Download Convective and Radiant Heat Transfer CHE 0201 Thursday A

Convective and Radiant Heat Transfer
CHE 0201
Thursday A-5
Bridget Csongradi
Karah Horbach
Melissa Lane
Catherine McElhinny
Gabrielle Schantz
Progress Report
Spring 2015
Table of Contents
Nomenclature .............................................................................................................. 2
1.0 Introduction and Background ...................................................................................... 2
2.0 Experimental Methodology ......................................................................................... 4
2.1 Equipment and Apparatus ................................................................................ 4
2.2 Experimental Procedures ................................................................................. 4
3.0 Results ........................................................................................................................ 5
4.0 Analysis and Discussion of Results ............................................................................. 6
5.0 Summary and Conclusions .......................................................................................... 6
6.0 Future Work ................................................................................................................ 7
7.0 References ................................................................................................................... 8
8.0 Appendix A-1.............................................................................................................. 9
9.0 Appendix A-2.............................................................................................................. 13
2
Progress Report
Spring 2015
Nomenclature
Symbol
Definition
Units
As
Surface Area
m2
D
Diameter of Heated Cylinder
m
F
Area or geometric factor
-
hc
Heat Transfer Coefficient, Natural Convection
W/ m2k
hf
Heat Transfer Coefficient, Forced Convection
W/ m2k
hr
Heat Transfer Coefficient, Radiation
W/ m2k
I
Heater Current
A
k
Thermal Conductivity
W/mk
Nu
Nusselt Number
-
Pr
Prandtl Number
-
Qc
Heat Transfer, Natural Convection
W
Qf
Heat Transfer, Forced Convection
W
Qr
Heat Transfer, Radiation
W
Qtot
Total Heat Loss
W
Re
Reynolds Number
-
T1
Temperature of Surrounding Air
°C/K
T2
Temperature of Heater Surface
°C/K
Ua
Air Velocity
m/s
Ue
Effective Air Velocity
m/s
V
Voltage to Heater
V
𝜎
Stefan Boltzmann Constant for Radiation
-
𝜀
Emissivity of cylinder
-
v
Dynamic Viscosity of Air
m2/s
3
Progress Report
Spring 2015
1.0 Introduction and Background
There are three main types of heat transfer: conduction, convection, and radiation. These
types of heat transfers occur naturally and constantly throughout the universe and can be used
in practical applications any time a room, object, or process requires heat. Conduction is the
transfer of heat from a hot body to a cold body in which no material is flowing, usually
through a solid. Convection involves the transfer of heat from one location to another by the
movement of fluids, such as liquids or gases. The fluid carries energy with it as it moves
from a location of higher temperature to one of lower temperature. Lastly, radiation is the
transfer of heat by electromagnetic waves, or light. Thermal radiation is a type of heat
transfer since the electromagnetic radiation emitted from a central source carries energy as
the wave moves [1].
Convection is a type of heat transfer via moving fluids that can be utilized in process
equipment. Depending on how the flow begins, the convection can be natural or forced.
Natural convection is any fluid movement by natural means such as warmer fluid moving
upward and cooler fluid moving downward. A type of driving force would also be a
difference in density between two locations, resulting in the heat of one fluid being absorbed
by another fluid. Natural convection can be found throughout nature, such as in earth’s
oceans and atmosphere, which are heated by this force [1].
Forced convection occurs when a fluid flows over a surface by induced external forces,
like a pump, fan, or mixer. The motion of the fluid increases heat transfer; there is a direct
relationship between velocity and heat transfer- higher velocity equals more heat transfer [2].
A practical example of this type of heat transfer would be home heating systems which heat
the air by force. Air in this equipment is heated by a type of furnace and blown by fans into a
room. The fan acts as the driving force for the fluid to move into the room and transfer the
heat gained by the furnace into the room [1].
Radiation can also be used in process equipment as a method of heat transfer of
electromagnetic waves. Waves, which contain energy, are emitted from a source and move
towards surrounding objects. These objects or surfaces absorb the energy possessed by the
waves. This energy increases the average kinetic energy of the objects’ particles, which
raises the temperature of the object or surface [1].
4
Progress Report
Spring 2015
The Combined Convection and Radiation Heat Transfer Unit allows one to investigate
both natural convection with radiation and forced convention. The surface temperature can
be varied throughout the experiment in order to study the effect of increasing temperature on
radiant heat transfer. Air velocity flowing through the cylinder on the equipment can also be
varied to measure its effects on forced convection. This machine takes advantage of natural
convection and radiation as well as forced convection, when the blower is turned on. First,
radiation and heat loss by natural convection can be measured. By increasing the temperature
of the equipment, radiation heats the air. Air transfers this heat to the environment as well as
the surface of the equipment via natural convection as the heat moves from the hotter air to
colder surfaces and fluids. Second, the equipment can be used to measure forced convection
by varying the air velocity using an air blower, transferring the heat as the fluid is blown
through the equipment.
With radiation and natural convection, the fraction of heat loss from the heating element
that occurs by radiation and natural convection to the air at steady-state can be calculated
using experimental data found by testing the equipment. The proportion lost due to radiation
depends on the surface temperature of the cylinder and the surroundings, the type of material
of the cylinder, and the emissivity (efficiency with which a surface emits thermal energy) of
the cylinder and the surroundings.
The heat of natural convection can be calculated as follows:
Qc=hcAs(Ts-Ta) (1)
Where hc is the overall heat transfer coefficient due to natural convection, A s is the heated
surface area of the cylinder, Ts is the temperature of the cylinder, and T a is the temperature of
the surrounding air.
The heat transfer of radiation is found using:
Qr=hrAs(Ts-Ta) (2)
Where hr is the overall heat transfer coefficient due to radiant convection.
The total heat transfer from the cylinder is determined as follows:
Qtot=Qr+Qc (3)
For forced convection, the air velocity’s effect on the surface temperature as well as the rate
of convective heat transfer can be studied from the heat transfer unit as well.
5
Progress Report
Spring 2015
Similar to the previous equations, the heat loss due to forced convection can be found
using:
Qf=hfAs(Ts-Ta) (4)
Combining this with the same heat transfer equation found for the radiant component
gives an equation for total heat transfer for the forced convection analysis:
Qtot=Qr+Qf (5)
By comparing the relationship between the total heat transfer of forced convection as well as
the surface temperature of the heating unit, the effect of air velocity on heat transfer can be
determined.
This experiment involved the testing of the radiant and natural convection heat transfer
by varying the temperature and finding the amount of heat lost through convection and
radiation. The second part of the experiment involved varying the air velocity into the
cylinder as the air blower was on and analyzing the relationship between air velocity and
forced convection heat transfer.
2.0 Experimental Methodology
2.1 Equipment and Apparatus – The Combined
Convection and Radiation H112D heat transfer
module (see Figures 1 & 2 below) was used in
this experiment to determine the heat loss due to
radiant heat transfer and natural convection of
the air as well as the effect of air velocity on the
heating element and the rate of convective heat
transfer from the heating element to the air. This
unit has two displays for temperatures T1 and
T2; T2 is the heating surface temperature and T1
is the air temperature thermocouple. A knob is
located on the unit adjusts the voltage and
measures and displays the current in amperes.
6
Fig. 1. Combined Convection and
Radiation H112D with its heat transfer
module.
Progress Report
Spring 2015
The combined convection and radiation
equipment is a tall cylindrical duct with an
inlet to allow air to flow and also contains a
heated cylinder at the top of the duct. The
anemometer is attached to the middle of the
duct and records the air velocity, which is
displayed digitally at the bottom of the device.
The air is brought through the fans and is
controlled by a damper that determines how
much space is open for the air to flow through.
2.2 Experimental Procedures –
In order to start the procedure, the voltage
Fig. 2. Labeled schematic of the Combined
Convection and Radiation Unit
must be turned to zero and the air flow must
be fully opened. The initial air velocity was steadied at 0 m/s and the temperatures at T1 and
T2 were at room temperature. While completing the first technical objective, the air blower
remained off and the voltage dial was adjusted until the T2 temperature reached
approximately 100°C. After this temperature reached steady state, when temperature and air
velocity were constant for 5 seconds, measurements of the range of T1, T2, current, and
voltage were recorded.
T2 was increased in increments of approximately 100°C by
increasing the voltage, and T1, T2, current, and voltage were recorded after reaching steady
state. The final measurement was read at a temperature of 400°C.
Technical objective two started with the air damper fully opened and the voltage set to
approximately 200 V before adjusting the damper to allow for an air velocity of 0.5 m/s.
Once T1, T2, and air velocity reached steady state, measurements were recorded for T1, T2,
current, and voltage. The air velocity was increased in intervals of 0.5 m/s, and the steady
state temperatures and air velocity were recorded. The final data was recorded when the
maximum air velocity was attained with the damper fully open.
7
Progress Report
Spring 2015
3.0 Results–
0.65
From the data collected without the
100 °C, the calculated fractional heat loss due
to natural convection (Qc/Qtot) decreased as
the temperature of the heater surface (T2)
increased. The fractional heat loss from
radiant heat transfer (Qr/Qtot) increased as
the
temperature
of the
heater
surface
increased. Figures 3 and 4 show the results
Qx/Qtot (x= c or r)
blower on and at incremental temperatures of
0.6
0.55
0.5
Qr/Qto
t
0.45
Qc/Qto
t
0.4
0.35
0.3
0
200
400
600
T2 (K)
Fig 3. Fractional Heat Loss versus Temperature
of the heater surface (T2), Trial 1.
from the two trials.
When the blower was running and
0.65
temperature of the heater surface decreased
as the effective air velocity (Ue) increased.
Additionally, the rate of forced convective
heat transfer (Qf) increased as the effective
air velocity increased.
Qx/Qtot (x=c or r)
voltage and current were held constant, the
0.6
0.55
Qr/Qtot
0.5
0.45
Qc/Qtot
0.4
0.35
0.3
These results are
0
200
displayed in Figures 5 and 6, respectively.
400
600
T2 (K)
Fig 4. Fractional Heat Loss versus Temperature of
the heater surface (T2), Trial 2.
60
Qf (W)
50
40
30
Trial 1
Trial 2
20
10
Fig 5. Temperature of the heater surface (T2) versus
Effective Air Velocity (Ue), Trials 1, 2.
0
5
10
15
Ue (m/s)
Fig 6. Rate of Forced Convective Heat Transfer
(Qf) versus Effective Air Velocity (Ue), Trials 1, 2
8
Progress Report
Spring 2015
4.0 Analysis and Discussion of Results–
The first objective was to collect data on natural convection and radiation by measuring
various temperatures, voltage and current. From these recorded values one can apply
equations 2 and 4 in Appendix A-2 to calculate convective and radiant heat transfer. The total
heat transfer can then be calculated by Equation 5 in Appendix A-2. Due to radiant and
convective heat transfer a linear relationship is found when the temperature of the heater
surface is graphed with the fractional heat transfer. As expected, when the fractional heat loss
due to radiant heat transfer is large, at a specific temperature, the fractional heat transfer due
to convective heat transfer is small. There is no other energy transfer happening, therefore the
fractional heat transfer due to radiation must equal 1-(fractional heat transfer due to
convection). This can be explained by the first law of thermodynamics, which states that
energy cannot be created or destroyed but can change forms.
As the temperature of the heater surface increases, the fractional heat transfer due to
radiation increases and the fractional heat transfer due to convection decreases. Convection is
a slower method of heat transfer compared to radiation. Radiation is inefficient but does not
require material-to-fluid contact as convection does, therefore as the temperature increases
the amount of radiation increases since radiation has a faster transfer time than convection.
Objective two was obtained by varying the air velocity while the current and voltage
stayed constant to investigate the effect of forced convection. As the air velocity increased,
the temperature at thermocouple two decreased in both trials. This relationship allows for the
conclusion that the surface cools as the air velocity is increased, the higher the air velocity
the cooler the surface. As the air velocity increases past 8 m/s the temperature at the heater
surface starts to plateau. One can make an inference that if the velocity could be increased
past 10.5 m/s that the temperature at the heater surface would remain fairly constant at
around 500K.
As the air velocity increased the rate of forced convective heat transfer (Qf) increased,
but the rate of increase of forced convective heat transfer slows as the air velocity is
increased. As seen in Appendix A-2, Equation 16, the forced convective heat transfer is only
dependent on the difference in temperature between the surface and the surrounding air. As
the air moves faster through the cylinder, the temperature at the heater surface will decrease.
While the temperature at the heater surface decreases, the temperature of the surrounding air
9
Progress Report
Spring 2015
remains fairly constant. This phenomenon will result in a smaller slope as the velocity
increases.
Errors in results may have stemmed from the door to the laboratory being opened and
shut as students left. When the door was opened the air velocity would fluctuate, thus making
it difficult to record accurate values for velocity and temperature. Velocity also fluctuated
when people walked by briskly. As the air velocity became constant the temperature
fluctuated and vice versa.
5.0 Summary and Conclusions–
The project involved the investigation of natural convection and radiation as well as
forced convection. For Objective 1, the goal was to determine the fraction of total heat loss
that occurs by radiation and natural convection to the air at steady-state. The difference
between natural and forced convection is that natural convection does not utilize external
forces such as, in the case of this experiment’s equipment, an air blower. Natural convection
is caused by forces from density differences caused by temperature (i.e. warmer air moving
upward and cooler air moving downward) variations in the air. Therefore, Objective 1
required that the damper be fully opened and the air blower remain completely off. To
measure the variation of radiant heat transfer, the surface temperature of the equipment was
increased. Two trials on two separate occasions were conducted and the results indicated the
fractional heat loss from natural convection (QC/Qtotal) decreased as the temperature increased
the heat loss. Ranging from 0.6 down to 0.39 while the temperature ranged from 100 K to
400 K. Additionally the fractional heat loss from radiant heat transfer increased as the
temperature increased. This is a logical conclusion because of radiant heat’s direct
correlation with temperature to the fourth power. The results from both trials were consistent
with one another. The conclusion can be made that the proportion of heat lost at relatively
low cylinder temperatures is predominantly due to convection and the proportion of heat lost
at raised cylinder temperatures is lost from the surface due to radiation.
Technical Objective 2 involved the evaluation of the effect of air velocity on the heating
element surface temperature. A change had to be made to the steadied velocity of the
damper which, in Objective 1, was set to 0.5 m/s and increased at intervals of additional an
0.5 m/s throughout the objective to completion. Forced convective heat transfer directly
10
Progress Report
Spring 2015
correlates to velocity because the motion of the fluid increases heat transfer. The air velocity
is controlled using the damper, as air velocity increased the rate of heat transfer also
increased. In support of this statement the results indicated that when the blower was running
at steady state (constant voltage and current), the air velocity increased as did the rate of
forced convective heat transfer by the air blower.
6.0 Future Work –
To further the work and fine tune the results of this particular experiment the changes that
would need to be made are few. The results from Trials 1 and 2 were consistent with one
another and with the samples in the lab book. In terms of error the biggest source of it during
this experiment was the inconsistencies in air flow when other students would walk past the
apparatus or the lab door would open and shut. Improvements can be made simply by
moving the equipment to an area with lower human traffic or conducting the experiment
during a time when the lab is not so crowded. The movement of external forces that were not
caused by the air blower made the recording of data more time consuming because it was
more difficult to get cylindrical duct values steady at the same time as the heating transfer
unit.
7.0 References–
[1] “Methods of Heat Transfer.” (2015). The Physics Classroom.
http://www.physicsclassroom.com/class/thermalP/Lesson-1/Methods-of-Heat-Transfer
[2] M. Bahrami. “Forced Convection Heat Transfer.” Simon Fraser University.
http://www.sfu.ca/~mbahrami/ENSC%20388/Notes/Forced%20Convection.pdf
11
Progress Report
Spring 2015
Appendix A-1
Table 1. Trial 1: Data to determine the fractional heat loss from radiant heat transfer and convective heat transfer.
T1
T1
T1
T2
T2
T2
I
I
I
(°C)
(°C)
(°C)
(°C)
(°C)
(°C)
(Amps (Amps) (Amps V Low V High V Ave.
Low
High
Ave.
Low
High
Ave.
) Low
High
) Ave.
(Volts) (Volts) (Volts)
21.0
21.3
21.0
21.3
21.0
21.3
99.6
202.5
99.6
202.6
99.6
202.5
0.072
0.134
0.074
0.137
0.073
0.135
49
84
50
97
49.5
90.5
21.7
22.0
21.7
22.0
21.7
22.0
302.4
397.5
302.5
397.6
302.4
397.5
0.179
0.23
0.184
0.233
0.181
0.231
113
143
115
145
114
144
Table 2. Trial 1. Radiant and Convective Heat Transfer Coefficients, Radiant and Conductive Heat Transfer, Total
Heat Transfer, Input Heat, and Fractional Heat Loss due to radiation and convection.
T1 (°C)
Ave.
T2 (°C)
Ave.
hr (W/m^2 K)
Qr (W)
hc (W/m^2 K)
Qc (W)
Qtot (W)
QInput
(W)
Qr/Qtot
Qc/Qtot
21.0
99.60
8.52
1.47
12.50
2.16
3.63
3.61
0.405
0.594
21.3
202.55
13.66
5.44
15.35
6.12
11.57
12.26
0.470
0.529
22.7
302.45
20.64
12.74
17.11
10.57
23.32
20.29
0.546
0.453
22.0
397.55
29.40
24.29
18.39
15.20
39.49
33.33
0.615
0.384
Table 3. Trial 2. Data to determine the fractional heat loss from radiant heat transfer and convective heat transfer
T1
T1
T1
T2
T2
T2
(°C)
(°C)
(°C)
(°C)
(°C)
(°C)
I(Amp I(Amp I(Amp V Low V High V Ave.
Low
High
Ave.
Low
High
Ave.
s) Low s) High s) Ave. (Volts) (Volts) (Volts)
21.6
21.8
22.3
22.4
21.6
21.8
22.3
22.4
21.6
21.8
22.3
22.4
100.1
203.4
300.1
400.4
100.2
204.4
300.5
400.6
100.15
203.9
300.3
400.5
0.071
0.136
0.186
0.235
0.074
0.139
0.187
0.239
0.0725
0.1375
0.1865
0.237
51
89
115
147
52
90
116
149
51.5
89.5
115.5
148
Table 4. Trial 2. Radiant and Convective Heat Transfer Coefficients, Radiant and Conductive Heat Transfer, Total
Heat Transfer, Input Heat, and Fractional Heat Loss due to radiation and convection.
T1 (°C)
T2 (°C)
hr
hc
Ave.
Ave.
(W/m^2
(W/m^2
QInput
K)
Qr (W)
K)
Qc (W) Qtot (W)
(W)
Qr/Qtot
Qc/Qtot
21.6
100.15
8.56
1.48
12.50
2.16
3.64
3.73
0.406
0.593
21.8
203.90
13.76
5.51
15.37
6.15
11.67
12.30
0.472
0.527
22.3
300.30
20.50
12.53
17.07
10.44
22.98
21.54
0.545
0.454
22.4
400.50
29.73
24.73
18.43
15.33
12
40.068
35.07
0.617
0.382
Progress Report
Spring 2015
Table 5. Trial 1. Data collected to Calculate Effect of Air Velocity on Temperature and Convective Heat Transfer
T1 (°C) Ave.
T2 (°C) Ave.
24.8
25.7
26.3
26.6
26.9
26.9
26.9
26.9
26.8
26.6
375.25
365.30
355.90
336.05
321.45
313.75
306.45
298.45
290.70
274.25
Ua (m/s) Ave.
0.51
1.01
1.47
2.00
2.52
3.07
3.53
4.07
4.44
4.91
I (Amps) Ave.
0.327
0.332
0.332
0.330
0.331
0.333
0.334
0.332
0.334
0.333
V (Volts)
200.0
201.5
200.0
200.0
200.5
200.0
202.5
202.0
201.5
200.5
26.3
26.1
263.55
247.45
5.42
5.90
0.335
0.329
202.5
200.0
25.9
25.8
236.65
230.15
6.58
7.00
0.331
0.330
199.5
199.5
25.6
25.8
25.8
25.7
25.8
224.05
222.45
220.45
220.05
221.25
7.47
8.16
8.48
9.18
9.48
0.329
0.329
0.331
0.330
0.333
198.5
198.5
198.5
198.5
199.5
Table 6. Trial 1. Calculated Variables for the Calculation of Forced Convective Heat Transfer.
T1 (K)
T2 (K)
Ue (m/s)
v(m^2/s)
Pr
k (W/mK)
Re
Nu
297.8
298.7
299.3
299.6
648.25
638.30
628.90
609.05
0.62
1.23
1.79
2.44
1.5462E-05
1.5548E-05
1.5609E-05
1.5639E-05
0.708
0.708
0.708
0.708
0.0260
0.0261
0.0261
0.0262
402.41
792.51
1152.86
1564.06
10.40
14.69
17.84
20.93
299.9
299.9
299.9
299.9
299.8
299.6
299.3
299.1
298.9
298.8
594.45
586.75
579.45
571.45
563.70
547.25
536.55
520.45
509.65
503.15
3.08
3.75
4.31
4.96
5.41
5.99
6.61
7.19
8.03
8.54
1.567E-05
1.567E-05
1.567E-05
1.567E-05
1.566E-05
1.5639E-05
1.5614E-05
1.5589E-05
1.5568E-05
1.5558E-05
0.708
0.708
0.708
0.708
0.708
0.708
0.708
0.708
0.708
0.708
0.0262
0.0262
0.0262
0.0262
0.0262
0.0262
0.0261
0.0261
0.0261
0.0261
1965.87
2394.08
2752.22
3168.76
3459.07
3830.20
4234.92
4617.48
5160.31
5493.02
23.62
26.24
28.28
30.52
32.01
33.85
35.76
37.51
39.89
41.31
298.6
298.8
497.05
495.45
9.11
9.95
1.5538E-05
1.5558E-05
0.708
0.708
0.0261
0.0261
5869.25
6398.73
42.86
44.99
298.8
298.7
298.8
493.45
493.05
494.25
10.34
11.19
11.25
1.5558E-05
1.5548E-05
1.5558E-05
0.708
0.708
0.708
0.0261
0.0261
0.0261
6649.66
7203.27
7402.45
45.97
48.09
48.84
13
Progress Report
Spring 2015
Table 7. Trial 1. Temperatures of Surrounding Air (T1) and Heater Surface (T2), Effective Air Velocities, Forced
Convective Heat Transfer Coefficient, and Forced Convective Heat Transfer.
T1 (K)
T2 (K)
Ue (m/s)
hf (W/M^2*K)
Qf (W)
297.8
648.25
0.62
27.14
20.92
298.7
638.30
1.23
38.41
28.69
299.3
628.90
1.79
46.71
33.87
299.6
609.05
2.44
54.86
37.34
299.9
594.45
3.08
61.97
40.15
299.9
586.75
3.75
68.84
43.44
299.9
579.45
4.31
74.20
45.63
299.9
571.45
4.96
80.07
47.84
299.8
563.70
5.41
83.96
48.74
299.6
547.25
5.99
88.72
48.33
299.3
536.55
6.61
93.68
48.88
299.1
520.45
7.19
98.18
47.81
298.9
509.65
8.03
104.36
48.38
298.8
503.15
8.54
108.02
48.56
298.6
497.05
9.11
112.03
48.91
298.8
495.45
9.95
117.65
50.89
298.8
493.45
10.34
120.22
51.48
298.7
493.05
11.19
125.73
53.76
298.8
494.25
11.25
127.72
54.92
Table 8. Trial 2. Data collected to Calculate Effect of Air Velocity on Temperature and Convective Heat Transfer
T1 (°C) Ave.
T2 (°C) Ave.
Ua (m/s) Ave.
I (Amps) Ave.
24.3
25.5
26.3
27.0
378.4
373.3
355.7
329.8
0.45
1.00
1.49
2.09
0.328
0.331
0.331
0.330
199.5
202.0
202.5
201.5
27.1
27.3
27.1
26.8
27.0
26.7
26.4
26.2
25.9
25.7
322.4
311.7
300.0
294.1
278.7
274.3
256.1
244.9
232.9
229.5
2.45
2.93
3.45
4.10
4.48
4.91
5.50
5.91
6.55
6.87
0.330
0.331
0.329
0.331
0.329
0.330
0.331
0.330
0.330
0.332
199.5
200.5
199.5
200.0
199.5
200.0
200.5
199.5
199.5
199.5
25.6
25.7
25.7
25.6
227.0
223.7
223.0
220.9
7.60
8.00
8.54
9.00
0.333
0.329
0.329
0.329
199.0
199.5
197.5
198.5
14
V Ave. (Volts)
Progress Report
Spring 2015
Table 9. Trial 2. Calculated Variables for the Calculation of Forced Convective Heat Transfer.
T1 (K)
T2 (K)
v (m^2/s)
Pr
k (W/mK)
Re
Nu
297.3
298.5
299.3
300.0
300.1
300.3
300.1
299.8
300.0
299.7
651.45
646.35
628.75
602.85
595.45
584.75
573.05
567.15
551.75
547.35
1.54057E-05
1.55276E-05
1.56089E-05
0.00001568
1.56902E-05
1.57105E-05
1.56902E-05
1.56597E-05
0.00001568
1.56495E-05
0.708
0.708
0.708
0.708
0.707
0.707
0.707
0.708
0.708
0.708
0.0260
0.0261
0.0261
0.0262
0.0262
0.0262
0.0262
0.0262
0.0262
0.0262
360.32
789.62
1164.59
1630.03
1908.90
2279.17
2682.57
3198.08
3485.71
3827.72
9.84
14.66
17.93
21.39
23.25
25.56
27.89
30.68
32.15
33.83
299.4
299.2
529.10
517.90
1.5619E-05
1.55987E-05
0.708
0.708
0.0261
0.0261
4299.94
4622.30
36.06
37.53
298.9
298.7
505.95
502.50
1.55682E-05
1.55479E-05
0.708
0.708
0.0261
0.0261
5132.88
5390.68
39.78
40.88
298.6
298.7
298.7
298.6
500.00
496.75
496.05
493.95
1.55378E-05
1.55479E-05
1.55479E-05
1.55378E-05
0.708
0.708
0.708
0.708
0.0261
0.0261
0.0261
0.0261
5967.39
6281.29
6705.01
7066.65
43.26
44.52
46.19
47.58
Table 10. Trial 2. Temperatures of Surrounding Air (T1) and Heater Surface (T2), Effective Air Velocities, Forced
Convective Heat Transfer Coefficient, and Forced Convective Heat Transfer.
T1 (K)
T2 (K)
Ue (m/s)
hf (W/M^2*K)
Qf (W)
297.3
298.5
299.3
300.0
300.1
300.3
300.1
299.8
300.0
651.45
646.35
628.75
602.85
595.45
584.75
573.05
567.15
551.75
0.55
1.22
1.81
2.55
2.99
3.58
4.20
5.00
5.46
25.64
38.319
46.96
56.13
61.04
67.12
73.22
80.45
84.36
19.97
29.32
34.03
37.39
39.66
42.00
43.97
47.32
46.72
299.7
299.4
299.2
298.9
298.7
298.6
547.35
529.1
517.9
505.95
502.5
500.0
5.99
6.71
7.21
7.99
8.38
9.27
88.71
94.48
98.27
104.05
106.86
113.07
48.33
47.74
47.28
47.39
47.91
50.10
298.7
298.7
298.6
496.75
496.05
493.95
9.76
10.42
10.98
116.40
120.75
124.34
50.71
52.42
53.44
15
Progress Report
Spring 2015
Appendix A-2: Example Calculations
Constant Data
𝜀 = 𝑒𝑚𝑖𝑠𝑠𝑖𝑣𝑖𝑡𝑦 = 1.0
𝐹 = 𝑆ℎ𝑎𝑝𝑒 𝐹𝑎𝑐𝑡𝑜𝑟 = 1.0
𝜎 = 𝑆𝑡𝑒𝑓𝑎𝑛 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 5.67 ∗ 10−8 𝑊/𝑚𝐾
𝐴𝑠 = 𝐻𝑒𝑎𝑡𝑒𝑑 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 0.0022 𝑚2
D= diameter of cylinder= .01 m
Calculation for Radiant Heat Transfer Coefficient
ℎ𝑟 = 𝜀 ∗ 𝐹 ∗ 𝜎 (𝑇24 − 𝑇14 )/(𝑇2 − 𝑇1 )
ℎ𝑟 = 1.0 ∗ 1.0 ∗ 1.0
(1)
372.754 − 294.154
𝑊
= 8.52 2
372.75 − 294.15
𝑚 𝐾
Calculation for Radiant Heat Transfer
𝑄𝑟= ℎ𝑟 ∗ 𝐴𝑠 ∗ (𝑇2 − 𝑇1 )
(2)
𝑄𝑟= 8.52 ∗ .0022 ∗ (372.75 − 294.15) = 1.47 𝑊
Calculation for Convective Heat Transfer Coefficient
ℎ𝑐 = 1.32 (
ℎ𝑐 = 1.32 (
𝑇2−𝑇1 .25
𝐷
)
372.15 − 294.15
)
. 01
(3)
.25
= 12.50
𝑊
𝑚2 𝐾
Calculation for Convective Heat Transfer
𝑄𝑐 = ℎ𝑐 ∗ 𝐴𝑠 ∗ (𝑇2 − 𝑇1 )
(4)
𝑄𝑐 = 12.50 ∗ .0022 ∗ (372.15 − 294.15) = 2.16 𝑊
Calculation for Total Heat Transfer
𝑄𝑡𝑜𝑡 = 𝑄𝑐 + 𝑄ℎ
𝑄𝑡𝑜𝑡 = 2.16 + 1.27 = 3.63 𝑊
16
(5)
Progress Report
Spring 2015
Calculation for Fractional Heat Loss due Radiation
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐻𝑒𝑎𝑡 𝐿𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑄𝑟 /𝑄𝑡𝑜𝑡
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐻𝑒𝑎𝑡 𝐿𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 =
(6)
1.47
= 0.405
3.63
Calculation for Fractional Heat Loss due Natural Convection
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐻𝑒𝑎𝑡 𝐿𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝐶𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 = 𝑄𝑐 /𝑄𝑡𝑜𝑡
𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐻𝑒𝑎𝑡 𝐿𝑜𝑠𝑠 𝑑𝑢𝑒 𝑡𝑜 𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝐶𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 =
(7)
2.16
= 0.594
3.63
Calculation for Effective Air Velocity
𝑈𝑒 = 1.22 ∗ 𝐴𝑖𝑟 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
(8)
𝑈𝑒 = 1.22 ∗ .51 = .62 𝑚/𝑠
Calculation for Dynamic Viscosity of Air
𝑣 = (1.016 ∗ 10−7 )𝑇1 − 1.48 ∗ 10−7
(9)
𝑣 = (1.016 ∗ 10−7 )297.8 − 1.48 ∗ 10−7 = 1.546 ∗ 10−5
𝑚2
𝑠
Calculation for Prandtl Number
𝑃𝑟 = (−2.2 ∗ 10−4 )𝑇1 + .774
(10)
𝑃𝑟 = (−2.2 ∗ 10−4 )297.8 + .774 = .708
Calculation for Thermal Conductivity Constant
𝑘 = (7.58 ∗ 10−5 )𝑇1 + .0035
𝑘 = (7.58 ∗ 10−5 )297.8 + .0035 = .0260
(11)
𝑊
𝑚𝐾
Calculation for Reynolds Number
𝑅𝑒 =
𝑅𝑒 =
𝑈𝑒 ∗𝐷
𝑣
. 62 ∗ .01
= 402.41
1.546 ∗ 10−5
17
(12)
Progress Report
Spring 2015
Calculation for Nusselt Number
(.62∗𝑅𝑒 .5 ∗𝑃𝑟 .33 )
𝑁𝑢 = .3 +
.25
.4 .66
) )
𝑃𝑟
× (1 + (
(1+(
𝑁𝑢 = .3 +
(. 62 ∗ 402.41.5 ∗. 708.33 )
. 4 .66
(1 + (. 708) )
.25
𝑅𝑒
282000
.5
) )
(13)
. 708 .5
× (1 + (
) ) = 10.40
282000
Calculation for Forced Convective Heat Transfer Coefficient
𝑘
ℎ𝑓 = 𝐷 ∗ 𝑁𝑢
ℎ𝑓 =
(14)
. 0260
𝑊
∗ 10.40 = 27.14 2
. 01
𝑚 𝐾
Calculation for Forced Convective Heat Transfer
𝑄𝑓 = ℎ𝑓 ∗ 𝐴𝑠 ∗ (𝑇2 − 𝑇1 )
𝑄𝑟= 27.14 ∗ .0022 ∗ (372.75 − 294.15) = 20.92 𝑊
18
(15)