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LIBRARY
Michigan §tate
Universrty

This is to certify that the

thesis entitled

Theoretical, Experimental and

Computer Model for Package
R-value Using Regular

Ice and Dry Ice»

presented by

Napawan Kositruangchai

has been accepted towards fulfillment
of the requirements for

 

 

m degree in M
l gm 6 wzgml/
ajor professor
Date 4-8-03
c»7639

MS U is an Affirmative Action/Equal Opportunity Institution

 

 

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6/01 cJCIRC/DaieDue.pss-p.15

THEORETICAL, EXPERIMENTAL AND COMPUTER MODEL FOR PACKAGE R-
VALUE USING REGULAR ICE AND DRY ICE

By

Napawan Kositruangchai

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

School of Packaging

2003

ABSTRACT
THEORETICAL, EXPERIMENTAL AND COMPUTER MODEL FOR PACKAGE R-
VALUE USING REGULAR ICE AND DRY ICE
By

Napawan Kositruangchai

Temperature sensitive products need specific containers to prevent heat loss or
gain. Several containers (corrugated box, EPS cooler, molded polyurethane and VIP
panels) were studied for their package insulating ability (R-value). Regular ice and dry
ice were compared by performing an ice melt test at three different conditions (72°F and
50%RH, 72°F and 85%RH, and 104°F and 50% RH). Experimental R-values were
compared to a fitted equation and to computer model results using a BASIC program.

The higher the R-value, the better the insulating container. The VIP was focused
to be a better insulating container than molded polyurethane, followed by an EPS cooler
and a corrugated box.

The experimental R-values using dry ice were higher than for regular ice because
the temperature of dry ice is much lower than regular ice. This lowers the thermal
conductivity and the convection heat transfer coefficient. The surface area also affects the
R-value. The higher the storage temperature and relative humidity, the lower the R-value.
The fitted equations agreed the experimental R-value for regular ice but not for dry ice.

On the other hand, the computer model R-values fit better for dry ice.

ACKNOWLEDGMENT

I would like to express my gratitude to Dr. Gary Burgess, my research and thesis
advisor, for his invaluable advice, support, and encouragement through my tough time. I
would also like to thank my committee, Dr. Janice Harte and Dr. Harold Hughes, for their
guidance and kindness.

I would also like to thank Dr. Bruce Harte for his advice and the staff and friends
in the School of Packaging.

Furthermore, I would like to acknowledge the Thai Embassy throughout my
program. Finally, my parents and family deserve a special mention for providing support

and encouragement throughout my life.

iii

TABLE OF CONTENTS

LIST OF TABLES ........................................................................ vi
LIST OF FIGURES ..................................................................... viii
Chapter 1: Introduction ..................................................................... 1
Chapter 2: Literature Review .............................................................. 7
2.1. Insulating Packages ............................................................ 8
Expanded Polystyrene Foam (EPS Foam) ............................. 8
Modified Corrugated Board Lined EPS Foam ....................... 10
Molded'Polyurethane .................................................... 10
Vacuum Insulation Panels or VIP ...................................... 12
2.2. Refrigerants .................................................................... 1 5
Regular Ice .............................................. ' .................. l 5
Dry Ice ..................................................................... 16
Gel Packs .................................................................. 1 8
2.3. Heat Transfer .................................................................. l9
Conduction Heat Transfer .............................................. 19
Heat transfer through a wall ................................... 20
Heat transfer through two walls .............................. 22
Heat transfer through multilayered walls .................... 24
Convection ................................................................. 25
Combined thermal resistance both conduction and convection
(no radiation) .............................................................. 26
Heat transfer through a wall contacted with air on
both sides ........................................................ 27
Heat transfer through two flat materials connected with
inside and outside air ........................................... 28
Heat transfer through multilayered walls in contact with
inside and outside air ........................................... 29
Radiation ................................................................... 3 1
2.4.Thermal Testing Approaches for Insulated Packaging ................... 34
ASTM D3103-92 ......................................................... 34
Other testing methods .................................................... 35
Chapter 3: Experimental Design and Results ................................................. 36
3.1 . Materials ....................................................................... 37
Box Samples .............................................................. 37
3.2. Methods ....................................................................... 39
Preconditioning ............................................................ 39
Procedure ................................................................... 40
3.3. Results .......................................................................... 44

iv

R-values from ice melt test using‘regular ice and dry ice at 72°F

and 50%RH .............................................................. 45

The effect of the relative humidity on the R-value .................. 47

The effect of the temperature on the R-value ......................... 48

Chapter 4: Analysis and Discussion ..................................................... 50
Radiation Balances ....................................................... 52

Energy Balances .......................................................... 53

Chapter 5: Conclusions and Future Studies ............................................. 63
5.1 . Conclusions .................................................................... 64

5.2. Future studies .................................................................. 65
Appendix ..................................................................................... 66
References ................................................................................... 7O

LIST OF TABLES

Table1.Distribution conditions and their temperatures ........................................... 3

Table2. Insulation performance of vacuum insulation panel (VIP) and conventional

insulations ................................................................................... 14
Table 3. Summary specifications for insulation materials ..................................... 39

Table 4. R-values of insulating packages from regular ice melt test at 72°F and 50%RH

............................................................................................... 45

Table 5. R-values of insulating packages from dry ice melt test at 72°F and 50%RH
................................................................................................. 46
Table 6. Thermal conductivity of materials ....................................................... 46

Table 7. R-value of C-flute corrugated box at 72°F and 85%RH using regular ice ....... 47

Table 8. R-value of C-flute corrugated box at 72°F and 85%RH using dry ice ............ 47
Table 9. R-value for regular ice at 104°F at 50% RH in a corrugated box .................. 48
Table 10. R-value of dry ice at 104°F at 50% RH in a corrugated box ...................... 48
Table 11. Summary of the R—values for three conditions (regular ice) ...................... 49
Table 12. Summary of the R-values for three conditions (dry ice) ........................... 49
Table 13. Properties of air at 284 and 293 (Holman, 1986) .................................... 54
Table 14. Constants for use in the equation 38 .................................................. 55
Table 15. Summary ofk and h at length 10 and 14 inch at 284 K of Air .................... 56
Table 16. Summary of k and h at length 10 and 14 inch at 293 K of Air .................... 56

Table 17. Results of R-value from the computer model of regular ice at T5 (72 °F)

and 50%RH at k=0.035, 0.025 and 0.0144 Btu/hr.ft°F ......................... '....57

vi

Table 18. Results of R-value from the computer model of regular ice at T5 (72 and

104°F) and 50%RH ...................................................................... 58
Table 19. Properties of Carbon Dioxide at 245 and 254 K .................................... 59
Table 20. Summary of k and h at length 10 and 14 inch at 245 K of Carbon Dioxide....59
Table 21. Summary of k and h at length 10 and 14 inch at 254 K of Carbon Dioxide. ...60
Table 22. The results of effect of the thermal conductivity of dry ice to the R-value. .....60
Table 23. R-values from the computer model using dry ice at 72 and 104°F and 50%RH

. .............................................................................................. 61
Table 24. Experimental, computer model and predicted R values for regular ice at 72°F

and 50%RH .................................................................................. 62

Table 25. Experimental, computer model and predicted R values for dry ice at 72°F and

50%RH ...................................................................................... 62

vii

LIST OF FIGURES

Figure 1. Expanded polystyrene foam (EPS foam) .......................................... 8
Figure 2. Molded polyurethane ............................................................... 10
Figure 3. Vacuum insulation panels (VIP) .................................................. 12
Figure 4. Components of VIP ................................................................. 13
Figure 5. Forms of dry ice: block and pellet ................................................ 16
Figure 6. Gel packs ............................................................................. 18
Figure 7. Diagram for heat transfer through a wall ....................................... 20
Figure 8. Diagram for heat transfer through two walls .................................... 22
Figure 9. Diagram of heat transfer through multilayered walls .......................... 24

Figure 10. Diagram for heat transfer of a wall connected with inside and
outside air ........................................................................... 27
Figure 11. Diagram for heat transfer of two walls connected with inside and outside
air .................................................................................... 28
Figure 12. Diagram of heat transfer through multilayered walls in contact with inside

and outside air ...................................................................... 30

Figure 13. Radiation incident on a surface can be absorbed, reflected and

transmitted .......................................................................... 32
Figure 14. Molded polyurethane with open cell foam ..................................... 38
Figure 15. VIP in a corrugated box ........................................................... 38
Figure 16. The procedure for the regular ice melt test ..................................... 42
Figure 17. The procedure for the dry ice melt test .......................................... 43
Figure 18. Performing ice melted test of ice or dry ice in a insulating container ...... 51

viii

Figure 19. Dry ice rod ......................................................................... 58

ix

Chapter 1

Introduction

1. Introduction

Temperature sensitive products such as biological materials, blood and some
organ parts, pharmaceuticals, frozen foods, fresh-products, diary products and delivery
pizza need specific containers (insulating packages) that can prevent the loss of heat and
either keep the temperature constant during transportation or keep products from melting,
thawing, or freezing. This is done by using thermal insulation. Thermal insulation is any
material or combination of materials that retard heat transfer by conduction, convection
or radiation (ASHRAE 1993). By retarding heat flow, thermal insulation conserves
energy by reducing heat loss or gain by the product.

The most commonly used insulation packages are expanded polystyrene (EPS),
rigid high-density polyurethane and reflecting surface materials (radiant barrier films)
(FedEx, 2002). Sometimes, refrigerants such as gel packs or dry ice are placed in the
container, especially for perishable products, to maintain the range of temperatures (cold
or frozen temperature). Normally gel packs are suitable for refrigerating products in the
range of 30°F (-1°C) to 60°F (16°C). Dry ice is a solid form of carbon dioxide. It has a
surface temperature of —108.4°F (-78°C) and is used to keep products frozen. Regular ice
is not normally used since it has many disadvantages including weight, thawing, leaking
and the need for expensive water resistant packages (FedEx, 2002).

Environmental temperature and moisture affect the thermal resistance of
insulating containers. When the temperature or the relative humidity increases, the
thermal resistance of packages decreases (FedEx, 2002). Not only environmental

temperature but changing temperature during transportation affect products as shown in

Table 1 (Panyarjun, 2002). The number of steps in the distribution process also affects
products.

There are two main users of insulating packages: perishable food industries, who
use specific containers for each product, and e-commerce businesses, who use several
standard packages to pack all products. The distribution process starts from the factory —>
storage room —> trucks or airplanes —) warehouse —) stores —> customers. In e-
commerce, companies will directly pack products using standard size packages, adding
dunnage such as scrap paper or peanut foam to protect products and sending it to

customers using FedEx, UPS or USPS.

Table 1. Distribution conditions and their temperatures

 

 

 

 

 

Conditions Temperature
(°C)
Freezer -18 to -35
Refi'igerator l to 4
Freezer truck, rail car or airplane -18 or below
efi'igerated truck, rail car or airplane O to 5

 

 

 

 

Each temperature sensitive product has specific characteristics. For instance,
frozen products need to maintain the frozen temperature to prevent melting or thawing.
For fresh fruits or fresh vegetables that still respite, releasing heat and increasing
temperature into the package would cause faster decay. On the other hand, pizza delivery
needs insulation packages to maintain hot temperatures (Fava, 1999). The designs and
materials for insulation packages have to meet these minimum requirements and also be

inexpensive.

The thermal resistance (R-value) of packages and the amount of refrigerants has
to be optimized to design for temperature sensitive products. The thermal resistance of
packages varies widely. It ranges from a low R-value for a corrugated box, medium R-
value for an EPS or rigid high-density polyurethane, to a high R-value for Vacuum
Insulation Panels (VIP). The optimal amount of refi-igerants to keep products frozen or
cold depends on several factors including the product mass, inside surface area of the
package, the wall thickness, and the time of transportation.

Insulating packages need to have high R-values. There are several methods to
determine the insulating ability of containers: ASTM D3103 and the ice —melt test
(Burgess, 1999). ASTM D3103: Standard Test Method for Thermal Insulation Quality of
Packages, uses temperature indicating devices such as thermocouples to assemble the
temperature profiles and determine the R-value.

In this work, the ice melt test method (Burgess, 1999) was used because it is very
simple, easy to test, inexpensive (do not need any machine or sensors) and is an effective
method to determine the insulating ability of a package. This method is similar to the

ASTM D3103 but the R-value is calculated from equation 1. By definition,

R-value= AXAT (1)
meltrate x latentheat

 

fi2.F.hr

Where R-value = thermal resistance of container wall ( Bt
u

)

A = inside surface area of package (ftz)

AT = temperature different (°F) between outside air and refrigerant used
Melt rate: the rate that ice melts during the experiment and is equal to the weight

of the ice melted divided by the melt time (lb/hr)

Latent heat = 144 Btu/lb for regular ice and 240 Btu/lb for dry ice

From equation 1, the ratio of AT/latent heat, depends on what refrigerant is used
and the outside air temperature. Burgess (1999) fitted experimental R-values to the
equation 2,

R —- value = 3.9th + 1.5np + 3.2nf(i20%accuracy) (2)
Where th = the average wall thickness (inch)

up = the number of plain surfaces

nf = the number of aluminium foil surfaces

In this equation, the R-value is separated into three parts: the effects of
conduction, convection and radiation. Conduction depends mainly on the wall
construction (th). Convection refers to heat transfer between air and surfaces so the R-
value depends upon the number of surfaces in contact with air (np). Radiation refers to
the emission or reflection of infrared waves and can be a significant factor. The number
of aluminium foil surfaces (nf) is therefore important to R-value.

In the design of insulating packages, equation 2 can be used to determine the R-
value needed. The requirement for refrigerants is found using equation 1. The R-value
calculated from equation 2 is only i20% accurate. The equation is linear but radiation is
known to be a strong function of temperature. The thermal conductivity of containers also
varies with temperature and relative humidity and this equation does not correct for the
relative humidity. Equation 2 uses an average of thermal conductivity (k) of 0.022
Btu/h.ft.°F but k varies for different materials. For example, corrugated board is about
0.035 Btu/h.ft.°F. The thermal conductivity of materials at lower temperatures (such as

dry ice) should have lower R-value due to lower k in lower temperature.

The objectives of this work were to investigate the effect of temperature and
relative humidity in order to reduce the error in determining the R-value of insulating

containers for each specific package and condition. There are three main objectives in

this work.

1. To compare the R-value between dry ice and regular ice using the ice
melted test method.

2. To compare thermal resistant (R-value) between the theory parts of heat
transfer using computer simulation of the insulated containers with the
ice melted test results.

3. To determine the effect of temperature and relative humidity of the

corrugated box on R-value

Chapter 2

Literature Review

2. Literature Review

2.1 Insulating Packages

There are several insulating packages that are used for temperature sensitive
products: Expanded Polystyrene Foam (EPS foam), Modified Corrugated board lined
.EPS Foam, Molded Polyurethane, corrugated with liners or blankets, and Vacuum

Insulation Panels (VIP).

Expanded Polystyrene Foam (EPS Foam)

 

Figure l. Expanded polystyrene foam (EPS foam)

EPS foam (Figure 1) is the most common packaging foam used for low
temperature distribution. It is known to show excellent insulation capability because of its
air-filled cells, EPS foam is an excellent insulator since air has particularly low heat

conductivity. This is the reason why EPS foam is being used for insulation in temperature

sensitive products. It is typically the least expensive packaging foam available. EPS foam
is relatively chemically inert and acceptable for use in food packaging for meat and
produce. EPS foam is lightweight, reusable and stackable (Hernandez, 2000). It can be
produced by standard technology with no special equipment. EPS foam is produced from
PS granules or beads that are impregnated with hydrocarbon blowing agent (such as
pentane 5-8%) and sometimes with flame-retardants during suspension polymerization.
The blowing agent produces bubbles in the plastics and forming cells. The properties of
foam depend upon the amount of blowing agent and process conditions. The PS beads are
pre-expanded heating to 85-96°C (185-205°F) to vaporize the pentane. The PS beads
usually expand about 25 to 40 times of their original size. The density of foam varies

from 13 to 48 kg/m3, with 24 lrg/m3 most common used for cushion packaging and 16

kg/m3 for insulation. These pre-expanded PS beads are aged to reach equilibrium then
packed into a mold under heat and pressure that cause these beads to fuse together. The
mold produces the final shape desired. The problem with EPS foam is that it is hard to

dispose after use.

Modified Corrugated Board Lined EPS Foam

Modified corrugated board lined EPS foam consists of a corrugated core layer,
which is an EPS foamed sheet, covered on both sides with a paperboard liner (Sasaki,
1999). EPS foams are used to improve heat insulation by stopping the movement of air
and decrease the absorption of water. The continuous process for mass production starts
from polystyrene foam. A sheet is heated and is passed through a pair of matching
toothed rolls to make corrugated. Next, adhesive is applied to the top of the corrugated
foamed sheet and this sheet is passed between two rolls to be formed into a cardboard

sheet. This material is light and can be recycled separately from paperboard.

Molded Polyurethane

 

Figure 2. Molded polyurethane.

V2

CE

(7

Molded Polyurethane (Figure 2) is designed with a layer of high performance
urethane foam injected between layers of corrugated fiberboard. Urethane gives a high R-
value (twice the R-value per inch compared to EPS foam) insulated shipping container
capable of meeting the most demanding temperature controlled requirements such as for
pharmaceutical products (Cold Ice, 2002). Urethane requires less- refrigerant than EPS
foam. This can be valuable for re-use options, where only some of the components, such
as the outer corrugate, needs to be replaced. It is also environmentally preferable where
components can be sorted for recycling. In addition, polyurethane can be injected
between two corrugated boards to increase the strength of the box and also improve

stability (Panyarjun, 2002).

ll

Val

Vacuum Insulation Panels or VIP

 

Figure 3. Vacuum insulation panels (VIP)

Vacuum Insulation is an advanced thermal insulation technology that significantly
outperforms insulation materials like closed-cell foams, foam beads or fiber blankets. It
removes the gases within the insulating space (Dow Chemical, 2002).

The vacuum insulation panel (VIP) in Figure 3 (Glacier Bay, 2002) is a
technologically advanced product that combines a high R-value in a relatively thin panel.
VIP in Figure 6 (Dow Chemical, 2002) consists of a core panel enclosed in an air-tight
envelope, for which a vacuum is applied. This product provides an insulative value of

three to seven times EPS (Table 2) (Dow Chemical, 2002). Figure 4 shows a filler

material that is 100 percent open cell, microoellular polystyrene called a core that is

encapsulated by a thin, super-barrier film, a membrane film, getters and desiccants.

Desiccant(s) INSTILL vacuum
and/or Getter(s) insulation core

./ \

/\ \

Super-Barrier Vacuum
Film 0.001-1.0 torr

(metallized polymer (1 .0 torr - 1/760
and/or polymeric or atmosphere)
metal foil structures)

 

 

Figure 4. Components of VIP

The core material has two main functions. First, it provides physical support to
the membrane or barrier film envelope so that it does not collapse in on itself when the
vacuum is applied. Second, the core material acts to disrupt the flow of the molecules of
gas which still remain in the evacuated space, thus reducing transfer heat between the

walls of the VIP.

The membrane films are the materials which form the walls of the VIP. It is the
job of the membrane film to provide an effective barrier to retard all gases and moisture
so that the vacuum can be maintained. Impermeable membrane materials are glasses and
metals. Unfortunately glass is too fragile. Metal can be used but significantly reduces

the average insulation value of the finished panel due to the conductance of heat around

13

the edges where the walls are joined. Also a pure metal membrane is higher in cost to

form and weld to the panel.

Getters and desiccants are chemicals that absorb gases and moisture respectively.
Getters and desiccants are used to extend the life of VIPs by absorbing unwanted gases

and moisture to prevent a rise in pressure within the evacuated space. In order to be
effective, the getters and desiccants must be thoroughly matched to the type and quantity
of gas/moisture they will be expected to absorb. They must be able to effectively absorb
and hold the gasses and moisture at the low pressures inside the VIP.

The insulative values are dependent upon the vacuum. The foil is fairly durable,
but it is fragile. The vacuum insulated panel cannot be punctured during installation or
use. Therefore, the panel must be protected by a material that is resistant to puncture such
as corrugated boxes. This requires that the VIP be contained within another product such

as a corrugated box or protective covering during transportation.

Table 2. Insulation performance of vacuum insulation panel (VIP) and conventional

 

 

insulations
Insulations Thermal Resistance
(ft2.hr.°F/Btu.in)
Vacuum insulation panel with 25-30

instill core @ 0.1 torr or 0.13 bar

Rigid Polyurethane (close cell)

Extruded Polystyrene/STYROFOAM brand
Expanded Polystyrene (EPS)(Beadboard)
Fiberglass Batting

 

 

 

 

AhUiQ

 

 

 

 

VIP has been tested with an R-30 forl inch of thickness. The cost of VIP is about

$3 - $5 per square foot (Toolbase, 2002) depending on the volume of the order and the

14

precise use and life parameters. The cost of this material may be adjusted to a more
competitive level. The VIP is ideal for applications which require a higher R-value but
without sufficient room for conventional materials. The VIP completes the insulative

envelope, where it has been difficult to insulate in the past.

2.2 Refrigerants

There are several refrigerants to keep products frozen or cold such as regular ice,

dry ice or gel pack. In this experiment, regular ice and dry ice were used to determine the

R-value.
Regular Ice

Regular ice consists of water molecules joined together by hydrogen bonds in a
regular arrangement. It appears that there is considerable empty space between the
molecules. The density of ice is 0.92 g/cm3 at 0°C (McCabe, 1993) and the latent heat is

144 Btu/lb.

15

Dry Ice

   

Block 3.425“ ‘. Pellet
Figure 5. Forms of dry ice: block and pellet

Dry Ice is frozen carbon dioxide (C02). Carbon dioxide is a colorless, odorless,
tasteless gas, about 1.5 times as heavy as air. The specific volume at atmospheric
pressure (100 kilopascals) and 70°F (21°C) is 8.74 fi3 /lb (0.546 m3 /kg). Under normal
conditions, it is stable, inert, and nontoxic.

Dry ice is available in two common forms (Figure 5): blocks and pellets (AAA
Ice, 2002). Block dry ice weighs, on the average, 55 pounds. It can be cut in a variety of
smaller square sizes to accommodate specific needs. Pellet ice is generally easier to
handle and is the preferred choice for special effect needs. The block is better for
longevity.

Dry ice is much denser and colder than traditional ice (ABC Ice house, 2002). In
addition, dry ice doesn't melt. It sublimates. Sublimation is the process of a solid going
directly to a gas. It is nontoxic and noncorrosive and leaves no residue. At atmospheric

pressure, it sublimes at -108.4°F (-78°C), absorbing its latent heat of 240 Btu/1b.

The manufacture of dry ice from carbon dioxide gas is a chemical process (Carl,

2002). The gas is liquefied by compressing it to 900-1000 lb/in2 (6.2-6.9 megapascals) in

three stages using reciprocating compressors and then condensing it in water-cooled
condensers. The liquid is expanded to atmospheric pressure where its temperature is
below the triple point (- 69.9°F or -56.6°C). In the expansion there is a flash separation
(at —lO8.4°F or -78°C) of some solid carbon dioxide "snow" which is very porous. The
snow is removed from the expansion chamber and mechanically compressed into
standard 50-lb (23-kg) blocks in which measure 10 by 10 by 10 in. (25 by 25 by 25 cm).
Dry Ice will sublimate (change from a solid to a gas) at a rate of 10 pounds every 24—
hours in a standard insulated container (ABC Ice house, 2002 ):

Dry ice needs to store in an insulated container (ABC Ice house, 2002 ). The
thicker the insulation, the slower it will sublimate. Never store dry ice in a absolutely
airtight container. The sublimation of dry ice to carbon dioxide gas will cause the
container to expand and probably explode. Dry ice should be stored using proper air
ventilation. The sublimated carbon dioxide gas will flow to low areas and substitute
oxygen. This can cause suffocation if breathed exclusively. Never store dry ice in a
refrigerator freezer. The extremely cold temperature will cause the thermostat to turn the
freezer off. It will keep everything frozen in the freezer but it will be used up at a faster
rate.

The pick up time of dry ice should be as close as possible to the time it is needed.
It sublimates at 10%, or 5 to 10 pounds every 24 hours (ABC Ice house, 2002). Carry it
in a well-insulated container such as an ice chest or cooler. If it is transported inside a car

or van for more than 15 minutes, make sure there is fresh air. For disposal, unwrap and

17

 

lean

[03

shi

CIE

shi

110

iii"

leave dry ice at room temperature in a well-ventilated area. It will sublimate from a solid

to a gas. Do not leave dry ice unattended around children.

The primary use of dry ice is to keep perishables fresh, especially during
shipment. Solid carbon dioxide made its debut as the packing material of choice for ice
cream and frozen food. (Cool Quiz network, 2002). Products such as fish and meat can be
shipped thousands of miles, and arrive at their destinations in good condition. Even
florists take advantage of the preservative property of dry ice and can prevent flower buds
from opening for up to three days by putting them in a solid carbon dioxide atmosphere.

Essentially, dry ice temporarily freezes the aging process to suit our purposes.

Gel Packs:

"capitol Muhollng laukun t... In.

 

 

Figure 6. Gel packs

Gel Packs in Figure 6 (FDC Packaging, 2002) are made from a non-toxic; super-
absorbent powder that, when combined with water, forms a viscous gel-like product

(Marko Foam Products, 2002).

Gel refrigerants are good for maintaining constant temperature for perishable
products. Normally gel refrigerants are suitable for refrigerating products in the range of
30°F (-1°C) to 60°F (16°C) (FedEx, 2002). Temperature maintenance and duration are
dependent on several factors such as the quality and wall thickness of the molded
container, the temperature at which the product is packaged and ambient temperature.
The standard ratio for gel-packs is 1 lb of gel for every 5-7 lbs of product (Marko Foam
Products, 2002). Gel packs are used in floral, dairy products, meat & poultry, prepared

foods, or seafood.

2.3 Heat Transfer

Heat transfer is the transport of thermal energy from one region to another,
normally from the hotter to the colder. The net rate of heat flow is in the direction of
decreasing temperature (Hagen, 1999). Heat transfer can occur by three distinct modes:

conduction, convection and radiation.

Conduction Heat Transfer:

Conduction is the transfer of thermal energy in a solid or a liquid from higher
temperature to an adjacent point of lower temperature. Equation 3 is called Fourier’s law
of heat conduction. The heat flux (q) is proportional to the temperature difference and
inversely proportional to the thickness. The proportionality coefficient (k) is called the

thermal conductivity (Middleman, 1998).

19

Heat transfer through a wall

 

 

 

 

 

T2
T] R
q -’ *q
4—»
Ax

Figure 7. Diagram for heat transfer through a wall

From Figure 7, conduction heat transfer through a wall at steady state in one

dimension follows equation 3. Based on the principle of heat transfer, the heat flows from

higher temperature T1 through the surface of wall to surface T2 (lower temperature).

(3)

q = heat flux (W/m2 or Btu/h.ft2)

k = thermal conductivity (W/m.K or Btu/h.ft°F)

AT = temperature different (°F or K)

Ax = thickness (ft or m)

Thermal conductivity (k) is the amount of heat transmitted through a unit area of
material in a unit time through its total thickness with a unit of temperature difference

between the two opposite sides (Turner, 1981) and indicates how well a material

conducts thermal energy (high for metal, low for plastic) (McCabe, 1993). Thermal

20

conductivity of metals has a broad range of values, about 10 Btu/fi.h.°F for stainless steel
to 240 Btu/fi.h.°F for silver. Water has a thermal conductivity of 0.3-0.4 Btu/ft.h.°F.
Gases have the lowest thermal conductivity; for air k, is about 0.014 Btu/fth.°F. Solids
having low k values are used as insulators to minimize the rate of heat flow. Examples
are polystyrene foam, urethane foam, and fiberglass. These porous insulating materials
act by entrapping air and thus eliminate convection with in the cell if the diameter is less
than 4 mm (Brody, 1997). Their k values are about equal to that for air.

Thermal conductivity is a function of temperature and relative humidity. When
temperature increases, k increases therefore insulation value decreases as shown in
equation 5. In the case of cellular foam, when the relative humidity increases, all or part
of the air or gas in the foam is filled with water so the thermal conductivity of the foam

will be the thermal conductivity of water (4.1 Btu.in/ft2.hr°F at 70°F) instead ofair (0.17
Btu.in/ft2.hr°F at 70°F) (Turner, 1981), so k increases

Rearranging the equation 3:

 

T-T
_ 1 2
q— A: (4)
k
Ax
R=— 5
k ()

R = thermal resistance (fi2.°F.h/Btu) is the material property that opposes the

passage of energy or heat through a material (Turner, 1981). In heat flow, the driving
force is the temperature difference between two bodies from one region to another region.
The thermal resistance of a homogeneous body of uniform cross-section is the reciprocal

of conductance as shown in equation 5 (Turner, 1981). High thermal resistance indicates

21

more effective insulation (ASHRAE, 1993). There are several pertinent properties of
insulation as well as thermal conductivity of structures: the service temperature, the
relative humidity, surface emissivity, reflectivity and absorptivity, density, form and
water transmission.

The reasons for using insulation materials (low thermal conductivity or high
thermal resistance) are to conserve energy by reducing heat loss or gain, control surface
temperature, facilitate temperature control of a chemical process in products, prevent
vapor condensation at the surface with a temperature below the dew point of the
surrounding atmosphere, or reduce temperature variations within a products (ASHRAE,
1993)

Heat transfer through two walls

For two walls, the heat transfer diagram is shown in Figure 8. Heat flows from T1
(higher temperature) through the wall] (k1, x1), past the contact between wall, and wallz

(T2), then through wall; (k2, x2) to surface T3.

T2

 

x1 x2

 

 

 

 

 

Figure 8. Diagram for heat transfer through two walls

Where q = heat flux (W/m2 or Btu/h.ft2)

22

x1 and x2 = thickness of wall] and wallz (ft or m)

k; and k2 = thermal conductivity of wall] and wall; (Btu/hr.ft.°F or W/m.K),
respectively

At steady state, the heat flux is constant so

q = qconduct ,walll = qconduct ,wall 2

kl k2
q=_(T1—T2)=—(T2-T3) (6)
x1 x2

Rearranging equation 6,

qx]
T —T ——
l 2 kl
9352
T -T =———
2 3
k2
Adding these two equations,
qx qx x x
T143 k1 7}: (if if)
1 2 2 2

 

‘E E ‘7’

The R-value for conduction heat transfer through two walls is a function of sum

of thicknesses divided by the thermal conductivities.

23

Heat transfer through multilayered walls

For multilayered walls, the heat transfer diagram is shown in Figure 9.

The heat transfers from T1: higher temperature through wall] (kl , x1) then passes through

the contact of wall] and wall; at T; and flows through wall structure until walln and

through the surface of walln at Tn“.

 

 

 

 

 

 

 

 

 

 

T
T; T3 / 4
I
T. / /
X1 X2 X3 X4 "m... xn
q

 

 

Tn+1

 

 

 

 

 

q= Tl—Tn =T1-Tn
x x x R
A+i+—3+...+i'i
kl k2 k3 kn
x x x
R=—i+—2——+—3+...+f—"—
kl k2 k3 kn

24

(8)

(9)

The R-value for conduction heat transfer through multilayer flat materials is the

sum of thicknesses divided by thermal conductivities.

Convection:

Convection is the mechanism by which thermal energy is transferred between a
solid surface and a fluid moving over the surface. Heat transfer by convection is
proportional to the temperature difference and surface area and is known as Newton’s
law of cooling (Hagen, 1999), shown in equation 10,

q = hAAT (10)
h = heat transfer coefficient (Btu/h.ft2.°F or W/m2.K)

The force that creates convection flow in fluids has two types (Hagen, 1999):
natural convection and forces convection. If currents result from buoyancy forces
generated by differences of density and these densities are caused by temperature
gradients in the fluid mass, the action is called natural convection. If the currents are set
in motion by the action of mechanical device such as fan, pump or agitator so that the
flow is independent of density gradients, this is called forced convection.

The heat transfer coefficient (h) is not a thermal property (Hagen, 1999). It can be
determined by several factors: type of fluid (liquid or gas), flow condition (laminar or
turbulent), forced or natural convection or phase change, free-stream velocity, surface
geometry and roughness, position along the surface and temperature dependence of fluid
properties.

Rearranging equation 10,

25

 

= 11
q l ( )
h
The thermal resistance R for convection is
R - l (12)
h

In convection heat flow, the driving force is the temperature difference between
two bodies. The thermal resistance is reciprocal of heat transfer coefficient as shown in

equation 12 (Turner, 1981).

Combined thermal resistance both conduction and convection (no radiation)

In this system, a wall of thickness x with conductivity k is in contact with air on

both sides as shown in Figure 10. The heat flows from higher temperature (T;) from
inside air to the surface of the wall (T;) by convection and conduction through the wall to

surface T3 and convection from the wall to the air (T4).

26

Heat transfer through a wall contacted with air on both sides

 

Air inside at T1

 

 

 

 

4—»
x,k

Air outside at T4

Figure 10. Diagram for heat transfer of a wall connected with inside and outside air

At steady state, the heat flux is constant so

q = qconvect,inside : qconduct : qconvectputside

k
q=h(T1—T2)=;(T2 —T3)=h(T3 -T4)

Rearranging equation 13,

Adding these three equations,

 

2q qx 2
T T—— —— —
1 4 h k (h
qul— 4 :T1_T4
3,: R

27

(13)

2 x
R=—+— 14
h k ( )

The R-value for combined heat transfer through a wall in contact with inside and
outside air depends upon the number of surfaces in contact with air, 2, the number of

walls, 1, the type of walls (k), and the wall thickness (x) as shown in equation 14.

Heat transfer through two flat materials connected with inside and outside air:

In the system shown in Figure 11, there are two walls in contact with the air

inside and outside.

 

 

 

 

T
'5 / 4
A' . .
1r rnsrde T] q k; k;
X1 X2 _’ q
—>
Air outside T5

 

 

 

 

Figure 1 1. Diagram for heat transfer through two walls connected with inside and outside

air

 

q= = (15)

28

R=——+—+— (16)

The R-value for combined conduction and convection heat transfer through two

walls in contact with inside and outside air depends upon the number of surfaces in

contact with air, 2, number of walls, 2, the type of walls (k], k;), and the wall thickness

(x1, x;) is shown in equation 16.

Heat transfer through multilayered walls in contact with inside and outside air

In multiple walls, there are "11" connected walls in which wall] is in contact with

inside air and walln is in contact with outside air. Heat flows from higher temperature

(Ta) to wall] by convection at surface T1 then conduction from wall] through walln until

surface of walln and convection from walln to air outside at Tb.

29

 

 

in f /‘ T“
/

 

 

 

 

 

 

 

 

 

 

 

 

T1 / Tn+l
Air inside Ta . .
X1 x; x3 x4 .......... ’ X“ Air outsrde Tb
k1 k2 k3 k4 kn ——"
—>
C1
C1

Figure 12. Diagram of heat transfer through multilayered walls in contact with inside and

outside air

 

T -—T T —T
(1:2 x x0 xb x = 0R b (17)
l+—2+—3+ + ”
h k1 k2 k3 kn
x x x
R=3+—l+—2+—l+...+x—" (18)
h kl k2 k3 k,,

The R-value for heat transfer through multiple walls in contact with inside and

outside air depends upon the number of surfaces in contact with air, 2, the number of
walls, n, the type of walls (k1, k;,. . .,kn), and the wall thickness (x1, x;,. . .,xn) is shown in

equation 18.

30

Radiation
Radiation is thermal energy transfer by electromagnetic waves at the speed of

light (2.998x108 m/s). It does not require a medium, so radiation can transmit energy
through a vacuum (Middleman, 1998). Thermal radiation lies in the range of wavelengths
from about 0.1 to 100 um. (Hagen, 1999)
All bodies with a surface temperature above absolute zero emit thermal radiation.
The maximum radiation which a body can emit is given by the Stefan-Boltzmann law
(Hagen, 1999) as shown in equation 19.
E b =0'T4 (19)

where Eb = blackbody emissive power (Btu/h.fi2)

o = Stefan-Boltzmann constant =0.1714x10'8 Btu/h.fi2.°R

T = absolute temperature (°R)

A body that emits radiation according to equation 19 is an ideal radiator. Real
surfaces do not emit radiation following the Stefan-Boltzmann law. Instead, the emission
power (E) is given by equation 20

E = £0T4 (20)

E = emissive power

a = emissivity = E3 (21)
b

Emissivity (e) is a surface property that is used to compare the emissive power of
a real surface with a blackbody at the same temperature. Emissivity varies between 0 to l

and it is dimensionless (Hagen, 1999). Emissivity is a function of surface temperature

31

and the wavelength and direction of the emitted radiation. Emissivity is a strong fimction
of the surface condition so emissivities of polished metals are low, in the range 0.03 to
0.08. For nonmetals such as paper, boards and building materials, emissivity is in the

range of 0.65 to 0.95 (McCabe, 1993).

Incident
radiation

(G)

Reflected radiation
(Gref)

 

Absorbed radiation (Gabs) Semltransparent material

\

Transmitted radiation
(Gtr)

 

Figure 13. Radiation incident on a surface can be absorbed, reflected and transmitted

Irradiation (G) is the radiant heat flux upon a surface (Btu/h.ft2) as shown in
Figure 13 (Hagen, 1999). Thermal radiation is principally a surface phenomenon. When
radiation impinges on a surface, a fraction of the incident radiation can be absorbed,
_ called absorptivity (or), reflected, called reflectivity (p), and/or transmitted, called
transmissivity (t). The amounts absorbed, reflected and transmitted depend on material

properties and may vary with the wavelength or frequency of the incident radiation.

32

These radiation properties can be defined as fractions of the irradiation as shown in the

following equations 22-24:

a = —"”S (22)

p = — (23)

r = J- (24)

where Gabs, Gref, G, are the absorbed, reflected and transmitted portions of the

irradiation G, respectively
Absorptivity, reflectivity and transmissivity are dimensionless numbers that are
range between 0 to 1. According to the first law of thermodynmics, energy is conserved
so the radiation incident on the surface must equal to the sum of the radiation absorbed,
reflected and transmitted (Hagen, 1999) as shown in equation 25,
G=Gabs +Gref +6” (25)
Dividing equation 25 by G:
a + p + 2' =1 (26)
If the material is opaque, t = 0 and the equation 26 becomes,
a + p =1 (27)
The radiation properties defined by absorptivity, reflectivity and transmissivity
are total hemispherical properties that are averaged over all wavelengths and directions.

Under steady state conditions, thermal equilibrium exits between objects and their

enclosure. The energy emitted and absorbed by the object is

33

As(T)aT4 = Aa(T)oT4 (28)
Dividing equation 28 by AoT4,

8(T) = a(T) (29)

Equation 29 is known as Kirchhoff 3 law (Hagen, 1999). It states that the total
hemispherical emissivity of a surface at temperature T equals to the total hemispherical

absorptivity for radiation arriving from a surface at the same temperature.

2.4 Thermal Testing Approaches for Insulating Packages

There are several procedures to estimate the rate of heat transfer for insulated
packages. One is the American Society for Testing and Materials (ASTM) D3103-92
(ASTM, 1997). Some researchers and companies have developed other testing for

insulated containers.

ASTM D3103-92

This standard is used to determine the thermal insulation capability of a package
with or without interiors or refrigerants. Temperature indicating devices such as
thermocouples, thermistors, bi-metals recorders, or portable recorders are used to
determine temperatures. For small packages (volume is less than 1 ft3), at least three
thermocouples are used to measure the temperature profile. For packages with interior
walls (separate product and refrigerant) more thermocouples may be desired to

adequately measure temperature. For the bigger packages (inside volume more than one

34

R3), at lease 10 thermocouples need to determine the locations of the coldest and warmest
spots to get the temperature profile.

Using refrigerants, the weights of refiigerants and temperatures are recorded
before being put into the package and during all testing until the temperatures are above
the maximum or below the temperature dictated by the product. This package is sealed
tightly and put on a wooden shelf at the test condition. The date, hour of the start and

finish of testing are collected.

Other testing methods

The ice melt test (Burgess, 1999) was used in this research to estimate the thermal
resistance (R-value) of insulating packages varying in the size between 0.5 to 5 ft3 .
Regular ice was used as the refrigerant for testing. R-values were calculated using
equation 1. The data was converted into equation 2. Equation 2 is used to estimate the R-
value of an insulating packaging at an accuracy about i20%. This is an easy method to
determine the R-value of containers and to approximate the amount of ice required to

maintain temperature in the packages.

35

Chapter 3

Experimental Design and Results

36

3. Experimental Design and Results

3.1 Materials:

1.

2.

3.

4.

Ice (cubed) from Quality Dairy (QD)
Dry ice (cubed) from Chemistry Department, Michigan State University
Gray paint buckets made from HDPE, 8"x6.5" (diameterxheight)

Mass Balance: Mettler PM 2000

Box Samples:

1.

Regular Slotted Container (RSC) C-flute corrugated box, inside dimensions
(LxWxD) = l3"x9"x14"

Regular Slotted Container (RSC) C-flute corrugated box, inside dimensions =
15"x12"x10"

Expanded polystyrene cooler (EPS cooler), 1.5 inch thick and inside dimensions =
12"x9"x10"

Expanded polystyrene cooler (EPS cooler), linch thick and inside dimensions =
14"x1 1"x8.5"

Molded Polyurethane box: one inch polyurethane injected between two
corrugated boards, inside dimensions = 10"x10"x10" and one open cell foam
cover as shown in Figure 14.

Vacuum Insulation Panel (VIP) in C-flute corrugated box: six VIP panels placed

inside box, inside dimensions = 10"x10"x10"(LxWxD) as shown in Figure 15.

37

   
 

Open cell foam
cover

Figure 14. Molded polyurethane with open cell foam

VIP

   

Figure 15. VIP in a corrugated box

38

Table 3. Summary specifications for insulation materials

 

 

 

 

 

 

 

 

 

 

 

 

 

Samples Inside Dimensions Thickness Inside Surface
(inxinxin) (in) Area (fiz)

1.RSC (C-flute) 13x9x14 0.156 5.90
2.RSC (C-flute) 15x12x10 0.156 6.25
3.EPS Cooler 12x9x10 1.500 4.42
Li. EPS Cooler 14x1 1x8.5 1.000 5.09
5.Molded polyurethane (PU) 10x10><10 1.312 4.17

.Vacuum Insulation Panel (VIP) lelelO 1.156 4.17
3.2 Methods

Three conditions were used for this work:
1. Standard conditions: 72°F and 50 % RH
2. 72°F and 85%RH (to compare %RH at the same temperature)
3. 104°F and 50 % RH (to compare temperature at the same %RH)
Standard conditions were tested for all sample boxes (Table 3). Conditions 2 and
3 were tested for only C-flute corrugated boxes (samples 1 and 2) to compare the effect
of board moisture content and temperature.
In this experiment, the ice melt test (Burgess 1999) in Figure 16 was performed to
estimate the thermal resistance (R-value) of these insulating packages. There were three

replicates for each package per condition.

Preconditioning
Box samples were put in the conditioning room for at least 24 hrs prior to testing.
A quantity of regular ice (as much as possible to reduce the error associated with water

on ice left from preconditioning) was placed into a bucket. The package was loosely

39

closed and stored in the conditioning room for several hours to ensure that all the surface
of the ice was covered by water. This was done to make the temperature of the ice equal

to 32°F (not the store temperature of ice of about 0°F). The ice will hold this temperature
until all the ice is melted. The water was drained and discarded after this step. Dry ice did

not need to be preconditioned because it evaporates to a gas at —108.4°F.

Procedure:

The ice melt test is shown in Figure 16. The bucket is placed back into the center
of the box and the box flaps are tightly sealed with tape. Sealing is important so no heat
loss occurs through openings. The box sits until almost all the ice melts (to reduce the
error from preconditioning that it has some water left on the surface of the ice). Finally,
the amount of water that melts is measured and the melt time is recorded.

The dry ice test is shown in Figure 17. Dry ice is put into the bucket and weighed.
The box is sealed tightly with tape. The box sits until almost all of the dry ice sublimates.
Finally, the remaining dry ice is weighed and subtracted from the original weight. This
would be the amount of dry ice sublimated. The time is noted. The experimental R-value

is then calculated from the experimental results using equation 2, as shown below

_ A x AT
meltrate x latentheat

 

exp

fi2.F.hr

Where RC = thermal resistance of container wall (
XP Btu

)

A = inside surface area of box (ftz)

AT = temperature difference (°F) between the outside air and the melting point of

the refrigerant used

40

Melt rate: the rate that ice melts during the experiment (equal to the weight of the
ice melted divided by the melt time (lb/hr))

Latent heat = 144 Btu/lb for regular ice and 240 Btu/lb for dry ice

The predicted R-value (Burgess, 1999) was calculated from the following fitted equation

R P = 3.9th +1.5np + 3.2nf (i20% accuracy)

Where th = the average wall thickness (inch)
up = the number of plain surfaces

nf = the number of aluminium foil surfaces

41

Put the sample box in the conditioning room

At least 24 hours

Put ice in the bucket

l Sit for several hours
Drain water out and discard

Put the bucket back
into the sample box

Seal the sample box

1

Place sample box in the conditioning storage

Measure melted water and note exposure time

Figure 16. The procedure for the regular ice melt test

42

 

—

Precondition

Procedure

Put the sample box in the conditioning room

At least 24 hours

Put dry ice into the bucket
Weigh dry ice and bucket and put into the sample box
1
Seal the sample box
Place sample box in the conditioning room
1

Open box, check weight and note the time

Figure 17. The procedure for the dry ice melt test

43

3.3 Results

The predicted R-value (RP) and the experimental R-values (chp) were compared.

The results in Tables 4 and 5 show the experimental results for regular ice and dry ice
respectively. There was no significant difference between the theoretical R-values using
regular ice under the three different conditions of the relative humidity and temperature.
The R-value for the corrugated box is less than the R-value for the EPS cooler, which is
less than the R-value for molded polyurethane, which is less than the R-value for the VIP.
The results for VIP for regular ice from ice melt test in Table 4 are lower than the R-
value (12.66) that the company quotes (25-30). The results for VIP for dry ice from the
ice melt test in Table 5 have an average about of 26 which does match with what the
company quotes. These prove that the same container can have different R-values using
different refrigerants. The temperature and the surface area of the product affect the R—
value too. Different materials used in these insulating packages can affect the R-value
depending on the thermal conductivity of the materials as shown in Table 6 (type,

structure or density), heat transfer coefficient, surface area and thickness.

44

R-values from ice melt test using regular ice and dry ice at 72°F and 50%RH

Table 4. R-values of insulating packages from regular ice melt test at 72°F and 50%RH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Samples Ice Melt Wt Melting Time R-value Average Stdev R-value % Error
(lb) (hr) Experimental Predicted R-value
l 4.39 14 5.23 5.44 0.19 5.11 -6.07
4.19 14 5.48
4.1 14 5.60
2 4.14 13 5.45 5.68 0.21 5.11 -9.98
3.94 13 5.73
3.86 13 5.85
3 4.74 39.25 10.17 9.86 0.50 10.35 5.01
4.76 39.25 10.12
5.29 40 9.28
4 4.37 25.5 8.25 8.25 0.27 8.4 1.86
4.32 26 8.51
4.25 24 7.98
5 4.06 38 10.84 10.73 0.11 9.62 -10.37
4.25 39 10.63
4.32 40 10.73
6 4.59 51 12.87 12.66 0.36
4.32 48 12.87
4.54 48 12.25

 

45

 

Table 5. . R-values of insulating packages from dry ice melt test at 72°F and 50%RH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Samples Ice Melt Wt Melting Time R-value Average Stdev R-value % Error
(lb) (hr) Experimental Predicted R-value
l 7.54 22 12.94 12.92 0.36 5.11 -60.44
7.07 20 12.55
6.69 20 13.26
2 4.91 10.5 10.04 12.37 2.04 5.11 -62.69
7.09 20 13.25
6.8 20 13.82
3 6.02 29 16.00 17.16 1.13 10.35 -39.67
5.79 30 17.21
6.55 36 18.26
4 6.83 23 12.88 14.06 1.13 8.4 -40.27
6.74 25 14.19
7.34 29 15.12
5 5.61 28.33 15.83 16.23 0.84 9.62 -40.73
6.00 30 15.67
5.43 30 17.19
6 8.19 72.5 27.75 26.43 1.34
5.00 40 25.08
8.17 69 26.47

 

 

Table 6. Thermal conductivity of materials

 

 

 

 

 

 

 

 

Materials Thermal Conductivity (Btu/hr.ftz.°F) at 70°F
Air 0.015
Water 0.346
Corrugated board 0.035
EPS foam 0.022
olyurethane 0.01 8

 

 

46

 

 

 

The effect of the relative humidity on the R-value

The relative humidity also affects the R-value since thermal conductivity is a

function of relative humidity, especially for insulating packages using corrugated boxes

that trap humidity with air. The thermal conductivity will be between the thermal

conductivity of pure water (0.346 Btu /fi.hr°F at 70°F) and air (0.015 Btu /fl.hr°F at

70°F) (Hagen, 1999). So the conductivity should increase as the relative humidity

increases. When the relative humidity increases as shown in Tables 7 and 8, the R-values

of packages are decreased.

Table 7. R-value of C-flute corrugated box at 72°F and 85%RH using regular ice

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Samples Ice Melt Wt Melting Time R-value Average Stdev R-value % Error
(lb) (hr) Experiment Predicted R-value
1 3.31 7 3.47 3.35 0.52 5.11 52.39
3.02 7 3.80
4.7 8 2.79
2 3.28 7 3.71 3.52 0.41 5.11 45.17
3.2 7 3.80
4.56 8 3.05
Table 8. R-value of C-flute corrugated box at 72°F and 85%RH using dry ice
Samples Ice Melt Wt Melting Time R-value Average Stdev R-value % Error
(lb) (hr) Experiment Predicted R-value
1 8.27 18.5 9.92 10.71 0.79 5.11 -52.29
8.17 19.75 10.72
7.62 19.75 11.49
2 8.66 20 10.85 11.42 0.63 5.11 -55.24
8.21 19.75 11.30
7.67 19.75 12.10

 

 

 

 

 

 

 

 

47

 

 

The effect of temperature on the R-value

Thermal conductivity is a function of temperature. The thermal conductivity of

insulating materials normally increases with increasing temperature (ASHRE, 1993) so

when the temperature increases, the R-value decreases as shown in Tables 9 and 10.

Dry ice has a much lower temperature (-108.4°F) than regular ice (32°F). The

thermal conductivity of the material and the heat transfer coefficient for carbon dioxide

decreases so the R-values for dry ice are somewhat higher than for regular ice in the same

containers.

Table 9. R-value for regular ice at 104°F at 50% RH in a corrugated box

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Samples Ice Melt Wt Melting Time R—value Average Stdev R-value % Error
(1b) (hr) Experiment Predicted R-value
1 4.19 5.5 2.15 2.18 0.03 5.11 134.40
4.12 5.5 2.19
4.10 5.5 2.20
2 3.95 5.0 2.20 2.18 0.05 5.11 134.05
4.30 5.5 2.22
4.48 5.5 2.13
Table 10. R-value of dry ice at 104°F at 50% RH in a corrugated box
Samples Ice Melt Wt Melting Time R-value Average Stdev R-value % Error
(lb) (hr) Experiment Predicted R-value
1 5.71 9.67 7.51 7.36 0.23 5.11 -30.57
5.99 9.58 7.09
5.63 9.50 7.48
2 6.11 9.83 7.55 7.70 0.16 5.11 -33.61
6.12 10.00 7.68
5.83 9.75 7.86

 

 

 

 

 

 

 

 

 

48

 

 

 

 

Both the relative humidity and temperature affect R-values. The higher the

relative humidity and storage temperature, the lower the R-value as shown in Tables 11

and 12.

Table 11. Summary of the R-values for three conditions (regular ice)

 

 

 

Samples R-value R-value R-value
72°F, 50%RH 72°F, 85%RH 104°F, 50%RH
1 5.44 3.35 2.18
2 5.68 3.52 2.18

 

 

 

 

Table 12. Summary of the R-values for three conditions (dry ice)

 

 

 

Samples R-value R-value R-value
72°F, 50%RH 72°F, 85%RH 104°F, 50%RH
1 12.92 10.71 7.36
2 13.70 11.42 7.70

 

 

 

 

49

 

 

Chapter 4

Analysis and Discussion

50

The predicted and experimental R-values of insulating containers were compared
in Chapter 3. The simulation model shown in the Appendix was developed using BASIC
to predict the R—value. The model considers conduction, convection and radiation to solve
the heat transfer problem shown in Figure 18.

The theory of heat transfer by conduction, convection and radiation requires
radiation balances and energy balances. Solving these equations to determine the heat

transfer rate allows us to calculate the R-value.

 

 

- Air outside T5

 

 

 

 

 

Figure 18. Performing ice melted test of ice or dry ice in a insulating container

The following equations are used in the energy and radiation balances:
T1 = temperature of product (using ice =32°F or dry ice =-108.4°F)
T; = temperature of air between product and inside wall (°F or K)
T3 = temperature of inside wall (°F or K)

T4 = temperature of outside wall (°F or K)

51

T5 = temperature of air outside (°F or K)
A] = surface area of product (ft2 or m2)
A3 = surface area of inside wall (ft2 or m2)

A4 = surface area of outside wall (ft2 or m2)

Q = input energy

G1, G3, G4 = irradiation on product surface, inside box surface and outside box
surface respectively

6 = Stefan-Boltzmann constant =0.1714x10'8 Btu/h.ft2.°R

e], 82, 83 = emissivity of product, inside box surface and outside box surface
respectively

011, 01;, a3 = absorptivity of product, inside box surface and outside box surface
respectively

p], p;, p3 = reflectivity of product, inside box surface and outside box surface

respectively
k’s = the thermal conductivity of wall

h’s = heat transfer coefficient of air

Radiation balances:

1. At the product surface (A1):

A
4 1 4

52

2. On the inside surface of the box (A3)
GA— AT4 GA AT4 G A‘
3 3‘51011 +91 1 1+(5303 3 +P3 3A3)(1“A—)
3

_ 4 4 4

3.0n the outside surface of the box (A4)

A
4
Energy balances:
4. Product:
Q+GiAla1=£1A1UT14 +hA1(T1—T2) (33)

5. Inside box wall
k3A3 4
hA3(T2 —T3)+a3G3A3 =_Ax—(T3 —T4)+.<:30'A3T3 (34)
6. Outside box wall

k3A3T T GA-hAT T AT4 35

"AT(3‘4)+0‘444— 4(4‘5)+54‘744 ()
7. Air between product and inside box wall

hAl (Tl — T2) = hA3(T2 — T3) (36)

In these equations, G1, G3, G4, Q, T;, T3 and T4 are unknown. These unknowns

were found using an iteration method in BASIC. The R-value is determined as the

temperature different, T5 -T1, divided by Q times A3.

53

 

The thermal conductivity, heat transfer coefficient and surface area of products
(regular ice and dry ice) are needed for the computer model. The heat transfer coefficient

is a function of temperature. The temperature of air for regular ice at the wall is taken as

the average or “film”temperature (Tf) between ice (32°F) and the storage temperature
(72°F), which is equal to 52°F (284 K). When the storage condition is 104 °F, the
average temperature (Tf) is equal to 68°F (293 K). From these temperatures, the

properties of air at atmospheric pressure are as shown in Table 13.

Table 13. Properties of air at 284 and 293 K (Holman, 1986)

 

 

 

 

 

 

 

T v><106 k Pr
(K) (m2/s) (W/m°C)
284 14.29 0.025 0.712
293 15.08 0.026 0.71

 

The heat transfer coefficient (h) of air was estimated from the product of the Grashof and

Praudtl numbers, GrPr. Gr can be calculated from equation 37 and Pr from Table 13.

Then, Nu f is calculated using equation 38 and h was found in equation 39.

g = acceleration velocity (9.8 m/sz) and l = height of container (m). For 1 = 10 in =

0.254 m. h can calculate from the following equations:

 

GrzgxfixATxl3

V2

(37)

9.8 x L x (295 — 273) x 0.2543
= 284

 

(14.29 x1043)2

G,=6.1x107

54

G,Pr =6.1x107x0.712 =4.34x107

From Table 14 (Holman) using vertical planes and cylinders at G,Pr between 104-

10

X517=C(Grf Pr f)“ (38)

1 1
Nu f = 0.59(G,P,)4 =0.59(4.34 x 107 )4 = 47.9

 

Nu xk

f
l

h:

 

(39)
= 47.9 x 0.025
0.254

h=4.71 —W—=0.83 —13’—”—— for air

m2K ft2 .hr.°F

Table 14. Constants for use in the equation 38.

 

 

 

 

Geometry GrfPrf_ C m

Vertical planes and cylinders 104-1g 0.59 1/4
103-101‘ 0.021 2/5

10’40" 0.1 1/3

 

 

 

 

 

 

In this experiment, the height of samples 1 and 2 (corrugated boxes) are 14 and 10
inches respectively and k and h are shown in Tables 15 and 16 for 284 and 293 K
respectively. The heat transfer coefficient depends on thermal conductivity and length

(k/l) and temperature difference.

55

Table 15. Summary of k and h at length 10 and 14 inches at 284 K for air

 

 

 

 

 

 

 

L Al k h
(in) (ftz) (Btu/fi.hr.°F) (Btu/ft2.hr.°F)
14 5.90 0.0144 0.77

10 6.25 0.0144 0.83

 

Table 16. Summary of k and h at length 10 and 14 inch at 293 K for air

 

 

 

L A1 k h
(in) (£12) (Btu/fi.hr.°F) (Btu/fi2.hr.°F)
14 5.90 0.015 0.89

10 6.25 0.015 0.97

 

 

 

 

 

 

 

The results for the R-value show that the thermal conductivity affects the R-value

very little as shown in Table 17. When the thermal conductivity changes from 0.035 to

0.0144 Btu/hr.ft.°F, R slightly changes from 21.58 to 22.22. On the other hand, the R-
value rapidly decreases when the product surface area increases from 0.2 to 0.4 fiz. It

goes from 21.58 to 12.09.
The product (ice) surface area affects the R-value more than the thermal
conductivity and heat transfer coefficient. The surface area for the regular ice used in the

experiment can be calculated from the surface area of the bucket that had a size dxh =
8x6.5 inxin, so the surface area = 2nr2+2rtrh = 1.83 fiz. When the regular ice melts, water

and regular ice are mixed so the surface area does not change during melting.

56

Table 17. Results for R-value from the computer model using regular ice at T5 (72 °F)

and 50%RH at k=0.035, 0.025 and 0.0144 Btu/hr.ft°F.

 

 

 

 

 

 

 

 

 

 

A3 (5.9 £12) A3 (6.25 it")
A; R R R R R R
fiz (k=0.035) (k=0.025) (k=0.0144) (k=0.035) (k=0.025) (k=0.0144)
()5 21.58 21.76 22.22 20.98 21.16 21.62
1,0 12.09 12.26 12.72 11.78 11.96 12.42
15 9.02 9.20 9.66 8.68 8.86 9.32
2.0 7.45 7.62 8.08 7.20 7.38 7.84
25 6.48 6.65 7.11 6.28 6.46 6.92
30 5.86 6.03 6.49 5.66 5.84 6.30
3.5 5.43 5.61 6.07 5.21 5.38 5.84
40 5.10 5.27 5.73 4.88 5.06 5.52

 

 

 

 

 

 

 

 

 

The R-values from the computer model show the effect of storage conditions (72
and 104°F). The higher temperature (104°F) has a lower R-value than at 72°F as shown

in Table 18. The higher the thermal conductivity, the faster the heat transfer, so the lower

the R-values.

57

Table 18. Results for R-value from the computer model for regular ice at 72 and 104°F

 

 

 

 

 

 

 

 

 

 

and 50%RH
A3 (5.90 82) A3 (6.25 112)

A1 R-value R-value R-value R-value
82 T5 =72°F T5 =104°F T5=72°F T5 =104°F
0.5 21.58 17.24 20.98 16.69
1.0 12.09 9.84 11.78 9.48
1.5 9.02 7.33 8.68 7.06
2.0 7.45 6.11 7.20 5.86
2.5 6.48 5.35 6.28 5.16
3.0 5.86 4.86 5.66 4.67
3.5 5.43 4.52 5.21 4.32
4.0 5.10 4.25 4.88 4.07

 

 

 

 

 

 

 

The surface area, the thermal conductivity and the heat transfer coefficient were
varied for dry ice. The surface area of dry ice can be calculated from the total weight of
dry ice put into the bucket divided by the weight of dry ice for one rod shown in Figure
19. We assume that the dry ice rods were packed as a block (L=W=D).

Weight of dry ice per rod = 14 g

1 cm Size of dry ice = dxh = 1cmx3 cm

<—>

4 v = nrzh = 3.142x0.52x3 = 2.36 cm3

V= Volume per rod

 

 

 

3 cm Weight of dry ice put into the bucket = 3800 g
So how many rods were put in the bucket
v V =3800g/14g =271 rods
Figure 19. Dry ice rod Volume for all dry ice = 271 X236 = 640 cm3

58

Volume for all dry ice = (L x W x D) = L3
L3 = 640 cm3
L = 8.62 cm

Surface area ofdry ice = 6x(8.62x8.62) = 446 cm3 = 0.48 fiz

When the dry ice sublimates, the size of the dry ice decreases, so the surface area

also decreases. The R-value therefore increases when the size of dry ice decreases.

The properties of carbon dioxide gas at 245 K (Tf =(l95+295)/2) and 254
(Tf‘:(195+313)/2) K are shown in Table 19. The results for the heat transfer coefficient

for carbon dioxide gas at the storage temperatures 72°F and 104 °F are shown in Tables

20 and 21. The calculations are the same as for air.

Table 19. Properties of carbon dioxide gas at 245 and 254 K

 

 

 

 

 

 

 

T vx106 k Pr
(K) (mZ/s) (W/m°C)
245 5.59 0.0125 0.797
254 6.01 0.0132 0.791

 

Table 20. Summary of k and h at length 10 and 14 inch at 245 K of carbon dioxide gas

 

 

 

 

 

 

 

L A1 k h
(in) (ftz) (Btu/fi.hr.°F) (Bttl/fi2.hr.°F)
14 5.90 0.0072 0.99

10 6.25 0.0072 0.96

 

59

 

 

 

Table 21. Summary of k and h at length 10 and 14 inch at 254 K of carbon dioxide gas

 

 

 

 

L A1 k h

(in) (112) (Btu/fi.hr.°F) (Btu/ftz.hr.°F)
14 5.9 0.0083 1.04

10 0.0072 0.0083 1

 

 

 

 

 

The surface area of the dry ice affects the R-value more than the thermal

conductivity as shown in Table 22. The R-value rapidly decreases when the surface area

increases from 0.2 to 0.4 it2 at k = 0.0211 Btu/hr.ft.°F, that is 33.13 to 17.67. The surface

area of dry ice from the experiment is about 0.5 it2 that the results from the computer

model at k = 0.0211 Btu/hr.ft.°F are about 15.09. This is better than the predicted result

calculated from equation 2.

Table 22. Effect of the thermal conductivity of dry ice on the R-value

 

 

 

 

 

 

 

 

 

 

 

A3 (5.9 82) A3 (6.25 82)
AI R R R R R R
ftz (k=0.035) (k=0.0211) (k=0.0072) (k=0.035) (k=0.022) (k=0.0072)
0,; 32.85 33.09 34.30 35.79 36.04 37.26
04 17.41 17.66 18.84 18.92 19.17 20.39
0,6 12.25 12.50 13.70 13.28 13.53 14.74
0,3 9.68 9.92 11.12 10.46 10.71 11.93
1 8.13 8.38 9.59 8.77 9.02 10.24
1.2 7.11 7.36 8.56 7.65 7.90 9.12
1.4 6.37 6.62 7.82 6.85 7.10 8.32
1,6 5.82 6.07 7.27 6.25 6.50 7.71
1,3 5.39 5.64 6.83 5.78 6.03 7.24

 

 

 

 

 

 

 

60

 

The R-values from the computer model show little effect of the storage conditions
(72 and 104°F). The higher temperature (104°F) has a lower R-value than at 72°F as

shown in Table 23. The higher the thermal conductivity and heat transfer coefficient, the

lower the R-values.

Table 23. R-values from the computer model using dry ice at 72 and 104°F and 50%RH

 

 

 

 

 

 

 

 

 

 

 

A3 (5.90 fil) A3 (6.25 tt‘)

A1 R-value R-value R-value R-value
ft2 T5 =72°F T5 =104°F T5=72°F T5 =104°F
0.2 33.09 31.43 36.04 34.54
0.4 17.66 16.76 19.17 18.33
0.6 12.50 11.85 13.53 12.94
0.8 9.92 9.40 10.71 10.23

1 8.38 7.95 9.02 8.62
1.2 7.36 6.97 7.90 7.54
1.4 6.62 6.26 7.10 6.77
1.6 6.07 5.74 6.50 6.19
1.8 5.64 5.33 6.03 5.74

 

 

 

 

 

 

 

The R-values from experiment fit well with the predicted R-values for regular ice
as shown in Table 24 since the predicted R-values were obtained using regular ice. The

computer model still needs to be modified to have a better fit for regular ice.

61

Table 24. Experimental, computer model and predicted R values for regular ice at 72°F

 

 

 

and 50%RH
Samples R-value R-value R-value
Experiment Computer model Predicted
1 5 .44 8.16 5. 1 1
2 5.68 7.88 5.1 1

 

 

 

 

 

The R-values from the computer model fits better for dry ice than the predicted R-

value as shown in Table 25. The surface area of dry ice affected the R-value the most.

This is not mentioned in the predicted R—value.

Table 25. Experimental, computer model and predicted R values for dry ice at 72°F and

 

 

 

50%RH
Samples R-value R-value R-value
Experiment Computer model Predicted
1 12.92 15.09 5.11
2 13.70 16.36 5.11

 

 

 

 

 

 

 

The relative humidity also affected to the R-values in the experiment (Tables 11
and 12). The disadvantage of the computer model is that it does not include the relative

humidity, so this model still needs to be developed. The predicted R-value also does not

include the relative humidity.

62

Chapter 5

Conclusions and Future Studies

63

5.1 Conclusions

The experimental results from the ice melt tests showed that the sizes of the R—
value were VIP>molded polyurethane>EPS cooler>corrugated box. The higher the R-
value, the better the insulating package.

Thermal conductivity is function of temperature and relative humidity. When the
temperature and the relative humidity increase, the thermal conductivity increases.
Therefore, the R-value decreases. The convection heat transfer coefficient is also a
function of temperature. As the temperature increases, it increases and the R-value
decreases. The surface area of the product affects the R-value more than the thermal
conductivity and the heat transfer coefficient.

The experimental R-values for dry ice were much higher than for regular ice since
the temperature of dry ice is much lower than regular ice. These lower both the thermal
conductivity and convection heat transfer coefficient, which raises the R-value. But the
effect of product surface area is even greater.

The experimental R-values fit well with the predicted R-values for regular ice
since the predicted R-values were based on regular ice. The computer model still needs to
be developed to get an even better fit for regular ice.

The R-values from the computer model fit better for dry ice than the predicted R-
value, because the computer model accounts for the surface area of dry ice. The surface
area was not accounted for the predicted R-value.

The computer model needs to be modified to include the effect of relative
humidity. The theoretical R-value also needs to be developed to include both the relative

humidity and the surface area of products.

64

5.2 Future Studies

It is recommended that further studies on more variety and sizes of packages
using different materials such as Al-foil or combination of materials. The effect of
openings in the packages should be considered. Other refrigerants such as gel packs
should be investigated too. The R-value should be compared with the regular ice and dry
ice.

The computer model should be developed to include the relative humidity and
also to optimize the properties of containers with cost. The warm up times also should be

studied the simulation model.

65

Appendix

66

The computer model was developed using the BASIC program to calculate for the
R-value of the insulating containers. The program has 73 lines from 10-730 as shown
below. There are six categories: first is input data from lines 20-120 (giving data). Second

is “find k value” from lines 140-280 (depending on the structure of containers). Third is

guesses and variable parameters from lines 300-350 (varied T1, T5 and A1). Fourth, is the

setup system of equations from lines 370-430 (equations 30-36). Fifth, is the solved
system from equations 450-570 (made in matrix form and solved using iteration). Finally,
are the solutions from line 590-730 (got some results and printing statement). G1, G3,

G4, T2, T3, T4, Q and R were solved from this model.

Computer Model

10 REM Solve for G1, G3, G4, T2, T3, T4, Q and R

20 REM: Users might change to line 50,60,80,l 10,120,140,190,320,330,340
30 '**************input data******#**#****

40 DIM C(7,8) 'matrix to get R-value by iteration

50 A3: 4.42 'ft"2

60 A4= 7.375 'ft"2

70 S=l.7l4*10"-9 'Stefan-Boltzman constant (btu/h.ft"2.R"4)
80 EM1=.985: M3=.9 : EM4=.9 'emmissivities of surfaces 1,3,4
90 AB1=EM1 : AB3=EM3 : AB4=EM4 'absorptivities

100 RE1=l-EM1 : RE3=l-EM3 : RE4=l-EM4 'reflectivities
110 X=.125 'overall thickness of wall (ft)

120 H=1 'convection heat transfer coeff (btu/h.ft"2.F)

130 '*********************find k value*******************
140 N=l

150 DIM A(N), B(N)

160 SUM=0

170 FOR M=1TO N

180 READ A(M), B(M)

190 DATA 0.125, 0.022

200 SUM=SUM+A(M)/B(M)

210 NEXT M

220 K==SUM

230 SUMA=0

240 FOR M=1 TO N

67

250 SUMA=SUMA+A(M)

260 NEXT M

270 L=SUMA

280 K3=L/K

290 '*********guess and variable parameters******

300 T20LD=0: T3OLD=0: T4OLD=0

3 10 G1 OLD=0: G3 OLD=0: G4OLD=0: QOLD=Q

320 FOR A1 = 2 TO 4 STEP .5

330 FOR T1=460-108 TO 460+32 STEP 140

340 FOR T5=460+72 TO 460+104 STEP 32

350 T2=460+43: T3=460+47 :T4=460+63

360 '*************Semp System ofequations********

370 C(1,1)=A1:C(1,2)=-RE3*A1:C(1,5)=-EM3*S*A1*T3"3

380 C(2,1)=-RE1*A1:C(2,2)=A3-RE3*A3+RE3*A1: C(2,5)=EM3*S*T3"3*(A1-
A3):C(2,8)=EM1*S*A1*T1"4

390 C(3,3)=A4: C(3,8)=EM4*S*A4*T5"4

400 C(4,1)=A1*AB1: C(4,4)=H*A1: C(4,7)=1: C(4,8)=EM1*A1*S*T1"4+H*A1*Tl
410 C(5,2)=AB3*A3: C(5,4)=H*A3: C(5,5)=-
(H*A3+(K3*(A3+A4)/2/X)+EM3*S*A3*T3"3): C(5,6)=K3*(A3+A4)/2/X
420 C(6,3)=AB4*A4: C(6,5)=K3*(A3+A4)/2/X: C(6,6)=-
((K3*(A3+A4)/2/X)+H*A4+EM4*S*A4*T4"3): C(6,8)=-H*A4*T5
430 C(7,4)=-H*(A1+A3): C(7,5)=H*A3: C(7,8)=-H*A1*T1

440 '*************Solve System ofequations*********

450 FOR I=1 TO 7 : PVT=I

460 FOR K=I+1 TO 7

470 IF ABS(C(K,I))>ABS(C(PVT,I)) THEN PVT=K

480 NEXT K

490 FOR J=I TO 8

500 CHG=C(I,J) : C(I,J)=C(PVT,J) : C(PVT,.I)=CHG

510 NEXT J

520 FOR K=1 TO 7 : IF K=I THEN 550

530 F=C(K,I)/C(I,I)

540 FOR J=I+1 TO 8 : C(K,J)=C(K,J)-F*C(I,J) : NEXT J

550 NEXT K

560 NEXT I

570 FOR K=1 TO 7 : C(K,8)=C(K,8)/C(K,K) : NEXT K

580 ' ********done With solutions************

590 T2=C(4,8): T3=C(5,8): T4=C(6,8)

600 G1=C(1,8):G3=C(2,8): G4=C(3,8): Q=C(7,8)

600 G1=C(1,8):G3=C(2,8): G4=C(3,8): Q=C(7,8)

610 R=A3*(Tl-T5)/Q

620 QTY=ABS(T2-T20LD)+ABS(T3 -T3 OLD)+ABS(T4-T4OLD)+ABS(G1 -
G1 OLD)+ABS(G3-G3 OLD)+AB

S(G4-G4OLD)+ABS(Q-QOLD)

630 T20LD =T2: T3OLD=T3z T4OLD=T4

640 GlOLD=G1: G3OLD=G3: G4OLD=G4: QOLD=Q

68

650 IF QTY<=.1 THEN GOTO 690

660 FOR I=l TO 7 :FOR J=1 TO 8: C(I,.l)=0

670 NEXT J: NEXT I

680 GOTO 370

690 'PRINT:PRINT"T1=";T1-460" T2=";T2-460;"T3=";T3-460;"T4=";T4-
460;"T5=";T5-460

700 'PRINT:PRINT"G1=";G1; "G3=";G3; "G4=";G4; "Q=";Q ;"R=";R
710 ‘PRINT : PRINT Al T1 T5 R

720 PRINT"A1=";A1, "T1="; T1-460 ,"T5="; T5-460 , "R=";R

730 NEXT T5: NEXT T1: NEXT A1

69

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70

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72

   

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