Prediction of WVTR with General Regression Models - · PDF filePrediction of WVTR with General...

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

Paper x.y Speakers name 2

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.


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.


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 −==


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−=


2. Materials and methodsPilot line


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)


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


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%


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


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


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)


• 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


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)


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


• 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.


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


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


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 ++++=


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


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


Report of the WVTR calculation program


5. Acknowledgements

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

Stora EnsoBorealis PolymersTopas Advanced Polymers

Thank you! Questions please…

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