contain the maximum amount of water. The relative permeability varies between zero and one for each phase depending on the water content within the fibre walls. The total permeability is, hence, the product of the intrinsic and relative permeability.
Pore space in the paper model The penetrating airflow is modelled with the continuity and Navier stokes equations in COMSOL Multiphysics 6.1, see Eqs. (2) and (3),
{ 𝑘 𝑔 = 𝑘 𝑔𝑖 𝑘 𝑔𝑟 𝑘 𝑤 = 𝑘 𝑤𝑖 𝑘 𝑤𝑟
𝜕(𝜌𝑢 𝑖 ) 𝜕𝑥 𝑖
𝜕𝜌 𝜕𝑡
+
=0
(2)
(8)
𝜕(𝜌𝑢 𝑖 ) 𝜕𝑡
𝜕𝑢 𝑖 𝜕𝑥 𝑗
𝜕𝑝 𝜕𝑥 𝑖
𝜕 𝜕𝑥 𝑗
+𝜌𝑢 𝑗
=−
+
𝜏 𝑖𝑗 −𝜌𝑔 𝑖
The equations that describe the relative permeability for each phase are described in Eq. (9) and (10). [38]
(3)
where 𝜏 𝑖𝑗 is the deviatoric stress tensor which includes the bulk viscosity due to compressibility in the air, see Eq. (4).
{ 𝑘 𝑔𝑟 =1−1.1𝑆 𝑤 ,𝑖𝑓 𝑆 𝑤 < 1 1.1 𝑘 𝑔𝑟 =0 , 𝑖𝑓 𝑆 𝑤 > 1 1.1
(9)
3
𝑆 𝑤 −𝑆 𝑖𝑟 1−𝑆 𝑖𝑟
𝑘 𝑤𝑟 =(
)
,𝑖𝑓 𝑆 𝑤 > 𝑆 𝑖𝑟 , 𝑖𝑓 𝑆 𝑤 < 𝑆 𝑖𝑟
𝜕𝑢 𝑗 𝜕𝑥 𝑖
𝜕𝑢 𝑖 𝜕𝑥 𝑗
2 3
𝜕𝑢 𝑘 𝜕𝑥 𝑘
{
𝜏 𝑖𝑗 = 𝜇 (
+
−
𝜇
𝛿 𝑖𝑗 )
(10)
(4)
𝑘 𝑤𝑟 =0
Turbulence modelling was excluded from the simulations due to small length scales in the order of 10e -6 m and, hence, leading to low Reynolds numbers. The moisture transport in the air is described with a modified advection diffusion model shown in Eq. (5),
The intrinsic permeability for wood fibres are used from [38].
Fluid transport in and between wood fibres is described with the following equation
𝜕𝑊(𝜑 𝑣 ) 𝜕𝑡
𝜕𝜔 𝑣 𝜕𝑥 𝑗
𝜕𝑔 𝑤 𝜕𝑥 𝑗
𝜕𝜌 𝑙 𝜕𝑥 𝑗
𝜕𝑔 𝑙𝑐 𝜕𝑥 𝑗
+𝜌 𝑔 𝑢 𝑔,𝑗
+
+𝜌 𝑔 𝑢 𝑙,𝑗
+
= 𝐺
(11)
𝜕 2 𝑐 𝜕𝑥 𝑗
𝜕𝑐 𝜕𝑡
𝜕𝑐 𝜕𝑥 𝑗
𝑀 𝑣
+𝑀 𝑣 𝑢 𝑗
−𝑀 𝑣 𝐷 𝑣
2 = 𝐺
(5)
where 𝑔 𝑤 (kg/(m·s)), see Eq. (12), is the vapor transport in the gaseous phase in the wood fibre. As the Millington and Quirk equation is used calculate the effective diffusivity in see Eq. (12). The capillary flux term 𝑔 𝑙𝑐 (kg/(m·s)), see Eq. (13), is described as the relative humidity gradient of the moisture content and an added diffusivity term 𝐷 𝑤 (m 2 /s), see Eq. (14). [38] The moisture content, 𝑊(𝜑 𝑣 ) , is a function of the relative humidity, see Eq. (15).
where 𝑀 𝑣 (kg/mol) is the molar mass of water vapour, 𝐷 𝑣 (m 2 /s) is the vapor diffusion in air coefficient and 𝐺 (kg/m 3 s) is the moisture source or sink term. The concertation term in in Eq. (5) is related to the relative humidity with the vapor saturation concentration.
𝑐 = 𝜑 𝑣 𝑐 𝑠𝑎𝑡
(6)
𝜕𝜔 𝑣 𝜕𝑥 𝑗
𝑔 𝑤 =−𝜌 𝑔 𝐷 𝑒𝑓𝑓
(12)
Porous media modelling in wood fibre The wood fibres are typically hygroscopic materials and, as such, should be modelled as a porous media. The fibres contain two phases in equilibrium, which is the liquid water and the moist air. The driving force for transportation of each phase is vapor diffusion and convection in air, as well as capillary flow and convection for water. Assuming that the inertial forces in the fibre s are neglectable, Darcy’s correction model is used to relate pressure gradients to the volume averaged superficial velocity
𝜕𝑊(𝜑 𝑣 ) 𝜕𝜑 𝑣
𝜕𝜑 𝑣 𝜕𝑥 𝑗
𝑔 𝑙𝑐 =−𝐷 𝑤
∙
(13)
2𝑤(𝜑 𝑣 ) (1−𝜀𝑝)𝜌𝑓
(−2.8+
)
−8 𝑒
𝐷 𝑤 =1 ∙ 10
(14)
𝑊(𝜑 𝑣 )= 𝜀 𝑝 (𝜌 𝑙 𝑠 𝑙 +𝜌 𝑔 𝜔 𝑔 (1−𝑠 𝑙 ))
(15)
The liquid saturation, 𝑠 𝑙 , is a dimensionless parameter and it describes the amount of water left within the pores, 𝜀 𝑝 is the porosity of the wood fibres and 𝜔 𝑔 is vapour mass fraction. Model simplifications Some of the major assumptions this model relies on is: • Aligned fibre positions due to 2- dimensional representation of the paper sheet
𝜅 𝑖𝑗 𝜇
𝜕𝑝 𝜕𝑥 𝑖
〈𝑢 𝑖 〉 =−
(7)
where 𝜅 𝑖𝑗 (m 2 ) is permeability coefficient and 𝜇 (Pa·s) is the dynamic viscosity. The permeability is typically decomposed into intrinsic permeability and relative permeability for multiphase flow in porous media according to [38]. The intrinsic parameter represents the permeability of liquid or gas in an entirely saturated state, i.e. the pores in the fibre
3
Made with FlippingBook Online newsletter maker