unsteady heat conduction

Dublin Institute of Technology

Bolton Street

Transient Heat Transfer

Student Name: Shiyas Basheer

Student Number: D10119909

Date: 14/02/2014

Class: DT 022/4

Module: Heat Transfer

Shiyas Basheer D10119909 Unsteady Heat Conduction

TABLE OF CONTENTS

Objective..................................................................................................................................2

Introduction..............................................................................................................................3

Theory......................................................................................................................................4

Method..................................................................................................................................... 5

Results......................................................................................................................................6

Calculations..............................................................................................................................7

For Brass Cylinder................................................................................................................7

For Stainless Steel cylinder...................................................................................................9

For Brass Sphere.................................................................................................................10

Discussion..............................................................................................................................11

Conclusion............................................................................................................................. 12

References..............................................................................................................................13

Table 1 Experimental Results...................................................................................................7

Table 2 Experimental results 2.................................................................................................7

Table 3 Results.......................................................................................................................11

Figure 1 Schematic of Armfield HT10X..................................................................................3

Figure 2 Chart of Solid Sphere.................................................................................................4

Figure 3 Biot number for Brass Cylinder.................................................................................8

Figure 4 Biot number for Stainless steel cylinder.....................................................................9

Figure 5 Biot number for brass sphere...................................................................................10

Plot 1 Temperature vs Time.....................................................................................................7

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ABSTRACT

This lab investigated the transient heat transfer due to conduction through a brass cylinder, a

stainless steel cylinder and a brass sphere. The heat transfer coefficient (h) value for the

sphere was calculated to be 806.83W/m2K whilst the h value for the cylinder was calculated

to be 620.33W/m2K. The experimental thermal conductivity (k) value for stainless steel was

determined to be 30.71W/mK compared to a referenced value of 16W/mK. It also shows heat

transfer change with time.

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OBJECTIVE

To investigate and observe unsteady state heat conduction of two different solid geometries

when a step change is applied to the temperature at the surface of the shape. The three pieces

which were tested includes:

A brass cylinder and stainless steel cylinder

A brass sphere

Of the two geometries, the brass cylinder and brass solid sphere will be used to determine the

h value for each geometries. This will be then used to determine the k value for stainless steel

cylinder.

INTRODUCTION

This experiment was carried out using an Armfield experimental apparatus HT17 and a

measurement unit HT10X which can be seen in figure 1.

Figure 1 Schematic of Armfield HT10X

With the supplied three simple shapes such as solid cylinder, solid sphere of 15 mm radius

and the rectangular brass sphere of 25mm radius, three test were carried out. Measurements

taken on a shape in one material can be used to confirm the conductivity of a similar shape

constructed from a different material. Transient-temperature/ heat flow charts are supplied for

each of the shapes.

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The apparatus consists of a 30 litres volume insulated water bath. At the end of the bath is an

electric heater controlled by a thermostat so that a constant bath temperature can be obtained.

A small pump is located near the side of the water bath and is used to circulate the water

inside the bath. The pump speed is controlled by setting the voltage (0-24 V) on the HT-10X

control console. The circulation of the water in the bath ensures that the temperature of the

water in the vicinity of the test specimen is constant. The water temperature is controlled by a

rotary switch located on the front of the bath. The temperature of the water in the bath is

indicated by a thermocouple. Another thermocouple measures the temperature embedded in

the centre of the test specimen [1].

THEORY

Heat transfer often occurs in an unsteady state conditions or a transient state. It simply means

a function of time and the analytical solution are available for the temperature distribution

and heat flow of various solid shapes which are subjected to sudden convection with a fluid at

a constant temperature. Solving these types of problems often involves using unsteady heat

transfer charts such as the one shown in Figure 2 for a long cylinder of radius b, where the

whole surface is subjected to a change in temperature:

Figure 2 Chart of Solid Sphere

The horizontal axis τ represents the Fourier number or dimensionless time, the vertical axis is

dimensionless temperature θ and the slanted lines represents the inverse of the Biot number

(Bi). Each can be identified by a formulae as follows:

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θ=T (r ,t )−T ∞T i−T ∞

Bi=hbk

τ=αtb2

Where:

k = Thermal conductivity (Wm-1°C-1)

α= The thermal diffusivity (m2s-1)

h = The heat transfer coefficient (Wm-2°C-1)

t = Time step (s)

T(0,t) = Temperature at the centre of the cylinder (°C)

Ti = Initial temperature of the cylinder (°C)

b = Radius of the cylinder (m)

T ∞= Temperature of the water bath (°C)

The following were given:

α for brass = 3.7x10-5 m2s-1

α for stainless steel = 0.6x10-5 m2s-1

k for brass = 121Wm-2°C-1

METHOD

The following procedures were done to conduct this experiment:

The water heater was first checked to be filled with water and then the electrical

supply was turned on to heat the water.

The red light was checked to ensure that the electrical power was connected to the

unit and the thermostat on the water heater was set to position 4.

The voltage was set to 12V for the circulating pump.

The temperature of the water was allowed to stabilize between 80-90°C.

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The temperature of the geometry was obtained and allowed to stabilize at room

temperature before being immersed in the water bath.

The initial temperature of the water bath and the center of the geometry was recorded

The shape was then immersed into the water bath

The temperature was then obtained for every 5 second interval till the center reached

80°C.

This was then repeated for the other geometries and materials

RESULTS

The results obtained from the experiment can be seen in Table 1 & Table 2, and a plot of the

temperature against time for all three materials can be seen in Plot 1:

Temperature (0C)Time (s) Brass Sphere Brass Cylinder Stainless Steel Cylinder

0 24.5 19.3 19.45 36.8 36.4 37.910 45.2 48.4 39.515 53.4 57.5 50.820 59.2 63.8 55.225 64.1 68.6 60.130 67.5 71.3 64.135 70 73.6 6740 71.8 75.5 69.345 73.6 76.6 71.450 75 77.4 72.855 76 78.4 7460 76.8 78.6 7565 77.4 79.6 75.870 78.1 79.3 76.575 78.7 79.6 7780 79 79.8 77.485 79.4 80 78.190 79.7 78.495 80 78.7100 78.8105 79110 79.2115 79.3120 79.4125 79.6130 79.6135 79.6140 79.7145 79.8

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150 79.9155 79.9160 80

Table 1 Experimental Results

Brass Sphere Brass Cylinder Stainless Steel Cylinder

Time taken to reach 800C (s) 96.31 87 158

Water Bath Temp (0C) 84.3 82.3 82.5

Table 2 Experimental results 2

0 20 40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

80

90

Temperature against Time

Brass Sphere Brass Cylinder Stainless Steel Cylinder

Time (s)

Tem

per

atu

re (

oC)

Plot 1 Temperature vs Time

CALCULATIONS

For Brass Cylinder

Using the equations mentioned in the theory section following can be calculated:

θ=T (r ,t )−T ∞T I−T∞

= 80−82.319.3−82.3

=0.0365

τ=αtb2 =

(3.7∗10−5 )(87)0.0152 =14.30

Now, the inverse of the Biot number can be calculated using the above values and the chart

for unsteady heat transfer for a long cylinder;

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Figure 3 Biot number for Brass Cylinder

From the above Figure it can be seen that

1Bi

=13

Bi=0.0769

Now, by rearranging the equation for Biot number the following can be obtained

h=Bi∗kb

h=0.0769∗1210.015

h=620.33W

m2 °C

Therefore, the h value of brass cylinder is 620.33W

m2 °C

For Stainless Steel cylinder

Re-applying the same as above:

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θ=T (r ,t )−T ∞T I−T∞

= 80−8219.4−82

=0.0319

τ=αtb2 =

( 0.6∗10−5 )(158)0.0152 =4.21

Now for the Biot number

Figure 4 Biot number for Stainless steel cylinder


1Bi

=3.3

Bi=0.303

Using h value obtained for the brass cylinder due to the fact its unknown and has the same

geometry as brass cylinder and rearranging for k:

k=hbBi

k=620.33∗0.0150.303

=30.71Wm°C

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Therefore, the thermal conductivity of stainless steel cylinder is 30.71Wm°C

For Brass Sphere

Again, same as above:

θ=T (r ,t )−T ∞T I−T∞

= 80−84.324.5−84.3

=0.0719

τ=αtb2 =

(3.7∗10−5 )(96.31)0.0252 =5.70

Now for the Biot number

Figure 5 Biot number for brass sphere


1Bi

=6.0

Bi=0.1667

Now, by rearranging the equation for Biot number the following can be obtained

h=Bi∗kb

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h=0.1667∗1210.025

h=806.83W

m2 °C

Therefore, the h value of Brass Sphere is 806.83W

m2°C

DISCUSSION

The results obtained can be summarised as follows:

Time taken to reach 800C

(s)

k Value (W/m

°C)

h Value (W/m2

°C)

Brass Sphere 96.31 121 806.83

Brass Cylinder 87 121 620.33

Stainless Steel

Cylinder158 30.71 620.33

Table 3 Results

It can be seen from the table above that the k value of Stainless steel cylinder is significantly

smaller than that of brass cylinder that also has a similar geometry. It can also be seen that the

Stainless steel cylinder took the longest to reach the target temperature of 80 oC. These

differences might be due to the fact that the Stainless steel has low thermal conductivity than

that of brass. It can also be noted that the brass Sphere has a high heat transfer coefficient

than that of brass cylinder and also it takes longer to reach the target temperature of 80 oC.

This could be due to the sphere having a lower surface area than the cylinder. From Plot 1

earlier in the results section, it can be seen that the graph doesn’t have liner lines but curved

ones, which shows that the unsteady state conditions exists.

However, there is a considerable difference between the experimental thermal conductivity of

stainless steel cylinder of 30.71 W/mK and the referenced thermal conductivity of 16 W/mK

[1]. This error might be due to the following reasons:

Error in measurement of temperature and time

Equipment error

Inaccuracies in using the chart

Human error

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The stainless steel used in the apparatus might have a different composition to the one

used to calculate the referenced k value.

CONCLUSION

The experiment demonstrated unsteady or transient heat transfer. Based on the results the

following can be concluded:

Unsteady heat transfer exists and it depends on both the geometry and the material

used

Same material with different geometries have different heat transfer coefficient under

same conditions

Different materials with same geometry behave differently under the same conditions

The experimental k value for stainless steel was determined to be 30.71W/mK

compared to a referenced value of 16W/mK.

Unsteady heat transfer changes with time and is nonlinear

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REFERENCES

[1] “Instruction manual,” [Online]. Available:

http://www.share-pdf.com/444302be79f84be9a1b25848e9926b1f/411_lab_2____2HT17_

Issue_11_Instruction_.htm.

[2] “Engineering toolbox,” [Online]. Available:

http://web.eng.fiu.edu/~wbao/EML_4906L/EML4906L_TransientHeatTransfer.htm..

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unsteady heat conduction

Documents

water temperature

solid cylinder

brass cylinder8figure

brass sphereof

constant bath temperature

rectangular brass sphere

heat transfer change

brass sphere10 plot