times me report

HEAT TRANSFER ANALYSIS OF FINS

CHAPTER 1

INTRODUCTION

1.1 Extended Surface

In the design & construction of various types of heat transfer equipment, simple shapes such as cylinders, bars & plates are used to implement the flow of heat between a source & a sink. They provide heat-absorbing or heat-rejecting surfaces, & each is known as a prime surface. When a prime surface is extended by appendages intimately connected with it, such as the metal tapes & spines on the tubes, the additional surface is known as extended surface. In some disciplines, prime surfaces & there extended surfaces are known collectively as extended surfaces to distinguish them from prime surfaces used alone. The elements used to extend the prime surfaces are known as fins. When the fin elements are conical or cylindrical, they may be referred to as spines or pegs.

Fig 1.1 – Some typical examples of extended surfaces: (a) longitudinal fin of rectangular profile; (b) cylindrical tube equipped with fins of rectangular profile; (c) longitudinal fin of trapezoidal profile; (d) longitudinal fin of parabolic profile; (e) cylindrical tube equipped with radial fin of rectangular profile; (f) cylindrical tube equipped with radial fin of trapezoidal profile; (g) cylindrical spine; (h) truncated conical spine; (i) truncated parabolic spine.

1.2 Fins

Three quarters of a century ago, a paper by Harper and Brown (1922) appeared as an NACA report. It was an elegant piece of work and appears to be the first really significant attempt to provide a mathematical analysis of the interesting interplay between convection and conduction in and upon a single extended surface. Harper and Brown called this a cooling fin, which later became known merely as a fin.

The convective removal of heat from a surface can be substantially improved if we put extensions on that surface to increase its area. These extensions can take a variety of forms. Figure below, for

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example, shows many different ways in which the surface area can be extended with protrusions of a kind we call fins.

Fig 1.2 – Eight examples of externally finned tubing 1) & 2) typical commercial circular fins of constant thickness; 3) & 4) serrated circular fins & dimpled spirally-wound circular fins, both intended to improve convection; 5) spirally-wound copper coils outside & inside; 6) & 8) bristle fins, spirally wound & machined from base metal; 7) a spirally intended tube to improve convection & increase surface area.

1.3 Fin Materials

The target of this project is to find the Heat Dissipation rate of the fins having different shapes. The Heat Dissipation rate mainly depends on following factors

1. Temperature difference between source & sink of heat.

2. Thermal conductivity of material

3. Surface area of Fins

4. Shape of fins

Different materials have different thermal conductivities. The commonly used materials for fins are

1. Aluminum – It has a thermal conductivity of about 205 W/mK. The production of aluminum heat sinks is inexpensive. It is also very light.

2. Copper – It has a thermal conductivity of about 400 W/mK. Its weight is very high & it is also very expensive.

3. Alpha P7125 – It combines the advantages of both Aluminum & Copper. Here the area in contact with the heat source is made of copper, which helps lead the heat away to the outer parts of the heat sinks.

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1.4 Experiment

In a motorcycle, cooling of excess heat generated inside the engine is done with the help of air. The reason behind that is the lesser space available for the manufacturer to employ some other technique of cooling. But engine on its own cannot be cooled significantly by air-cooling because of lesser surface area exposed to air. Hence fins are used as an extended surface to increase the surface area of the engine exposed to air thus increasing the rate of heat transfer. Fins offers us a practical mean for achieving a large total heat transfer surface area without the use of an excessive amount of primary surface area for good heat transfer efficiency using Air Cooling.

The fins available today are of various profiles & different shapes. In order to achieve maximum amount of cooling/heat transfer, one should know about the efficiency of different shapes of the fins & the properties of different fin materials.

The project here is to study the effect of the fin shape on its heat transfer efficiency. The fins made of mild steel of rectangular & triangular shapes having radial profiles are used here to perform the experiment. Both the fin plates were TIG welded on mild steel cylinders & sealed from one side with asbestos. The heat storing liquid ethylene glycol was filled in both the cylinders. The ethylene glycol was then heated with a 300 W heater. During the heating of the ethylene glycol, a stirrer connected to AC motor of 30-36 rpm was used to rotate the ethylene glycol & to heat it uniformly. J-type thermocouple was used to measure the temperature of ethylene glycol & K-type thermocouples were used to measure the surface temperatures of fins. The heater was connected to a variable transformer to control the voltage supply to the heater. The Ammeter & Voltmeter were also used to show the voltage & current supplied to the heater. The ethylene glycol was heated until it reached a temperature of 120°C & then allowed to cool down to 100°C. When the temperature reached to 100°C, the temperatures at different points of both fin shapes were measured & recorded to find out which fin shape has higher heat dissipation capacity. Along with this, modeling of whole experiment was also done on the ANSYS software & the practical & analytical results were compared. Along with this, it was also tried to find out the solution in FEM using Free Galerikan Method.

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CHAPTER 2

EXPERIMENTAL EQUIPMENT & PROCEDURE

The test rig developed to perform the experiment consists of the following main components:-

1. Test Cylinders

2. Fins

3. Ethylene Glycol

4. Heat Insulator

5. Variable transformer

6. Motor

7. Stirrer

8. Heater

9. Thermocouples

10. Data Logger

11. Ammeter

12. Voltmeter

Figure shows the test rig setup to perform the experiment

Fig 2.1 – Test Rig Setup

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2.1 Test Cylinders

Two cylinders made of mild steel are used to perform the experiment. The dimensions of the cylinders are:-

Length = 200 mm

Thickness = 2.5 mm

Stroke volume = 780 to 825 cm3

Outer Dia. = 75 mm

Inner Dia. = 70 mm

Each cylinder consists of four fins. One cylinder consists of fins having triangular shape & other consists of fins having rectangular shape. The composition of the mild steel used to make the cylinders is:-

Carbon – 2%

Manganese – 1.65%

Copper – 0.6%

Silicon – 0.6%

& small amount of cobalt, chromium, niobium, molybdenum, titanium, nickel, tungsten, vanadium & zirconium. Its specifications are:-

Young’s Modulus – 210 GPa

Density – 4.56 Kg/m3

Thermal Conductivity – 15.5 W/mK

2.2 Fins

Four number of fins made of mild steel are used in each cylinder. Four fins are of triangular shape & four are of rectangular shape. The mild steel used for making fins is same as used for the cylinders. The fins are tightly fitted at the cylinders first & then they are TIG welded with the cylinders.

2.2.1 Triangular Fins

The dimensions of the triangular fins are:-

Length – 50 mm

Base thickness – 5 mm

Edge thickness – 2 mm

Slope Angle – 4°

Four triangular fins are used.

2.2.2 Rectangular Fins

The dimensions of the rectangular fins are:-

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Length – 50 mm

Thickness – 5 mm

Four rectangular fins are used.

2.3 Ethylene Glycol

Ethylene Glycol is an organic compound widely used as automotive antifreeze. In its pure form, it is an odorless, colorless, syrupy, sweet-tasting liquid. Ethylene Glycol is also toxic. Ethylene glycol was used in this experiment as heat storage liquid. The properties of Ethylene Glycol are:-

Freezing Pt. – -12°C

Boiling Pt. – 197°C

Molar mass – 62.068 g/mol

Density – 1113.2 kg/m3

2.4 Heat Insulator

Asbestos is used as heat insulating material at the top & the base of the cylinder. The asbestos is a good insulator of heat and can resist the temperatures up to 1500°C. As the heater was in direct contact with the insulation, the asbestos was used to resist the temperature of heater and to insulate the passage of heat from the cylinders. The asbestos sheet 22 mm in thickness was cut & machined to make the insulating covers for the cylinders.

2.5 Variable Transformer

The variable transformer was used here to control the voltage supplied to the heater & to maintain a constant heating temperature. The transformer is closed type variable transformer. The specifications of variable transformer are:-

Input Voltage – 240 V

Frequency – 50/60 Hz

Output Voltage – 0-270 V

Current limit – 0-4 A

2.6 Motor

AC motor was used in order to connect with the stirrer & to heat the Ethylene Glycol uniformly. The specifications of the motor are:-

Voltage – 220 V

Frequency – 50-60 Hz

Power – 4.5 W

RPM – 30-36 rpm

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2.7 Stirrer

The stirrer was used to rotate the ethylene glycol while it is heated & to make it uniformly heated. The stirrer was made from a hollow pipe of 10 mm dia. with a steel sheet of 1 mm welded with it. The stirrer was connected to the motor with a screw & hence rotated.

2.8 Heater

The heater was used to heat the ethylene glycol. The heater was inserted in the cylinder by making a hole in the insulation cover & inserted in the ethylene glycol inside the cylinder. The specifications of the heater are:-

Length – 6 in.

Power – 300 W

Type – Pencil Heater

2.9 Thermocouples

Thermocouples were used in this experiment to measure the temperature of ethylene glycol & the surface temperatures of fins. Two type of thermocouples were used in this experiment i.e. J-type & K-type. J-Type thermocouple was used to measure the temperature of the ethylene glycol inside the cylinders & K-type thermocouples were used to measure the temperature of the fin surface instantly. The specifications of J-type & K-type thermocouples are:-

2.9.1 J-Type

Type – Rod type

Temperature – 0 to 900°C

Accuracy – ±3°C

2.9.2 K-Type

Type – Wire type

Range – 1300°C

Accuracy – ±1°C

2.10 Data Loggers

Data loggers or we can say temperature indicators were used in this project to show & record the temperatures. Two data loggers were used here. One is connected to the J type thermocouple & one is connected to K type thermocouples. The 1st logger which is connected to the J type thermocouple indicates the changing temperature of the Ethylene Glycol inside the cylinders & the 2nd logger which is connected to the K type thermocouples indicates the surface temperature of the fins.

2.11 Ammeter

Ammeter is used in this project to display the amount of current supplied to the heater.

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Range – 0 to 5 Amp

2.12 Voltmeter

Voltmeter is used in this project to display the voltage supplied to the heater.

Range – 0 to 270 V

2.13 Procedure

Experiment was performed using the following steps.

1. The experimental equipment in fig 1 was set & the cylinder was filled with heat storage liquid.

2. The heat storage liquid was heated using the heater & the stirrer was operated.3. When the heat storage liquid reached a temperature of approximately 120°C, the heater was

turned off.4. The temperature of the heat storage liquid decreased to 100°C by cooling at room

temperature & the temperature was recorded.

The temperature was recorded until it reached room temperature. In order to measure the temperature on the fin surface junctions of K-type thermocouples were placed at the 0, 10, 20, 30, 40 & 50 mm from the fin root in radius.

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CHAPTER 3

ANSYS ANALYSIS

ANSYS was used as a tool to determine the temperature distribution in fin analytically.

Here different steps for modal analysis of fins are given.

3.1 ANSYS Analysis for Rectangular Fin

Step 1: Give the Analysis a Title

After starting the ANSYS program and have entered the GUI, we need to begin the analysis by assigning a title to it. To do so we perform the following tasks:

1. Choosing Utility Menu>File>Change Title. The change title dialog box appears as shown below:

Fig 3.1 – Give Analysis a Title Dialog Box

2. Enter the text Steady State Thermal Analysis of Rectangular Fin

3. Click on Ok.

Step 2: Set Measurement Unit

We need to specify unit of measurement for the analysis. Here we are using the British System of Units (based on inches). To do this type /UNITS,SI in the ANSYS input window and press ENTER.

Step 3: Set Preference for GUI Filtering

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1. Choose Main Menu>Preferences. A Preference for GUI Filtering dialog box appears. Check Thermal and h-Method.

Fig 3.2 – Set Preference dialog box

2. Click on OK.

Step 4: Define the Element Type

Here we are using thermal solid element. To define it we do the following:

1. Choosing Main Menu>Preprocessor>Element Type>Add/Edit/Delete. The element type dialog box appears.

Fig 3.3 – Element type dialog box

2. Click on Add. The Library of Element type dialog box appears.

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3. In the list on the Left, scroll down to pick (highlight) “Thermal Solid” and in the list on right, pick “Quad 8node 77”. As shown below:

Fig 3.4 – Element Library dialog box

4. Click on Ok.

5. Click on Close to close the Element Types dialog box.

Step 5: Defining Material Properties

To define the material properties we followed the following procedure:

1. Choosing Main Menu>Preprocessor>Material Props>Material Models. The Define Material Properties dialog box appears.

2. In the Material Model Available window double click on: Thermal, Density. A dialog box appears as shown below:

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Fig 3.5 – Enter the Value of Density dialog box

3. Enter the value 4.5657 for DENS, Click on OK.

4. In the Material Model Available window double click on: Thermal, Conductivity, Isotropic. A similar dialog box as above appears.

Fig 3.6 – Enter the Value of KXX dialog box

5. Enter the value 15.406 for KXX, Click on OK.

6. Choosing menu path Material>Exit to remove the Material Model Behavior dialog box.

Step 6: Building the Model Geometry:

We followed the following procedure to make the model geometry:

1. Choose Main Menu>Preprocessor> Modeling >Create>Area>Rectangle >By Dimension. A ‘Create Rectangle by Dimension’ dialog box appears as shown below:

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Fig 3.7 – Enter Dimension of Rectangle dialog box

2. Enter the values as 0 for “X1”, .0025 for “X2”, 0 for “Y1”, .2 for “Y2” and then click on Apply.

3. Ansys will Create a rectangle and the create Rectangle by Dimension dialog box will appear again, now enter the values as .0025 for “X1”, .0525 for “X2” .05 for “Y1” and .055 for “Y2”. Again click on Apply.

4. In Create Rectangle by Dimension dialog box, do not change the values for “X1” and “X2” and Enter the values as .08 for “Y1” and .085 for “Y2”. Click on Apply.

5. Again in Create Rectangle by Dimension dialog box, do not change the values for “X1” and “X2” and Enter the values as .110 for “Y1” and .115 for “Y2”.Click on Apply.

6. In Create Rectangle by Dimension dialog box, do not change the values for “X1” and “X2” and Enter the values as .140 for “Y1” and .145 for “Y2”. Click on OK. Finally we get the desired Geometry as shown in Fig below.

Fig 3.8 – Rectangle Geometry

Step 7: Add The Areas:

We followed the following procedure to ADD all the Areas:

1. Choose Main Menu>Preprocessor> Modeling >Operate>Booleans> Add>Areas. The Add Areas picking menu appears.

2. Click on Pick All button and then OK.

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Fig 3.9 – Add Area dialog box

Step 8: Mesh the Model

We followed the following procedure to mesh our Model:

1. Choose Main Menu>Preprocessor>Meshing>Size Cntrls>Manual Size> Global>Size. A Global Element Sizes dialog box appears as shown in Fig. below:

Fig 3.10 – Enter Element Length dialog box

2. Enter the value .00125 for Element Edge Length, and then click on OK.

3. Choose Main Menu>Preprocessor>Meshing>Mesh>Areas>Free. A Mesh Areas Picking menu appears as shown below:

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Fig 3.11 – Mesh Area picking Menu

4. Click on Pick All. In the Graphics Window, ANSYS builds the Meshed model

Fig 3.12 – Meshed Area

Step 9: Applying Loads

We followed the following procedure to Apply Loads on the Model Geometry

1. Choose Main Menu>Solution>Analysis Type>New Analysis. The New Analysis dialog box appears.

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Fig 3.13 – Analysis Type Selection dialog box

2. Click on OK to choose the default analysis type (Steady-state).

3. Choose Main Menu>Solution>Define Loads>Apply>Thermal> Temperature>On Lines. An Apply Temperature on Lines pick menu appears. Pick the inner most line as shown in Fig.

Fig 3.14 – Apply Temp. On Lines picking menu

4. Click OK, an Apply TEMP on Lines Menu appear. Enter the values as 373 for “Load TEMP Value” as shown in Fig.

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Fig 3.15 – Apply Value of Temp. dialog box

5. Click on OK.

6. Choose Main Menu>Solution>Define Loads>Apply>Thermal>Heat Flux> On Lines. An Apply HFLUX on Lines pick menu appears. Pick the inner most line of the Model as shown in Fig.

Fig 3.16 – Apply Heat flux on Lines picking menu

7. Click OK, an Apply HFLUX on Lines Menu appear. Enter the values 87.732 for “Heat Flux.” Click on OK.

8. Choose Main Menu>Solution>Define Loads>Apply>Thermal>Convection >On Lines. An Apply Convection on Lines pick menu appears. Pick the Outer Lines of the Model as shown in Fig.

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Fig 3.17 – Apply Convection on Lines picking menu

9. Click on OK. An Apply Convection on Lines dialog box appears. Enter the values as 68.57 for “Film Coefficient” and 298 for “Bulk Temperature”. Click on OK.

Fig 3.18 – Enter Value of Conv. Dialog box

Step 10: Solving the Model

1. Choose Main Menu>Solution> Solve >Current LS

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Fig 3.19 – Solve Current LS dialog box

2. Choose Close to close the /STAT command window.

3. Click on OK in the Solve Current Load Step dialog box.

4. The solution run, when the Solution is done, Window appears, Click on OK.

Step 11: Reviewing the Nodal Temperature Results

1. Choose Main Menu> General Postproc > Plot result >Contour Plot>Nodal Solution. The Contour Nodal Solution Data dialog box appears.

2. For “Item to be contoured,” Pick “DOF solution” from the list on the left, and then click “Nodal Temperature”.

Fig 3.20 – Plot Results dialog box

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3. Click on OK. The Graphics window displays a contour plot of the nodal temperature results.

Fig 3.21 – Nodal Temperature Plot

4. Choose Main Menu> General Postproc > List results>Nodal solution. The List Nodal Solution dialog box appears.

5. For “Item to be Listed,” Pick “DOF solution” from the list on the left, and then click “Nodal Temperature”.

6. Click on OK. The Graphics window displays a list of the nodal temperature results.

Fig 3.22 – Nodal Temperature List

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3.2 ANSYS Analysis for Triangular Fin

Step 1: Give the Analysis a Title

After starting the ANSYS program and have entered the GUI, we need to begin the analysis by assigning a title to it. To do so we perform the following tasks:

1. Choosing Utility Menu>File>Change Title. The change title dialog box appears as shown below:

Fig 3.23 – Give Analysis a Title Dialog Box

2. Enter the text Steady State Thermal Analysis of Rectangular Fin

3. Click on Ok.

Step 2: Set Measurement Unit

We need to specify unit of measurement for the analysis. Here we are using the British System of Units (based on inches). To do this type /UNITS,SI in the ANSYS input window and press ENTER.

Step 3: Set Preference for GUI Filtering

1. Choose Main Menu>Preferences. A Preference for GUI Filtering dialog box appears. Check Thermal and h-Method.

2. Click on OK.

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Fig 3.24 – Set Preference dialog box

Step 4: Define the Element Type

Here we are using thermal solid element. To define it we do the following:

1. Choosing Main Menu>Preprocessor>Element Type>Add/Edit/Delete. The element type dialog box appears.

Fig 3.25 – Element type dialog box

2. Click on Add. The Library of Element type dialog box appears.

3. In the list on the Left, scroll down to pick (highlight) “Thermal Solid” and in the list on right, pick “Quad 8node 77”. As shown below:

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Fig 3.26 – Element Library dialog box

4. Click on Ok.5. Click on Close to close the Element Types dialog box.

Step 5: Defining Material Properties

To define the material properties we followed the following procedure:

1. Choosing Main Menu>Preprocessor>Material Props>Material Models. The Define Material Properties dialog box appears.

2. In the Material Model Available window double click on: Thermal, Density. A dialog box appears as shown below:

Fig 3.27 – Enter the Value of Density dialog box

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3. Enter the value 4.5657 for DENS, Click on OK.

4. In the Material Model Available window double click on: Thermal, Conductivity, Isotropic. A similar dialog box as above appears.

Fig 3.28 – Enter the Value of KXX dialog box

5. Enter the value 15.406 for KXX, Click on OK.

6. Choosing menu path Material>Exit to remove the Material Model Behavior dialog box.

Step 6: Building the Model Geometry:

We followed the following procedure to make the model geometry:

1. Choose Main Menu>Preprocessor> Modeling >Create>Area>Rectangle >By Dimension. A create Rectangle by Dimension dialog box appears as shown in Fig 3.29.

2. Choose Main Menu>Preprocessor> Modeling >Create>Key points>On Working Plane. A create KPs on WP dialog box appears as shown in Fig 3.30.

3. Select the option as Global Cartesian and enter the values as .0025, .05 in the picker box.

4. Click on Apply. The ANSYS program will make a point in GUI.

5. Similarly put the values as .0025, .055. Click Apply.

6. Put values as .0525, .0515. Click on Apply.






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17. Put values as .0525, .145. Click on Apply


19. Put values as .0025, .1435. Click on OK.

Fig 3.29 – Enter Dimension of Rectangle dialog box

Fig 3.30 – Create KP on WP dialog box

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Fig 3.31 – Key points on WP

20. ChooseMainMenu>Preprocessor>Modeling>Create>Area>Rectangle>Arbitrary> Through KP. Choose KP 5,7,8,6 as shown in Fig below:

Fig 3.32 – Create Area With KP dialog box

21. Click on Apply.

22. Now similarly choose KPs 9,11,10,12. Click Apply.

23. Choose KPs 13,15,14,16. Click Apply.

24. Choose KPs 17,19,20,18. Click OK. Finally we get the desired geometry as shown below:

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Fig 3.33 – Model Geometry

Step 7: Add The Areas:

We followed the following procedure to ADD all the Areas:

1. Choose Main Menu>Preprocessor> Modeling >Operate>Booleans >Add>Areas. The Add Areas picking menu appears.

2. Click on Pick All button and then OK.

Fig 3.34 – Add Area dialog box

Step 8: Mesh the Model

We followed the following procedure to mesh our Model:

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1. Choose Main Menu>Preprocessor>Meshing>Size Cntrls>Manual Size> Global>Size. A Global Element Sizes dialog box appears as shown in Fig. below:

Fig 3.35 – Enter Element Length dialog box

2. Enter the value .00125 for Element Edge Length, and then click on OK.

3. Choose Main Menu>Preprocessor>Meshing>Mesh>Areas>Free. A Mesh Areas Picking menu appears as shown below:

Fig 3.36 – Mesh Area picking Menu

4. Click on Pick All. In the Graphics Window, ANSYS builds the Meshed model

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Fig 3.37 – Meshed Area

Step 9: Applying Loads

We followed the following procedure to Apply Loads on the Model Geometry

1. Choose Main Menu>Solution>Analysis Type>New Analysis. The New Analysis dialog box appears.

2. Click on OK to choose the default analysis type (Steady-state).

3. Choose Main Menu>Solution>Define Loads>Apply>Thermal> Temperature>On Lines. An Apply Temperature on Lines pick menu appears. Pick the inner most line.

Fig 3.38 – Apply Temp. On Lines picking menu

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4. Click OK, an Apply TEMP on Lines Menu appear. Enter the values as 373 for “Load TEMP Value” as shown in Fig.

Fig 3.39 – Apply Value of Temp. dialog box

5. Click on OK.

6. Choose Main Menu>Solution>Define Loads>Apply>Thermal>Heat Flux> On Lines. An Apply HFLUX on Lines pick menu appears. Pick the inner most line of the Model as shown in Fig.

Fig 3.40 – Apply Heat flux on Lines picking menu

7. Click OK, an Apply HFLUX on Lines Menu appear. Enter the values 87.732 for “Heat Flux.” Click on OK.

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8. Choose Main Menu>Solution>Define Loads>Apply>Thermal>Convection >On Lines. An Apply Convection on Lines pick menu appears. Pick the Outer Lines of the Model as shown in Fig.

Fig 3.41 – Apply Convection on Lines picking menu

9. Click on OK. An Apply Convection on Lines dialog box appears. Enter the values as 68.57 for “Film Coefficient” and 298 for “Bulk Temperature”. Click on OK.

Step 10: Solving the Model

1. Choose Main Menu>Solution> Solve >Current LS

Fig 3.42 – Solve Current LS dialog box

2. Choose Close to close the /STAT command window.

3. Click on OK in the Solve Current Load Step dialog box.

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4. The solution run, when the Solution is done, Window appears, Click on OK.

Step 11: Reviewing the Nodal Temperature Results

1. Choose Main Menu> General Postproc > Plot result >Contour Plot>Nodal Solution. The Contour Nodal Solution Data dialog box appears.

2. For “Item to be contoured,” Pick “DOF solution” from the list on the left, and then click “Nodal Temperature”.

Fig 3.43 – Plot Results dialog box

3. Click on OK. The Graphics window displays a contour plot of the nodal temperature results.

Fig 3.44 – Nodal Temperature Plot

4. Choose Main Menu> General Postproc > List results>Nodal solution. The List Nodal Solution dialog box appears.

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5. For “Item to be Listed,” Pick “DOF solution” from the list on the left, and then click “Nodal Temperature”.

6. Click on OK. The Graphics window displays a list of the nodal temperature results.

Fig 3.45 – Nodal Temperature List

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CHAPTER 4

FEM ANALYSIS

4.1 FEM Analysis of Heat Transfer through Fins

The purpose of the present analysis is to describe the use of finite element method for the solution of engineering problems in well defined areas of heat transfer. In particular, we will concentrate on finite element computational procedures for the areas of conduction and convection heat transfer.

In the remainder of this chapter, we will provide a resume of the partial differential equations that describe heat transfer in the continuum. The boundary conditions for typical problems are also presented.

In the present study we are interested in the conductive transport of thermal energy in a material region,. The equilibrium of a continuous medium is governed by global conservation principles.

There are two alternative descriptions used to express the conservation laws in analytical form.

1. Eulerian description or spatial description

2. Lagrangian description or material description

Second description (i.e. lagrangian) focuses attention on a set of fixed material particles, irrespective of their spatial locations. So we use this description for our analysis.

The major steps in the finite elements analysis of a typical problem are:

1. Discretization of the domain into a set finite elements (mesh generation).

2. Weighted – integral or weak formulation of the differential equation to be analyzed.

3. Development of the finite element model of the problem using its weighted-integral or weak form.

4. Assemly of finite elements to obtain the global system of algebraic equations.

5. Imposition of boundary conditions.

6. Solution of equations.

7. Postcomputation of the solution and quantities of interest.

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Heat transfer (Conduction):

………………………(1)

Model Differential Equation:

Consider the problem of finding the steady-state temperature T(x,y) distribution ina two-dimensional orthotropic medium Ω, with boundary Γ. The equation governing the temperature distribution is given by setting the time to zero and x1=x , x2=y in Eq (1).

……………..(2)

Where kxx and kyy are conductivities in the x & y directions, respectively, and Q(x,y) is the known internal heat generation per unit volume. For a nonhomogeneous conducting medium, the conductivities kxx and kyy are functions of position (x,y). For an isotropic medium, we set

Kxx = kyy = k in Eq (2) and obtain the Poisson equation

……………………(3)

Equation 3 must be solved in conjunction with specified boundary conditions of the problem. The following two types of boundary conditions are assumed in the following development :

……………………………….(4)

…………………(5)

Where ΓT and Γq are disjoint portions of the boundary Γ such that

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Γ= ΓT U Γq, q(c) refers to the convective component of heat flux

……………………………………(6)

And ( nx , ny ) denote the direction cosines of the unit normal vector on the boundary. In Eq (6), h(c) denotes the convective heat transfer coefficient.

Weight-Integral Statements & Weak Forms

In the case of weighted-residual method, we seek to satisfy the governing differential equation in a weighted-integral sense. Since we are working with a typical element , we satisfy the differential equation in a weighted-integral sense over the element Ωc .This process leads to n algebraic equations among the nodal values . The process leads to n algebraic equations among the nodal values ( Te

1, Te2, …Te

n ) . The set of equations is termed a finite element model of the original differential equation. The type of finite element model depends on the weighted-integral form used to generate the algebraic equations. Thus, if one uses a variational form, also called a weak form, the resulting model will be different from those obtained with a weighted-residual statement in which the weight function can be of several choices. Throughout the present study, we will be primarily concerned with the weak-form finite element models, in which the weight functions are selected to be the same as the approximation functions (the so called Ritz-Galerkin models).

The weak form of a differential equation is a weighted-integral statement that is equivalent to both the governing differential equation as well as the associated natural boundary conditions. We shall develop the weak from of Eq(5) & Eq(2) over the typical element Ω(c).

There are three steps in the development of a weak form.

The first step is to take all nonzero expressions in Eq(2) to one side of the equality, multiply the resulting equation with a weight function w, and integrate the equation over the element domain Ω(c):

……………..(7)

The expression in the square brackets of the above equation represents a residual of the approximation of the differential equation Eq 2, because Te(x,y) is only an approximation of T(x,y). Therefore, Eq 7 is called the weighted-residual statement of Eq 2. For every choice of weight function w(x,y), we obtain an algebraic equation from Eq 7 & among the nodal values Te

j. For n independent choices of w, we obtain a set of n linearly independent algebraic equations. This set is called a weighted-residual finite element model.

In the second step, we distribute the differentiation among T & w equally, so that both T & w are required to be differentiable only once with respect to x & y. To achieve this we use integration-by-parts (or the Green Gauss theorem) on the first two terms in Eq.7. First we note the following identities for any differentiable functions w(x,y), F1(x,y) and F2(x,y).

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…………..(8)

…………...(9)

Next, we recall the component form of the gradient (or divergence) theorem,

……………………(10)

……………………(11)

Where nx and ny are the components (i.e. the direction cosines) of the unit normal vector

…………………….(12)

On the boundary Γe, and ds is the arc length of an infinitesimal line element along the boundary. Using Eq 8, 9, 10, 11 with

We obtain,

…………….. (13)

For each choice of w we obtain an algebraic relation among (Te1, Te

2,….Ten). We label the algebraic

equation resulting from substitution of in the equation:-

………………..(14)

Where the coefficients Keij, Qe

i, and qei are defined by

………………...(15)

37


………………..(16)

………………...(17)

In the matrix notation, Eq (14) takes the form

…………………(18)

The matrix [ke] is called the coefficient matrix, or conductivity matrix in the present context. We note that Ke

ij = Keji (i.e., [ke] is symmetric). Equation 18 is called the finite element model of Eq 2.

In this article, a mesh less element free Galerkin method has been used to obtain discrete equations for two-dimensional heat transfer in the fins. In this method, the function over the solution domain requires only a set of nodes. It does not require element connectivity. The integration over the solution domain requires only simple integration of cells to obtain the solution. The variational method has been used for the discretization of the governing equation for two dimensional heat transfers in the fins. Lagrange multipliers are used to enforce the essential boundary conditions.

4.2 The EFG Method

The element free Galerkin method is a mesh less method because only a set of nodes and a description of the model’s boundary are required to generate the discrete equations. The discretization of the governing equation by the EFG method requires moving least-square approximants, which are made up of three components: a weight associated with each node, a polynomial basis, and a set of non-constant coefficients.

The Moving Least-Square Approximants

The unknown function T(x) is approximated by moving least-square approximants Th(x). In two dimensions, for a linear basis, Th(x) can be written as

……………………..(19)

Where

XT = [x y]

pT(x) = [1 x y] (monomial basis function)

m = 3 ( number of terms in the basis)

The unknown coefficients aj(x) at any given point are determined by minimizing the functional J,

……………………………(20)

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Where n is the number of nodes in the neighborhood of x for which the weight function w(x-x i) ≠0, & Ti is the nodal parameter of T at x = xI. The stationary value of J in Eq. (2) with respect to a j(x) leads to the following set of linear equations:

……………………………………………(21)

where

…………………(22)

………………….(23)

………………….(24)

By substituting Eq 10 in Eq 8, the MLS approximants can be defined as

…………………………….(25)

Where the shape function ФI(x) is defined by

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……………(26)

The derivative of the shape function is given as

………………(27)

Fig 4.1 – Two dimensional model of rectangular fins for steady & transient analysis

4.3 The Discrete Equation

A general form of energy for two- dimensional heat transfer in the fins is given as

………..(28a)

The boundary conditions are

At edge Γ1;

T = Te ………………(28b)

At edge Γ3;

…………………(28c)

The weighted integral form of Eq. (28a) is given as

40


………………(29)

The weak form of Eq. (29) will be

…………………(30) The functional I(T) is obtained as

…………………….(31)

Using Lagrange multiplier technique to enforce essential boundary conditions, the functional I(T) is obtained as

…………………..(32)

Using the variational method, Eq. (32) reduces to

……………..……(33)

Since δT and δl are arbitrary in preceding equation, the following relations are obtained using Eqs. (25) and (33):

……………(34a)

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…………….(34b)

Where

……………..(35a)

…………..(35b)

…………..(35c)

…………..(35d)

……………(35e)

Where

Using Crank–Nicolson technique for time approximation, Eq. (18) can be written as

…………(36)

Where

.…….…(37a)

………..(37b)

Equation (36) is called the finite element model of equation (28a).

This completes the finite element model development for the fins.

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CHAPTER 5

RESULTS AND DISCUSSION

The limiting assumptions, which are considered during performing the experiment,are:

1. The heat flow in the fin and its temperatures remain constant with time.

2. The fin material is homogeneous; its thermal conductivity is the same in all directions, and it remains constant.

3. The convective heat transfer coefficient on the faces of the fin is constant and uniform over the entire surface of the fin.

4. The temperature of the medium surrounding the fin is uniform.

5. The fin thickness is small, compared with its height and length, so that temperature gradients across the fin thickness and heat transfer from the edges of the fin may be neglected.

6. The temperature at the base of the fin is uniform.

7. There is no contact resistance where the base of the fin joins the prime surface.

8. There are no heat sources within the fin itself.

9. The heat transferred through the tip of the fin is negligible compared with the heat leaving its lateral surface.

10. Heat transfer to or from the fin is proportional to the temperature excess between the fin and the surrounding medium.

5.1 Heat Release from the Cylinder

Heat release from the cylinder was obtained by multiplying the Mass and heat storing capacity of the Ethylene Glycol by the difference between 100°C and the temperature after 10 minutes from the start of recording, then dividing it by the measurement time. The experiment was carried out at an ambient temperature of 25°C, because ambient temperature has a significant effect on temperature measurement of heat storing liquid.

Mass of Ethylene glycol = 0.66792 Kg

Heat capacity of Ethylene Glycol = 3789.054 J/Kg

Temperature after 10 minutes = 48°C

Total Time Taken = 1500 Seconds

Heat Release from the Cylinder = Mass*Heat capacity*(100-48)/Time

= 0.66792*3789.054*52/1500 J/Sec

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= 87.732 Joules/Second

5.2 Temperature Distribution of Rectangular & Triangular Fins

Distance from

Base (mm)

Rectangular Fin

Temp. (°C)

Triangular Fin

Temp. (°C)

0 62 59

10 57 52

20 54 48

30 51 45

40 47 42

50 44 39

Table 5.1 – Temp. Distribution of Rectangular & Triangular fin

5.3 Graphs

Fig 5.1 – Temperature Distribution in Rectangular fins

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Fig 5.2 – Temperature Distribution in Triangular Fins

Fig 5.3 – Comparison of Both Fins

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CONCLUSION

The experimental cylinder for an Air-Cooled engine was developed and the effect of The Fin profile

was investigated experimentally and through ANSYS analysis. From the results obtained above

from, Practical Experiment & ANSYS Analysis, we can conclude that the Heat Transfer Capacity &

Effectiveness of a Triangular Fin is better than Rectangular Fin. So to increase the cylinder cooling

the Fins should be made of triangular profile.

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REFERENCES

[1].William R. Hamburgen,” Optimal Finned Heat Sinks”, Digital Equipment Corporation Western Research Laboratory, Palo Alto, 28 October 1986.

[2]. Singh I. V., Sandeep K. and Prakash Ravi ,”Heat Transfer Analysis of Two-Dimensional Fins Using Mesh less Element Frees Galerkin Method”, 2003,Numerical Heat Transfer, Part A: Applications by Taylor and Francis, 44:1, 73 — 84.

[3]. Hassani, Vahab, Dickens, James and Bell, Kennett J,” The Fin-on-Plate Heat Exchanger: A New Configuration for Air-Cooled Power Plants”, 2005, Heat Transfer Engineering by Taylor and Francis, 26:6,7 — 15.

[4]. A.R.A. Khaled, “Maximizing Heat Transfer through Joint Fin Systems” February 2006, Journal of Heat Transfer by ASME, Vol. 128, p-203-206.

[5]. B. kundu, B. Maiti and P. K. Das,” Performance Analysis of Plate Fins Circumscribing Elliptic Tubes”, 2006, Heat Transfer Engineering by Taylor and Francis, 27:3, 86-94.

[6]. Hans Dieter Baehr · Karl Stephan, “Heat and Mass Transfer” Springer-Verlag Berlin Heidelberg, 2006.

[7]. Yunus A. Cengel , “Heat and Mass Transfer” Tata McGraw Hill, 2007.

[8]. R. L. Shilling, P. M. Bernhagen, V. M. Goldschmidt, P. S. Hrnjak, David Johnson, and Klaus D. Timmerhaus, “Perry’s Chemical Engineers Handbook” Section 11 Heat-Transfer Equipment, Tata McGraw Hill, 2008.

[9].Kevin D. Rafferty & Gene Culver, “HEAT EXCHANGERS” Geo-Heat Center Klamath Falls, Oregon.

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