Numerical methods in Transient-heat-conductiontmuliya
This file contains slides on Numerical methods in Transient heat conduction.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Finite difference eqns. by energy balance – Explicit and Implicit methods – 1-D transient conduction in a plane wall – stability criterion – Problems - 2-D transient heat conduction – Finite diff. eqns. for interior nodes – Explicit and Implicit methods - stability criterion – difference eqns for different boundary conditions – Accuracy considerations – discretization error and round–off error - Problems
This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
This file contains slides on NUMERICAL METHODS IN STEADY STATE 1D and 2D HEAT CONDUCTION – Part-I.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students.
Contents: Why Numerical methods? – Advantages – Finite difference formulation from differential eqns – 1D steady state conduction in cartesian coordinates – formulation by energy balance method – different BC’s – Problems
Numerical methods in Transient-heat-conductiontmuliya
This file contains slides on Numerical methods in Transient heat conduction.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010.
Contents: Finite difference eqns. by energy balance – Explicit and Implicit methods – 1-D transient conduction in a plane wall – stability criterion – Problems - 2-D transient heat conduction – Finite diff. eqns. for interior nodes – Explicit and Implicit methods - stability criterion – difference eqns for different boundary conditions – Accuracy considerations – discretization error and round–off error - Problems
This chapter contains:-.
Analytical Methods of two dimensional steady state heat conduction
Finite difference Method application on two dimensional steady state heat conduction.
Finite difference method on irregular shape of a system
This file contains slides on NUMERICAL METHODS IN STEADY STATE 1D and 2D HEAT CONDUCTION – Part-I.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students.
Contents: Why Numerical methods? – Advantages – Finite difference formulation from differential eqns – 1D steady state conduction in cartesian coordinates – formulation by energy balance method – different BC’s – Problems
Heat transfer from extended surfaces (or fins)tmuliya
This file contains slides on Heat Transfer from Extended Surfaces (FINS). The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Governing differential eqn – different boundary conditions – temp. distribution and heat transfer rate for: infinitely long fin, fin with insulated end, fin losing heat from its end, and fin with specified temperatures at its ends – performance of fins - ‘fin efficiency’ and ‘fin effectiveness’ – fins of non-uniform cross-section- thermal resistance and total surface efficiency of fins – estimation of error in temperature measurement - Problems
Introduction to transient Heat conduction, Lamped System Analysis, Approxiamate Analytical and graphical method and Numerical method for one and two dimensional heat conduction by using Explicit and Implicit method
Definition and Requirements
Types of Heat Exchangers
The Overall Heat Transfer Coefficient
The Convection Heat Transfer Coefficients—Forced Convection
Heat Exchanger Analysis
Heat Exchanger Design and Performance Analysis
introduction of condensation, what is it types etc. horizontal condenser, vertical condenser, process aplications, all examples related to the process,
It is basic information about what is critical thickness and why we should we know this. Then there is critical thickness formula for cylindrical pipe and spherical shell.
Heat transfer from extended surfaces (or fins)tmuliya
This file contains slides on Heat Transfer from Extended Surfaces (FINS). The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
Contents: Governing differential eqn – different boundary conditions – temp. distribution and heat transfer rate for: infinitely long fin, fin with insulated end, fin losing heat from its end, and fin with specified temperatures at its ends – performance of fins - ‘fin efficiency’ and ‘fin effectiveness’ – fins of non-uniform cross-section- thermal resistance and total surface efficiency of fins – estimation of error in temperature measurement - Problems
Introduction to transient Heat conduction, Lamped System Analysis, Approxiamate Analytical and graphical method and Numerical method for one and two dimensional heat conduction by using Explicit and Implicit method
Definition and Requirements
Types of Heat Exchangers
The Overall Heat Transfer Coefficient
The Convection Heat Transfer Coefficients—Forced Convection
Heat Exchanger Analysis
Heat Exchanger Design and Performance Analysis
introduction of condensation, what is it types etc. horizontal condenser, vertical condenser, process aplications, all examples related to the process,
It is basic information about what is critical thickness and why we should we know this. Then there is critical thickness formula for cylindrical pipe and spherical shell.
I am Rachael W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Massachusetts Institute of Technology, USA
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
Numerical Solution of Diffusion Equation by Finite Difference Methodiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
We understand that you're a college student and finances can be tight. That's why we offer affordable pricing for our online statistics homework help. Your future is important to us, and we want to make sure you can achieve your degree without added financial stress. Seeking assistance with statistics homework should be simple and stress-free, and that's why we provide solutions starting from a reasonable price.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
if you are struggling with your Multiple Linear Regression homework, do not hesitate to seek help from our statistics homework help experts. We are here to guide you through the process and ensure that you understand the concept and the steps involved in performing the analysis. Contact us today and let us help you ace your Multiple Linear Regression homework!
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
Finite Volume Method Advanced Numerical Analysis by Md.Al-AminMd. Al-Amin
Finite Volume Method Advanced Numerical Analysis by Md.Al-Amin
Imam Hasan Al-Amin, professionally known as MD Al-Amin, He was born on December 25th, 1999, and brought up in Pirojpur. He is a Bangladeshi entrepreneur and mathematician. He graduated from Khulna University, Khulna, Bangladesh also Bangladesh University of Engineering and Technology(BUET) in mathematics. He is the co-founder and CEO of Juhod Shop-যুহদ শপ, which is mainly an online shop in Bangladesh. Here, you can buy products online with a few clicks or convenient phone calls. Also, he is the founder and CEO of Juhod IT-Care, a full-service digital media agency that partners with clients to boost their personal and business outcomes. His expertise in marketing has allowed him to help a number of businesses increase their revenue by tremendous amounts. From childhood, he wanted to do something different that would be fruitful for mankind. He started working as a vocal artist when he was only 18 years old.
With rising crude prices and depleting quality of crude, however, the level of wastewater pollutants in petroleum wastewater is at new high. Such conditions are forcing refineries to use a more advanced water treatment, water recovery methods, and robust processes that work well under a variety of conditions and can handle the changing refinery effluent flow rates. Finally a process that is economical in overall life time cost is needed to make all of this feasible. Aquatech has experience working with these refinery effluent pollutants in the refinery market and offers the advanced petroleum wastewater treatment and recovery technology necessary for the refinery’s needs.
An internal combustion engine (ICE) is an engine where the combustion of a fuel occurs with an oxidizer (usually air) in a combustion chamber that is an integral part of the working fluid flow circuit. In an internal combustion engine the expansion of the high-temperature and high-pressure gases produced by combustion apply direct force to some component of the engine. The force is applied typically to pistons, turbine blades, or a nozzle. This force moves the component over a distance, transforming chemical energy into useful mechanical energy.
Design Considerations for Plate Type Heat ExchangerArun Sarasan
A plate heat exchanger is a type of heat exchanger that uses metal plates to transfer heat between two fluids. This has a major advantage over a conventional heat exchanger in that the fluids are exposed to a much larger surface area because the fluids spread out over the plates. This facilitates the transfer of heat, and greatly increases the speed of the temperature change. Plate heat exchangers are now common and very small brazed versions are used in the hot-water sections of millions of combination boilers. The high heat transfer efficiency for such a small physical size has increased the domestic hot water (DHW) flowrate of combination boilers. The small plate heat exchanger has made a great impact in domestic heating and hot-water. Larger commercial versions use gaskets between the plates, whereas smaller versions tend to be brazed.
The environmental impact of paper is significant, which has led to changes in industry and behavior at both business and personal levels. With the use of modern technology such as the printing press and the highly mechanized harvesting of wood, disposable paper has become a cheap commodity. This has led to a high level of consumption and waste. With the rise in environmental awareness due to the lobbying by environmental organizations and with increased government regulation there is now a trend towards sustainability in the pulp and paper industry.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
2. • Due to the increasing complexities
encountered in the development of modern
technology, analytical solutions usually are not
available.
• For these problems, numerical solutions
obtained using high-speed computer are very
useful, especially when the geometry of the
object of interest is irregular, or the boundary
conditions are nonlinear.
3.
Numerical methods are necessary to solve many practical
problems in heat conduction that involve:
– complex 2D and 3D geometries
– complex boundary conditions
– variable properties
An appropriate numerical method can produce a useful
approximate solution to the temperature field T(x,y,z,t); the
method must be
– sufficiently accurate
– stable
– computationally efficient
4. General Features
A numerical method involves a discretization process, where
the solution domain is divided into subdomains and nodes
The PDE that describes heat conduction is replaced by a
system of algebraic equations, one for each subdomain in
terms of nodal temperatures
A solution to the system of algebraic equations almost always
requires the use of a computer
As the number of nodes (or subdomains) increase, the
numerical solution should approach the exact solution
Numerical methods introduce error and the possibility of
solution instability
5. Types of Numerical Methods
1. The Finite Difference Method (FDM)
– subdomains are rectangular and nodes form a
regular grid network
– nodal values of temperature constitute the
numerical solution; no interpolation functions are
included
– discretization equations can be derived from
Taylor series expansions or from a control volume
approach
6. 2. The Finite Element Method (FEM)
– subdomain may be any polygon shape, even with
curved sides; nodes can be placed anywhere in
subdomain
– numerical solution is written as a finite series sum
of interpolation functions, which may be linear,
quadratic, cubic, etc.
– solution provides nodal temperatures and
interpolation functions for each subdomain
7. • In heat transfer problems, the finite difference method
is used more often and will be discussed here.
• The finite difference method involves:
Establish nodal networks
Derive finite difference approximations for the
governing equation at both interior and exterior nodal
points
Develop a system of simultaneous algebraic nodal
equations
Solve the system of equations using numerical schemes
8. The Nodal Networks
The basic idea is to subdivide the area of interest into sub-volumes with the distance
between adjacent nodes by Dx and Dy as shown. If the distance between points is small
enough, the differential equation can be approximated locally by a set of finite difference
equations. Each node now represents a small region where the nodal temperature is a
measure of the average temperature of the region.
Example:
Dx
m,n+1
m-1,n
m,n
m+1, n
Dy
m,n-1
x=mDx, y=nDy
m+½,n
m-½,n
intermediate points
9. Finite Difference Approximation
q 1 T
Heat Diffusion Equation: T
,
k t
k
where =
is the thermal diffusivity
C PV
2
No generation and steady state: q=0 and
0, 2T 0
t
First, approximated the first order differentiation
at intermediate points (m+1/2,n) & (m-1/2,n)
T
DT
x ( m 1/ 2,n ) Dx
T
DT
x ( m 1/ 2,n ) Dx
( m 1/ 2,n )
Tm 1,n Tm ,n
Dx
( m 1/ 2,n )
Tm ,n Tm 1,n
Dx
10. Finite Difference Approximation (cont.)
Next, approximate the second order differentiation at m,n
2T
x 2
2T
x 2
m ,n
m ,n
T / x m 1/ 2,n T / x m 1/ 2,n
Dx
Tm 1,n Tm 1,n 2Tm ,n
( Dx ) 2
Similarly, the approximation can be applied to
the other dimension y
2T
y 2
m ,n
Tm ,n 1 Tm ,n 1 2Tm ,n
( Dy ) 2
11. Finite Difference Approximation (cont.)
Tm 1,n Tm 1,n 2Tm ,n Tm ,n 1 Tm ,n 1 2Tm ,n
2T 2T
x 2 y 2
2
( Dx )
( Dy ) 2
m ,n
To model the steady state, no generation heat equation: 2T 0
This approximation can be simplified by specify Dx=Dy
and the nodal equation can be obtained as
Tm 1,n Tm 1,n Tm ,n 1 Tm ,n 1 4Tm ,n 0
This equation approximates the nodal temperature distribution based on
the heat equation. This approximation is improved when the distance
between the adjacent nodal points is decreased:
DT T
DT T
Since lim( Dx 0)
,lim( Dy 0)
Dx x
Dy y
12. A System of Algebraic Equations
• The nodal equations derived previously are valid for all interior
points satisfying the steady state, no generation heat equation.
For each node, there is one such equation.
For example: for nodal point m=3, n=4, the equation is
T2,4 + T4,4 + T3,3 + T3,5 - 4T3,4 =0
T3,4=(1/4)(T2,4 + T4,4 + T3,3 + T3,5)
• Derive one equation for each nodal point (including both
interior and exterior points) in the system of interest. The result is
a system of N algebraic equations for a total of N nodal points.
13. Matrix Form
The system of equations:
a11T1 a12T2 a1N TN C1
a21T1 a22T2
a2 N TN C2
a N 1T1 a N 2T2
a NN TN CN
A total of N algebraic equations for the N nodal points and the system can be
expressed as a matrix formulation: [A][T]=[C]
a11 a12
a
a22
21
where A=
aN 1 aN 2
a1N
T1
C1
T
C
a2 N
, T 2 ,C 2
aNN
TN
C N
14. Numerical Solutions
Matrix form: [A][T]=[C].
From linear algebra: [A]-1[A][T]=[A]-1[C], [T]=[A]-1[C]
where [A]-1 is the inverse of matrix [A]. [T] is the solution vector.
• Matrix inversion requires cumbersome numerical computations and is not efficient if
the order of the matrix is high (>10).
• For high order matrix, iterative methods are usually more efficient. The famous
Jacobi & Gauss-Seidel iteration methods will be introduced in the following.
15. Iteration
General algebraic equation for nodal point:
i 1
a T
j 1
ij
j
aiiTi
N
aT
j i 1
ij
j
Ci ,
(Example : a31T1 a32T2 a33T3
a1N TN C1 , i 3)
Rewrite the equation of the form:
N
aij ( k 1)
Ci i 1 aij ( k )
(k )
Ti T j T j
aii j 1 aii
j i 1 aii
Replace (k) by (k-1)
for the Jacobi iteration
• (k) - specify the level of the iteration, (k-1) means the present level and (k) represents
the new level.
• An initial guess (k=0) is needed to start the iteration.
• By substituting iterated values at (k-1) into the equation, the new values at iteration
(k) can be estimated
• The iteration will be stopped when maxTi(k)-Ti(k-1), where specifies a
predetermined value of acceptable error
16. CASE STUDY
Finite Volume Method Analysis of Heat Transfer in
Multi-Block Grid During Solidification
Eliseu Monteiro1, Regina Almeida2 and Abel Rouboa3
1CITAB/UTAD - Engineering Department of
University of Tr´as-os-Montes e Alto Douro, Vila Real
2CIDMA/UA - Mathematical Department of
University of Tr´as-os-Montes e Alto Douro, Vila Real
3CITAB/UTAD - Department of Mechanical Engineering and
Applied Mechanics of University of Pennsylvania,
Philadelphia, PA
1,2Portugal
3USA
17. The governing differential equation for the solidification problem may
be written in the following conservative form
∂ (ρCPφ)
∂t
= ∇· (k∇φ) + q˙
(1)
where ∂(ρCPφ) ∂t represents the transient contribution to the
conservative energy equation (φ temperature); ∇· (k∇φ) is the
diffusive contribution to the energy equation and q˙ represents the
energy released during the phase change.
27. Concluding remarks
A multi-block grid generated by bilinear interpolation was successfully applied in
combination with a generalized curvilinear coordinates system to a complex
geometry in a casting solidification scenario. To model the phase change a
simplified two dimensional mathematical model was used based on the energy
differential equation. Two discretization methods: finite differences and finite
volume were applied in order to determine, by comparison with experimental
measurements, which works better in these conditions. For this reason a coarse
grid was used. A good agreement between both discretization methods was
obtained with a slight advantage for the finite volume method. This could be
explained due to the use of more information by the finite volume method to
compute each temperature value than the finite differences method. The multiblock grid in combination with a generalized curvilinear coordinates system has
considerably advantages such as:
28. • better capacity to describe the contours through a lesser
number of elements, which considerably reduces the
computational time;
• - any physical feature of the cast part or mold can be
straightforwardly defined and obtained in a specific zone of
the domain;
• - the difficulty of the several virtual interfaces created by the
geometry division are easily overcome by the continuity
condition