1. 1
Convective and Radiant Heat Transfer
University of Pittsburgh
ChE 0201 Section 1030
Foundations of Chemical Engineering Laboratory
Authors:
Kaitlin Muzic, Kaylene Kowalski, Kayla Williams,
Andrew Fogal, Elena Ream, and Dominique Chavis
Thursday A-3

2. 2
Table of Contents
Nomenclature 3
1.0 Introduction and Background 4
2.0 Experimental Methodology 7
2.1 Equipment and Apparatus 7
2.2 Experimental Procedures 8
3.0 Results 8
3.1 Technical Objective 1 Results 8
3.2 Technical Objective 2 Results 10
4.0 Analysis and Discussion of Results 13
5.0 Summary and Conclusions 15
References 17
Appendix A-1 18
Appendix A-2 19
Appendix A-3 22

3. 3
Nomenclature
Variable Description Units
Ts Temperature of Heated Surface K
Ta Temperature of Surroundings K
D Diameter of cylinder m
u Air velocity m/s
As Surface area of the heated cylinder m2
𝑣 Viscosity m2
s-1
hc Overall heat transfer coefficient due to natural convection W m-2
K-1
hr Overall heat transfer coefficient due to radiation W m-2
K-1
hf Overall heat transfer coefficient due to forced convection W m-2
K-1
Qc Heat loss due to natural convection W
Qr Heat loss due to radiation W
Qf Heat loss due to forced convection W
Qtot Total heat loss W
𝜎 Stephan Boltzmann Constant W m-1
K-1
𝜉 Emissivity constant No units
k Thermal Conductivity W m-1
K-1
Nu Nusselt number No units
Re Reynold’s number No units
Pr Prandtl number No units

4. 4
1.0 Introduction and Background
The most important aspect of any machine is efficiency. The ideal efficiency for a
machine is to have 100% of the net work be harnessed. Any energy loss in a process due to the
non-ideality of nature (e.g. friction, heat loss) results in lower efficiency and cost effectiveness.
Heat loss can occur through different forms of heat transfer like radiation, convection, and
conduction. Radiation is the emission of energy as electromagnetic waves (light) or as moving
subatomic particles through space. Convection is a type of heat transfer associated with motion
between fluids or gases due to a difference in temperature. Conduction is the transfer of heat
between two parts of a stationary system which, like convection, is caused by a temperature
difference between components [1]. An illustration of these forms of heat transfer can be seen
below in Figure 1. All of these types of heat transfer can have a dramatic effect on the efficiency
of a machine.
FIGURE 1. Illustration of different types of heat transfer [2]
A practical example of convective heat transfer that most people are familiar with is a
water heater. The overall efficiency of a water heater can be indicated by its Energy Factor (EF),
which is based on the amount of hot water produced per unit of fuel consumed over a time period
of one day [3]. A standard water heater has an EF between 0.54 and 0.58, which implies that
almost half of the energy used to heat the water is being lost [3]. Thus, there is a large industry in
researching materials and improving efficiency of heat and energy transfer units. If researchers

5. 5
can find materials that minimize energy loss, then a machine’s efficiency can be increased, thus
saving the consumer money.
In subjects such as physics and chemistry, matter is most stable when at a state with the
lowest amount of energy. To test this phenomenon, this experiment utilizes a heating element,
which provides heat and raises the temperature of the material it is put in contact with. For this
experiment, the heating element was placed on the top of a large cylindrical metal tube. Two
thermocouples were used to measure temperatures where one is near the top around the heating
element and one is in the area at the bottom of the tube. When the heating element increased in
temperature, the material’s molecules became excited and higher in energy. In order to return to
a lower level of energy, the molecules released heat to the surroundings. Natural convection is
one of the modes of heat transfer that allows for this flow of matter from higher to lower energy
levels. Entropy, the energetic degree of disorder of a system, also plays a role in driving heat
transfer [4]. The change in entropy of the universe is always greater than or equal to zero. The
heating element’s molecules become excited and more disordered, thus in an effort to increase
disorder, heat is transferred to the surroundings. In this experiment, the surroundings include the
area near the bottom of the cylindrical tube which is cooler and more energetically ordered
compared to the heated top area. Therefore, a convective current is created between the hot and
cold regions of the cylindrical tube. From this difference in temperature at the top and bottom of
the tube, the heat transfer coefficient due to natural convection can be calculated (hc). This
coefficient is the material’s ability to lose heat via convection. From this coefficient, the total
heat loss from convection (Qc ) can be determined. Other factors can also affect this flow such as
forced convection which was tested in this experiment by using a fan to increase the flow of air
through the cylindrical tube. This new air flow had a great impact on the convective motion
through the system [4].
There was also thermal radiation when the top of the cylindrical tube was heated, which
caused charged molecules--specifically protons and electrons--to move more rapidly. This
movement in the particles caused a release of electromagnetic radiation away from the heated
area. In some instances, though not in this experiment, the movement in the particles is so great
that electromagnetic waves in the visible light region are emitted causing the object to glow [5].
This is because, as previously stated, matter moves from high energy states to low energy states.
To calculate the heat transfer coefficient (hr) and heat loss from radiation (Qr), different

6. 6
properties of the material, such as emissivity, are considered. Emissivity is the effectiveness of a
material to emit thermal radiation. This value is a literature accepted value for each specific
material. Another variable in the calculation is the Stefan-Boltzmann constant. This constant was
derived from quantum mechanics and is the intensity of radiation over all wavelengths as the
temperature increases. These variables and constants are used in all thermal radiation
calculations [6].
No real-life heat transfer system is under ideal vacuum conditions so different non-ideal
conditions need to be considered. In the real world, machines are subjected to many factors that
all have an effect on its ability to transfer heat. The effects of these varying conditions on a
system require careful consideration, since industry is financially focused, the least amount of
energy lost means more money being gained.
The following experiment was conducted to determine the effects of temperature and air
velocity on different modes of heat transfer by varying the input voltage and increasingly
opening the air flow damper. The difference in temperature at the top of the heating element
from the bottom allows the heating coefficients of the system to calculated, whether subjected to
forced convection or not. Ultimately, the heat losses due to radiation, natural convection, and
forced convection were used to determine the total amount of heat loss from the system.

7. 7
2.0 Experimental Methodology
2.1 Equipment and Apparatus
The Combined Convection and Radiation H112D was used to complete the objectives of
this lab. The apparatus used in the lab is illustrated in Figure 2.
FIGURE 2: Annotated diagram of the Combined Convection and Radiation H112D
The main part of the apparatus consists of a cylindrical duct mounted over a centrifugal
fan base. The fan can be turned on and off with the fan switch located under the base of the duct.
At the top of the duct is a heated cylinder, or heating element. The power supplied to the heating
element is provided by the Heat Transfer Service Unit H112, which is pictured to the right of the
cylindrical duct in the figure. The electric current and voltage are measured and displayed on the
Heat Transfer Service Unit H112, and a knob is used to control how much power is supplied.
There are two thermocouples on the cylindrical duct, one near the heated cylinder that
measures the heated cylinder surface temperature and one toward the bottom of the cylindrical
duct that measures the temperature of the surroundings. The temperatures are shown on the
center display of the Heat Transfer Service Unit H112.

8. 8
In the middle of the duct, there is a Hot-Wire anemometer that measures air velocity. The
air velocity is displayed under the base of the cylindrical duct. At the base of the cylindrical duct,
there is a throttle butterfly, or air flow damper, which is used to control the air velocity.
2.2 Experimental Procedures
The first technical objective was to determine the fraction of total heat loss from the
heating element, due to radiation and natural convection, to the surroundings at steady state using
various temperatures. For this objective, the fan was turned off so that convective heat loss
would be due to natural rather than forced convection. Also, the air flow damper was completely
open throughout this objective.
Once the power was turned on, the power was adjusted to achieve steady state at a heated
cylinder surface temperature near 100ºC. At steady state, the temperature of the heated cylinder
surface, the temperature of the surroundings, the voltage required, and the electric current
required were measured. This process was repeated at a heated cylinder surface temperature near
200ºC, 300ºC, and 400ºC. For each of these temperatures, the system reached steady state and
the same properties were measured and recorded.
The second technical objective was to evaluate the effect of air velocity on the heating
element surface temperature and on the rate of convective heat transfer from the heating element
to the surroundings at steady state. The fan was turned on for this objective. The air flow damper
was initially completely open. Voltage to the heating element was set to about 200 volts.
The air flow damper was adjusted until an air velocity around 0.5 m/s was achieved.
Once the temperatures stabilized and steady-state was achieved, the actual air velocity, the
electric current, the voltage, and the temperatures of both the heating element and the air were
measured and recorded. This process was repeated at increments of around 0.5 m/s until the air
damper was completely open, where the air velocity was around 9 m/s. For each increment, the
system achieved steady-state and the same properties were measured and recorded each time.
3.0 Results
3.1 Technical Objective 1 Results
While completing Technical Objective 1, the following were observed: (1) the voltage
was adjusted multiple times to reach the correct target temperature, and (2) the voltage was

9. 9
adjusted generally after reaching a steady state where the temperature of the heating element was
consistently too high compared to the target temperature. Reaching the target temperature by
adjusting the voltage was completed using an educated trial-and-error process.
Technical Objective 1 employed analyzing the comparison of natural convection and
radiation. As seen in Figure 3, when the heat losses for each of these forms of heat transfer were
plotted, the plot looks like an “x.” The fraction of total heat loss due to natural convection with
varying temperatures consistently decreased to what appeared to be a negatively sloped linear
trend-line. By contrast, the fraction of heat loss due to radiation appeared to be a positively
sloped linear trend-line. The curves intersected near the midway point between the highest and
lowest temperatures of the heated surface element.
FIGURE 3. Fraction of total heat loss due to natural convection and fraction of total heat
loss due to radiation vs. temperature of the heating element
As seen in Table 1, the temperature of the surroundings was very consistent with little
deviation while both heat transfer coefficients and both forms of heat loss (natural convection
and radiation) were observed to increase as the surface temperature of the heating element
increased. The fraction of heat loss for both forms of heat loss stayed within a similar range from
0.604
0.542
0.468
0.3950.396
0.458
0.532
0.605
0.350
0.400
0.450
0.500
0.550
0.600
373 423 473 523 573 623 673
FractionofHeatLoss
Temperature of Heating Element (K)
Heat Loss due to Natural Convection and
Radiation vs. Temperature
Heat Loss due to Natural Convection Heat Loss due to Radiation

10. 10
about 0.4 to 0.6. Additionally, the total heat loss increased as the surface temperature of the
heating element increased, which indicated more heat was lost at higher temperatures. This is
consistent with the laws of thermodynamics as the system interacts with its surroundings, which
will be discussed in detail in the proceeding section of this paper.
Table 1. Data and calculated variables for Technical Objective 1
Ts (K) Ta (K) hc (
𝑾
𝒎 𝟐 𝑲
) Qc (W) hr (
𝑾
𝒎 𝟐 𝑲
) Qr (W) Qtot (W) Qc / Qtot Qr / Qtot
374.9 294.7 12.49 2.20 8.20 1.45 3.65 0.604 0.396
473.2 295.0 15.25 5.98 12.87 5.04 11.02 0.542 0.458
572.1 295.2 17.03 10.37 19.36 11.79 22.17 0.468 0.532
673.1 295.5 18.40 15.29 28.19 23.42 38.71 0.395 0.605
3.2 Technical Objective 2 Results
While completing Technical Objective 2, the following were observed: (1) the air
velocity was altered using the air flow damper, which seemed temperamental and highly
sensitive to the environment, (2) the system did not take long to reach steady state when there
were few observable changes in the surroundings, and (3) if someone opened the door or walked
behind the system, the temperature spiked but quickly returned to steady state.
Technical Objective 2 required analyzing the effect of air velocity on temperature of the
heating element and the rate of heat loss due to forced convection. It was decided to include the
comparison of fraction of heat loss due to radiation and forced convection to better analyze
Technical Objective 1 (see Figure 3). Compared to Figure 3, Figure 4 looks very different and
does not exhibit an “x” shaped property. Interpreted from Table 2, the intersection of heat loss
due to forced convection and heat loss due to radiation occurred at the highest recorded
temperature of the heating element. This coincides with the lowest air velocity.

11. 11
FIGURE 4. Fraction of total heat loss due to forced convection and fraction of total heat
loss due to radiation vs. temperature of the heating element
Table 2. Data and calculated variables for Technical Objective 2
u (m/s) Ts (K) Ta (K) hf (
𝑾
𝒎 𝟐 𝑲
) Qf (W) hr (
𝑾
𝒎 𝟐 𝑲
) Qr (W) Qtot (W) Qf / Qtot Qr / Qtot
0.48 636.5 298.6 23.75 17.66 24.90 18.51 36.17 0.488 0.512
1.01 624.5 299.2 34.57 24.74 23.86 17.07 41.82 0.592 0.408
1.56 605.3 299.6 43.24 29.08 22.23 14.95 44.03 0.660 0.340
1.98 594.7 299.9 48.95 31.74 21.38 13.86 45.61 0.696 0.304
2.51 581.8 300.2 55.43 34.34 20.36 12.62 46.96 0.731 0.269
3.1 575.4 300 61.99 37.56 19.86 12.03 49.59 0.757 0.243
3.47 565.6 300 65.82 38.46 19.11 11.17 49.63 0.775 0.225
3.95 559.4 299.9 70.55 40.28 18.65 10.65 50.92 0.791 0.209
4.45 546.6 299.6 75.22 40.88 17.71 9.62 50.50 0.809 0.191
4.97 535.4 299.4 79.85 41.46 16.92 8.79 50.24 0.825 0.175
5.49 522.1 299.1 84.29 41.35 16.01 7.86 49.21 0.840 0.160
5.97 508.8 299 88.23 40.72 15.15 6.99 47.72 0.853 0.147
6.46 498.3 298.8 92.13 40.43 14.49 6.36 46.80 0.864 0.136
6.97 493.2 298.6 96.06 41.12 14.18 6.07 47.19 0.871 0.129
7.45 490.9 298.4 99.65 42.20 14.03 5.94 48.14 0.877 0.123
8.05 489.5 298.3 104.02 43.75 13.94 5.87 49.62 0.882 0.118
8.5 490.3 298.4 107.21 45.26 14.00 5.91 51.17 0.885 0.115
9.06 493.2 298.3 111.09 47.63 14.16 6.07 53.71 0.887 0.113
0.100
0.300
0.500
0.700
0.900
485 505 525 545 565 585 605 625
FractionofHeatLoss
Temperature of Heating Element (K)
Heat Loss due to Forced Convection and
Radiation vs. Temperature
Heat Loss due to Forced Convection Heat Loss due to Radiation

12. 12
Also required for proper analysis of the system was to observe the effects of air velocity
on the temperature of the heated element. In Figure 5, the temperature of the heated element was
generally negatively linear until the target air velocity neared 7 m/s (see Table 2). After this
point, the temperature leveled out, but during the final two recorded velocities, the temperature
increased slightly.
FIGURE 5. Temperature of Heated Element vs. Air Velocity
Additionally, Technical Objective 2 necessitated observing the effect of air velocity on
the rate of convective heat transfer. In Figure 6, the relationship between the aforementioned
exhibits somewhat of a polynomial trend-line with a local maximum and minimum in the curve
with a turning point. As observed in Table 2, these dips occurred at air velocities of 4.97 m/s,
yielding a local maximum rate of heat transfer (41.46 W), and 6.46 m/s, yielding a local
minimum rate of heat transfer (40.43 W).
485
505
525
545
565
585
605
625
645
0 1 2 3 4 5 6 7 8 9 10
TemperatureofHeatedElement(K)
Air Velocity (m/s)
Temperature of Heated Element vs. Air
Velocity

13. 13
FIGURE 6. Rate of Forced Convective Heat Transfer vs. Air Velocity
4.0 Analysis and Discussion of Results
In Technical Objective 1, the goal was to determine the effect of temperature on the
fraction of heat loss due to natural convection versus the fraction of heat loss due to radiation.
The hypothesis was that as the temperature of the heating element increased, the fraction of heat
loss due to radiation would increase while the fraction of heat loss due to natural convection
would consequently decrease. The fraction of heat loss due to radiation was predicted to be
higher because, as air velocity remains constant, heat loss due to natural convection via air
currents becomes less significant to the total heat loss compared to radiation. In the equations
used for calculating heat loss due to radiation and convection, it can be seen that Qc and Qr both
increase with temperature. However, Qr is more highly dependent on temperature. It is
proportional to T4
, while Qc is proportional to √T
4
. This was proven by the data collected during
the experiment and can be seen in Figure 3. The fraction of convective heat transfer was
negatively proportional to temperature of the heating element; as the temperature of the heating
element increased, the fraction of heat loss due to natural convection decreased. The fraction of
heat loss due to natural convection decreased because, throughout the experiment, the damper
15
20
25
30
35
40
45
50
0 2 4 6 8 10
RateofConvectiveHeatTransfer(W)
Air Velocity (m/s)
Forced Convective Heat Transfer vs.
Air Velocity

14. 14
was kept open but the fan was turned off. With the fan off, the air velocity remained constant,
and as the heating element increased in temperature, the natural convection could not remove as
much heat. As seen in Table 1, when the heating element was at 374.9 K, the fraction of heat loss
due to natural convection was 0.604 and the fraction of heat loss due to radiation was 0.396. The
fraction of heat loss due to radiation steadily increased while heat loss to natural convection
decreased until the final recorded temperature was reached at 673.1 K, where the fractions of
heat loss were 0.605 and 0.396, respectively (Table 1).
In Technical Objective 2, the goal was to evaluate the effect of air velocity on the rate of
convective heat transfer from the heating element to air at steady state. The hypothesis for this
objective was that the fraction of heat loss due to the forced convective heat transfer would
increase as velocity increased. The heat loss due to forced convection was predicted to be
consistently greater than the fraction of heat loss due to radiation as the air velocity increased.
Increasing air velocity allowed more air to flow across the heating element, and thus caused heat
to be removed from the element through forced convection at a faster rate. However, the fraction
of heat loss due to forced convection increased as the temperature of the heating element
decreased, and consequently, the fraction of heat loss due to radiation decreased. Heat loss due to
radiation decreased because, with a decreased temperature of the heating element, the molecules
contained less energy to release to the surroundings. This means the molecules had less energy to
move from hot to cold air by means of radiation. The data showed that forced convection was
more of a contributing factor to the overall heat loss than radiation as velocity of the air
increased.
Another factor that caused the fraction of heat loss due to radiation to decrease as
velocity increased was keeping the voltage constant in the second objective. With a constant
voltage, the only effect on the temperature came from some method of heat transfer. By
increasing the air velocity, which decreases the temperature of the heating element, it was
observed that forced convection was the major contributing form of heat loss (Figure 4). The
fraction of heat loss due to forced convection was numerically much higher than that in objective
one and consistently remained higher while the fraction of heat loss due to radiation decreased.
Because of the effective heat loss due to forced convection, the temperature of the heating
element decreased. Figure 6 shows that more heat was lost to forced convection as air velocity

15. 15
was increased, and Figure 5 shows that the temperature of the heating element decreased with
heat loss due to forced convection.
5.0 Summary and Conclusions
This experiment investigated two different objectives by varying either temperature or air
velocity to understand how they affected heat transfer. The main concepts behind this project are
that the change in entropy of the universe is always greater than or equal to zero for any process
and that heat flows in such a way that molecules can exist at the lowest state of energy possible
by reaching a state of equilibrium. The first objective was to see how heat loss due to natural
convection compared to heat loss due to radiation at different temperatures. The data showed as
temperature increased, more heat was lost to radiation than to natural convection. This is because
the heat can be removed from the heating element more efficiently through radiation. Since the
fluid (air) is flowing at a negligible rate around the heating element, it does not provide a suitable
means of removing heat through natural convection, thus forcing heat to be lost in the form of
radiation. Looking at the equations supports this notion. The heat due to natural convection is
proportional to √T
4
, which appears in the overall heat transfer coefficient for natural convection.
By contrast, the heat due to radiation is proportional to T4
, which appears in the overall heat
transfer coefficient for radiation. Thus, heat transfer due to radiation is drastically more
dependent on temperature than heat transfer due to natural convection.
The second objective was to see how varying air velocity affected the rate of heat transfer
due to forced convection as well as the temperature of the heating element. According to the
results, as air velocity increased, the heated cylinder surface temperature decreased. For this
objective, air is no longer moving at a negligible rate and thus provides a more effective means
of heat removal with increasing velocity. Initially, heat loss is equal for radiation and forced
convection, but as the air velocity increases, the heat loss due to radiation diminishes as forced
convection can more quickly dissipate the heat.
Heat transfer is important in industry to cool processes that generate a lot of heat. The
underlying principles of this project are thus applied greatly in industry, especially forced
convection. Examples of cooling processes in industry that utilize forced convection include:
shell and tube exchanger, brazed plate heat exchanger, and plate and frame exchanger. The heat

16. 16
exchanger most commonly used for cooling processes is the shell and tube exchanger [7].
Evidently, forced convection is an important thermodynamic concept for industrial applications.

17. 17
References
[1] Tufts School of Engineering, “Chapter 1: Overview of Heat Transfer,” 2002. [Online].
Available: http://www.tufts.edu/as/tampl/en43/lecture_notes/ch1.html. [Accessed: March
1, 2016].
[2] “Modes of Heat Transfer -- Conduction, Convection, & Radiation,” 2015. [Online].
Available: http://www.spectrose.com/modes-of-heat-transfer-conduction-convection-
radiation.html. [Accessed: March 1, 2016].
[3] U.S. Department of Energy, “Estimating Costs and Efficiency of Storage, Demand, and
Heat Pump Water Heaters,” [Online]. Available:
http://energy.gov/energysaver/estimating-costs-and-efficiency-storage-demand-and-heat-
pump-water-heaters. [Accessed: March 1, 2016].
[4] Engineers Edge, “Convective Heat Transfer Convective Equation and Calculator,” 2016.
[Online]. Available: http://www.engineersedge.com/heat_transfer/convection.htm.
[Accessed: March 1, 2016].
[5] Thermal Fluids Central, “Radiation,” July 2010. [Online]. Available:
https://www.thermalfluidscentral.org/encyclopedia/index.php/Radiation. [Accessed:
March 1, 2016].
[6] E. Narimanov and I. Smolyaninov, “Beyond Stefan-Boltzmann Law: Thermal Hyper-
Conductivity,” Sept 2011. [Online Document]. Available:
http://arxiv.org/pdf/1109.5444.pdf. [Accessed: March 1, 2016].
[7] Cooling Technology, “Heat Transfer Equipment,” 2015. [Online]. Available:
http://www.coolingtechnology.com/about_process_cooling/heat-transfer/default.html.
[Accessed: March 1, 2016].

18. 18
Appendix A-1
Experimental Data
Table 1. Technical Objective 1
Time (s) V
(Volts)
i (Amps) Ta (oC) Target
Ts (oC)
Ts (oC) Ta (K) Ts (K)
11.07 50 0.71 21.7 100 101.9 294.7 374.9
11.34 83 0.133 22 200 200.2 295 473.2
12.13 114 0.182 22.2 300 299.1 295.2 572.1
14.49 146 0.235 22.5 400 400.1 295.5 673.1
Table 2. Technical Objective 2
Target u
(m/s)
u (m/s) i (Amps) V (Volts) Ta (o
C) Ts (o
C) Ta (K) Ts (K)
0.5 0.48 0.325 198 25.6 363.5 298.6 636.5
1 1.01 0.325 197 26.2 351.5 299.2 624.5
1.5 1.56 0.325 197 26.6 332.3 299.6 605.3
2 1.98 0.326 198 26.9 321.7 299.9 594.7
2.5 2.51 0.325 197 27.2 308.8 300.2 581.8
3 3.1 0.326 198 27 302.4 300 575.4
3.5 3.47 0.328 199 27 292.6 300 565.6
4 3.95 0.329 199 26.9 286.4 299.9 559.4
4.5 4.45 0.327 199 26.6 273.6 299.6 546.6
5 4.97 0.328 199 26.4 262.4 299.4 535.4
5.5 5.49 0.328 198 26.1 249.1 299.1 522.1
6 5.97 0.328 198 26 235.8 299 508.8
6.5 6.46 0.328 197 25.8 225.3 298.8 498.3
7 6.97 0.327 195 25.6 220.2 298.6 493.2
7.5 7.45 0.325 196 25.4 217.9 298.4 490.9
8 8.05 0.324 195 25.3 216.5 298.3 489.5
8.5 8.5 0.328 198 25.4 217.3 298.4 490.3
9 9.06 0.328 198 25.3 220.2 298.3 493.2

19. 19
Appendix A-2
Equation Sheet
The equation for overall heat transfer coefficient due to natural convection for the first technical
objective is
ℎ 𝑐 = 1.32 (
𝑇𝑠−𝑇𝑎
𝐷
)
.25
, (1)
where 𝑇𝑠 is the temperature of the heated surface, 𝑇𝑎 is the temperature of the surrounding area,
and 𝐷 is the diameter of the cylinder.
The equation for determining heat loss due to natural convection is
𝑄 𝑐 = ℎ 𝑐 × 𝐴 𝑠 × (𝑇𝑠 − 𝑇𝑎), (2)
where ℎ 𝑐 is the overall heat transfer coefficient due to natural convection, 𝐴 𝑠 is the surface area
of the heated cylinder, 𝑇𝑠 is the temperature of the heated surface, and 𝑇𝑎 is the temperature of
the surrounding area.
The equation for overall heat transfer coefficient due to radiation is
ℎ 𝑟 = 𝜎 × 𝜉 × (
𝑇𝑠
4−𝑇𝑎
4
𝑇𝑠−𝑇𝑎
), (3)
where 𝜎 is the Stephan Boltzmann Constant which is 5.67×10-8
W m-1
K-1
, 𝜉 is the emissivity of
the surface which is .95, 𝑇𝑠 is the temperature of the heated surface, and 𝑇𝑎 is the temperature of
the surrounding area.
The equation for determining heat loss due to radiation is
𝑄 𝑟 = ℎ 𝑟 × 𝐴 𝑠 × (𝑇𝑠 − 𝑇𝑎), (4)
where ℎ 𝑟 is the overall heat transfer coefficient due to natural convection, 𝐴 𝑠 is the surface area
of the heated cylinder, 𝑇𝑠 is the temperature of the heated surface, and 𝑇𝑎 is the temperature of
the surrounding area.
The equation to determine total heat loss for the first technical objective is
𝑄𝑡𝑜𝑡 = 𝑄 𝑐 + 𝑄 𝑟, (5)
where 𝑄 𝑐 is heat loss due to natural convection and 𝑄 𝑟 is heat loss due to radiation.

20. 20
The expression to determine the fraction of total heat loss due to natural convection is
𝑄 𝑐
𝑄 𝑡𝑜𝑡
, (6)
where 𝑄 𝑐 is heat loss due to natural convection and 𝑄𝑡𝑜𝑡 is the total heat loss.
The expression to determine the fraction of total heat loss due to radiation is
𝑄 𝑟
𝑄 𝑡𝑜𝑡
, (7)
where 𝑄 𝑟 is heat loss due to radiation and 𝑄𝑡𝑜𝑡 is the total heat loss.
The equation for overall heat transfer coefficient due to forced convection for the second
technical objective is
ℎ𝑓 =
𝑘×𝑁𝑢
𝐷
, (8)
where k is the thermal conductivity which was determined to be .0264 W m-1
K-1
from the
provided k vs. temperature graph, 𝐷 is the diameter of the cylinder, and 𝑁𝑢 is the Nusselt
number.
The equation for the Nusselt number is
𝑁𝑢 = .3 + (
.62×𝑅𝑒.5×𝑃𝑟.33
(1+(
.4
𝑃𝑟
)
.66
)
.25 ) × (1 + (
𝑅𝑒
282000
)
.5
), (9)
where 𝑅𝑒 is Reynold’s number and 𝑃𝑟 is the Prandtl number which was determined to be .708
from the provided Pr vs. temperature graph.
The equation for Reynold’s number is
𝑅𝑒 =
𝑢×𝐷
𝑣
, (10)
where 𝑢 is the effective air velocity, 𝐷 is the diameter of the cylinder, and 𝑣 is the kinematic
viscosity which was determined to be .000016 m2
s-1
from the provided v vs. temperature graph.
The equation for determining heat loss due to forced convection is
𝑄 𝑓 = ℎ𝑓 × 𝐴 𝑠 × (𝑇𝑠 − 𝑇𝑎), (11)

21. 21
where ℎ 𝑟 is the overall heat transfer coefficient due to forced convection, 𝐴 𝑠 is the surface area
of the heated cylinder, 𝑇𝑠 is the temperature of the heated surface, and 𝑇𝑎 is the temperature of
the surrounding area.
The equation to determine total heat loss for the second technical objective is
𝑄𝑡𝑜𝑡 = 𝑄 𝑓 + 𝑄 𝑟, (12)
where 𝑄 𝑓 is heat loss due to forced convection and 𝑄 𝑟 is heat loss due to radiation.
The expression to determine the fraction of total heat loss due to forced convection is
𝑄 𝑓
𝑄 𝑡𝑜𝑡
, (13)
where 𝑄 𝑓 is heat loss due to forced convection and 𝑄𝑡𝑜𝑡 is the total heat loss.

22. 22
Appendix A-3
Example Calculations
Equation 1:
Using Technical Objective 1 at a target temperature of 100o
C
ℎ 𝑐 = 1.32 (
374.9 − 294.7
. 01
)
.25
= 12.49 𝑊 𝑚−2
𝐾−1
Equation 2:
Using Technical Objective 1 at a target temperature of 200o
C
𝑄 𝑐 = 15.25 × .0022 × (473.2 − 295.0) = 5.98 𝑊
Equation 3:
Using Technical Objective 1 at a target temperature of 300o
C
ℎ 𝑟 = (5.67 × 10−8
) × .95 × (
572.14
− 295.24
572.1 − 295.2
) = 19.36 𝑊 𝑚−2
𝐾−1
Equation 4:
Using Technical Objective 1 at a target temperature of 400o
C
𝑄 𝑟 = 28.19 × .0022 × (673.1 − 295.5) = 23.42 𝑊
Equation 5:
Using Technical Objective 1 at a target temperature of 100o
C
𝑄𝑡𝑜𝑡 = 2.20 + 1.45 = 3.65 𝑊
Equation 6:
Using Technical Objective 1 at a target temperature of 200o
C
𝑄 𝑐
𝑄𝑡𝑜𝑡
=
5.98 𝑊
11.02 𝑊
= .542

23. 23
Equation 7:
Using Technical Objective 1 at a target temperature of 200o
C
𝑄 𝑟
𝑄𝑡𝑜𝑡
=
5.04 𝑊
11.02 𝑊
= .458
Equation 8:
Using Technical Objective 2 at a target velocity of 0.5 m/s
ℎ𝑓 =
. 0264 × 8.98
. 01
= 23.75 𝑊 𝑚−2
𝐾−1
Equation 9:
Using Technical Objective 2 at a target velocity of 0.5 m/s
𝑁𝑢 = .3 +
(
. 62 × 300.5
×. 708.33
(1 + (
. 4
. 708
)
.66
)
.25
)
× (1 + (
300
282000
)
.5
) = 8.98
Equation 10:
Using Technical Objective 2 at a target velocity of 0.5 m/s
𝑅𝑒 =
. 48 × .01
. 000016
= 300
Equation 11:
Using Technical Objective 2 at a target velocity of 9 m/s
𝑄 𝑓 = 111.09 × .0022 × (493.2 − 298.3) = 47.63 𝑊
Equation 12:
Using Technical Objective 2 at a target velocity of 8.5 m/s
𝑄𝑡𝑜𝑡 = 45.26 + 5.91 = 51.17 𝑊

24. 24
Equation 13:
Using Technical Objective 2 at a target velocity of 1.0 m/s
𝑄 𝑓
𝑄𝑡𝑜𝑡
=
24.74 𝑊
41.82 𝑊
= .592