simulation record

1

Sl.

No.

Name of Exercise Page

Number

Remarks

THERMAL ANALYSIS using ANSYS

1 Study of Introduction to Finite Element Analysis

2 Simulation of One Dimensional Heat Conduction Flow

3 Simulation of One Dimensional Heat Flow with Internal Heat Generation

4 Simulation of Two Dimensional Steady State Heat Conduction Flow

5 Simulation of Two Dimensional Steady State Heat Convection Flow

6 Simulation of Two Dimensional Unsteady State (Transient) Heat

Conduction

7 Simulation of Heat Flow on Two Dimensional Axi-Symmetric Plane

8 Simulation of Thermal Stress Analysis of Stepped Shaft

9 Simulation of Coupled Field Structural And Thermal Analysis

COMPUTATIONAL FLUID DYNAMICS using ANSYS

10 Simulation of Steady State Fluid Flow Analysis

HEAT EXCHANGER ANALYSIS using MATLAB

11 Study of basics of MATLAB

12 Determination of Matrix Operations

13 Determination of Eigen Values and Eigen Vectors

14 Solution of Simultaneous Equations

15 Plot 2-D and 3-D Graphs

16 Simulation of Heat Exchanger – NTU Method

17 Simulation of Heat Exchanger – LMTD Method

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EX.NO.01: STUDY OF INTRODUCTION TO FINITE ELEMENT ANALYSIS

DATE:

AIM:

To study of introduction to finite element analysis.

INTRODUCTION:

Finite element analysis has now become an integral part of computer aided engineering

and is being extensively used in the analysis and design of many complex real life systems.

FEA consists of a computer model of a material or design that is stressed and analyzed

for specific results. It is used in new product design and existing product refinement. Modifying

an existing product or structure is utilized to qualify the product or structure for a new service

condition. In case of structural failure, FEA may be used to help determine the design

modifications to meet the new condition.

There are generally two types of analysis that are used in industry, 2D modeling and

3D modeling. While 2D modeling conserues simplicity and allows the analysis to be run on a

relatively normal computer, it tends to yield less accurate results. 3D modeling however

produces more accurate results while specifying and sacrificing the ability to run on all but the

fastest computers effectively.

Within each of these modeling schemes the programmer can inset numerous algorithms

(functions) which may make the system behave linearly or non – linearly. Linear systems are for

less complex and generally do not take into account plastic deformation, and many also are

capable of testing a material all the way to fracture.

HOW DOES FEA WORKS?

FEA uses a complex system of points called nodes which make a grid called a mesh. The

mesh is programmed to contain the material and structural properties which define how the

structural will react to certain loading conditions. Nodes are assigned at a certain density

throughout the material depending on the anticipated stress usually have a higher node density

than those which experience little or no stress.

Points of interest may consist of fracture point of previously tested material, fillets,

corners, complex details and high stress areas. The mesh acts like a spider web in that from each

node, there extends a mesh element to each of the adjacent nodes. This (element) web of vectors

is what carries the material properties to the object creating many elements.

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A wide range of objective function are available for minimization (or) maximization.

Mass, Volume, Temperature.

Strain energy, Stress-strain.

Force, Displacement, Velocity, Acceleration.

Synthetic [User Defined].

There are multiple loading conditions which may be applied to a system.

Point, pressure, thermal, gravity and centrifugal static loads.

Thermal loads from solution of heat transfer analysis.

Heat flux and conviction.

Point, pressure and gravity dynamic loads.

Each FEA program may come with an element library, or one in constructed overtime.

Some sample elements are

Bar elements

Beam elements

Plate / Shell / Composite elements

Solid elements

Spring elements

Mass elements

Viscous damping elements

Many FEA programs also equipped with the capability to use multiple materials within

the structure such as,

Isotropic, Identical throughout.

Orthotropic, Identical at 90 degree.

General anisotropic, different throughout.

TYPES OF ENGINEERING ANALYSIS

STRUCTURAL ANALYSIS:

Structural analysis consists of linear and non-linear models. Linear models use simple

parameters and assume that the material is not plastically deformed. Non-linear models consist

of stressing the material post its elastic capabilities. The stresses in the material then vary with

the amount of deformation.

VIBRATIONAL ANALYSIS

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Vibrational analysis is used to test a material against random variation vibrations, shock

and impact. Each of these incidences may act on the natural vibrational frequency of the

material, which in turn, may cause resonance and subsequent failure.

FATIGUE ANALYSIS

Fatigue analysis helps designers to predict the life of a material or structure by showing

the effects of cyclic loading on the specimen. Such analysis can show the areas where crack

propagation is most likely to occur. Failure due to fatigue may be also show the damage

tolerance of the material.

HEAT TRANSFER ANALYSIS

Heat transfer analysis models the conductively or thermal fluid dynamics of the material

or structure. This may consist of a steady- state or transient transfer. Steady-state, transfer refers

to constant thermo properties in the material that yield linear heat diffusion.

STEPS INVOLVED IN FINITE ELEMENT ANALYSIS

Step 1: Discretization of the structure.

This first step involves dividing the structure or domain of the problem into small

divisions or elements. The analyst has to decide about the type, size and arrangement of the

elements.

Step 2: Selection of proper interpolation (or displacement) model.

A simple polynomial equation (liner/quadratic/cubic) describing the variation of state

variable within an element is assumed. This model generally is of the interpolation shape

function type. Certain conditions are to be satisfied by this model in order that the results are

meaningful and converging.

Step 3: Derivation of element stiffness matrices and load vectors

Response of an element to the loads can be represented by element equation of the form where,

=

= Element stiffness matrix.

= Element response matrix or element nodal displacement vector,

= Element load matrix or element nodal vector.

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From the assumed displacement model, the element properties, namely stiffness matrix and the

load vector are deriving Element stiffness matrix is characteristics property of element and

depends on geometry as well as material. There are three approaches for deriving element

equations. They are,

(a) Direct approach,

(b) Variational approach (Piece wise Rayleigh Ritz method)

(c) Weighted residual approach (e.g., Galerkin method).

(a) Direct approach:

Direct approach:

In this method, direct physical reasoning is used to establish the element properties

(Stiffness matrices and load vectors) in terms of pertinent variables. Although this approach is

limited to simple types of elements, it helps to understand the physical interpretation of the finite

element method.

Variational approach:

This approach can be adopted when variational theorem (extreme principle) that governs

the physics of the problem is available. The method involves minimizing a scalar quantity known

as functional that is typical of the problem at hand (e.g., potential energy in stress analysis

problems)

Weighted residual approach:

This approach is more general in the sense that it applicable to all situations where

governing differential equation of the problem available. The method involves minimizing error

resulting from substituting trial solution in to the differential equation.

Step 4: Assembling of element equations to obtain the global equations

Element equations obtained in step 3 are assembled to form global equation of the form

=

That describes the behaviour of entire structure.

Where, is the global stiffness matrix

Is the vector of global nodal displacements and

Is the global load vector of nodal forces for the complete structure.

Step 5: Solution for the unknown nodal solution displacements

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The global equations are to be modified to account for the boundary conditions of the

problem. After specifying the boundary conditions, the equilibrium equations can be expressed

as,

=

For linear problems, the vector r can be solved very easily.

Step 6: Computation of element strains and stresses

Form the known nodal displacements the element strains and stresses can be computed

by using predefined equations for structure.

The terminology used in the previous six steps has to be modified if we want to extend the

concept to other fields. For example, field variable in place of displacement, characteristics

matrix in place of stiffness matrix and element resultants in place of element strains.

METHODS OF PRESCRIBING BOUNDARY CONDITIONS

There are two methods of prescribing boundary conditions.

This method is useful while performing hand calculations and poses difficulties in

implementing in software. This method has been used in this book for solving the problems by

finite element method using hand calculations. This method is explained below in brief. Consider

following set of global equations,

=

Let be prescribed i.e., u= s. This condition is imposed as follows:

(1) Eliminate the row corresponding to u

(2) Transfer the column corresponding to u to right hand side after multiplying it by„s‟.

These steps result in the following set of modified equations,

(3) = - s

This set of equation now may be solved for non trivial solution.

RESULT: Thus, the introduction to finite element analysis and steps involved in finite

element analysis are studied.

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Fig.2.1 PROBLEM DESCRIPTION

Fig.2.2 MESHED MODEL

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EX.NO.02: SIMULATION OF ONE DIMENSIONAL HEAT CONDUCTION FLOW

DATE:

AIM:

To perform simulation of one dimensional heat conduction flow of given component

using ANSYS software and plot the results.

PROCEDURE:

Step 1: Set the analysis title

Utility Menu > File > Change the title > simulation of one dimensional heat flow > ok.

Step 2: Define the type of the problem

Main Menu > Preference> Turn on Thermal>ok

Step 3: Define the Type of Elements

Main Menu > Preprocessor > Element Type > Add/Edit/Delete > Add > Link > 3D

conduction 33 > ok>close.

Step 4: Define the Real constant

Main Menu > Preprocessor > Real constant > Add/Edit/Delete > Add > Link > ok >Enter

cross sectional area = 1> ok >close.

Step 5: Define the Material Properties

Main Menu > Preprocessor > Material Props > Material Models > Thermal >

Conductivity > Isotropic > KXX = 10 (Thermal conductivity) > ok >close.

Step 6: Create the Key points

Main Menu > Preprocessor > Modeling > Create > Key points > In Active CS>

Key point

Number

X y Z

1 0 0 0

2 1 0 0

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Fig.2.3 LOADS AND BOUNDARY CONDITIONS ARE APPLIED

Fig.2.4 TEMPERATURE DISTRIBUTION

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Step 7: Create the line

Main Menu > Preprocessor > Modeling > Create > Lines > Lines>Straight Line > pick

the Key Points 1 and 2>ok

Step 8: Mesh Size

Main Menu > Preprocessor > Meshing > Size Cntrls > Manual Size > Lines> All Lines >

Enter Element Edge Length =0.05 > ok.

Step 9: Mesh

Main Menu > Preprocessor > Meshing > Mesh > Lines > Free > Pick All

Step 10: Define Analysis Type

Solution > Analysis Type > New Analysis > Steady-State > ok.

Step 11: Apply the Boundary conditions and loads

Solution > Define Loads > Apply > Thermal > Temperature > On Key points> Select

Key point 1 > ok>select TEMP>Enter Temperature Value = 373.

Solution > Define Loads > Apply > Thermal > Temperature > On Key points> Select

Key point 1 > ok>select TEMP>Enter Temperature Value = 273.

Step 12: Solve the System

Solution > Solve > Current LS > ok >close.

Step 13: Reviewing the Result

Main Menu > General Postproc > Plot Results > Contour Plot > Nodal Solu > DOF

solution > Nodal Temperature >ok

RESULT:

Simulation of one dimensional heat conduction flow of the given component using

ANSYS software are performed and the results are plotted.

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12

EX.NO.03: SIMULATION OF ONE DIMENSIONAL HEAT FLOW WITH INTERNAL

HEAT GENERATION

DATE:

AIM:

To perform simulation of one dimensional heat flow with internal heat generation of the

given component using ANSYS software and plot the results.

PROCEDURE:


1. Choose menu path Utility Menu > File > Change Title

2. Type the text thermal analysis of one dimensional heat flow and click on OK.

Step 2: Define the Type of problem

Main Menu > Preferences > select Thermal > ok

Step 3: Define the element types

Main Menu > Preprocessor > Element Type > Add/Edit/Delete > Add > Link > 2D

conduction 32 > ok > Add > Link > 3D conduction 34 > ok > Close.

Step 4: Define the real constants

Main Menu > Preprocessor > Real constants > Add/Edit / Delete > Add > select link 32 >

ok > Enter the Cross-sectional area AREA > Enter 1 > ok > Add > Click on Link 34 > ok > Enter

the Cross-sectional area AREA > Enter 1 > ok

Step 5: Define the material properties

Main Menu>Preprocessor>Material Props> Material Models

Material Model Number 1:

Click Thermal > Conductivity > Isotropic

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Enter KXX = 0.75 > ok

Material Menu > New Model > ok


Click Thermal > Convection or Film Coef

Enter HF = 15 > ok > close

Step 6: Create the Nodes

Main Manu > Preprocessor > Modeling > Create > Nodes > In Active CS > Enter Co-

ordinates of Nodes > ok

Node locations

Node number X co-ordinate Y co-ordinate

1 0 0

2 0.1 0

3 0.2 0

4 0.3 0

Step 7: Create the elements

Main Menu > preprocessor > Modeling > Create > Elements > Elem Attributes > ok >

Auto Numbered > Thru nodes > Pick 1st and 2

nd node > Apply > then Pick the 2

nd and 3

rd node >

ok

Elem Attributes > Change the element type to Link 34 > Change the material number to 2

> change the Real constant set number to 2 > ok > Auto Numbered > Thru nodes > Pick 3rd

and

4th

node > ok

Step 8: Apply boundary conditions and loads

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Apply Loads

Main Menu > Preprocessor > Loads > Define Loads > Apply > Thermal > Temperature >

On Nodes > Pick the 4th

node > Apply > Click on TEMP and Enter Value = 35 > ok

Apply temperature load

Main Menu > Preprocessor > Loads > Define Loads > Apply > Thermal > Heat Generat

> On Nodes > Pick the 1st, 2

nd and 3

rd nodes > Apply > Enter HGEN Value = 6000 > ok

Step 9: Solve the model

Main Menu > Solution > solve > Current LS > ok > close

Step 10: Review the results

Temperature Distribution


Solution > Temperature > ok

Main Menu > General Postproc > List Results > Nodal Solu > Select Temperature > ok

RESULT:

Simulation of one dimensional heat flow with internal heat generation of the given

component using ANSYS software are performed and the results are plotted.

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EX.NO.04: SIMULATION OF TWO DIMENSIONAL STEADY STATE HEAT CONDUCTION FLOW

DATE:

AIM:

To perform simulation of two dimensional steady state heat conduction flow of the


PROCEDURE:


Utility Menu > File > Change the title > Conductive Heat Transfer Analysis of a 2D

Component > ok.


Main Menu > Preprocessor > Element Type > Add/Edit/Delete > Add > Solid > Quad

4Node 55 > ok.



Conductivity > Isotropic > KXX = 10 (Thermal conductivity) > ok >close.

Step 4: Create the Geometry

Main Menu > Preprocessor > Modeling > Create > Areas > Rectangle > By 2 Corners >

X=0, Y=0, Width=1, Height=1 > ok

Step 5: Mesh Size

Main Menu > Preprocessor > Meshing > Size Cntrls > ManualSize > Areas > All Areas >

0.05 > ok.

Step 6: Mesh

Main Menu > Preprocessor > Meshing > Mesh > Areas > Free > Pick All


Solution > Analysis Type > New Analysis > Steady-State > ok.

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Solution > Define Loads > Apply > Thermal > Temperature > On Nodes > Click the Box

option and draw a box around the nodes on the top line > ok.

Select the Temperature > Enter the constant temperature 500 > ok

Using the same method, constrain the remaining 3 sides to a constant value of 100



Step 10: Reviewing the Result


solution > Nodal Temperature >ok

RESULT:

Simulation of two dimensional steady state heat conduction flow of the given


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EX.NO.05: SIMULATION OF TWO DIMENSIONAL STEADY STATE HEAT CONVECTION FLOW

DATE:

AIM:

To perform simulation of two dimensional steady state heat convection flow of the


PROCEDURE:


Utility Menu > File > Change the title > Convective Heat Transfer Analysis of a 2D

Component > ok.


Main Menu > Preprocessor > Element Type > Add/Edit/Delete > Add > Solid > Quad

4Node 55 > ok.



Conductivity > Isotropic > KXX = 1.5 (Thermal conductivity) > ok >close.


Main Menu > Preprocessor > Modeling > Create > Areas > Rectangle > By 2 Corners >

X=0, Y=0, Width=0.4, Height=0.6 > ok

Step 5: Mesh Size

Main Menu > Preprocessor > Meshing > Size Cntrls > Manual Size > Areas > All Areas

> 0.005 > ok

Step 6: Mesh

Main Menu > Preprocessor > Meshing > Mesh > Areas > Free > Pick All


Solution > Analysis Type > New Analysis > Steady-State > ok

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Apply Conduction Constraints

Solution > Define Loads > Apply > Thermal > Temperature > on line > Select the top

and bottom side of the block > ok > Select the Temperature > Enter the constant temperature =

180 > ok

Apply Convection Boundary Conditions

Main Menu > Solution > Define Loads > Apply > Thermal > Convection > On Lines >

Select the right side of the block > ok > this will specify a convection of 50 W/m2*C and an

ambient temperature of 25 corresponding VALI and Bulk temperature > ok.

Apply Insulated Boundary Conditions

o Main Menu > Solution > Define Loads > Apply > Thermal > Convection > On

Lines > Select the left of the block > ok.

o Enter a constant Film coefficient (VALI) of 0 > ok.



Step 10: Reviewing the Results


solution > Nodal Temperature > ok

RESULT:

Simulation of two dimensional steady state heat convection flow of the given


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Fig6.2 MESHED MODEL

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EX.NO.06: SIMULATION OF TWO DIMENSIONAL UNSTEADY STATE (TRANSIENT) HEAT CONDUCTION

FLOW.

DATE:

AIM:

To perform simulation of two dimensional unsteady state heat conduction flow of the


PROCEDURE:

1. Give example a Title

Utility Menu > File > Change Title>Transient Thermal Conduction

2. Open preprocessor menu

ANSYS Main Menu > Preprocessor

3. Create geometry

Preprocessor > Modeling > Create > Areas > Rectangle > By 2 Corners

X=0, Y=0, Width=1, Height=1

4. Define the Type of Element

Preprocessor > Element Type > Add/Edit/Delete... > click 'Add' > Select Thermal

Mass Solid, Quad 4Node 55

5. Element Material Properties

Preprocessor > Material Props > Material Models > Thermal > Conductivity >

Isotropic > KXX = 5 (Thermal conductivity)

Preprocessor > Material Props > Material Models > Thermal > Specific Heat > C

= 2.04

Preprocessor > Material Props > Material Models > Thermal > Density > DENS =

920

6. Mesh Size Preprocessor > Meshing > Size Cntrls > ManualSize > Areas > All Areas > 0.05

7. Mesh Preprocessor > Meshing > Mesh > Areas > Free > Pick All

Solution Phase: Assigning Loads and Solving

1. Define Analysis Type

Solution > Analysis Type > New Analysis > Transient

2. Set Solution Controls

Solution > Analysis Type > Sol'n Controls

A) Set Time at end of loadstep to 300 and Automatic time stepping to ON.

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B) Set Number of substeps to 20, Max no. of substeps to 100, Min no. of

substeps to 20.

C) Set the Frequency to Write every substep.

Click on the NonLinear tab at the top .

D) Set Line search to ON .

E) Set the Maximum number of iterations to 100.

3. Apply Constraints

For thermal problems, constraints can be in the form of Temperature, Heat Flow,

Convection, Heat Flux, Heat Generation, or Radiation. In this example, 2 sides of the

block have fixed temperatures and the other two are insulated.

o Solution>DefineLoads>Apply >Thermal > Temperature > On Nodes

o Click the Box option and draw a box around the nodes on the top line and then

click OK.

o Fill the window in as shown to constrain the top to a constant temperature of 500

o Using the same method, constrain the bottom line to a constant value of 100

Orange triangles in the graphics window indicate the temperature contraints.

4. Apply Initial Conditions

Solution > Define Loads > Apply > Initial Condit'n > Define > Pick All

Fill in the IC window as follows to set the initial temperature of the material to

100 K:

5.Solve the System

Solution > Solve > Current LS

SOLVE

Post processing: Viewing the Results

1. Results Using ANSYS

Plot Temperature

General Postproc > Plot Results > Contour Plot > Nodal Solu ... > DOF solution,

Temperature TEMP

RESULT:

Simulation of two dimensional unsteady state heat conduction flow of the given


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EX.NO.07: SIMULATION OF HEAT FLOW ON TWO DIMENSIONAL AXI-SYMMETRIC PLANE

DATE:

AIM:

To perform simulation of heat flow on two dimensional axi symmetric plane of the given

component using ANSYS software and plot the results.

PROCEDURE:


1. Choose menu path Utility Menu > File > Change Title

2. Type the text thermal analysis of 2d axi-symmetric component > ok.

Step 2: Define the Type of problem

Main Menu > Preferences > select Thermal > ok


Main Menu > Preprocessor > Element Type > Add/Edit/Delete > Add > Solid > Quad 8

node 77 > ok > close.



Conductivity > Isotropic

Enter KXX = 0.031 > ok


Main Manu > Preprocessor > Modeling > Create > Areas > Circle > Partial Annulus >

Enter the data > ok

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31

Wp x 0

Wp y 0

Rad – 1 125

Theta -1 0

Rad -2 250

Theta -2 90

Step 6: Meshing

Main Menu > Preprocessor > Meshing > Mesh tool > Click on the Smart Size > Move

the Slider available below the Smart Size to 2 (towards Fine side) > Close

Main Menu > Preprocessor > Meshing > Mesh > Area > Free > Pick the quadrant > ok


Main Menu > Preprocessor > Loads > Define Loads > Apply > Thermal > Temperature >

On Nodes > Select the line on the interior > Apply > Click on TEMP and Enter Value = 300 >

ok

Main Menu > Preprocessor > Loads > Define Loads > Apply > Convection > On Lines >

Select the lines defining the outer surface > Enter the convection coefficient = 0.0994 and the

fluid (Bulk temp) temp = 280 > ok

Main Menu > Preprocessor > Loads > Apply > Heat Flux > On Lines > Select the

vertical and horizontal lines of symmetry > Enter the heat Flux = 0 > ok



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Temperature Distribution


Solution > Temperature > ok

Main Menu > General Postproc > List Results > Nodal Solu > Select Temperature > ok

RESULT:

Simulation of heat flow on two dimensional axi symmetric plane of the given component using


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Fig.8.2 DISPLACEMENT PLOT

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EX.NO.08: SIMULATION OF THERMAL STRESS ANALYSIS OF STEPPED SHAFT

DATE:

AIM:

To perform simulation of thermal stress analysis of the given component using ANSYS

software and plot the results.

PROCEDURE:


1. Choose menu path Utility Menu>File>Change Title

2. Type the text Stepped Bar Analysis > ok.


Main Menu>Preprocessor>Element Type>Add/Edit/Delete>Add>Structural>Link>2D

spar>OK>Close.

Step 3: Define the real constants

Main Menu>Preprocessor>Real constants>Add/Edit>Delete>Add>OK

1st element: Cross-sectional area AREA>Enter 2400>OK>Add>OK

2nd

element: Cross-sectional area AREA>Enter 1200>OK>Add>OK

3rd

element: Cross-sectional area AREA>Enter 600>OK>Add>OK>close.


Main Menu>Preprocessor>Material Props> Material Models


Click Structural > Linear > Elastic > Isotropic

Enter EX = 0.83E5 and PRXY 0.34 > OK

Enter the coefficient of thermal expansion α

Click Structural > Thermal expansion > Secant coefficient > Isotropic

Enter ALPX = 18.9E-6 >OK

Material Menu > New Model > OK

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Fig.8.3 STRESS PLOT

36



Enter EX = 0.75E5 and PRXY 0.35 > OK



Enter ALPX = 23E-6 >OK



Enter EX = 2E5 and PRXY 0.3 > OK



Enter ALPX = 11.7E-6 >OK

Step 5: Create the Nodes

Main Manu > Preprocessor> Modeling > Create > Nodes > In Active CS >Enter Co-

ordinates of Nodes

Node locations

Node number X co-ordinate Y co-ordinate

1 0 0

2 800 0

3 1400 0

4 1800 0

Step 6: Create elements

Main Menu > preprocessor > Modeling > Create > Elements > Elem Attributes > ok >

Auto Numbered > Thru nodes > Pick 1st and 2

nd node > ok

Elem Attributes > Change the material number to 2 > change the Real constant set

number to 2 ok > Auto Numbered > Thru nodes > Pick 2nd

and 3rd

node > ok

Elem Attributes > Change the material number to 3 > change the Real constant set

number to 3 ok > Auto Numbered > Thru nodes > Pick 3rd

and 4th

node > ok

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Constraint

Main Menu > Preprocessor > Loads > Define Loads > Apply > structural > Displacement

> On Nodes > Pick the 1st node and 4

th node > Apply > All DOF = 0 > ok

Apply Loads

Main Menu > Preprocessor > Loads > Define Loads > Apply > structural >

Force/Moment > On Nodes > Pick the 2nd

node > ok > Force/Moment value = -50e3 in FX

direction > ok > Force/Moment > On Nodes > Pick the 3rd

node > ok > Force/Moment value = -

75e3 in FX direction > ok

Apply temperature load

Main Menu > Preprocessor > Loads > Define Loads > Apply > structural > Temperature

> On Elements > Pick the 1st element, 2

nd element, 3

rd element > ok > Enter Temperature at

location N =75 > ok




Displacement


Solution > Displacement vector sum > ok

Stress

Main Menu > General Postproc > Element Table > Define Table > Add > Select By

sequence num and 1 after LS > ok

Main Menu > General Postproc > Plot Results > Contour Plot > Elem Table > Select LS1

> ok

RESULT:

Simulation of thermal stress analysis of the given component using ANSYS software are

performed and the results are plotted.

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Fig9.2 MESHED MODEL

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EX.NO.09: SIMULATION OF COUPLED FIELD STRUCTURAL AND THERMAL ANALYSIS

DATE:

AIM:

To perform simulation of coupled field structural and thermal analysis of the given component

using ANSYS software and plot the results.

PROCEDURE:

Thermal Environment - Create Geometry and Define Thermal Properties

1. Give example a Title

Utility Menu > File > Change Title >Enter Coupled Field Thermal Stress

2. Open preprocessor menu

ANSYS Main Menu > Preprocessor

3. Define Keypoints

Preprocessor > Modeling > Create > Keypoints > In Active CS

Keypoint Coordinates (x,y,z)

1 (0,0)

2 (1,0)

4. Create Lines

Preprocessor > Modeling > Create > Lines > Lines > In Active Coord

Create a line joining Key points 1 and 2, representing a link 1 meter long.

5. Define the Type of Element

Preprocessor > Element Type > Add/Edit/Delete

For this problem we will use the LINK33 (Thermal Mass Link 3D conduction)

element.

6. Define Real Constants

Preprocessor > Real Constants... > Add...

In the 'Real Constants for LINK33' window, enter the following geometric

properties:

i. Cross-sectional area AREA: 4e-4

This defines a beam with a cross-sectional area of 2 cm X 2 cm.

7. Define Element Material Properties

Preprocessor > Material Props > Material Models > Thermal > Conductivity >

Isotropic

In the window that appears, enter the following geometric properties for steel:

i. KXX: 60.5

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41

8. Define Mesh Size

Preprocessor > Meshing > Size Cntrls > ManualSize > Lines > All Lines > Ement

edge length = 0.1

9. Mesh the frame

Preprocessor > Meshing > Mesh > Lines > click 'Pick All'

10. Write Environment

The thermal environment (the geometry and thermal properties) is now fully

described and can be written to memory to be used at a later time.

Preprocessor > Physics > Environment > Write

In the window that appears, enter the TITLE Thermal and click OK.

11. Clear Environment

Preprocessor > Physics > Environment > Clear > OK

Doing this clears all the information prescribed for the geometry, such as the

element type, material properties, etc. It does not clear the geometry however, so

it can be used in the next stage, which is defining the structural environment.

Structural Environment - Define Physical Properties

Since the geometry of the problem has already been defined in the previous steps, all that is

required is to detail the structural variables.

12. Switch Element Type

Preprocessor > Element Type > Switch Elem Type > Choose Thermal to Struc

from the scoll down list.

This will switch to the complimentary structural element automatically. In this

case it is LINK 8. For more information on this element, see the help file. A

warning saying you should modify the new element as necessary will pop up. In

this case, only the material properties need to be modified as the geometry is

staying the same.

13. Define Element Material Properties

Preprocessor > Material Props > Material Models > Structural > Linear > Elastic

> Isotropic

In the window that appears, enter the following geometric properties for steel:

i. Young's Modulus EX: 200e9

ii. Poisson's Ratio PRXY: 0.3

Preprocessor > Material Props > Material Models > Structural > Thermal

Expansion Coef > Isotropic

iii. ALPX: 12e-6

14. Write Environment

The structural environment is now fully described.

Preprocessor > Physics > Environment > Write

In the window that appears, enter the TITLE Struct

Solution Phase: Assigning Loads and Solving

42

15. Define Analysis Type

Solution > Analysis Type > New Analysis > Static

ANTYPE, 0

16. Read in the Thermal Environment

Solution > Physics > Environment > Read

Choose thermal and click OK.

If the Physics option is not available under Solution, click Unabridged Menu at the

bottom of the Solution menu. This should make it visible.


Solution > Define Loads > Apply > Thermal > Temperature > On Keypoints

Set the temperature of Keypoint 1, the left-most point, to 348 Kelvin.

18. Solve the System


SOLVE

19. Close the Solution Menu

Main Menu > Finish

It is very important to click Finish as it closes that environment and allows a new

one to be opened without contamination. If this is not done, you will get error

messages.

The thermal solution has now been obtained. If you plot the steady-state temperature on

the link, you will see it is a uniform 348 K, as expected. This information is saved in a

file labelled Jobname.rth, were .rth is the thermal results file. Since the jobname wasn't

changed at the beginning of the analysis, this data can be found as file.rth. We will use

these results in determing the structural effects.

20. Read in the Structural Environment

Solution > Physics > Environment > Read

Choose struct and click OK.


Solution > Define Loads > Apply > Structural > Displacement > On Keypoints

Fix Keypoint 1 for all DOF's and Keypoint 2 in the UX direction.

22. Include Thermal Effects

Solution > Define Loads > Apply > Structural > Temperature > From Therm

Analy

As shown below, enter the file name file.rth. This couples the results from the

solution of the thermal environment to the information prescribed in the structural

environment and uses it during the analysis.

23. Define Reference Temperature

Preprocessor > Loads > Define Loads > Settings > Reference Temp

For this example set the reference temperature to 273 degrees Kelvin.

43

Fig.9.5 DISPLACEMENT PLOT

Fig.9.6 COMPRESSIVE STRESS

44

24. Solve the System


SOLVE

Postprocessing: Viewing the Results

1. Get Stress Data

Since the element is only a line, the stress can't be listed in the normal way.

Instead, an element table must be created first.

General Postproc > Element Table > Define Table > Add

Fill in the window as shown below.

Eff NU for EQV strain = 0

User lable for item = CompStr

Select >By Sequence Num > LS

Enter > LS,1

2. List the Stress Data

General Postproc > Element Table > List Elem Table > COMPSTR > OK

3. Hand Calculations

Hand calculations were performed to verify the solution found using ANSYS:

As shown, the stress in the link should be a uniform 180 MPa in compression.

RESULT:

Simulation of coupled field structural and thermal analysis of the given component using


45



46

EX.NO.10: SIMULATION OF STEADY STATE FLUID FLOW ANALYSIS

DATE:

AIM:

To perform simulation of steady state fluid flow analysis of the given component using

ANSYS software and plot the results.

PROCEDURE:


. Utility Menu>File>Change Title>Enter simulation of steady state fluid flow

Step 2: Define type of problem

Main Menu > Preferences, then select FLOTRAN CFD > OK


Main Menu > Preprocessor > Element Type > Add/Edit/Delete > Add > FLOTRAN CFD

> 2D FLOTRAN 141 > OK

Step 4: Create Rectangle

Main Menu > Preprocessor > Modeling > Create > Areas > Rectangle > By 2 Corners

Enter (Lower left corner) WP X = 0.0 and Width = 2, Height = 1 > OK

Step 5: Create solid circle

Main Menu > Preprocessor > Modeling > Create > Areas > Circle > Solid Circle. Enter

WP X = 1, WP Y = 0.5 and Radius = 0.2 > OK

Step 6: Subtract Circle from Rectangle

Utility Manu > Plot controls > Numbering > Turn on Areas

Now subtract the circle (Numbered 2) from the rectangle (Numbered 1). (Read the

messages in the window at the bottom of the screen as necessary)

Main Manu > Preprocessor > Modeling > Operate > Booleans > Subtract > Areas > Enter

1>OK>Enter 2 > OK

47


Fig.10.4 VELOCITY VECTOR PLOT

48

Step7: Meshing

Create a mesh of quadrilateral elements over the area.

Main Manu > Preprocessor > Meshing > Mesh Tool

The Mesh Tool dialog box appears. Close the Mesh Tool box.

Main Menu > Preprocessor > Meshing > Mesh > Areas > Free > Pick the area > OK

Step8: Apply Boundary Conditions and Constraints

Apply the velocity boundary conditions and pressure.

Main Menu > Preprocessor > Loads > Define Loads > Apply > Fluid/CFD > Velocity >

On Lines Pick the left edge of the plate > OK > Enter VX = 1 > OK

Main Menu > Preprocessor > Loads > Define Loads > Apply > Fluid/CFD > Velocity >

On Lines Pick the edges around the cylinder > OK > Enter VX = 0 and VY = 0 > OK

Main Menu > Preprocessor > Loads > Define Loads > Apply > Fluid/CFD > Pressure

DOF > On Lines Pick the top, bottom and right edges of the plate > OK > OK

Step9: Solution.

1. Main Menu > Solution > FLOTRAN Set Up > Fluid Properties > A dialog box appears in

that select against density as liquid and against viscosity as liquid. > OK

Then another dialog box appears in that enter the value of density = 1000 and viscosity

value = 0.001 > OK

2. Main Menu > Solution > FLOTRAN Set Up > Execution Ctrl > a dialog box appears in

that Enter in the first row “Global iterations EXEC” = 200

3. Main Menu > Solution > Run FLOATRAN

When the solution is complete, close the „Solution is Done!‟ window.

Step10: Post processing

We can now plot the results of this analysis and also list the computed values.

Main Menu > General Postproc > Read Results > List Set

General Postproc > Plot Results > Contour Plot > Nodal Solu

Select DOF Solution and Fluid Velocity and click OK

49

Fig.10.5 VELOCITY FLOW PATTERN

Fig.10.6 PRESSURE FLOW PATTERN

50

Next, go to Main Menu > General Postproc > Plot Results > Vector Plot > Predefined.

General Postproc > Plot resulst > Contour Plot > Nodel Sol

Select DOF Solution and Pressure and Click OK.

RESULT:

Simulation of steady state fluid flow analysis of the given component using ANSYS

software are performed and the results are plotted.

51

EX.NO.11: STUDY OF BASICS OF MATLAB

DATE:

AIM:

To study the basic operations, commands and codes of MATLAB.

INTRODUCTION

This chapter is a brief introduction to MATLAB ( an abbreviation of MATrix LABoratory)

basics, registered trademark of computer software, version 4.0 or later developed by the Math

Works Inc. The software is widely used in many of science and engineering fields, MATLAB is

an interactive program for numerical computation and data visualization. MATLAB is supported

on unix, macintosh, and windows environments. for more information on MATLAB, contact the

Math Work.com. a windows version of MATLAB is assumed here. The syntax is very similar

for the dos version.

MATLAB integrates mathematical computing, visualization, and a powerful language to provide

a flexible environment for technical computing. The open architecture makes it easy to use

MATLAB and its companion products to explore data, create algorithms, and create custom

tools that provides early insights and competitive advantages.

Known for its highly optimized matrix and vector calculations, MATLAB offers on intuitive

languages for expressing problems and their solutions both mathematically and visually.

Typically uses include:

o Numeric computation and algorithm development

o Symbolic computation (with the built-in symbolic math function)

o Modeling, simulation, and prototyping

o Data analysis and signal processing

o Engineering graphics and scientific visualization

In this chapter, We will introduce the MATLAB environment, We will learn how to create , edit,

save, run, and debug m-files (ASCII files with series of MATLAB statements). We will see how

to create arrays (matrices and vectors), and explore the built-in MATLAB linear algebra

functions for matrix and vector multiplication, do not cross products, transpose, determinates,

and inverses and for the solution linear equations. MATLAB is based on the language c, but is

generally much easier to use. We will also see how to program logic constructs and oops in

MATLAB, how to use subprograms and functions, and how to use comments (%) for explaining

the programs and tabs for easy readability, and how to print and plot graphics both two and three

dimensional MATLAB's functions for symbolic mathematics are presented. use of these

functions to perform symbolic operations, to develop closed from expression for solution to

52

algebraic equations, ordinary differential equations , and system equation for presented .

Symbolic mathematics can be used to determine analytical expression for the derivative and

integral of an expressions.

Starting and Quitting MATLAB

To start MATLAB click on the MATLAB icon or type in MATLAB , followed by pressing the

enter or return key at the system prompt. the screen will produce the MATLAB prompt >>(or

EDU>>), which indicates that MATLAB is waiting for a command to be entered.

in order to quit MATLAB, type quit or exit after prompt , followed by pressing the enter or

return key.

Display Windows

MATLAB has three display windows. They are

1. A command Windows which is used to enter commands and data to display plots and graphs.

2.A Graphic Windows which is used to display plots and graphs

3. A Edit window which used to crate and modify m-files. m-flies are the files that contain a

program are script of MATLAB commands.

Entering Commands

Every command has to be followed by a carrier return <cr> (enter key) in order that the

command can be executed. MATLAB command are case sensitive and lower case letters are

used throughout.

To execute an M-files (such as Project_1.m ). simply enter the name of the file without its

extension (as in project_1).

MATLAB expo

In order to see some of the, MATLAB capabilities , enter the demo command this will initiate

the MATLAB expo , MATLAB Expo is a graphical demonstration environment that shows the

some of the different type of operations which can be conducted with MATLAB.

Abort

In order to abort a command in MATLAB , hold down the control key and press c to generate a

local abort with MATLAB.

The Semicolon (;)

53

If a semicolon (;) is typed at the end of the command , the output of the command is not

displayed.

Typing %

When percent symbol (%) is typed in the beginning of a line, the line is designated as a

command . W.....hen the enter key is pressed , the line is not executed.

The clc Command

The typing clc command and pressing enter cleans the command windows. once the clc

command is executed a clear windows is displayed.

Help

The MATLAB has a host of bulit in fuctions. For complete list , refer to MATLAB user's guide

or refer to online Help . To obtain the help on the particular topic in the list, eg. inverse , type

help inv.

Statements and Variables

Statements have the form

>> variable = expression

the equal sign implies the asssign ment of the expression to the variable. for instance to enter a

2x2

matrix with a variable name A, we write

>> A= ={1 2 ; 3 4 } <ret>

the statement is executed after the carriage return (or enter ) key is pressed to display

A=

1 2

3 4

ARITHMATIC OPERATION

The symbols for arithmetic operations with scalars are summarized below in tables 2.1

Table

54

Arithmatic operation Symbol Example

Addition + 6+3=9

Subtraction - 6-3=3

Multiplication x 6*3=18

Right division / 6/3=2

Left division \ 6\3=3/6=1/2

Exponentiation ^ 6^3(63=261)

Table Common Math Functions

Function Description

abs(x) Computes the absolute value of x

sqrt(x) Computes the square root of x

round(x) Rounds x to nearest integer

fix(x) Rounds (or truncates ) x to the nearest integer

floor(x) Rounds x to the earest integer toward -∞

ceil(x) Rounds x to the earest integer toward ∞

sign(x) Returns a value of -1 if x is less than 0, a value

of 0 if equals 0, and a value of 1 otherwise

rem(x,y) Returns the remainder of x/y. for example,

rem(25,4) is 1, and rem (100,21) is 16. This

function is also called a modulus function

exp(x) Computes ex, where e is the base for natural

logarithms , or approximately 2.718282

log(x) Computes ln x, the atural logarithm of x to the

base e.

log10(x) Computes log10 x, the common logarithm of x

to the base 10

55

Table Exponential Functions


exp(x) Exponential (ex)

log(x) Natural logarithm

log10(x) Base 10 logarithm

sqrt(x) Square root

Table trigonometric and hyperbolic functions


sin(x) Computes the sine of x, where x is in radians

cos (x) Computes the cosine of x, where x is in

radians

tan(x) Computes the tangent of x, where x is in

radians

asin(x) Computes the arcsine or inverse sine of x,

where x is must be between -1 and 1. the

function returns an angle oin radians between

- and

acos(x) Computes the arccosine or inverse cosine of

x, where x is must be between -1 and 1. the

function returns an angle oin radians between

0 and π

atan(x) Computes the arctangent or inverse tangent

of x. the function returns an angle oin radians

between - and

56

atan2(y,x) Computes the arctangent or inverse tangent

of y/x. the function returns an angle oin

radians between x and y

sinh(x) computes the hyperbolic sine of x,which is

equal to ex-e

-x/2

cosh(x) computes the hyperbolic cosine of x,which is

equal to ex+e

-x/2

tanh(x) compute the hyperbolic tangent of x, which is

equal to sinhx/coshx

asinh(x) compute the hyperbolic sine of x, which is

equal to x+

acosh(x) compute the hyperbolic cosine of x, which is

equal to x+

atanh(x) compute the inverse hyperbolic tangent of x,

which is equal to

Table online help

function description

help lists topics on which help is available

helpwin opens the interactive help window

helpdesk opens the web browser based help facility

helptopic provides help on topic

lookfor string lists help topics containing string

demo runs the demo program

Workplace information

function description

57

who lists variables currently in the workplace

whos lists variables currently in the workplace with

their size

what lists m, mat, and mex- files on the disk

clear clears the workplace, all variables are

removed

clear x y z clears only variables x, y, and z.

clear all clears all variables and functions from

workplace

mlock fun looks fun so that clear cannot remove it

munlock fun unlocks function fun so that clear can remove

it

clc clear command window, command histo0ry

is lost

home same as clc

clf clears figure window

ARRAYS

An array ia a list of numbers arranged in rows and or columns. a one dimensional arrray is a row

or a column of numbers and a two dimensional array has a set of numbers arranged in rows and

columns. an array operation is performed element by element.

ROW VECTOR

A vector is a row or column of elements.

In a row vector the elements are entered with a space or a comma between the elements inside

the square brackets. for example,

x=[7 -1 2 -5 8]

COLUMN VECTOR

58

In a column vector the elements or enterd with asemicolun between the elements inside the

square brackets. for example,

x= [7; -1; 2; -5; 8;]

MATRIX

A matrix is a two dimensional array which has numbers in rows and columns. A matrix is a

enterd a row wise with consecutive elements of a rows sepearted by a space or a comma, and the

rows seperated by a semi colon or carriage returns, the entire matrix is enclosed with in square

brackets, in the elements of matrix may be real numbers or comples numbers. for example to

enter the matrix

A=

The mat lab input command is

A=[1 3 -4; 0 -2 8]

similarly for complex number elements of a matrix B

B=

The matlab input command is

B=[-5*x log(2*x)+7*sin(3*y); 3i 5-13i ]

ADDRESING ARRAYS

A colon can be used in mat lab to addres a range of elements in a vector or a matrix

COLON FOR A VECTOR

Va(:) - refers to all the elements of the vector Va (either a row or a column vector)

Va(m:n) - refers to a elements m through n of the vector Va

For instance

>> V =[2 5 -1 11 8 4 7 -3 11]

>> U= V(2:8)

59

U= 5 -1 11 8 4 7 -3 11

COLUMN FOR A MATRIX

Table gives the use of colon in addresing arrays in a matrix\

colon use for a matrix

Command Description

A(:,n)

A(n,:)

A(:,m:n)

A(m:n, :)

A(m:n, p:q)

refers to the elements in alll the rows of a coloumn n of the matrix A

Refers to the elements in all the columns of row n of the matrix A

Refers to the elements in all the rows between coloumns m and n of the

matrix A

Refers to the elements in all the coloumns between rows m and n of the

matrix A

Refers to the elements in all the rows m through n and coloumns p through

q of the matrix A

Adding elements to a vector or a matrix

A variable that exists as a vector, or a matrix, can be changed by adding elements to it. Addition

of elements is done by assigining values of the additional elements, or by appending existing

varibales . rows and or coloumns can be added to an existing matrix by assigning values to the

new rows or columns.

DELETING ELEMENTS

An element, or a range of elements. of an existing variable can be deleted by reassigning blanks

to these elements. this is done simply by the use of square brackets with nothing typed in

between them.

BUILT-IN FUNCTIONS

Some of the built-in functions available in MATLAB for managing and handling arrays as listed

Built-in functions for handling arrays


60

Length(A) Returns the number of elements in the vector A.

Size(A) Returns a row vector [m,n], Where m and n are the size mxn of the array A.

Reshape(A,m,n) Rearrange a matrix A that has r rows and a coloumns to have m rows and n

columns r times s must be equal to m times n.

Diag(v) When v is a vector, create a square matrix with the elements of v in the

diagonal

Ia (A0 When A is a matrix, creates a vector from the diagonal of A.

2.10 OPERATIONS WITH ARRAYS

We consider here matrices that have than one row and more than one column

ADDITION AND SUBTRACTION OF MATRICES

The addition ( the sum) or the subtraction (the difference) of the two arrays is obtained by adding

or subtracting their corresponding elements. These operations are performed with arrays of

identiacal size (same number of rows and columns.)

for example if A and B are two arrays (2x3 matrices)

A = and B=

then, the matrix addition (A+B) is obtained by adding A and B is

DOT PRODUCT

The product is a scalar computed from two vectors of the same size. the scalar is the sum of the

products of the values in corresponding positions in the vectors.

for n elements in the vectors A and B

dot product = A.B =

61

dot (A,B) computes the dot product of A and B. if A and B are matrices , the dot product is a

vector containing the dot products for the corresponding columns of A and B

ARRAY MULTIPLICATION

The value in positon ci,j of the product C of two matrices , A and B, is the dot product of row I of

the first matrix and column j of the second matrix:

Ci,j =

Array division

The division operation can be explained by means of the identity matrix and the inverse matrix

operation .

Identity matrix

An identity matrix is a square matrix in which all the diagonal elements are 1‟ds, and the

remaining elements are 0‟s. if a matrix A is square, then it can be multiplied by the identity

matrix, I, from the left or from the right

AI=IA=A

Inverse of a matrix

The matrix B is the inverse of the matrix A if when the two matrices are multiplied the product is

the identity matrix. Both matrices A and B must be square and the order of the multiplication can

be AB or BA

AB=BA=I

Transpose

The transpose of a matrix is a new matrix in which the rows of the orginal matrix are the

columns of the new matrix. The transopse of a given matrix A is denoted by AT. in MATLAB,

the transpose of the matrix A is denoted by A‟

Determinant

A determinant is a scalar computed from the entries in a square matrix. For a 2x2 matrix A, the

determinant is

|A| = a11a22-a21a12

62

MATLAB will compute determinant of a matrix using the det function:

Det (A) computes the determinant of a suare matrix A .

Array Division

MATLAB has two types of array divisions, which are the left division and the right

division.

Left division

The left division is used to solve the matrix equation Ax=B where x and B are column

vectors.

Multiplying both sides of this equation by inverse of A, A-1

, we have

A-1

Ax=A

-1 B

or Ix = ᵪ = A-1

B

In MATLAB, the above equation is written by using the left division character:

ᵪ = A\B

Right Division

The Right division is used to solve the matrix equation xA = B where x and B are row

vectors.

Multiplying both sides of this equation by the inverse of A, A-1

, we have

ᵪ.A A-1

= B.A-1

ᵪ = B . A-1

or

In MATLAB, this equation is written by using the right division character:

ᵪ = A / B

Eigenvalues and Eigenvectors

Consider the following equation,

63

AX = λX

Where A is an n X n square matrix, X is a column vector with n rows and λ is a scalar.

The values of λ for which X are nonzero are called the eigenvalues of the matrix A, and

The corresponding values of X are called the eigenvalues of the matrix A .

Equation (2.1) can also be used to find the following equation

(A – λI) X = 0

Where IIs an n x n identity matrix. Equation (2.2) corresponding to a set of homogeneous

Equation and has nontrivial solutions only if the determinant is equal to zero, or

│A – λI│ = 0

Equation (2.3) is known as the characteristic equation of the matrix A. The solution to

Eq. (2.3) gives eigenvalues of the matrix A.

MATLAB determines both the eigenvalues and ei8genvectors for a matrix A.

eig (A) Computes a column vector containing the eigenvalues of A.

[Q,d] = eig(A)Computes a square matrix Q containing the eigenvectors of A as columns

And a square matrix d containing the eigenvalues (λ) of A on the diagonal.

The values of Qand d are such thatQ*Qis the identity matrix and A*X

Equals (λ) times X.

Element – by element operations

Arithmetic operations

Matrix operators

Array operators

+ Addition

+ Addition

64

- Subtraction

* Multiplication

^ Exponentiation

/ Left division

\ Right division

- Subtraction

.* Array Multiplication

.^ Array Exponentiation

./ Array Left division

.\ Array Right division

Matrix Division

The solution to the matrix equation AX = B is obtained using matrix division , or X =

A/B. The vector X then contains the values of x

Matrix Inverse

For the solution of the matrix equation AX = B , we premultiply both sides of the

equation by A-1

A-1

AX = A-1

B

Or IX = A-1

B

Where I is the identity matrix

Hence X = A-1

B

In MATLAB , we use the command x = inv (A)*B. Similarly , for XA = B, we use the

command x = B*inv (A)

65

The basic computional unit in MATLAB is the matrix . A matrix expression is enclosed

in square brackets , [ ].Blanks or commas seprate the column elements , and semisolons or

carriage return separate the rows.

Matrix operations require that the matrix dimensions be compatible . If A is an n * m and

B is a p * r then A ± B is allowed only if n = p and m = r . similarly , matrix product A*B is

allowed only if m = p.

SCRIPT FILES

A script is a sequence of ordinary statement and functions used at the command prompt

level. A Script is invoked at the command prompt level by typing the file-name or by using the

pull down menu. Script can also invoke other scripts.

The commands in the command window cannot be saved and executed again. Also, the

command window is not interactive. To overcome these difficulties, the procedure is first to

create a file with a list of command, save it, and then run the file. In this way the commands

contained are executed in the order they are listed when the file is run. In addition, as the need

arises, one can change or modify the commands in the file, the file can be saved and run again

the file that are used in this fashion are known as script file. Thus, a script file is a text file that

contains a sequence of MATLAB commands. Script file can be edited (corrected and/or

changed)and executed many times.

Creating and Saving a script File

Any text editor can be used to create script files. In MATLAB script file are created and edited in

the Editor/Debugger window .this window can be opened from the command window, select file,

New, and then M-file. Once the window is open, the commands of the script file are typed line.

The commands can also be typed in any text editor or word processor program and then copied

and pasted in the edit/debugger window. The second type of M-File is the function file. Function

file enables the user to extend the basic library function by adding one‟s own computational

procedures. Function M-files are expected to return one or more results. Script files and function

files may include reference to other MATLAB toolbox routines.

MATLAB function file begins with a header statement of the form:

Function (name of result or results)=name (argument list)

Before a script file can be executed it must be saved. All script files must be saved with the

extension “.m”. MATLAB refers to them as m-files. When using MATLAB M-files editor, the

files will automatically be saved with the “.m” extension. If any other text editor is used, the file

must be saved with the “.m” extension, or MATLAB will not be able to find and run the script

file. This is done by choosing save As… from the file menu, selecting a location, and entering a

66

name for the file. The names of user defined variables, predefined variables, MATLAB

commands or functions should not be used to name script files.

Running a script file

A script file can be executed either by typing its name in the command window and then

pressing the enter key, directly from the Editor Window by clicking on the Run icon. File The is

assumed to be in the current directly, or in the search path.

Input to a script file

There are three ways of assigning a value to variable in a script file.

1. The variable is defined and assigned value in the script file

2. The variable is defined and assigned value in the command windows

3. The variable is defined in the script file, but a specified value

Output commands

There are two commands that are commonly used to generate output. They are the disp

and fprintf commands.

1. The disp command

The disp command displays the elements of a variable without displaying the name of the

variable, and displays text

2. The fprintf command

The fprint command displays output (text and data) on the screen or saves it to a file.

The output can be formatted using this command

PROGRAMMING IN MATLAB

One most significant of MATLAB is its extendibility through user-written programs such as the

M-files. M-files are ordinary ASCII text files written in MATLAB language. A function file is a

subprogr

Relational and logical operators

A relational operator compares two numbers by finding whether a comparison statement is true

or false. A logical operator examines true/false statements and produces a result which is true or

false according to the specific operator. Relational and logical operators are used in mathematical

67

expressions and also in combination with other commands, to make decisions that control the

flow of a computer program.

MATLAB has six relational operators as shown in table

Relational operators

Relational operators

Interpretation

<

<=

>

>=

==

=

Less than

Less than or equal

Greater than

Greater than or equal

Equal

Not equal

The logical operators in MATLAB are shown in table

Logical operators

Logical operator Name Description

&

Example: A&B

AND Operators on two operands (A and B). If both are true, the result

is true (1), otherwise the result is false (0).

I

Example: AIB

OR Operators on two operands (A and B). If either one, or both are

true, the result is true (1), otherwise (both are false) the result is

false (0).

˞

Example A

NOT Operators on one operand (A). Give the opposite of the

operand. True (1) if the operand is false, and false (0) if the

operand is true.

68

Order of precedence

The following table shows the order of precedence used by MATLAB.

Precedence Operation

1 (highest)

2

3

4

5

6

7

8 (lowest)

Parentheses (If nested parentheses exist, inner have precedence).

Exponentiation

Logical NOT ( )

Multiplication, Division.

Addition, Subtraction.

Relational operators ( , , =, =,==, =)

Logical AND (&).

Logical OR (I)

2.16.3 Built-in logical Functions

The MATLAB built-in functions which are equivalent to the logical operators are:

And (A, B) Equivalent to A & B

Or (A, B) Equivalent to A I B

Not (A) Equivalent to A

List of the MATLAB logical built-in functions are described in table

GRAPHICS

MATLAB has many commands that can be used to create basic 2-D plots, overlay plots,

specialized 2-D plots, 3-D plots, mesh, and surface plots.

Basic 2-D plots

The basic command for producing a simple 2-D plots is

69

Plot(xvalue y values, „style option‟)

Where x value and y values are vectors containing the x-and y-coordinates of points on the

graph.

Style option is an optional argument that specifies the color, line-style, and the point-

marker style.

The style option in the plot command is a character string that consists of 1,2 or 3 characters that

specify the color and/or the style. The different color, line-style and marker style option are

summarized in table

color, line-style, and marker-style options

Color style-option Line style-option Marker style-option

Y yellow

M magenta

C cyan

R red

G green

B blue

W white

R black

- Solid

- - dashes

; dotted

-. Dash-dot

+ plus sign

0 circle

* asterisk

X x-mark

. point

^up triangle

S square

D diamond, etc.

Specialized 2-D plots

There are several specialized graphics function available in MATLAB for 2-D plots. The list of

functions commonly used in MATLAB for plotting x-y data are in Table.

List of function for plotting x-y data


70

Area

Bar

Barh

Comet

Compass

Contour

Contourf

Errobar

Feather

Fill

Fplot

Hist

Loglog

Pareto

pcolor

Creates a filled area plot.

Creates a bar graph.

Creates a horizontal bar graph.

Makes an animated 2-D plots.

Creates arrow graph for complex number.

Makes contour plots

Makes filled contour plots.

Plots a graph and puts error bars.

Makes a feather plot.

Draws filled polygons of specified color.

Plots a function of a single variable.

Makes histograms.

Creates plots with log scale on both x and y axes.

Makes pareto plots.

Makes pseudo color plot matrix

command Description

Pie

Plotyy

Plotmatrix

Polar

Quiver

Rose

Scatter

Creates a pie chart.

Makes a double y-axis plot.

Makes a scatter plot of a matrix.

Plots curves in polar coordinates.

Plots vector fields.

Makes angled histograms.

Creates a scatter plot.

71

Semilogx

Semilogy

Stairs

stem

Makes semilog plot with log scale on the x-axis.

Makes semilog plot with log scale on the y-axis.

Plots a stair graph.

Plots a stem graph.

Overlay plots

There are three ways of generating overlay plots in MATLAB, they are

(a)Plot command

(b)Hold command

(c)Line command

(a)Plot command. Example E2.7(a) shows the use of plot command used with matrix

argument, each column of the second argument matrix plotted against the corresponding column

of the first argument matrix.

(b)Hold command . invoking hold on at any point during a session freezes the current plot in

the graphics window. All the next plots generated by the plot command are added to the exiting

plot. See Example E2.7(a).

(c)Line command. The line command takes a pair of vectors (or a triplet in 3-D) followed by

a parameter name/parameter value pairs as argument. For instance, the command: line (x data, y

data, parameter name, parameter value) adds lines to the existing axes. See Example E2.7(a).

3-D plots

MATLAB provides various options for displaying three-dimensional data. They include line and

wire, surface, mesh plots, among many others. More information can be found in the Help

window under plotting and Data visualization. Table lists commonly used function.

Functions used for 3-D graphics

command Description

Plot3

Plots three-dimensional graph of the trajectory of a set of thee

parametric equations x(t), y(t), and z(t) can be obtained using

plot3(x,y,z).

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Meshgrid

Mesh(X,Y,z)

If x and y are two vectors containing a range of points for the evaluation

of a function, [X,Y]=meshgrid(x,y) returns two rectangular matrices

containing the x and y values at each point of a two-dimensional grid.

If X and Y are rectangular arrays containing the values of the x and y

coordinates at each point of a rectangular grid, and if z is the value of a

function evaluated at each of these points, mesh(X,Y,z) will produce a

three-dimensional perspective graph of the points. The same results can

be obtained with mesh(x,y,z) which can also b used.

command Description

Mesch,meshz

Surf

Surfc

Colormap

If the xy grid is rectangular, these two function are merely variations of

the basic plotting program mesh, and they operate in an identical

fashion. Meshc will produce a corresponding contour plot drawn on the

xy plane below the three dimensional figure, and meshz will add a

vertical wall to the outside features of the figures drawn by mesh.

Produces a three-dimensional perspective drawing. Its use is usually to

draw surfaces, as opposed to plotting functions, although the actual

tasks are quite similar. The output of surf will be a shaded figure. If

row vectors of length n are defined by x=y cos and y=r sin , with 0_<

2they correspond to a circle of radius r. if r is column vector equal to

r=[0 1 2]‟;then z=r*ones(size(x)) will be a rectangular, 3 x n,arrays of

0‟s and 2‟s, and surf(x,y,z) will produce a shaded surface bounded by

three circles; i.e., a cone.

This function is related to surf in the same way that meshc is related to

mesh.

Used to change the default coloring of a figure. See the MATLAB

reference manual or the help file.

Controls the type of color shading used in drawing figures. See the

MATLAB reference manual or the help file.

View(az,el) controls the perspective view of a three-dimensional plot.

The view of the figure is from angle ”el” above the xy plane with the

coordinate axes (and the figure) rotated by an angle “az” in a clockwise

direction about the z axis. Both angles are in degrees. The default

73

View

Axis

Contour

Plot3

values are az=371/2 and el=30

Determines or changes the scaling of p plot. If the coordinate axis

limits of a two dimensional of three-dimensional graph are contained in

the row vector r=[] axis will return the values in this vector, and axis(r)

can b used to alter them. The coordinate axes can be turned on and off

with axis(„on‟) and axis(„off‟). A few other string constant inputs to

axis and their effects are given below:

Axis(„equal‟) x and y scaling are forced to be the same.

Axis(„square‟) The box formed by the axes is square.

Axis(„auto‟) Restores the scaling to default settings.

Axis(„normal‟) Restoring the scaling to full size, removing any effects

of

square or equal settings.

Axis(„image‟) Alters the aspect ratio and the scaling so the screen

pixels are

square shaped rather than rectangular.

The use is contour(x,y,z). A default value of N=10 contour lines will be

drawn. An optional fourth argument can be used to control the number

of contour lines that are drawn. Contour(x,y,z,V), if V is a vector

containing values in the range of z values, will draw contour lines at

each value of z=V.

Plots line or curves in three dimensions. If x,y, and z are vectors of

equal length, plot3(x,y,z)will draw, on a three-dimensional coordinate

axis system, the lines connecting the points. A fourth argument,

representing the color and symbols to b used at each point can be added

in exactly the same manner as with plot.

Grid on adds grid lines to a to-dimensional or three-dimensional graph;

grid off removes them.

Draws “slices” of a volume at a particular location within the volume.

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Grid

slice

Result :-

Thus the basic commands and codes of MATLAB were studied.

75

EX.NO.12: DETERMINATION OF MATRIX OPERATIONS

DATE:

AIM:

Consider two matrices A = and B = Using MATLAB, determine

the following, (a) A+B, (b) AB, (c) A2, (d) A

T, (e)B

-1, (f) B

TA

T, (g) A

2+B

2-AB, and (h)

determinant of A, determinant of B and determinant of AB

SOLUTION

>> A=[1 0 1;2 3 4;-1 6 7]

A =1 0 1

2 3 4

-1 6 7

>> B=[7 4 2;3 5 6;-1 2 1]

B =7 4 2

3 5 6

-1 2 1

>> C=A+B

C =8 4 3

5 8 10

-2 8 8

>> D=A*B

D =6 6 3

19 31 26

76

4 40 41

>> E=A^2

E =0 6 8

4 33 42

4 60 72

>> %Let F=transpose of A

>> F=A'

F = 2 -1

0 3 6

1 4 7

>> H=inv(B)

H = 0.1111 0.0000 -0.2222

0.1429 -0.1429 0.5714

-0.1746 0.2857 -0.3651

>> J=A'*B'

13 7 2

24 51 12

37 65 14

>> I=B'*A'

I =6 19 4

6 31 40

3 26 41

>> K=A^2+B^2-A*B

K =3 52 45

77

15 51 58

-2 28 42

>> det(A)

ans = 12

>> det(B)

ans = -63.0000

>> det(A*B)

ans = -756.0000

Result: - (a) A+B =

(b) AB =

(c) A2=

(d) AT

=

(e)B-1

=

(f) BTA

T =

(g) A2+B

2-AB =

(h) Determinant of A=

Determinant of B =

Determinant of AB =

78

EX.NO.13: DETERMINATION OF EIGEN VALUES AND EIGEN

VECTORS

DATE:

AIM:

To determine the eigenvalues and eigenvectors of A and B using MATLAB

A = and B =

Solution

>> A=[4 2 -3; -1 1 3; 2 5 7]

A =4 2 -3

-1 1 3

2 5 7

>> B=[1 2 3; 8 7 6; 5 3 1]

B =1 2 3

8 7 6

5 3 1

>> eig(A)

ans =0.5949

3.0000

8.4051

>> lamda=eig(A)

lamda = 0.5949

3.0000

79

8.4051

>> [v,d]=eig(A)

v = -0.6713 0.9163 -0.3905

0.6713 -0.3984 0.3905

-0.3144 0.0398 0.8337

d = 0.5949 0 0

0 3.0000 0

0 0 8.4051

Result:-

Eigenvalues =

Eigenvectors =

80

EX.NO.14: SOLUTION OF SIMULTANEOUS EQUATIONS

DATE:

AIM:

Solve the following set of equations using MATLAB

x1+2x2+3x3+5x4=21

-2x1+5x2+7x3-9x4=18

5x1+7x2+2x3-5x4=25

-x1+3x2-7x3+7x4=30

Solution

[A][x] = [B]

x =

[x] = [A]-1

B

>> A=[1 2 3 5; -2 5 7 -9; 5 7 2 -5; -1 3 -7 7]

A= 1 2 3 5

-2 5 7 -9

5 7 2 -5

-1 3 -7 7

>> B=[21; 18; 25; 30]

B = 21

18

25

30

81

>> X=inv(A)*B

X = -1.5130

5.9108

0.3762

1.9126

>> check=A*X

check =

21.0000

18.0000

25.0000

30.0000

Result:-

=

83

EX.NO.15: PLOT 2-D AND 3-D GRAPHS

DATE:

AIM:

To plot the function z= 2-1.5

over the domain -4 ≤ x ≤ 4 and -4 ≤ y ≤ 4.

(a) Mesh plot

(b) Surface plot

(c) Mesh curtain plot

(d) Mesh and contour plot

(e) Surface and contour plot

(f) Surface plot with lighting

(g) Water fall plot

(h) 2-D contour plot

(i) 3-D contour plot

Solution :-

>> %(a) Mesh Plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;

>> [x,y]=meshgrid(x,y);

>> z=2^(-1.5*sqrt(x^2+y^2))*cos(0.5*y)*sin(x);

>> mesh(x,y,z)

84

(b) Surface Plot

(c) Mesh curtain plot

85

>> %(b) Surface Plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> surf(x,y,z)

>> %(c) Mesh curtain plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> meshz(x,y,z)

86

(d) Mesh and contour plot

(e) Surface and contour plot

87

>> %(d) Mesh and contour plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> meshc(x,y,z)

>> %(e) Surface and contour plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> surfc(x,y,z)

88

(f) Surface plot with lighting

(g) Water fall plot

89

>> %(f) Surface plot with lighting

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> surfl(x,y,z)

>> %(g) Water fall plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> waterfall(x,y,z)

90

(h) 2-D Contour plot

(i) 3-D Contour plot

91

>> %(h) 2-D Contour plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> contour(x,y,z,15)

>> %(i) 3-D Contour plot

>> x=-4:0.25:4;

>> y=-4:0.25:4;



>> contour3(x,y,z,15)

Result:-

The following plots were drawn Mesh plot, Surface plot, Mesh curtain plot, Mesh and

contour plot, Surface and contour plot, Surface plot with lighting, Water fall plot, 2-D

contour plot and 3-D contour plot

92

EX.NO.16: SIMULATION OF HEAT EXCHANGER – NTU METHOD

DATE:

AIM:

A 4 Kg/s product stream from a distillation column is to be cooled by a 3Kg/s water

stream in a counter flow heat exchanger. The hot and cold stream inlet temperatures are 400 and

300 K, respectively and the heat transfer area of the heat exchanger is 30 m2. If overall heat

transfer co efficient is estimated to be 820 W/m2K, determine the product stream outlet

temperature. The specific heat of product stream can be taken to be 2500 J/KgK

Matlab program

%input is

u=820; % Overall heat transfer coefficient in W/m2/k

a=30; % Area of heat exchanger in m2

m_c=3; % Mass flow rate of cooling fluid in Kg/s

m_h=4; % Mass flow rate of hot fluid in Kg/s

c_ph=2500; % Specific heat of hot fluid in Kg/Kj/K

c_pc=4180; % Specific heat of cold fluid in Kg/Kj/K

t_cin=300; % Temperature of Cooling inlet in K

t_hin=400; % Temperature of Hot inlet in K

% Formula is

c_hmin=m_h*c_ph; % Minimum specific heat in Kg/Kj/K

c_cmax=m_c*c_pc; % Maximum specific heat in Kg/Kj/K

r_c=c_hmin./c_cmax; % Capacity ratio

n_tu=u*a./c_hmin; % Number of transfer unit

E=(1-exp((-n_tu)*(1-r_c)))./(1-(r_c)*exp((-n_tu)*(1-r_c))); % Effectiveness of heat

exchanger

t_hout=t_hin-(E*c_hmin*(t_hin-t_cin))./c_hmin; % Temperature of Hot outlet in K

93

t_cout=t_cin+(c_hmin*(t_hin-t_hout))./c_cmax; % Temperature of Cooling

outlet in K

% Output is

c_hmin = 10000

c_cmax = 12540

r_c = 0.7974

n_tu = 2.4600

E = 0.7613

t_hout = 323.8736

t_cout = 360.7069

Result:-

(a) Number of transfer unit =

(b) Effectiveness of heat exchanger =

(c) Temperature of Hot outlet in K =

(d) Temperature of Cooling outlet in K =

94

EX.NO.16: SIMULATION OF HEAT EXCHANGER – NTU METHOD

DATE:

AIM:

The flow rates of hot and cold water streams running through a parallel flow heat

exchanger are 0.2 Kg/s and 0.5 kg/s respectively. The inlet temperatures on the hot and cold

sides are 75oC and 20

oC respectively. The exit temperature of hot water is 45

oC. If the individual

heat transfer coefficients on both sides are 650 W/m2o

C, calculate the area of heat exchanger.

Matlab program

>> %input is

m_h=0.2; % Mass flow rate of cooling fluid in Kg/s

m_c=0.5; %Mass flow rate of cooling fluid in Kg/s

t_hin=75; % Temperature of Hot inlet in K

t_hout=45; %Temperature of Hot outlet in K

t_cin=20; %Temperature of Cooling inlet in K

h_in=650; %heat transfer coefficient inlet in w/m2/oC

h_out=650; % heat transfer coefficient inlet in w/m2/oC

c_ph=4187; % Specific heat of hot fluid in Kg/KJ/oC

c_pc=4187; %Specific heat of cold fluid in Kg/KJ/ oC

%formula is

q=m_h*c_ph*(t_hin-t_hout); % heat transfer in watts

t_cout=t_cin+(m_h*c_ph*(t_hin-t_hout))./(m_c*c_pc);% Temperature of Cooling outlet in oC

l_mtdpf=(t_hin-t_cin)-(t_hout-t_cout)./log((t_hin-t_cin)./(t_hout-t_cout)); %log mean temp. diff.

u=(h_in*h_out)./(h_in+h_out); %Overall heat transfer coefficient in W/m2/k

a=q./(u*l_mtdpf); %area of heat exchanger in m2

% Output is

95

q = 2.5122e+04

t_cout = 32

l_mtdpf = 45.9871

u = 325

a =21.6809

Result:-

(a) heat transfer in watts =

(b) Temperature of Cooling outlet in oC =

(c) LMTDPF =

(d) Overall heat transfer coefficient in W/m2/k =

(e) area of heat exchanger in m2 =

simulation record

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