Prediction of WVTR with General Regression Models Kimmo Lahtinen - - PowerPoint PPT Presentation

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Prediction of WVTR with General Regression Models Kimmo Lahtinen - - PowerPoint PPT Presentation

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TAMPERE UNIVERSITY OF TECHNOLOGY I 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 Lahtinen Session 9.1 1. Introduction Target TARGET: To establish a practical, fast and

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TAMPERE UNIVERSITY OF TECHNOLOGY

I 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 Lahtinen Session 9.1

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1. Introduction

Target

  • TARGET: To establish a practical, fast and easy-to-use

computer-aided prediction model for water vapour barrier of extrusion coated paper

  • Computer-aided prediction model creates a base for

material selection cost estimation

  • ptimization
  • f a new packaging material.

Already existing packages: Modelling eases the load

  • f experimental testing.
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Regression models

  • Results in this study are based on statistical findings.

Experimental tests Regression 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

dx dc D J − =

( ) ( )

l p p P l p p DS l c c D J s

1 1 1

) ( − = − = − =

L p p P dt dQ A WVTR ) ( 1

1 2 −

= =

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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 Arrhenius

relationship as follows:

L p p P dt dQ A WVTR ) ( 1

1 2 −

= =

) / exp( RT E P P

p

− =

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  • 2. Materials and methods

Pilot line

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Materials

  • Modelled polymers

LDPE, density 923 kg/m3 HDPE, density 941 kg/m3 PP COC

  • 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. conditions

Series 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

R

2 = 0,9888

10 20 30 40 50 10 20 30 40 50 60 coating weight (g/m

2)

WVTR (g/m

2/24h)

LDPE

Standard tropical conditions 38°C, RH 90%

Applied method

  • Power law of

regression

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  • 3. Results

WVTR 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

  • f water (kg) and the

amount of dry air (kg).

  • T and RH determine

mixing ratio from the h,ω diagram 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)

( ) ( )

T p RH p T p

h h

' ' − = µ ω

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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 ratio

20 g/m2 LDPE coating

23°C y = 267,24x + 0,5836 R2 = 0,9887 30°C y = 404,66x - 0,2514 R2 = 0,9953 38°C y = 532,34x - 2,1323 R2 = 0,9961 45°C y = 659,17x - 6,5855 R2 = 0,9916

5 10 15 20 25 30 35 0,02 0,04 0,06 0,08 Mixing ratio WVTR (g/m

2/24h)

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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 + b4x1

2 + 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 + b4x2 2

0,279 0,170 0,222 0,320

y = b0 + b1x1 + b2x2 + b3x1x2 + b4x1

2 + b5x2 2

0,291 0,170 0,232 0,331

y = b0 + b1x1 + b2x1

2 + b3x2 2

0,519 0,463 0,434 0,313

y = b0 + b1x2 + b2x1

2 + 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 + b2x2 2

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 11 13 10 8 9 76 4 5 2 3 1

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

Reliability indicators SSE 0,857234 S 0,279160 S2 0,077930 R2 0,999216

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Step 2

  • Influence of coating weight

Coating weight has an inverse proportion on WVTR Thus for LDPE

( ) ( )

2 1 3

, 20 x x f x y =

( )

2 2 4 2 1 3 2 2 1 1 3

20 x b x b x b x b b x y + + + + =

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Step 3

  • Influence of multilayers
  • Provided that

All the P-values of the layers are independent of pressure and concentration There 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

P L P L P L P L

tot tot

+ + =

... 1 1 1 1 + + =

3 2 1

WVTR WVTR WVTR WVTRtot

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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

  • f 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 Enso Borealis Polymers Topas Advanced Polymers

Thank you! Questions please…