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Transcript of 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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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
From the above Figure it can be seen that
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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
From the above Figure it can be seen that
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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Shiyas Basheer D10119909 Unsteady Heat Conduction
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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