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Helical Coil Heat Exchanger
Thermal Design of Heat Exchangers
ME 436
Fall 2007
Department of Mechanical Engineering
Faculty Advisor: Dr. Abdelmessih
Team Leader: Joel Parker
Team Members: S. Cummings, M. Jorgenson
Helical Coil Heat Exchanger Page 2
Table of Contents:
Helical Coil Heat Exchanger Design Report
The Problem Statement. .............. ...... ............ ........ ........ ......3
Acknowledgements............ ........ ........ ............ ........ ........ .. ...3
Background and Scope............. .. ....... . ............ ........ ........ ... ..4
Prel iminary Design............ ........ ........ ............ ........ ........ ......5
Prel iminary Calculations for the Heat Exchanger... ........ .... .. ..6
Final Design..... ..... ............ ........ ........ .......... .. ........ ... ..... .....7
The Apparatus....... ............ ........ ..... ... ............ ........ ........8 -12
Flow Chart and Description ......... ...... ............ ........ ....... 13-14
Revised Calculations for the Heat Exchanger......... .... .... . ....15
Results of the Test Run....... ......... ..... .. .......... ........ ........ ....16
Final Calculations for the Heat Exchanger........ ........ .... ... ....17
Discussion..... ....... ............ ........ ........ ............ ........ .... .... ....18
Concluding Remarks......... ......... ........ ............ ........ ..... ... ...19
Bibl iography......... . ............ ........ ........ ............ ........ ..... ... ...20
Nomenclature....... . ............ ........ ........ ............ ........ ........ ... 21
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Appendices.... ....... ... ......... ........ ........ ............ ........ ........ ..... ..22-37
A-1) Budget......... .. .... ........ ........ ........ ............ ........ ........ . ..22
A-2) Laboratory Manual Sample....... ... ............ ........ .... ...23-29
A-3) Contribution of Team Members.... . ........... ........ ........ ...30
A-4) Experimental Data....... .... ..... ..... ............ ........ ........ ....31
A-5) Calculat ions by Hand..... ........ ..... ............ ........ ...... 32-37
Prel iminary..... ...... ............ ........ ........ ........ .... .... ...32-34
Revised....... ........ . ............ ........ ........ ............ ..... ... ...35
Final........ ......... .... ............ ........ ........ ............ ... ...36-37
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Helical Coil Heat Exchanger Design Report :
The Problem Statement:
The thermal design of heat exchangers class received the
challenge of designing and building a second heat exchanger
apparatus for the thermal engineering laboratory. The heat
exchanger apparatus currently in the thermal engineering laboratory
is the 1-4 * shell and tube heat exchanger designed and built in
spring semester of 2002. The heat exchanger designed and built by
the thermal design of heat exchangers class wil l give future
mechanical engineering students an opportunity to work with a
helical coil in a hands-on manner. In testing this apparatus, future
students will be able to use their data to calculate Nusselt number,
heat transfer coeff icients and pressure drops for a helical coil.
Acknowledgements:
The Design Team would like to thank the following people for
their assistance in the completion of this project . First, we would
like to thank the school for funding this design project. We would
like to extend a thank you to Ms. Hopie Lopez for her assistanc e
with processing paperwork. We must also thank Otto Jorgenson, as
he has agreed to help make the plaque that we will proudly display
on the apparatus; it is a donation from Jorgenson Manufacturing in
Auburn.
* 1-4 means 1 shell with 4 tube passes
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Background & Scope:
Up unti l now, the students in heat transfer classes offered at
Saint Mart in’s University would not have been motivated to study
helical coils and their related equations and correlations. Now
however, they will not only be motivated to study these things, but
there exists a means by which to apply them practically in a
laboratory setting. By test ing our apparatus, they wil l gain
knowledge and experience that may benefit them in their careers.
As a term project, a f inned helical coi l was presented to the
team. The team was instructed to design a laboratory apparatus
featuring the helical coil. Then it was built, tested and an
experiment was created based on it. One of the main concerns for
this experiment was safety based on the maturity levels of the
students who wil l be testing the apparatus in future years . The
goals of the experiment are as follows ; students should be able to
calculate Nusselt number, calculate the heat transfer coeff icient,
calculate the friction factor, and calculate the pressure drop for the
helical coil.
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Prel iminary Design:
The original design for the apparatus involved the use of PVC
tubing because it is l ight, inexpensive, resistant to corrosion and
fouling, and easy to maintain. However, i t was dif f icult to f ind
f itt ings compatible with the thermocouples and the selected
industrial gauges.
The helical coi l is housed in a f ive-gallon plastic bucket. The
thermal characteristics of the plastic were considered as hot water
is being used. The risks of deformations over t ime from loading on
the lid of the bucket due to the weight of the helical coi l were also
considered.
The choice was made to work with single -phase condit ions for
the heat exchanger apparatus. Water was chosen as the working
f luid for both the hot side and the cold side for safety reasons. It is
a very safe f lu id to work with. Touching it does not cause chemical
burns although hot water can cause serious scalds and burns if
caution is not taken to avoid contact. It wil l not stain clothing, nor
will it destroy the actual fabric. It is not l ikely to cause i l lness if
ingested and it is not harmful if the vapor is inhaled.
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Prel iminary Calculations for the Heat Exchanger:
In this section, a summary of the calculations performed to
size components needed to build the heat exchanger is presented .
For further information, please refer to the Calculat ions section in
the appendices (A-5).
Table 1: Summary of Preliminary Calculations
Parameter Amount Units
Mass Flow Rate 2732.4 lb/hr
Mean Velocity 32346.8 ft/hr
Reynolds Number 53074
Prandtl Number 4.308
Dean Number 21667.4
Radius of Curvature 6.0
Nusselt Number
Straight 276.94
Coil 475.09
Heat Transfer Coefficient
Straight 2429.25 BTU/hr*sq. ft.*F
Coil 4167.38 BTU/hr*sq. ft.*F
Friction Factor
Straight 0.005169
Coil 0.02038
Pressure Drop 3.407 PSI
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Final Design:
We mounted the apparatus on a ut il ity cart for the purpose of
mobility and ease of storage. For the f inal design copper tubing and
brass f itt ings were chosen. We housed the helical coi l in a f ive -
gallon plast ic bucket . Furthermore, we took into consideration the
thermal characteristics of the plastic considering we are using hot
water. We also considered the risks of deformations over t ime from
loading on the lid of the bucket due to the weight of the helical coil .
For thermal data acquisit ion, we received two two-channeled Fluke
meters. They process the electrical dif ferential of the type -K
thermocouples and display a temperature reading .
Figure 1: The Complete Apparatus in i ts Final State
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The Apparatus:
The following is a piecewise breakdown of the other main
components that went into construct ion of the heat exchanger
apparatus. The prices are noted where applicable. For further
information on the budget please refer to appendix A -1.
The Helical Coil
This is the main component in our apparatus. The helical coi l
is the focus of the heat exchanger; our apparatus wil l be unique in
this aspect.
Figure 2: The Helical Coil Figure 3: CAD drawing of the Helical
Coil with Dimensions
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The Hot Water Heater:
The hot water heater is a 3.85 gallon 110 VAC water heater.
The hot water heater has a recovery rate of seven gallons per hour.
It has a maximum pressure of 150 PSI and a temperature range of
65-145˚F. It was purchased at Home Depot for $182.30.
Figure 4: The Hot Water Heater
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The Pump:
The pump is a one-half horsepower 110 VAC centrifugal pump
made by Chicago Electric Power tools. It was purchased at Harbor
Freight for $32. It is oversized for the design but due to budget
constraints, options were l imited. I t has a maximum flow rate of 330
Gallons per hour and 115 feet of l if t.
Figure 5: The Pump
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The Volumetric Flow Meters:
The volumetric f low meters are from Cole -Palmer. One is a
f if ty-f ive to three hundred gallon per hour f low meter (f igure 5a) for
use with the cooling water and it cost $72. The other is a zero to
sixty gallon per hour f low meter (f igure 5b) for the hot water and it
cost $110. Each Volumetric f low meter has a metering valve built
into it. The volumetric f low meters were chosen based on init ial f low
calculations for each individual f low source.
Figures 6a (left) and 6b (r ight): Volumetric Flow Meters
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Pressure Gauges
The Differential Pressure Gauge:
The differential pressure gauge (f igure 7a) is also from Cole-
Palmer, although it is an Ashcroft product. It cost $100. It has a
zero to f if teen pound per square inch dif ferential chosen based on
early calculat ions. After running the experiment, it was discovered
that the dif ferential was too large. With a calculated pressure drop
of 0.18 PSI a smaller dif ferential gauge would work better.
The Standard Pressure Gauges:
These were procured at Grainger for $86.40 for the set of two.
They are the solution to the init ial option not functioning properly.
Figure 7a: The Differential Pressure Gauge
Figure 7b: The Set of Two Standard Pressure Gauges
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Flow Chart and Description:
Figure 8: Flow Chart for the Final Design
The f igure above shows the f low paths of the hot water and
coolant, which are the working f luids for the heat exchanger
apparatus. The coolant, which is ordinary tap water, f lows from the
tap through a volumetric f low meter and then into the heat
exchanger that houses the coil. The water entering the system
spli ts off into two f lows, one to the hot water tank and one to the
volumetric f low meter. The cold water enters the exchanger shell
and discharges out the bottom of the exchanger to a drain. The hot
water runs directly through the helical coil into the reservoir, at the
reservoir where it drains to the pump. The f low splits again to the
reservoir and back to the water tank when valve 2 is closed creating
a closed circuit . The configuration dependent f low shown in green
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in the above f igure is dependent on whether valve 2 is open or
closed. When valve 2 is open, it allows cold tap water to enter the
hot water tank. When valve 2 is closed, it creates a closed circuit
recycl ing the hot water that is pumping though the system.
Table 2: Valves
Valve 1 Master Drain Valve
Valve 2 Source Inlet to Hot Water Tank
Valve 3 Pump Outlet
Valve 4 Reservoir Overflow Valve
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Revised Calculations for the Heat Exchanger:
In the table below is a summation of the results of calculating
with improved measurements. These measurements were taken to
aid in making the CAD drawing of the helical coil. The orange f i l l in
the table below shows what parameters changed. For further
information, please refer to the Calculations section in the
appendices (A-5).
Table 3: Summary of Revised Calculat ions
Parameter Amount Units
Mass Flow Rate 2732.4 lb/hr
Mean Velocity 32346.8 ft/hr
Reynolds Number 53074
Prandtl Number 4.308
Dean Number 18882.9
Radius of Curvature 7.9
Nusselt Number
Straight 276.94
Coil 457.48
Heat Transfer Coefficient
Straight 2429.25 BTU/hr*sq. ft.*F
Coil 4012.90 BTU/hr*sq. ft.*F
Friction Factor
Straight 0.005169
Coil 0.02038
Pressure Drop 3.59 PSI
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Results of the Test Run:
A test run was completed on the apparatus. Once all of the
components were in place, the system was checked thoroughly for
leaks. After f ixing the leaks, the apparatus was prepared for testing.
The test run commenced with the apparatus being tested under
laboratory condit ions. Data was recorded every f ive minutes unti l
the apparatus reached steady state. The hot temperatures fell as
expected; the cold temperatures seemed to be more unpredictable
in one instance rising six degrees in f ive minutes and then on the
next reading fall ing three degrees. The apparatus took 150 minutes
to reach steady state, which can vary based on operating condit ions.
Readings were taken until the three -hour mark; however, the data
became inconsistent, so a steady state set was determined based
on proximity of the readings. The raw data is included in the
appendices for reference (A-4). Full numerical results are
presented in the appendices (A-5) and a summation is given in the
section below. The apparatus is in working order and ready for use
in the laboratory with a few small exceptions (see discussion).
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Final Calculations for the Heat Exchanger:
In the tables below, we present our init ial condit ions and a
summation of calculations. We computed these calculat ions after
testing the apparatus under laboratory condit ions.
Table 4: Init ial Conditions
T-hot in T-hot out T-cold in T-cold out (dV/dt)hw (dV/dt)cw P-drop
93.0 84.0 55.5 61.5 58 52 DNF*
Table 5: Summary of Final Calculations
Parameter Amount Units
Mass Flow Rate 481.77 lb/hr
Mean Velocity 5685.5 ft/hr
Reynolds Number 7908
Prandtl Number 5.2
Dean Number 2814
Radius of Curvature 7.9
Nusselt Number
Straight 182.6
Coil 292.5
Heat Transfer Coefficient
Straight 1569 BTU/hr*sq. ft.*F
Coil 2153.00 BTU/hr*sq. ft.*F
Friction Factor
Straight 0.008414
Coil 0.03286
Pressure Drop 0.18 PSI
* Did Not Function
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Discussion:
These problems were encountered during the course of
working on the design. A reservoir bucket cracked during the
building process and a replacement was purchased. The apparatus
sprung a couple of leaks during the f irst and second trials. The hot
water heater had to be repositioned, as it would not function
properly in the previous position.
After testing the apparatus, some possible modif icat ions to the
system are suggested. One such suggestion is to procure a better
pressure gauge that f its our f inal design , one that can detect a
pressure drop of 0.18 PSI. This problem was so lved by using two
standard pressure gauges one for the inlet and one for the outlet.
Another suggestion is at some point replacing the probe
thermometer with a thermocouple to obtain the cold entrance
temperature more accurately. The thermocouple has been procured
and has been mounted and functioned properly .
The calculat ions went through mult iple runs, as new
information was discovered that changed what the results calculated
previously. A majority of the needed information for the calculations
was found in textbooks for previous courses or current courses.
The Konakov correlation for frict ion factor for the helical tube was
found in volume two of the Heat Exchanger Design Handbook.
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Concluding Remarks:
The future students will benef it from the hands-on study of the
helical coil heat exchanger apparatus. This apparatus will be a
valuable addition to the thermal engineering laboratory for years to
come.
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Bibl iography:
Heat Exchanger Design Handbook, Begell House, 2002 Ed. Volume
2 Section 5 Equation 42
Heat Exchangers: Select ion, Rating and Thermal Design, Kakac &
Liu 2nd Ed. Pgs 95 and 119
Fundamentals of Fluid Mechanics, Munson et . al. 5 t h Ed. Appendix
B Table 1
Fundamentals of Heat and Mass Transfer, Incropera, DeWit t,
Bergman & Lavine, 6 th Ed. Page 949,Table A.6 and Conversion
Factors, End pages
Thermal Engineering Laboratory Manual, Abdelmessih, 6 t h Ed. Heat
Exchanger, Chapter 17, Pages 58-63
Thermodynamics: An Engineering Approach, Boles and Cengel, 5 th
Ed. Page 938 Table A-3E
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Nomenclature:
Roman Symbols:
D= Coil diameter, inches
d= Tube diameter, inches
(dV/dt)cw=Cold Water Volumetric f low rate, Gallons per Hour
(dV/dt)hw= Hot Water Volumetric f low rate, Gallons per Hour
LL= long straight leg of the coil, inches
SL= short straight leg of the coil, inches
P-drop= pressure drop, PSI
Greek Symbols:
λ= curvature ratio (D/d)
Subscripts:
c= Coiled tube property
cw= cold water
hw= hot water
i= inner dimension
o= Outer dimension
s= Straight tube property
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Appendix A-1
Budget:
Our init ial budget allotted was $400. We ended up spending
approximately $830, which was definitely in excess of our init ial
budget. The approximated amounts were paid in cash and the
receipts have been submitted to the secretary of the Engineering
department. The exact budget is dif f icult to compute at this t ime, as
some of the receipts are not in the team’s possession.
Table 6: List of Expenditures
Part Source Unit Cost Quantity Total Cost
Ariston Water Heater Home Depot $ 182.30 1 $ 182.30
Galvanized Steel Bucket Lowe's $ 8.00 1 $ 8.00
Clearwater Pump Harbor Freight $ 32.00 1 $ 32.00
Differential Pressure Gauge Cole-Palmer $ 100.00 1 $ 100.00
Volumetric Flow meter (55-300 GPH) Cole-Palmer $ 72.00 1 $ 72.00
Volumetric Flow meter (0-60 GPH) Cole-Palmer $ 110.00 1 $ 110.00
Tax from Cole-Palmer purchases Cole-Palmer $ 23.69 1 $ 23.69
Copper tubing Lowe's $ 17.00 2 $ 42.00
Blue Bucket Lowe's $ 4.00 1 $ 4.00
Ball Valves Lowe's $ 7.00 4 $ 28.00
Compression Fit Spigot Lowe's $ 8.00 1 $ 8.00
Utility Cart Harbor Freight $ 43.35 1 $ 43.35
Clear 1/16th inch plexiglass DK Boos Glass $ 5.42 1 $ 5.42
Thermocouple shield Omega $ 22.50 1 $ 22.50
Pressure Gauge Grainger $ 43.20 2 $ 86.40
Sundry Fittings and Miscellany Various $ 174.66
Total $ 942.32
Table 7: List of Donations
Part Source
Duct Tape Peter Jorgenson
Commemorative Plaque Jorgenson Manufacturing
White Bucket Saint Martin's University Maintenance
Surge Protector Saint Martin's University
Helical Coil Thermal Engineering Laboratory
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Appendix A-2
(Draft)
Chapter ___
Helical Coil Heat Exchanger Experiment
Foreword:
In fall semester of 2007, Thermal Design of Heat Exchangers
class (Jorgscumpark Industries) created this laboratory experiment
for not only your enlightenment but also your entertainment. We
would like to take this opportunity to remind you to keep safety in
mind as you work through this experiment. So, remember have fun
and be safe.
Problem Statement:
The Heat Transfer Company is seeking your participation in a
program for the investigation of the Helical Coil Heat Exchanger
designed and built by Jorgscumpark Industries in the fall of 2007.
They donated their t ime creating this apparatus for the Thermal
Engineering Laboratory as well as for the generations of future
employees of the Heat Transfer Company. It is now up to you to
test it.
Purpose:
From your experimental data, you should be able to make
various calculations. You can calculate Nusselt number, heat
transfer coeff icient, and pressure drops for the helical coi l. You
should be able to f ind all the information you require in your Heat
Transfer textbook or within this laboratory manual. If you should
need any other information, you may perform a literature search for
it and cite the sources you used.
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Equipment:
Gauges:
Three Thermocouples and a probe thermometer
Two Volumetric f low meters- Cole Palmer
One Differential pressure meter- Ashcroft
Pump:
Clearwater Pump- Chicago Electr ic Power tools Model #01479
Buckets:
Five gallon white bucket- housing for the coil
Three-gallon bucket- overf low reservoir and trapped air removal
method
Hot Water Heater:
Four-gallon water heater- Ariston- It is imperative that you avoid
gett ing excess water on the hot water heater.
Data Acquisit ion:
2 two channeled Fluke meters
Miscellany:
A mop and mop bucket- for any spil ls and the leaky fresh water
supply
Rags or paper towels- also for spi l lage
Reservoir Cover
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Dimensions of the Helical Coil:
Figure 1: Dimensions of the Helical Coil
Experimental Procedure:
Experimental Phase:
1. First, as to not damage the circuit
breaker, turn off any equipment you
are not using for this laboratory
exercise.
2. Perform a safety check on the
apparatus. Check the device for
exposed wires and other such
dangers. This is also a good time to
check the connections on the
apparatus. Make sure that al l valves
are closed. Plug in the surge
protector.
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3. If the pump has not been used in
more than f ive days, remove the
cover and turn the fan with a
screwdriver.
4. Connect the source hose to the
source spigot, making sure that the
spigot is closed. Place the
discharge hose into the f loor drain.
5. Open valve 2 and Turn on source
water. Al low the water level to r ise
to a point where the helical coil is
fully submerged before. Use the
view port on the heat exchanger lid
to verify, visually, the water level.
While the heat exchanger is f i l l ing
up, turn on the hot water heater.
6. Once desired temperature is
reached, open the metering valve on
hot water volumetric f low meter and
begin f i l l ing the reservoir
approximately halfway.
7. Note: Make sure that the valves are
opened in the proper order specif ied.
8. Close valve 2
9. Open valve 3 and turn on pump by
plugging into the surge protector
(Make sure water in reservoir is at
least halfway full as fail ing to do so
will burn out the pump). This will
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create a closed circuit for the hot
water.
10. Adjust hot water f low rate as
desired.
11. Adjust valve 1 and cold water
f low rate (via the volumetric f low
meter) such that the cold water
entering the heat exchanger is in
synch with the cold water exit ing out
the discharge.
12. Connect the three
thermocouples to the Fluke meters.
Make note of thermocouple
orientat ion.
13. Insert the type-K temperature
probe into the hole on the top of the
heat exchanger until it makes
contact with the cold-water inlet f low.
Then connect it to the Fluke meter
port.
14. Begin taking readings at
desired intervals until steady state
conditions.
15. Keep an eye on water level in
the reservoir, if overf low is imminent
open valve 4 to discharge.
Table 1: Note on valves:
Valve 1 Master Drain Valve
Valve 2 Source Inlet to Hot Water Tank
Valve 3 Pump Outlet
Valve 4 Reservoir Overflow Valve
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Post Experimental Phase:
1. Secure the apparatus; close al l
valves unplug the pump, hot water
heater and anything else you
plugged in. Also, please leave the
thermal engineering lab as clean as
you found it.
2. From the volumetric f low rates,
calculate mass f low rates. Do this
with your steady state data
3. Predict the fouling factors for the
apparatus.
4. Begin your calculat ions for the
following…
a. Nusselt numbers
b. Heat transfer coeff icients
c. Pressure drops
Useful Correlations:
Petukhov’s Correlation for Nusselt number (2)
Nus=((f/2)*Re*Pr)/(1.07+12.7*(f/2)^(1/2)*(Pr^(2/3) -1))
Where f=(1.58*ln(Re)-3.28)^-2
Schmidt’s Correlat ion for Nusselt number (2)
Nuc=(1.0+3.6*(1-(1/ λ))*(1/ λ)^0.8)* Nus
Konakov’s Correlat ion for frict ion factor (1)
f=(1.8*log(Re)-1.5)^-2
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References:
(1) Heat Exchanger Design Handbook, Begell House, 2002 Ed.
(2) Heat Exchangers: Selection, Rating and Thermal Design, Kakac
& Liu 2nd Ed.
(3) Thermal Engineering Laboratory Manual, Abdelmessih, 6 th Ed.
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Appendix A-3:
Contribution of Team Members:
J. Parker: Team Leader & Treasurer
Overall design and construction of heat exchanger apparatus
Development of experimental procedure
Maintained budget records
General research
S. Cummings: Production Assistant & Draftsman
Construct ion of thermocouple units
Computer Drafting of f low chart and helical coil
Assisted in design and construction of the apparatus
Photography
General research
M. Jorgenson: Secretary & Principal Photographer
Calculat ions
Correlat ion research
Photography
Compiled f inal report
Scholar’s Day Liaison
Helical Coil Heat Exchanger Page 32
Appendix A-4:
Exper imental Data
Volumetr ic f low rates: Hot: 58 GPH Cold: 52 GPH
Table 8: Data Spreadsheet
Time (sec) time (min) T-hot in T-hot out T-cold in T-cold out
0 0 93.9 84.0 55.5 61.5
300 5 93.7 83.9 54.5 61.5
600 10 93.8 84.1 53.4 61.7
900 15 93.5 83.9 53.1 61.6
1200 20 93.2 83.7 53.4 61.6
1500 25 93.3 83.7 54.0 61.2
1800 30 93.1 83.5 52.5 61.2
2100 35 93.3 83.6 53.3 61.4
2400 40 93.3 83.5 53.4 61.6
2700 45 92.9 83.1 54.3 61.9
3000 50 93.1 83.3 54.3 61.7
3300 55 92.8 83.1 54.5 61.5
3600 60 92.9 83.1 54.9 61.6
3900 65 92.6 82.7 55.2 61.9
4200 70 92.8 82.9 54.0 61.9
4500 75 92.5 82.8 54.5 61.5
4800 80 92.7 82.8 55.4 61.6
5100 85 92.7 82.9 56.4 61.5
5400 90 92.7 82.7 56.9 61.6
5700 95 92.6 83.1 51.5 61.1
6000 100 92.7 83.3 51.8 61.4
6300 105 93.0 82.8 57.0 61.7
6600 110 92.7 82.7 55.3 62.3
6900 115 92.5 82.9 51.7 61.4
7200 120 92.5 82.9 51.9 61.4
7500 125 92.7 82.9 51.9 61.4
7800 130 92.9 83.0 51.9 61.5
8100 135 92.8 83.0 51.4 61.1
8400 140 92.8 83.1 51.6 61.4
8700 145 92.9 83.3 51.5 61.4
9000 150 92.9 83.3 51.5 61.4
9300 155 93.1 83.7 51.5 60.9
9600 160 93.3 83.6 51.5 61.7
9900 165 93.4 83.4 51.5 61.7
10200 170 93.3 83.3 51.5 61.7
10500 175 93.1 82.7 51.5 61.9
10800 180 92.9 82.5 51.5 61.9
The yel low highl ight indicates the values used for steady state calculat ions.