thermal modelling

Department of Mechanical Engineering, GEC, Thrissur 1

CHAPTER- I

INRODUCTION

Modeling and simulation enables designers to test whether design

specifications are met by using virtual rather than physical experiments. The use

of virtual prototypes significantly shortens the design cycle and reduces the cost of

design. It further provides the designer with immediate feedback on design

decisions which, in turn, promises a more comprehensive exploration of design

alternatives and a better performing final design. Simulation is particularly

important for the design of multi-disciplinary systems in which components in

different disciplines (mechanical, electrical, embedded control, etc.) are tightly

coupled to achieve optimal system performance. This project surveys the current

state of the art in modeling and simulation and examines to which extent current

simulation technologies support the design of engineering systems.

One of the most basic requirements for simulations in the context of design

is that the modeling language be sufficiently expressive to model the non-linear,

multi-disciplinary, hybrid continuous-discrete phenomena encountered in the

design prototypes. Over the years, many modeling and simulation languages have

been developed, but only a few of these languages are well suited for modeling of

multi-disciplinary systems. The earliest simulation languages, based on CSSL

(Continuous System Simulation Language), were procedural and provided a low-

level description of a system in terms of ordinary differential equations. From

these languages emerged two important developments: declarative (or equation-

based) modeling, and object-oriented modeling.

Current research further builds on these developments by moving towards

component-based modeling and by providing support for hybrid (mixed

continuous-discrete event) systems. Another requirement is that simulation


models are easy to create and reuse. Creating high-fidelity simulation models is a

complex activity that can be quite time-consuming. Object-oriented languages

provide clear advantages with respect to model development, maintenance, and

reuse. In addition, to take full advantage of simulation in the context of design, it

is necessary to develop a modeling paradigm that is integrated with the design

environment, and that provides a simple and intuitive interface that requires a

minimum of analysis expertise.

OBJECT ORIENTED PROGRAMMING (OOP)

Object oriented programming is a technique for programming – a

paradigm for writing ‘‘good’’ programs for a set of problems. If the term ‘‘object

oriented programming language’’ means anything it must mean a programming

language that provides mechanisms that support the object oriented style of

programming well. Over the years, the emphasis in the design of programs has

shifted away from the design of procedures towards the organization of data.

Among other things, this reflects an increase in the program size. A set of related

procedures with the data they manipulate is often called a module.

SINGLE-DOMAIN SIMULATION

Simulation modeling defined for a single domain such as mechanical or

electrical systems is a mature area, withseveral companies offering robust

simulation packages.

HYDRAULIC AND THERMAL SYSTEMS

Hydraulic and thermal systems are often modeled as interacting with each

other and with mechanical components. The behavior of both thermal and

hydraulic systems depends strongly on the geometry of the components and their

physical configurations. As for mechanical systems, a tight integration with the

3D design environment is essential. From the review above, it is clear that many

single-domain simulation environments are closely integrated with design tools.


This trend is currently expanding towards the simulation and design of multi-

disciplinary systems in general.

VISUAL BASIC

Visual Basic is a high level object oriented program developed by the

Microsoft Corporation during the 90’s. It is the advanced version of the BASIC

language. It was mainly intended to make web and database applications. But

recently, the astonishing changes brought in the VB helped programmers to use it

in almost in every field.

NUMERICAL METHODS

Many problems in engineering require solution of nonlinear algebraic

equations In what follows we will show that the most popular numerical methods

for solving such equations involve linearization which leads to repeatedly solving

linear systems of the form Ax=b

Solution of nonlinear equations always requires iteration . That is unlike

linear systems where if solutions exist they can be obtained exactly with Gaussian

elimination with nonlinear equations only approximate solutions are obtained.

However in principle the approximations can be improved by increasing the

number of iterations. Some of the important numerical methods are Newton

Raphson method and Gauss Seidel iterative method.


CHAPTER -II

PROJECT DEVELOPMENT

NATURE OF WORK

The projects consist of methods of solving using computational iterations such as

in Newton Raphson and Gauss Seidel Method. Both the methods are applied on

typical mathematical representations of problems underlying in mechanical

engineering based. The projects are developed using Microsoft Visual Basic of

Microsoft Visual Studio Professional Edition, one of the powerful object oriented

programs using today.

Each project consists of objects such as buttons, labels, textboxes etc. to

manipulate or handle data more efficiently. The main advantages using this

language are the freedom given to the developer to make more meaningful

programs, their implementation, error detection etc. The methods are clearly

being told one by one.

In Newton Raphson Method , the following objects are being used in the

development

1. A windows form to equip all controls or objects to

manipulate data supplied by the user.

2. Buttons and labels to give more understanding about the

whole program.

3. Timers to control the looping that takes place during

iterations

4. Sub Procedures and Functions to do discrete amount of

mathematical works alone. Etc.


In Gauss Seidel the same objects used for the application development in Newton

Raphson are also being used.

The following are the main tools of the Visual Studio to develop standalone

programs

1. A Code editor window , which helps to contain all the

codes written according to the syntax

2. A Designer window, which primarily shows the form ,

where pic and place methods are used to populate the form

with objects

3. A solution explorer to give the details about the current

project

4. A properties window to show all the properties of each and

every objects used on the application.

5. A Compiler which converts the codes written into

executable application , if correct syntax is provided. Etc.

NEWTON RAPHSON METHOD EXAMPLE

A trunnion has to be cooled before it is shrink fitted into a steel hub.

Figure 2.1. A Trunnion and Steel Hub


The equation that gives the temperature Tf to which the trunnion has to be cooled

to obtain the desired contraction is given by

F(Tf)= (-0.50598 * 10-10

x Tf3) + (0.38292 * 10

-7 x Tf

2) + (0.74363 x 10

-4 x Tf) +

(0.88318 x 10-2

) ..........1

Use the Newton-Raphson method of finding roots of equations to find the

temperature Tf to which the trunnion has to be cooled. Conduct three iterations to

estimate the root of the above equation. Find the absolute relative approximate

error at the end of iteration.

SOLUTION

The root of the equation can be found out from the equation given under

At nth iteration the root is given by:

...........................................................2

Where,

= Guessed value

= Function value

= Differentiated function value

The absolute relative approximate error at the end of each iteration is given by:


| (

) .........................................................................3

APPLICATION DEVELOPMENT IN VISUAL STUDIO

The form is created and all the necessary objects or controls are placed on

the form in a specific manner to give the user more easy to understand the

application utilization.

The following are the screen shots of the project work being developed on

a Microsoft Windows 7 Operating system running 64-bit memory processor.

System Info is given under:

OS Name Microsoft Windows 7 Ultimate

Version 6.1.7600 Build 7600

Other OS Description Not Available

OS Manufacturer Microsoft Corporation

System Name VISHNU R

System Manufacturer Dell Inc.

System Model Inspiron N5010

System Type x64-based PC

Processor Intel(R) Core(TM) i3 CPU M 350 @ 2.27GHz,

2261 MHz, 2 Core(s), 4 Logical Processor(s)

BIOS Version/Date Dell Inc. A04, 5/10/2010

The User interface is as follows


Figure 2.2 VB Integrated Development Environment

It consists of the following objects

1. Buttons

2. Group box

3. Labels

4. Textboxes

5. Timer

6. Windows Form

Along with useful functions, subs that are preserved or encapsulated in the code

module


The Code editor is as follows

Figure 2.3 Code Editor of Form Class

The module (iterative and error finder) code editor is as follows

Figure 2.4 Code Editor of Module


PROGRAM CODE

********* DESIGNED AND DEVELOPED BY VISHNU.R ***********/'

'****************NEWTON RAPHSON METHOD*********************'

Option Explicit On

Option Infer On

Imports System.Windows.Forms

Imports System.Math

Public Class Form1

Public temp(), dftemp(), nrsolution() As Single

Dim tempfirst As Decimal

Public i As Short = 0

Public reqstring As String

Private Sub btnNR_Click(ByVal sender As Object, ByVal e As

System.EventArgs) Handles btnNR.Click

Try

tempfirst = (-1 * (Val(txtAssume.Text)))

reqstring = ""

txtFinal.Text = ""

Select Case txtLimit.TextLength

Case Is > 0

ReDim temp(Val(txtLimit.Text) - 1)

ReDim dftemp(Val(txtLimit.Text) - 1)


ReDim nrsolution(Val(txtLimit.Text) - 1)

Timer1.Enabled = True

Case Else

MessageBox.Show("Something has went wrong", "Critical Error",

MessageBoxButtons.OK, MessageBoxIcon.Error)

End Select

Catch ex As Exception

MsgBox(ex.Message, MsgBoxStyle.Critical)

End Try

End Sub

Private Sub Timer1_Tick(ByVal sender As Object, ByVal e As

System.EventArgs) Handles Timer1.Tick

Try

Select Case i

Case Is < Val(txtLimit.Text)

Call timercall(tempfirst, i, temp, dftemp, reqstring, nrsolution)

Case Is = Val(txtLimit.Text)

Timer1.Enabled = False

txtFinal.Text = txtFinal.Text & reqstring

Form2.TextBox1.Text = Form2.TextBox1.Text & vbNewLine &

vbNewLine & reqstring

Form2.StartPosition = FormStartPosition.CenterScreen


Form2.Show()

reqstring = ""

End Select

If i > 0 Then

If percenterror(i, nrsolution, tempfirst) < 0.1629 Then

Timer1.Enabled = 0

txtFinal.Text = txtFinal.Text & reqstring


vbNewLine & reqstring

Form2.StartPosition = FormStartPosition.CenterScreen

Form2.Show()

reqstring = ""

End If

End If

i = i + 1



Form2.Enabled = 0

End Try

End Sub

Private Sub btnClearall_Click(ByVal sender As System.Object, ByVal e As

System.EventArgs) Handles btnClearall.Click


reqstring = ""

Form2.TextBox1.Clear()

Timer1.Stop()

txtFinal.Clear()

txtFinal.Enabled = 0

txtAssume.Clear()

txtLimit.Clear()

End Sub

End Class


CODE ENCAPSULATED IN MODULE

'/********* DESIGNED AND DEVELOPED BY VISHNU R***********/'

'****************NEWTON RAPHSONMETHOD*****************'

Imports System.Math

Module Iterartive_and_error_finder

Public Opstring As String

Public Sub timercall(ByRef tempfirst As Integer, ByRef i As Integer, ByRef

temp() As Single, ByRef dftemp() As Single _

, ByRef reqstring As String, ByRef nrsolution() As Single)

Try

temp(i) = (-0.50598 * 10 ^ (-10) * (tempfirst ^ 3)) + (0.38392 * (10 ^ (-7))

* (tempfirst ^ 2)) _

+ (0.74363 * (10 ^ (-4)) * tempfirst) + (0.88318 * (10 ^ (-2)))

dftemp(i) = (-1.51974 * (10 ^ (-10)) * (tempfirst ^ (-2))) + (0.76584 * (10

^ (-7)) * tempfirst) _

+ (0.74363 * (10 ^ (-4)))

nrsolution(i) = tempfirst - (temp(i) / dftemp(i))

reqstring = CStr(reqstring & "**ITERATION NUMBER :" & (i + 1) &

vbCrLf & "TEMPERATURE = " & CDec((temp(i))) & vbCrLf &

"DIFFERENTIATED TEMPERATURE VALUE = " & CDec((dftemp(i))) _

& vbNewLine & "THE ESTIMATED ROOT IS = " &

CDec((nrsolution(i))) & _


vbNewLine & "THE PERCENTAGE OF ERROR IS= " & percenterror(i,

nrsolution, tempfirst) & "%" & vbNewLine & vbNewLine)

tempfirst = nrsolution(i)



End Try

End Sub

Public Function percenterror(ByRef i As Integer, ByRef nrsolution() As Single,

ByVal tempfirst As Single) As Single

Dim returnsoln As Single

If i >= 0 Then

returnsoln = ((nrsolution(i) - tempfirst) / nrsolution(i)) * 100

If returnsoln < 0 Then

returnsoln = (-1 * returnsoln)

Else

returnsoln = returnsoln

End If

End If

Return returnsoln

End Function

End Module


GAUSS SEIDEL METHOD

The relation between temperatures at three different points of a thermal

system is shown under in the form of nonlinear equations. Our aim is to correct

the equations based on a guessed value. This is being carried out with the help of

iterations. This method is known as Gauss Seidel Method

Consider three nonlinear equations whose unknowns need to be found.

5Ta + 2Tb - Tc = 8 .................................................................4

Ta + 4Tb + Tc = 10 ................................................................5

Ta - Tb + 2Tc = 9 ................................................................6

It is rewritten as

Ta = (8 - 2Tb + Tc)/5 .............................................................7

Tb = (10 - Ta – Tc)/4 .............................................................8

Tc = (9 – Ta + Tb)/2 ..............................................................9

The above mentioned equations are solved simultaneously using a

computer program where a set of predetermined iterations are implemented. The

iteration stops at the point where the required convergence is met.

APPLICATION DEVELOPMENT IN VISUAL STUDIO

The form is created and all the necessary objects or controls are placed on

the form in a specific manner to give the user more easy to understand the

application utilization.

The following are the screen shots of the project work being developed on

a Microsoft Windows 7 Operating system running 64-bit memory processor.


System Info is given under:

OS Name Microsoft Windows 7 Ultimate

Version 6.1.7600 Build 7600

Other OS Description Not Available

OS Manufacturer Microsoft Corporation

System Name VISHNU R

System Manufacturer Dell Inc.

System Model Inspiron N5010

System Type x64-based PC

Processor Intel(R) Core(TM) i3 CPU M 350 @ 2.27GHz,

2261 MHz, 2 Core(s), 4 Logical Processor(s)

BIOS Version/Date Dell Inc. A04, 5/10/2010

THE USER INTERFACE

Figure 2.5 VB IDE for Gauss Seidel


THE CODE EDITOR

Figure 2.6 VB Code editor of form class

MODULE CODE EDITOR

Figure 2.7 VB Code editor of Module


SOURCE CODE (FORM CLASS)

'/**********DESIGNED AND DEVELOPED BY VISHNU.R*************/'

'******************GAUSS SEIDEL METHOD***********************'

Public Class Form1

Dim AssumTb, AssumTc As Single

Dim finalText As String = ""

Dim Ta() As Single = New Single() {}

Dim Tb() As Single = New Single() {}

Dim Tc() As Single = New Single() {}

Dim i As Integer

Private Sub Button1_Click(ByVal sender As Object, ByVal e As

System.EventArgs) Handles btnClick.Click

Try

ReDim Ta(CType(txtiter.Text, Single) - 1)

ReDim Tb(CType(txtiter.Text, Single) - 1)

ReDim Tc(CType(txtiter.Text, Single) - 1)

AssumTb = CType(txtSecondTemp.Text, Single)

AssumTc = CType(txtThirdTemp.Text, Single)

Timer1.Enabled = 1

Timer1.Interval = 1

Form2.StartPosition = FormStartPosition.Manual

Form2.Location = New Point(250, 250)



Timer1.Stop()

MsgBox("ERROR DETECTED " & vbNewLine & ex.Message)

Finally

txtSecondTemp.TabStop = 1

End Try

End Sub

Private Sub Timer1_Tick (ByVal sender As Object, ByVal e As

System.EventArgs) Handles Timer1.Tick

Call TempSolver(AssumTb, AssumTc, Ta, Tb, Tc, i, finalText)

If i >= 1 Then

If Tc(i - 1) - Tc(i) <= 0.000001 AndAlso Ta(i - 1) - Ta(i) < 0.000001 _

AndAlso Tb(i - 1) - Tb(i) < 0.000001 Then

Form2.Show()


vbNewLine & finalText


vbNewLine _

& " **The required condition is met at iteration number : " & (i + 1) &

"**"

Timer1.Stop ()

ElseIf i = (Val(txtiter.Text) - 1) Then

Form2.Show ()

Form2.TextBox1.Text = finalText


Timer1.Stop ()

End If

End If

i = i + 1

End Sub

Private Sub btnClear_Click(ByVal sender As Object, ByVal e As

System.EventArgs) Handles btnClear.Click

txtFirstTemp.Clear()

txtSecondTemp.Clear()

txtThirdTemp.Clear()

txtiter.Clear()


Form2.TextBox1.Clear ()

Form2.Close ()

End Sub

Private Sub Form1_Load(ByVal sender As Object, ByVal e As

System.EventArgs) Handles Me.Load


End Sub

End Class


CODE ENCAPSULATED (MODULE)

'/**********DESIGNED AND DEVELOPED BY VISHNU.R*************/'

'*****************GAUSS SEIDEL METHOD**********************'

Module Module1

Public Sub TempSolver(ByRef AssumTb As Single, ByRef AssumTc As

Single, ByRef Tempf() As Single, ByRef TempS() As Single , ByRef TempT()

As Single, ByRef i As Integer, ByRef finalString As String)

TempS(i) = AssumTb

TempT(i) = AssumTc

Tempf(i) = ((8 - 2 * (TempS(i)) + TempT(i)) / 5)

TempS(i) = ((10 - Tempf(i) - AssumTc) / 4)

TempT(i) = ((9 - Tempf(i) + TempS(i)) / 2)

AssumTb = TempS(i)

AssumTc = TempT(i)

finalString = finalString & "* The Tc value is " & TempT(i) & _

" and " & vbNewLine & " The Tb value is " & TempS(i) _

& vbNewLine & " The Ta value is " & Tempf(i) & vbNewLine & "

The solution obtained at Iteration number : " _

& (i + 1) & vbNewLine & vbCrLf

End Sub

End Module


CHAPTER -III

RESULTS AND DISCUSSION

NEWTON RAPHSON METHOD

APPLICATION WITH INPUTS GIVEN

Figure 3.1 VB IDE with inputs given for Newton Raphson method

APPLICATION DEVELOPED OUTPUTS

Figure 3.2 VB code compiled and executed


OUTPUT WINDOW SEPERATED

Figure 3.3 VB Output Window of NR Method

OBTAINED OUTPUT

Assumed value of temperature: 100oC

Iterations required : 5 no’s

Iteration number: 1

After substitution

Temperature= .001830018

Differentiated temperature= 0.0000667046

The estimated root of the equation= -127.437

The percentage of Approximate error = 21.5281%….

Iteration number: 3

After substitution


Temperature= -0.00001352756

Differentiated temperature= 0.00006448366

The estimated root of the equation= -128.7902

The percentage of Approximate error = 0.1628833%

At the third iteration the predetermined condition i.e. the percentage of

approximate error was being set to .169, and an instruction was given to stop

looping if the error value comes below the predetermined value.

The results are tabulated below

Iteration Temperature Diff.Temperature Root(oc) Error %

1 .001830018 0.0000667046 -127.4347 21.52841%

2 0.0001105676 0.00006463684 -128.7106 1.329029%

3 -0.00001352756 0.00006448366 -128.7902 0.1628833%

Table 3.1 Tabulated Results of NR Method

GAUSS SEIDEL METHOD

APPLICATION WITH INPUTS GIVEN

Figure 3.4 VB IDE with inputs given for Gauss Seidel Method


OUTPUT WINDOW

Figure 3.5 VB Output Window of GS Method

OBTAINED OUTPUTS

The guessed value at point b= 0oC

The guessed value at point c= 0oC

The iterations given = 100

Required conditions were

1. Tc(i - 1) - Tc(i) <= 0.0001 (older - latest)

2. Ta(i - 1) - Ta(i) < 0.0001

3. Tb(i - 1) - Tb(i) < 0.0001

The satisfied solution was obtained at iteration no. 6

Iteration Ta 0C Tb

0C Tc

0C

1 1.6 2.1 4.75

2 1.71 0.885 4.0875

3 2.0635 0.96225 3.949375

4 2.004975 1.011412 4.003219

5 1.996079 1.000176 4.002048

6 2.00034 0.999403 3.999532

Table 3.2 Tabulated Results of GS Method


CONCLUSIONS

This work depicts the state-of-the-art in modeling and simulation of

engineering systems. Simulation is a very broad area that comprises many

research issues that are not included in this work. I have limited my attention to

issues that are particularly important with respect to system-level modeling in

support of design. The modeling of mechatronic systems requires a language

capable of describing physical phenomena in multiple energy domains, in

continuous time as well as in discrete time. Recent advances in modeling have

resulted in several modular, object-oriented languages that satisfy these

requirements. To further simplify the modeling process and avoid unnecessary

duplication of data entry, it is critical that the simulation environment be

integrated with the design environment. In single-domain simulation

environments, this is already common practice. Current research is expanding this

integration towards simulation of multi-disciplinary systems. Finally, due to the

multi-disciplinary nature of mechatronic systems, the design requires a team of

experts with different backgrounds. Systems modeling, therefore, must support

collaborative modeling, including support for standardized languages, model

management tools, and model abstraction tools.

While using the Newton Raphson numerical method computationally, it

has been found out the estimated root of the equation obtained obeying all

prescribed conditions was -128.7902 0C and the iteration at which the

convergence reached was at 3

While using the Gauss Seidel numerical method computationally, it has

been found out the estimated values of the equation obtained obeying all

prescribed conditions was Ta= 1.60C, Tb= 0.999403

0C, Tc= 3.999532

0C


REFERENCES

[1] http://numericalmethods.eng.usf.edu

[2] Ascher, U. M. and Petzold, L. R., Computer Methods for Ordinary

Differential Equations and Differential-Algebraic Equations. Philadelphia,

Pennsylvania: SIAM, 1998.

[3] Design of Thermal Systems by Stoecker

[4] MODELING AND SIMULATION METHODS FOR DESIGN OF

ENGINEERING SYSTEMS, Rajarishi Sinha , Vei-Chung Liang,

Christiaan J.J. Paredis,Pradeep K. Khosla, Carnegie Mellon University,

Pittsburgh, PA 15213, USA

http://numericalmethods.eng.usf.edu/

thermal modelling

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