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This is to certify that the

thesis entitled

MOISTURE SHELF LIFE OF PACKAGED MILK POWDER

presented by

KRITTIKA TANPRASERT

has been accepted towards fulﬁllment
of the requirements for

 

 

MASTER degree in PACKAGING

Q“; ' J... J'ﬁ

Major professor

 

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

 

LIBRARY

Michigan State
University

 

 

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DATE DUE

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MOISTURE SHELF LIFE OF PACKAGED MILK POWDER
By

Krittika Tanprasert

A THESIS
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
MASTER OF SCIENCE
School of Packaging

1 999

ABSTRACT
MOISTURE SHELF LIFE OF NON-FAT DRY MILK
By

Krittika Tanprasert

As new packaging materials become available for non-fat dry milk, there is a need
for shelf life simulation to reduce the time and cost needed. In order to perform a
simulation, information about product and package characteristics was obtained. GAB
equation was employed to describe the moisture sorption isotherm of the product. The
barrier properties of packages were described by the permeance of the material since the
package is leak proof.

Shelf life simulation consisted of shelf life modeling, model analysis, and model
validation. Shelf life modeling was based on mass transfer through the package wall and
the GAB equation. From sensitivity analysis of the model, shelf life is most sensitive to
the ratio of area to product dry weight and least sensitive to parameter C of GAB
equation. Model validation is necessary to ensure the applicability of the model. The
shelf life model based on GAB equation is applicable to predict the shelf life of non-fat
dry milk packaged in F -ﬂute corrugated board/plastic boxes except for non-fat dry milk

package in the poorest barrier package.

Dedicated to my beloved Grandfather,
Mr. Preecha Tanprasert.

iii

ACKNOWLEDGEMENTS

I would like to express my great appreciation to my major professor, Dr. Ruben J
Hernandez, for his precious advice, patience, and kindness. I also would like to extend
my appreciation to Dr. Jack R Giacin (School of Packaging) and Dr. Zeynep Ustlmol
(Department of Food Science and Human Nutrition) for their valuable comments.

I would like to express my gratitude to Stone Container Corp. for the ﬁnancial
and material support, Amy Placon (Stone Container Corp.) for her kind help and
cooperation, and Dr. Michael Rich (Composite Center) for his help on using the optical
microsc0pe.

I would like to express my thank to my parents, Dr. Pravit and Dr. Kunnikar
Tanprasert for their great inspiration, love, encouragement, and support. Without both of
you, I would not be able to accomplish this.

I would like to thank Manoch Srinangyam, Jaemin Choi, Cengiz Caner, and
Robert Hurwitz for their tremendous help on my thesis. Lastly, thanks to my old good
friend, Pinrat Charubhumi, and other friends at School of Packaging who always be of

great help whenever I ask for.

iv

TABLE OF CONTENTS

List of Tables ........................................................................................ vii
List of Figures ......................................................................................... x
Nomenclatures ..................................................................................... xiii
Introduction .......................................................................................................................... 1

Chapter 1 Literature Review

Non-fat Dry Milk ..................................................................................................... 3
Changes in Non-Fat Dry Milk During Storage ........................................................ 6
Packaging for Non-F at Dry Milk ............................................................................. 8
Factor Affecting Shelf Life Moisture Sensitive Product ......................................... 9
Water Vapor Transmission Model ......................................................................... 11
Equations for Sorption Isotherm ............................................................................ 13
Shelf Life Modeling of Moisture Sensitive Product .............................................. 22
Model Analysis ...................................................................................................... 28
Chapter 2 Material and Methods ........................................................................................ 30

Chapter 3 Product and Package Characteristics

Initial Moisture Content ......................................................................................... 42
Moisture Sorption Isotherm ................................................................................... 43
GAB Equation Analysis ......................................................................................... 52
Water Vapor Transmission of Packaging Material ................................................ 57
Package Integrity ................................................................................................... 64

Chapter 4 Shelf Life Simulation

Shelf life modeling ................................................................................................. 69

Shelf life model analysis ........................................................................................ 75

Model validation .................................................................................................... 81
Conclusion ......................................................................................................................... 93
Appendices

Appendix A Parameter Estimation of GAB Equation ........................................... 95

Appendix B Experimental and Calculated Moisture Sorption Isotherms .............. 99

Appendix C Potential Error and Proposed Correction of Parameter

C in GAB Model .............................................................................. 102

Appendix D Water Vapor Transmission and Permeance of Packaging
Materials ......................................................................................... 107
Appendix E Sensitivity Analysis of Shelf Life Model ....................................... 109
Appendix F Error Analysis of Shelf Life Model ................................................ 124
Bibliography .................................................................................................................... 130

vi

LIST OF TABLES

Table 1. Typical compositional range of non-fat dry milk (NFDM) , instant

non-fat dry milk (INFDM), and dry whole milk (DWM) ...................................... 3
Table 2. Food product application and function of non-fat dry milk ................................... 5
Table 3. Description of packaging material ....................................................................... 33
Table 4. Package description ............................................................................................. 33
Table 5. Salt solutions and their corresponding relative humidities at 20, 30
and 40°C .............................................................................................................. 35
Table 6. Experimental design for model validation experiment ........................................ 41
Table 7. Initial moisture content of non-fat dry milk ......................................................... 42
Table 8. Experimental moisture sorption isotherm for non-fat dry
milk at 20, 30 and 40°C ....................................................................................... 43
Table 9. GAB equation describing moisture sorption isotherm of
non-fat dry milk .................................................................................................. 44
Table 10. Oswin equation describing moistm'e sorption isotherm of
non-fat dry milk ................................................................................................. 44
Table 11. Henderson equation describing moisture sorption isotherm of
non—fat dry milk ................................................................................................. 44
Table 12. Cubic polynomial model describing moisture sorption isotherm
of non-fat dry milk ............................................................................................ 45
Table 13. GAB constants and their standard deviation at 20, 30, and 40°C ...................... 45
Table 14. Permeances of packaging materials at 25, 30, and 40°C ................................... 57

Table 15. Experimental and calculated weight gain of Bellow and FOL style
boxes made with material E and ﬁlled with desiccant at 40°C,
74i1 %RH ......................................................................................................... 65

Table 16. Experimental and calculated weight gain of material of Bellow style boxes
made with material A and ﬁlled with desiccant at 40°C, 73:1:2% RH ............... 66

vii

Table 17. Parameters used for shelf life modeling of packaged non-fat

dry milk at two storage conditions ..................................................................... 74
Table 18. Conditions used in sensitivity coeﬁicicnt calculation. ...................................... 77

Table 19. Sensitivity coefﬁcients of parameters in shelf life model .................................. 79

Table 20. Experimental and predicted moisture content as a function of
time of non-fat dry milk in three diﬁ‘erent packages stored at
40°C, 83% RH ................................................................................................... 83

Table 21. Experimental and predicted moisture content as a function of
time of non-fat dry milk in three different packages stored at
25°C, 67% RH ................................................................................................... 84

Table 22. Percent error employed in construction of x-axis error bar ............................... 85

Table 23. Calculated shelf life of non-fat dry milk by the shelf life model
based on GAB equation .................................................................................... 86

Table 24. Experimental and calculated (with GAB equation) moisture
sorption data for isotherm non-fat dry milk at 20°C ........................................ 100

Table 25. Experimental and calculated (with GAB equation) moisture
sorption isotherm data for non-fat dry milk at 30°C ........................................ 100

Table 26. Experimental and calculated (with GAB equation) moisture
sorption isotherm data for non-fat dry milk at 40°C ........................................ 101

Table 27. Predicted moisture content at water activity of 0.01 using GAB
equation in Table 9 of Chapter 3 ...................................................................... 104

Table 28. GAB parameter at 20°C derived from the experimental data and
the additional data point in low water activity region ..................................... 104

Table 29. GAB parameter at 30°C derived from the experimental data and
the additional data point in low water activity region ...................................... 104

Table 30. GAB parameter at 40°C derived from the experimental data and
the additional data point in low water activity region ...................................... 105

Table 31. The comparison of moisture content calculated from original and
corrected C value at 20 ,30, and 40°C. ............................................................ 105

Table 32. Combined water vapor transmission rate, WVTR, and permeance, R,
of packaging materials at 40°C, ...................................................................... 108

viii

Table 33. Combined water vapor transmission rate, WVTR, and permeance, R,

of packaging material A and E at 30°C, 75:1:1 %RH ....................................... 108
Table 34. Combined water vapor transmission rate, WVTR, and permeance, R,

of packaging material A and E at 25°C, 67% RH ........................................... 108
Table 35. Sensitivity coefﬁcient of C at different factor combinations ........................... 114
Table 36. Sensitivity coefﬁcient of k at different factor combinations ........................... 115
Table 37. Sensitivity coemcient of Wm at different factor combinations ....................... 116
Table 38. Sensitivity coefﬁcient of Mi at diﬂ'erent factor combinations ......................... 117
Table 39. Sensitivity coefﬁcient of Wu at diﬁ‘erent factor combinations ........................ 118
Table 40. Sensitivity coefﬁcient of A at different factor combinations .......................... 119
Table 41. Sensitivity coefﬁcient of A/Wd at diﬂ‘erent factor combinations .................... 120
Table 42. Sensitivity coefﬁcient of R at different factor combinations ........................... 121
Table 43. Sensitivity coefﬁcient of p, at different factor combinations .......................... 122
Table 44. Sensitivity coefﬁcient of RH at diﬁ‘erent factor combinations ........................ 123

Table 45. Percent error of parameters in shelf life model for packages at
40°C, 83% RH and 25°C, 67% RH. ................................................................ 125

Table 46. Percent error of shelf life for non-fat dry milk in three diﬂ‘erent
packages stored at 40°C, 83% RH. .................................................................. 126

Table 47. Percent error of shelf life for non-fat dry milk in three different
packages stored at 25°C, 67% RH. .................................................................. 126

ix

LIST OF FIGURES

Figure l. The steady state mass transfer in product—package—

environment system ............................................................................................ 23
Figure 2. Bellow style box ................................................................................................. 31
Figure 3. FOL style box ..................................................................................................... 32
Figure 4. Experimental and calculated (with GAB) moisture sorption

isotherm of non-fat dry milk at 20°C .................................................................. 46
Figure 5. Experimental and calculated (with GAB equation) moisture

sorption isotherm of non-fat dry milk at 30°C .................................................... 47
Figure 6. Experimental and calculated (with GAB equation) moisture

sorption isotherm of non-fat dry milk at 40°C ................................................... 48
Figure 7. Calculated moisture sorption isotherm of non-fat dry milk at

20, 25, 30, and 40°C ........................................................................................... 51
Figure 8. Plot of sensitivity coemcient of parameter C as a function of

water activity ....................................................................................................... 54
Figure 9. Plot of sensitivity coeﬁicient of parameter k as a function of

water activity ....................................................................................................... 55
Figure 10. Plot of sensitivity coefﬁcient of parameter Wm as a function of

water activity ..................................................................................................... 56

Figure l 1. Permeances of packaging materials at 40°C, 75% RH ..................................... 58
Figure 12. Permeance of material A as a function of temperature .................................... 60
Figure 13. Arrhenius plot of material E ............................................................................. 60
Figure 14. Cross sectional area of material A .................................................................... 62
Figure 15. Cross sectional area of material E .................................................................... 63

Figure 16. Experimental and calculated weight gain of material E boxes
ﬁlled with desiccant at 40°C, 74:1:1 %RH ........................................................ 67

Figure 17. Experimental and calculated weight gain of small Bellow style
boxes (material A) at 40°C, 73:1:2 %RH .......................................................... 68

Figure 18. The plot of xc as a function of C at different factor combinations .................. 78

Figure 19. Predicted and experimental moisture content as a function of time
of non-fat dry milk in small boxes made from material A and
stored at 40°C, 83% RH (HiAS) ....................................................................... 87

Figure 20. Predicted and experimental moisture content as a function of time
of non-fat dry milk in large boxes made from material A and
store at 40°C, 83% RH (HiAL) ......................................................................... 88

Figure 21. Predicted and experimental moisture content as a function of time
of non-fat dry milk in small boxes made ﬁ'om material E and
stored at 40°C, 83% RH (HiES) ....................................................................... 89

Figure 22. Predicted and experimental moisture content as a function of time
of non-fat dry milk in small boxes made from material A and
stored at 25°C, 67% RH (LoAS) ....................................................................... 90

Figure 23. Predicted and experimental moisture content as a ﬁmction of time
of non-fat dry milk in large boxes made from material A and
stored at 25°C, 67% RH (LoAL) ...................................................................... 91

Figure 24. Predicted and experimental moisture content as a function of time
of non-fat dry milk in small boxes made from material E and

stored at 25°C, 67% RH (LoES) ...................................................................... 92
Figure 25. The plot of XC as a function of C at different factor combinations ................ 114
Figure 26. The plot of )0, as a ftmction of k at different factor combinations ................. 115
Figure 27 . The plot of mm as a function of Wm at different factor combinations ......... 116
Figure 28. The plot of m as a fImction of M at diﬁ‘erent factor combinations ............. 117
Figure 29. The plot of xwa as a function of Wd at diﬂ‘erent factor combinations ............ 118
Figure 30. The plot of m as a function of A at different factor combinations ................ 119
Figure 31. The plot of do as a function of A/Wd at diﬁerent factor combinations ..... 120
Figure 32. The plot of n as a function of R at different factor combinations ................ 121

xi

Figure 33. The plot of )0», as a ﬁmction of ps at different factor combinations ............... 122
Figure 34. The plot of m as a function of RH at different factor combinations ............ 123

Figure 35. Contribution of error from each model parameters to total error
in shelf life of non-fat dry milk packaged in small boxes made
from material A and stored at 40°C, 83% RH (HiAS) ................................... 127

Figure 36. Contribution of error ﬁ-om each model parameters to total error
in shelf life of non-fat dry milk packaged in large boxes made
from material A and stored at 40°C, 83% RH (HiAL) .................................. 127

Figure 37. Contribution of error from each model parameters to total error
in shelf life of non-fat dry milk packaged in small boxes made
from material E and stored at 40°C, 83% RH (HiES) ................................... 128

Figure 38. Contribution of error from each model parameters to total error
in shelf life of non-fat dry milk packaged in small boxes made
from material A and stored at 25°C, 67% RH (LoAS) ................................... 128

Figure 39. Contribution of error from each model parameters to total error
in shelf life of non-fat dry milk packaged in large boxes made
from material A and stored at 25°C, 67% RH (LoAL) ................................. 129

Figure 40. Contribution of error from each model parameters to total error
in shelf life of non-fat dry milk packaged in small boxes made

from material E and stored at 25°C, 67% RH (LoES) ..................................... 129

xii

NOMENCLATURES

P/E

"U

Pa

Mi

permeance

permeability coefﬁcient
thickness

gas constant

temperature

relative humidity

activation energy

constants

water vapor transmission rate
water activity

moisture content, dry basis
moisture content, wet basis
product dry weight

surface area

saturated water vapor pressure
time

initial moisture content

ﬁnal moisture content
headspace volume

sensitivity coefﬁcient

xiii

INTRODUCTION

Non-fat dry milk, a product with many applications in the food industry, is very
sensitive to moisture. Its shelf life and physical and chemical properties depend on its
moisture content. Non-fat dry milk requires proper package to maintain its quality and
control the uptake of water from the environment. Diverse packaging materials with a
range in barrier values have been available for non-fat dry milk. F-ﬂute corrugated board
in combination with barrier layer is a new alternative that is of interest due to its strength
and barrier properties.

As new materials become available, there is a need to develop expertise to
evaluate shelf life of packaged food product in general and dry milk in particular. The
experimental testing of shelf life of food products is costly and time consuming
especially for the product with long shelf life such as non-fat dry milk. Therefore, shelf
life modeling is of great importance for package development and optimization. With the
right model and the use of computer, shelf life analysis can be performed rapidly and
conveniently.

The shelf life models of moisture sensitive product are based on the mass transfer
principles applied to the packaging material and moisture sorption isotherm equation. In
order to apply the shelf life model, information about moisture sorption isotherm of a
product and barrier characteristic of a package must be obtained. The accuracy of the
predicted shelf life depends on the accuracy of the model selected, the accuracy of

parameters of the model and the satisfactory of assumptions. Veriﬁcation is necessary to

ensure the validity of the model. Sensitivity analysis of parameters in the model is

essential to determine the impact of each parameter in the model on the shelf life value.

The objectives of this study were:

A) To determine moisture sorption isotherm of non-fat dry milk.

B) To determine the barrier character of a barrier layer/F-ﬂute corrugated boxes

C) To apply a shelf life model for non-fat dry milk based on moisture sorption isotherm

equation that best described its sorption characteristics.

D) To perform sensitivity analysis of the shelf life model.

B) To validate Shelf life model at two isothermal storage condition.

CHAPTER 1

LITERATURE REVIEW

N on-fat Dry Milk

Non-fat dry milk is deﬁned by federal law under standard of identity as: the
product resulting ﬁ'om the removal of fat and water from milk, and containing the lactose,
milk protein, and milk mineral in the same relative portions as the fresh milk from which

made. It contains not over 5 percent of moisture by weight. The fat content shall not

exceed 1‘/2 percent by weight (USDA, 1999).

Instant non-fat dry milk is non-fat dry milk that has been produced in a manner to
substantially improves its dispersing and reconstitution characteristics. Table 1 shows the

composition in non-fat dry milk, instant non-fat dry milk, and dry whole milk.

Table 1. Typical compositional range of non-fat dry milk (NFDM) , instant non-fat dry
milk (INFDM), and dry whole milk (DWM)

 

 

 

 

 

 

 

 

 

 

Composition Percentage

NFDM INFDM DWM
Protein 34.0 — 37.0 34.0 — 37.0 24.5 — 27.0
Lactose 49.5 — 52.0 49.5 —- 54 36.0 —38.5
Fat 0.6 — 1.25 0.6 — 1.25 26.0 — 28.5
Ash 8.2 — 8.6 8.2 — 8.6 5.5 - 6.5
Moisture 3.0 - 4.0 3.5 —- 4.5 2.0 — 4.5

 

 

 

Source: American Dairy Product Institute (1998a)
Non-fat dry milk provides advantages of distribution, application and shelf life
over ﬂuid milk. Removal of water minimizes the weight and volume resulting in

reduction of shipping and storage cost. Furthermore, non-fat dry milk can be stored at

ambient temperature but lower than 27°C. It also makes the addition to dry concentrated
mixes possible. Due to its minimal fat content, chemical deterioration mechanisms are
reduced, thus increasing shelf life (Ogden, 1993).

Since non-fat dry milk is a product of uniform composition with excellent
qualities, low handling cost, and consistent supply, it has various applications in the food
industry. Table 2 shows application and flmction of non-fat dry milk. In addition, non-
fat dry milk is also found as an ingredient in dairy beverages, custard, gravies, sauces,
frozen foods, packaged dry mixes, soups, infant formulas, snack foods, cosmetics
(American Dairy Product Inst., 1998a), margarine, salad dressing, and healthy foods
(Duxbury, 1992).

Non-fat dry milk is classiﬁed as moisture sensitive product because its shelf life
depends on its moisture content. The moisture uptake in non-fat dry milk leads to series
of physical and chemical changes resulting in reduced quality and consumer acceptance.
Since non-fat dry milk must be maintained at very low moisture content there is a large
water vapor gradient across the package which requires high moisture barrier package.
The low fat content of non-fat dry milk make it more prone to absorb water making it
more sensitive to moisture while less sensitive to oxygen. This makes non-fat dry milk
highly prone to moisture while not sensitive to oxygen. The critical moisture content of
non-fat dry milk is about 10% at room temperature (Thompson, 1997). It has a shelf life
of about 1/2 to IV: years when kept at the condition below 80°F and 65% RH (American

Dairy Products Inst., 1998a).

Table 2. Food product application and function of non-fat dry milk

 

Application

Function

 

Reconstituted ﬂuid milk

- Fortiﬁes and standardizes
milk composition

 

Frozen dessert (ice cream, custard,
ice milk, sherbet and frozen yogurt

- Increase palatability

- Increase food value

- Provide economical
source of serum solid

 

Cottage cheese

- Increase yields up to 5%

 

Yogurt

- Improves texture
- Irnparts smooth

appearance

 

Bread

- Enlarges loaf volume

- Irnparts soft and tender
texture

- Improves crust and crumb
color

 

Bakery products

- Improves water adsorption

- Irnparts smooth texture

- Improves product
uniformity

 

Meat products

 

 

- Enhances nutrition

- Enhances slicing quality

- Minimizes crumbling and
shrinkage

- Provides excellent binding
properties

- Provides desirable ﬂavors

 

Adapted from American Dairy Product Inst., 1998b,c and Duxbury, 1992.

 

Changes in N on-fat Dry Milk During Storage

During storage, the quality of non-fat dry milk decreases due to series of physical,
chemical, and biological changes trigged by an increase of moisture content, for example,
lumping or caking, collapsing, Maillard reaction, loss of nutrient and protein
solubilization and microbiological deterioration. Browning, oxidation and loss of
nutrient occur over wide range of water activity. The gain of moisture above the BET
monolayer, which is a water activity corresponding to sattuation of all primary sorption
sites by one water molecule, increases the rate of these mechanisms (Labuza, 1982).
Microbial spoilage and loss of ﬂow properties require minimum water activity. Below
certain water activity, the reaction will not take place (Mannheim et al., 1994). Potential
changes as a ftmction of moisture content in non-fat dry milk are discussed.

a) Physical changes

The moisture adsorption of non-fat dry milk promotes the formation of
interparticle liquid bridges. Further moisture uptake will cause caking, collapse and
crystallization of lactose, an amorphous carbohydrate of non-fat dry milk. Caking or
clumping reduces ﬂow and leads to poor rehydration and dispersibility. Collapse occurs
when a matrix cannot support its own weight resulting in volume reduction and sticking.
Lactose crystallization is observed as loss of sorbed water or discontinuity of adsorption
isotherm.

Chuy and Labuza (1994) studied the eﬂ'ect of storage condition on physical
changes of dairy-based food powders by using surface caking temperature, T5,, and
advanced caking temperature, T”. Tsc is a measurement of initial clumping while Tao is a

‘ measurement of an advanced stage of collapse. They found that T“ and Tac decreased

with increasing moisture content due to the plasticizing effect of water. The effect of
storage temperature was random. Lai and Schmidt (1990) studied the lactose
crystallization in skim milk powder during two week storage at 20°C. At water activity
0.43 or below, no crystallization was observed. Lactose crystallization occurred after 2.2
days at water activity 0.54. At water activity above 0.54, the crystallization process
occurred at the beginning of the sorption process.
b) Nonenzymatic browning

Nonenzymatic browning or Maillard reaction is a predominant deterioration
mechanism in dried products and products containing lactose. It is the reaction of
reducing sugar and amino group. This reaction results in ﬂavor change, discoloration,
nutritive loss, reduced rehydration, and the formation of mutagenic compounds. In non-
fat dry milk, Maillard reaction causes the yellowing. Lueng (1987) revealed that at 20°C,
the rate of Maillard reaction in skim milk powder increased with increasing water activity
in the range of water activity 0.23 to 0.8. At water activity below 0.23, the reaction was
detected after a month storage. Tsai et al. (1991) studied the effect of water activity and
temperature on nonenzymatic browning in simulated model system of amino acid,
glucose, celite and phosphate buffer. The reaction of amino acids can be classiﬁed into
three types. Type 1 (lysine, histidine, and glycine) showed high rate over broad range of
water activity. Type 2 (cysteine and methionine) showed intermediate browning rates
and multiple maximum rates. Type 3 (glutaminc, valine, and tryptophan) had low rate
with single maxima in the water activity range of 0.75 to 0.9. The reaction rate increased
as temperature increased from 20 to 33°C, below that, the reaction was retarded. The

activation energy was 10 to 35 kcal/mole and Q10 was in the range of 1 to 8.

c) Loss of vitamin A

In addition to loss of nutrient due to nonenzymatic browning, loss of vitamin A
due to oxidation is important for nutritionally purpose especially in vitamin A fortiﬁed
food. Arya and Thakur (1990) reported that vitamin A degradation in wheat ﬂour as a
function of water activity depended on types of wheat ﬂour. In regular wheat, the rate of
degradation was lowest at water activity 0.0 and increased with an increasing water
activity while in gluten, starch and enzyme inactivated wheat ﬂour, the rate was highest
at water activity of 0.0 and decreased as water activity increases. In microcrystalline
cellulose system, the minimum rate was at aw 0.33 and increased both below and above
this water activity.
d) Microbiological spoilage

Microbial deterioration takes place when non-fat dry milk is stored in equilibrium
with high relative humidity. The growth of microﬁmgi causes loss of dry matter.
Microbial deterioration takes place at water activity over 0.6. Pisecky (1992) stated that,
in general, minimum water activity for bacterial and fungi were 0.9 and 0.88-0.80,
respectively. The reduction of water activity can inhibit microbial growth by increasing
lag phase and decreasing growth rate and maximum level of development of

microorganism (Molard et al., 1993).

Packaging for N on-fat Dry Milk

Non-fat dry milk is normally packaged either in bulk container (25 kg) or in
package for retail sales. The common package for bulk containers are multiwall Kraft

bag with a polyethylene liner, tote bin, and big bag or bag in box (Tokley and Gronborg,

1995). The retail packages that are currently in used are multi-quart carton with
aluminum foil wrap or individual pre-measured envelopes in a cardboard box. Most
retail packs contain instant non-fat dry milk (Carpentier and Clark, 1998).

There is an interest in the development of sift-proof retail package using plastic
coated or laminated on F-ﬂute corrugated board. The good glue pattern along with the
development in glue application system can solve sifting problem. A plastic coating or
lamination improves barrier properties, thus, eliminating the need of liner and wrapper

but yet retain the desired shelf life.

Factor Affecting Shelf Life of Moisture Sensitive Product

Shelf life can be taken as the length of time that a packaged product will remain in
acceptable condition under the speciﬁed storage condition (Sacharow, 1986). The
acceptance judgement of food product is subjective based on visual inspection followed
by organoleptic evaluation after purchase. Factors affecting shelf life of the moisture
sensitive product can be classiﬁed in two groups: a) factors affecting the product
characteristics and b) factors affecting barrier characteristics of a package.

a) Processing and some product compositions inﬂuenced the sorption capacity of
the moisture sensitive product. For example, drying temperature considerably affected
the sorption isotherm of dried beef. The beef dried at higher temperature had lower
sorption capacity (Iglesias et al., 1977). The particle size distribution of product had no
effect on the isotherm (Iglesias and Chirife, 1982). Fat, which acts as water repellant,
reduced sorption capacity. The whole egg powder absorbed less water than albumen

(Passy and Mannheim, 1982) and ground sunﬂower nut meat absorbs less water than its

defatted products. Non-protein of defatted sunﬂower had higher sorption capacity than
protein material at high water activity (Mok and Hettiarachchy, 1990). Desugarization
did not have any effect on sorption capacity (Passy and Mannheim, 1982).

b) Package integrity had inﬂuence on the barrier characteristics of a package.
Cardoso and Labuza (1983) reported a large diﬂ‘erence between the experimental water
vapor permeability of the complete paperboard package with creases and end openings
and the calculated result based on permeability of the material measured from the dish
method. Labuza (1982) presented the table of water vapor leak through pinholes of
packages that pass the leak test at different level. Pires et al. (1988) reported substantial
difference in shelf life of oral solid drugs in blister packages and in multiple unit
containers. The difference was mame due to water vapors entering the multiple unit
container when the container was open to obtain the dose.

c) Storage condition described by temperature and relative humidity had effect on
both product and a barrier property of a package. Wang and Brennan (1991) reported
that, at constant temperature, equilibrium moisture content increased with increasing of
relative humidity. At speciﬁc relative humidity, equilibrium moisture content increased
with decreasing of temperature. Sopade and Ajisegiri (1994) reported that the effect of
temperature on moisture sorption isotherm was generally more pronounced at lower
water activity. Pisecky (1992) stated that temperature had little inﬂuence on sorption

isotherm of non-fat dry milk above moisture content of 18%.

10

Water Vapor Transmission Model

Determination of the packaging material permeability coefﬁcient value for every
single storage condition is costly, time consuming, and not practical. Therefore, the
mathematical model describing the relationship between moisture permeability value and
environmental factors becomes useful. It allows the calculation of the permeability value
within the range of condition that the model was based on. Four water vapor
transmission models for commonly used packaging materials are discussed.

Cardoso and Labuza (1983) described models to predict the permeance of
polyethylene and polypropylene as a function of temperature and relative humidity. The
models were based on Arrhenius relationship, eqn. 1, with linear relationship of

activation energy and relative humidity, eqn. 2, and the permeance pre exponential

constant, (P/Z)o as a function of relative humidity as eqn 3.

new]

(3}) = k.+k.(RH) (2)
RH
Ep .. Eo+[k3xl—03) (3)

where (P/Z)o = permeance pre exponential constant, T = absolute temperature, K, and E0
= activation energy at 0% RH.

Samaniego-Esguera and Robertson (1991) developed a similar model for
permeance of LDPE, PET and a laminate at 20-40°C, 55-90 %RH. The model was also
based on Arrhenius equation but had different approach to the relationship of permeance
pre-exponential constant and relative humidity, eqn. 4, and the relationship of activation

11

energy and relative humidity, eqn. 5. The model is presented in eqn. 6. A good

agreement between experimental data and predicted value were obtained.

P g It;
(21, - *1 elm] <1
k.
Ep — k3+RH (5)
P _ E. _ .131. _1__
(F) —- [heprHJexp (k3+RHxRT] (6)

where PM = permeance, k1, k2, kg, and k4 = constants, RH = relative humidity, R = gas
constant, and T = absolute temperature, K.

Piergiovanni et a1 (1995) presented the mathematical model for the estimation of
water vapor transmission rate of PET, PVC, EVA and LDPE in the range of 1 to 45°C
and 11 to 100% RH. They claimed that the effect of temperature on driving force of
water vapor was more important than the effect of temperature on the diﬂ‘usion of water
vapor through the material. The author describes the effect of absolute temperature, T,

on water vapor pressure by a Clausius Clapeyron’s relationship:

. p[2&H....,[1 1]]
WVP = WVTR ex T —- (7)

 

T T'
where WVP = water vapor pressure, WVTR' = water vapor transmission rate at
temperature T‘, and AHmp = enthalpy of water evaporation (45.05 KJ/mol at 273 K), and

T = absolute temperature

The model based on Clausius Clapeyron’s relationship is presented as

12

 

WVTR,“ = WVTRouexp[-54l8.6[ 1 — l ]][ARHMW] (3)
T Told Al{Hold '

where subscript “old” refers to the condition of known WVTR and script “new” refers to
the condition that the WVTR will be predicted.

The authors reported a good agreement between observed and predicted water
vapor transmission rate of PET and LDPE ﬁlm. The model overestimated water vapor
transmission rate of PVC and EVA at low temperature.

Cardoso and Labuza (1983) constructed a model to predict permeability of
paperboard box under temperature of 30-45°C. Since its permeability did not follow

Arrhenius behavior, they ﬁtted the experimental data by a second-degree polynomial.

P = k1+k2T+k3RH+k4T2 +k5RH2 +k6[$j+k7RHT (9)

Equations for Sorption Isotherm

Moisture sorption isotherm represents the equilibrium relationship between the
moisture content and the water activity at constant pressure and temperature. It is a
fundamental characteristic necessary for food processing, such as drying and packaging.
It provides an easy way to evaluate physical, chemical, and microbiological parameters
necessary for processing condition, package design, and shelf life determination (Leiras
and Iglesias, 1991). Fitting sorption isotherm data to suitable equation is vital step in

order to use sorption isotherm. The major considerations in equation selection are type of

13

food, ease of evaluation, simplicity, and suitability for application over the range of water

activity of interest (Sopade and Ajisegiri, 1994).

Kuhn equation (Boquet et al., 1978)

 

 

k
M = ‘ +k 10
haw 2 ( )
Mizrahi equation (Boquet et al., 1978)
a... = k,+M (11)
k2+M

Young and Nelson equations (Boquet et al., 1979)

 

 

 

M, = k,(6+a)+k2(p (12)
Md = A(6+a)+ﬂ6awu (13)
where 9 = a“
aw+(l-aw)k3
a) = aw6
2
- 1‘3““ + 1‘3 1n[k3 “‘3 1)aw:|—(k3+1)ln(l-aw)
k3-(k3-1)a.. (k3—1) k.

5 refers to adsorption, (1 refers to desorption, and A, [3 and E = constants

Chen equation (Boquet et al., 1979)

aW = exp(k3+k,e"’M) (14)
This equation can be rewritten as
1 1
M = ——ln — Ina —k 15

14

Hailwood and Horrobin equation (Boquet et al., 1979)

W

aw
M = k,+k2aw—k3a 2 (16)

Oswin equation (Mok and Hettiarachchy, 1990)

M = k[ 3“] (17)

l-aw

 

Henderson equation (Mok and Hettiarachchy, 1990; Sopade and Ajisegiri, 1994)

l—aw = exp(k,TM“=) (18)
This equation can be rearranged as
_1 %2
M = —ln 1- 19
[le ( a.)] ( )

Chung and Pfost equation (Mok and Hettiarachchy, 1990; Sopade and Ajisegiri, 1994)
k
In a = - —' -k M 20
w RT exp( 2 ) ( )

This equation can be rewritten as

M = 1n(§k!T—)-ln(-lnaw)

 

 

21
k, ( )
Chen and Clayton equation (Mok and Hettiarachchy, 1990)
aw = exp(— k1T"’ exp(— k,T"‘M)) (22)
This equation can be rewritten as
M ___ lnkl+k21nT-1n(-lnaw) (23)

k3T"‘

15

Hasley equation (Sopade and Ajisegiri, 1994)

K.
M = [ k1 ] (24)

 

Iglesias and Chirife equation (Mok and Hettiarachchy, 1990)
This model is a modiﬁcation of Hasley multilayer adsorption equation

 

aw = exp(-exp((k,T+k,)M"‘=) (25)
It can be rewritten as
%c.
M = [exp(k,T+k2)] (26)
—lnaw

Bradley equation (Sopade and Ajisegiri, 1994)
In aw = klk;M (27)
Caurie equation (Sopade and Ajisegiri, 1994)

100

111M = ki-E—
2 w

(28)

Brunauer Emmer and Teller (BE 7) equation

It was developed for localized physical adsorption of inert gases on solid surfaces
based on monolayer concept. The ﬁrst molecule, or monolayer, on site has stronger
interaction with the sorbent than following molecules, or multilayer, on the site (Van den
Berg, 1983).

_M_ : Cbkaw (29)
W (l-anI—aw+Cbaw)

 

16

where Wm = monolayer moisture content which is the water content corresponding to the
saturation of all primary adsorption site, and Cb = B.E.T constants of adsorption.

BET equation was applicable only up to water activity of 0.5 (Schuchmann et al., 1990).
More parameter(s) were added to the equation to extend range of applicable water
activity. Van den Berg (1983) had summarized three- and four-parameter equations that
were derived from BET equation and evaluated their applicable water activity range.
Three-parameter equations substantially extended the applicable water activity range

compare to the original two-parameter BET equation.

Guggenheim, Anderson and de Boer (GAB) equation
GAB equation is one of the three-parameter equation derived ﬁom BET equation
that was found to be as good as or better than most sorption isotherm equations:

M = Ckaw (30)
w (l-kanI—kaw +Ckaw)

 

 

where Wm = water content corresponding to saturation of all primary adsorption or
monolayer in BET model, C = the Guggenheim constant which can be expressed as a
function of temperature as c'exp [(IL-Hm)/RT], k = a factor correcting properties of the
multilayer water molecules with respect to bulk liquid and can be expressed as a ﬁmction
of temperature as k'exp[(Hl-Hq)/RT], H. = heat of condensation of pure water vapor, Hq
= total heat of sorption of the multilayer water molecule, and Hm = total heat of sorption
of the ﬁrst layer on primary sites.

The parameter k describes the energetic state of the average multilayer molecules.

Parameter C determines the sigmoid shape and describes the temperature effect on food

17

isotherms over a range of at least 40 degree Celsius (Van den Berg, 1983). The GAB

equation can be rewritten in quadratic form as (Schar and Ruegg 1985):

 

 

(in?) = mi+ﬂaw+r (31)
where a = —k——[—l—-1] (32)
W, C
l 2
’3 " w. [“6] ‘33)
- 1 (34)
7 ' w Ck

Bizot (1991) claimed that GAB was the best model for moisture sorption isotherm
of food and had been adopted by European cooperation in the ﬁeld of scientiﬁc and
technical research subgroup on water activity.

Schuchmann et al. (1990) wrote GAB equation in general form as:

Claw

= (35)
(1+ Czawxcs " aw)

 

where C1 = Cka, C2 = C-k, and C3 = l/k. The attempt to extend the prediction
capability up to the water activity of 0.98-0.99 was made by replacing aw in eqn. 35 with
either ln[1/(l-aw)] or a“/(l-aw). The log transformation yielded better ﬁt. The constant
C1 described the sorptive capacity at low water activity, the constant C2 was a rough
measurement of shoulder, and the constant C3 was a measurement of steepness at high
water activity region but none of them had kinetic signiﬁcant.

Boquet et al. (1978, 1979) evaluated the capability of two- and three- parameter

sorption isotherm equations in describing water sorption isotherm of various food

18

products. They concluded that the use of third parameter did not necessarily improve the
goodness of ﬁt over the two-parameter equation. For milk products, the most suitable
equation was Hailwood and Horrobin equation, which is mathematically equivalent to
GAB equation. They suggested that this equation was versatile and easy to handle
mathematically.

Larnaruo et al. (1985a, 1985b) compared the capability of Hasley, Oswin, Iglesias
and Chirife, and GAB equation for products including fruits, vegetables, meat products,
milk, coffee, tea, nuts, oilseeds, spices and starchy foods. GAB equation gave a better ﬁt
over wide range of water activity for most sigmoid shape isotherm.

Mok and Hettiarachchy (1990) compared percent root mean square of four
sorption isotherm equations containing a parameter accounting for temperature eﬂ‘ect.
Chen-Clayton equation gave the best ﬁt for sorption isotherm of ground sunﬂower
nutmeat and its product.

Wang and Brennan (1991) reported that GAB and Oswin equation gave the best
ﬁt to the moisture sorption isotherm data of potatoes in water activity range of 0 up to
0.88 while BET equation gave the best ﬁt only for water activity range of 0.05 to 0.40.

Sopade and Ajisegiri (1994) studied the sorption isotherm for maize and sorghum.
Henderson model gave the best ﬁt either when temperature effect was excluded or
included.

Several researchers used general mathematical model to describe sorption
isotherm. Lazarides (1990) reported that the sorption data of Basturma, an intermediate
moisture meat product, can be well-described with eqn. 36 for both desorption and

adsorption isotherm.

19

M = k,e"’" (36)

Mok and Hettiarachchy (1990) used cubic polynomial equation to describe
sigmoid isotherm at the speciﬁc temperature.
M = k. + kzaw + 113a,.2 + km? (37)
They reported that percent root mean square of cubic polynomial equation was
comparable to GAB equation but signiﬁcantly less than Oswin equation.
Sharma and Nath (1991) studied the sorption isotherm of dehydrated rings of

onion which was J-type and found that eqn. 38 gave 3 highest correlation coefficient

M = 19a: (38)

Peleg (1993) used semi-empirical double power law four-parameter to ﬁt sigmoid

moisture sorption isotherm:

M = klafj+k3atf (39)

This equation had the same or better ﬁt than GAB equation for agar-agar, carageenan,
gelatin, pectin, wheat bran, raisins, casein, potato starch, dextrin, and coffee.
Furthermore, this equation is not based on the assumption of the existence of well-
deﬁned monolayer.
Evaluation the goodness of ﬁt

The goodness of ﬁt is an important tool to evaluate the capacity of sorption
isotherm equation to describe the experimental data. It measures the deviation of the
calculated isotherm from the experimental data. Several methods to evaluate the

goodness of ﬁt have been proposed.

20

a) Sum of squares of the residuals, S (Schar and Ruegg, 1985)

s = i<M.,—M....>’ (40)

b) Variance, V (Pisecky, 1992)

 

n M —M
V = §( wen-10¢): (41)

c) Mean relative percentage deviation in modulus, PD (Boquet et al., 1979; Wang and

Brennan, 1 991 )

 

 

n In -hd
PD = BEEP WM “1“] (42)

n 1:] exp

d) Percent root mean square, PRMS (Schar and Ruegg, 1985)

 

2
:[MW — Mm]
s: M
‘ ""’ x100 (43)
II

where Mm, is experimental moisture content, Mme is calculated moisture content, and n

PRMS:

 

is number of observations.

PRMS and PD take into account the number of observations and they yield the
relative deviation to the experimental data. S and V yield the absolute deviation from
experimental data and only V takes into account the number of observations. All of these
methods can be employed to compare the goodness of ﬁt for the same experimental data.
Only PRMS and PD can be used to compare the goodness of ﬁt from the different

experimental data.

21

The acceptable limit of deviation ﬁom experimental data can be established based
on the experimental error. Lerias and Iglesias (1991) presented errors of 9% at low water
activity and 2-3 % at high water activity. Since the reproducibility of sorption isotherm
was slightly dependent on the type of product, they set the acceptable limit at 9%

deviation.

Shelf Life Modeling of Moisture Sensitive Product

The complete experimental testing of shelf life is costly and time consuming.

It might not be even possible in some situations. Therefore, shelf life simulation becomes
necessary for package design and optimization. Gyeszly (1980) stated several other
reasons to use shelf life simulation: the uneconomy of over- or under packaging, the
possibility of establishing shelf life to packaging cost relationship, and the necessity of
short development time due to the competitive nature of the industry. Reliable shelf life
models are of prime important for shelf life simulation.

Many researchers have developed mathematical models and computer programs
to calculate moisture gain, predict shelf life, and optimize the package for moisture
sensitive product. Most models were basically based on the steady state mass transfer as
expressed by the permeability coefﬁcient, P. It is assumed that all the resistance to
moisture transport is in the ﬁlm and so there is no internal resistance. All the moisture
permeated into the package are absorbed by the product immediately. The schematic of

the steady state mass transfer in product-package-environment system is illustrated in
Figure 1.

22

 

PAPs (aw, - aWi kit

 

 

 

 

 

x
A Environmental condition (T, RH)
dW
/ Package
wd°dM Surface area
Pr (1 t
0 uc Barrier
Sorption isotherm characteristics
Initial & Final
moisture content
Product weight

 

 

 

 

 

 

Figure 1. The steady state mass transfer in product-package-environment system

23

The change in the amount of water in a product is equal to the amount of water

permeated into a package in a given time,

 

deM = E’—:‘%(awo—awl.)c1t (44)
t M

[dt = we i dM (45)
0 PAP, Mi (awo —awi)

where dM is a change in the amount of water in the product or the amount of water
permeated into the package in time dt, am is water activity outside the package, and aw, is
water activity inside the package which relates to moisture content of the product by the
sorption isotherm equations. Various isotherm equations can be employed.

Iglesias et al. (1977) studied the moisture gain of dried beef as a ﬁmction of time
based on the integration of eqn. 45 using three diﬂ‘erent moisture sorption isotherm

equations including straight line (eqn. 46), BET equation (eqn. 47), and Hasley equation.

t = [delnMe—Mini . (46)
PAP: Me -Mt

 

where b is slope of straight line portion of sorption isotherm

at--£—1VA[[1 dp ]+[1 (Cb-1)dp H wm (47)

PA 15.-(1-(13/13.»2 P—s(1+(C.,‘-1)(I>/p.))2 Pom-P

where Cb and Wm are constants for BET equations.

 

The model based on straight line gave signiﬁcant error while analysis based on both BET
and Hasley equation gave good agreement with experimental data when narrow moisture
content interval (4-9%) was considered. For wider moisture content range (4-13%),

Hasley equation-based analysis gave a better prediction of moisture change. In 1979,

24

they developed computer program for prediction of moisture transfer in the mixture of
packaged dried food based on the application of BET equation and concept of additivity
of isotherm.

Cliﬂ'ord et al (1977) predicted internal relative humidity of three types of cereal as
a function of time, RHi(t), by using the concept of steady state mass transfer and taking
water vapor in the headspace into account as presented in eqn. 48 and 49. The authors
found that the difference between calculated and experimental data was less than 10%

except one type of cereal in the package with the poorest barrier property.

 

RHi(t) = RH,-RH,(I=0)‘“ (48)
para
a = - “-00 w (49)
(1 —a)+%e‘

where RHi(t) is relative humidity inside the package at time t, RHe is equilibrium relative
humidity which is equal to the storage relative humidity, RHi(t=0) is initial relative
humidity inside the package, V is headspace volume, and b is the slope of straight line
portion of moisture sorption isotherm.

Cardoso and Labuza (1983) used a computer iterative technique based on the
steady state mass transfer and linear isotherm (eqn. 46) to predict moisture gain of
packaged pasta subjected to the sine wave of temperature and relative humidity. Good
predictions were obtained for pasta packaged in laminated polypropylene bags. The
technique failed to give a good prediction for pasta packaged in paperboard boxes
because the assumption of all resistance to moisture transfer is in the package was not
satisﬁed for this type of package.

Chuzel and Zakhia (1991) used eqn. 46 to calculate the potential storage time of

gari Imder three isothermal storage conditions. The equilibrium moisture content, M, of
25

product when exposed to atmosphere outside the package was calculated by iteration
from the GAB equation that used to describe isotherm at the storage temperature.

Mannheim et al. (1994) stated that the eqn. 46 is valid only for hydrophobic
packaging material. For hydrophilic material, the effect of different driving force must
be taken into account. The authors described model based on the assumption of second
order polynomial relationship between the permeability value and the driving force but
no experimental validation was performed.

1w,

t = MEAPps(aw. —aw(NI’T))2 (50)

 

Diosady et a1. (1996) developed shelf life model of canola meal based on the mass
transfer equation and the assumption that moisture sorption isotherm was described by

GAB equation. The model was valid only when C is large so that 4WmC >> 2WmC2.

 

 

 

_ 1 _ 2wmc (ﬂ+C)Mﬁn —2w,,,c
t ' (ﬁ+C)¢[M‘ M‘ + (,6+C)ln[(,6+C)Mim. -2me]] (51)
where C = c'exp[M] (52)
RT
k = k'exp[.ai_-_HJQ] (53)
RT
_ 2 ps A100
¢ "' e 21<(1—C)wd (54)

 

 

 

h
l

_(3).)“.m-)I.)

26

M _ Wkaaw
° (l—kanI—kaw +Ckaw)

 

(56)

The model is adequate to describe moisture gain as a function of time of canola meal
packaged in Propaﬁlm C and Melinex 813 for temperature lower than 40°C.

Alves et al. (1996) applied mathematical model as shown in eqn. 45 to predict
shelf life of cream cracker biscuit using linear, Hasley and GAB equation to describe the
product sorption isotherm. Eqn. 45 gave the good prediction up to the product moisture
content of 11% regardless of the isotherm equation. At higher moisture content only
GAB and Hasley equations yielded the good prediction.

Rudolph (1986) suggested a simple method to determine shelf life and optimize
the packaging of moisture sensitive food by plotting a graph of a = (WV TR)At/W 8
versus M, where Q is package transmission rate and t], is shelf life. Ifthe initial moisture
content is given, the origin of the graph will change from (0,0) to (Mini, QaWd). It was
claimed that this model could also be applied to non- reactive gases and vapors other than
moisture. Dock et al. (1998) validated Rudolph’s method and reported that, for pretzel,
the method was valid only at some temperatures.

Kim et al. (1998) developed a model and a computer program to estimate the
unsteady state moisture transfer through blister package using ﬁnite difference method.
The model is based on diffusion coefﬁcient of water in the product and the packaging
material and the solubility of water and the packaging material. The shelf life value of a
circular plate shaped product obtained from this model agreed well with the value
obtained from the analytical solution for the case that the diﬂirsion coefﬁcient of the

product is signiﬁcantly lower than the diffusion coefﬁcient of the packaging material.

27

Model Analysis

Any mathematical model can be evaluated by using the so-called sensitivity
analysis (Beck and Arnold, 1977). A sensitivity analysis indicates the magnitude of the
change in the dependent variable due to the alteration in the values of the parameters. To
perform a sensitivity analysis, we must calculate the sensitivity coefﬁcient which are the

ﬁrst derivative of model with respect to each parameter in the model.

_ ﬂ
1 — 6,6 (57)

where n = n(x, t, [3), x and t are independent variables, and B is a parameter

A sensitivity coefﬁcient provides information about the linearity of parameters. If
all sensitivity coefﬁcients are not ﬁmctions of the parameters, the model is linear in terms
of its parameters. If the sensitivity coefficients over the range of the observations are not
linearly dependent, parameters of the model can be estimated. For parameter, the linear

dependence occurs when eqn. 58 holds for all i observations and not all Cj values equal to

zero.

C,-a—n“-+C2-a—77i+ ...... 4C9ﬂ = o (58)
613. 6.62 64.

Liu et al (1997) analyzed the sensitivity for hygrostress crack formation in
cylindrical food during drying. They evaluated the trend of time, moisture ratio, average
temperature, and crack formation as a function of initial moisture content, relative
humidity and temperature of dried air, convective surface, mass and heat transfer

coeﬁcients, initial cylindrical diameter, and ratio of assumed and reference moisture

28

diffusivity. For the effects of relative humidity and temperature of dry air on temperature
at crack formation were further examined by constructing three—dimensional plot of Biot

number, food moisture concentration at surface, and surface temperature.

29

CHAPTERZ

MATERIALS AND METHODS

Product-Package System
Non-fat dry milk

Instant non-fat dry milk fortiﬁed with vitamin A and D was obtained from Stone
container Corp (Chicago, IL). Its ingredients were non-fat dry milk, vitamin A palmitate,
and vitamin D. Instant non-fat dry milk composed of carbohydrate (52%), protein (35%),
sodium (0.5%), cholesterol (less than 0.02%), calcium, and vitamin D, A, and C as stated

on the label of the package.

Package

Six types of packaging material were supplied by Stone Container, Corp.
(Chicago, IL). Materials were conditioned at 72°F, 50% RH for 72 hours prior testing.
The description of each material is presented in Table 3. The two most suitable
packaging materials were chosen to construct boxes. Boxes were closed by two different
glue patterns, identiﬁed as Bellow, and full overlap, FOL, as shown in Figure 2 and 3,

respectively. There were two sizes of boxes as described in Table 4.

30

 

Figure 2. Bellow style box

31

 

Figure 3. FOL style box

32

Table 3. Description of packaging material

 

Type

Description

 

AA coated F—ﬂute corrugated board

 

B coated F-ﬂute corrugated board

 

BB coated F-ﬂute corrugated board

 

Unw>

Single-face polyliner F-ﬂute corrugated board

 

Double-face polyliner F-ﬂute corrugated board

 

Control

 

 

Paperboard

 

Table 4. Package description

 

 

 

 

 

 

 

Character Large Small
Dimension 32.5 x 21.5 x 8.7 cm 25 x 17 x 6 cm
Surface area, cm7 2315 1354
Product capacity, g 350 290

 

Initial Moisture Content Determination

The initial moisture content (IMC) of non-fat dry milk was determined by Karl
Fischer Titration (Brinkmann Instruments, Inc., Westbury, NY). Half a gram of the
sample were transferred to a weighing spoon and placed into a titration beaker containing
fresh solution of methanol. The sample was homogenized by a Brinkmann apparatus for
40 seconds. Then, water molecules in the sample were titrated with Hydranal Composite
5 (Aldrich Chemical Company, Inc., Milwaukee, WI) consisting of Diethylene glycol
monoethyl ether, irnidazole, sulfur dioxide, iodine, and hydriodic acid. The moisture

content was reported on percent wet basis, M... The percent moisture content (dry basis),

M, was calculated as

33

 

 

 

M = w (59)

Moisture Sorption Isotherm Determination

Moisture sorption isotherms were determined gravimetrically by exposing a
sample of non-fat dry milk of known weight and initial moisture content to humid air in a
closed container. The equilibrium moisture content of the sample were evaluated in three
series of nine relative humidity values ranging from 8% to 95% RH. Each series were
maintained at a diﬂ‘erent constant temperature of 20, 30 and 40°C. The equilibrium
moisture content, expressed as dry basis, was calculated based on moisture gain of a
sample at equilibrium.

The relative humidity value in the container was obtained by placing in 500 ml
crystal bowls selected saturated salt solution. The containers were a 3.5-gallon plastic
bucket tightly-closed. The saturated salt solutions were prepared from analytical pure
salts and distilled water. Salts employed and their corresponding measured relative
humidity values are listed in Table 5. The relative humidity in each bucket was
monitored by humidity sensor (Hygrodynamics, Inc., Silver Spring, MD) which was
mounted to the lid of each bucket. From relative humidity, water activity can be

calculated by

RH
= — 60
a 100 ( )

Approximately 2 grams of non-fat dry milk were weighed into a petridish with diameter

of 5.5 cm. Three replicates and two controls (empty petridish) were used at each

34

condition of temperature and relative humidity. At predetermined time intervals, lidded

petridishes were weighed on AB 160 Mettler Analytical Balance. The procedure was

repeated until a constant weight was obtained. The equilibrium moisture content (dry

basis), M, was calculated by eqn. 62.

Table 5. Salt solutions and their corresponding relative humidities at 20, 30 and 40°C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Saturated salt Relative humidity at Source of salt
”mm“ 20°C 30°C 40°C
Lithium chloride 8.4 8.9 9.2 Aldrich Chemical Company, Inc., WT
Potassium acetate 23.1 23.2 23.8 J .T. Baker, NJ
Magnesium chloride 34.0 34.2 34.3 J.T. Baker, NJ
Potassium carbonate 47.2 46.6 45.6 Mallinckrodt Baker, Inc., KY
Potassium nitrite 50.4 48.7 47.6 EM Science, NJ
Sodium nitrite 64.9 63.6 62.7 Columbus Chemical Industries, Inc., WI
Sodium chloride 75.6 75.9 79.6 J .T. Baker, NJ
Ammonium sulfate 81.3 81.5 77.5 Columbus Chemical Industries, Inc., WI
Potassium nitrate 94.5 93.5 91.3 EM Science, NJ
M = [W—l]x100 (61)
where M = equilibrium moisture content, % dry basis

M, = initial moisture content, % dry basis

W, = weight at equilibrium, g

W, = initial weight, g

GAB, Oswin, Henderson, and cubic polynomial equations were employed to ﬁt

the experimental moisture sorption isotherm. Parameters in each equation were estimated

as the following:

35

 

GAB equation

M = Ckaw (62)
w (l-kanl—kaw4-Ckaw)

 

 

In order to estimate parameters, the equation was transformed into a quadratic form as

 

 

a 2
where a = iG-I) _ 1 (1—c)
Wm C WmC
1 2 1
= — 1-— = C—2
’6 WEI c) wmc( )
1
7 = —

The quadratic regression was performed by Microsoft Excel 97-SR1. GAB constants C,
Wm, and k were calculated by eqn. 64, 65, and 66, respectively. The detail of obtaining

those equations are presented in Appendix A.

 

 

C = Giulia-46 (64)
£2
h 19 = 4+——
W ere (—a)y
1
w = ——C—2 65
.. ﬁc( ) ( )
k = l (66)

36

Oswin Equation

 

M = k,[ ”W J (67)

Parameter A0 and B0 were estimated by non-linear regression performed by STATPRO, a
statistical software. The ﬁrst derivatives with respect to each parameter were required to

be entered into the program

 

6M aw k’
.11

L“ = I.“ 3“ sz( “w J] (69)
6k2 l—aw l-aw

 

 

Henderson Equation

l—aw = exp(k,M"’) (70)

“ln (l-aw)/T” was considered as dependent variable, y, and “M” was considered as an
independent variable, x. Non-linear regression was performed by STATPRO to estimate

parameters. The ﬁrst derivatives with respect to each parameter needed to be entered are
by = M"= (71)

517.
3y— = kle’ lnM (72)
5k:

37

Cubic Polynomial Equation
M = k13w3 + kzaw2 + ksaw +De (73)
The polynomial regression was performed by Microsoft Excel 97-SR1
The goodness of ﬁt for each isotherm equation was evaluated based on the

minimum value of percent root mean square, PRMS.

i[Mexp —Mu,c]
,. M
‘ °’° x100

where Mexp = experimental moisture content, % dry basis, Marc = calculated moisture

 

      

 

PRMS =

 

(74)

content, % dry basis, and N = number of data point

Standard deviations of parameters in the equation that best-described the
experimental moisture sorption isotherms were determined for further use in the error
analysis of shelf life calculation. For GAB equation, since its parameters were estimated
using transformed equation, the standard deviations were determined using Gauss’ Law
of Error Propagation which stated that an error of ﬁnal results derives from the error in
each number used to get that results. The equation employed for calculation of standard

deviation of GAB constants are presented in Appendix A.

Determination of' Water Vapor Transmission of' Packaging Material

The dish method in ASTM E96-93, Standard Test Methods for Water

Transmission of Materials, was employed to determine water vapor transmission rate,
WVTR, and permeance, P/l, of the packaging material (ﬂat and unscored). Six types of

packaging materials, A, B, C, D, E, and control, were tested in four replicates at 40°C,

38

75% RH. Each aluminum dish was cleaned thoroughly and approximately 20 grams of
desiccant (8 mesh anhydrous calcium sulfate) was added to cover the bottom of the dish.
A specimen was cut into circle with diameter of 8.25 cm and sealed to the mouth of the
test dish with the hot parafﬁn wax.

The inner side of actual package was placed toward desiccant in order to simulate
water vapor ﬂow direction. Dish sets were weighed in a AB 160 Mettler Analytical
balance and placed in an environmental chamber at controlled humidity. Then they were
reweighed at predetermined intervals. One extra sample of each type of packaging
material was mounted to an aluminum dish containing no desiccant and tested in the
same manner as other samples. This was done to compensate the error from moisture

sorption of test material. The water vapor transmission rate and permeance were

calculated based on the slope of graph of weight gain as a ﬁmction of time using eqn. 76

 

and 77.
WVTR = §£ (75)
A
P = WVTR (76)
l Ap

where SL = slope of the weight gain versus time graph, g/day, A = 0.5346 m2, pin =
partial pressure inside the package, mmHg, and pout = partial pressure outside the
package, mmHg

Two packaging materials with the lowest permeance values were selected for the

additional testing at 25°C and 30°C, 75% RH.

39

Evaluation of' Package Integrity

The moisture gain of FOL and Bellow style boxes were experimentally
determined in accordance with the ASTM D895-79, Test Method for Water Vapor
Permeability of Packages. Packages were ﬁlled with approximately 454 grams of
desiccant (8 mesh anhydrous calcium sulfate) and sealed by using ITW Dynatec
Adhesive Application System (Glenview, IL). At each end and side of package, extra
adhesive was applied to ensure the sift proof. The sealed package was weighed and
placed in the environmental chamber at 40°C, 75% RH. In order to eliminate the eﬂ‘ect
of moisture sorption of packaging material and ﬂuctuation of storage environment, one
package for each glue pattern without desiccant was used as a control. Control packages
were sealed and tested with the same manner as test packages. The control and test
packages were reweighed at predetermined intervals always with the same sequence.
The experimental total moisture gain was determined and compared to the calculated

moisture gain obtained as

Moisture gain = (P/l)-A-Ap-t (77)

The box style with better barrier characteristics were constructed using material
A and were tested at the same manner to ensure the package integrity of package made

ﬁ'om material A.

Model Validation

The two most suitable package based on the previous experiment was chosen for

shelf life validation experiment. Boxes ﬁlled with non-fat dry milk were closed in the

40

same manner as in the experiment to evaluate package integrity. Boxes were weighed
and placed in the environmental chamber and reweighed every day. Two storage
conditions including 40°C, 83% RH and 25°C, 67% RH were employed. Controls
(empty boxes) were tested along with other samples. The experimental design is

described in Table 6.

Table 6. Experimental design for model validation experiment

 

 

 

 

 

 

 

Condition Material Size
40°C, 75% RH A Small
A Large

E Small
25°C, 75% RH A Small
A Large
E Small

 

 

 

 

 

41

CHAPTER3

PRODUCT AND PACKAGE CHARACTERISTICS

Initial Moisture Content

Initial moisttu'e content of non-fat dry milk determined by Karl Fischer was 4.86

:t 0.04 % (dry basis). The experimental data are presented in Table 7.

Table 7. Initial moisture content of non-fat dry milk

 

 

 

 

 

 

 

 

 

 

Replicate Moisture content, % (dry basis)
1 4.86
2 4.79
3 4.84
4 4.92
5 4.86
6 4.85
7 4.91
Average 4.86
Standard deviation 0.04

 

 

 

 

42

Moisture Sorption Isotherm
The equilibrium moisture content on a dry basis was determined at 20, 30 and
40°C, for each value of relative humidity. The experimental values at all temperatures

are presented in Table 8.

Table 8. Experimental moisture sorption isotherm for non-fat dry milk at 20, 30 and 40°C

20°C 30°C 40°C
RH M, % (dry basis) RH M, % (dry basis) RH M, % (dry basis)
0.0 0.00 r 0.00 0.0 0.00 :1: 0.00 0.0 0.00 :1 0.00
8.4 4.23 r 0.01 8.9 3.88 :1: 0.03 9.2 3.67 r 0.02
23.1 4.84 i 0.02 23.2 4.58 r 0.03 23.8 4.44 s: 0.02
34.0 6.04 :1: 0.03 34.2 6.09 1: 0.03 34.3 5.90 r 0.08
47.3 8.97 r 0.07 46.6 6.54 r 0.04 45.6 6.00 r 0.02
50.4 8.03 1; 0.01 48.7 7.13 i 0.02 47.8 6.43 i 0.01
64.9 11.58 i 0.03 63.6 11.59 i 0.02 62.8 9.61 r 0.01
75.6 14.35 :1 0.02 75.9 14.01 :1: 0.03 77.5 13.68 1: 0.01
81.3 16.96 i 0.07 81.5 16.51 :1: 0.01 79.6 16.42 i 0.02
94.5 38.45 :i: 0.51 93.5 33.48 at 0.06 91.3 27.3 :1: 0.04

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As indicated in Chapter 1, several equations were available to describe the
experimental data of moisture sorption isotherm. In this work, GAB, Oswin, Henderson,
and cubic polynomial equations were employed to ﬁt the experimental moisture sorption
data of non-fat dry milk. Equations and their percent root mean square, PRMS, are
presented in Table 9, 10, 11, and 12, respectively.

As seen from Table 9 to 12, the GAB equation showed the best ﬁt since it
produced the lowest percent root mean square for the experimental sorption isotherm of
non-fat dry milk at all temperatures. The result agreed with the one reported by Boquet

et al. (1979). The authors claimed that Hailwood and Horrobin equation, or transformed
43

Table 9. GAB equation describing moisture sorption isotherm of non-fat dry milk

 

Temperature, °C GAB equation PRMS

20 M _ 4.25sw 7.03
" (1 - 0.93sw )(1 - 0.93sw + 96.24aw)

 

 

 

30 M _ 4.67aw 6.10
" (1 — 0.9511w )(1- 0.954w +118.45aw)

 

40 M _ 3.35sw 7.78
- (1 - 0.96aw)(1— 0.96aw + 91.6411“)

 

 

 

 

 

Table 10. Oswin equation describing moisture sorption isotherm of non-fat dry milk

 

 

 

 

 

 

 

Temperature, °C Oswin Equation PRMS
20 a 056 17.44
M = 0.0813[ “’ )
l-aw
30 0-55 15.37
M = 0.0785[ " J
-aw
40 a 056 13.83
M = 00744( w J
l-aw

 

 

 

 

 

Table l 1. Henderson equation describing moisture sorption isotherm of non-fat dry milk

 

 

 

 

 

 

Temperature, °C Henderson equation PRMS
20 M = _ 1'10 _ 3w) 096 39.99
30 0.33T 26.01
40 T is temperature, °C 32°47

 

 

 

 

 

* Since Henderson model includes the effect of temperature, only one equation is
presented for the three temperatures.

44

Table 12. Cubic polynomial model describing moisture sorption isotherm of non-fat dry

 

 

 

 

milk
Temperature, °C Cubic polynomial equation PRMS
20 M=1.64aw3-l.82aw2+0.65aw-0.0045 19.76
30 M=1.40aw3-l.50aw2+0.54aw-0.0019 13.66

40 M=1.1law3-1.15aw2+0.43aw-0.0009 7.23

 

 

 

 

 

GAB equation, gave the best ﬁt for the isotherm of dairy product. The values of

experimental and calculated (with GAB equation) moisture sorption isotherm of non-fat
dry milk at 20, 30, and 40°C are presented in Appendix A and the plots are presented in
Figure 4, 5, and 6, respectively. The GAB constants and their standard deviation for all

temperatures are presented in Table 13.

Table 13. GAB constants and their standard deviation at 20, 30, and 40°C

 

 

 

 

 

 

 

 

 

 

 

Temperature, °C C sc k 8;, Wm, g/ g 3%» g/ g
20 103.55 0.05 0.93 0.1 1 0.0441 0.0057
30 124.63 0.03 0.95 0.10 0.0394 0.0046
40 95.46 0.06 0.96 0.1 1 0.0366 0.0044

 

 

Parameter C changed signiﬁcantly when temperature changes but had no pattern.
The similar result was observed in the adsorption isotherm of potato in the temperature
range of 40 to 70°C (Wang and Brennan, 1991) and the desorption isotherm of maize in
the temperature range of 20 to 40°C (Sopade and Ajisegiri, 1994). This error would only
contribute to the very low region of water activity (0.01 to 0.04) and will not have much
impact on the shelf life prediction. The error of parameter C is discussed and the

correction is proposed in Appendix C.

45

 

45~

 

4O _ 0 Experimental
-— Calculated (GAB) o

 

 

 

N
0|
1

M. %(dTY basis)
N
o

.3
0|

10*

 

 

 

0 20 4O 60 80 100
Relative humidity. %

 

 

 

Figure 4. Experimental and calculated (with GAB) moisture sorption isotherm of non-fat
dry milk at 20°C

46

 

45~

40~

 

0 Experimental
— Calculated (GAB)

 

 

 

354

30*

25~

20‘

M, %(dry basis)

15~

10~

 

O I T r
20 4O 60 80 100

Relative humidity, %

 

 

 

Figure 5. Experimental and calculated (with GAB equation) moisture sorption isotherm
of non-fat dry milk at 30°C

47

 

45-

401

 

0 Experimental
35 4 —Calculated (GAB)

 

 

 

M, %(dry basis)

 

 

T

O 1 1
40 60 80 100

0 20
Relative humidity, %

 

 

 

Figure 6. Experimental and calculated (with GAB equation) moisture sorption isotherm
of non-fat dry milk at 40°C

48

As seen in Table 13, parameter k slightly increased with increasing temperature.
The relationship between parameter k and temperature, T, was described by an Arrhenius

equation (Van den Berg, 1983)

(73)

k = k'exp[ RT
where IL is heat of condensation of pure water, H,I is total heat of sorption of the
multilayer, and k' = pre-exponential k constant.

From this study, (HI-Hg) was — 0.29 kCal/mol (r2 = 0.9697). Since the heat of
condensation of pure water in the range of experimental temperature was approximately
10.4 kCal/mol (Lide and Frederikse, 1996), total heat of sorption of the multilayer was
found to be approximately 10.6 kCal/mol.

Parameter k indicate the property of bulk liquid. If k = l, multilayers will have
the property of bulk liquid (Diosady et al., 1996). The parameter k of non-fat dry milk
indicated that multilayer water in non-fat dry milk had properties between monolayer and
bulk liquid but closer to the bulk liquid and had trend toward behaving more like bulk
liquid as temperature increases.

The monolayer moisture content, Wm, depends on the number of adsorption sites
which in principle are constant and therefore independent of temperature. However, the
monolayer moisture content of non-fat dry milk decreased with increasing temperature
from 0.0441 g/g at 20°C to 0.0366 g/g at 40°C. Similar result was reported for moisture
adsorption and desorption isotherms of maize, sorghum, and millet (Sopade and Ajisegiri,
1994). The result could be explained by the reduction of sorption sites due to

physicochemical changes induced by an increasing of temperature. The temperature

dependency of Wm was expressed by Arrhenius relationship.
49

 

r E.
W... - (W...) exp[- RT) (79)

where Wm' is pre-exponential Wm, and E. is monolayer binding energy

In this study, monolayer binding energy for non-fat dry milk in the temperature
range of 20 to 40°C was —1.70 kCal/mole (r2 = 0.9898).

Since the shelf life model validation was performed at 40°C and 25°C, the
information on the moisture sorption isotherm at 25°C was necessary. Lagrange
interpolation method, as shown in eqn. 80, was employed to obtain those information

based on isotherm at 20, 30, and 40°C.

 

 

_ (TX —T2XTx —T3) (TX—TleX -T3) (Tx—TIXTX_T2)
Mx - (Tl—szTl—TB) 1+('I‘z‘TrX'I‘z-‘I‘J 2+(T3"'I‘IX'I‘3-'I.2)Ms (80)

where Mx = % equilibrium moisture content (dry basis) at Tx = 25°C, M1 = %
equilibrium moisture content (dry basis) at T1 = 20°C, M2 = % equilibrium moisture
content (dry basis) at T2 = 30°C, and M3 = % equilibrium moisture content (dry basis) at
T3 = 40°C

The GAB constants at 25°C were found to be C = 120.16 :l:0.05, Wm = 0.0419 :I:
0.0049 or 4.19 :l: 0.49 %, and k = 0.94 i 0.11. The moisture sorption isotherm of 20, 25,
30, and 40°C are presented in Figure 7.

Equilibrium moisture content increased with increasing temperature at constant
relative humidity (Figure 7). The eﬁ‘ect of temperature was more pronounced in the
intermediate relative humidity range (20-80%). The upper limit of 80% corresponds to
17-19% moisture content (dry basis). At relative humidity higher than 80%, all isotherms

tended to collapse into one line. This agreed with Pisecky (1992) who reported the

50

 

 

45-

 

 

 

 

—4OC
40 1 +300
+200
35 4
30 a
'3 25 3
3
Z’
'3
32
. 20 ~
2

 

 

o T T 7 —T
0 20 40 60 80 100

Relative humidity. %

 

 

Figure 7. Calculated moisture sorption isotherm of non-fat dry milk at 20, 25, 30 and
40°C

51

intersection of isotherm at relative humidity of 60% which corresponds to 18% moisture

content (dry basis).

GAB Equation Analysis

Sensitivity coemcient, x, of each GAB parameters were determined and plot
against the independent variables, aw. The sensitivity coefﬁcients of parameter C, k, and

Wm are presented in eqn. 81, 82, and 83, respectively.

Q)

M kaaw

 

 

= _ = 81
1C ac (l—kaw+Ckaw)2 ( )
aM Cka
= —— = “’ +1 2 2 +1 82
1“ 6k [(1--kanl—kaw+Ckaw)]2[(c )ka" 1 ( )
Cka
[W M w (83)

 

~ = aw,n = (l-kanl-kaw+Ckaw)

As seen in eqn. 81, 82, and 83, )(c was a function of aw, C, k, and Wm; 30, was a function
of aw, C, k and Wm; and xwm was a function of aw, C, and k. The equation was not linear
in term of its parameters. Sensitivity coefﬁcients of each parameter at 20, 30, and 40°C
were plotted against water activity as shown in Figure 8, 9, and 10.

The plot of xc against water activity in Figure 8 indicates that the moisture
content was very sensitive to parameter C at low water activity region (0-0.04 aw) and the
sensitivity is greater at 20°C. The sensitivity to parameter C was about the same at 20
and 40°C. In higher water activity, the sensitivity of moisture to C for all temperatures
were minute. However, ﬁ'om Figure 8 to 10, the sensitivity of the equilibrium moisture

content to parameter C was much smaller than those to Wm and k.

52

The graph of 30‘ versus water activity in Figure 9 indicates that parameter k had
little or no eﬂ'ect below water activity of 0.8. Above that, sharp rise, indicating the
increase in sensitivity of moisture content to k, was noticed. This agreed with the fact
that k is the property of multilayer water because k did not have any effect at low water
activity range which is monolayer. The sensitivity of equilibrium moisture content to
parameter k was the least at 20°C and the most at 40°C. The impact of error due to an
error from parameter R was expected to be larger at high water activity especially at high
temperature.

The graph of 1%, versus water activity in Figure 8 indicated that the equilibrium
moistm'e content in the range of high water activity (0.8-1 aw) was very sensitive to
parameter Wm. The sensitivity was the greatest at 40°C and least at 20°C. At low water
activity region, the sensitivity of equilibrium moisture content to parameter Wm were
small and not signiﬁcantly diﬁ‘erent for all temperature. The impact of error due to an
error in parameter Wm was expected to be large at high water activity especially at high

temperature.

53

 

 

12-

 

 

 

 

 

 

 

 

 

10 ( ____::
I -'--«m
8
“:9
x6“
>2
4-
21 \
o l 1 l l
0 02 Q4 06 08 1

 

Figure 8. Plot of sensitivity coemcient of parameter C as a function of water activity

54

 

 

25.

 

 

 

 

 

 

 

 

—3oc ;
" " '40C I
201 I
I
I
I
15 . g
2
x l
X I
I
I
101 I.
l
l
54
o f 1 l
0 0.2 04 1
3w

 

 

 

Figure 9. Plot of sensitivity coemcient of parameter k as a ﬁmction of water activity

55

 

 

 

 

 

 

 

 

 

 

30]
-——BOC
25~ - - -400
201
><£153
10—
5s
0 TI f T T 1
O 02 0.4 06 08 1

 

Figure 10. Plot of sensitivity coefﬁcient of parameter Wm as a function of water activity

56

 

Water Vapor Transmission of Packaging Material

Permeance of six packaging materials were determined at approximately 40°C,
75% RH. The averages of permeance of packaging materials are presented in Figure 1 1.
The data were analyzed by Minitab Release 11.12. Analysis of variance showed that all
the data were signiﬁcantly different at 95% conﬁdent level. Then a Fisher Least
Signiﬁcant Difference was employed for multiple comparison. Control material was
excluded from the multiple comparison test due to its extremely high permeance. The
statistical analysis indicated that permeance of all materials were signiﬁcantly different at
conﬁdent level of 95%. Packaging material type A and E had the lowest permeance or
highest barrier properties. Therefore, they were chosen for the additional testing at 30°C,
75% RH and 25°C, 67% RH. The averages of permeance of all materials at all
conditions are presented in Table 14. The experimental values of WVTR and permeance

are presented in Appendix D.

Table 14. Permeances of packaging materials at 25, 30, and 40°C

 

 

 

 

 

 

 

 

Material 25°C 30°C 40°C
RH. % P/e‘ RH. % P/e‘ RH. % PM
A 67i0 l.8:l:0.2 75:1 l.6:l:0.l 733:3 l.2:l:0.3
B - - - - 73 :l: 3 1.7 d: 0.4
C - - - - 74 :l: 3 2.4 d: 0.2
D - - - - 74 d: 3 4.2 :t 0.3
E 67 :1: 0 0.44 :1: 0.07 75 :l: 1 0.46 :l: 0.1 74 i 3 0.49 :l: 0.06
Control - - - - 75 :1: 3 55 :1: 1.3

 

 

 

 

 

 

 

 

.P/I.’ is permeance, g/(m2.day.mmHg)

57

 

 

 

P/I, gl(m2.day.mmHg)

40-

 

201

l
1
10 1
1L-1ﬁ-7 ,If __,__

A B C D E CONTROL

01

 

Figure 11. Permeances of packaging materials at 40°C, 75% RH

58

 

Permeance of material A increased with decreasing temperature, which indicated
that permeance as a function of temperature did not follow Arrhenius equation. Based on

analysis of variance and Fisher Least Signiﬁcant Difference, at 95% conﬁdent level,
although the permeance of material A at 25 was lower than at 30°C, they were not
signiﬁcantly diﬂ‘erent but both of them were signiﬁcantly different ﬁ'om the permeance at
40°C. The permeance of material was plotted against the temperature as in Figure 12.
Quadratic regression was performed by Microsoﬁ Excel 97-SR1 to obtain the equation
describing the effect of temperature on the permeance of material A in the range of 25 to
40°C as

(PM) = 0.00413 + 0.3T — 3.8 (84)
where T = temperature, °C, and (PM) = permeance, g/(m2.day.mmHg)

Permeance of material E increased with increasing of temperature, following
Arrhenius equation. Analysis of variance and Fisher Least Signiﬁcance Difference at
95% conﬁdence indicated no signiﬁcant difference in Permeance of material E in the
temperature range of 25 to 40°C. Arrhenius plot was constructed, as in Figure 13, and

Arrhenius equation was obtained from the linear regression of Arrhenius plot.

loge-j = -%‘93+1.4351 (85)

where T is temperature, K and (P/Z) = permeance, g/(m2.day.mmHg)
The activation energy was calculated using eqn. 86 and was found to be 0.97

KCal/mol. Low activation energy supported the result of little or no temperature effect.

59

 

 

 

2.5 —

 

 

 

 

 

 

 

 

 

 

2 4
a
E -
>~
G
1!
Th: 1 -
B:
E
0.5 «
o T 1 1 1 ﬁ
20 25 30 35 40 45
Tenperature, °C
Figure 12. Permeance of material A as a ﬁmction of temperature
1 _
I: f
.3.’
0.1 T . ﬂ
0.00310 0.00320 0.00330 0.00340
1/T, 1/K

 

Figure 13. Arrhenius plot of material E

60

 

_ _sx2.3xR
' 1000

where E, is activation energy, KCal/mol, s is slope of Arrhenius plot obtained by linear

(86)

regression, and R is gas constant, 1.987 KCal/mol.

Cardoso and Labuza (1983) reported non-Arrhenius behavior in permeability of
paperboard. They provided an explanation of such behavior by assuming a reduction of
mean pore size of paperboard at high temperature, resulting in decreasing diﬂ‘usion rate
because diﬂ'usion rate is partly proportional to the pore radius. The effect of diffusion
was greater than the eﬁ‘ect of solubility, which increases with increasing temperature.
Since material A and E consist of F-ﬂute corrugated board, this concept should be
applicable to both materials but it is only applicable to material A.

The cross sectional area of material A and B were magniﬁed by optical
microscope with the magniﬁcation of 37.5 to examine their structure and provide
potential explanation on their barrier property as a function of temperature. The structure
of material A and E are presented in Figure 14 and 15, respectively. Both materials
consisted of three layers from outside to inside the package: paper liner, corrugated
paper, and paper and plastic liner. Two outer layer of both materials were same. The
plastic layer in the inner liner of material A was difﬁth to locate from Figure 14 but
with the visual and tearing inspection, the plastic layer was the innermost layer. As seen
in Figure 15, it was obvious that the plastic layer of material E was incorporated in the
middle of the inner liner.

The diﬂ‘erence in the inner liner should be accounted for the diﬂ‘erence in the
barrier property as a function of temperature. In material A, the plastic and the paper

layers acted as two separate layers. The paper layers, which faces high relative humidity

61

Inner liner

—1

 

Figure 14. Cross sectional area of material A

62

Inner liner

_

 

Figure 15. Cross sectional area of material E

63

environment, limited the amount of moisture exposed to the plastic barrier layer. Thus,
permeability of the paper layer affected the driving force across the plastic layer. As
temperature increased, permeability of the paper layer as well as the driving force
decreased. Unlike material A, the plastic molecule in material E which disperse into
voids between ﬁbers restricted the change of pore size. Therefore, the permeability as a
function of temperature of material E depends on the behavior of plastic which follows

Arrhenius equation.

Package Integrity

Weight gain of small size FOL and Bellow style boxes made with material E and
ﬁlled with desiccant was monitored as a function of time at 40°C, 74:1:1% RH to compare
the performance of each style. The experimental moisture gain was then compared with
the calculated moisture gain to evaluate package integrity in each box style. The
experimental and calculated data are presented in Table 15.

The calculated and experimental weight gain of FOL and Bellow style boxes were
plotted as a function of time as shown in Figure 16. The calculated data was represented
by a band instead of a single line because it had incorporated an error in relative humidity
ﬂuctuation (1.4%) and the uncertainty in permeance (12.2%). The performance of
Bellow style boxes were not signiﬁcantly different from the performance of FOL style
boxes.

Weight gain of large size bellow style boxes made ﬁ'om material A was
monitored as a function of time. The experimental data was compared to calculated data

in Table 16. Graphical comparison is presented in Figure 17. The calculated bar

64

incorporating the error of relative humidity ﬂuctuation (4.1%) and the tmcertainty in
permeance (25%).

After about 10 days for packages made from material E and 3 days for packages
made from material A, the experimental data deviated ﬁ'om the calculated data. It was

due to the moisture building up inside the package after desiccant were saturated. This

made Ap in eqn. 77 variable and no longer equal to pressure outside the package.

Table 15. Experimental and calculated weight gain of Bellow and F 0L style boxes made
with material E and ﬁlled with desiccant at 40°C, 74:1:1 %RH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time, day Weigh_tgain,
Bellow FOL Calculated

0.0 0.00 0.00 0.00

1.0 2.31 2.41 2.72

2.0 5.16 5.15 5.43

3.0 8.00 7.90 8.15

4.0 10.88 10.68 10.86
5.0 13.69 13.48 13.58
6.0 16.46 16.20 16.30
7.0 19.22 18.93 19.01
8.0 21.94 21.63 21.73
9.0 24.54 24.18 24.45
10.0 27.13 26.72 27.16
1 1.0 29.43 29.1 1 29.88
12.1 31.74 31.47 32.82
13.0 33.27 33.00 35.31
14.0 34.80 34.43 38.03
15.0 36.16 35.85 40.74
16.0 37.60 37.26 43.46
17.0 38.92 38.56 46.18
18.0 40.08 39.75 48.89
19.0 41.17 40.83 51.61
20.0 42.20 41.87 54.32

 

 

 

 

 

 

65

 

Table 16. Experimental and calculated weight gain of material of Bellow style boxes
made with material A and ﬁlled with desiccant at 40°C, 73:1:2% RH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time, day Weightgain, g
Experimental Calculated

0.0 0.00 0.00

0.8 9.34 8.41

1.8 20.04 19.63
2.3 26.00 25.24
2.8 30.88 31.09
3.3 35.41 36.46
3.8 38.96 42.07
4.3 41.66 47.68
4.8 44.76 53.29
5.8 50.89 64.51
6.8 57.29 75.73

 

 

Since, for both packaging materials, experimental data was within the calculated
bar, it can be concluded that the package was leak-proof. Water vapor permeated into the
package only through the package wall. Therefore, permeance of package can be

calculated ﬁom permeance of packaging material.

66

 

701

 

 

 

A Bellow
o FOL
60 — Calculated
50
U)
.6 4° ‘
a
D
E
.9
2 so—
: .3:
1 J 2 ﬂ 5 i
.0 ‘
. .
1o - ‘
e
a
I
o 1 1 l I
O 5 10 15 20
Time, day

 

25

 

Figure 16. Experimental and calculated weight gain of material E boxes ﬁlled with

desiccant at 40°C, 74:1:1 %RH.

67

 

 

 

 

 

e Sample1
A Sample 2

Calculated

. ...u..q.»._....u......q....,
Lhm_dmv....u~...... m.“ H ..n ,, ,
a........—m. .ck...‘................._.._..

1.N.q._u_»..n.u.._._u_.mu_..».a“.auw2.". A

.--u-u-uuuuuuug..-~.-u.~.uv-~u-...V..

....,Laaaw......‘.........aa..~..m...w..a.a“...

..w.uuu_u—.._....._._._...“...—..“._gun—un.“ma.
.u____—._-.~u___-~..~__.u.k..~.~._~_—__ nu.
.....w.......v.y.... ........... ..~a . ...
....m..“..mmw.m_.1...‘..n..mmu.‘_um‘a..u.._
nuu.uumanna.“___~u_unu~.u_u_uuuauu-uuu.u..u
... u,.va.m1._y.m_v.a..w:..waa....am..__._.

.h._C«~___—_~____-u—..~___~ _~_H_.
.:5:....z..:....:.:.:..:::2...
......._._wu_-.-__..-_u_nun-u...___.

.,u.~.u.--.~..u.~...-.uu~.euu..p.
I . ...._..W.www“y........am._an
a
:.s.:....:::..:::::.
....2..u..a‘.....‘»....ak..._

..hN—u»——~u--~—.

um..um

q_~_n~_M—uu«_.nu_qm.

._m...ma_.u..

_ .

1.~H...1.

 

 

120 —

 

 

 

 

 

 

 

f small Bellow style boxes (material

gaino

ght

tal and calculated wei

perimen

A) at 40°C, 73:2 %RH

17. Ex

Figure

68

CHAPTER 4

SHELF LIFE SINIULATION

Shelf life simulation consisted of shelf life modeling, model analysis, and model
validation. All calculations performed in this chapter were based on the data obtained

from Chapter 3.

Shelf Life Modeling

GAB equation was the best equation to ﬁt the moisture sorption isotherm of non-
fat dry milk. Therefore, the shelf life model was developed based on the mass balance of
water permeated into the package and the amount of water sorbed by non-fat dry milk
and GAB equation.

Assumptions for the shelf life modeling were:

1. The moisture permeated into the package is calculated based on permeance of
packaging material because the package has no discontinuities.

2. All the moisture permeated into the package is absorbed by the product and
distributed to the whole product. Therefore, the quantity of permeant in the
headspace is negligible.

3. The equilibrium between product and headspace can be achieved within a short

period of time.

69

4. The water activity inside the package is a ﬁmction of moisture content of the product

expressed by GAB equation.

The shelf life model for non-fat dry milk was developed as follow (Diosady et a1. 1996):

Water permeated into the package is given by

P

dW = i-A-Ap-dt (87)
and the water sorbed by the product is
dM
dW = W — 88
a 100 ( )
Substituting eqn. 87 into eqn. 88, gives
P dM
—AA dt = W — 89
g P d 100 ( )
t M!
BAAp Idt = E1. 1dM (90)
I! o 100 M1

The value of Ap = po-pi is the difference in partial water vapor pressure across the
package wall being po the outside value and pi the inside value. The partial pressure
inside the package is a function of moisture content. In order to integrate eqn. 90, Ap

need to be expressed as po-pi(M) and eqn. 90 can be rewritten as

P ‘ w M dM
e p! 1001p.-p.(M) ( )

M1

Partial pressure outside the package, p0, is a function of relative humidity and saturated

water vapor pressure as

70

RH
= ° 92
p0 p! 100 ( )

 

Water activity in the package headspace directly relates to moisture content of non-fat
dry milk by GAB equation. GAB equation (eqn. 62) can be rewritten in the form of

water activity as a function of moisture content as follows:

Wm-C'k'aw : 1-kaw+Ckaw-kaw+kza:,+Ck233v (93)

 

 

k2(1-C)af,+k[C[l—‘Z&"]-2]aw +1 = 0 (94)
Since eqn. 94 is in quadratic form, there are two solutions. Only the positive solution

will be used and can be rewritten as

2111.11.11.11-4...

 

 

 

 

 

 

 

— 95
8" 2k(1- c) ( )
Therefore, the value of pi = p,;.aW can be expressed as a ﬁmction of M as
2
2+[W'n —1)C— 2+(yl—IJC —4+4C
M M (96)
‘ 7 ' 2k(1- c)
Substituting eqn. 92 and 96 into eqn. 91
t M:
10015123131 = 1' (M (97
I Wd 0 M,

11111} 2+(%-1F-J(2+(%-1)C)’-4+4c

ﬂ 2k(1-C)

71

)

 

P A-p 1 ‘
1 — - ' dt =
001:) w, 2k(l-C)5l

 

 

 

“1 dM (98)
”12110—0 E11-2{film—10+“\/[§“£"—1(4c-2(:’)+[E'11zcz +c2
100 M M M
. _ B .A-p, 1
Assrgn a - 1001!) wd 2k(1—C) (99)
RH
p _ 12k(1-C)-[m)]—2+C (100)

Then, eqn. 97 can be rewritten as

at = MI “W (101)

 

 

IfC value is much larger than 2, 4C will be much smaller than 2C2 and can be neglected.
In this study, the minimum C value is about 95, which make (-2C2) differ from (4C-2C2)
for approximately 2%. Therefore, 4C can be disregarded and eqn. 101 can be rewritten

as

 

 

 

 

 

 

at = I M 2 (102)
~.,_1:_v._1.+.11w.1 41:)“
M M M
2 2
Since [Wm] —2(-&]+1 =[1—[wm]:1 ,eqn.102canberewrittenas
M M M
at = 1 M (103)

 

72

(104)

 

”1 MdM
at =
M. M(B + c)- 2wmc

Making a change variable, U = M(B+C)—2WmC, and dU/dM = B+C. Eqn. 104 can be

 

 

 

 

 

rewritten as
at ___ Edam-2w: U + 2WmC dU _1_
mm..- (,6 + c) (p + c) U
(105)
1 M,(p+c)—2w_c
at = [U + 2wm C(ln U))] (106)
(13 'l' C)2 M,(p+c)—2w,,c
_ 1 _ _ (M,(0+c)-2w,£)
at - (13 + (2)2103 + c)(M,. Mi) 2wmc In (M «3+ c)— 2W..¢)l (107)
Assign a = p + c = (1 - c112k1%11— 21 (108)

Substituting eqn. 108 into eqn. 107

(109)

   

 

t__1_1_|__,,,,2\11;c(M19-2w:c)1

(19 (M 0— 2w (3)

 

wherea=1 030(JAW1’Wd 3.2k(11- C)

(19] )-]

Q
ll

73

Shelf life model at 40 °C and 25 °C

The shelf life model of packaged non-fat dry milk at a speciﬁc condition was
developed based on the shelf life model and the results ﬁom Chapter 3. Parameters
employed in shelf life modeling of non-fat dry milk packaged in small and large size
Bellow style packages made ﬁ'om material A and E and stored at two storage condition

are presented in Table 17.

Table 17. Parameters used for shelf life modeling of packaged non-fat dry milk at two
storage conditions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Condition
HiAS HiAL HiES LoAS LoAL LoES
Temperature, °C 40 40 40 25 25 25
RH, % 83 83 83 67 67 67
Packaging material A A E A A E
Pack_age size Small large small small large small
C 95.4 95.4 95.4 120.16 120.16 120.16
K 0.96 0.96 0.96 0.94 0.94 0.94
_‘Ya’ % dry basis 3.66 3.66 3.66 4.19 4.19 4.19
_Px mmHg 55.324 55.324 55.324 23.756 23.756 23.756
Mi: % dry basis 4.86 4.86 4.86 4.86 4.86 4.86
A, m2 ' 0.1354 0.2315 0.1354 0.1354 0.2315 0.1354
WM 273.56 333.96 276.72 276.88 334.09 277.07
pm, g/(mz-daX-mmHj) 1.2 1.2 0.49 1.8 1.8 0.44

 

 

Hi is high storage temperature and relative humidity,
Lo is low storage temperature and relative humidity,
A is package made from material A,

E is package made from material E,

S is small package, and

L is large package

Shelf life models at eachspeciﬁc condition are:

HiAS' (High temperature and relative humidity, small package made from material A)

t = 1 _1Mf_4.86+18.20.h[38.39M,498.7711

 

- (110)
0.70 -512.19

74

HiAL (High temperature and relative humidity, large package made ﬁ'om material A)

t = -L- M,—4.86+18.20-11.138‘391VI‘”698'77 (111)
0.97 -512.19

 

HiES (High temperature and relative humidity, small package made from material E)

1

t = -—-
0.28

 

(112)

[Mr — 4.86 + 18.20 - 111138'39Mr - 698-7711

- 512.19
LoAS (Low temperature and relative humidity, small package made from material A)
1

t = -——. M,-4.86+11.43-ln
0.82

 

88.23M, 4006.941] (113)

- 578.14
LoAL (Low temperature and relative humidity, large package made from material A)

t = --—1—- M,-4.86+11.43~1n[88'23M‘-1006'94 (114)
1.16 —578.14

 

LoES (Low temperature and relative humidity, small package made from material E)

 

t = -—l - Mf -4.86 + 1 1.43-1:1(88'23Mf 400634) (115)
0.20 - 578.14
Shelf Life Model Analysis

In the shelf life model based on GAB equation, nine parameters are needed to
calculate shelf life based on the ﬁnal equilibrium moisture content. The parameters can
be classiﬁed into three groups: parameters describing product characteristics, parameters
describing package characteristics, and parameters describing storage conditions.
Parameters describing product characteristics include GAB constants (C, Wm, and k) and

M5. Parameters describing package characteristics include A, Wd, and PM. Parameters

75

describing storage condition include RH and p,. Temperature is included in the shelf life
model through saturated water vapor pressure, p,, sorption isotherm, and permeance or
permeability of package. In packaging design and optimization, it is necessary to learn
how each parameter in the model inﬂuences the global outcome of the model. One way
to rmderstand the impact of each parameter on the independent variable t or shelf life is
by performing the sensitivity analysis of the model.

Sensitivity coemcient, 1, allows the evaluation of the sensitivity of the model to
each parameter. The plus or minus sign of the sensitivity coefﬁcient indicates whether
the change in the parameter will shorten or lengthen the shelf life. The shelf life will be
longer when the parameter which has positive sensitivity coefﬁcient increases or when
the parameter whose sensitivity coefﬁcient is negative decreases. The absolute value
indicates the magnitude of the impact. When the parameter with higher absolute value of
sensitivity coefﬁcient changes, the shelf life will be altered more than when the parameter
with a lower value of sensitivity coefﬁcient changes.

The sensitivity of coefﬁcient of each parameter are presented in Appendix E.
From the sensitivity coefﬁcient, it can be said that the shelf life model was not linear in
its parameter because a sensitivity coefﬁcient of one parameter was a ftmction of other
parameters. To analyze the sensitivity of one parameter, the value of other parameters
must be assigned. Sensitivity coefﬁcients were calculated at the certain combination of
factors. The factors were described by assigned values which are divided into 1: storage

condition, I]: permeance, and 111: size of package. The calculation was performed at 8
factor combinations based on full 23 factorial design. The calculation of XCP/e) and Md

were performed at 4 factor combinations based on 22 factorial design (I and III) and (I

76

and II), respectively. The description of parameter in each group and its assigned values
are presented in Table 18. Table 19 summarize the value of sensitivity coeﬁicient at each
condition. The values and the plot of sensitivity coefﬁcient as a function of the parameter
are presented in Appendix E. The example of the plot of parameter C against its

sensitivity coefﬁcients are presented in Figure 18.

Table 18. Conditions used in sensitivity coeﬁicient calculation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Condition Storage condition Permeance Size of ‘ ckage
High Low High Low High Low
SH SL RH RL Ps PL
Description 40°C, 25°C, High Low Small Large
83% RH 67% RH permeance permeance
C 95.46 120.16 - - - -
K 0.96 0.94 - - - -
Wm, % dry basis 3.66 4.19 - - - -
p11L mmHg 55.324 23.756 - - - -
RH, % 83 67 - - - -
P/e‘ - - 1.8 0.44 - -
A. m7 - - - - 0.1354 0.2315
Wd, g - - - - 275 333
M; % dry basis 4.86 4.86 - - - -
_FM2 % dry basis 17.82 11.26 - - - -
g/(m2.day.mmHg)

77

 

 

 

0.9 4
0.8 ~
0.7 4
0.6 4
,2 0.5 4
0.4 1
0.3 1
0.2 «

0.1 1

 

 

”on. Sh, Rh, P8

-o--- Sh. Rh, PI
"-11-.- Sh, Rl. Ps
”ox... Sh, RI, Pl

—e—Sl. Rh, Ps
——e—Sl. Rh. Pl
-—e——Sl. RI, Pa
-—x-—Sl. RI, PI

 

 

 

 

 

 

 

 

70

110

120 130 140

150

 

Figure 18. The plot of xc as a function of C at different factor combinations

78

 

Table 19. Sensitivity coemcients of parameters in shelf life model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SH, RH, 311, RH, 311, RL, 511, RL, SL. RH, SL, RH, 31., RL. 51., RL.
PS PL, PS PL, P3 P1, PS PL
x; Min 0.08 0.06 0.36 0.25 0.05 0.04 0.22 0.13
Max 0.23 0.17 0.95 0.67 0.16 0.11 0.66 0.47
m Min -11 -8 —45 -32 -9 -7 -39 27
Max -990 -701 -4048 -2868 -150 -141 -610 432
103 Min -238 -169 -974 -690 -127 -90 -520 368
H Max -4818 -3412 -19700 -13900 -1373 -970 -5615 3978
7041 Min 019 -0.13 -O.78 -0.56 -0.43 -0.3 -1.7 1.2
Max -037 -0.26 -1.5 -1.06 -094 -O.66 -3.8 2.7
W“ 0.21 0.12 0.86 0.5 0.16 0.09 0.67 0.38
701‘ Min -372 -158 -1520 -644 -283 -120 -ll60 492
Max -591 -205 -2417 -837.3 .451 -156 -1845 639
W; Min -0.06 -0.02 -0.004 .0.02
x107 Max -2.84 -11.6 -2.16 -8.86
70: Min -04 -0.3 -1.5 -1.1 -0.1 -0.1 -0.5 0.3
Max -38 -27 -154 -109 -12 -8 -48 34
TR? Min 29 -2 -11.7 -8.3 -5.7 -4 -23 16
Max -68 -481 -2778 -197 -456 -322.8 4864.8 1320
70? Min 83, PS -29 SH, PL -20 sL, P3 -22 $1,, PL -16
Max -649 -460 -495 -350

 

 

' Sensitivity coeﬂicients decrease with increasing parameter

°° Sensitivity coemcients increase with increasing parameter

m Sensitivity coemcients are independent of magnitude of parameter

From the sensitivity analysis, parameters had either positive or negative values of

sensitivity coefﬁcient. Parameters with negative value of sensitivity coeﬁcient include

k, Wm, M,, A, A/Wd, P/Z, ps, and RH. As these parameters increase, shelf life of non-fat

dry milk decreases. Parameters with positive value of sensitivity coefﬁcient included C,

Mf, and Wd. As these parameters increase, shelf life of non-fat dry milk increases. This

means that the shelf life can be increased by either reduction of parameter k or Wm of

GAB equation, increasing parameter C, lowering initial moisture content (Mi), reduction

of surface area of package (A), reduce the ratio of surface area to product weight, ﬁlling

79

1110

abs

the

the

CO

136

01
SE

te

hi

a1

10

more product in each package (W 6). using packaging material with better barrier property
(PM), or store the product at lower temperature and relative humidity (ps and RH).

The impact of each parameter on the shelf life were compared by comparing the
absolute value of the sensitivity coeﬁicient. When consider each single parameter, k had
the highest sensitivity coefﬁcient which infer that parameter k had the highest impact on
the shelf life, then Wm, A, Mf, RH, PM, p, and M5, in this order. The impact of parameter
C and Wd on shelf life was minute. The reduction of parameter k could signiﬁcantly
increase shelf life while the increasing of parameter C may insigniﬁcantly or may not
increase the shelf life. Change in product dry weight may not seem to be signiﬁcant
when consider only the value of xwa but it does has high impact on shelf life because it
will change the ratio of surface area and product dry weight which had higher sensitivity
coefﬁcient than n.

The change in any parameter for NFDM packaged in small box made from low
permeance package altered shelf life more than the change in the same parameter for
other boxes. When consider the same type of package, the shelf life model was more
sensitive to the change of Wm, Mi, and RH when the product was stored at low
temperature and relative humidity (or low temperature in case of parameter RH) than at
high temperature and relative humidity. The model was more sensitive to the change of

C, k, Mf, A, P/Z, Wd, p3, and W4 when non-fat dry milk was stored at high temperature

and relative humidity (or high relative humidity in case of p3) than at low temperature and
relative humidity.
The impact of parameter on the shelf life changed as the parameter increased or

decreased except for Wd that its sensitivity coefﬁcient at the certain factor combination

80

was independent of the change in product weight. When GAB constants (C, Wm, and k),
saturated water vapor pressure, p,, surface area, A, and permeance, P/t’, increased, they

had less impact on shelf life. The impact of initial moisture content and relative humidity

increased as the parameter increased.

Model Validation

The validation of the shelf life model was conducted with boxes of two sizes at
two isothermal storage conditions. The predicted moisture content was calculated using
shelf life models presented in eqn. 110 to 115. The experimental and predicted moisture
content of the packaged milk powder as a function of time at 40°C and 25°C are
presented in Table 20 and 21, respectively. Iteration method performed by computer
software, namely Mathematica, was employed to determine the value of moisture content
as a ftmction of time.

Figure 19 to 24 show the graph of predicted and experimental moisture content as
a function of time. Predicted values are presented as a band incorporating uncertainties
of the parameters. The uncertainties of parameters are presented in Appendix E. The
predicted band was constructed using x-axis error bar to describe error in the independent
variable t. The width of the band was equal to the percent error of each speciﬁc package
at certain storage condition. Percent errors were calculated based on Gauss’ Law of Error
Propagation. All parameters and Mf value must be speciﬁed in order to calculate the
percent error. Therefore, values of percent error were calculated at certain values of
variable My and are presented in Appendix F. The average of the percent error of central

Mf values were chosen to represent the error bar and are presented in Table 22.

81

Total error for certain type of package and storage condition at each value of ﬁnal
moisture content consists of uncertainties ﬁom GAB constants, initial moisture content,
product dry weight, permeance of packaging material, relative humidity, and saturated
water vapor pressure. Graphs of error contributed from each parameter relative to the
total percent error at each speciﬁc storage condition are presented in Appendix F. The
contribution of an uncertainty of each parameter to the total error related to the sensitivity

coemcient of that parameter and the magnitude of its uncertainty. The error was mainly

ﬁom PM, Wm, M,, and k. The error due to M was signiﬁcant only when Mf was close to

M. The error ﬁom Wm and PM were higher when Mf was close to M; and became less as

Mr is closer to equilibrium moisture content of that particular storage condition. The
error due to parameter k increased as Mf was approaching equilibrium moisture content.
At 40°C, the error due to relative humidity was more signiﬁcant than at 25°C and equally
contributes to the total error for all Mf values.

As seen in Figure 19 to 21, the experimental moisture contents of non-fat dry milk
as a function of time were within the predicted band upto 16% moisture content (dry
basis) or 0.8 aw where the growth of microorganism was observed outside boxes. The
decreasing of moisture content of non-fat dry milk in boxes made from material A was
not because of the loss of moisture but the loss of dry mass due to the growth of
microorganism. The shelf life model based on GAB equation was applicable to predict
the moisture content of non-fat dry milk at this condition regardless of permeance of

packaging material but the model was unusable when there is a growth of microorganism.

82

Table 20. Experimental and predicted moisture content as a function of time of non-fat
dry milk in three different packages stored at 40°C, 83% RH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time, Moisture content, % (dry basis)
day Material A-small Material A-large Material E-small
Experimental Calculated. Experimental Calculated Experimental Calculated
0.0 4.86 :1; 0.00 4.86 4.86 :1; 0.00 4.86 4.86 :1: 0.00 4.86
0.8 6.03 :1; 0.11 6.05 6.10 :1; 0.07 6.44 5.10 :1: 0.03 5.39
2.3 8.01 :1; 0.22 7.72 8.32 : 0.13 8.49 6.02 j; 0.03 6.26
2.8 8.49 :1; 0.24 8.17 8.94 :1; 0.16 9.00 6.27 :1: 0.02 6.52
3.8 9.43 :1; 0.32 8.94 9.93 :1: 0.18 9.88 6.71 : 0.11 6.99
4.3 9.79 :1; 0.35 9.28 10.27 1; 0.17 10.27 6.88 :1: 0.11 7.21
5.8 10.89 j; 0.37 10.16 11.40 :I: 0.19 11.25 7.37 : 0.12 7.80
6.8 11.52 :1; 0.39 10.67 12.01 : 0.28 11.79 7.70 :1; 0.11 8.15
8.8 12.87 :1: 0.41 11.54 13.32 :1; 0.19 12.70 8.35 :1; 0.11 8.79
9.7 13.32 : 0.40 11.82 13.79 :1; 0.17 13.00 8.63 :1: 0.11 9.00
12.7 14.07 :1; 0.34 12.78 14.43 ;1; 0.20 13.98 9.33 : 0.13 9.78
14.8 14.12 :1; 0.29 13.37 14.39 : 0.19 14.55 9.78 ;|; 0.13 10.28
16.8 14.60 i 0.27 13.83 14.85 :1; 0.18 14.98 10.13 : 0.14 10.68
17.8 14.94 : 0.28 14.03 15.20 :1: 0.21 15.17 10.34 : 0.14 10.87
20.8 15.86 :1; 0.29 14.57 16.11 :1: 0.22 15.66 10.90 :1: 0.14 11.39
21.8 16.32 : 0.26 14.73 16.55 : 0.22 15.80 11.26 :t 0.14 11.55
23.8 16.41 :1; 0.22 15.02 16.58 ;1; 0.20 16.06 11.53 :1; 0.15 11.86
24.8 16.51 :1; 0.21 15.16 16.65 3; 0.21 16.17 11.68 :I: 0.15 12.00
26.8 16.64 :1; 0.18 15.41 16.72 1: 0.23 16.38 11.95 :1; 0.14 12.27
27.8 16.70 1; 0.17 15.52 16.78 :1; 0.23 16.47 12.05 :1; 0.15 12.40
29.8 16.95 :1; 0.17 15.74 17.09 i 0.23 16.65 12.32 :1; 0.15 12.65
31.8 17.16 : 0.15 15.93 17.28 ;1; 0.24 16.80 12.54 : 0.15 12.88
34.8 17.95 :1; 0.20 16.18 18.16 1; 0.26 16.99 13.05 :1; 0.14 13.20
42.8 18.41 :1: 1.03 16.71 18.10 ;|; 0.99 17.37 14.01 :1; 0.16 13.94
48.7 16.12 :1; 1.58 16.99 15.31 :1; 2.59 17.55 14.57 : 0.18 14.38
51.8 15.00 :1: 1.70 17.12 14.20 1; 3.32 17.63 14.75 :1: 0.18 14.60
54.8 12.34 :1; 1.68 17.20 11.69 :1; 4.85 17.69 15.10 :1: 0.18 14.79
56.8 15.26 :1; 0.18 14.91
61.8 15.64 :1; 0.18 15.19
65.8 15.91 :1; 0.19 15.39
67.8 16.06 :1; 0.19 15.66
73.8 16.51 :1: 0.17 15.82

 

83

 

Table 21 . Experimental and predicted moisture content as a flmction of time of non-fat
dry milk in three different packages stored at 25°C, 67% RH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time, Moisture Content, % dry basis
day Material A-small Material Am Material E—small
Experimental Calculated Experimental Calculated Experimental Calculated
0.0 4.86 :1; 0.00 4.86 4.86 j; 0.00 4.86 4.86 : 0.00 4.86
0.8 5.22 :1; 0.00 5.59 5.40 :1; 0.04 5.84 4.97 :I: 0.01 . 5.05
1.7 5.67 :1; 0.05 6.17 6.13 :1; 0.13 6.57 5.10 : 0.02 5.23
2.7 6.10 :1; 0.06 6.79 6.89 :1; 0.16 7.31 5.26 :1; 0.02 5.46
3.8 6.69 1; 0.12 7.42 7.66 :1; 0.21 8.02 5.44 : 0.04 5.73
4.7 7.02 :1: 0.11 7.73 8.14 : 0.17 8.36 5.57 :1: 0.04 5.87
5.7 7.45 :1; 0.17 8.09 8.70 : 0.20 8.75 5.73 :1; 0.05 6.05
6.7 8.21 1; 0.21 8.41 9.48 :1: 0.15 9.07 6.03 : 0.06 6.23
7.7 8.75 :1; 0.49 8.69 9.65 :1; 0.09 9.35 6.20 : 0.07 6.39
8.7 8.92 :1; 0.25 8.93 9.78 : 0.08 9.59 6.35 : 0.07 6.54
9.7 9.22 1; 0.23 9.15 9.86 :t 0.06 9.79 6.48 :1; 0.08 6.69
10.8 9.49 :1; 0.15 9.39 9.90 i 0.07 10.01 6.57 : 0.14 6.89
11.7 9.65 :1; 0.10 9.51 9.94 : 0.04 10.13 6.74 : 0.09 6.96
12.7 9.77 :1; 0.10 9.67 10.00 :t 0.03 10.27 6.87 :1: 0.09 7.08
13.7 9.86 :1; 0.09 9.81 10.04 :1: 0.01 10.39 6.99 : 0.10 7.20
14.7 9.95 :1; 0.11 9.94 10.07 :1; 0.02 10.50 7.11 : 0.11 7.32
16.7 10.14 :1;0.ll 10.16 10.19:0.04 10.68 7.36 :1; 0.12 7.53
18.7 10.30 :1: 0.11 10.35 10.25 : 0.01 10.82 7.61 : 0.13 7.73
19.8 10.45 :1: 0.12 10.45 10.37 : 0.04 10.89 7.83 : 0.14 7.85
21.8 10.58 :1; 0.13 10.59 10.48 :1; 0.07 10.99 8.04 : 0.14 8.02
23.7 10.67 :1: 0.11 10.69 10.54 :1: 0.07 11.06 8.27 :1: 0.15 8.17
27.7 10.88 :1; 0.10 10.88 10.66 :1: 0.04 11.18 8.67 :1; 0.17 8.47
29.7 10.98 i 0.10 10.96 10.69 : 0.02 11.22 8.83 i 0.14 8.60
31.7 11.11 :1; 0.09 11.02 10.74 : 0.01 11.26 8.99 :1; 0.11 8.73
34.7 11.12 :1; 0.04 11.10 10.74 : 0.03 11.30 9.06 :1: 0.12 8.90
37.8 11.23 :1; 0.05 11.17 10.77 :1; 0.03 11.33 9.19 :1: 0.11 9.08
39.8 11.27 : 0.05 11.20 10.84 :1; 0.03 11.35 9.21 :1: 0.17 9.18
44.8 11.31 :1; 0.04 11.26 10.78 :1; 0.07 11.37 9.38 : 0.12 9.41
46.8 11.31 :1: 0.01 11.28 10.77 :1: 0.03 1 1.38 9.42 :1: 0.12 9.49
48.8 11.30 :1: 0.02 11.30 10.79 : 0.04 1 1.39 9.47 :1; 0.13 9.57
54.8 11.32 i 0.00 11.34 11.01 : 0.02 11.40 9.63 :1: 0.13 9.79
56.8 11.30 :1; 0.01 11.35 11.02 :1: 0.01 11.40 9.70 :1; 0.12 9.85

 

 

 

 

 

 

 

84

 

Table 22. Percent error employed in construction of x-axis error bar

 

 

 

 

 

 

 

Condition Error, %
HiAS 38
HiAL 37
HiES 33
LoAS 30
LoAL 30
LoES 33

 

 

 

 

As seen in Figure 22 and 24, the experimental moisture contents of non-fat dry
milk in small box made ﬁ'om material A and B were within the predicted band. As seen
ﬁ'om Figure 23, the experimental moisture contents of non-fat dry milk in large box made
from material A were within the predicted band up to moisture content of 10%. Beyond
this point, the model over predicted the moisture content. This could results ﬁom the
violation of assumption of equilibrium between product and headspace can be achieved
within a short period of time. This assumption could be satisﬁed only when the diffusion
coemcient of water in the product is higher than the diﬁ‘usion coemcient of water in the
packaging material. Large box made from material A bad the highest permeation rate
because of high permeance value of material A and its large surface area.

With the shelf life model based on GAB model, shelf life values of non-fat dry
milk in different package at each storage conditions were calculated based on critical
moisture content of 10% and are presented in Table 23. Shelf life of non-fat dry milk in
the large box made from material A was not calculated since the model was not

applicable.

85

Table 23. Calculated shelf life of non-fat dry milk by the shelf life model based on GAB

 

 

 

 

 

 

equation
Packgg description Calculated shelf life, day
Material Size 40°C, 83% RH 25°C, 67% RH
A Small 5.3 15
A Largg 3.8 -
E Small 13.2 61.5
E Large 7.7 36

 

 

 

 

 

 

86

 

20-

18—

16:

14—

_L
N
l

M, %(dry basis)
3

 

 

 

 

 

 

 

 

 

8 a
6
Calculated
4 0 Experimental
2
o 1 r 1 r I r 1
0 20 40 60 80 100 120
time, days

 

 

 

Figure 19. Predicted and experimental moisture content as a function of time of non-fat
dry milk in small boxes made from material A and stored at 40°C, 83% RH

(HiAS)

87

 

M. %(dry basis)

 

 

 

 

 

 

 

 

 

 

 

 

1’ Calculated
4 i 0 Experimental
2 1
0 Y T 7 T l . _—
0 20 40 60 80 100 120
time, days

 

 

Figure 20. Predicted and experimental moisture content as a function of time of non-fat
dry milk in large boxes made ﬁom material A and stored at 40°C, 83% RH

(HiAL)

88

 

18~

M, %(dry basis)

 

 

 

 

 

 

 

 

4 + ——Calculated
9 Experimental
2 -
0 V T I I T j
0 20 40 60 80 100 120

time, days

 

 

Figure 21. Predicted and experimental moisture content as a function of time of non-fat
dry milk in small boxes made from material E and stored at 40°C, 83% RH

(HiES)

89

 

 

12-

 

 

 

 

 

 

 

 

 

b 6
3
a?
2

4 4

Calculated
2 d 0 Experimental
0 T T T T T j
0 20 40 60 80 100 120

time, days

 

 

 

Figure 22. Predicted and experimental moisture content as a ftmction of time of non-fat
dry milk in small boxes made ﬁ'om material A and stored at 25°C, 67% RH

(LoAS)

90

 

 

14-

121

 

M. %(dry basis)

 

 

 

 

 

 

 

Calculated
0 Experimental
2
O I T T T ’1
0 1O 20 30 4O 50 60 70 80 90

time. days

 

 

Figure 23. Predicted and experimental moisture content as a function of time of non-fat
dry milk in large boxes made from material A and stored at 25°C, 67% RH

(LoAL)

91

 

 

121

 

 

 

 

 

 

 

 

i?
'3
3
b
3
33
2
4 J
Calculated
2 J 0 Experimental
0 ‘T T T r T T
O 20 40 60 80 100 120

time. days

 

 

Figure 24. Predicted and experimental moisture content as a function of time of non-fat
dry milk in small boxes made from material E and stored at 25°C, 67% RH

(LoES)

92

CONCLUSION

The moisture sorption isotherms of non-fat dry milk at 20, 30, and 40°C were best
described by GAB equation. The information of moisture sorption isotherm at
temperature between 20 and 40°C were obtained using Lagrange interpolation method
based on moisture sorption isotherm at 20, 30, and 40°C.

Barrier characteristics of Bellow and FOL style boxes were insigniﬁcantly
different and both of them were free of discontinuity. Therefore, the moisture permeated
into the package only through the package wall and the barrier characteristics of package
were represented by the permeance of a packaging material.

The shelf life model based on GAB equation and permeance of packaging
material were applied to predict moisture content as a function of time or to predict shelf
life of non-fat dry milk. The sensitivity analysis for each parameter indicated that the
shelf life was the most sensitive to the change in parameter k and the change of ratio of
surface area and product dry weight while least sensitive to the change in parameter C.
The predicted moisture content as a function of time at two isotherm storage condition,
40°C, 83% RH, and 25°C, 67% RH of non-fat dry milk packaged in three types of
packages were validated with the actual experiment. At least up to the critical moisture
content of 10%, the experimental data were within the predicted band of moisture
content, which incorporated uncertainties of parameters. Therefore, the model was
applicable to predict shelf life of non-fat dry milk in packages made ﬁom the

combination of barrier layer and f-ﬂute corrugated board.

93

APPENDICES

94

 

APPENDIX A

PARANIETER ESTIMATION OF GAB EQUATION

95

In order to easily estimate GAB parameters, GAB model was transformed into

 

 

 

 

 

 

 

 

 

quadratic form by the following:
M = Ckaw (116)
W“I (l-kanI—kaw+Ckaw)
ﬁ—Sm—w = l-kaw+Ckaw—kaw+k2aw2-ck2aw2 (117)
a_w = k (i—IJaw2+ l (l—ZJaw+ 1 (118)
M W“n C Wm C Wka
Assign
k l l
a = —-1 = l-C 119
“(C ] Wmc( ) ( )
1 2 l
= — l—— = C—2 120
ﬂ WA C) Wmc( ) < >
- l (121)
7 Wka
Eqn. 118 canbe rewrittenas
a 2
_1 = aa. +ﬂa.+r (122)

Constants a, [3, and 7 were obtained from the quadratic regression and were employed to
calculate GAB constants, C, Wm, and k as follow

Substituting eqn. 119 into eqn. 120 gives

— (123)

Substituting eqn. 123 into eqn. 120 gives

96

_1_ _ -9_7_(C-2)C
W... — .6 (C-1) (124)

 

Substitute eqn. 124 into eqn. 120 gives

[3’ ((32 —4c +4)

 

 

 

 

— = (125)
(-a)r (C-l)
ﬂZ
Assign X = and eqn. 125 can be rewritten as
(—a)r
C2—(4+X)C+(4+X) = o (126)
ﬂZ
AssignO = 4+X = 4+ ( )7. The solution of eqn. 126 are
—a
2 _-
c = 6* “92 40 (127)

Since the equation is in quadratic form, there are two possible solutions for C. Only the
solution that gives positive values for all GAB constants was the right answer. Parameter
Wm and k can be obtained by substitution of C into eqn. 120 and 121, respectively.

The standard deviations of C, k and Wm (Sc, 81,, 8%,) was estimated using the
Gauss’ Law of Error Propagation and standard error of a, B, and 7 (8.12, $52, and syz) as

(Taylor, 1982)

 

2 2 2
sc = \I(§) s: +[EJ 82(6—(3) s: (128)
aa 6,6 or

 

where

 

 

 

 

 

 

 

 

 

 

 

 

 

6k 2 2 6k 2 , 6k 2 2

vi.) we) 1.) s,
g = 1
a“ .62-407
.53 = L __ﬂ__
5,5 27 ‘/ﬂ2—4ay
6k 1 2 2
— = J -4 +2 —
57 2y2‘/ﬂ2—4ay(p ﬂ a7 a7 ﬂ)

_ awm’z awmzzavvﬁz
5“ ’ l/(aa ) “l aﬂ ] ”l 67]?” (13°)
6W"!i = 27
6a \/('Bz_4a7)3
awm = J26
6" x/(ﬂz-‘larl
6W 2a

67 = M 4057)]

 

 

98

APPENDIX B

EXPERIMENTAL AND CALCULATED MOISTURE SORPTION ISOTHERMS

 

99

Table 24. Experimental and calculated (with GAB equation) moisture sorption isotherm

 

 

 

 

 

 

 

 

 

 

 

 

data for non-fat dry milk at 20°C
Relative Humidity Moisture content, % (dry basis)
1 2 3 Average Calculated
0.0 :l: 0.00 0.0000 0.0000 0.0000 0.00 :l: 0.00 0.00
8.4 i 0.05 4.2399 4.2282 4.2220 4.23 :l: 0.01 4.26
23.1 i 0.05 4.8680 4.8303 4.8349 4.84 i 0.02 5.42
34.0 1: 0.25 6.0755 6.0240 6.0311 6.04 :I: 0,033 6.24
47.3 i 0.25 8.9686 8.9797 8.9617 8.97 i 0.07 7.75
50.4 :I: 0.25 8.0227 8.0413 8.0294 8.03 :l: 0.01 8.16
64.9 :l:0.25 11.6198 11.5736 11.5514 11.58 :1: 0.03 11.10
75.6 :t 0.25 14.3704 14.3408 14.3350 14.35 :I: 0.02 14.54
81.3 :t 0.25 17.0415 16.9302 16.9120 16.96 i: 0.07 17.84
94.5 i: 0.25 38.8087 37.8571 N/A 38.45 :t 0.51 35.05

 

 

 

 

 

 

 

Table 25. Experimental and calculated (with GAB equation) moisture sorption isotherm
data for non-fat dry milk at 30°C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Humidity Moisture content, % (dry basis)
1 2 3 Average Calculated
0.0 :l: 0.00 0.0000 0.0000 0.0000 0.00 :t 0.00 0.00
8.9 i 0.05 3.8475 3.8736 3.9110 3.88 :l: 0.03 3.97
23.2 i 0.05 4.5957 4.6010 4.5447 4.58 i 0.03 4.90
34.2 :1: 0.25 6.0869 6.1267 6.0680 6.09 :t 0.03 5.73
4.66 :t 0.25 6.5827 6.5054 6.5405 6.54 i: 0.04 6.93
48.7 :t 0.25 7.1250 7.1564 7.1063 7.13 :I: 0.02 7.31
63.6 d: 0.25 11.5924 11.5683 11.5989 11.59 i: 0.02 10.01
75.9 i 0.25 14.0061 13.9898 14.0472 14.01 i003 13.67
81.5 :1: 0.25 16.4967 16.5089 16.5141 16.51 :t 0.01 16.39
93.5 :1: 0.25 33.4394 33.5232 N/A 33.48 :1: 0.06 33.81

 

100

 

 

Table 26. Experimental and calculated (with GAB equation) moisture sorption isotherm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

data for non-fat dry milk at 40°C
Relative Humidity Moisture content, % (dry basis)
1 2 3 Average Calculated
0.0 :l: 0.00 0.0000 0.0000 0.0000 0.00 i 0.00 0.00
9.2 i 0.05 3.2139 3.8736 3.9110 3.67 :l: 0.02 3.61
23.8 :1: 0.05 4.4300 4.4168 4.4668 4.44 i 0.02 4.53
34.3 :1: 0.25 5.8055 5.9436 5.9473 5.90 i 0.08 5.32
45.6 d: 0.25 5.9936 5.9738 6.0182 6.00 i 0.02 6.36
47.6 3: 0.25 6.4217 6.4261 6.4432 6.43 i: 0.01 6.59
62.8 :I: 0.25 9.6050 9.6113 9.6262 9.61 i 0.01 8.99
77.5 i 0.25 13.6980 13.6740 13.6780 13.68 i 0.01 13.99
79.6 :I: 0.25 16.3945 16.4363 16.4294 16.42 :I: 0.02 14.53
91.3 i 0.25 27.338 27.3216 27.2586 27.31 :1: 0.04 28.94

 

101

 

APPENDIX C

POTENTIAL ERROR AND PROPOSED CORRECTION

OF PARAMETER C IN GAB MODEL

102

As mentioned in Chapter 2, parameter C of GAB model is temperature dependent
and can be correlated with temperature by the Arrhenius equation. Parameter C in GAB
equation of non-fat dry milk at 20, 30, and 40°C showed the temperature dependency but
the behavior did not followed the Arrhenius equation. As shown in the sensitivity
analysis, GAB equation is very sensitive in the region of low water activity where it is
difﬁcult to get equilibrium moisture content value. The error could result from the
insufﬁcient number of experimental data in low water activity region. The possible cause
of error and the attempt to correct the value of C parameter are discussed below.

Since it has been known from the sensitivity analysis of GAB equation that
parameter C describes the moisture sorption isotherm in low water activity region, the
main reason for an error in C value is the insufﬁcient of experimental data in low water
activity region. The highest sensitivity coeﬂicient of C is at 0.01 aw but the lowest
experimental data was obtained at water activity of about 0.08. Therefore, parameter C
was estimated based mostly on the extrapolated data. In order to correct the C value,
more experimental data in the range of 0 to 0.08 water activity would be included in the
isotherm. This would result in different value of parameter C, but not parameter k and
Wm which have very low sensitivity coefﬁcient in the low water activity range.

From the value of C at 20, 30, and 40°C in Table 13, three possibilities are:
parameter C at 30°C was too high, or parameter C at 20 or 40°C was too low. In order to
have a correct trend with the estimated new values, corrected values of moisture content
at 0.01 aw for three temperatures have been included into the experimental data and the
new GAB parameters at each temperature were estimated. The estimated values of

moisture content were obtained by lowering (for moisture sorption isotherm at 30°C) or

103

 

raising (for moisture sorption isotherm at 20 or 40°C) the predicted moisture content at
water activity of 0.01. Table 27 shows the predicted moisture content at water activity of
0.01 obtained from GAB equation presented in Table 9 of Chapter 3. Table 28 to 30
present the corrected moisture content and the GAB parameters estimated from the

experimental data with the addition of corrected data.

Table 27. Predicted moisture content at water activity of 0.01 using GAB equation in

 

 

 

 

Table 9 of Chapter 3 F
Temperature, °C Moisture content, %
20 2.19
30 2.18
40 1.78

 

 

 

 

 

Table 28. GAB parameter at 20°C derived from the experimental data and the additional
data point in low water activity region

 

 

 

 

 

 

Parameter Corrected moisture content at aw 0.01 added to the experimental data
none 2.4% 2.6% 2.8% 3.0%
C 103.55 109.14 115.37 121.24 126.85
Wm 0.04 0.04 0.04 0.04 0.04
k 0.93 0.93 0.93 0.93 0.93

 

 

 

 

 

 

data point in low water activity region

 

 

 

 

 

 

 

 

 

Parameter Corrected moisture content at aW 0.01
added to the ex rimental data
none 2.10% 1 .90% 1 .70%
C 124.63 120.29 110.73 100.79
Wm 0.04 0.04 0.04 0.04
k 0.95 0.95 0.95 0.95

 

 

 

104

Table 29. GAB parameter at 30°C derived from the experimental data and the additional

 

Table 30. GAB parameter at 40°C derived ﬁom the experimental data and the additional
data point in low water activity region

 

 

 

 

 

 

Parameter Corrected moisture content at aw 0.01 added to the experimental data
none 1.90% 2.00% 2.10% 2.20% 2.30%
C 95.46 111.58 116.21 120.73 124.94 129.39
Wm 0.04 0.04 0.04 0.04 0.04 0.04
K 0.96 0.96 0.96 0.96 0.96 0.96

 

 

 

 

 

 

 

 

As seen, the additional of data point in low water activity region only affected
parameter C but not parameter Wm and k. In order to increase the value of parameter C at
20°C to be higher than the value of parameter C at 30°C, the corrected moisture content
at 0.01 aw had to be increased upto at least 3.0%. To lower the value of parameter C at
30°C to be in between the value of parameter C at 20 and 40°C, the corrected moisture
content at 0.01 aw had to be decreased down to 1.7% or lower. To increase the value of
parameter C at 40°C to be greater than parameter C at 20 and 30°C, the corrected
moisture content at 0.01 aw had to be increased upto the minimum of 2.3%. With the
corrected C parameter in the last column of Table 28 to 30, the moisture content at 0.01

aw was calculated and compared with the original data as presented in Table 31.

Table 31 The comparison of moisture content calculated from original and corrected C
value at 20 ,30, and 40°C.

 

 

 

 

 

Temperature, Moisture content, %, at 0.01 aw obtained from GAB using_
°C Original C value Corrected C value
20 2.19 2.50
30 2.18 1.97
40 1.78 4.73

 

 

 

 

 

105

 

The evaluation of possible error is made by comparing the corrected moisture
content with the rest of original moisture content based on two assumptions: 1) only
parameter C at one temperature was incorrect and 2) at the speciﬁc water activity,
moisture content of the product subjected to low temperature is higher than those
subjected to high temperature. It is obvious that the corrected C value at 40°C is too high.
This reduces the possible error to either the parameter C at 20°C is too low or the
parameter C at 30°C is too high. Since, in most cases, parameter C decreases with an
increasing of temperature, the error of C at 20°C is likely to be incorrect and the
discussion on the proposed correction is discussed below.

The proposed method to correct the value of parameter C at 20°C is based on the
Arrhenius equation obtained from parameter C at 30 and 40°C. The possible value of
parameter C at 20°C was obtained ﬁ'om the extrapolation of the Arrhenius equation was
approximately 165. The additional data of moisture sorption isotherm at water activity in
the range of 0 to 0.08 must be obtained to derive the exact value of parameter C.

From the sensitivity analysis of shelf life model, parameter C has very low
sensitivity coefﬁcient. Therefore, the change in parameter C would not have much

impact on predicted shelf life.

106

APPENDIX D

WATER VAPOR TRANSMISSION AND PERIVIEANCE

OF PACKAGING MATERIALS

 

rim-w

107

Table 32. Combined water vapor transmission rate, WVTR, and permeance,

 

 

 

 

 

 

 

 

 

R, of packaging materials at 40°C,
Material RH, % Combined WVTR, g[(day.m2) R,
1 2 3 4 Average 8/(m2-day-
MEL
A 73 :1: 3 62.85 38.16 50.73 42.65 49 :t 11 1.2 i 0.3
B 73 r 3 92.93 75.43 53.86 52.98 69 j: 19 1.7 :l: 0.4
C 74 :I: 3 95.23 98.92 87.46 102.92 96 :l: 7 2.4 r 0.2
D 74 i 3 168.36 157.14 167.91 186.77 170 :1: 12 4.2 i 0.3
E 74 i 3 16.74 20.30 22.32 21.82 20 :1: 2.5 0.49 r 0.06
Control 75 :t 3 2296.45 2295.56 2319.35 2165.36 2300 i 70 55 r 1.3

 

 

 

 

 

 

 

 

Table 33. Combined water vapor transmission rate, WVTR, and permeance, R, of
packaging material A and E at 30°C, 753:1 %RH

 

 

 

 

 

 

 

 

 

 

Material Combined WVTR, g/(daymz) R,
1 2 3 4 Average 8/(m2-day-
mmHL
A 36.28 38.19 42.72 - 39 r 3 1.6 :l: 0.1
E 10.50 14.56 8.35 - 11 :l: 3 0.46 :I: 0.1

 

 

 

Table 34. Combined water vapor transmission rate, WVTR, and permeance, R, of
packaging material A and E at 25°C, 67% RH

 

 

 

 

 

 

 

 

 

 

Material Combined WVTR, g/(daymz) R,
1 2 3 4 Average 8/ (1112113)”
mmHjL
A 28.81 27.38 30.88 24.67 27 i 2 1.8 i 0.2
E 5.41 7.64 8.12 6.21 6.8 :1: 1.2 0.44 :t 0.07

 

 

 

108

 

 

APPENDIX E

SENSITIVITY ANALYSIS OF SHELF LIFE MODEL

109

 

 

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Table 35. Sensitivity coefﬁcient of C at different factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Factor combination 1C at C value of
60 100 140
83, RH, Ps 0.23 0.13 0.08
83, RH, PL 0.17 0.09 0.06
SH; RL, Ps 0.95 0.53 0.36
s“, RL, P1, 0.67 0.38 0.25
SL, RH_, PS 0.16 0.08 0.05
81,, R11, PL 0.11 0.06 0.04
S1,, RL, Ps 0.66 0.35 0.22
‘ S1,, RL, PL 0.47 0.21 0.13
1 -
A-
”, ---c--- Sh, Rh. Ps
“ ---o---Sh,Rh,Pl
0.8, ---A---Sh,Rl,Ps
......x Sb, RI, Pl
0 7 —e—Sl, Rh, Ps
' l —e—Sl. Rh, Pl
0 —-A——Sl, RI, Pa
'6 ‘ -—x—— 31. RI, Pl
,2 0.5-
0.4 J
0.3 4
0.2 «
0.1 -
o 4 A . -
50 so 70 so so 100 110 120 130 140
c

 

 

 

Figure 25. The plot of age as a flmction of C at different factor combinations

114

Table 36. Sensitivity coefﬁcient of k at different factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Factor combinations . at k value of

0.94 0.96 0.97 0.98 1.00

83, RH, PS "' -48l8 * -517.1 -238

Sn, RH, PL * ~3412 * -366.2 -169

SHLRL, PS * -19700 " -2115.5 -974

Sat, RL, PL "‘ -13900 "' -1498.3 -690

SL, Rugs _1373 * -267 * -127

SL, RH, PL -970 * -189 "' -90

S1,, R1,, PS -5615 * -1094 * -520

, SL, RL, PL -3978 * -775 * -368

* No data

0 4
-5000 .
-1oooo -

’5 We sn, Rh, P8
1 x -.-o--.Sh,Rh,PI
'50001 ---A---Sh,Rl,Ps

”x... Sb. RI, Pl

.- —e——Sl, Rh. Ps

2m A: +S', Rh, Pl

' ‘ —a—SI. RI, Ps

—x—Sl, RI. Fl
-25000 4 e 4 4 4 4 4 4
0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
k

 

 

 

 

Figure 26. The plot of )0, as a function of k at different factor combinations

115

Table 37. Sensitivity coefﬁcient of Wm at different factor combinations

 

 

 

 

 

 

 

 

Factor combinations at W value of
3.6 4 4.2 4.5 4.6
-990 -30 * -11 *
-701 -21 "‘ -8 "‘
-4048 -123 * -45 *
-2868 -87 -32 "'
~20
-14
-84
-59
-1000 4 e'
-15“) 4
-2000 4
3 2500 3' .....o. sn, Rh, Ps
' ‘ .....e sn. Rh. Pl
,3' .....a sn, RI, Fe
m 4 :0 ...x...Sh,RI,Pl
—o—SI, Rh, Pa
43500 - —e—SI. Rh, Pl
+s1, RI, P8
4000 q A ——-x—Sl, RI, PI
3 3 5 4 4 5 5
wm. %

 

 

 

 

Figure 27. The plot of 3mm as a function of Wm at different factor combinations

116

Table 38. Sensitivity coemcient of M, at diﬁerent factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Factor combinations , XMi at M (%) value of
3 4 5

83, RH, Ps -0.19 -0.27 -0.37

SH, RH, PL -0.13 -0.19 -0.26

SH,RL, Ps -0.78 -1.10 -1.50

SH_, RL, PL -0.56 -0.79 -1.06

S1,, RH}PS -0.43 -0.65 -0.94

SL, RHLPL -0.30 -0.46 -0.66

81,, RL, Ps -1.70 -2.60 -3.80

, S1,, RL, PL -1 .20 -1.80 -2.70

 

 

 

 

 

 

 

 

 

    

 

.25 _ ---o—-- Sh, Rh, Ps
---o---Sh, Rh, Pl
mam Sh, RI, Ps
4H ......x Sh, RI. Pl
—e——Sl, Rh, Ps
.35 - —e—Sl, Rh, Pl
—a—Sl, RI, Ps
——x—-Sl. RI, Pl
.4 , -
3 3.5 4.5

 

 

Mi, %

 

 

 

Figure 28. The plot of m; as a function of M, at diﬂ‘erent factor combinations

117

Table 39. Sensitivity coefﬁcient of Wd at different factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Factor combinations Xv“ at Wd (g) value of Wd
275 425 575 725 300 800 1300 1800
SH, RH, Ps 0.21 0.21 0.21 0.21 "‘ * * "’
83, RH, PL "' "‘ "‘ * 0.12 0.12 0.12 0.12
SH, RL, P§ 0.86 0.86 0.86 0.86 * * * *
SHLRL, PL * * "‘ * 0.50 0.50 0.50 0.50
SL,RH_,PS 0.16 0.16 0.16 0.16 * * * *
S1,, R11, PL * * "' * 0.09 0.09 0.09 0.09
S1,, R1,, Ps 0.67 0.67 0.67 0.67 * * * "'
- SL, RL, PL * "' "‘ * 0.38 0.38 0.38 0.38
* No data
1 4
mon- Sh, Rh, Fe
0.9- ---o---Sh,Rb,Pl
A """" A """ A """" A ---a---Sh-Rl.Ps
0.8 4 ~-x---Sh,Rl.Pl
+81, Rh, P8
0.7 ~ —e—Sl, Rh, Pl
F a a :- -—-e-—Sl, RI, P8
0.6 4 —-)(-—SI, RI, Pl
s 0.5 a x ....................... x ....................... x ....................... x
0‘4 3 x x x x
0.3 —
0.2 . o ------ o ------ o ------ o
o——e——o—e
011 g----------............; ...................... .3: ................... g
o I T I T r T T 1
200 400 600 800 1000 1200 1400 1600 1800
Wd! 9

 

 

 

Figure 29. The plot of 1% as a function of Wd at different factor combinations
118

Table 40. Sensitivity coefﬁcient of A at different factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Factor 30.. at A (m2) value of
combinations 0.115 0.125 0.135 0.145 0.215 0.225 0.235 0.245
SH,RH_,PS -590.7 -500.0 -428.0 -371.6 * * * *
Sail-141R * r * * -204.6 -186.8 -171.2 -157.6
s--_, R1,, Ps -2416.5 -2045.8 -1753.8 -1520.3 * r r *-
sn, RL,PL * * * * -837.3 -764.5 -700.9 -644.8
SL,RH_,PS -451.0 -381.7 -3273 -283.7 * r * *
S1,,RH_,PL * * * * -156.2 -142.6 -130.7 -120.3
SL,RL,PS -1845.5 -1561.9 -1339.0 -1160.7 * * * *
81,, R1,, PL * r * * -639.2 -583.7 -535.0 4923
‘No data
0-
..0
-5004 6:» M
..x"'x
x.,.x'
.1000.
,2 4500—
-o--Sh.Rh,Ps
-O--Sh,Rh,Pl
-2000. --ar--Sh.Rl.Ps
3° --x--Sb,R1,PI
' —e—Sl.Rh,Ps
A! —e—SI.Rh,PI
am 4 +31. RI, Ps
—x——SI,RI,PI
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
111.012

 

 

Figure 30. The plot of 70., as a function of A at different factor combinations

119

 

Table 41. Sensitivity coefﬁcient of A/Wd at different factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Factor combinations mm at A/Wd, mZ/g value of
0.0001 0.0003 0.0005 0.0007
SL1, RH -2.84 -0.03 -0.01 -0.01
SH_, R, -11.6 -0.13 -0.05 -0.02
81., R11 -2.16 -0.02 -0.01 0004
SL, R1. -8.86 -0.1 -0.03 -0.02
0 ‘ k c
-2 4
.4 a
9
NE .6 4 ’ ' 0 ' 'Sh. Rh
90‘ - - o - -Sh, RI
‘- —0-Sl, Rh
% -8 , +31, RI
x
-10 « .
d
-12 -1
-14 r - .
0 4 6 8
ANde10", mzlg

 

Figure 31. The plot of do as a function of A/Wd at different factor combinations

120

 

 

Table 42. Sensitivity coefﬁcient of R at different factor combinations

 

 

 

 

 

 

 

 

 

Factor combinations n at R (g/mjdaymmHg) value of
0.4 0.9 1.4 1.9

SH, P3 -649 -128 -53 -29

SH_, PL -460 -91 -38 -20

SL, PS -495 -98 -40 -22

81,, PL -350 -69 -29 -16

 

 

 

 

 

 

 

 

 

 

 

 

0 4
-100 _.
-200 -1
.3003 ---o---Sh-P8
ﬁ.....oSb.1=1
—o—Sl, Fe
m - —e—SI, Pl
-500 4
8
.7m T T I 1
0 0.5 1 1.5 2
R, g/(day.m2.mmHg)

 

 

 

Figure 32. The plot of m as a function of R at different factor combinations

121

 

Table 43. Sensitivity coefﬁcient of p, at diﬂ‘erent factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100

 

Factor xp, at ps (mmHg) value of
comblnatlons 9.209 17.535 31.824 55.324 92.51
(10°C) (20°C) (30°C) (40°C) (50°C)
8H, RH, P3 -37.6 -104 -32 -1.0 -0.4
8“, RH, P1, -26.6 -7.4 -2.2 -0.7 -0.3
SH_, RL, PS -153.9 -42.4 -12.8 -4.3 -1.5
$3, RL, PL -109.0 -30.0 -9.1 -3.0 -1.1
sL, Rn4 P3 -11.8 -3.2 -1.0 -0.3 01
S1,, RHLPL -8.3 -2.3 -0.7 -0.2 -0.1
S1,, RL, P3 -48.4 -13.3 -4.0 -1.3 -0.5
S1,, R1,, PL -34.2 -9.4 -2.8 -1.0 -0.3
0 4
-20
.40 4
.60 4
.80 4
:- ..o--.Sh,Rh.Ps
>2. 3 ..e.. sn, Rh, Pl
"00 4 E. ------A 811. R1, Ps
x ---x---Sh.Rl.Pl
-120 - : —e—-Sl, Rh, Ps
; +81, Rh, Pl
_140 , —e—SI, Rl. P8
5 —-)(—SI, RI, Pl
A
-160 4
-180 - - 4 -
0 20 40 60 80
pa: mmHg

 

 

 

Figure 33. The plot of 70:, as a function of ps at diﬁ‘erent factor combinations

122

Table 44. Sensitivity coefﬁcient of RH at diﬂ'erent factor combinations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Factor combinations m at RH (%) value of
50 60 70 80 90
, 83, RH, P3 -2.9 -8.7 —2.4 -101.3 -679.1
SH, RH, PL -2.0 -6.2 -17.0 -71.7 -480.9
SH; RL, P3 -11.7 -35.5 -98.3 -414.4 -2778.1
SHLRL, PL -8.3 -25.1 -69.6 -293.5 -1967.5
S1,, R0, P3 -5.7 -12.8 -31.5 -95.4 -455.8
S1,, RH_, PL -4.0 -9.1 -22.3 -67.5 -322.8
SL, R1,, P3 -23.2 -52.7 -129.2 -390.4 -1864.8
SL, R1,, 111, -16.4 -37.3 -91.5 -276.5 -1320.7
0 I 3‘ m ‘_
-500 4 ’
“>0
-1000 4
I ---c---Sh,Rh,Ps
>1 450°“ ---o---Sh.Rh,Pl
”A... sn, RI, Ps ~.
...x...Sh,RI.PI ‘~
.2000 ~ +SI, Rh, Ps x
—e—Sl. Rh, Pl
—-e—SI, Rl, Ps '-
_2500 ﬁ —-x—Sl, RI, Pl
8
m I f V
40 50 60 70 80 90
RH, %

 

 

Figure 34. The plot of m as a function of RH at diﬂ‘erent factor combinations

123

I
r
:.

 

APPENDIX F

ERROR ANALYSIS OF SHELF LIFE MODEL

 

124

and 25°C, 67% RH.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Error, %

HiAS HiAL HiES LoAS LoAL LoES

C 0.06 0.06 0.06 0.04 0.04 0.04

WE, 1.09 1.09 1.09 1.19 1.19 1.19

k 11.46 11.46 11.46 11.7 11.7 11.7

ps 0.81 0.81 0.81 0.29 0.29 0.29
RH 2.81 2.81 2.81 0.16 0.16 0.16
M, 0.82 0.82 0.82 0.82 0.82 0.82
Wd 0.44 0.21 0.28 0.06 0.17 0.1 1

R 25 25 12.24 11.11 11.11 15.91

 

 

Table 45. Percent error of parameters in shelf life model for packages at 40°C, 83% RH

The absolute error of shelf life value was calculated by Gauss’ Law of Error Propagation

based on the error of model parameters as follow:

(pi-(X?) +[ at ~6Wm] +(2t-dt) +[at bps) +
(6C 6W 0k 6p,
61 = (141)

2 2 2 2
($an + i«SMi + at «5W, + ﬁrm)
aMi 6Wd 6R

Where 5 represents absolute error

 

 

 

 

 

 

z::
[a

With Gauss’ Law of Error Propagation, the percent error of shelf life for non-fat dry milk

at certain ﬁnal moisture content were calculated as:

Error,% = {—T-XIOO (142)
and percent error of predicted shelf life for non-fat dry milk packaged in different
packages at 40°C, 83 %RH and 25°C, 67 %RH are presented in Table 46 and 47,

respectively.

125

Table 46. Percent error of shelf life for non-fat dry milk in three different packages

 

 

 

 

 

 

 

 

 

stored at 40°C, 83% RH.
Mf Error, %
HiAS HiAL HiES
5 34.79' 33.01' 33.63'
7 3280‘ 32.80' 24.86‘
9 38.23' 38.13' 31.28'
11 49.49‘ 47.59' 42.48T
13 66.52 66.38 62.67
15 107.40 107.62 105.00
17 258.35 258.40 257.69

 

 

 

 

 

‘ data used for construction x-axis error bar

Table 47. Percent error of shelf life for non-fat dry milk in three diﬂ‘erent packages stored

 

 

 

 

 

 

 

at 25°C, 67% RH.
Mf LoAS LoAL LoES

5 38.4 38.46 38.18

6 26.19' 26.31' 28.43'

7 29.86' 2968‘ 32.22‘

8 36.40‘ 36.51' 38.40'

9 48.92 49.15 50.14

10 77.33 77.35 78.34

 

 

 

 

 

 

 

 

. o a
data used for constructlon x-ax15 error bar

The error that each parameter contributed to the total error at certain moisture content of

each package and storage condition are graphically presented in Figure 35 to 40.

126

 

 

100% -

 

 

Contribution to total error

 

 

 

Mf, %

 

 

 

Figure 35. Contribution of error ﬁom each model parameters to total error in shelf life of
non-fat dry milk packaged in small boxes made from material A and stored at

40°C, 83% RH (HiAS)

 

 

 

 

Contribution to total error

 

 

 

 

Mf,%

 

 

 

Figure 36. Contribution of error ﬁom each model parameters to total error in shelf life of
non-fat dry milk packaged in large boxes made from material A and stored at

40°C, 83% RH (HiAL)

127

 

rm—

 

 

100%

 

80% ~

60%~

40%-

Contribution to total error

20% 4

 

 

 

 

0% I T I I T
5 7 9 11 13 15 17

M, %

 

 

 

 

Figure 37. Contribution of error from each model parameters to total error in shelf life of
non-fat dry milk packaged in small boxes made from material E and stored at

40°C, 83% RH (HiES)

 

 

 

 

 

Contribution to total error

 

 

 

 

Mf,%

 

 

 

Figure 38. Contribution of error from each model parameters to total error in shelf life of
non-fat dry milk packaged in small boxes made ﬁ'om material A and stored at

25°C, 67% RH (LoAS)

128

 

 

 

100%

 

 

 

 

E 80%-
i a...-
.9
g 40%—
'5
§ 20% Wm
0% - 4 4
1 2 3 4 5 e 7

Mf,%

 

 

 

Figure 39. Contribution of error ﬁ'om each model parameters to total error in shelf life of
non-fat dry milk packaged in large boxes made ﬁ'om material A and stored at

25°C, 67% RH(LoAL)

 

 

 

 

 

 

 

 

 

100%
'- 80% 4
E
3
5 00%4
.9-
S
E 40% 4
'5

Wm
5 26%-
0% - m -
1 2 3 4 5 6 7
Mf, %

 

 

 

Figure 40. Contribution of error from each model parameters to total error in shelf life of
non-fat dry milk packaged in small boxes made from material E and stored at

25°C, 67% RH (LoES)

129

 

 

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