FINAL_201 Thursday A-3 Convective and Radiant Heat Transfer

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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

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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

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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

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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

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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

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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.

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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.

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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

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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

Fra

ctio

n o

f H

eat

Lo

ss

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

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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

𝒎𝟐𝑲) Qc (W) hr (

𝑾

𝒎𝟐𝑲) 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.

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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

𝒎𝟐𝑲) Qf (W) hr (

𝑾

𝒎𝟐𝑲) 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

Fra

ctio

n o

f H

eat

Lo

ss

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

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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 10Tem

per

atu

re o

f H

eate

d E

lem

ent

(K)

Air Velocity (m/s)

Temperature of Heated Element vs. Air

Velocity

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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 √T4

. 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

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20

25

30

35

40

45

50

0 2 4 6 8 10Ra

te o

f C

on

vec

tiv

e H

eat

Tra

nsf

er (

W)

Air Velocity (m/s)

Forced Convective Heat Transfer vs.

Air Velocity

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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

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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 √T4

, 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

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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.

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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].

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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 (oC) Ts (oC) 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

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Appendix A-2

Equation Sheet

The equation for overall heat transfer coefficient due to natural convection for the first technical

objective is

ℎ𝑐 = 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

𝑄𝑐 = ℎ𝑐 × 𝐴𝑠 × (𝑇𝑠 − 𝑇𝑎), (2)

where ℎ𝑐 is the overall heat transfer coefficient due to natural convection, 𝐴𝑠 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

ℎ𝑟 = 𝜎 × 𝜉 × (𝑇𝑠4−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 equation for determining heat loss due to radiation is

𝑄𝑟 = ℎ𝑟 × 𝐴𝑠 × (𝑇𝑠 − 𝑇𝑎), (4)

where ℎ𝑟 is the overall heat transfer coefficient due to natural convection, 𝐴𝑠 is the surface area



The equation to determine total heat loss for the first technical objective is

𝑄𝑡𝑜𝑡 = 𝑄𝑐 + 𝑄𝑟 , (5)

where 𝑄𝑐 is heat loss due to natural convection and 𝑄𝑟 is heat loss due to radiation.

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The expression to determine the fraction of total heat loss due to natural convection is

𝑄𝑐

𝑄𝑡𝑜𝑡, (6)

where 𝑄𝑐 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

𝑄𝑟


where 𝑄𝑟 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

𝑄𝑓 = ℎ𝑓 × 𝐴𝑠 × (𝑇𝑠 − 𝑇𝑎), (11)

21

where ℎ𝑟 is the overall heat transfer coefficient due to forced convection, 𝐴𝑠 is the surface area



The equation to determine total heat loss for the second technical objective is

𝑄𝑡𝑜𝑡 = 𝑄𝑓 + 𝑄𝑟 , (12)

where 𝑄𝑓 is heat loss due to forced convection and 𝑄𝑟 is heat loss due to radiation.

The expression to determine the fraction of total heat loss due to forced convection is

𝑄𝑓


where 𝑄𝑓 is heat loss due to forced convection and 𝑄𝑡𝑜𝑡 is the total heat loss.

22

Appendix A-3

Example Calculations

Equation 1:

Using Technical Objective 1 at a target temperature of 100oC

ℎ𝑐 = 1.32 (374.9 − 294.7

. 01).25

= 12.49 𝑊 𝑚−2 𝐾−1

Equation 2:


𝑄𝑐 = 15.25 × .0022 × (473.2 − 295.0) = 5.98 𝑊

Equation 3:


ℎ𝑟 = (5.67 × 10−8) × .95 × (

572.14 − 295.24

572.1 − 295.2) = 19.36 𝑊 𝑚−2 𝐾−1

Equation 4:


𝑄𝑟 = 28.19 × .0022 × (673.1 − 295.5) = 23.42 𝑊

Equation 5:


𝑄𝑡𝑜𝑡 = 2.20 + 1.45 = 3.65 𝑊

Equation 6:


𝑄𝑐𝑄𝑡𝑜𝑡

=5.98 𝑊

11.02 𝑊= .542

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Equation 7:


𝑄𝑟𝑄𝑡𝑜𝑡

=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−2𝐾−1

Equation 9:


𝑁𝑢 = .3 +

(

. 62 × 300.5 ×. 708.33

(1 + (. 4. 708)

.66

)

.25

)

× (1 + (

300

282000).5

) = 8.98

Equation 10:


𝑅𝑒 =. 48 × .01

. 000016= 300

Equation 11:

Using Technical Objective 2 at a target velocity of 9 m/s

𝑄𝑓 = 111.09 × .0022 × (493.2 − 298.3) = 47.63 𝑊

Equation 12:


𝑄𝑡𝑜𝑡 = 45.26 + 5.91 = 51.17 𝑊

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Equation 13:


𝑄𝑓

𝑄𝑡𝑜𝑡=24.74 𝑊

41.82 𝑊= .592

FINAL_201 Thursday A-3 Convective and Radiant Heat Transfer

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Transcript of FINAL_201 Thursday A-3 Convective and Radiant Heat Transfer