Thermal Analysis of Single Walled and Double Walled Beverage
Containers
by
Peter Tu
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Professor Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2014
i
© Copyright 2014
by
Peter Tu
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
SYMBOLS ....................................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. INTRODUCTION ......................................................................................................... 1
2. Theory & Methodology ................................................................................................. 3
2.1 Theory ................................................................................................................... 3
2.2 Methodology ......................................................................................................... 4
3. Results & Discussion ..................................................................................................... 7
3.1 COMSOL Heat Transfer in Solids Model – Single Walled Cup............................... 13
4. Conclusions.................................................................................................................. 39
5. References.................................................................................................................... 40
6. Appendices .................................................................................................................. 43
iii
LIST OF TABLES
iv
LIST OF FIGURES
Figure 1: Left - Single walled cup; Right - Double walled cup
Figure 2: Left – Plastic with straw, Center – Titanium with handle, Right – Ceramic
Figure 3: Schematic of the heat transfer process: 1. Thermal conduction through inner
wall, 2. Convection via trapped gas, 3. Thermal conduction through outer wall, 4.
Energy lost to environment
v
SYMBOLS
𝑄
𝑡
= heat transferred over time
𝑘 = thermal conductivity of the material
𝐴 = area of the interface
𝑇 = temperature
𝑑 = thickness of the interface
vi
ACKNOWLEDGMENT
I would like to acknowledge my family for their constant support and encouragement
throughout my career and studies. I would also like to acknowledge Professor Ernesto
Gutierrez-Miravete for his guidance and council as I conducted this final study.
vii
ABSTRACT
This project documents the analytical simulations that were performed along with the
collection of empirical data to confirm the validity of the claim that double walled
beverage cups keep hot liquids hotter for longer periods of time than single walled cups.
viii
1. INTRODUCTION
The traditional cup is typically a single walled container formed to hold liquids which
result can result in a rapid loss or gain of thermal energy depending on the geometry and
materials used. The need for cups that were less prone to rapid heat transfer has been
addressed by the invention of double walled cups.
The first double walled cup patent was issued on July 22, 1969 under number 3456860
to William L. Janninck. The original design featured plastic inner and outer walls
supported by circumferential and axial ribs that maintained structural integrity of the cup
while minimizing heat transfer from the inner to outer wall. Modern day manufacturers
now make double walled cups from a variety of materials, shapes, and sizes to
distinguish their products in terms of aesthetics and functionality.
Figure 1 depicts a single walled cup next to a double walled cup.
Figure 1: Left - Single walled cup; Right - Double walled cup
1
Figure 2 provides examples of products from various manufacturers to illustrate diverse
range of double walled cups.
Figure 2: Left – Plastic with straw, Center – Titanium with handle, Right – Ceramic
Based on similarity of double walled beverage cups to double paned windows, double
walled cups are expected to retain heat longer than single walled cups. The feature that
makes double walled cups effective at slowing the transfer of thermal energy is the layer
of trapped gas that modifies the path of transfer from only conduction to two instances of
conduction separated by convection.
This project characterizes analytically quantifies how much more effective a double
walled cup is when compared to a single walled cup.
2
2. Theory & Methodology
Double walled beverage cups are promoted as being able to keep hot drinks hot and cold
drinks cold for longer periods of time than single walled beverage cups. Much like
double pane windows, double walled mugs capture an insulating layer of air between
two layers of material and prevent thermal energy from being readily conducted through
to the other side.
2.1 Theory
The process of heat transfer can be simplified as follows:
1. The inner most wall of the window obtains thermal energy and transfers the energy to
towards the second wall via thermal conduction
2. The gas in between the two walls then transfer the thermal energy to the second wall
through convection
3. The energy is then conducted through the second wall
4. Lost to the environment via convection
Figure 3 is a schematic of this heat transfer process.
Figure 3: Schematic of the heat transfer process: 1. Thermal conduction through inner
wall, 2. Convection via trapped gas, 3. Thermal conduction through outer wall, 4.
Energy lost to environment
3
2.2 Methodology
Heat transfer is the process where thermal energy is exchanged between neighboring
systems in response to a temperature difference. This is in accordance to conservation of
energy theory where the total energy of an isolated system remains constant and the first
law of thermodynamics where the change in internal energy of a system is equal to the
heat added to the system minus the work done by the system. Heat transfer is further
categorized into three modes: conduction, convection, and radiation.
Heat conduction is heat transfer across a medium without any motion of the material as a
whole. In a solid, the mechanism of conduction is the atomic activity in the form of
lattice vibration which contribution of the translational motion of electrons if the solid is
electrically-conducting. Heat conduction in a liquid or gas is due to the random motion
and interaction of the molecules. Examples of conductive heat transfer: the transfer of
heat energy down the axis of a metal rod when one end is a higher temperature than the
other; and the transfer of heat energy from a stove element through a metal pan into the
contents within. The rate of heat conduction can be expressed as follows:
𝑄 𝑘𝐴(𝑇ℎ𝑜𝑡 − 𝑇𝑐𝑜𝑙𝑑 )
=
𝑡
𝑑
Where:
𝑄
𝑡
= heat transferred over time (watts)
𝑘 = thermal conductivity of the material (𝑊 ⁄𝑚 ∙ 𝐾)
𝐴 = area of the interface (𝑚2 )
𝑇 = temperature (K)
𝑑 = thickness of the interface (m)
Convective heat transfer occurs between systems by moving fluid past a wall of a
different temperature. Convection can be forced or natural - for the purposes of this
discussion and project, only natural will be described and used in the analysis model.
4
Natural convection occurs in accordance to the Ideal Gas Law, where a liquid that
increases in temperature also decreases in density and rises. Due to the resulting internal
buoyancy force, circulating currents then transport the heat energy away from the energy
source.
The convective heat transfer model in this study will be simplified into natural
convection from a horizontal and vertical plate. The assumption is that the cup is on a
level surface and the cup geometry can be approximated by a vertical wall. For a time
dependent study, the natural convection across a vertical or horizontal plate is
represented by the following equation:
−𝑛 ∙ (−𝑘∇𝑇) = ℎ ∙ (𝑇𝑒𝑥𝑡 − 𝑇)
Where:
ℎ = ℎ𝑎𝑖𝑟 (𝐿, 𝑝𝐴 , 𝑇𝑒𝑥𝑡 ) (𝑊 ⁄𝑚2 ∙ 𝐾)
L = plate height, length, or diameter (m)
𝑝𝐴 = Absolute pressure (Pa)
𝑇𝑒𝑥𝑡 = Externals temperature (K)
Thermal radiation is a form of energy emitted by matter at a nonzero temperature and
can be considered to be the propagation of electromagnetic waves or particles. For the
purpose of this project, this form of heat transfer will be considered insignificant.
The following inputs and assumptions were used:
1. Initial temperature of the glass, air, and surrounding environment: 70 degrees
Fahrenheit or 21 degrees Celsius or 294.261 Kelvin
2. Initial temperature of the water: 212 degrees Fahrenheit or 100 degrees
Celsius or 373.15 Kelvin
3. Thermal conductivity of water @ 300 Kelvin: 0.609 𝑊 ∙ 𝑚−1 ∙ 𝐾 −1
4. Thermal conductivity of glass @ 300 Kelvin: 0.8 𝑊 ∙ 𝑚−1 ∙ 𝐾 −1
5
5. Natural convection heat transfer of a horizontal plate in air at 300 Kelvin:
11.26 𝑊 ∙ 𝑚−2 ∙ 𝐶 −1
6. Natural convection heat transfer of a vertical plate in air at 300 Kelvin: 7.03
𝑊 ∙ 𝑚−2 ∙ 𝐶 −1
6
3. Results & Discussion
The test specimen obtained for this project is the Kiran Tea Glass distributed exclusively
by Teavana. The cup is double walled, made from borosilicate glass, and holds 8 ounces
or 235 milliliters of liquid. The cup can be procured from any Teavana retail store or
from the company’s internet website. Figure 4 is a picture of the cup.
Figure 4: Kiran Tea Glass by Teavana
The reason for selecting this cup model is due to its design being simple, axisymmetric,
and being made from a clear glass material. This simplifies not only the analysis, but
also eliminates complications from the analysis model.
A two-dimensional sketch and three-dimensional model of the cup was created using
NX 6.0, the computer-aided design software package formerly known as NX
7
Unigraphics. The inputs used to model the cup were gathered using a scale, digital
caliper, and radius gages. All dimensions were gathering in the United States customary
system units then converted to metric using NX. Figure 5 is a picture of the tools used to
gather the dimensions of the cup being analyzed.
Figure 5: Scale, digital caliper, and radius gages
8
Figure 6 is a picture of the two-dimensional sketch of the cup created using NX.
Figure 6: 2D Sketch of the Kiran Tea Glass
Figure 7 is a picture of the three dimensional model of the cup created using NX.
Figure 7: 3D model of the Kiran Tea Glass
9
To facilitate the analysis model, a two-dimensional sketch and three-dimensional model
of the water and air were also created using the curves from the two-dimensional cup
sketch. As with the cup, the air and water were created in the United States customary
system units then converted to metric using. Figure 8 is a picture of the two-dimensional
of the air model created using NX.
Figure 8: 2D sketch of the air within the Kiran Tea Glass
10
Figure 9 is a picture of the two-dimensional sketch of the water model created using NX.
Figure 9: 2D Sketch of the water within the Kiran Tea Glass
Figure 10 is a picture of the three-dimensional model of the air created using NX.
Figure 10: 3D model of the air within the Kiran Tea Glass
11
Figure 11 is a picture of the three-dimensional model of the water created using NX.
Figure 11: 3D Model of the water within the Kiran Tea Glass
Figure 12 is a picture of the cup, air, and water three-dimensional models nested and
cross-sectioned using NX.
Figure 12: The 3D models of the cup, air, and water nested and cross-sectioned
12
While the three-dimensional models of the cup, air, and water was not necessary for the
study, it was useful for predicting what the two-dimensional-axisymmetric thermal
model would look like based on the two-dimensional NX model.
COMSOL Multiphysics 4.2a was then used to create and solve the finite element
analysis model. The geometry created using NX was imported into COMSOL via a
Drawing Exchange Format (DXF) file. The heat transfer module was used to perform
the heat transfer simulations.
3.1 COMSOL Heat Transfer in Solids Model – Single Walled Cup
The first thermal model created in COMSOL was a time-dependent Heat Transfer in
Solids study. The cup, air, and water geometry were imported into COMSOL then united
and repaired to compensate for any approximations that were made when the NX files
converted measurement systems and file formats. The default relative repair tolerance
function in COMSOL was utilized to perform the geometry repair. The repair tolerance
was set to 1e-3 millimeters. Figure 13 is a picture of the cup, air, and water after being
imported and repaired in COMSOL.
13
Figure 13: The 2D sketches after importing and repairing in COMSOL
Note that the same models created for the double walled cup will be used. For this
singled walled cup study, the geometry that would normally be air domain was set to the
same properties as the cup domain thereby creating what would be a single walled cup
with the same profile as the double walled cup. This was done to eliminate any potential
discrepancies a single walled cup of different geometry would have when drawing
comparison conclusions between single and double walled cups.
The material properties for both the glass and water were added to their respective
domains from the built-in library in COMSOL. Silica glass and liquid water were the
two library entries selected for this study. Figure 14 is a picture of the cup, air, and water
where domains 1 and 2 are both set to silica glass and domain 3 set to liquid water.
14
Figure 14: COMSOL Built-in Materials applied to the cup, air, and water domains
The thermal model is two-dimensional axisymmetric. The axis of symmetry is shown in
Figure 15 by the solid blue line. Figure xyz is of the thermal model with the axis of
symmetry highlighted.
15
Figure 15: COMSOL model with the axis of symmetry highlighted
A thermal insulation boundary condition was added to the bottom of the cup, where it
would normally rest on a surface. In this thermal model, zero heat energy was lost to the
surrounding environment. Figure 16 is of the thermal model with the thermal insulation
boundary condition highlighted.
16
Figure 16: COMSOL model with the thermal insulation boundary highlighted
In heat transfer in solids models, any curves between materials are set to conductive heat
transfer by default. The heat transfer properties were all set to the default values built
into the COMSOL library.
The natural convection boundary conditions were the applied. Figure 17 is of the thermal
model with the natural convection past a vertical wall applied to the highlighted cup
surfaces.
17
Figure 17: COMSOL model with the natural convection past a vertical wall boundaries
highlighted
Figure 18 is of the thermal model with the natural convection past a horizontal wall
applied to the water surface.
18
Figure 18: COMSOL model with the natural convection past a horizontal plate boundary
highlighted
The initial temperature value of the water was set to 373.15 Kelvin. The initial
temperature value of the cup was set to 294.26 Kelvin.
An extra fine mesh of the thermal model was created. The mesh consists of 5917
elements. Figure 19 is of the mesh of the thermal model.
19
Figure 19: Extra fine mesh of the thermal model in COMSOL
A stepped time dependent study was performed. The duration was for thirty minutes and
with a minute for each step. Figure 20 is of the results of the thermal model at time =
zero minutes.
20
Figure 20: Thermal model at time = zero minutes
21
Figure 21 is of the results of the thermal model at time = ten minutes.
Figure 21: Thermal model at time = ten minutes in COMSOL
22
Figure 22 is of the results of the thermal model at time = twenty minutes.
Figure 22: Thermal model at time = 20 minutes in COMSOL
23
Figure 23 is of the results of the thermal model at time = thirty minutes.
Figure 23: Thermal model at time = 30 minutes in COMSOL
After thirty minutes, the water temperature dropped approximately 40 degrees and the
cup temperature rose approximately 40 degrees. Also notice that the cup temperature
was at its hottest at 20 minutes, being approximately 50 degrees higher than its initial
temperature. The thermal analysis model appears to be well behaved and utilizes the
applied boundary conditions appropriately.
3.2 COMSOL Heat Transfer in Solids Model – Double Walled Cup
The second thermal model created in COMSOL was another time-dependent Heat
Transfer in Solids study. The same cup, air, and water geometry used in the single
walled study were imported into COMSOL then united and repaired. The default relative
24
repair tolerance function in COMSOL was utilized to perform the geometry repair. The
repair tolerance was set to 1e-3 millimeters.
The material properties for the glass, air, and water were added to their respective
domains from the built-in library in COMSOL. The cup, air, and water are domains 1, 2,
and 3 respectively. Figure 24 is a picture of the cup, air, and water where domains 1, 2,
and 3 are set to silica glass, air, and liquid water, respectively.
Figure 24: COMSOL Built-in Library Materials applied to the cup, air, and water
domains
The thermal model is two-dimensional axisymmetric. Figure 25 is of the thermal model
with the axis of symmetry highlighted.
25
Figure 25: Double Walled thermal model with the axis of symmetry highlighted
A thermal insulation boundary condition was added to the bottom of the cup, where it
would normally rest on a surface. In this thermal model, zero heat energy was lost to the
surrounding environment. Figure 26 is of the thermal model with the thermal insulation
boundary condition highlighted.
26
Figure 26: Thermal model with the thermal insulation boundary condition highlighted
In heat transfer in solids models, any curves between materials are set to conductive heat
transfer by default. The heat transfer properties were all set to the default values built
into the COMSOL library.
The natural convection boundary conditions were the applied. Figure 27 is of the thermal
model with the natural convection past a vertical wall applied to the highlighted cup
surfaces.
27
Figure 27: Thermal model with the natural convection past a vertical wall boundary
conditions highlighted
28
Figure 28 is of the thermal model with the natural convection past a horizontal wall
applied to the water surface.
Figure 28: Thermal model with the natural convection past a horizontal wall boundary
condition highlighted
The initial temperature value of the water was set to 373.15 Kelvin. The initial
temperature value of the cup was set to 294.26 Kelvin.
An extra fine mesh of the thermal model was created. The mesh consists of 5917
elements. Figure 29 is of the mesh of the thermal model.
29
Figure 29: Extra fine mesh of the thermal model
A stepped time dependent study was performed. The duration was for thirty minutes and
with a minute for each step. Figure 30 is of the results of the thermal model at time =
zero minutes.
30
Figure 30: Double walled cup thermal model at time = zero minutes
31
Figure 31 is of the results of the thermal model at time = ten minutes.
Figure 31: Double walled cup thermal model at time = ten minutes
32
Figure 32 is of the results of the thermal model at time = twenty minutes.
Figure 32: Double walled cup thermal model at time = 20 minutes
33
Figure 33 is of the results of the thermal model at time = thirty minutes.
Figure 33: Double walled cup thermal model at time = 30 minutes
After thirty minutes, the water temperature dropped approximately 10 degrees and the
cup temperature rose approximately 15 degrees. The thermal analysis model appears to
be well behaved and utilizes the applied boundary conditions appropriately.
3.2 Measured Data
Boiled water was added to the Kiran Tea Glass on hand and measured incrementally as it
cooled off. The data is available in Appendix 1. A graph of the measured temperature
versus time can be seen in Figure 34.
34
Temperature (K)
370.00
360.00
350.00
340.00
Temperature (K)
330.00
320.00
310.00
300.00
Time (s)
Figure 34: Graph of the measured temperature of boiled water in the Kiran Tea Glass
The double walled thermal model was then updated to replicate the measured data by
changing the ambient and initial temperatures. The results are as follows.
35
Figure 35: Double walled thermal model updated inputs at time = 10 minutes
36
Figure 36: Double walled cup with updated inputs at time = 20 minutes
37
Figure 37: Double walled cup with updated inputs at time = 30 minutes
38
4. Conclusions
The water in the double walled cup exhibited a lower rate of thermal energy loss to the
surrounding environment than the single walled cup. The non-wetted surfaces of the
double walled cup also gained thermal energy slower than the single walled cup for the
duration of the analysis.
The measured data also verified that the thermal model was well behaved and a close
representation of the physical specimen.
In conclusion, double walled cups are indeed more effective at keeping hot liquids hot
for longer periods than single walled cups. The layer of air between two walls of glass
allow the double walled cups to perform in a similar fashion to double pane windows.
Also, single walled cups meant for hot liquids most often feature a handle which aligns
with the need for the user to have a way to hold the cup without being exposed to hot
surfaces.
39
5. References
Janninck, W.L. (1969). U.S. Patent No. 3456860. Washington, DC: US Patent and
Trademark Office.
http://www.google.com/patents?id=qJ9ZAAAAEBAJ&printsec=abstract&zoom=4#v=o
nepage&q&f=false
The big one 2004. (2003). Catering Update, , 25-26,29-30. Retrieved from
http://search.proquest.com/docview/223152167?accountid=37764
Emily, B. Y. (2012, Mar 22). McDonald's testing paper cups for hot drinks. McClatchy Tribune Business News. Retrieved from
http://search.proquest.com/docview/929381998?accountid=37764
Young, C. (2011). Building a double wall. Building Design & Construction, Retrieved
from http://search.proquest.com/docview/1278116856?accountid=37764
Swiss inventor develops ornamental design for double wall cup. (2008, May 23). US Fed
News Service, Including US State News. Retrieved from
http://search.proquest.com/docview/472009520?accountid=37764
Donberg, D. (2003). Insulated cup a high-flying success. Paper, Film and Foil
Converter, 77(10), 14-14. Retrieved from
http://search.proquest.com/docview/211371519?accountid=37764
By, R. G. (1999, Feb 22). Starbucks plans to test a paper cup that insulates hands from
hot coffee. Wall Street Journal. Retrieved from
http://search.proquest.com/docview/398688536?accountid=37764
By, S. K. (1998, Mar 24). These people search for a cup that suits the coffee it holds --the current models can burn fingers, come unglued; mr. sadlier hawks `Insulair'. Wall
40
Street Journal. Retrieved from
http://search.proquest.com/docview/398614978?accountid=37764
New Duo Shield[TM] double wall Cup. (2010, March). Food Trade Review, 80, 177.
Retrieved from http://go.galegroup.com.colelibprxy.ewp.rpi.edu/ps/i.do?id=GALE%7CA228269410&v=2.1&u=22507&it=r&p=GPS&
sw=w
"Skin-Care Packaging." Global Cosmetic Industry June 2000: 7. Vocations and Career
Collection. Web. 29 Mar. 2013.
Document URL
http://go.galegroup.com.colelibprxy.ewp.rpi.edu/ps/i.do?id=GALE%7CA63411694&v=2.1&u=22507&it=r&p=GPS&s
w=w
"Mighty Jo travel mug -- 16 oz." Specialty Coffee Retailer June 2002: 54. Small Business
Collection. Web. 29 Mar. 2013.
Document URL
http://go.galegroup.com.colelibprxy.ewp.rpi.edu/ps/i.do?id=GALE%7CA87692510&v=2.1&u=22507&it=r&p=GPS&s
w=w
Wolff, S. K. (1994). Double walled paper cup Retrieved from
http://search.proquest.com/docview/34737338?accountid=37764
Singaporean inventors develop double-walled cup. (2009, Sep 18). Indian Patents News.
Retrieved from http://search.proquest.com/docview/443239946?accountid=37764
Reflecting quality. (2006). Catering Update, , 30-30. Retrieved from
http://search.proquest.com/docview/223134574?accountid=37764
Kiran Tea Glasses at Teavana | Teavana. Retrieved from http://www.teavana.com/teaproducts/tea-cups-mugs/glass-tea-cups/p/kiran-tea-glasses-8oz
41
R. Nave. Heat Transfer. Retrieved from http://hyperphysics.phyastr.gsu.edu/hbase/thermo/heatra.html#c1
https://www.thermalfluidscentral.org/encyclopedia/index.php/Heat_and_Mass_Transfer
42
6. Appendices
6.1 Appendix 1: Measured temperature data from boiled water in the double walled cup
Time (s)
0
60
120
180
240
300
360
420
480
540
600
660
720
780
840
900
960
1020
1080
1140
1200
1260
1320
1380
1440
1500
1560
1620
1680
1740
1800
Temperature
(F)
61
194
187
182
177
173
169
165
162
159
155
153
150
147
145
143
141
139
137
135
134
132
131
129
127
126
125
123
122
121
120
Temperature
(K)
289.26
363.15
359.26
356.48
353.71
351.48
349.26
347.04
345.37
343.71
341.48
340.37
338.71
337.04
335.93
334.82
333.71
332.59
331.48
330.37
329.82
328.71
328.15
327.04
325.93
325.37
324.82
323.71
323.15
322.59
322.04
43