S
Fo
MONASH UNIVERSITY – UNWAY CAMPUS
Free and rced
Convectio n CHE2163 Laboratory Report
9/17/2010
Free and Forced Convection I.
ii
Title Page
2010
Free and Forced Convection II.
2010
Table of Contents
I.
Title Page ........................................................................................................................ii
II.
Table of Contents ...................................................................................................... iii
III.
Table of figures ...........................................................................................................
v IV.
List of Graphs..............................................................................................................
v V.
List of Tables...............................................................................................................
v
VI.
Summary
....................................................................................................................vi 1.
1.1.
Background information .......................................................................................... 1
1.2.
Relevant theories and Key equations....................................................................... 2
1.3.
Motivation for study ................................................................................................ 4
1.4.
Intended Scope ........................................................................................................ 4
2.
Aims ................................................................................................................................... 5
3.
Experimental Work ............................................................................................................ 6
4.
4
Introduction........................................................................................................................ 1
3.1.
Safety issues ............................................................................................................ 6
3.2.
Description of apparatus .......................................................................................... 7
3.3.
Diagram of apparatus............................................................................................... 9
3.4.
Experimental Procedure ........................................................................................ 12
Results and Discussion .................................................................................................... 14 4.1.
Calculated Data...................................................................................................... 14
4.2
Discussion of trends and interpretation of graphs ................................................. 32
4.3
Comparison of Results........................................................................................... 34
4.4
Comparison with expectations............................................................................... 34
4.5
Errors ..................................................................................................................... 37
4.6
Difficulties/Limitations ......................................................................................... 38
Free and Forced Convection 5.
5
2010
Appendices....................................................................................................................... 39
Free and Forced Convection
2010
Appendix 1....................................................................................................................... 39
6
6.
References ........................................................................................................................ 40
7.
Nomenclature list ............................................................................................................. 41
Free and Forced Convection III.
2010
Table of figures
Figure 1: Heat Convection example: Transfer of heat from a fire to the body.......................... 1 Figure 2: Heat Convection example: boiling of water ............................................................... 1 Figure 3: Free convection example: Water cycle ...................................................................... 2 Figure 4: Forced Convection example: Blowing a hot cup of coffee ........................................ 2 Figure 5: Free and forced convection apparatus (LS-17004) .................................................... 9 Figure 6: Finned Plate .............................................................................................................. 10 Figure 7: Digital Handheld Anemometer ................................................................................ 10
IV.
List of Graphs
Graph 1: Free Convection - Temperature Profile versus Time................................................ 15 Graph 2: Free convection - Surface temperature vs. Time ...................................................... 16 Graph 3: Forced convection - Temperature Profile versus Time (v = 0.6m/s) ........................ 20 Graph 4: Forced Convection - Surface temperature vs. Time (v = 0.6 m/s) ........................... 21 Graph 5: Forced convection - Temperature Profile versus Time (v = 0.9 m/s) ....................... 25 Graph 6: Forced convection - Surface temperature vs. Time (v = 0.9 m/s) ............................ 25 Graph 7: Forced convection - Temperature Profile versus Time (v = 1.2 m/s) ....................... 29 Graph 8: Forced Convection - Surface temperature vs. Time (v = 1.2 m/s) ........................... 29
V.
List of Tables
Table 1: Free convection: Temperature profile vs. Time ........................................................ 14 Table 2: Forced convection - Temperature profile vs. Time (v = 0.6 m/s) ............................. 20 Table 3: Forced Convection - Temperature Profile versus Time (v = 0.9 m/s) ....................... 24 Table 4: Forced Convection – Temperature Profile vs. Time (v = 1.2 m/s) ............................ 29 Table 5: Forced Convection - Comparison of results .............................................................. 34
7
Free and Forced Convection VI.
8
Summary
2010
Free and Forced Convection
2010
1. Introduction
1.1. Backg rou n d in f orm ation Heat transfer is the movement of heat from one place to another. When an object is at a different temperature from its surrounding temperature, the heat will move from the higher temperature to the lower temperature until the object and the surrounding have the same temperature. There are many modes of heat transfers, but the one that will be considered in the following experiment will be heat transfer through convection. Heat convection is the transfer of heat between an object and its surrounding due to fluid movement. An example of this phenomenon is the cooling of hot water; where hot water vapour is released into the atmosphere until it reaches the surrounding temperature. More examples include the heat obtained by a fireplace, the boiling of water, the transfer of heat from a hot water radiator etc. A few heat convection methods are illustrated below.
Figure 1: Heat Convection example: Transfer of heat from a fire to the body
Invalid source specified.
9
Figure 2: Heat Convection example: boiling of water
Free and Forced Convection
2010
Heat convection can be divided into two categories. They are free convection and forced convection. Free convection is the movement of fluid due to the density difference
10
in the fluid and the surrounding. Forced convection is fluid motion which is generated by an external source such as a pump or a fan. The figures below include a few examples of free and forced convection.
Invalid source specified. Invalid source specified.
Figure 3: Free convection example: Water cycle
Figure 4: Forced Convection example: Blowing a hot cup of coffee
1.2. Rele van t th eori es an d Ke y equ ation s This experiment was divided into two parts; namely, free and forced convection. Using the results obtained, the Nusselt number (Nu), Rayleigh number (Ra), and Prandtl number (Pr) will first be calculated for free convection. This will be followed by the calculation of the Reynolds number (Re), Nusselt and Prandtl number for forced convection. The convection heat coefficient
was calculated using Newton’s law of
cooling. The equation is as follows:
Where ,
Equation (1)
The Prandtl Number (Pr) relates the kinematic viscosity with the specific heat
, dynamic viscosity
and thermal diffusivity
, and the thermal conductivity
.
It is a dimensionless number and since it does not change with the fin or flat plate length, it is not subscripted with the length scale. In heat transfer problems the Prandtl number controls the relative thickness of momentum and the thermal boundary layers. I.e. when the Pr is small, the heat diffuses quickly compared to the velocity (momentum). The equation for the Prandtl number is as follows: (Prandtl Number, 2010)
The Nusselt number,
Equation (2)
relates the convective heat transfer coefficient,
thermal conductivity of the fluid,
and the characteristic length
Prandtl number, the Nusselt number is also dimensionless. A
the
. Just as the which is close to
unity, i.e. convection and conduction of similar magnitude, is characteristic of a laminar flow. The larger the Nusselt number, it corresponds to a more active convection with turbulent flow. The equation of the Nusselt number is as follows: (Nusselt Number, 2010)
The Rayleigh Number,
Equation (3)
is a dimensionless number related to the buoyancy
driven flow which is also known as free convection. This number is the multiplication of the Grashof number,
and the Prandtl Number. The Grashof number is the
measure of the ratio of buoyancy forces to viscous forces and relates the acceleration due to gravity,
, The thermal expansion coefficient,
fluid temperature
, the characteristic length
, the surface temperature, and the kinematic viscosity,
, the .
When the Rayleigh number is below the critical number of that fluid, heat transfer is primarily in the form of conduction. When it exceeds the critical value, heat transfer is
primarily in the form of convection. The equation of the Rayleigh number is as follows: (Rayleigh Number, 2010)
The Reynolds number
is a dimensionless number which gives the ratio of the
inertia and viscous forces. This number relates the density, velocity, viscosity
Equation (4)
and the characteristic length
dynamics viscosity,
,
. It can also be related to the kinematic
. The equation of the Reynolds number is given below. (Reynolds Number,
2010)
1.3.
Equation (5)
Motivati on f or stu d y
The main motivation to conduct this experiment was the opportunity to apply our theoretical knowledge on fins practically. By doing this we were able to analyse and study the phenomena of free and forced convection. Furthermore, we managed to compare the coefficient of heat transfer using the equation of a flat plate to the expected value for a fin. Also by calculating the numbers mentioned above we were able to get a better idea of the type of flow of the system.
1.4. In ten de d S c op e The scope of this experiment is to calculate the heat transfer coefficients with temperature profiles and a heat flux in a rectangular air duct fitted with a finned plate surface. By calculating the experimental heat transfer coefficient and the theoretical heat transfer coefficient we are able to analyse the comparison and errors of the experiment. The rest of the report includes the objectives, experimental procedure, results and discussion and the conclusion of our experiment.
2. Aims
The aim of this experiment is to make a comparison between the theoretical and experimental value obtained for the convection heat transfer coefficient For free convection, we have compared the Nusselt number
so as to determine the
percentage error of the experiment in comparison to the theory. In this case the rate of heat transfer will be constant throughout the experiment. Therefore we will obtain one value for each of the numbers. For forced convection, we have found the convection heat transfer coefficient through the Reynolds number
as well as the Nusselt Number. Just as before the rate of heat
transfer will be constant but the velocities will change. Therefore, we will obtain more than one value for each of the numbers. As a result, the values will be plotted in order to obtain a pattern in the data.
3. Experimental Work 2. 3. 3.1. S afet y is su es It is an undisputable fact that safety comes first in every aspect of our lives, particularly when an experiment is being conducted. Professionalism must prevail in every undertaking to elude any possible hazard. Therefore, a few precautions and golden rules have to be adhered duly at all times. Firstly, only students with proper attires are permitted into the lab. Laboratory dress code is strictly practiced; all students must don up in lab coats and covered shoes as these attires are designed to protect students should any accidents occur. Besides that, locations of safety aids are identified prior to the conduction of the experiment. Usage of fire extinguishers, emergency bells, emergency shower, emergency route and first aid kit must be integrated into one’s mind. Experiments as such require undivided attention and patience as it subjects to cooling and heating of an element to steady state. Often, some individuals may perceive it as dull and time consuming, causing them to lose concentration and further exposed to underlying dangers. Therefore, air circulation in the lab must be kept at a comfortable level with the help of fans and air conditioners. Besides that, ample space must be provided in the workspace for students’ mobility. In accordance to that, the workspace must be kept neat and number of students permitted into the lab must be limited at all times. Moreover, students must master the theory of free and forced convection before the commencement of the lab session. Also, the setup of the experimental apparatus must be done appropriately. A good setup of the experimental apparatus is arranged systematically so that nothing intertwines or impedes the progress of the experiment. Therefore, all experimental apparatus, such as the stopwatch was positioned at a strategic location. Besides that, the nuts and bolts (M) securing the heat plate must be tightly secured to reduce any possible inaccuracies such as heat loss, posed. Whilst conducting the experiment, students are prohibited from
touching the heat plate (P) or air duct (B) as it may inflict burns and blisters. Besides that, the fan (A) of the apparatus (LS-17004) was kept out of reach to avoid potentially harmful cuts. First aid are carried out immediately should any misfortune occurs. Any injury must be reported instantaneously, so that any possible threats are under medical attention. Apart from that, the fan (A) is switched off when conducting the free convection experiment. It is imperative to only remove heat plate from chamber after the heat plate has been cooled down to avoid burns from occurring. Moreover, the handheld digital anemometer (Q) should be handled dexterously, with extreme care as the probe (R) is exceptionally fragile and tremendously costly to replace. The handheld digital anemometer is positioned suitably once the temperature and air velocity is noted down. Before the temperature readings are recorded, the temperature probe must be in contact with the heat plate surface.
Besides this, the electrical power source should be
disconnected when the heat socket is connected to the power source in order to prevent a short circuit. Consequently, students should never attempt to change the setting of the digital power meter.
3.2. Des cripti on o f ap par atu s
In this experiment, the free and forced convection apparatus (LS-17004 Lotus) is developed for the demonstration of free and forced convection phenomena. The apparatus mainly consists of a vertical air duct and a control panel.
Inside the
convection chamber, there is a compartment where different heated surfaces can be fitted in to determine its heat convection coefficients. Under our case study, we are given the task to scrutinize finned plate as shown in Figure 3. The finned plate is attached to an electrical heating element which in this scenario functions as the heat source. In the free convection experiment, a handheld digital anemometer was used to measure the initial temperature of the heater.
Then, probes of the free and forced
convection apparatus (LS-17004 Lotus) were inserted into its respective hole to
calculate its respective temperature point from time to time. The digital screen of control panel displays the temperature points at that specific time. For forced convection, air is fed into the duct by the fan which is placed at the top of the air ducts. Air flow or rather air circulation is generated when the fan is switched on as air is drawn out concurrently from the top of the air duct. Before the commencement of forced convection, the handheld digital anemometer was used to measure the initial temperature of the heater and the velocity of air flow which functions as the manipulated variable. The air velocity is adjusted using the fan speed regulator function found in the control panel.
3.3. Diagram o f ap paratu s
A
N B
M C L K
J
I
H D
E
F
Figure 5: Free and forced convection apparatus (LS-17004)
G
O
P
Figure 6: Finned Plate
Q
R
Figure 7: Digital Handheld Anemometer
Legend
A
=
Fan
B
=
Air duct
C
=
Temperature probes
D
=
Timer
E
=
Fan speed regulator
F
=
Fan switch
G
=
Temperature switch (T1, T2, T3)
H
=
ON/OFF Switch
I
=
Power regulator
J
=
Power meter
K
=
L
=
Temperature points (T1, T2, T3) indicator Temperature (Ts) indicator
M
=
Bolts and buts
N
=
Heater Placement
O
=
Fins
P
=
Heater Plate
Q
=
Digital handheld Anemometer
R
=
Digital handheld Anemometer probe
3.4. Experim en tal Pro ce d u re
3.4.1. Free Convection Experiment 1.
The LS-17004-FFC Free and forced convection apparatus was placed on a level table. The adjustable levelling feet was adjusted if necessary.
2.
Three pins plug to the 240VAC was plugged to the main power supply. The power supply was activated
3.
The power supply unit in front of the control panel was switched on.
4.
The shape and the dimensions of the finned plate(O) was measured and recorded.
5.
The finned plate was fixed tightly to the heater placement(N) with the bolts and nuts(M) provided.
6.
The finned plate heater cable was connected to the heater socket which was located at the back of the control panel.
7.
The digital handheld temperature probe and meter(Q) was used to measure the initial temperature of the heater by putting the temperature probe into temperature point. It was ensured that the probe was touching the surface of the heater and the fan was switched off before the readings were taken down.
8.
The power supplied was regulated to 100W by turning the power regulator.
9.
For every 5 minutes elapsed, the heater plate surface temperature and the temperature points was recorded.
10.
The experiment was continued until steady state was achieved.
11.
A graph of time against temperature difference was plotted for the different temperature point. T1, T2, T3 respectively.
12.
The overall coefficient for the heater was calculated.
13.
The Nusselt number, Nu, Rayleigh number, Ra and Prandtl number Pr was calculated.
3.4.2. Forced Convection Experiment 1.
The same apparatus set up was assembled as the one in free convection experiment.
2.
The power supplied was regulated to 100W by turning the power regulator.
3.
The digital handheld anemometer probe(R) was inserted into the side opening of the air duct where the temperature points were. The initial temperature of the heater was taken down with probe touching the heater surface. Then, the desired wind speed was adjusted with the help of the handheld anemometer probe(Q).
4.
For every 30 seconds elapsed, the temperature at different points T1, T2 ,T3 was taken down.
5.
The experiment was continued until steady state is achieved.
6.
The experiment was repeated with wind speed regulated to 0.9m/s and 1.2 m/s respectively, by repeating steps 4 and 5.
7.
The overall coefficient for all the heaters was calculated.
8.
The Nusselt number, Nu, Rayleigh number, Ra and Prandtl number Pr for all the cases was determined and a graph of Nusselt number versus Rayleigh number was plotted.
4. Results and Discussion
4. 4.1. Calcu lat ed Data Free Convection Power = 100W = 24.1°C = 297.1 K Time (min) 0
36 48 39 .8 .7 .8 40 40 43 .3 .5 .3 43 43 46 .0 .1 .4 45 45 48 .6 .4 .8 47 47 50 .8 .5 .7 49 49 52 .6 .6 .6 50 50 55 .6 .5 .0 51. 51 55 81 .5 .8 51 51 56 .5 .7 .7 51 52 57 .8 .3 .1 Table 1: Free convection: Temperature profile vs. Time
5 10 15 20 25 30 35 40 45
50 .0 55 .4 60 .1 64 .2 67 .7 71 .3 74 .0 76 .2 77 .4 77 .7
90 80 70
T1
60
T2 T3
50 40 Tsurface Log.
30
(T2) Log. (T2)
20
Linéaire (T3)
10 0 0
10
20
30
40
50
Graph 1: Free Convection - Temperature Profile versus Time
Surface temperature vs. Time 80
75
St ea
dy State 70
Time (min) 65
60
55
500
5
10
15
20
25
30
35
40
Surface Temperature (K) Graph 2: Free convection - Surface temperature vs. Time
4.1.1. Assumptions The following assumptions were assumed for this experiment: •
A vertical flat plate was assumed to be used
•
Steady state condition
45
•
Negligible radiation effect
•
Negligible heat conduction from the air duct and the plate
•
Ideal gas behaviour
4.1.2. Experimental Values The experimental value of h can be calculated using the following formula: Using Equation (1);
Total surface area of fins exposed is: A = 6 [ 2(8.7 × 0.6 ) + 2 (10.8 × 8.7 ) + 2 (0.6 × 10.8) ]= 1267.97
= 0.127
Assuming the total surface area of the fins exposed is equal to a vertical plate area, since the assumption taken in these calculations is that a vertical flat plate was used. The average value of
was calculated using Microsoft Excel,
= 67.4 °C = 340.4 K Thus, calculating for the value of h gives :
4.1.3. Calculation of the Nusselt Number, Rayleigh Number and Prandtl Number (Incropera, 2007) Thermo physical properties of Air at Atmospheric Pressure At
=
= 318.75 K
From Table A.4, using interpolation method at temperature of
:
W/m.K
Density of air, ρ = 1.099 kg/ Assuming that air is an ideal gas and calculated using the following formula:
is the absolute temperature, β can be
Substituting into Equation (3) gives:
Using Equation (2);
Using Equation (4);
4.1.4. Theoretical calculations The Rayleigh Number was found to be . Thus, it is laminar flow. For laminar flow, C = 0.59 and n = ¼. Thus, the following equation is used as Ra Using equation (9.27)
which was in the interval of
Where,
Thus,
Calculating percentage error of
= = 23.45 %
:
Forced Convection
A) Velocity = 0.6m/s = 24.3°C = 297.3 K Power = 100W Time (s) 0
40.0
27.8
29.3
31.1
30
40.2
27.8
29.2
31.2
60
40.4
27.8
29.5
31.3
90
40.5
27.8
29.5
31.3
120
40.7
27.9
29.4
31.4
150
40.9
27.9
29.7
31.5
180
41.0
28.1
29.7
31.6
210
41.3
28.1
29.8
31.7
240
41.3
28.2
29.9
31.8
270
41.5
28.2
29.9
31.8
300
41.5
28.3
30.0
31.9
330
41.7
28.4
30.1
32.0
360
41.7
28.4
30.1
32.0
Table 2: Forced convection - Temperature profile vs. Time (v = 0.6 m/s)
45 40 35 30 25 20 15 10 5
0 0 100 200 300 400
Tsurface
Linéaire
T1
(Tsurface)
T2
Linéaire (T1)
T3
Linéaire (T2) Linéaire (T3)
Graph 3: Forced convection - Temperature Profile versus Time (v = 0.6m/s)
Surface temperature vs. Time 41.8 41.6 41.4
Steady State
41.2
41 Time (min) 40.8 40.6 40.4 40.2 40 0
50
100
150
200
250
300
350
400
Surface Temperature (K) Graph 4: Forced Convection - Surface temperature vs. Time (v = 0.6 m/s)
4.1.5. Experimental Values Using Equation (1);
Total surface area of fins exposed is, A = 6 [ 2(8.7 × 0.6 ) + 2 (10.8 × 8.7 ) + 2 (0.6 × 10.8) ]= 1267.97
= 0.127
Assuming the total surface area of the fins exposed is equal to a vertical plate area, since the assumption taken in these calculations is that a vertical flat plate was used. The
average
value
of
= 40.9 °C = 313.98 K Thus, calculating for the value of h gives
was
calculated
using
Microsoft
Excel,
4.1.6. Calculation of Prandtl Number, Nusselt Number and Reynolds Number Thermo physical properties of Air at Atmospheric Pressure: From Appendix .. Table A.4, Using interpolation at a v = 0.6 m/s and
Pr number = 0.7062 Using the equation below taken from Chapter 7 section 7.4.1 of the prescribed text book, Reynolds number is calculated:
Using Equation (5);
Using Equation (3);
Theoretical Calculations Using equation 7.52 below, Nusselt number is calculated:
Where from table 7.3 of Appendix 2, following the assumption of a vertical flat plate, C = 0.228 and m = 0.731
To obtain
, =
86.17 = = 21.43 Calculating Percentage Error : Percentage error of Nusselt number =
= = 121.61 % Percentage error of h =
= = 120.30 %
B) Velocity = 0.9m/s = 22.0°C = 295 K Power = 100W Time (s) 0
31.8
28.8
26.5
31.3
30
32.0
28.8
26.5
31.4
60
32.2
28.8
26.6
31.6
90
32.6
28.8
26.7
31.8
120
32.6
29.0
26.8
32.1
150
32.8
29.0
26.7
31.9
180
33.0
29.1
27.0
32.5
210
33.2
29.2
27.0
32.6
240
33.4
29.4
27.1
32.8
270
33.6
29.4
27.2
33.0
300
33.7
29.7
27.3
33.2
330
33.8
29.7
27.3
33.5
360
33.8
29.7
27.3
33.5
Table 3: Forced Convection - Temperature Profile versus Time (v = 0.9 m/s)
40 35 30 25 20 15 10
5 0 0 100 200 300 400
Tsurface T1 T2 T3 Linéaire (T1) Linéaire (T2) Linéaire (T3) Linéaire
(T3)
Graph 5: Forced convection - Temperature Profile versus Time (v = 0.9 m/s)
Surface temperature vs. Time
33.8 33.6 33.4
tate
33.2 Time (min) 33 32.8 32.6 32.4 32.2 32 31.80
50
100
150
200
250
300
350
Surface Temperature (K) Graph 6: Forced convection - Surface temperature vs. Time (v = 0.9 m/s)
400
4.1.7. Experimental Values The experimental value of h can be calculated using the following formula Using Equation (1);
Total surface area of fins exposed is, A = 6 [ 2(8.7 × 0.6 ) + 2 (10.8 × 8.7 ) + 2 (0.6 × 10.8) ]= 1267.97
= 0.127
Assuming the total surface area of the fins exposed is equal to a vertical plate area, since the assumption taken in these calculations is that a vertical flat plate was used. The average value of
was calculated using Microsoft Excel,
= 32.9 °C = 305.96 K Thus, calculating for the value of h gives:
4.1.8. Calculation of Prandtl Number, Nusselt Number and Reynolds Number Thermo physical properties of Air at Atmospheric Pressure: From Table A.4, of Appendix 1 Using interpolation at a v = 0.9 m/s and
Pr number = 0.7069 Using the equation below taken from Chapter 7 section 7.4.1 of the prescribed text book, Reynolds number is calculated Using Equation (5);
Using Equation (3);
4.1.9. Theoretical Calculations Using equation 7.52 below, Nusselt number is calculated:
Where from table 7.3, following the assumption of a vertical flat plate, C = 0.228 and m = 0.731
To obtain the
, =
119.12 = = 29.00
Calculating Percentage Error : Percentage error of Nusselt number =
= = 147.65 % Percentage error of h =
= = 147.72 %
C) Velocity = 1.2m/s = 22.2°C = 295.2 K Power = 100W Time (s) 0
31.8
26.4
25.9
29.2
30
32.1
26.5
25.9
29.4
60
32.2
26.6
26.0
29.4
90
32.5
26.7
25.9
29.7
120
32.6
26.8
26.1
29.9
150
32.8
26.8
26.2
30.0
180
32.9
27.0
26.3
30.2
210
33.1
27.1
26.4
30.3
240
33.2
27.2
26.4
30.5
270
33.5
27.3
26.5
30.6
300
33.5
27.4
26.5
30.8
330
33.7
27.5
26.6
30.9
360
33.9
27.5
26.6
31.1
390
34.2
27.8
26.6
31.6
420
34.3
27.9
27.0
31.8
Table 4: Forced Convection – Temperature Profile vs. Time (v = 1.2 m/s)
40 35
Tsurface
30
T1 T2
25
T3
20
Linéaire
15
(Tsurface) Linéaire (T1)
10
Linéaire (T2)
5
Linéaire (T3)
0 0 400
100 500
200
300
Graph 7: Forced convection - Temperature Profile versus Time (v = 1.2 m/s) Surface temperature vs. Time 34.5
34 Steady State 33.5
Time (min) 33
32.5
32
31.50
50
100
150
200
250
300
350
400
450
Surface Temperature (K) Graph 8: Forced Convection - Surface temperature vs. Time (v = 1.2 m/s)
4.1.10. Experimental Values The experimental value of h can be calculated using the following formula: Using Equation (1)
Total surface area of fins exposed is : A = 6 [ 2(8.7 × 0.6 ) + 2 (10.8 × 8.7 ) + 2 (0.6 × 10.8) ]= 1267.97
= 0.127
Assuming the total surface area of the fins exposed is equal to a vertical plate area, since the assumption taken in these calculations is that a vertical flat plate was used. The average value of
was calculated using Microsoft Excel,
= 33.11 °C = 306.11 K Thus, calculating for the value of h gives,
Calculating for Prandtl Number, Nusselt Number and Reynolds Number: Thermo physical properties of Air at Atmospheric Pressure: From Table A.4, of Appendix 1 Using interpolation at a v = 0.9 m/s and
Pr number = 0.7069 Using the equation below taken from Chapter 7 section 7.4.1 of the prescribed text book, Reynolds number is calculated, Using Equation (5);
4.1.11. Theoretical Calculations Using equation 7.52 below, Nusselt number is calculated,
Where from table 7.3, following the assumption of a vertical flat plate, C = 0.228 and m = 0.731
To obtain the
, =
147.0 =
31
= 35.80
31
Calculating Percentage Error Percentage error of Nusselt number =
Percentage error of h = = 1 2 3 4 4.1 4.2 Dis cu s sion o f tr en ds an d in ter pret ation o f graph s
For free convection, the rate of heat transfer was held constant while the temperature of the surface of the fin was measured with time. The surface temperature of the fin increased exponentially with time. Therefore we can say that the surface temperature is proportional to time. Once the temperature reached its steady state, it was used to calculate all the dimensionless groups which were dependent on the fin length, i.e. Nusselt Number, Rayleigh Number and Reynolds Number. Graph 1 is the increment of the temperature profiles, T1, T2, T3 and surface temperature with time. This shows that temperature is directly proportional to time during free convection. The surface temperature changes more rapidly with time whereas the temperatures – T1, T2 and T3 increase less rapidly with a similar pace. As the graph illustrates, 45
This is because Tsurface is nearest to the vertical plate and T3, T2 and T1 follow respectively. This is shown in the figure below.
This pattern of temperature increments are seen during force convection as well. The progress of the surface temperature with time is seen more clearly in Graph 2. Here the surface temperature increases exponentially until it reaches the steady state. The time taken for the free convection experiment spans from 0 – 50 minutes and the temperatures range from approximately 30 – 80 K. Graph 3 shows the temperature patterns during forced convection at a velocity of 0.6 m/s. Here the surface temperature starts at a higher level in comparison to T1, T2 and T3 even though it increases in a similar pace. All the temperatures increase linearly but due to various types of errors the experimental values do not produce a perfectly linear graph. This is seen in Graph 4 where the surface temperature ultimately reaches steady state. The time taken for the two forced convection graphs span from 0 – 400 seconds and the temperatures range from 25 – 45 K. The surface temperature in Graph 5 increases linearly at a similar pace as the rest of the temperature values. Here the temperatures are taken during forced convection at a velocity at 0.9 m/s. This shows that as velocity increases the values of the temperature tend to produce similar patterns. Graph 6 explains the increment of surface temperature and shows how it 46
enters steady state clearly. The temperatures of these two graphs span from 25 – 35 K and the time taken is between 0 and 400 seconds. Graph 5 shows the same temperature patterns for a velocity of 1.2 m/s as seen previously for the forced convection velocities at 0.6 and 0.9 m/s. Graph 6 shows a linear increment of surface temperature until it reaches its steady state. The temperatures of these two graphs span from 25 – 35 K and the time taken is between 0 and 400 seconds. 4.3 ts
Com parison of Resu l
Velocity
Re
Pr
(m/s) 0.6
21.43
109.96
0.9
29.00
295.00
1.2
35.80
296.98
Table 5: Forced Convection - Comparison of results
4.4
Com parison with e xp ect ation s
Graph of heat transfer coefficient against 90
47
velocity of air
0.706 2 0.706 9 0.706 9
Heat Transfer coefficient, h 80 70 60 50 40 30 20 10 0 0
0,2
0,4 0,6 1,2 1,4 Velocity of air (m/s)
0,8
Graph 9: Heat transfer coefficient vs. velocity of air
48
1
As the above graph shows, the experiment values of the heat transfer coefficient do no coincide perfectly with the expect values. Theoretically, the graph of heat transfer coefficient against the velocity of air should ne linear but due to certain unavoidable errors this is not so.
Graph of log (Nu/Pr) against log (Re)
3 2,5 log (Nu/Pr) 2 1,5 1 0,5 0 3,55
3,6
3,65
3,7 3,9
3,75
3,8
3,85
3,95 log Re
Graph 10: log (Nu/Pr) vs. log (Re)
Theoretically even though the log values of (nu/Pr vs. The log value of Re should be linear the experimental results do not give a perfectly linear graph.
Graph of log (Nu exp) against log (Re) 49
3 2,5 2 1,5 log (Nu) 1 0,5 0
3,55
50
3,6
3,85
3,65
3,9 log (Re)
3,7
3,95
3,75
3,8
Graph 11: log (Nu exp) vs. log (Re)
The experimental results of the above graph are almost similar to the expected theoretical results. But due to certain errors the obtained points do not contribute to a linear graph. But since we use the best fit graph at instances such as these we could achieve the graph we’re R² = 0,99
expecting.
Graph of theoretical nusselt number against reynolds number
160 140
y = 0,0144x + 30,049 92
120 Nusselt number 100 80 60 40 20 0 0 10000
2000
4000
6000
Reynolds number
Graph 12: Theoretical Nusselt number vs. Reynolds Number
51
8000
Graph of experiment nusselt number against 350
reynolds number 300 Nusselt number 250 200 150 100 50 0 0
1000
2000 7000
3000 8000 Reynolds Number
4000 9000
5000
6000
Graph 13: Experimental Nusselt number vs. Reynolds number
Graph 12 and 13 show the comparison between the expected and the obtained graph using the experimental results.
Graph 12 was calculated using the theoretical value of
, whereas, the experimental value was calculated using . This equation uses the experimental heat transfer coefficient. Therefore, the experimental graph does not give perfectly linear points.
4.5
Errors Errors are inevitably prevalent in any experiment. Nevertheless, it is best to keep it as
minimal as possible. The discrepancies of data are due to a few assumptions made in the calculation of theoretical forces. These assumptions made paradoxically do not happen in the real world. One of the many assumptions made was the negligence of heat conduction through the air duct. Contrary to that theory, heat loss to surrounding would hugely influence the final result as huge amount of energy might have been lost to the surrounding before the calculation was done. Consequently, affects the convection heat transfer coefficient. In addition, heat loss may have happened at the back of the heat plate as the outer layer may not be perfectly insulated. Not to forget, the opening present at the temperature points of the wall
of air duct may disrupt the data accuracy. This is because; the openings may affect the temperature and the properties of the air, especially when forced convection was carried out. This error could be reduced by closing any opening present as tight as possible. Insulation of the system could also be improved. Besides that, the system was assumed to reach steady state when the slightest visible temperature could still be observed. The effect of such assumptions cannot be totally ignored, as it might contribute to discrepancies of results. Moreover, the experiment conducted may have been different from the standard conditions where the table of coefficient was set up. The constantly changing humidity and temperature in the air conditioned lab may affect the consistency of data yielded. Factors stated above may clarify the discrepancies of experimental and theoretical data.
4.6 ns
Dif ficu lti es/Lim itatio Contrasting from errors, limitations are mistakes present due to governing parameters
beyond our control which further explains the differences between the experimental values and theoretical. One of the main reasons is human limitation. This problem surfaced due to time lag. Time lags sets in as human do not have that most immediate and accurate response to time the stop watch and record down the data, concurrently. Nevertheless, this error can be reduced by repeating the experiments and conducting them with the same method, consistently. Another limitation present was the air humidity. The humidity of air might be different at any time of the experiments especially the free and forced convection experiments. This humidity although not taken into great importance, does affect the results because temperature gradients exist. Consequently, defy the notion to set Tinfinity as a constant as temperature distribution was occurring rapidly from time to time. The key to tackle these limitations is to perform the experiment several times consistently in a closed lab with stagnant air of constant air humidity and air velocity.
5. Appendices Appen di x 1
6. References
Incropera, F. D. (2007). In Fundamentals of Heat and Mass Transfer (6th Edition ed., pp. 371-377). New Jersy: John Wiley & Sons (Asia) Pte Ltd. Nusselt Number. (17 08, 2010). Retrieved 13 09, 2010, from Wikipedia, the free encyclopedia: http://en.wikipedia.org/wiki/Nusselt_number Prandtl Number. (04 09, 2010). Retrieved 13 09, 2010, from Wikipedia, the free encyclopedia: http://en.wikipedia.org/wiki/Prandtl_number Rayleigh Number. (03 09, 2010). Retrieved 13 09, 2010, from Wikipedia, the free encyclopedia: http://en.wikipedia.org/wiki/Rayleigh_number Reynolds Number. (10 09, 2010). Retrieved 13 09, 2010, from Wikipedia, the free encyclopedia: http://en.wikipedia.org/wiki/Reynolds_number
7. Nomenclature list
