Thermal combustion and oxygen chemisorption of wood exposed to low temperature long term heating 3

Wood is treated as a porous slab, and two heat and mass transfer models are presented for low temperature heating of wood.. This porous model is developed based on conservation equations

Chapter Three: Mathematical Formulations

3 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

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

Figure 3.1: Salient features of porous models for low-temperature long-term heating

in wood

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

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 one-dimensional flow where it could be schematically represented as follow

Figure 3.2 The one-dimensional flow in porous model

The following assumptions are introduced alongside the development of the models in order to make the mathematical treatment tractable

1 Wood is modeled as a porous slab However, all structural changes such as swelling, shrinkage, crack formation, are negligible during drying

2 Liquid and gaseous transport is dominant and responsible for internal heat transfer, through diffusion and convection

3 Pyrolysis is negligible at low temperature drying

Mw

Mv

X

Exposed surface

Insulated surface

Convective heat loss

Radiative heat loss

e

q′′

4 Water and vapour fluxes are combined into a total moisture flux (M with ) effective diffusivity, which includes two phases and two transport mechanisms

5 Evaporation of moisture is sufficiently rapid to attain thermodynamic equilibrium

6 Movement of both water (W and combined moisture flow () M are taken into ) account using Darcy’s law

7 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

( w ) ( w w w)

( V ) ( v v v)

Combining Equations (3.1) and (3.2) leads to

( w ) ( v ) ( w w w) ( v v v)

where the evaporation term ( )I has been eliminated Total moisture content as mass of water per unit mass of dry matter Mis defined as

( ) / s

and a total moisture flux (diffusive and convective) is written as

( m ) ( m m m)

A new parameterD , known as the effective diffusivity is introduced, combining water m

diffusivity D and vapour diffusivity w D The concept of effective transport property v

lump the two phases and different transport mechanisms together as one

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 Cartesian-tensor form of equations for one-dimensional flow

Initial Drying Phase (IDP) Model

3.1.3.1 Vector representation

Energy Balance

( cT) (u w w c T) k eff T

∂

∂



(3.6)

where k eff is the effective thermal conductivity, ( )ρ is the average heat capacity of the c

solid and fluid mixture medium The effective thermal conductivity is defined as

(1 )

The average heat capacity is defined as

(1 )

s s w w

Mass Balance

W

t

ϕ∂ = − ∇ ⋅ϕ + ∇ ⋅ϕ ⋅∇

∂



(3.9)

where the internal term of the rate of evaporation is omitted

Initial and boundary conditions for energy equation

Boundary condition (x = 0)

4 4

eff e

T

Boundary condition (x →∞)

0

eff

T k x

∂

Initial condition (t = 0, x ≥ 0)

Initial and boundary conditions for conservation of mass

Boundary condition (x = 0)

,

W

x

Boundary condition (x →∞)

0

w

W D t

∂

Initial condition (t = 0, x ≥ 0)

,0 ,

w w

3.1.3.2 Cartesian representation

Energy Balance

Mass Balance

2

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

Extended Drying Phase (EDP) Model

3.1.4.1 Vector representation

Energy Balance

( cT) (u m m c T) k eff T (1 ) s( H R ev ev)

∂



(3.18)

where k eff is the effective thermal conductivity, ( )ρ is the average heat capacity of the c

solid and fluid mixture medium The effective thermal conductivity is defined as

(1 )

The average heat capacity is defined as

(1 )

Mass Balance

M

t

ϕ∂ = − ∇ ⋅ϕ + ∇ ⋅ϕ ⋅∇ − −ϕ ρ

∂



(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

Initial and boundary conditions for energy equation

Boundary condition (x = 0)

4 4

eff e

T

Boundary condition (x →∞)

0

eff

T k x

∂

Initial condition (t = 0, x ≥ 0)

Initial and boundary conditions for conservation of mass

Boundary condition (x = 0)

,

M

x

Boundary condition (x →∞)

m

M D t

∂

Initial condition (t = 0, x ≥ 0)

,0 ,

3.1.4.2 Cartesian representation

Energy Balance

( cT) (k eff T) ( m m c u T m)

Mass Balance

2

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

Equilibrium approach is adopted in the low-drying model, and the equilibrium between liquid and vapour is assumed to reach instantaneously, so that

( )

v sat

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:

/

( ) B R vexp( )

sat

v

A

R T

−

3.18 10 kJ kg

2.5kJ kg

6.05 10 Nm

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

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 6 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

pressure drop and the fluid velocity as u k D P

x

∂

= − ∂ , where k is the Darcy’s coefficient D

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

u

t

µ

κ

∞

∂



(3.32)

0

u

where pis 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

1

u

t

µ

∂



(3.34)

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:

2

1 2 0

D

p

x

∂

Forchheimer’s modification in the one dimensional flow as k D p a u1 a u2 2

x

∂

Equation (3.35) can be conveniently formalized for two and three dimensions as

0

m m m

p ρ g µ v ρ bu u

where bdenotes 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

[ ( ] m m

u

t

∞

∂

∂



(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

2 2

1

2

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 , 2 1 2

C ρ , is the inertial loss term u C is the 2

inertial resistance factor in the inertial loss term Fluent therefore is able to provide for modification to Darcy’s law through the constant C that providing a correction for 2

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 21 2

C ρ is u

simply “switched off” by taking C to be zero Various methods of computing 2 C are 2

given in the Fluent 6.3 user guide

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 2 has been applied onto the porosity of green wood to derive the porosity of pre-burn wood, since pre-pre-burn wood in this study has a 50% degree of pre-pre-burn Therefore, the pre-burn wood has a porosity of2 0.4× =0.8; a full solid slab would have porosity 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