Thermal combustion and oxygen chemisorption of wood exposed to low temperature long term heating 4

Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments Chapter Four: Spontaneous Heating and Calorimeter Ignition Tests and Experiments Cone Introduction This chapter presents the methodology and experiments for measuring ignition and combustion of wood in bench-scale tests: spontaneous heating of wood cubes and ignition testing in Cone Calorimeter For spontaneous heating, the mathematical considerations of thermal explosion were first considered as it laid the groundwork for the experimental design For ignition testing in Cone Calorimeter, the preparation of both green wood and preburn were discussed The determination of thermophysical properties for green wood, preburn wood, and moisture was also highlighted An analytical method was proposed for correlation of ignition data in both green and preburn wood In the assessment on self-heating propensity in wood, two main methods have been used: heat-based method and the non-heat based method (Wang, Dlugogorski and Kennedy, 2006) The measurement of thermal runaway of wood cubes in preheated isothermal oven is a heat-based method Heat-based method measures heat generation and hence the critical temperature for autoignition as a basis for evaluating the tendency of the material to self-heat The non-heat based method on the other hand assesses oxygen adsorption, or the capacity of the material to consume oxygen, as a measure of the propensity to selfignition The chemical aspect of self-heating in wood by oxygen adsorption is discussed separately in Chapter 72 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments 4.1 Spontaneous Heating According to Frank-Kamenetskii theory, spontaneous ignition involves a size effect Ignition occurs when heat generated by exothermic processes exceeds heat dissipated to the surrounding; in this case, heat generation is related to the cube of dimension while heat dissipation is related to either dimension or square of dimension (Walker 1967, Cuzzillo 1997) This section discusses the principle underlying the critical F-K parameter δ c in determining the critical size of wood cube and the experiment design used to study spontaneous ignition 4.1.1 Method to determine the size for thermal runaway This method was developed by Bowes and Cameron, based on the FrankKamenetskii model for thermal ignition of packed solids (Bowes and Cameron 1971, Bowes 1984) The mathematical considerations that underlie the experimental design of spontaneous heating of wood slab are shown below Mathematical consideration For a slab of half-width r, the general energy conservation equation is expressed as ∂T ∂ 2T E = + ρ QA exp(− ρc k ) ∂t ∂x RT (4.1) 73 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments where it is subjected to the following boundary conditions: ∂T = −k ∂x −k = at x ∂T =h(T − T∞ ) ∂x x =r at (4.2) (4.3) where r represents the radius or half-thickness of the slab, h denotes the convective heat transfer coefficient at the solid surface, and T∞ is the temperature of the ambient At steady-state, the non-dimensional steady state form of the general energy conservation in Equation (4.1) is d 2θ dθ + + δ exp(θ ) = dz dz (4.4) where the non-dimensional quantity z represents x / r , and θ is defined as = θ E (T − T∞ ) RT∞2 (4.5) and δ in Equation (4.6) denotes the Frank-Kamenetskii parameter, δ = ρ QA Er k ∞ RT exp(− E ) RT∞ (4.6) The boundary conditions for Equation (4.6), as deduced from the definition of z and that of Equation (4.7), correspond to 74 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments = ∇θ = and θs (4.7) Taking natural logarithm and rearranging Equation (4.6) yields δ T2 E ρ QAE = ln ln c 2∞ − r kR RT∞ (4.8) δ c is known as the critical Frank-Kamenetskii parameter, which is primarily a function of geometric shape of the solid and the average surface Biot number Beever (1995) gives an excellent review of critical parameters for various geometries and the corrections required for proper application T∞ , the critical ignition temperature is to be determined for each sample size (i.e varying r ) Based on experimental measurements obtained for a series of sample sizes, ln(δ cT∞2 / r ) can be plotted as a function of the reciprocal temperature1/ T∞ , permitting a least squares fitting of equation (4.8) The least square fitting yields an estimate of the activation energy A and pre-exponential factor E Equation (4.8) can then be used to predict critical temperatures T∞ as a function of size r of the wood slab However, this method can be cumbersome because of the lengthy experimental iteration process to determine the kinetic parameters, involving first the evaluation of a series of critical ignition temperature T∞ for different sample sizes A number of trial and error experiments have to be carried out to “bracket” the right critical ignition temperature for 75 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments a given sample size As a result, a large number of experiments are required just to get the kinetic parameters Chen and Chong (1995, 1998) introduced the “crossing-point” method which reduces the number of experiments required to determine the kinetic parameters For a sample undergoing self-heating, there will be some point in time where the temperature profile at the centreline becomes locally flat At the unique time where ∂ 2T ∂x crosses zero, x =0 Equation (4.1) at the centre plane, x = , becomes  ∂T QA E  = exp  −  ∂t x =0 c  RTctr  (4.9) where Tctr is the centre-plane temperature at which the conduction term vanishes, i.e the centre region of the slab is adiabatic Taking logarithm of both sides of Equation (4.9) yields ∂T QA E − ln = ln ∂t x =0 c RTctr A plot of ln −E RTctr ∂T ∂t x =o (4.10) versus yields a straight line The slope of the straight line, Tctr , gives the activation energy and its intercept gives the pre-exponential as 76 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments ln QA , the log of the theoretical maximum adiabatic rate of temperature rise c In terms of experimental work, only the Tctr needs to be determined, not the critical ignition temperature T∞ Tctr is the centre plane temperature where the conduction term vanishes In practice, two thermocouples are placed at a short distance away from the centre line of the sample, and the temperature is determined when the temperatures of the two thermocouples cross each other In Chen’s method, every experiment is able to yield a Tctr ; in Frank-Kamenetskii method, many experiments are however required just to produce one T∞ In experimental determination of the critical size relating to spontaneous combustion for a given wood slab, and/or its associated kinetic parameters, the crossing point method (Chen and Chong 1995, Chen and Chong 1998) or simply known as Chen’s method has clear advantages over Frank-Kamenetskii model First, it reduces the number of experiments required to determine the kinetic parameters Since the shape of the material and the Biot number hr / k constitute the most important parameters affecting δ c (Bowes 1984, Cuzzillo 1997), the exclusion of critical Frank-Kamenetskii parameter eliminates the direct Biot number effects on data interpretation Heat transfer coefficient h in the oven and conductivity k of the wood slab need not therefore be known during the experiments Chen’s method certainly offers a much simpler and neat experimental method 77 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments 4.1.2 Self-heating experiment Cuzzillo (1997) has used Chen’s method to derive a critical size of 89mm wood cube corresponding to a critical ignition temperature of 200ºC In this study, the same critical size and critical ignition temperature is used: Kapur wood was sized into 89mm cubes and placed in an isothermal oven maintained at 200°C to investigate the effects of self-heating The purpose of adopting the same sample size and critical temperature is to enable comparison of results to be made between these two studies directed at investigating self-heating in wood cubes The 89 mm (3.5”) cube was to be heated symmetrically in the isothermal oven at 200oC until thermal runaway was observed To study spontaneous heating, thermocouples typeT were inserted at the centre of the cube and connected to a data logger Yokogawa DAQ station DX230 A total of 18 thermocouples were used Holes were drilled to reach half the depth of wood cube i.e 45mm and sixteen thermocouples were inserted The details on the location of thermocouples are discussed in Section 4.1.2.2 If the centre temperature exhibited thermal runaway, the temperature would show a “peak” on the thermocouple readings The wood cube experimental design was discussed below 4.1.2.1 Specimen preparation and orientation The cube was oriented so that the two side-grain surfaces (i.e grain of wood is perpendicular to heat flux) facing two heating modules of the oven The wood cube was insulated at the end-grain faces (grain wood is parallel to heat flux), as well as the top and 78 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments bottom faces with calcium silicate wrapped with rock wool and aluminium foil to prevent escape of volatiles through these faces Wood cube preparation was shown diagrammatically as in Figures 4.1 and 4.2 Transverse (z-axis) Side grain face Convective heat loss Thermocouples’ locations Radiative heat loss Mw Insulation at end grain faces, as well as top and bottom faces Radial (y-axis) Mv z x Longitudinal (x-axis) Figure 4.1: The three principal axes of flow in wood cube Symmetry of thermally thick wood b Figure 4.2: Schematic domain on plan view The purpose of insulation was two-fold First, it prevented the escape of volatiles, mainly through the end-grain faces in order to minimise the non-isotropic effects on the spontaneous combustibility of wood Secondly, the insulation enabled the wood domain to be treated as one-dimensional which was the case for consideration in this study 79 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments 4.1.2.2 Location of thermocouples Eighteen type-T thermocouples (operating temperature: -200°C to 350°C) were used to measure the temperature history of the wood cube throughout the heating regimes The thermocouples in use were grouped into groups: Group 1: thermocouples were positioned along transverse axis of the cube (01, 02, 03, 04, 05, 06, 07, 08, 09) Transverse axis is the line connecting two center points of any two opposite side-grain surfaces as shown in Figure 4.3 Table 4.1 shows the locations of Group thermocouples with respect to the side grain face Group 2: thermocouples were positioned along longitudinal axis of the cube (09, 10, 11, 12, 13, 14, 15, 16, 17) Longitudinal axis (or axial axis) is the line connecting two center points of two end-grain surfaces, as shown in Figure 4.3 Table 4.2 shows the locations of Group thermocouples with respect to the side grain face Group 3: thermocouple no.18 in the oven to measure the oven air temperature The location and distribution of thermocouples are shown in Figure 4.3 and Figure 4.4 respectively 80 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments Table 4.1 Group Thermocouples Thermocouple Distance to nearest side grain face 01, 08 02, 07 03, 06 04, 05 09 mm (on side-grain surface) mm 13 mm 25 mm 45 mm (center) Table 4.2 Group Thermocouples Thermocouple Distance to nearest end grain face 10,17 11,16 12,15 13,14 09 mm (on end grain face) mm 13 mm 25 mm 45 mm (center) 18 16 15 14 Side Grain Face 13 12 Side Grain Face Heat 11 End Grain 10 Axial (Longitudinal) Heat Tangential (Transverse) Transverse Centerline Longitudinal Centerline Figure 4.3 Location of thermocouples in isometric view 81 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments ignition for heat fluxes > 40 kWm-2 Flaming ignition occurred almost immediately upon observation of glowing surfaces The same observation was also noted by Boonmee and Quintiere (2002) in their study of autoignition of Redwood Their study also found that for high irradiance above 40kWm-2, the sample started glowing, and then flaming ignition occurred almost immediately However, when the external heat fluxes were lowered below 40kWm-2, sustained ignition did not occur For samples heated between 39kWm-2 to 35kWm-2, even though each experimental run was extended beyond the stipulated 20 minutes cut-off, no sustained ignition occurred But wood samples in Boonmee and Quintiere (2002) study were observed to glow and transition to flaming combustion within 40 minutes when heated between 28kWm-2 and 40kWm-2 Different sample sizes, species variation, moisture content and other experimental conditions might have accounted for the different experimental observations between the two studies here Table 4.6: Ignition data for spontaneous ignition of green wood above 30kWm-2 Sample Density (kgm-3) Time to flaming ignition (hr:min:s) 0:0:43 Surface temperature at ignition (°C)a Observation 564 External heat Flux (kW m-2) 50 448 640 40 0:2:26 457 512 39 NA@1hr -a 520 38 -a 536 37 NA@1hr 16min NA@ 52min 632 35 NA@1hr 18min -a Immediate flaming Immediate flaming No sustained ignition No sustained ignition No sustained ignition No sustained ignition -a a Surface temperatures were only recorded for cases when sustained ignition occurred Surface temperatures were not recorded when the surfaces were noted for glowing ignition 103 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments For heat fluxes below 30kWm-2, only glowing ignition was noted As shown in Table 4.7, the surface temperatures spanned the range of 230°C to 395°C, which was similar to the surface temperatures of piloted ignition for green wood that ranged from 270°C to 375°C The glowing ignition noted at these low heat fluxes accounted for the similar span of surface temperatures at ignition observed for autoignition and piloted ignition tests (Babrauskas 2001) Table 4.7: Ignition data for spontaneous ignition of green wood below 30kWm-2 Sample Density (kgm-3) External heat Flux (kW m-2) 508 540 580 556 584 540 30 20 15 12 11 10 Time to glowing ignition (hr:min:s) 0:2:50a 0:14:12 0:40:28 1:30:54 3:30:42 NA@6hrs Surface temperature (°C) 394.6 319.8 280.5 253.2 230.7 - a No sustained flaming ignition was noted for green wood, even though it was exposed to 30kWm-2 up to one hour and 50 minutes Two ignition modes therefore were noted for spontaneous ignition of green wood When wood samples were heated at high heat fluxes above 40kWm-2, the samples glowed and almost immediately proceeded to flaming ignitions The glowing zone acted as a high temperature pilot to ignite the flammable air-mixtures, deriving the additional energy from the char oxidation on sample surface (Babrauskas 2001, Boonmee and Quintiere 2002) No sustained ignition was observed for intermediate heat fluxes for 40kWm −2 < &e < 30kWm −2 Samples heated below 40kWm-2 were noted to glow q′′ without flaming ignitions at all 104 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments On the other hand, two ignition modes occurred for spontaneous ignition of preburn wood: flaming combustion for heat fluxes > 45kWm-2 and glowing ignition for heat fluxes < 40kWm-2 The surface temperatures for spontaneous ignition of preburn wood were higher than that of green wood, displaying the same trend as noted for piloted ignition For spontaneous ignition of preburn wood, the time to ignition and surface temperatures were tabulated in Table 4.8 and Table 4.9 Table 4.8: Ignition data for spontaneous ignition of preburn wood above 40kWm-2 Sample Density (kgm-3) Time to ignition (hr:min:s) Surface temperature (°C) Observations 539 External heat Flux (kW m-2) 50 0:0:27 426.1 478 45 0:1:01 537.4 467 42 0:0:42 416.8 456 41 - 550 40 NA@1hr 5min NA@ 45min Immediate flaming Immediate flaming No sustained ignition No sustained ignition No sustained ignition Table 4.9: Ignition data for spontaneous ignition of preburn wood below 20kWm-2 Sample Density (kgm-3) External heat Flux (kW m-2) 526 511 555 467 20 10 Time to glowing ignition (hr:min:s) 0:3:54 1:45:4 2:13:40 NA@6hrs Surface temperature (°C) 312.1 222.2 197.2 - 105 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments The surface temperatures and time to ignition collected for piloted and spontaneous ignitions of both green and preburn wood would be used for analyzing thermal combustion models as well as derivation of critical heat flux in Chapter 4.3.3.3 Thermal thickness For specimen that was greater than 50mm thick, it could be readily seen that for almost any realizable combination of thermophysical properties and incident radiant fluxes, a 50mm specimen was thermally thick, and increasing thickness would not change the ignition times (Weatherford and Sheppard 1965, Wesson, Welker and Sliepcevich 1971) For specimen of any other sizes, the formula below was used to determine the thermal thickness of the material for a given density and heat flux combination  L > (0.6 ρ / q′′)(mm) (4.30)  where L is the thickness (mm), ρ is the density (kgm-3) and q ′′ is the heat flux (kWm-2) The proportionality of the required thickness was derived from classical heat conduction theory by equating the time for the front surface to reach the ignition temperature to the time for the rear surface temperature to begin to rise and assuming the thermal conductivity was proportional to the density (Babrauskas and Parker 1987) Janssens (1991b) has pointed out 25mm thick oven-dry green wood is thermally thick for the range of incident heat fluxes used in this study For partially charred wood, the thermal thickness has been computed and shown in Table 4.10 below The results also suggested the partially charred wood remained thermally thick 106 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments Table 4.10: Thermal thickness for piloted ignition of pre-burn wood Material Nyatoh density ρ = 442.16 kg/m3 L = 25mm External radiant flux,  ′′ qe (kW/m2) 50 40 30 25 15 Estimated pre-experimental thermal thickness  ′′ L > (0.6 ρ / qe )(mm) 5.31 6.63 8.84 10.61 17.69 4.3.4 Determination of thermophysical properties for green wood The thermophysical properties of green wood such as thermal conductivity and specific heat capacity were determined according to Wood Handbook (1999) Thermal conductivity was first calculated as a function of specific gravity and subsequently determined as a linear function of temperature as k =G ( B + CM ) + A (4.31) k (T ) = + 0.002(T − T0 )] k0 [1 (4.32) where G is the specific gravity based on oven dry weight and was found to be 0.55 for Nyatoh wood (Futonlife 2006) Since the specific gravity was greater than 0.3 and the oven-dry moisture content ( M ) of green wood was found to be 13.86%, the three constants in Equation (4.13) were taken to be A = 0.001864 , B = 0.1941 and C = 0.004064 according to Wood Handbook 107 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments The specific heat was calculated for green wood above and below fiber saturation point (fsp) The specific heat capacity of dry wood above fiber saturation point (c> fsp ) was given as a function of temperature: = 0.1031 + 0.003867T c> fsp (4.33) Below fiber saturation point, the specific heat capacity was computed as the sum of the heat capacity of green wood (c> fsp ) and that of water (cw ) , plus an additional adjustment factor Ac that accounted for the additional energy in the wood-water bond: c< fsp = c> fsp + 0.01M ⋅ cw ) / (1 + 0.001M ) + Ac ( (4.34) Ac = M (b1 + b2T + b3 M ) (4.35) where the specific heat of water cw was taken to be 4.19 kJ/kg K The three constants in Equation (4.34) were b1 = −0.06191 ,= 2.36 ×10−4 and b3 = × 10−4 b2 −1.33 The thermal conductivity, k and the specific heat capacity, c of green Nyatoh wood in this study were found to be k = 0.125W/mK, and c = 2040J/kgK 4.3.5 Determination of thermophyiscal properties of partially charred wood Unlike green wood, the thermophysical properties of partially charred wood depend on the degree of conversion from virgin wood to char, α (Janssens 1991b, Moghtaderi and 108 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments Kennedy 1998, Yuen 1998, Janssens 2004, Boonmee 2004) According to the preburn experiments, the degree of preburn was found to be in average 0.50, as shown in Table 4.11 This section discussed the methods used for the determination of thermophysical properties for the partially charred wood Table 4.11 Mass fraction of virgin wood Material Density of virgin solid, ρw Density of pyrolysing solid, ρs Density of final char, ρf Mass fraction of virgin wood α = ( ρ s − ρ f ) /( ρ w − ρ f ) Nyatoh 735.50 543.19 537.57 352.43 0.50 0.48 4.3.5.1 Specific heat capacity of partially charred wood For partially charred wood, the specific heat capacity varied with the degree of conversion from virgin wood to char during pyrolysis (Chan, Kelbon and Krieger 1985) This study adopted the widely proposed formulation for specific heat capacity of partially charred wood (Chan et al 1985; Yuen, 1998) which assumed that the volumetric specific heat capacity varies linearly between the virgin wood and final char, given as ρ= αρ wcw + (1 − α ) ρ f c f s cs (4.36) α =ρ f ) / ( ρw − ρ f ) (ρs − (4.37) where α is the mass fraction of virgin wood (i.e unconverted wood) The specific heat capacity of dry wood was found to be cw = 2.04kJ/kg K , as computed in the previous 109 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments section The specific heat capacity of final char assumed the value of c f = 0.672kJ/kg K according to the study by Fredlund (1988) 4.3.5.2 Thermal conductivity of partially charred wood Similarly, the thermal conductivity of partially charred wood was also assumed to vary linearly between that of virgin wood and that of final char (Alves and Figueiredo 1989, Yuen 1998) Thermal conductivity of partially charred wood was calculated as follow: k s= α kw + (1 − α )k f (4.38) Where kw = 0.125W/m K as computed in previous section for green wood and the thermal conductivity for final char was given as k f = 0.189W/m K (Yuen, 1998); α is the mass fraction of virgin wood defined in Equation (4.37) above 4.3.6 Determination of thermophysical properties for water and vapour For porous model, the concept of effective thermal conductivity (keff ) and average heat capacity ( ρ c) were obtained by interpolating the values between the liquid and the solid medium, the formulations of which have been presented in Chapter This chapter discussed the determination of the specific heat capacity and thermal conductivity of water and vapour, which could be used to compute the required data for the porous model Other than the average heat capacity and effective thermal conductivity, the use of porous model also introduced the concept of combined diffusivity ( Dm ) (Zhang and 110 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments Datta 2004) For the total moisture transport of liquid water and vapour, the combined diffusivity was treated as the average of the water diffusivity ( Dw ) and vapour diffusivity ( Dv ) For specific heat capacity of liquid water (cw ) , Rogers and Mayhew (1980) showed that the specific heat capacity ranges from 4.181 to 4.219 kJ kg-1 K-1 for temperature below 100°C This study took on the value of cw = 4.19kJ kg −1 K −1 proposed by Alves and Figueiredo (1989) which fell in the range suggested by Rogers et al (1980) For vapour, Fredlund (1988) proposed a temperature-dependent range of specific heat capacity for vapour, which varied linearly between 1.86 to 1.90 kJ kg-1 K-1 The proposed cv = 1.88kJ kg −1 K −1 by Alves and Figueiredo (1989), which was equivalent to the average of the upper and lower limit suggested by Fredlund (1988), was adopted in this study For thermal conductivity, this study used the following values: kw = 0.58W/m K measured at 298K and kv = 0.016W/m K measured at 398K respectively Motosuke, Nagasaka, and Nagashima (2004) measured thermal diffusivity of water through sol-gel transition using an optical technique known as the forced Rayleigh scattering method, which has the possibility of applying to the real-time monitoring of the thermal diffusivity The thermal diffusivity of distilled water at the temperature range from 298 to 323 K was measured and this study adopted the value of thermal diffusivity Dw = 1.45m 2s −1 measured at 300K 111 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments Lee and Wilke (1954) obtained the diffusion data for air-vapour through experiment where one of the components, in the liquid state, was placed in the bottom of a vertical tube With the tube and liquid maintained at constant temperature, the second component, a gas, was passed over the top of the tube at a rate sufficient to keep the partial pressure of the vapor there at a value essentially corresponding to the initial composition of the gas During diffusion the liquid level fell as vaporization proceeded and the rate of evaporation was not constant Assuming quasi-steady-state conditions, it can be shown that the rate of evaporation may be taken as the arithmetic average of values at the beginning and end of the diffusion period and used in conjunction with the over-all average rate of diffusion to give correct values of Dv by numerical solution They obtained the diffusion data for air-vapour at 298K at one atmospheric pressure and found that Dv = 2.6E-05m 2s −1 , which was adopted in this study 4.3.7 Correlation of time-to-ignition data Correlation of time to ignition data at various irradiance levels with thermal properties of wood can be considered by using a thermal model and a critical surface temperature criterion for ignition In this work, correlation of the time to ignition data was analysed using a mathematical model developed from exact solution via Laplace transforms of the linearised simple thermal model as discussed in Section 4.3, assuming constant thermophysical properties 112 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments 4.3.7.1 Analytical model for correlation of ignition data Following the discussion in Section 4.3, the energy conservation for a simple thermal can be deduced to a simple expression as ∂  ∂T k ∂x  ∂x ∂T   = ρc ∂t  (4.39) Exact solution of the above heat balance can be done by Laplace transforms To solve the time dependent energy equation, Equation (4.38) is first re-arranged ∂ 2T ∂T = ∂x λ ∂t where λ is the thermal diffusivity, λ = (4.40) k So, ρc ∂ 2T ( x, t ) ∂T ( x, t ) − = ∂x λ ∂t (4.41) Applying Laplace transform, Equation (4.41) becomes T ∂ T ( x, s ) s − T ( x, s ) + ∞ = ∂x λ λ (4.42) 113 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments The thermal problem can be simplified by linearising the heat losses from the exposed surface where x = The boundary condition ( x = 0) pertinent to an opaque material has been formulated in Section 4.3 as −k ∂T  ′′ = α qe − hc (T − T∞ ) − εσ (T − T∞4 ) ∂x (4.43) Heat losses from the exposed surface are linearised by assuming α= ε= and introducing a constant heat loss coefficient, h The boundary condition at x = becomes −k ∂T q′′ =e − h(T − T∞ ) ∂x (4.44) Applying Laplace transform to Equation (4.43) and rearranging yields −k  ′′ qe hT ∂T ( x, s ) + hT ( x, s ) = ∞ + ∂x s s (4.45) T ( x, s ) can be expressed as follows = T ( x, s ) C1e −x s λ + T∞ s (4.46) Differentiating Equation (4.46) 114 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments ∂T ( x, s ) s −x = −C1 e ∂x λ s λ (4.47) Substituting Equations (4.46) and (4.47) into Equation (4.45) yields C1 as C1 = Substituting Equation (4.48) and H =  ′′ qe (4.48)  s  + h s k  λ  h into Equation (4.46), then T ( x, s ) can be k expressed as = T ( x, s )  ′′  qe  h s  ( H λ s+H λ ) e −x s λ   + T∞  s  (4.49) Taking inverse Laplace transform of T ( x, s ) , T ( x, t ) is found to be hx h t  h 2t +  ′′ qe  x x  erfc T (= x, t ) − e k k ρ c erfc  +  + T  k ρ c λt   ∞ h  λt    (4.50) At the surface of solid where x = h t  ′′ qe  h 2t  1 − e k ρ c erfc  T (0, t ) = + T∞ h k ρc    (4.51) 115 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments At t = tig , T (0, t ) becomes the surface ignition temperature Tig Equation (4.51) for Tig can be expanded in a series as Tig = + T∞    ′′ qe   1  1− − + −    h  π  τ ig2 2τ ig2 4τ ig2    (4.52) where τ ig = h 2tig / k ρ c According to (Lawson and Simms 1952), the series in Equation (4.52) can be truncated to  ′′ the first term for determination of critical heat flux qcr The expression can therefore be written as Tig = + T∞  ′′ qe  k ρ c  1 −  h  h π tig    (4.53)  ′′ Using Equation (4.53), the ignition data of incident heat flux qcr can be plotted against  ′′ the time to ignition as qe / tig for green and preburn wood where the critical heat flux  ′′ qcr is to be found as the intercept with the ordinate 116 Chapter 4: Spontaneous Heating and Cone Calorimeter Ignition Tests and Experiments 4.3.8 Concluding remarks for Cone Calorimeter ignition testing Cone Calorimeter testing provided the experimental data on time to ignition and surface temperature at ignition The experimental data permitted the evaluation of moisture effects in thermal combustion of wood using thermal models developed in Chapter The correlation of ignition data for derivation of critical heat flux was intended to assess the effects of different thermophysical and moisture contents of wood on ignition using green and preburn wood The methodology framework for evaluating the moisture effects in thermal combustion models using Cone Calorimeter experimental data was summarised by the schematic framework below CONE CALORIMETER DATA ANALYTICAL MODEL THERMAL MODELS PURE THERMAL MODEL REVISED THERMAL MODEL ANALYTICAL CORRELATION MODEL - Simulate Tig to compare with measured Cone Tig - Analyse ignition in green and preburn wood - Calculate Tig to compare with measured Cone Tig - Analyse the effects of moisture in thermal combustion - Correlate experimental time to ignition with incident heat flux - Derive critical heat flux of green and preburn wood Figure 4.12: Methodology framework for analysis of Cone Calorimeter Data 117 ... λ s λ (4. 47) Substituting Equations (4. 46) and (4. 47) into Equation (4. 45) yields C1 as C1 = Substituting Equation (4. 48) and H =  ′′ qe (4. 48)  s  + h s k  λ  h into Equation (4. 46), then... 10 644 6 24 640 640 636 6 24 708 652 628 6 64 External heat flux &e q′′ (kW m-2) 50a 40 a 30a 25a 20a 15b 11c 10c 9c 8c Time to ignition tig (hr:min:s) 0:0:26 0:0: 54 0:1 :48 0:5: 24 0:12 :44 0:53 :42 ... content ( M ) of green wood was found to be 13.86%, the three constants in Equation (4. 13) were taken to be A = 0.0018 64 , B = 0.1 941 and C = 0.0 040 64 according to Wood Handbook 107 Chapter 4: Spontaneous