TAMPERE UNIVERSITY OF TECHNOLOGYI n s t i t u t e o f P a p e r C o n v e r t i n g

Prediction of WVTR with General Regression Models

Kimmo LahtinenSession 9.1

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1. IntroductionTarget

• TARGET: To establish a practical, fast and easy-to-use computer-aided prediction model for water vapourbarrier of extrusion coated paper

• Computer-aided prediction model creates a base formaterial selectioncost estimationoptimization

of a new packaging material.Already existing packages: Modelling eases the load of experimental testing.

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Regression models

• Results in this study are based on statistical findings.Experimental testsRegression analysis

• Regression models are sort of “black-box type” models.No theoretical linkages between variables

• In technology, regression models are used when more deterministic models are not efficient due to complexity and disturbances.

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Background of water vapour permeation

• Mathematical treatment of water vapour transmission rate (WVTR)

Fick’s first law: Steady state diffusion → D does not depend on penetrant’s concentration.The product DS is called coefficient of permeation (P)Henry’s law: c = Sp →

→ The determination of WVTR: Unit: g/m2/24h

dxdcDJ −=

( ) ( )l

ppPl

ppDSl

ccDJ s

010101 )( −=

−=

−=

LppP

dtdQ

AWVTR

)(1 12 −==

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Three external factors influencing moisture barrier of polymer film

• temperature; effect on P• humidity; effect on (p2-p1)• thickness; effect on L

• The effect of temperature is controlled by the Arrheniusrelationship as follows:

LppP

dtdQ

AWVTR )(1 12 −==

)/exp(0 RTEPP p−=

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2. Materials and methodsPilot line

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Materials

• Modelled polymersLDPE, density 923 kg/m3

HDPE, density 941 kg/m3

PPCOC

• Paper One-side pigment coated paper to offer a smooth substrate for coating polymer (practically no influence on WVTR)

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WVTR test method

• Cup method (SCAN-P22:68)

• The advantage: capable to carry multitude of samples at the same time

• Accurate enough for a statistical study

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Test series

• Regression modelling requires extensive experimental testing for statistics.

5 set points with different coating weights for each coating.4 parallel measurements with each coating weight giving 20 results total for each polymer.16 different atmospheric conditions (T and RH):

1. conditions 2. conditions 3. conditions 4. conditionsSeries 1 23°C 50% 30°C 50% 38°C 50% 45°C 50%Series 2 23°C 63% 30°C 63% 38°C 63% 45°C 63%Series 3 23°C 77% 30°C 77% 38°C 77% 45°C 77%Series 4 23°C 90% 30°C 90% 38°C 90% 45°C 90%

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WVTRs were measured for exact 20 g/m2 coating weight to achieve an accurate comparison between the results in different atmospheric conditions.

y = 423,67x-1,0349

R2 = 0,9888

0

10

20

30

40

50

0 10 20 30 40 50 60

coating weight (g/m2)

WVT

R (g

/m2 /2

4h)

LDPEStandard tropical conditions

38°C, RH 90%

Applied method• Power law of

regression

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3. ResultsWVTR as a function of T and RH

3D Surf ace Plot (Spreads heet2.s ta 13v *16c)

WVTR = Distance Weighted Least Squares

35 30 25 20 15 10 5

50% RH 63% RH 77% RH 90% RH 23°C 3,03 3,45 4,30 4,89 30°C 5,11 6,86 8,16 9,74 38°C 9,41 12,13 15,37 19,08 45°C 14,74 18,92 25,73 32,59

WVTR results for 20 g/m2 LDPE coating

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Definition of mixing ratio• Relative humidity is not

the actual water concentration of surroundings.

• Mixing ratio (ω) is defined as the ratio of the amount of water (kg) and the amount of dry air (kg).

• T and RH determine mixing ratio from the h,ωdiagram of humid air. (basic thermodynamics)

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• Mixing ratio as a function of T and RH:

where µ = MH20 / Mair = 18,015/28,964 = 0,6220, p = normal air pressure = 1 bar and ph’(T) = saturated vapour pressure (function of temperature)

( )( )TpRH

pTp

h

h

'

'

−= µω

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WVTR as a function of T and ω

Observations:1) Linear correlation

between WVTR and mixing ratio

2) Temperature influences slightly on the slope of the WVTR-mixing ratio curve

3) Most likely suitable for regression estimation

WVTR vs. mixing ratio20 g/m2 LDPE coating

23°Cy = 267,24x + 0,5836

R2 = 0,9887

30°Cy = 404,66x - 0,2514

R2 = 0,9953

38°Cy = 532,34x - 2,1323

R2 = 0,9961

45°Cy = 659,17x - 6,5855

R2 = 0,9916

0

5

10

15

20

25

30

35

0 0,02 0,04 0,06 0,08

Mixing ratio

WVT

R (g

/m2 /2

4h)

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Model development

• Step by step scheme for calculations

Regression

1)

2)

The influence of:

i. Temperature

ii. Mixing ratio

WVTR of 20 g/m2 single layer

The influence of iii. coating weight

3)The influence of multilayers

WVTR of a single layer

WVTR of a multilayer structure

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• We define temperature, mixing ratio and coating weight as independent variables (x1, x2 and x3, respectively) and WVTR as a dependent variable (y)

• Step 1:Several first- and second-order models were tested to obtain results for 20 g/m2 single layer.Equation including all possible terms:

y = b0 + b1x1 + b2x2 + b3x1x2 + b4x12 + b5x2

2

We apply a spreadsheet or statistical computer program to solve the b-values and reliabilities of different models.

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List of the tested models and the corresponding standard errors (the best values bolded)

Model Std error LDPE

Std error HDPE

Std error PP

Std error COC

y = b0 + b1x1 + b2x2 0,897 0,590 0,590 0,649 y = b0 + b1x1 + b2x2 + b3x1x2 0,689 0,435 0,355 0,447 y = b0 + b1x1 + b2x2 + b3x1

2 0,908 0,589 0,623 0,626 y = b0 + b1x1 + b2x2 + b3x2

2 0,353 0,203 0,234 0,310 y = b0 + b1x1 + b2x2 + b3x1x2 + b4x1

2 0,440 0,298 0,295 0,346 y = b0 + b1x1 + b2x2 + b3x1x2 + b4x2

2 0,284 0,162 0,223 0,323 y = b0 + b1x1 + b2x2 + b3x1

2 + b4x22 0,279 0,170 0,222 0,320

y = b0 + b1x1 + b2x2 + b3x1x2 + b4x12 + b5x2

2 0,291 0,170 0,232 0,331 y = b0 + b1x1 + b2x1

2 + b3x22 0,519 0,463 0,434 0,313

y = b0 + b1x2 + b2x12 + b3x2

2 0,410 0,238 0,214 0,313 y = b0 + b1x1 + b2x1x2 + b3x2

2 0,660 0,520 0,320 0,310 y = b0 + b1x2 + b2x1x2 + b3x2

2 0,476 0,286 0,278 0,316 y = b0 + b1x1

2 + b2x22 0,782 0,627 0,483 0,303

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Results of step 1

Model: WVTR=b0+b1*Temp+b2*Mix+b3*Temp*Temp+b4*Mix*Mix

z=(-6,9493)+(,427809)*x+(203,756)*y +(- ,00454)*x*x+(5101,53)*y *y

40 35 30 25 20 15 10 5

16

15

14

12

1113

10

8

9

76

4

5

23 1

WVTR as a function of T and ω for 20 g/m2 LDPE coating

Reliability indicatorsSSE 0,857234S 0,279160S2 0,077930R2 0,999216

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Step 2

• Influence of coating weightCoating weight has an inverse proportion on WVTRThus

for LDPE

( )( )213

,20 xxfx

y =

( )224

21322110

3

20 xbxbxbxbbx

y ++++=

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Step 3

• Influence of multilayers• Provided that

All the P-values of the layers are independent of pressure and concentrationThere are no barriers to diffusion due to interfacial phenomena between layers

Multilayer film obeys the equation

• As partial pressure difference stays as a constant in the WVTR test

...3

3

2

2

1

1

PL

PL

PL

PL

tot

tot ++=

...1111++=

321 WVTRWVTRWVTRWVTRtot

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4. The end result• A Labview based WVTR estimation computer program

User-selected input values:– Temperature (T) – Relative humidity

(RH)– Polymers of layers

1-5 and the corresponding coating weights

Computer aided results:– WVTR of chosen

structure in selected conditions

– 3D graphs; WVTR of chosen structure in different conditions

– WVTR of chosen structure in standard conditions

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Report of the WVTR calculation program

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5. Acknowledgements

• Many thanks to the companies that kindly arranged their materials for the study

Stora EnsoBorealis PolymersTopas Advanced Polymers

Thank you! Questions please…