This document describes a numerical model that couples mechanical and heat and mass transfer processes during manufacturing of medium density fiberboard (MDF) using hot pressing. The model includes: 1) Heat and mass transfer equations accounting for air, water vapor, energy, and resin polymerization kinetics. 2) A mechanical model using an aging linear elastic constitutive law to model compression of the mat. 3) Coupling of the mechanical and heat/mass transfer models using a finite element method with implicit time stepping. 4) Numerical results from the model show good agreement with laboratory experiments and provide temperature, moisture content, and density profiles throughout the pressing process.

1. COUPLED MECHANICAL
AND
HEAT AND MASS TRANSFER
NUMERICAL MODEL
FOR MDF HOT PRESSING PROCESS
Zanin Kavazović
Ph.D. Jean Deteix
Prof. Alain Cloutier
Prof. André Fortin

2. Physics involved:
 Heat & Mass Transfer
 Mechanical Compression of the Mat
 Resin Polymerization

3. OUTLINE
Heat & Mass Transfer Model
Mechanical Model
Their Coupling & FEM & 3D Moving Domain
Numerical Results (Graphs & Movies)
Conclusions & Perspectives

4. Heat & Mass Transfer
Some of the Literature: Conservation Principles:
 Humphrey & Bolton  Mass of AIR
(1982, 1989a,b,c,d)  Mass of Water Vapor
 Thömen & Humphrey et al.
 Energy
(2000, 2003,2006, 2008)
 Zombori et al.
(2001,2003,2004)  Sorption Model (EMC-RH-T)
(Malmquist; Vidal&Cloutier 2005)
 Dai et al .
(2001,2004,2005,2006)  Resin Curing Kinetics
(Xing&Riedl 2004)
 Carvalho & Costa et al.
(2001,2003,2006)  Porosité  (Belley 2009)
 Pereira et al. (2006)  Local Thermodynamic
 Nigro & Storti (2006) Equilibrium

5. Heat & Mass Transfer Model
AIR D( a )   a   M  
     K p   P       a Deff   Pa   0
Dt  
     RT  
  
Bulk Flow Molecular Diffusion
D( v  )    M   D( M )
VAPOR      v K p   P       v Deff   Pv    OD
Dt       RT    Dt
Evaporation Rate
D D
 OD CMatT   CaT  a   CvT  v  
Dt Dt
 T  
   K T   T        a Ca   v Cv  K p    P 
   
ENERGY
 C M    C M  
    a a Deff    Pa       v v D eff    Pv 
 R    R  
D(M )
 Qr   H fg   Cbw  Cv  T   OD
  Dt

6. Heat & Mass Transfer Model
Solution Strategy
 The 3 conservation equations:
– form a coupled system of 3 PDEs
– are expressed in terms of 3 state variables (Pa, Pv, T)
 3D Moving Domain
 Space Discretization : Finite Element Method
 Time Discretization : Implicit Second Order Backward scheme
 Time Step of 0.5 s was used
 The 3 coupled equations were solved simultaneously as a system
 Newton’s method was employed
 All material properties were updated after each nonlinear iteration

7. Mechanical Model
• Ageing Linear Elastic Model
• Composite Constitutive Law (Bazant 1979, 1993; Dubois 2005)
 D DE
E: 0 (Hardening, Tangent Law)
D 
 Dt Dt

Dt  D DE DE
E:  : 0 (Softening, Hooke's Law)

 Dt Dt Dt
• Quasi-static Incremental Formulation
U 0  0;  0  0;  0  0
 d iv     d iv ( N )
U N 1  U N  U
 N 1   N  
      U     U       U   
1 T
2   N 1   N  
  E N 1 :    E  :  N

E  EN 1  EN

8. Heat & Mass Coupled with Mechanics
Beginning of a Time Step
 Heat & Mass model is solved first
• Pa, Pv, T, M, and Degree of Resin Cure are calculated
• Update of Mechanical Properties
• Displacement Increment is imposed at the top surface
 Mechanical model is solved
• U is calculated
• Update of the Mesh, Element Sizes and the Mat Density
• Update of Heat & Mass transfer Material Properties
End of a Time Step

9. 3D Moving Domain & Temperature

10. Comparison Lab - Numerical 3D:
Temperature
ERROR LESS than 10%

11. Comparison Lab - Numerical 3D:
Gas Pressure
STD DEV/MEAN = 6%
ERROR LESS than 15%

12. Numerical Results:
Predicted Oven-dry Density Profile

13. TEMPERATURE

14. MOISTURE CONTENT

15. Partial VAPOR PRESSURE

16. Conclusions & Perspectives
Coupling of Mechanical and Heat & Mass
Transfer Models gives good and consistent results
FEM & Implicit Time Scheme allow the use of
larger time steps (lower computational cost)
In the future:
Consider a Viscoelastic Mechanical Model
Take into account Press Opening (Venting)
Introduce a Plastic Component into the Model
Large Deformations

17. Département de mathématiques et de statistique

18. Programmes de fermeture
en réserve
 7 différents programmes de fermeture de presse
 Calculs effectués sur plusieurs maillages:
16x16; 32x16; 32x32; 48x32; 64x32; 128x64;

19. Programmes de fermeture

20. Programmes de fermeture