Chapter 3: Mathematical Formulations Chapter Three: Mathematical Formulations Introduction This chapter discusses the mathematical considerations and formulation of a heat and mass transfer in wood Wood is treated as a porous slab, and two heat and mass transfer models are presented for low temperature heating of wood The numerical implementation using Fluent® version 6.3 is also addressed 3.1 Heat and mass transfer for porous slab This study develops a porous model for heat and mass transfer in wood because wood by nature is a porous structure This porous model is developed based on conservation equations which are built on the assumptions that all phases are continuous, instead of discrete pore model, so that a better mechanistic understanding can be achieved of the mass, energy and momentum transport in porous model for wood drying than from solid slab In this work, Darcy’s law is modified by ways of changes made to the momentum equation in order to account for the effects of inertia and that of boundary on flow field Two porous models are proposed: First model considers liquid water transport only and with surface evaporation; the second model introduces a combined moisture flow of liquid water and vapour, assuming moisture-vapour equilibrium and contains an internal 78 Chapter 3: Mathematical Formulations evaporation term The first model represents the initial drying phase of low-temperature drying model while as the second model emulates the extended drying phase where the surface evaporation has retreated inwards, creating an internal evaporation zone The salient features of the porous model are summarised by schematic diagram below CONTINUED HEATING LOW TEMPERATURE LONG TERM HEATING IN WOOD POROUS MODEL AT INITIAL DRYING PHASE POROUS MODEL AT EXTENDED DRYING PHASE INITIAL DRYING PHASE FINDINGS  surface evaporation   liquid water movement only   slow and steady velocity profile   constant temperature profile  EXTENDED DRYING PHASE FINDINGS  internal evaporation   combined liquid and vapour  transport  moisture-vapour equilibrium   rapid and erratic velocity profile   S-curve temperature profile  COUPLED VELOCITY AND TEMPERATURE DEVELOPMENT IN SELF-HEATING Figure 3.1: Salient features of porous models for low-temperature long-term heating in wood 79 Chapter 3: Mathematical Formulations 3.1.1 Assumptions for heat and mass transfer in porous slab The heat and mass transfer model in the porous slab is modelled as onedimensional flow where it could be schematically represented as follow Figure 3.2 The one-dimensional flow in porous model Insulated surface Exposed surface Convective heat loss Mw Mv  ′′ qe Radiative heat loss X The following assumptions are introduced alongside the development of the models in order to make the mathematical treatment tractable Wood is modeled as a porous slab However, all structural changes such as swelling, shrinkage, crack formation, are negligible during drying Liquid and gaseous transport is dominant and responsible for internal heat transfer, through diffusion and convection Pyrolysis is negligible at low temperature drying 80 Chapter 3: Mathematical Formulations Water and vapour fluxes are combined into a total moisture flux ( M ) with effective diffusivity, which includes two phases and two transport mechanisms Evaporation of moisture is sufficiently rapid to attain thermodynamic equilibrium Movement of both water (W ) and combined moisture flow ( M ) are taken into account using Darcy’s law Escaping vapour is in thermal equilibrium with the solid matrix The out flowing vapour does not withdraw sufficient energy from the solid to affect the solid temperature In other words, the mass flux is slow 3.1.2 Mathematical considerations for liquid and vapour transport In this work, a combined moisture flow of liquid and vapour is introduced for the second porous model when the internal evaporation is considered Other than simplicity and convenience, there is also a consideration to model wood as a homogenous model though by nature it is heterogeneous Vortmeyer, Dietrich and Ring (1974) pointed out that a reliable representation of a heterogeneous media can be achieved by a homogeneous model with the introduction of effective transport coefficients This model embodies the concept of effective transport properties through the combined moisture flow approach, enabling the wood slab to at least attain pseudo-homogeneity To combine the liquid and vapour flow, a model for typical water ( W ) and vapour ( V ) conservation laws is first considered 81 Chapter 3: Mathematical Formulations ∂W ∂ ∂W ∂ = ( Dw ) + ( ρ wcwvw )T − I ∂t ∂x ∂t ∂x (3.1) ∂V ∂ ∂V ∂ = ( DV ) + ( ρv cv vv )T + I ∂t ∂x ∂t ∂x (3.2) Combining Equations (3.1) and (3.2) leads to ∂W ∂V ∂ ∂W ∂ ∂V ∂ ∂ = + ( Dw ) + ( Dv ) + ( ρ wcwvw )T + ( ρv cv vv )T ∂t ∂t ∂x ∂x ∂x ∂x ∂x ∂x (3.3) where the evaporation term ( I ) has been eliminated Total moisture content as mass of water per unit mass of dry matter M is defined as M (V + W ) / ρ s = (3.4) and a total moisture flux (diffusive and convective) is written as ∂M ∂ ∂M ∂ = ( Dm ) + ( ρ m cm vm )T ∂t ∂x ∂x ∂x (3.5) A new parameter Dm , known as the effective diffusivity is introduced, combining water diffusivity Dw and vapour diffusivity Dv The concept of effective transport property lump the two phases and different transport mechanisms together as one 82 Chapter 3: Mathematical Formulations 3.1.3 Mathematical model for liquid water transport with surface evaporation To account for heat and mass transfer in a porous slab for drying in wood, a mathematical model is developed after Gatica, Viljoen and Hlavacek (1989) for flow in a packed bed The first model is presented for modelling liquid water transport within the porous slab with surface evaporation i.e evaporation term is eliminated from the heat balance It is constructed to emulate the initial drying phase of low-temperature drying In this model of initial drying phase, evaporation is deemed to occur at evaporation temperature at 100°C The model is presented first in vector form, and then Cartesiantensor form of equations for one-dimensional flow Initial Drying Phase (IDP) Model 3.1.3.1 Vector representation Energy Balance  ∂ ( ρ cT ) = −ϕ∇ ⋅ (uρ wcwT ) + ∇ ⋅ keff ⋅∇T ∂t (3.6) ( ) where keff is the effective thermal conductivity, ρ c is the average heat capacity of the solid and fluid mixture medium The effective thermal conductivity is defined as keff = ϕ kw + (1 − ϕ )ks (3.7) 83 Chapter 3: Mathematical Formulations The average heat capacity is defined as ρ c = ρ s (1 − ϕ )cs + ρ wϕ cw (3.8) Mass Balance ϕ  ∂W = −ϕ∇ ⋅ ( M w u ) + ϕ∇ ⋅ Dw ⋅∇M w ∂t (3.9) where the internal term of the rate of evaporation is omitted Initial and boundary conditions for energy equation Boundary condition (x = 0) −keff ∂T  ′′ =qe − h(T − T∞ ) − εσ (T − T∞4 ) ∂x (3.10) Boundary condition (x →∞) −keff ∂T = ∂x (3.11) 84 Chapter 3: Mathematical Formulations Initial condition (t = 0, x≥ 0) T = T∞ (3.12) Initial and boundary conditions for conservation of mass Boundary condition (x = 0) − ρ w Dw ∂W hD (0, t ) = ( M w (0, t ) − M w,∞ (t )) ∂x (3.13) Boundary condition (x →∞) − Dw ∂W = ∂t (3.14) Initial condition (t = 0, x≥ 0) M w,0 = M w,∞ (3.15) 85 Chapter 3: Mathematical Formulations 3.1.3.2 Cartesian representation Energy Balance ∂ ∂ ∂T ∂ = ( ρ cT ) (keff ) + ϕ ( ρ wcwuw )T ∂t ∂x ∂x ∂x (3.16) Mass Balance ∂W ∂ 2W ∂ ∂W = ϕ Dw + (ϕ uw ) ∂t ∂x ∂x ∂x (3.17) The initial and boundary equations for the respective energy and mass balance are the same as that outlined in Equations (3.10) to (3.15), and will not be repeated here 3.1.4 Mathematical model for combined transport with internal evaporation The second model combines gaseous and liquid transport as one phase flow with effective diffusivity as discussed in Section 3.1.3 Extended drying phase is defined when the evaporation front recesses into the domain, as heating continues when the temperature has reached 100°C An internal evaporation term is introduced into the heat balance The model is presented first in vector form, and then Cartesian-tensor form of equations for one-dimensional flow 86 Chapter 3: Mathematical Formulations Extended Drying Phase (EDP) Model 3.1.4.1 Vector representation Energy Balance  ∂ ( ρ cT ) = −ϕ∇ ⋅ (uρ m cmT ) + ∇ ⋅ keff ⋅∇T + (1 − ϕ ) ρ s (−∆H ev Rev ) ∂t (3.18) ( ) where keff is the effective thermal conductivity, ρ c is the average heat capacity of the solid and fluid mixture medium The effective thermal conductivity is defined as keff= ϕ km + (1 − ϕ )k s (3.19) The average heat capacity is defined as ρ c = ρ s (1 − ϕ )cs + ρ mϕ cm (3.20) Mass Balance ϕ  ∂M = −ϕ∇ ⋅ ( M m u ) + ϕ∇ ⋅ Dm ⋅∇M m − (1 − ϕ ) ρ s Rev ∂t (3.21) An internal evaporation term is re-introduced into the energy balance and the rate of evaporation also now appears in the mass balance 87 Chapter 3: Mathematical Formulations Initial and boundary conditions for energy equation Boundary condition (x = 0) −keff ∂T  ′′ =qe − h(T − T∞ ) − εσ (T − T∞4 ) ∂x (3.22) Boundary condition (x →∞) −keff ∂T = ∂x (3.23) Initial condition (t = 0, x≥ 0) T = T∞ (3.24) Initial and boundary conditions for conservation of mass Boundary condition (x = 0) − ρ m Dm ∂M hD (0, t ) = ( M m (0, t ) − M m ,∞ (t )) ∂x (3.25) Boundary condition (x →∞) 88 Chapter 3: Mathematical Formulations − Dm ∂M = ∂t (3.26) Initial condition (t = 0, x≥ 0) M m ,0 = M m ,∞ (3.27) 3.1.4.2 Cartesian representation Energy Balance ∂ ∂ ∂T ∂ = ( ρ cT ) (keff ) + ϕ ( ρ m cmum )T ∂t ∂x ∂x ∂x (3.28) Mass Balance ∂M ∂2M ∂ ∂M = ϕ Dm + (ϕ um ) ∂t ∂x ∂x ∂x (3.29) The initial and boundary equations for the respective energy and mass balance are the same as that outlined in Equations (3.22) to (3.27), and will not be repeated here 89 Chapter 3: Mathematical Formulations Equilibrium approach is adopted in the low-drying model, and the equilibrium between liquid and vapour is assumed to reach instantaneously, so that Pv = Psat (T ) (3.30) in the presence of liquid water A simple analytical expression is used to relate the saturation pressure to temperature (Sahota and Pagni 1979) The expression has the following form: Psat (T ) = CT − B / Rv exp(− A ) RvT (3.31) where= 3.18 ×103 kJ kg −1 , B = 2.5kJ kg −1 and= 6.05 ×1026 Nm −2 A C 3.1.5 Numerical Implementation Numerical implementation of both heat transfer in the pure thermal model and the heat and mass transfer in the porous model were carried out using Fluent® version 6.3 Fluent is a general purpose Computational Fluid Dynamics (CFD) model, which solves the Navier-Stokes equations via control volume approach The main difference between a general purpose CFD model, such as Fluent and CFX developed by ANSYS, Inc and PHOENICS marketed by CHAM limited, as compared to the Fire Dynamics Simulator (FDS) which is a more specific fire field model, is that in FDS model, the equations governing the transport of mass, momentum, and energy by the fire-induced flows are 90 Chapter 3: Mathematical Formulations simplified to effectively and efficiently solve the fire scenarios of interest (McGrattan K 2004) Fluent is first applied in this study to solve for heat transfer in wood as a solid slab, where only heat conduction problem is solved; no flow equations are involved The temperature rise is simulated in order to derive the ignition temperature and critical heat flux The details on pure thermal model are discussed in Chapter 4; the data on ignition temperature is tabulated in Chapter 4, and the graphical derivation of critical heat flux is found in Chapter The heat transfer in this “pure thermal model” is modelled as simple heat conduction, omitting convection flux terms, as well as heat source terms such as pyrolysis and evaporation Fluent is chosen as a numerical implementation tool mainly because of its capability to deal with inertial losses in fluid flow in porous medium The importance and the capability of Fluent to handle Darcy’s law in porous medium arises from the need to address modification to Darcy’s law in the simulation of heat and mass transfer in the porous slab developed in this Chapter 3.1.5.1 Modifications to Darcy’s Law in Porous Medium Wood is treated as a porous medium in the study of low temperature heating Darcy’s law has been used extensively to predict flow through porous medium In a laminar flow, the flow distribution is assumed to be well represented by a linear relation between the 91 Chapter 3: Mathematical Formulations pressure drop and the fluid velocity as u = −k D ∂P ∂x , where k D is the Darcy’s coefficient relevant to the particular type of fluid However, Darcy’s law is insufficient to account for the effect of boundaries on the flow field and the increasing importance of the inertial effects as the flow speed increases Modifications to Darcy’s law are necessary to overcome the aforesaid problems It has been proposed that if fluid obeys the Boussinesq’s approximation, the flow field will indeed be governed by the Darcy-Oberbeck-Boussinesq model for flow through a porous medium (Joseph 1976) as   µ   ∂u = −∇p − ρ m [γ (T − T∞ ] g − m u ρm ∂t κ (3.32)  ∇ ⋅u =0 (3.33) where p is the static pressure To incorporate the effects of the boundaries on the flow field and to account for the inertial effects as flow speed increases, inertial effects are added as a sink term to the momentum transfer (Choudhary, Propster and Szekely 1976) The Darcy’s law therefore becomes modified as   µ   ∂u   ρ m ( + u ⋅∇u ) = −∇p − ρ m [γ (T − T∞ ] g − m u ∂t ϕ κ (3.34) 92 Chapter 3: Mathematical Formulations   where the inertial forces are represented by the term u ⋅∇u However, Beck (1972) found   that the inclusion of this term u ⋅∇u may lead to inconsistencies between boundary conditions and governing equations, even though this term arises from a formal volume averaging in the point field equations (Drew and Segel 1971) To resolve inconsistencies,   the inertial effect is accounted for by including a term in the form of u ⋅ u known as Forchheimer’s modification to Darcy’s law (Irmay 1958) Following Forchheimer (1901) proposal to adding higher order terms to the relation between pressure drop and fluid velocity, the Darcy’s law becomes “modified” as below: −kD ∂p + a1u + a2u = ∂x Forchheimer’s modification in the one dimensional flow as −k D (3.35) ∂p =u + a2u shown in a1 ∂x Equation (3.35) can be conveniently formalized for two and three dimensions as  µ  ρ   −∇p − ρ m g + m v + m bu u = κ κ (3.36) where b denotes a matrix structure property associated with inertia effects Introducing the above modified Darcy’s law in equation (3.36) into the Darcy-Oberbeck-Boussinesq flow equation as shown in equation (3.32), the overall momentum equation incorporating the inertial effects through modified Darcy’s law therefore becomes 93 Chapter 3: Mathematical Formulations   µ  ρ   ∂u = −∇p − ρ m [γ (T − T∞ ] g − m v − m bu u ρm ∂t κ κ (3.37) Fluent addresses the addition of inertial losses in a porous medium as a momentum source term The pressure drop, governed by Darcy’s law, is also treated as a momentum source term The overall source term therefore consists of two parts: a viscous loss term (Darcy’s law in porous medium) and an inertial loss term Therefore, in a simple homogenous porous medium representation, this momentum sink is represented as follow Si = ui + C2 −( k D ρ ui ) (3.38) Where the first term on the right hand side of equation (3.38) is the viscous loss term given by Darcy’s law, and the second term , C2 ρ ui , is the inertial loss term C2 is the inertial resistance factor in the inertial loss term Fluent therefore is able to provide for modification to Darcy’s law through the constant C2 that providing a correction for inertial losses in the flow through the porous medium In a laminar flow where the pressure drop is typically proportional to velocity, the inertial loss term C2 ρ ui is simply “switched off” by taking C2 to be zero Various methods of computing C2 are given in the Fluent 6.3 user guide 3.1.5.2 Porosity of wood 94 Chapter 3: Mathematical Formulations Porosity in wood slab is determined by designating the volume fraction of fluid within the porous region, or the open volume fraction of the medium According to a study by Usta (2003), the green wood has a porosity of 0.4; the porosity of preburn wood is determined by the degree of preburn When modelling preborn wood in this study, a factor of has been applied onto the porosity of green wood to derive the porosity of preburn wood, since pre-burn wood in this study has a 50% degree of pre-burn Therefore, the pre-burn wood has a porosity of × 0.4 = ; a full solid slab would have porosity 0.8 equal to 1.0 When the porosity is equal to 1.0, the solid portion of the medium will have no impact on heat transfer or the source terms in the medium Porosity is treated as a constant, but the effect of porosity on the time derivative terms has been accounted for in all scalar transport equations and the continuity equations in Fluent 95 ... and erratic velocity profile   S-curve temperature profile  COUPLED VELOCITY AND TEMPERATURE DEVELOPMENT IN SELF -HEATING Figure 3. 1: Salient features of porous models for low- temperature long- term. .. a factor of has been applied onto the porosity of green wood to derive the porosity of preburn wood, since pre-burn wood in this study has a 50% degree of pre-burn Therefore, the pre-burn wood. .. fraction of the medium According to a study by Usta (20 03) , the green wood has a porosity of 0.4; the porosity of preburn wood is determined by the degree of preburn When modelling preborn wood in