle variant, profiled in another part of this report, and
opened an all-new 10,000-square foot North American
Engineering and Innovation Center in Troy, Michigan, USA.The facility opened in March with 60 engineering
professionals, with the number expected to exceed 100 by the
end of the year.
4. Machinery Manufacturing
There are five major trends that are evident in the global
industrial machinery segment: agile product development; cost
fitness/fine-tuning the cost structure to fund growth;
leveraging IT/digitization; developing the BRIC markets and
other emerging regions; and impacting technological trends
connectivity, monitoring and emissions reduction.
For many industrial companies, developing the BRIC markets
and other emerging regions represent the best opportunities
for growth. With the machinery manufacturing sector
witnessing steady growth worldwide, Tata Technologies is wellpositioned to support various OEMs and tier 1 suppliers. Its
service offerings cover product design, mechanical, electrical
and embedded electronics and manufacturing engineering. The
aim is to be a full service provider to the machinery
-
8/22/2019 Heat Tranfer Project Report
22/97
Tata Technologies Limited2
manufacturing industry by leveraging our skills and
knowledge in engineering, Product Lifecycle Management
(PLM) and enterprise IT solutions. Its machinery
manufacturing domain comprises experts with extensiveknowledge in systems engineering, mechanical engineering,
product design, electrical, electronics and embedded design
and development. Its innovative and frugal engineering
approach helps organizations create products, at a faster pace
with a lower cost, delivering more value to end customers.
Human Capital
The Company continues to pursue its strategy of growing
returns on human capital. It strives to consistently sustain a
high performance culture to bring out the best in people in a
signature working experience. This supports its customers
with highly proactive and motivated teams to ensure delivery
on promises of best of breed solutions cost effectively.
Customer delight leads to Shareholder value.
Talent Acquisition
A focused branding of the Company both with Campuses and
the industry job market lies at the foundation of our strategy
for talent acquisition. The focus on acquiring skills, specific to
customer needs takes root with specialized courses designed by
us and offered to select institutions. The company addressesthe job market with several propositions for quick engagement
of exceptional talent. Its PFLE branding (Passionate Fun
Loving Engineer), a proactive team of recruiters supported by
imaginative IT enabled processes, ensures the highest
-
8/22/2019 Heat Tranfer Project Report
23/97
Tata Technologies Limited2
productivity on talent acquisition. Costs of recruitment
benchmark well with industry. Recruitment lead times are
continuously on the decline.
Fig. 1.5: Global initiatives to sustain and grow employee engagement
-
8/22/2019 Heat Tranfer Project Report
24/97
Tata Technologies Limited2
ERM
Definition
ERM is a process, effected by an entity's Board of Directors,management and other personnel, applied in strategy setting
and across the enterprise, designed to identify potential events
that may affect the entity, and manage risks to be within its
risk appetite, to provide reasonable assurance regarding the
achievement of entity objectives.
The Company has established a formal Enterprise Risk
Management (ERM). The Company has adopted the
recommendations on the Enterprise Risk Management
framework provided by the Committee of Sponsoring
Organizations of the Treadway Commission (COSO). As a
services focused Company, it is necessary for the Company to
manage risk at the individual transaction level and to consider
aggregate risk at the customer, industry and geographic,
where appropriate.
ERM Organization and Process
The Executive Management Team of the Company is
responsible for implementing the Risk Management
Framework under the direction of the Audit Committee of the
Company, and the Audit Committee provides periodical
updates to the Board of Directors of the Company. The Board
monitors the overall performance of the Risk Management
function.
-
8/22/2019 Heat Tranfer Project Report
25/97
Tata Technologies Limited2
Risk Management Activities
A disciplined approach to risk is important in an organization
such as ours in order to ensure that we are executing according
to our strategic objectives and this process is designed to
identify potential events that, if they occur, will affect our
Company. The management has identified the following top 10
risks, classified into external risk factors and internal risk
factors, to help achieve business objectives in a robust manner.
Fig. 1.6: Risk Management Process
-
8/22/2019 Heat Tranfer Project Report
26/97
Tata Technologies Limited2
Global delivery
Tata Technologies has fine-tuned its ability to gather its global
resources in combinations that reap the utmost benefit for its
clients. It can combine its scalable resourcesboth human and
physical with the proven technology and our accumulated
lengthy experience to address any client requirement. And it
can do it while providing the underlying assurance that all its
efforts are underpinned by a robust reputation for thorough
organization and first-class engineering. At each client
engagement, it balances its resources for maximum benefit.
With access to several thousand engineering and technology
specialists, it can also provide the right level of qualifications,
experience and skills to meet your exact requirements. In
addition, its global services delivery team is led by individuals
with a robust background in engineering.
It offers real-world education for:
1) Dassault Systmes
2) Siemens PLM
3) Autodesk (mechanical design applications)
4) MSC Software
Product-development IT
IT is the lifeblood of your business. And aligning your
enterprise systems, PLM, enterprise resource planning (ERP),
customer relationship management (CRM), information
lifecycle management (ILM), application lifecycle management
(ALM) and production systems to your business strategies is
-
8/22/2019 Heat Tranfer Project Report
27/97
Tata Technologies Limited2
key to creating successful products, satisfied customers and
enhanced profits. Tata Technologies is unique in that it look at
this area holistically, allying your business strategies to the
most appropriate software and systems.
The company has been engaged in SAP implementation
services for product-centric organizations in a wide variety of
industries around the world. Its SAP service offerings span all
phases of the lifecycle of your enterprise from planning to
implementation, customization, development, testing and post-
manufacturing support. And thats how it keeps the lifeblood of
your company flowing. Better CRM initiatives are not limited
to software installation, they involve considering the context,
support and understanding of professionals so that they can
learn and take full advantage of your information systems.
-
8/22/2019 Heat Tranfer Project Report
28/97
Tata Technologies Limited2
Software partners
Dassault Systmes
Tata Technologies is recognized as a Premier Partner withDassault Systmes, a global software alliance for the
CAD/CAM/CAE/PLM market.
1)CATIA products are based on the open and scalableDassault Systmes V5 and V6 architecture. CATIA is the
leading product development solution for all
manufacturing companies, from big OEMs to small
producers.
2)SMARTEAM brings affordable product datamanagement capabilities to small-to-medium companies.
3)ENOVIA is a leading enterprise solution for robust,collaborative product data management.
4)DELMIAs digital manufacturing solutions enable thecontinuous creation and validation of manufacturing
processes throughout the product lifecycle.
MSC Software
MSC Software is the leading global provider of integrated
enterprise simulation solutions.
1)SimOffice easy-to-use simulation enables engineers toverify design in the Microsoft Windows desktop
environment.
-
8/22/2019 Heat Tranfer Project Report
29/97
Tata Technologies Limited2
2)Nastran is the powerful all-purpose finite elementanalysis solution used by the worlds most admired
manufacturers.
3)Adams, the market-leading motion-simulation software,simulates system level and loads.4)Patran, with its universal graphics user interface enables
finite element modeling, analysis and data integration,
analysis simulation and visualization capabilities.
5)Marc is ideal for the simulation of non-linear physicalbehavior of material conditions under extreme stress.
Siemens PLM
Siemens PLM provides leading software solutions that help
manufacturers turn more ideas into successful products.
1)Teamcenter powers innovation and improvesproductivity by connecting your team with the product
and process knowledge you need to make good decisions
throughout the product lifecycle. Teamcenters open PLM
foundation powers end-to-end lifecycle process excellence.
2)NX Synchronous technology from Siemens PLMSoftware can make your design process up to 100 times
faster. With this breakthrough, you no longer have to
choose between constraint-driven or history-free
modeling, and you can use data from multiple CAD
systems.
3)Solid Edge with synchronous technology is a completefeature-based 2D/3D CAD system that combines the
speed and flexibility of direct modelling with precise
control of dimension-driven design.
-
8/22/2019 Heat Tranfer Project Report
30/97
Tata Technologies Limited3
Autodesk
Autodesk is a world-recognized leader in design, visualization
and documentation software products. Tata Technologies is an
Autodesk Premier Solutions Provider with the added
distinctions of being a Manufacturing Specialist and an
Authorized Training Center. Tata Technologies provides
clients with AutoCAD, AutoCAD Electrical, AutoCAD
Mechanical and Autodesk Inventor software and service-
based solutions.
-
8/22/2019 Heat Tranfer Project Report
31/97
Tata Technologies Limited3
Fig. 1.7: Some clients of Tata Technologies
-
8/22/2019 Heat Tranfer Project Report
32/97
Tata Technologies Limited3
Project in the Aerospace Department
Todays intelligent aerospace businesses understand
that the key to successful engineering and PLMcollaboration is finding the right strategic partner a
partner with credible experience in the aerospace
domain, integrated global design delivery centers,
innovative business processes, proven resource
scalability, and one who understands your unique
business requirements.
For over two decades Tata Technologies has been
providing the worlds foremost aerospace organizations
with complete design-to-build solutions and customized
answers to the most complex PLM challenges. Seeing
the problems better and delivering better solutions is
what we do well no matter what the challenge. Asglobal delivery and outsourcing become key strategies
for the aerospace industry, its approach helps
companies to achieve cost savings, time to market, and
gain competitive advantage. Tata Technologies brings
blended onshore and offshore delivery resources
combined with state-of-the-art design center facilities
and the largest concentration of PLM aerospace
technology expertise in the world. Its engineering
pedigree is its foundation which is better in every way
-
8/22/2019 Heat Tranfer Project Report
33/97
Tata Technologies Limited3
for aircraft design. Andbetter for the customers at
35,000 feet.
Fig. 1.8: Aerospace Design Procedure
-
8/22/2019 Heat Tranfer Project Report
34/97
Tata Technologies Limited3
Fig. 1.9: Airplane structure components
-
8/22/2019 Heat Tranfer Project Report
35/97
Tata Technologies Limited3
Project on 1 dimensional heat transfer
Project Statement
Create a simulation tool/calculation tool that will allowus to simulate a floor fire test and the behavior of the
material in the test. See sketch below for test setup.
- The tool shall be able to calculate the temperature at
the top of the flooring material inside the vehicle floor
- The tool shall have the ability to simulate thetemperature rise at the top of the flooring material
when the under said floor is subjected to a temperature
curve per ASTM E-119. The tool has to be able to
calculate the "top of floor" temperature at least every
minute for at least 30 minutes.
-
8/22/2019 Heat Tranfer Project Report
36/97
Tata Technologies Limited3
Fig. 2.1: Time-Temperature Curve
-
8/22/2019 Heat Tranfer Project Report
37/97
Tata Technologies Limited3
Fig. 2.2: Values of source temperature and time
-
8/22/2019 Heat Tranfer Project Report
38/97
Tata Technologies Limited3
CHAPTER II
1D analytical approach
In Simple Words
Fig. 2.3: Transient heat transfer through a series of blocks
The given problem is to be sampled for the base of an aero
plane which is linked to the engine and has a high heat
exposure. Since, exact experimental tests are not very feasible
for this problem, a theoretical study and a simulation model is
necessary for this very purpose. This model is then to be
simulated in reality and passed through required tests for the
aircraft to be considered risk-free for manufacturing purpose.
-
8/22/2019 Heat Tranfer Project Report
39/97
Tata Technologies Limited3
If the heat at the top of the surface is found to exceed danger
limits, then the experimental simulation will not pass the
given test. In any case, a theoretical model for this 1
dimensional heat transfer is highly essential.
Explanation
In the given model, an unknown number of blocks (to be
decided by the user) are placed on top of one another (all of
different materials and properties). The sides of the blocks are
insulated so that heat transfer takes place only from bottom to
base in 1 direction. The bottom of the 1st block is heated on an
imaginary oven which has varying temperature according to
the plot shown in figure 2.1 (and also the values printed in
figure 2.2). The ambient temperature is 293 K.
Assumptions
Heat flow is considered to be transient in nature The interfaces of the blocks are assumed to be in perfect
thermal contact so that temperature at the interface is
equal for both the blocks and thus no convection takes
place at the interface
Rate of heat conduction at the interfaces is assumed to bethe same for both the blocks
Radiation and convection effects within the blocks areneglected (convection on the top surface is however
considered)
-
8/22/2019 Heat Tranfer Project Report
40/97
Tata Technologies Limited4
The properties of the blocks like thermal conductivity,density etc. are assumed to be constant with temperature.
The heat transfer coefficient at the convection surface is
also assumed to be constant. The temperature at the bottom surface is approximated to
be a sixth order polynomial (varying with time).
The values of beta in the solution are taken only up to 10values as the effect of the latter values on the solution is
assumed to be negligible.
No heat generation in any of the blocks
Formulation of the differential equations
Boundary Conditions
-
8/22/2019 Heat Tranfer Project Report
41/97
Tata Technologies Limited4
Initial Conditions
Solution
Background:
non homogeneity in equations
knowledge of Greens function transformation of non-homogenous boundary conditions to
homogenous boundary conditions by superposition
principle
separation of variables for solving partial differentialequations
In the given problem, we have a non - homogeneity at the
last boundary condition which we attempt to remove. We
do this by transforming the non homogenous boundary
conditions to homogenous boundary conditions although it
-
8/22/2019 Heat Tranfer Project Report
42/97
Tata Technologies Limited4
will leave us with a non-homogenous set of differential
equations which can be solved through the greens
function.
The function will be obtained through the polyfit()function of MATLAB as a sixth order polynomial.
The functions (x) and (x) are described below.
Boundary Conditions
and
-
8/22/2019 Heat Tranfer Project Report
43/97
Tata Technologies Limited4
Boundary Conditions
Solution of these set of ordinary differential equations is
pretty simple and can be easily obtained through matrixmanipulations in MATLAB.
For (x,t), the following equations hold:
-
8/22/2019 Heat Tranfer Project Report
44/97
Tata Technologies Limited4
Boundary Conditions
Initial Conditions
( )
We first solve the homogenous system assuming gi(x,t)=0 .
From the method of sepration of variables,
-
8/22/2019 Heat Tranfer Project Report
45/97
Tata Technologies Limited4
For a 2 block system
The eigen value problem for 2 slabs then yields
=0 where a=a1, b=a2These equations are for a 2 block system and can be extended
to multi block by extending the determinant. After finding the
values of for each iteration, values of will be known.Thus, []=[ ]will give the values of and .The solution for for a 2 block system would then be | | | |
-
8/22/2019 Heat Tranfer Project Report
46/97
Tata Technologies Limited4
where
|
and The final solution is for any of the two blocks.
The code for the above solution has however been developed for
any number of slabs in MATLAB. The code developed is
printed inAppendix A.
-
8/22/2019 Heat Tranfer Project Report
47/97
Tata Technologies Limited4
Input Data
Enter the number of slabs: 3
Enter material of plate 1: aluminium
Enter thickness of plate 1: 0.005
Enter material of plate 2: brass
Enter thickness of plate 2: 0.005
Enter material of plate 3: stainless steel
Enter thickness of plate 3: 0.005
Enter value of coefficient h for given problem: 10
Graph 2.1: Variation of source and interface 1 temperature with time
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Temp
erature(K)
Time(minutes)
Source temperature
Interface 1
-
8/22/2019 Heat Tranfer Project Report
48/97
Tata Technologies Limited4
Graph 2.2: Variation of source and interface 2 temperature with time
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Temperature(K)
Time(minutes)
Source temperature
Interface 2
-
8/22/2019 Heat Tranfer Project Report
49/97
Tata Technologies Limited4
Graph 2.3: Variation of source and all interface temperatures with time
The graphs obtained from the program have been plotted into
EXCEL and shown above. The graphs show the variation oftemperature versus time at the different interfaces. As we can
see, the trend followed is similar to the source curve but the
graphs are not easily distinguishable. The code takes about a
minute to show up the solution.
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Tem
perature(K)
Time(minutes)
Source temperature
Interface 1
Interface 2
Interface 3
-
8/22/2019 Heat Tranfer Project Report
50/97
Tata Technologies Limited5
Improvements
converting the program into GUI The temperatures obtained are not fully correct as they
exceed the source temperature at some locations
The time taken to solve this problem can certainly bereduced
increasing options for user to input values likesurrounding temperature etc.
creating separate functions for some part of the code forbetter reusability
validating the answer obtained through experimentaltechniques
making the problem more and more generalized byreducing number of assumptions
Conclusion
Although this model gives a fair idea of the trend followed by
the interface temperatures, it is not a very accurate or fast
model, and thus I have shifted to solving this problem by the
finite difference scheme covered in Chapter 3 which is really
fast as well as accurate.
-
8/22/2019 Heat Tranfer Project Report
51/97
Tata Technologies Limited5
CHAPTER III
1D numerical approach
In Simple Words
Fig. 3.1: 1 dimensional heat transfer through a series of blocks
Explanation
In the given model, an unknown number of blocks (to bedecided by the user) are placed on top of one another (all of
different materials and properties). The sides of the blocks are
insulated so that heat transfer takes place only from bottom to
base in 1 direction. The bottom of the 1st block is heated on an
-
8/22/2019 Heat Tranfer Project Report
52/97
Tata Technologies Limited5
imaginary oven which has varying temperature according to
the plot shown in the figure 2.1 (and also the values printed in
figure 2.2).The ambient temperature is 293 K.
Assumptions
Heat flow is considered to be transient in nature The interfaces of the blocks are assumed to be in perfect
thermal contact so that temperature at the interface is
equal for both the blocks and thus no convection takes
place at the interface
Rate of heat conduction at the interfaces is assumed to bethe same for both the blocks
Radiation and convection effects within the blocks areneglected (convection on the top surface is however
considered)
The properties of the blocks like thermal conductivity,density etc. are assumed to be constant with temperature.
The heat transfer coefficient at the convection surface isalso assumed to be constant.
No heat generation in any of the blocks The block is given 5 partitions and the time step is taken
to be 0.1 minute although these values can be changed
any time before the program run.
-
8/22/2019 Heat Tranfer Project Report
53/97
Tata Technologies Limited5
Formulation of the differential equations
Boundary Conditions
Initial Conditions
-
8/22/2019 Heat Tranfer Project Report
54/97
Tata Technologies Limited5
Solution
Background
Fully implicit finite difference scheme of solving partialdifferential equations
Solving a matrix using Gauss-Seidel iterative methodsThe given set of equations is solved using the fully
implicit finite difference. The fully implicit scheme is
chosen due to its ease of implementation and because it isunconditionally convergent and stable, thereby
warranting any time step and number of divisions to be
chosen. In figure 3.1, each of the blocks is partitioned into
a given number of segments. The finite difference
equations would be applied at each of the segments. For
the position derivative, the central difference
approximation is used and for time derivative, thebackward difference approximation. The following
substitutions were made:
-
8/22/2019 Heat Tranfer Project Report
55/97
Tata Technologies Limited5
If the number of slabs is taken to be 3 and the number of
nodes per slab to be 5 as shown in figure 3.1, then the
following system of equations will hold starting from n=0:
An imaginary node
is taken here to satisfy the
conditions. Similarly, other such nodes will be taken when
required to form the given matrix although the values
obtained for these variables will hold no meaning.
Continuing with the system of equations:
-
8/22/2019 Heat Tranfer Project Report
56/97
Tata Technologies Limited5
The given system of equations has 17 variables and 17
equations to be solved which is then encoded into a matrix andsolved by MATLAB using the command A\b.
But in very high dimensional codes, the non-iterative codes
used by MATLAB may not succeed and thus an iterative
technique (the Gauss-Seidel method) has been worked out
which gives just the same solutions but takes a lot more time
inside the for loops.
In the Gauss-Seidel procedure, the above system of equations
would be written in the following manner:
-
8/22/2019 Heat Tranfer Project Report
57/97
Tata Technologies Limited5
and so on till each node is explicitly written in terms of other
nodes. A set of values for all 17 nodes is then assumed and the
1st node is then calculated from the above equation. The
modified 1st node and the remaining nodes are then put into
the 2nd node equation. The procedure keeps repeating till the
consecutive values of the same node converge. For the Gauss-
Seidel method, this convergence mostly happens if the given
matrix is tri-diagonal or diagonally dominant. In this case,
even though the formed matrix is neither, it is very close to
being a tri-diagonal matrix and thus the solution does
converge.
After the 17 values for n=0 are obtained either by direct matrix
solutions or iterative procedures, the matrix is solved again forn=1 and so on till all required values for all times are obtained.
The code for the above solution has however been developed for
any number of slabs and partitions in MATLAB. The code
developed is printed inAppendix B.
-
8/22/2019 Heat Tranfer Project Report
58/97
Tata Technologies Limited5
Input Data
Enter the number of slabs: 3
Enter material of plate 1: aluminium
Enter thickness of plate 1: 0.005
Enter material of plate 2: brass
Enter thickness of plate 2: 0.005
Enter material of plate 3: stainless steel
Enter thickness of plate 3: 0.005
Enter value of coefficent h for given problem: 10
Without Gauss-Seidel iteration
Graph 3.1: Variation of source and interface temperature with time by
direct solving of matrix
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Temperature(K)
Time(minutes)
Source temperature
Interface 1
Interface 2
Interface 3
-
8/22/2019 Heat Tranfer Project Report
59/97
Tata Technologies Limited5
With Gauss-Seidel iteration
Graph 3.2: Variation of source and interface temperature with time
(Gauss-Seidel iteration)
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Temperature(K)
Time(minutes)
Source temperature
Interface 1 gs
Interface 2 gs
Interface 3 gs
-
8/22/2019 Heat Tranfer Project Report
60/97
Tata Technologies Limited6
Comparison (with and without Gauss-Seidel)
Graph 3.3: Variation of source and interface temperature with time
(both direct by direct matrix and Gauss-Seidel iteration)
As we can see, both the non-iterative and the iterative
programs yield exactly the same results with the respective
graphs in both the methods overlapping. The iterative
procedure however takes up slightly more time and memoryspace although it guarantees solution for very large matrices
too.
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Temperature(K)
Time(minutes)
Source temperature
Interface 1 gs
Interface 2 gs
Interface 3 gs
Interface 1
Interface 2
Interface 3
-
8/22/2019 Heat Tranfer Project Report
61/97
Tata Technologies Limited6
Convergence of Gauss-Seidel procedure
Graph 3.4: Convergence of temperature at particular time by Gauss-
Seidel iterations
The above graph is shown for the convergence of temperatures
of the top-most surface of all the slabs at different time
intervals for the Gauss-Seidel method. This confirms that eventhough the formed matrix in this case does not rigorously
satisfy convergence criteria, but the implemented code does
happen to converge.
0
200
400
600
800
1000
1200
0 10 20 30 40 50 60 70 80 90
Temperatu
re(Kelvin)
Iterations
5min
10min
15min
20min
25min
-
8/22/2019 Heat Tranfer Project Report
62/97
Tata Technologies Limited6
Improvements
converting the program into GUI increasing options for user to input values like
surrounding temperature etc.
creating separate functions for some part of the code forbetter reusability and optimizing code to work faster
validating the answer obtained through experimentaltechniques
making the problem more and more generalized byreducing number of assumptions
Conclusion
The fully implicit finite difference method, both by with and
without Gauss-Seidel iteration, gives a seemingly good and
accurate solution which can now be verified by experimental
techniques. Now that a seemingly right methodology has beenaccomplished, the scope of expanding the project to higher
dimensions (2D has been included next) or including radiation
conditions always remains.
-
8/22/2019 Heat Tranfer Project Report
63/97
Tata Technologies Limited6
CHAPTER IV
2D numerical approach
In Simple Words
Fig. 4.1: 2 dimensional heat transfer through a series of blocks
Explanation
In the given model, an unknown number of blocks (to be
decided by the user) are placed on top of one another (all of
different materials and properties). The sides of the blocks are
now open to atmosphere so that heat transfer takes place in 2
directions, from bottom to top and through the sides. The
bottom of the 1st block is heated on an imaginary oven which
-
8/22/2019 Heat Tranfer Project Report
64/97
Tata Technologies Limited6
has varying temperature according to the plot shown in figure
2.1 (and also the values printed in figure 2.2). The ambient
temperature is 293 K.
Assumptions
Heat flow is considered to be transient in nature The interfaces of the blocks are assumed to be in perfect
thermal contact so that temperature at the interface is
equal for both the blocks and thus no convection takes
place at the interface.
Rate of heat conduction at the interfaces is assumed to bethe same for both the blocks
Radiation and convection effects within the blocks areneglected (convection on the top surface and the sides is
however considered)
The properties of the blocks like thermal conductivity,density etc. are assumed to be constant with temperature.
The heat transfer coefficient at the convection surface isalso assumed to be constant.
No heat generation in any of the blocks There are two values of interface temperatures found at
each time step which differ by a slight proportion. Thus,
the first has been taken as the valid temperature in such
a scenario.
-
8/22/2019 Heat Tranfer Project Report
65/97
Tata Technologies Limited6
Formulation of the differential equations
Boundary Conditions
-
8/22/2019 Heat Tranfer Project Report
66/97
Tata Technologies Limited6
Initial Conditions
Solution
Background
The alternating direction implicit (ADI) finite differencescheme of solving partial differential equations which is
well suited for a 2D heat transfer problem
The above set of equations is solved using a slight variant
of the fully implicit FDM scheme called the alternating
direction implicit method. A usual implementation of a
fully implicit FDM in this case would result in a near
penta-diagonal matrix which would have to be coded all
over again. Instead, the ADI scheme breaks a time step
into two halves. In the first half, the x-derivative is
written as an implicit central-difference approximation
and the y-derivative as an explicit central-difference
approximation. The reverse holds true for the second half
of the time step. Each of these halves result in matrices
-
8/22/2019 Heat Tranfer Project Report
67/97
Tata Technologies Limited6
very similar to tri-diagonal matrix, the results of which
are then combined to obtain the final solution.
Since the resulting matrices are close to tri-diagonal, thecode from Chapter III could be re-used and modified to
give the solution. This scheme also has the advantage of
being unconditionally stable just as the fully implicit
methods. The Gauss-Seidel iterative procedure for the
solving of the matrices has not been implemented here as
it has already been taken up in the previous chapter and
the implementation would exactly be the same.
This time the blocks were divided into segments in both x
and y directions such that it resulted in creation of point
nodes. At each of these nodes, the equations and
boundary conditions stated above are valid. The following
substitutions were made:
For (n+)th time step
-
8/22/2019 Heat Tranfer Project Report
68/97
Tata Technologies Limited6
For (n+1)th time step
If the number of slabs is taken to be 3 and the number of
segments per slab to be 5 in both x and y direction as
shown in figure 4.1, then the following system ofequations will hold starting from n=0:
( )
(
)
( ) ( ) And so on for the remaining slabs.
-
8/22/2019 Heat Tranfer Project Report
69/97
Tata Technologies Limited6
From the equations obtained a 1717 matrix is formed and
solved for corresponding temperatures. Once the temperatures
for j=1 have been obtained, we move on to j=2.
( ) ( ) ( ) ( )
And so on for the remaining slabs.
Thus, we solve the matrix for 1 j 5, and finally, we would
have solved the complete node system at n= .After having all the values, we move to the next half of the
solution method for finding values at n=1. The following
equations hold:
( ) ( )
( )
( ) ( )
-
8/22/2019 Heat Tranfer Project Report
70/97
Tata Technologies Limited7
and so on.
Similarly, here we form the matrix at the ith value and keep
solving for each i until we have the temperatures for each node
in the mesh at n=1.
We then move on to solving for n=2 by the same procedure
described above. The difference from the previous chapters is
that the temperatures at each node here can be different even
varying along the y direction although only slightly.
The code for the above solution has been developed for any
number of slabs and partitions in MATLAB. The code
developed is printed inAppendix C.
-
8/22/2019 Heat Tranfer Project Report
71/97
Tata Technologies Limited7
Input Data
Enter the number of slabs: 3
Enter material of plate 1: aluminium
Enter thickness of plate 1: 0.005
Enter material of plate 2: brass
Enter thickness of plate 2: 0.005
Enter material of plate 3: stainless steel
Enter thickness of plate 3: 0.005
Enter equal width of all plates: 0.01
Enter value of coefficent h for given problem: 10
Graph 4.1: Variation of node 1 temperatures of all interfaces with time
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Tem
perature(K)
node
Source temperature
Interface 1 node 1
Interface 2 node 1
Interface 3 node 1
-
8/22/2019 Heat Tranfer Project Report
72/97
Tata Technologies Limited7
The graph above has been shown for the corner nodes of all the
interfaces and follows similar trends as discussed in the
previous chapters.
Graph 4.2: Variation of temperature of all nodes at top surface withtime
The graph above shows temperature curves for all nodes of the
top surface. As we see, the temperature variation along the y
0
200
400
600
800
1000
1200
0 5 10 15 20 25 30 35
Temperature(K)
Time(minutes)
Top Surface
Source temperature
node 1
node 2
node 3
node 4
node 5
-
8/22/2019 Heat Tranfer Project Report
73/97
Tata Technologies Limited7
direction is not much and the temperature curves for all these
nodes overlap.
Graph 4.3: Temperature of all nodes at 30 min at top surface
This graph shows the temperature variation along the top
surface at time t=30 min. As expected, the temperature is
symmetric with respect to the middle node with heat flowing
from both sides of the middle node to the surroundings at
ambient temperature.
1092.2
1092.4
1092.6
1092.8
1093
1093.2
1093.4
1093.6
1093.8
0 1 2 3 4 5 6
Temperature(K)
node
Top Surface
30 min
-
8/22/2019 Heat Tranfer Project Report
74/97
Tata Technologies Limited7
Improvements
converting the program into GUI increasing options for user to input values like
surrounding temperature etc.
creating separate functions for some part of the code forbetter reusability and optimizing code to work faster
validating the answer obtained through experimentaltechniques
making the problem more and more generalized byreducing number of assumptions
can be extended to the 3 dimensional case
Conclusion
The ADI finite difference technique works for the 2
dimensional case and gives good results which can now be
validated using experimental techniques.
-
8/22/2019 Heat Tranfer Project Report
75/97
Tata Technologies Limited7
Appendix A
% This program executes the given project statement using methods mentioned
in Chapter II. It gives the output as the graphs of all interfaces and source
temperature varying with time
% The symbolic toolbox is must for the running of this program% Please refer to the theory mentioned for smooth understanding of thisprogram% The following user inputs are required:% 1) material of slabs (only following materials are allowed:% (i)brass (ii)alunminium (iii) stainless steel (iv) wood% 2) thickness of slabs (the user is requested to keep thickness small < 0.02
meters)% 3)heat transfer convection coefficient% "All values must be entered in SI Units"clcclearA={'*' 'k' 'rho' 'c' ; 'water' 0.617 996 4178 ; 'brass' 110 8530 380 ;
'stainless steel' 14.9 7900 477 ; 'soil' 0.4 3333.3 800 ; 'wood' 1.26 700
1700 ; 'aluminium' 237 2700 897}; %declares matrix of available materialswith physical propertiesA{1,5}='alpha';
for i=2:7A {i,5}=A{i,2}/(A{i,3}*A{i,4}); % calculates value of alpha for all
materialsendn=input('Enter the number of slabs: ');
mat=cell(1,30);L=zeros(1,30);k=zeros(1,30);rho=zeros(1,30);c=zeros(1,30);
alpha=zeros(1,30);for i=1:nfprintf('Enter material of plate %d',i);mat{i}=input(': ','s');if(i==1)
fprintf('Enter thickness of plate %d',i);
L(i)=input(': ');else
fprintf('Enter thickness of plate %d',i);L(i)=L(i-1)+input(': ');
endendh=input('Enter value of coefficent h for given problem: ');for j=1:n
confirm=0;for i=2:size(A,1)%disp('1');if (strcmp(A{i,1},mat{j})==1) %loop compares values entered with
values stored in matrix and assigns properties accordingly%disp('1');confirm=1;k(j)=A{i,2};rho(j)=A{i,3};
-
8/22/2019 Heat Tranfer Project Report
76/97
Tata Technologies Limited7
c(j)=A{i,4};alpha(j)=A{i,5};
endif(confirm==1)
break;end
endendphi=zeros(2*n-1,2*n-1);phi(1,1)=L(1);phi(1,2)=L(1)*-1;phi(1,3)=-1;for i=2:n-1
phi(i,2*i-2)=L(i);phi(i,2*i-1)=1; % creates left hand matrix for solution of phi(x)phi(i,2*i)=-1*L(i);phi(i,2*i+1)=-1;
endphi(n,1)=k(1);phi(n,2)=k(2)*-1;for i=n+1:2*n-2
phi(i,2*i-2*n)=k(i-n+1);phi(i,2*i-2*n+2)=-1*k(i-n+2);
endphi(2*n-1,2*n-2)=k(n)+h*L(n);phi(2*n-1,2*n-1)=h;ans1=zeros(2*n-1,1); % creates right hand matrix for solution of phi(x)ans1(1,1)=-1;coeffphi=phi\ans1; %coeffiecients obtainedsyms x;fphi=zeros(n,1);
fphi=sym(fphi);fphi(1,1)=coeffphi(1,1).*x+1;for i=2:n
fphi(i,1)=coeffphi(2*i-2,1)*x+coeffphi(2*i-1,1); %creates function(x)
using coefficientsendzi=phi;ans2=zeros(2*n-1,1);ans2(2*n-1,1)=h;coeffzi=zi\ans2; % coefficients for zhi(x) obtainedfzi=zeros(n,1);fzi=sym(fzi);fzi(1,1)=coeffzi(1,1).*x;for i=2:n
fzi(i,1)=coeffzi(2*i-2,1)*x+coeffzi(2*i-1,1); %creates function(x)using coefficientsendF=zeros(n,1);F=sym(F);for i=1:n
F(i,1)=293*(1-fphi(i,1)-fzi(i,1)); %inital condition for transformed
solutionend
-
8/22/2019 Heat Tranfer Project Report
77/97
Tata Technologies Limited7
syms x1;betamat=zeros(2*n-1,2*n-1);betamat=sym(betamat);betamat(1,1)=-1*sin((x1*L(1))/sqrt(alpha(1)));betamat(1,2)=sin((x1*L(1))/sqrt(alpha(2)));
betamat(1,3)=cos((x1*L(1))/sqrt(alpha(2)));betamat(n,1)=-
((k(1)/k(2))*sqrt(alpha(2)/alpha(1))*cos((x1*L(1))/sqrt(alpha(1))));betamat(n,2)=cos((x1*L(1))/sqrt(alpha(2)));betamat(n,3)=-(sin((x1*L(1))/sqrt(alpha(2))));for i=2:n-1
betamat(i,2*i-2)=sin((x1*L(i))/sqrt(alpha(i)));betamat(i,2*i-1)=cos((x1*L(i))/sqrt(alpha(i)));betamat(i,2*i)=-(sin((x1*L(i))/sqrt(alpha(i+1))));betamat(i,2*i+1)=-(cos((x1*L(i))/sqrt(alpha(i+1))));
endfor i=n+1:2*n-2
betamat(i,2*i-2*n)=(k(i-n+1)/k(i-n+2))*sqrt(alpha(i-n+2)/alpha(i-
n+1))*cos((x1*L(i-n+1))/sqrt(alpha(i-n+1)));betamat(i,2*i-2*n+1)=-((k(i-n+1)/k(i-n+2))*sqrt(alpha(i-n+2)/alpha(i-
n+1))*sin((x1*L(i-n+1))/sqrt(alpha(i-n+1))));betamat(i,2*i-2*n+2)=-(cos((x1*L(i-n+1))/sqrt(alpha(i-n+2))));betamat(i,2*i-2*n+3)=sin((x1*L(i-n+1))/sqrt(alpha(i-n+2)));
endbetamat(2*n-1,2*n-
2)=(((h*sqrt(alpha(n)))/(x1*k(n)))*sin((x1*L(n))/sqrt(alpha(n))))+cos((x1*L(n
))/sqrt(alpha(n)));betamat(2*n-1,2*n-
1)=(((h*sqrt(alpha(n)))/(x1*k(n)))*cos((x1*L(n))/sqrt(alpha(n))))-
sin((x1*L(n))/sqrt(alpha(n)));
y=det(betamat); %calculates determinant of matrix formed for getting values
of betay=simplify(y);y=matlabFunction(y);syms t;%a=[0 5 10 15 20 25 30];%a=[0 5 10 15 20 25 30 35 40 45 50 55 60 ];a=[0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 ];%b=[293 811 977 1033 1068 1094 1116];b=[293 811 977 1033 1068 1094 1116 1135 1151 1165 1178 1189 1200 ];f1temp=polyfit(a,b,6);%f1temp=polyfit(a,b,15);f1=f1temp(1).*(t).^6+f1temp(2).*(t).^5+f1temp(3).*(t).^4+f1temp(4).*(t).^3+f1
temp(5).*(t).^2+f1temp(6).*(t)+f1temp(7); %sixth-order polynomial
approximation, any of the other functions commented can also be usedf1(t)=f1;%f1=f1temp(1).*t.^6+f1temp(2).*t.^5+f1temp(3).*t.^4+f1temp(4).*t.^3+f1temp(5)
.*t.^2+f1temp(6).*t+f1temp(7);%put simulating time after iteration%f1=(1.7267*t)+293;%t/60 is 1.7267%f1=990.1098*exp((0.0027948556*t)/60) - 728.6*exp((-0.23*t)/60);%f1=990.10977*exp(0.0040060346*(t/60)) - 697.22292*exp(-0.25160733*(t/60));%f1=713.09*cos(0.16482*(t/60)) - 34.844*cos(0.2747*(t/60)) -
13.599*sin(0.2747*(t/60)) - 268.81*sin(0.16482*(t/60)) -
1128.9*cos(0.05494*(t/60)) + 1100.0*cos(0.10988*(t/60)) +
2.1099*cos(0.21976*(t/60)) + 2311.0*sin(0.05494*(t/60)) +
1211.9*sin(0.10988*(t/60)) - 235.36*sin(0.21976*(t/60)) - 358.39;
-
8/22/2019 Heat Tranfer Project Report
78/97
Tata Technologies Limited7
%f1=(0.0001513*t^5 - 0.02684*t^4 + 1.787*t^3 - 53.74*t^2 + 1965*t + 2613)
/(t + 8.92);%f1=2163*sin(0.06295*(t/60)-0.6615) + 1875*sin(0.1206*(t/60)+0.5249) +
1207*sin(0.1647*(t/60)+2.177) + 361.1*sin(0.1984*(t/60)+4.188);%f1= 2.501*10^-5*(t/60)^5 - 0.004423*(t/60)^4 + 0.2938*(t/60)^3 -
9.088*(t/60)^2 + 134*(t/60) + 304;f1der=diff(f1);g=zeros(n,1);g=sym(g);for i=1:n
g(i,1)=(fphi(i,1)*f1der); %function g(x,t) is calculated for all slabsendsyms xdtou;g=subs(g,x,xd);g=subs(g,t,tou);%Theta(x,t)=sym('Theta(x,t)');Theta=0;green(x,xd,t,tou)=sym('green(x,xd,t,tou)');%integral1(x,t,tou)=sym('integral1(x,t,tou)');%integralgen1(x,t,tou)=sym('integralgen1(x,t,tou)');
for z=1:ngreen=0*x;p=1;i=1;beta2=0;
while(p~=11)beta=fzero(y,0.1*i); %value of beta around 0.1*i is obtained by solving
y=0if(beta
-
8/22/2019 Heat Tranfer Project Report
79/97
Tata Technologies Limited7
rhsmat=zeros(2*n-2,1);rhsmat(1,1)=sin((beta*L(1))/sqrt(alpha(1)));
rhsmat(n,1)=(k(1)/k(2))*sqrt(alpha(2)/alpha(1))*cos((beta*L(1))/sqrt(alpha(1)
));solmat=coeffmat\rhsmat; %coefficients A and B for different zhi are
obtainedzhimat=zeros(n,1);zhimat=sym(zhimat);zhimat(1,1)=sin((x*beta)/sqrt(alpha(1)));for j=2:n
zhimat(j,1)=solmat(2*j-3,1)*sin((x*beta)/sqrt(alpha(j)))+solmat(2*j-
2,1)*cos((x*beta)/sqrt(alpha(j))); %zhi(x) is formed, not to be confused with
previous zhi(x) , this is part of theta(x) solutionendNn=(k(1)/alpha(1))*int(zhimat(1,1).^2,0,L(1)); %Nn evaluatedfor j=2:n
Nn=Nn+(k(j)/alpha(j))*int(zhimat(j,1).^2,L(j-1),L(j));end
green=green+((1/Nn)*(k(z)/alpha(z))*zhimat(n,1)*exp(-(beta.^2*(t-
tou)))*subs(zhimat(z,1),x,xd)); %green's function evaluatedgreen=vpa(green,6);i=i+1;p=p+1;
end%green=simplify(green);if(z==1)
integral1(x,t,tou)=int((green*subs(F(1,1),x,xd)),xd,0,L(1));integral1=integral1(x,t,0);integral1=vpa(integral1,6);integral2=int(int(green*g(1,1),xd,0,L(1)),tou,0,t);integral2=vpa(integral2,6);Theta=Theta+integral1+integral2;
Theta=vpa(Theta,6);Theta=simplify(Theta);Theta=vpa(Theta,6);
elseintegralgen1(x,t,tou)=int((green*subs(F(z,1),x,xd)),xd,L(z-1),L(z));integralgen1=integralgen1(x,t,0);integralgen1=vpa(integralgen1,6);integralgen2=int(int(green*g(z,1),xd,L(z-1),L(z)),tou,0,t);integralgen2=vpa(integralgen2,6);Theta=Theta+integralgen1+integralgen2; %final calculation of Theta(x,t)Theta=vpa(Theta,6);Theta=simplify(Theta);Theta=vpa(Theta,6);
% Theta=simplify(Theta);%input for calcualting temp. bet. a and b
endendTfinal(x,t)=Theta+fphi(n,1)*f1+fzi(n,1)*293; %The actual temperature
varying with time is finally obtained
Tfinal=vpa(Tfinal,6);%Tfinal=simplify(Tfinal);disp('Equation of temperature on surface is ');%pretty(Tfinal(L(n),t));%why is it giving 2 answers but it is 1*1 matrixezplot(f1(t*60),[0,30]);hold all;
-
8/22/2019 Heat Tranfer Project Report
80/97
Tata Technologies Limited8
Tfinalatsurf=zeros(n,1);Tfinalatsurf=sym(Tfinalatsurf);for i=1:nTfinalatsurf(i)=Tfinal(L(i),t); %recording temperature at interfacesendTfinalatsurf=vpa(Tfinalatsurf,6);Tfinalatsurf(t)=Tfinalatsurf;Tfinalatsurf=Tfinalatsurf(t*60);Tfinalatsurf=simplify(Tfinalatsurf);Tfinalatsurf=vpa(Tfinalatsurf,6);%Tfinalatsurf(t)=Tfinalatsurf;%disp(Tfinalatsurf);for i=1:nezplot(Tfinalatsurf(i),[0,30]);end% This program has certain pitfalls as it does not give exact variation of
temperature of interfaces with time but merely a correct trend.% Some other improvements include:% 1) optimization of code for faster running
% 2) Ambient temperature can be input by user
% 3) User can be asked if he/she wants the trend of temperature with time fora line within the block instead of interface in which case only values
varying between the given lengths will be accepted
-
8/22/2019 Heat Tranfer Project Report
81/97
Tata Technologies Limited8
Appendix B
Matrix solved directly
% This program executes the given project statement using methods mentionedin Chapter III. It gives the output as the graphs of all interfaces and
source temperature varying with time% Please refer to the theory mentioned for smooth understanding of this
program% The following user inputs are required:% 1) material of slabs (only following materials are allowed:% (i)brass (ii)alunminium (iii) stainless steel (iv) wood% 2) thickness of slabs (the user is requested to keep thickness small < 0.02
metres)% 3) heat transfer convection coefficient% "All values must be entered in SI Units"clc
clearA={'*' 'k' 'rho' 'c' ; 'water' 0.617 996 4178 ; 'brass' 110 8530 380 ;'stainless steel' 14.9 7900 477 ; 'soil' 0.4 3333.3 800 ; 'wood' 1.26 700
1700 ; 'aluminium' 237 2700 897};A{1,5}='alpha';for i=2:7
A {i,5}=A{i,2}/(A{i,3}*A{i,4}); % calculates value of alpha for all
materialsendn=input('Enter the number of slabs: ');mat=cell(1,n);L=zeros(1,n);k=zeros(1,n);rho=zeros(1,n);
c=zeros(1,n);alpha=zeros(1,n);for i=1:n
fprintf('Enter material of plate %d',i);mat{i}=input(': ','s');%if(i==1)
fprintf('Enter thickness of plate %d',i);L(i)=input(': ');
% else% fprintf('Enter thickness of plate %d',i);% L(i)=input(': ');
% endend
h=input('Enter value of coefficent h for given problem: ');for j=1:nfor i=2:size(A,1)%disp('1');if (strcmp(A{i,1},mat{j})==1) %loop compares values entered with values
stored in matrix and assigns properties accordingly%disp('1');k(j)=A{i,2};rho(j)=A{i,3};c(j)=A{i,4};
-
8/22/2019 Heat Tranfer Project Report
82/97
Tata Technologies Limited8
alpha(j)=A{i,5};
endend
enddeltax=zeros(1,n);deltat=0.1;r=zeros(1,n);N=5;a=[0 5 10 15 20 25 30];b=[293 811 977 1033 1068 1094 1116];for i=1:n
deltax(i)=L(i)/(N-1);
r(i)=(alpha(i)*deltat)/(deltax(i)^2);endfdmmat=zeros(n*N+n-1,n*N+n-1);fdmmat(1,1)=1+2*r(1);fdmmat(1,2)=-r(1);for i=2:(N-1)
fdmmat(i,(i-1))=-r(1);fdmmat(i,i)=1+2*r(1);
fdmmat(i,i+1)=-r(1);end%fdmmat(N,N-1)=1+2*r(2);%fdmmat(N,N+1)=-r(2);%fdmmat(N,N+2)=-r(2);j=1;%for i=N+1:2*N-1% fdmmat(i,i-2)=-r(2);% fdmmat(i,i+1)=1+2*r(2);% fdmmat(i,i+2)=-r(2);
%endfor i=N:n*N-1
if(mod(i,N)==0)j=j+1;fdmmat(i,i+j-3)=1+2*r(j);fdmmat(i,i+j-1)=-r(j);fdmmat( i,i+j)=-r(j);
elseif(mod(i,N)==1)fdmmat(i,i+j-4)=-r(j); %creating left hand side matrixfdmmat(i,i+j-1)=1+2*r(j);fdmmat(i,i+j)=-r(j);else
fdmmat(i,i+j-2)=-r(j);fdmmat(i,i+j-1)=1+2*r(j);fdmmat(i,i+j)=-r(j);
endend
endj=N-4;p=1;for i=n*N:n*N+n-2
fdmmat(i,j+2)=-k(p)/deltax(p);fdmmat(i,j+4)=k(p)/deltax(p);fdmmat(i,j+5)=k(p+1)/deltax(p+1);fdmmat(i,j+6)=-k(p+1)/deltax(p+1);j=j+N+1;p=p+1;
-
8/22/2019 Heat Tranfer Project Report
83/97
Tata Technologies Limited8
endfdmmat(n*N+n-1,n*N+n-3)=-k(n)/(2*deltax(n)*h);fdmmat(n*N+n-1,n*N+n-2)=1;fdmmat(n*N+n-1,n*N+n-1)=k(n)/(2*deltax(n)*h);rhsmat=zeros(n*N+n-1,1);rhsmat(1,1)=293+r(1)*((deltat/5)*(b(2)-b(1))+b(1));for i=2:n*N-1
rhsmat(i,1)=293; %creating right hand side matrixendrhsmat(n*N+n-1,1)=293;solmat=fdmmat\rhsmat; %first solution matrixsurfrec=zeros(301,n);for i=1:n
surfrec(1,i)=293;end%surfrec(2,2)=solmat(4.1);i=N-1;j=1;while(i=0 && q*deltat=5 && q*deltat=10 && q*deltat=15 && q*deltat=20 && q*deltat=25 && q*deltat
-
8/22/2019 Heat Tranfer Project Report
84/97
Tata Technologies Limited8
solmat=fdmmat\rhsmat; %these are the temperatures
obtained at given iteration(time)%surfrec(q+1)=solmat(n*N+n-2,1);%surfrec(q+1,1)=solmat(1,1);c=N-1;d=1;while(c
-
8/22/2019 Heat Tranfer Project Report
85/97
Tata Technologies Limited8
alpha=zeros(1,n);for i=1:n
fprintf('Enter material of plate %d',i);mat{i}=input(': ','s');%if(i==1)
fprintf('Enter thickness of plate %d',i);L(i)=input(': ');
% else% fprintf('Enter thickness of plate %d',i);% L(i)=input(': ');
% endendh=input('Enter value of coefficent h for given problem: ');% mat{1,1}='aluminium';% mat{1,2}='aluminium';% mat{1,3}='aluminium';% L(1)=0.005;L(2)=0.005;L(3)=0.005;% h=10;for j=1:n
for i=2:size(A,1)
%disp('1');if (strcmp(A{i,1},mat{j})==1) %loop compares values entered with values
stored in matrix and assigns properties accordingly%disp('1');k(j)=A{i,2};rho(j)=A{i,3};c(j)=A{i,4};alpha(j)=A{i,5};
endend
enddeltax=zeros(1,n);deltat=0.1;r=zeros(1,n);N=5;a=[0 5 10 15 20 25 30];b=[293 811 977 1033 1068 1094 1116];for i=1:n
deltax(i)=L(i)/(N-1);r(i)=(alpha(i)*deltat)/(deltax(i)^2);
endfdmmat=zeros(n*N+n-1,n*N+n-1);fdmmat(1,1)=1+2*r(1);fdmmat(1,2)=-r(1);for i=2:(N-1)
fdmmat(i,(i-1))=-r(1);fdmmat(i,i)=1+2*r(1);
fdmmat(i,i+1)=-r(1);end%fdmmat(N,N-1)=1+2*r(2);%fdmmat(N,N+1)=-r(2);%fdmmat(N,N+2)=-r(2);j=1;%for i=N+1:2*N-1% fdmmat(i,i-2)=-r(2);% fdmmat(i,i+1)=1+2*r(2);% fdmmat(i,i+2)=-r(2);
-
8/22/2019 Heat Tranfer Project Report
86/97
Tata Technologies Limited8
%endfor i=N:n*N-1
if(mod(i,N)==0)j=j+1;fdmmat(i,i+j-3)=1+2*r(j);fdmmat(i,i+j-1)=-r(j); %creating left hand side matrixfdmmat( i,i+j)=-r(j);
elseif(mod(i,N)==1)fdmmat(i,i+j-4)=-r(j);fdmmat(i,i+j-1)=1+2*r(j);fdmmat(i,i+j)=-r(j);else
fdmmat(i,i+j-2)=-r(j);fdmmat(i,i+j-1)=1+2*r(j);fdmmat(i,i+j)=-r(j);
endend
endj=N-4;p=1;
for i=n*N:n*N+n-2fdmmat(i,j+2)=-k(p)/deltax(p);fdmmat(i,j+4)=k(p)/deltax(p);fdmmat(i,j+5)=k(p+1)/deltax(p+1);fdmmat(i,j+6)=-k(p+1)/deltax(p+1);j=j+N+1;p=p+1;
endfdmmat(n*N+n-1,n*N+n-3)=-k(n)/(2*deltax(n)*h);fdmmat(n*N+n-1,n*N+n-2)=1;fdmmat(n*N+n-1,n*N+n-1)=k(n)/(2*deltax(n)*h);rhsmat=zeros(n*N+n-1,1);rhsmat(1,1)=293+r(1)*((deltat/5)*(b(2)-b(1))+b(1));for i=2:n*N-1
rhsmat(i,1)=293; %creating right hand side matrixendrhsmat(n*N+n-1,1)=293;solmat=zeros(n*N+n-1,1);for i=1:n*N+n-1
solmat(i,1)=293;endlarge=1;
while(large>=0.0001)
temp=solmat;k=1;for i=1:n*N-1
sum=rhsmat(i,1);change=1;for j=1:n*N+n-1if mod(i,N)~=0
if fdmmat(i,j)~=1+2*r(k)sum=sum-fdmmat(i,j)*solmat(j,1); %getting first solution
endelseif mod(i,N)==0
-
8/22/2019 Heat Tranfer Project Report
87/97
Tata Technologies Limited8
if j==1k=k+1;endif fdmmat(i,j)==-r(k)
if change==1change=0;
elseif change==0sum=sum-fdmmat(i,j)*solmat(j,1);end
endelseif fdmmat(i,j)~=-r(k)
sum=sum-fdmmat(i,j)*solmat(j,1);end
endend
endendfor j=1:n*N+n-1
if mod(i,N)~=0if fdmmat(i,j)==1+2*r(k)
sum=sum/fdmmat(i,j);end
elseif fdmmat(i,j)==-r(k)
sum=sum/fdmmat(i,j);break;
endend
endsolmat(i+k-1,1)=sum;
endsum=0;j=N-4;for i=n*N:n*N+n-2
sum=-
(fdmmat(i,j+2)*solmat(j+2,1)+fdmmat(i,j+5)*solmat(j+5,1)+fdmmat(i,j+6)*solmat
(j+6,1));solmat((i-n*N)*(N+1)+N)=sum/fdmmat(i,j+4);j=j+N+1;
endsolmat(n*N+n-1,1)=(rhsmat(n*N+n-1,1)-fdmmat(n*N+n-1,n*N+n-2)*solmat(n*N+n-2)-
fdmmat(n*N+n-1,n*N+n-3)*solmat(n*N+n-3,1))/fdmmat(n*N+n-1,n*N+n-1);
temp=solmat-temp;
large=max(temp);endsurfrec=zeros(301,n);
for i=1:nsurfrec(1,i)=293;
end%surfrec(2,2)=solmat(4.1);i=N-1;j=1;while(i
-
8/22/2019 Heat Tranfer Project Report
88/97
Tata Technologies Limited8
end
for q=2:300
iter=0;j=2;
if q*deltat>=0 && q*deltat=5 && q*deltat=10 && q*deltat=15 && q*deltat=20 && q*deltat=25 && q*deltat=0.0001)iter=iter+1;
temp=solmat;k=1;for i=1:n*N-1
sum=rhsmat(i,1);change=1;for j=1:n*N+n-1if mod(i,N)~=0
if fdmmat(i,j)~=1+2*r(k)sum=sum-fdmmat(i,j)*solmat(j,1);
endelseif mod(i,N)==0
if j==1k=k+1;
-
8/22/2019 Heat Tranfer Project Report
89/97
Tata Technologies Limited8
endif fdmmat(i,j)==-r(k)
if change==1change=0;
elseif change==0sum=sum-fdmmat(i,j)*solmat(j,1);end
endelseif fdmmat(i,j)~=-r(k)
sum=sum-fdmmat(i,j)*solmat(j,1);end
endend
endendfor j=1:n*N+n-1
if mod(i,N)~=0if fdmmat(i,j)==1+2*r(k)
sum=sum/fdmmat(i,j);end
elseif fdmmat(i,j)==-r(k)
sum=sum/fdmmat(i,j);break;
endend
endsolmat(i+k-1,1)=sum;
endsum=0;j=N-4;for i=n*N:n*N+n-2
sum=-
(fdmmat(i,j+2)*solmat(j+2,1)+fdmmat(i,j+5)*solmat(j+5,1)+fdmmat(i,j+6)*solmat
(j+6,1));solmat((i-n*N)*(N+1)+N)=sum/fdmmat(i,j+4);j=j+N+1;
endsolmat(n*N+n-1,1)=(rhsmat(n*N+n-1,1)-fdmmat(n*N+n-1,n*N+n-2)*solmat(n*N+n-2)-
fdmmat(n*N+n-1,n*N+n-3)*solmat(n*N+n-3,1))/fdmmat(n*N+n-1,n*N+n-1);if q==50
iter2=1;result(iter,iter2)=solmat(n*N+n-2,1);
elseif q==100iter2=2;result(iter,iter2)=solmat(n*N+n-2,1);
elseif q==150
iter2=3;result(iter,iter2)=solmat(n*N+n-2,1);
elseif q==200iter2=4;result(iter,iter2)=solmat(n*N+n-2,1);
elseif q==250iter2=5;result(iter,iter2)=solmat(n*N+n-2,1);
elseif q==250iter2=6;
-
8/22/2019 Heat Tranfer Project Report
90/97
Tata Technologies Limited9
result(iter,iter2)=solmat(n*N+n-2,1);
endtemp=solmat-temp;
large=max(temp);end%surfrec(q+1)=solmat(n*N+n-2,1);%surfrec(q+1,1)=solmat(1,1);c=N-1;d=1;while(c
-
8/22/2019 Heat Tranfer Project Report
91/97
Tata Technologies Limited9
Appendix C
% This program executes the given project statement using methods mentioned
in Chapter IV. It gives the output as the graphs of node 1 of all interfaces
and source temperature varying with time
% Please refer to the theory mentioned for smooth understanding of thisprogram% The following user inputs are required:% 1) material of slabs (only following materials are allowed:% (i)brass (ii)alunminium (iii) stainless steel
(iv)wood)% 2) thickness of slabs (the user is requested to keep thickness small < 0.02
metres)% 3) width of slabs (all slabs must be equal width)% 3) heat transfer convection coefficient% "All values must be entered in SI Units"
%"Do not take very large number of partitionsclcclear
A={'*' 'k' 'rho' 'c' ; 'water' 0.617 996 4178 ; 'brass' 110 8530 380 ;'stainless steel' 14.9 7900 477 ; 'soil' 0.4 3333.3 800 ; 'wood' 1.26 700
1700 ; 'aluminium' 237 2700 897};A{1,5}='alpha';for i=2:7
A {i,5}=A{i,2}/(A{i,3}*A{i,4});endn=input('Enter the number of slabs: ');%can ask user for outside temperature%n=3;mat=cell(1,n);L=zeros(1,n);k=zeros(1,n);rho=zeros(1,n);
c=zeros(1,n);alpha=zeros(1,n);for i=1:n
fprintf('Enter material of plate %d',i);mat{i}=input(': ','s');%if(i==1)
fprintf('Enter thickness of plate %d',i);%you have to introduce vpa
at many places. Check to see results for 3 or more slab right or not then use
vpa% L(i)=input(': ');
% else% fprintf('Enter thickness of plate %d',i);L(i)=input(': ');
% endendwidth=input('Enter equal width of all plates: ');h=input('Enter value of coefficent h for given problem: ');% mat{1,1}='aluminium';% mat{1,2}='brass';% mat{1,3}='stainless steel';% L(1)=0.005;L(2)=0.006;L(3)=0.007;L(4)=0.008;% h=10;% width=0.01;
-
8/22/2019 Heat Tranfer Project Report
92/97
Tata Technologies Limited9
for j=1:nfor i=2:size(A,1)%disp('1');if (strcmp(A{i,1},mat{j})==1)
%disp('1');k(j)=A{i,2};rho(j)=A{i,3};c(j)=A{i,4};alpha(j)=A{i,5};
endend
endN=5;deltax=zeros(1,n);deltay=width/(N-1);deltat=0.1;rx=zeros(1,n);ry=zeros(1,n);a=[0 5 10 15 20 25 30];b=[293 811 977 1033 1068 1094 1116];
for i=1:ndeltax(i)=L(i)/(N-1);rx(i)=(alpha(i)*deltat)/(2*deltax(i)^2);ry(i)=(alpha(i)*deltat)/(2*deltay^2);
endsolmat=zeros(n*N+n-1,N);fdmmat=zeros(n*N+n-1,n*N+n-1);rhsmat=zeros(n*N+n-1,1);for w=1:Nfdmmat(1,1)=1+2*rx(1);fdmmat(1,2)=-rx(1);for i=2:(N-1)
fdmmat(i,(i-1))=-rx(1);fdmmat(i,i)=1+2*rx(1);fdmmat(i,i+1)=-rx(1);
endj=1;for i=N:n*N-1
if(mod(i,N)==0)j=j+1;fdmmat(i,i+j-3)=1+2*rx(j);fdmmat(i,i+j-1)=-rx(j);fdmmat( i,i+j)=-rx(j);
elseif(mod(i,N)==1)fdmmat(i,i+j-4)=-rx(j);fdmmat(i,i+j-1)=1+2*rx(j);fdmmat(i,i+j)=-rx(j);
elsefdmmat(i,i+j-2)=-rx(j);fdmmat(i,i+j-1)=1+2*rx(j);fdmmat(i,i+j)=-rx(j);
endend
endj=N-4;p=1;for i=n*N:n*N+n-2
-
8/22/2019 Heat Tranfer Project Report
93/97
Tata Technologies Limited9
fdmmat(i,j+2)=-k(p)/deltax(p);fdmmat(i,j+4)=k(p)/deltax(p);fdmmat(i,j+5)=k(p+1)/deltax(p+1);fdmmat(i,j+6)=-k(p+1)/deltax(p+1);j=j+N+1;p=p+1;
endfdmmat(n*N+n-1,n*N+n-3)=-k(n)/(2*deltax(n)*h);fdmmat(n*N+n-1,n*N+n-2)=1;fdmmat(n*N+n-1,n*N+n-1)=k(n)/(2*deltax(n)*h);rhsmat(1,1)=293+rx(1)*((deltat/10)*(b(2)-b(1))+b(1));for i=2:n*N-1
rhsmat(i,1)=293;endrhsmat(n*N+n-1,1)=293;solmat(:,w)=fdmmat\rhsmat;endfdmmat2=zeros(N+2,N+2);rhsmat2=zeros(N+2,1);solmat2=zeros(n*N-1,N+2);
p=1;u=0;for w=1:n*N-1
if mod(w,N)==0p=p+1;
endfor i=1:N
fdmmat2(i,i)=-ry(p);fdmmat2(i,i+1)=1+2*ry(p);fdmmat2(i,i+2)=-ry(p);
endfdmmat2(N+1,1)=k(p)/(2*deltay*h);fdmmat2(N+1,2)=1;fdmmat2(N+1,3)=-k(p)/(2*deltay*h);fdmmat2(N+2,N)=-k(p)/(2*deltay*h);fdmmat2(N+2,N+1)=1;fdmmat2(N+2,N+2)=k(p)/(2*deltay*h);if w==1for i=1:N
rhsmat2(i,1)=(1-
2*rx(p))*solmat(w,i)+rx(p)*solmat(w+1,i)+rx(p)*((deltat/10)*(b(2)-
b(1))+b(1));end
elseif mod(w,N)==0rhsmat(i,1)=rhsmat(i-1,1);u=u+1;
else
for i=1:Nrhsmat2(i,1)=(1-
2*rx(p))*solmat(w+u,i)+rx(p)*solmat(w+u+1,i)+rx(p)*solmat(w+u-1,i);endend
endrhsmat2(N+1,1)=293;
-
8/22/2019 Heat Tranfer Project Report
94/97
Tata Technologies Limited9
rhsmat2(N+2,1)=293;solmat2(w,:)=fdmmat2\rhsmat2;
endsurfrec=zeros(301,n);surfrec2=zeros(301,N);surfrec(1,:)=293;surfrec2(1,:)=293;i=N-1;j=1;while(i=0 && q*deltat=5 && q*deltat=10 && q*deltat=15 && q*deltat=20 && q*deltat=25 && q*deltat
-
8/22/2019 Heat Tranfer Project Report
95/97
Tata Technologies Limited9
if mod(w,N)==0p=p+1;
endfor i=1:N
fdmmat2(i,i)=-ry(p);fdmmat2(i,i+1)=1+2*ry(p);fdmmat2(i,i+2)=-ry(p);
endfdmmat2(N+1,1)=k(p)/(2*deltay*h);fdmmat2(N+1,2)=1;fdmmat2(N+1,3)=-k(p)/(2*deltay*h);fdmmat2(N+2,N)=-k(p)/(2*deltay*h);fdmmat2(N+2,N+1)=1;fdmmat2(N+2,N+2)=k(p)/(2*deltay*h);if w==1for i=1:N
rhsmat2(i,1)=(1-
2*rx(p))*solmat(w,i)+rx(p)*solmat(w+1,i)+rx(p)*Tatiter;end
elseif mod(w,N)==0rhsmat(i,1)=rhsmat(i-1,1);u=u+1;
else
for i=1:Nrhsmat2(i,1)=(1-
2*rx(p))*solmat(w+u,i)+rx(p)*solmat(w+u+1,i)+rx(p)*solmat(w+u-1,i);endend
endrhsmat2(N+1,1)=293;rhsmat2(N+2,1)=293;
solmat2(w,:)=fdmmat2\rhsmat2;endc=N-1;d=1;while(c
-
8/22/2019 Heat Tranfer Project Report
96/97
Tata Technologies Limited9
References
Heat conduction by M Necati Ozisik http://highered.mcgraw-hill.com/sites/dl/free/0073129305/314124/cen29305_ch04.pdf
en.wikepedia.orgwww.mathworks.inwww.wolframalpha.orgComputational Fluid Dynamics by John D. Anderson
http://highered.mcgraw-hill.com/sites/dl/free/0073129305/314124/cen29305_ch04.pdfhttp://highered.mcgraw-hill.com/sites/dl/free/0073129305/314124/cen29305_ch04.pdfhttp://www.mathworks.in/http://www.mathworks.in/http://www.wolframalpha.org/http://www.wolframalpha.org/http://www.wolframalpha.org/http://www.mathworks.in/http://highered.mcgraw-hill.com/sites/dl/free/0073129305/314124/cen29305_ch04.pdf -
8/22/2019 Heat Tranfer Project Report
97/97
Thank You!!!