lab 5 - conduction finalize)

Thermalfluids lab EMD5M8A

Heat Conduction – Simple Bar

CONTENTS

1.0 INTRODUCTION……………………………………………………………………..2

2.0 OBJECTIVE………………………………………………………………………..….3

3.0 THEORETICAL BACKGROUND

4.0 APPARATUS……...………………………………………………………….………..7

5.0 PROCEDURE………..………………………………………………………..……….8

6.0 DATA AND CALCULATIONS……………………………………………………….10

6.1 SAMPLE CALCULATIONS………………………………………………….10

7.0 REFERENCES…...…………………………………………………………………....13

8.0 APPENDIX

8.1 DATA TAKEN FROM EXPERIMENT……………………………….……..14

8.2 INDIVIDUAL DISCUSSION AND CONCLUSION………………………...15

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1.0 INTRODUCTION

Conduction is defined as the transfer of energy from more energetic particles to adjacent

less energetic particles as a result of interactions between the particles. In solids, conduction is

the combined result of molecular vibrations and free electron mobility. Metals typically have

high free electron mobility, which explains why they are good heat conductors.

Conduction can be easily understood if we imagine two blocks, one very hot and the other cold.

If we put these blocks in contact with one another but insulate them from the surroundings,

thermal energy will be transferred from the hot to the cold block, as evidenced by the increase in

temperature of the cold block. This mode of heat transfer between the two solid blocks is termed

‘conduction’.

In heat transfer, conduction (or heat conduction) is the transfer of thermal energy between neighboring molecules in a substance due to a temperature gradient. It always takes place from a region of higher temperature to a region of lower temperature, and acts to equalize the temperature differences. Conduction takes place in all forms of matter, viz. solids, liquids, gases and plasmas, but does not require any bulk motion of matter. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons. In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion.

Conduction is the transfer of heat from one part of a substance to another part of the same

substance, or from one substance to another in physical contact with it, without appreciable

displacement of the molecules forming the substance. For example, the heat transfer in the metal

bar mentioned previously is by conduction.

Perhaps the simplest phenomenon that can be modeled by the heat equation is heat

conduction in a long uniform rod. In most instances heat conduction occurs in three dimensions -

a situation that is complicated to analyze. In the laboratory, we use an apparatus that exhibits

one-dimensional heat flow to demonstrate the basic concepts associated with the heat equation.

Thus, in our experiment is to investigate Fourier’s law for linear conduction of heat along a simple bar. More about Fourier’ law was explained in Theoretical Background part.

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http://en.wikipedia.org/wiki/Molecular_diffusion

http://en.wikipedia.org/wiki/Free_electron_model

http://en.wikipedia.org/wiki/Free_electron_model

http://en.wikipedia.org/wiki/Crystal_structure

http://en.wikipedia.org/wiki/Temperature_gradient

http://en.wikipedia.org/wiki/Thermal_energy



2.0 OBJECTIVE

The objective of this experiment is two-fold:

1. To investigate Fourier’s law for linear conduction of heat along a simple bar. To conduct a series of experiments in thermal conduction in order to deduce the relationship between the heat transfer rate, the temperature difference, the cross-sectional area, and the length of model across which conduction takes placed.

3.0 THEORETICAL BACKGROUND

Heat is a form of energy that can be transferred from one system to another as a result of

a temperature difference. Meanwhile, heat transfer carried a meaning of science that deals with

the determination of rates of energy transfer. There are three basic mechanisms of heat transfer

which are:

i) Conduction

ii) Convection

iii) Radiation

Conduction

Conduction is a transfer of energy from the more energetic particles to an adjacent substance

with less energetic particles. Conduction could take place in liquids, solids or gases. In solids, it

is due to vibrations of the molecules in their lattice while in gas, it is due to the collisions and

diffusion of the molecules due to their random motion.

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Heat Conduction – Simple BarFor steady state 1-Dimensional Heat Conduction, for problems where the temperature variation

is only 1-dimensional such as along the x-coordinate direction, Fourier's Law of heat conduction

simplies to the scalar equations:-

q=−kδTδx

, where q isthe heat flux .

Q̇=−kAδTδx

, whereQ̇ is theheat transfer rate .

In this formula the heat flux q depends on a given temperature profile, T and thermal

conductivity, k. The minus sign ensures that heat flows down the temperature gradient.

In the above equation,Q̇ represents the heat flow through a defined cross-sectional area A,

measured in watts,

Q̇=∫A

a

q . δA

Integrating the 1D heat flow equation through a material's thickness δx gives,

Q̇=−kA∆ x

(T 1−T2)

whereT 1and T 2 are the temperatures at the two boundaries.

The negative sign indicates that heat is transferred in the direction of decreasing

Temperature. More generally, Fourier's Law is a vector relationship which includes all

directions of heat transfer:

The thermal conductivity, kvaries between different materials and can be a function of

temperature, but it can be treated as a constant over small temperature ranges. Because of the

enhancement of heat transfer by free electrons, thermal conductivity is analogous to electrical

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Heat Conduction – Simple Barconductivity and as a result, metals that are good conductors of electricity are also good

conductors of heat. In this experiment we will investigate Fourier's Law by finding the

thermal conductivity, kfor brass. To do so we will calculate the cross sectional area, Ax of the

bar and the slope, dT/dx from a plot of measured temperatures vs. length. We can then use

these values in a rearranged version of Fourier's Law to find the thermal conductivity, k:

k= Q̇A

∆ x∆ T

units[ Wm. K ]

In this experiment we will investigate conduction in an insulated long slender brass bar

like the one in Figure 1. We will assume that the bar is of length L, a uniform hot temperature

This imposed on one end, and a cold temperature Tcis imposed on the other. We will also

assume, because the bar is insulated in the peripheral direction that all the heat flows in the axial

direction due to an imposed temperature differential along the bar.

Figure 1: Schematic of a Long Cylinder Insulated Bar

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If a plane wall of thickness (Δx) and area (A) and thermal conductivity (k) supports a

temperature difference (ΔT) then the heat transfer rate by conduction is given by the equation:

Q̇ = k A dTdx

Figure 2: Conduction Process

The thermal conductivity k varies between different materials and can be a function of

temperature, but it can be treated as a constant over small temperature ranges. Because of the

enhancement of heat transfer by free electrons, thermal conductivity is analogous to electrical

conductivity and as a result, metals that are good conductors of electricity are also good

conductors of heat.

In this experiment we will investigate Fourier's Law by finding the thermal conductivity k for

brass and comparing this value to the actual value from one or more references. To do so we will

calculate the cross sectional area Ax of the bar and the slope dT/ dx from a plot of measured

temperatures vs. length. We can then use these values in a rearranged version of Fourier's Law to

find the thermal conductivity, k.

Assuming a constant thermal conductivity throughout the material and a linear temperature

distribution, this is:

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Q̇ = k A ΔTΔx

4.0 APPARATUS

Figure 3: Simple Bar with Sample

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Heater

Cooler

Brass Sample



Figure 4: Apparatus for the Heat Conduction Control

Figure 5: Digital Clock

5.0 PROCEDURE

1. Firstly, the heater knob on the heat conduction control (control box) was turned fully counterclockwise (this is the OFF position).

2. Power supply and heat conduction controlwas turned ON.3. An intermediate position was selected for the heater power control (e.g. 10 W)

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Heat Conduction – Simple Bar4. The power was turned up on the heater to 10 Watts and allowed sufficient time about 10

minutes for a steady state to be achieved before recording the temperature (T) at all 9

sensor points and the input power reading on the wattmeter (Q̇).

5. Set up the time by using digital clock.

6. After 10 minutes, the reading has been taken of all nine temperature measurement power by using heater power control.

7. This procedure was repeated for other input powers (e.g. 20 W and 30 W) up to

maximum setting of the control.

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8. All the data was recorded in the table.

6.0 RESULT AND CALCULATION

X(mm) 0 10 20 30 40 50 60 70 80X(m) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Test Q̇(W)

T1

(℃)T2

(℃)T3

(℃)T4

(℃)T5

(℃)T6

(℃)T7

(℃)T8

(℃)T9

(℃)A 10 41.3 45.2 41.8 38.9 38.0 37.7 33.6 33.0 32.2B 20 58.0 64.8 58.3 50.6 48.3 47.7 38.1 36.6 34.8C 30 82.2 89.2 81.4 67.3 63.5 62.9 41.2 39.1 36.8

Table 1: Temperature result for test A, B and C

Heat Transfer Rate , Q̇ (W)Thermal Conductivity, k (

Wm. K

)

10 203.67

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HEATER COOLER

BRASS SAMPLE REGION


Heat Conduction – Simple Bar20 162.9730 142.13

Table 2: Rate of conduction and Thermal Conductivity

Diameter of the Cylinder = 25mm

Cross Sectional Area ofthe Cylinder = 4.91 × 10-4

Sample Calculation:

a) Cross Sectional Area Of The Cylinder

= π4

d2

= π4

(0.025)2

= 4.91 × 10 -4

7.0 Graph:

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-0.0199999999999999

9.36750677027476E-17

0.0200000000000001

0.0400000000000001

0.0600000000000001

0.08000000000000010

10

20

30

40

50

60

70

80

90

Temperature, T versus Length,L

Length, L (m)

Tem

pera

ture

, T (º

C)

Where;

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At 10W heat suppliedAt 20W heat suppliedAt 30W heat supplied



b) Since the value of area, and temperature distribution have been obtained, the value of thermal condtivity, k can be obtained by Fourier Equation.

k @10 W=QA

∆ x∆T

¿ 100.000491

0.04−0.0338.9−38.0

¿226.3 0W /m ∙K

k @20 W=QA

∆ x∆ T

¿ 200.000491

0.04−0.0350.6−48.3

¿177.10 W /m ∙K

k @30 W=QA

∆ x∆T

¿ 300.000491

0.04−0.0367.3−63.5

¿160.79 W /m ∙K

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10.0 References:

1. http://www.mcs.utulsa.edu/~class_diaz/cs4533/flowheat/node4.html2. http://www.engineeringtoolbox.com/thermal-conductivity-metals-d_858.html.3. Thermodynamics An Engineering Approach Sixth Edition (SI Units) by Yunus A. Cengel

And Michael A. Boles. (McGraw Hill)4. Fundamentals of heat and mass transfer (sixth edition) Incropera Dewitt Bergmann

Lavine

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http://www.mcs.utulsa.edu/~class_diaz/cs4533/flowheat/node4.html

lab 5 - conduction finalize)

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