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

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 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 � optimization of a new packaging material. � Already existing packages: Modelling eases the load of experimental testing. Paper x.y Speakers name 2

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. Paper x.y Speakers name 3

Background of water vapour permeation • Mathematical treatment of water vapour transmission rate (WVTR) dc = − J D � Fick’s first law: dx � Steady state diffusion → D does not depend on penetrant’s concentration. � The product DS is called coefficient of permeation (P) ( ) ( ) − − − D ( c c ) DS p p P p p � Henry’s law: c = Sp → = = = 1 0 1 0 1 0 J s l l l 2 − P ( p p ) � → The determination of WVTR: 1 dQ = = 1 WVTR A dt L � Unit: g/m 2 /24h Paper x.y Speakers name 4

Three external factors influencing moisture barrier of polymer film • temperature; effect on P 2 − 1 dQ P ( p p ) = = 1 WVTR • humidity; effect on (p 2 -p 1 ) A dt L • thickness; effect on L • The effect of temperature is controlled by the Arrhenius relationship as follows: = − P P exp( E / RT ) 0 p Paper x.y Speakers name 5

2. Materials and methods Pilot line Paper x.y Speakers name 6

Materials • Modelled polymers � LDPE, density 923 kg/m 3 � HDPE, density 941 kg/m 3 � PP � COC • Paper � One-side pigment coated paper to offer a smooth substrate for coating polymer (practically no influence on WVTR) Paper x.y Speakers name 7

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 Paper x.y Speakers name 8

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 23 ° C 50% 30 ° C 50% 38 ° C 50% 45 ° C 50% Series 1 23 ° C 63% 30 ° C 63% 38 ° C 63% 45 ° C 63% Series 2 23 ° C 77% 30 ° C 77% 38 ° C 77% 45 ° C 77% Series 3 23 ° C 90% 30 ° C 90% 38 ° C 90% 45 ° C 90% Series 4 Paper x.y Speakers name 9

WVTRs were measured for exact 20 g/m 2 coating weight to achieve an accurate comparison between the results in different atmospheric conditions. LDPE Standard tropical conditions 38 ° C, RH 90% Applied method 50 • Power law of 40 2 /24h) regression 30 WVTR (g/m -1,0349 y = 423,67x 20 2 = 0,9888 R 10 0 0 10 20 30 40 50 60 2 ) coating weight (g/m Paper x.y Speakers name 10

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 WVTR results for 20 g/m 2 LDPE coating 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 35 30 25 20 15 10 5 Paper x.y Speakers name 11

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, ω diagram of humid air. (basic thermodynamics) Paper x.y Speakers name 12

• Mixing ratio as a function of T and RH: ( ) p ' T ω = µ h ( ) p − p ' T RH h where µ = M H20 / M air = 18,015/28,964 = 0,6220, p = normal air pressure = 1 bar and p h ’(T) = saturated vapour pressure (function of temperature) Paper x.y Speakers name 13

WVTR as a function of T and ω Observations: WVTR vs. mixing ratio 1) Linear correlation 20 g/m 2 LDPE coating between WVTR and 35 45°C mixing ratio y = 659,17x - 6,5855 30 R 2 = 0,9916 2) Temperature 2 /24h) 25 influences slightly on 38°C 20 WVTR (g/m y = 532,34x - 2,1323 R 2 = 0,9961 the slope of the 15 WVTR-mixing ratio 30°C 10 y = 404,66x - 0,2514 R 2 = 0,9953 curve 5 23°C y = 267,24x + 0,5836 0 3) Most likely suitable R 2 = 0,9887 0 0,02 0,04 0,06 0,08 for regression Mixing ratio estimation Paper x.y Speakers name 14

Model development • Step by step scheme for calculations 1) The influence of: WVTR of 20 g/m 2 single layer i. Temperature Regression ii. Mixing ratio 2) WVTR of a single layer The influence of iii. coating weight 3) WVTR of a multilayer structure The influence of multilayers Paper x.y Speakers name 15

• We define temperature, mixing ratio and coating weight as independent variables (x 1 , x 2 and x 3 , respectively) and WVTR as a dependent variable (y) • Step 1: � Several first- and second-order models were tested to obtain results for 20 g/m 2 single layer. � Equation including all possible terms: 2 + b 5 x 2 2 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 x 2 + b 4 x 1 � We apply a spreadsheet or statistical computer program to solve the b-values and reliabilities of different models. Paper x.y Speakers name 16

List of the tested models and the corresponding standard errors (the best values bolded) Model Std error Std error Std error Std error LDPE HDPE PP COC y = b 0 + b 1 x 1 + b 2 x 2 0,897 0,590 0,590 0,649 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 x 2 0,689 0,435 0,355 0,447 2 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 0,908 0,589 0,623 0,626 2 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 2 0,353 0,203 0,234 0,310 2 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 x 2 + b 4 x 1 0,440 0,298 0,295 0,346 2 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 x 2 + b 4 x 2 0,284 0,223 0,323 0,162 2 + b 4 x 2 2 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 0,279 0,170 0,222 0,320 2 + b 5 x 2 2 y = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 1 x 2 + b 4 x 1 0,291 0,170 0,232 0,331 2 + b 3 x 2 2 y = b 0 + b 1 x 1 + b 2 x 1 0,519 0,463 0,434 0,313 2 + b 3 x 2 2 y = b 0 + b 1 x 2 + b 2 x 1 0,410 0,238 0,214 0,313 2 y = b 0 + b 1 x 1 + b 2 x 1 x 2 + b 3 x 2 0,660 0,520 0,320 0,310 2 y = b 0 + b 1 x 2 + b 2 x 1 x 2 + b 3 x 2 0,476 0,286 0,278 0,316 2 + b 2 x 2 2 y = b 0 + b 1 x 1 0,782 0,627 0,483 0,303 Paper x.y Speakers name 17

Results of step 1 WVTR as a function of T and ω for 20 g/m 2 LDPE coating 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 Reliability indicators 16 SSE 0,857234 15 S 0,279160 14 S 2 0,077930 12 13 11 R 2 0,999216 10 9 8 76 5 40 35 4 3 2 30 1 25 20 15 10 5 Paper x.y Speakers name 18

Step 2 • Influence of coating weight � Coating weight has an inverse proportion on WVTR � Thus 20 ( ( ) ) y = f x , x 1 2 x 3 � for LDPE ( ) 20 = + + + + 2 2 y b b x b x b x b x 0 1 1 2 2 3 1 4 2 x 3 Paper x.y Speakers name 19

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 L L L L = + + tot 1 2 3 Multilayer film obeys the equation ... P P P P tot 1 2 3 • As partial pressure difference stays as a constant in the WVTR test 1 1 1 1 = + + ... WVTR tot WVTR WVTR WVTR 1 2 3 Paper x.y Speakers name 20

4. The end result • A Labview based WVTR estimation computer program User-selected input Computer aided values: results: – Temperature (T) – WVTR of chosen structure in – Relative humidity selected conditions (RH) – 3D graphs; WVTR – Polymers of layers of chosen structure 1-5 and the in different corresponding conditions coating weights – WVTR of chosen structure in standard conditions Paper x.y Speakers name 21

Report of the WVTR calculation program Paper x.y Speakers name 22

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… Paper x.y Speakers name 23