Download Project no. FP6-018505 Project Acronym FIRE PARADOX Project

Project no. FP6-018505
Project Acronym FIRE PARADOX
Project Title FIRE PARADOX: An Innovative Approach of Integrated Wildland Fire
Management Regulating the Wildfire Problem by the Wise Use of Fire: Solving the Fire
Paradox
Instrument Integrated Project (IP)
Thematic Priority Sustainable development, global change and ecosystems
Deliverable 2.5-5-36
Criteria for building material ignition and for burn injuries in wildland
fires: final achievements
Due date of deliverable: February 2009
Actual submission date: November 2009
Start date of project: 1st March 2006
Duration: 48months
Organisation name of lead contractor for this deliverable: The University of Edinburgh
Revision (1000)
Project co-funded by the European Commission within the Sixth Framework Programme
(2002-2006)
Dissemination Level
PU
Public
PP
Restricted to other programme participants (including the Commission Services)
RE
Restricted to a group specified by the consortium (including the Commission Services)
CO
Confidential, only for members of the consortium (including the Commission Services)
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X
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Authors:
Pedro Reszka (P26: UE)
José Luis Torero (P26: UE)
Executive summary
In fire modelling, an accurate prediction of the ignition of solid fuels requires the
solution of solid- and gas-phase processes. Methods that decouple the solid from the
gas phase would result in significant savings in computational cost. The work described
herein presents a novel methodology for this decoupling. It is based on the observation
that the time to ignition can be scaled with the square of the time integral of the
incident heat flux. This relationship can be readily demonstrated for the classical
solutions for time to ignition which consider constant incident heat fluxes. However,
some fire applications, in particular situations of wild fires approaching the wildlandurban interface, present time-varying incident heat fluxes which render the classical
solutions inaccurate. A new analytical solution for obtaining the time to ignition for
ramping incident heat fluxes is presented. The proposed methodology completely
decouples the solid and gas phases and is accurate in the prediction of ignition times.
The methodology can be applied to both the new and classical analytical solutions. It
was validated with tests carried out on PMMA and PA6. The results presented here will
be useful for other teams developing forest fire models, especially the team at VTT
which is working on the problem of the wildland-urban interface.
The second part of the report is concerned with the development of a numerical model
for the assessment of the influence of moisture migration in the severity of burn injuries
affecting fire fighters. It was proven that this is an important factor affecting the severity
of burn injuries. This information will constitute the basis for a research on the
physiology of thermal skin burns, and will also help in the development of protective
clothing for fire fighters.
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Table of contents:
1.
A simple model for the assessment of building material response to
wildland fires: Final results. .................................................................................... 4
1.1.
Introduction ...................................................................................................... 4
1.2.
Solid Fuel Ignition Modelling ............................................................................... 7
1.2.1.
Solution for a Ramped Heat Flux ..........................................................................................9
1.2.2.
New Analytical Solution .......................................................................................................9
1.2.3.
Case when
1.2.4.
Comparison of Different Solutions ...................................................................................... 11
q 0
= 0........................................................................................................... 10
1.3.
Decoupling of Solid and Gaseous Phases ........................................................... 12
1.4.
Experimental Description .................................................................................. 15
1.5.
Experimental Results and Discussion ................................................................. 18
1.5.1.
Validity of Semi-Infinite Solid Assumption ............................................................................ 18
1.5.2.
Benchmark Tests for Constant Incident Heat Flux ................................................................ 20
1.5.3.
Ramping Heat Fluxes ........................................................................................................ 20
1.6.
Effect of Material Thermophysical Properties ...................................................... 23
1.7.
Example of the use of the decoupling methodology ............................................ 24
1.8.
Concluding Remarks ........................................................................................ 26
1.9.
References ...................................................................................................... 27
2.
Criteria for the assessment of wildland fire effect on burn injuries:
Model description and experimental results. ........................................................ 29
2.1.
Introduction .................................................................................................... 29
2.2.
Experimental Procedure ................................................................................... 30
2.3.
Experimental Results........................................................................................ 31
2.4.
Modelling ........................................................................................................ 37
2.5.
Numerical Results ............................................................................................ 38
2.6.
Conclusions ..................................................................................................... 41
2.7.
Nomenclature.................................................................................................. 41
2.8.
References ...................................................................................................... 41
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1. A simple model for the assessment of building material response to
wildland fires: Final results.
1.1.
Introduction
The prediction of flaming ignition of solid fuels is necessary in many applications of fire
safety engineering, ranging from fire modelling to risk assessments and flame spread.
Flaming ignition is the result of a series of phenomena occurring simultaneously in the
gas and solid phases [1]. To accurately model this process in a practical way, a series of
simplifications and assumptions are normally done, conditioning the validity of the
results to specific situations.
In fire modelling, the differences in scales of the several physical and chemical
processes involved makes their resolution become computationally expensive [2, 3].
These scale differences are particularly evident between the solid and gas phases (see
Error! Reference source not found. below). A method for decoupling both phases
would lessen the computational cost and permit the application of CFD codes to complex
geometries. This section will report the work undertaken to develop a methodology for
the decoupling of solid and gas phases in the modelling of flaming ignition, and on a
new simplified analytical solution of the ignition process, with emphasis on the wildlandurban interface. The objective is to provide a mechanism to assess the potential for
ignition without adding an excessive computational burden to CFD or other fire models.
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Type
Time Scale (s)
Vertical Scale (m)
Horizontal
(m)
Combustion
0.0001 – 0.01
0.0001 – 0.01
0.0001 – 0.01
Fuel particles
-
0.001 – 0.01
0.001 – 0.01
Fuel complex
-
1 – 20
1 – 100
Flames
0.1 – 30
0.1 – 10
0.1 – 2
Radiation
0.1 – 30
0.1 – 10
0.1 – 50
Conduction
0.01 – 10
0.01 – 100
0.01 – 0.1
Convection
1 – 100
0.1 – 100
0.1 – 10
Turbulence
0.1 – 1,000
1 – 1,000
1 – 1,000
Spotting
1 – 100
1 – 3,000
1 – 10,000
Plume
1 – 10,000
1 – 10,000
1 – 100
scale
Table Error! No text of specified style in document.-1: The varying temporal and
spatial length scales of the major processes occurring in a forest fire [2].
An ever increasing problem in forest management is the occurrence of fires within the
wildland-urban interface (WUI). Here, forest fires interact with buildings, representing a
threat to people and property, and the presence of buildings can redefine fire
propagation. Major WUI fires have been reported in several countries [4]. It is therefore
important to understand the interaction between forest fires and building materials. An
overview of this problem has been presented by Rehm [5].
It has become clear that due to the high intensities of the heat fluxes caused by some
wildfires, radiant ignition of objects located at the order of tens of metres away from the
fire front can be achieved [6]. An approaching fire will impose a time-varying heat flux
on a stationary target [4, 6]. The shape of the heat flux can vary, but can normally be
regarded as an exponential function of time, as can be seen in the following figures.
However, for practical purposes, and given the accuracy level that is being dealt with in
the problem, these time-varying heat fluxes can be approximated as linear functions of
time after the initial growth stage.
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Figure Error! No text of specified style in document..1. Measured total heat flux at the
walls of a shed at the WUI for an approaching prescribed fire (that developed into
a crown fire) at New Jersey, USA. Taken from Manzello et al. [4].
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Figure Error! No text of specified style in document..2. Variation in total incident heat
flux with time during the crown fires of the International Crown Fire Modelling
Experiment [6] for one of the burnt plots.
Given the complexity of this problem, it has become evident that CFD fire models are
necessary to study the interaction of wildland fires and buildings. As previously stated, if
the gas and solid phases were completely decoupled, ignition times could be predicted
without the need to completely resolve the solid phase. This is particularly important in
the use of atmospheric wildfire models, where the grid cells are of the order of tens of
metres and any building material would lie within the subgrid scale. The decoupling can
be readily accomplished using standard analytical solutions to the problem of solid fuel
ignition [7, 1], as will be shown in this report.
However, the use of these solutions under conditions different to the ones they were
obtained for can introduce significant errors [1]. This is the case when modelling the
ignition of building materials from approaching wildland fires, because of the time
variability of the imposed heat fluxes. The next sections will discuss the modelling of
solid fuel ignition and the development of a method for the decoupling of solid and gas
phases. Finally, a comparison with experimental results will be presented.
1.2.
Solid Fuel Ignition Modelling
Classical approaches to ignition [7, 8] obtain analytical solutions to the heat diffusion
equation. By considering an ignition temperature at the surface (typically equal to the
pyrolysis temperature), times to ignition can be calculated. These methods consider a
one-dimensional, semi-infinite, inert solid subjected to a constant incident heat flux and
convective heat losses at the exposed surface; the heat diffusion equation being
expressed as
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(1)
with the boundary and initial conditions given by
and
The convective coefficient can be made to include radiative losses following standard
heat transfer techniques. If the heat flux to the surface is a constant the analytical
solution to this equation for the surface temperature is [9, 10]
(2)
For high incident heat fluxes (i.e. short ignition times), the time to ignition is obtained by
performing a Maclaurin series expansion and by making the surface temperature reach a
pyrolysis or ignition temperature (Ts=Tpyrolysis). Common solutions are of the form [8,
10]:
(3)
Where the a is the surface absorptivity, usually taken as unity, k the thermal
 
conductivity, the density, cp the specific heat, q i the incident heat flux, TP the pyrolysis
temperature, T∞ the ambient temperature and tP the pyrolysis time, usually considered
as being equal to the ignition time [1, 9]. A natural consequence of Equation (3) is that
if each material is assumed to ignite at a constant temperature, then it follows that there
is a linear dependency between
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and
, or that:
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1.2.1. Solution for a Ramped Heat Flux
First, the case of a ramped imposed heat flux will be considered. If no heat losses at the
surface are considered, and the incident flux is defined as,
(4)
a solution for Equation (1) under the new boundary conditions has been presented by
Carslaw & Jaeger [11, pp. 75-76]. The solution for x = 0, expressed in a similar form as
Equation (3) becomes,
(5)
1.2.2. New Analytical Solution
A similar expression can be obtained if the same approach is followed for a variable
incident heat flux, but this time considering heat losses at the surface. Assuming that
the imposed heat flux grows linearly with time, it can be defined as
(6)
A solution for this problem is not readily available. To obtain it, the Convolution Theorem
was applied to the solution of the differential equation in the Laplace space. The new
analytical solution for x = 0 is
(7)
The previous solution was approximated by using a Maclaurin series expansion (for small
times):
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(8)
Equation (8) constitutes a general solution, incorporating the terms corresponding to a
ramping incident heat flux (first term on the RHS of the equation), a constant incident
heat flux (second term; cf. with Equation (3)), and the terms multiplied by the
convective coefficient, corresponding to the heat losses. It is important to note that in
the series expansion that is carried out to obtain the solution presented in Equation (3),
the terms containing h are cancelled. This is not the case in the present solution, but it
must be pointed out that for the situation of high heat fluxes and short times, as was
considered in the solution leading to Equation (3), the terms containing h are not
significant, and therefore the solutions presented in equations (3) and (8) are
equivalent.
1.2.3. Case when q 0 = 0
For simplicity and the sake of clarity, the analysis will continue assuming that q 0 = 0.
Under this assumption, the solution presented in Equation (8) becomes
(9)
The idea now is to have an expression of a similar form as Equations (3) and (5). So, if
in the solution a net heat flux of the form
(10)
can be generated, then the solution can be expressed in the desired way. To do this, a
reference temperature is defined as:
(11)
The reference temperature represents a time-varying characteristic temperature. A
constant reference (or characteristic) temperature cannot be obtained due to the nature
of the problem, specifically the time-varying imposed heat flux. Factorizing and dividing
by
, the solution then becomes
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(12)
taking
. An expression for an equivalent net heat flux can then be generated:
(13)
Finally, the solution for the time to ignition considering a ramped incident heat flux and
convective surface losses is expressed as
(14)
1.2.4. Comparison of Different Solutions
The following figure presents a comparison between the different solutions to the
ignition problem that have been discussed in this work. A numerical solution of the onedimensional conduction problem with ramping incident heat flux and convective losses
at the surface is also plotted. For demonstration purposes, this solution is regarded as
being an “exact” solution.
The results show the influence of the convective terms. The analytical solution in
Equation (7) closely matches the exact solution for all times. The approximate solution,
Equation (9), shows good agreement for short times, diverging away from the exact
solution at longer times, where the approximation does not hold. The curve
corresponding to Equation (5), i.e. the solution for a ramped heat flux and no convective
losses, overestimates surface temperatures in a similar way than the solution for a
constant heat flux. The constant heat flux intensity used corresponds to the ramp
intensity at 100 s., showing that good results can be obtained using this solution if a
proper radiant intensity is selected.
In the case of h = 0, Equation (7) diverges, but interestingly enough, its approximation
(Equation (9)) does not, and becomes equal to Equation (5). If m = 0, Equation (7)
behaves in a similar way to the constant heat flux solution, but presents lower surface
temperatures due to the presence of terms containing h.
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1000
900
800
Temperature (K)
700
600
Numerical Solution
500
Analytical - Constant (Eq. 3)
400
Analytical - Ramped - No Losses
(Eq. 5)
300
Analytical - Ramped - Heat
Losses - 1st Order (Eq. 9)
200
Analytical - Ramped - Heat
Losses - No Approx. (Eq. 7)
100
0
0
50
100
150
200
250
300
350
400
Time (s)
Figure Error! No text of specified style in document..3. Comparison of different
analytical solutions. The numerical solution is regarded as being the exact
solution in this example. The curves for the numerical solution and Equation (8)
are superposed, showing the good agreement of the full analytical solution.
Thermophysical properties similar to those of wood were used. In all cases, the
ramp used is 83.3 W·m-2·s-1 and the total heat transfer coefficient was 35 W·m -2·s1
. The constant incident heat flux used in Equation (3) was of 8,330 W·m -2, i.e. the
level of irradiance of the ramping heat flux at 100 s.
1.3.
Decoupling of Solid and Gaseous Phases
Beyond all criticism, the methodology originally described in ASTM-E-1321 [12] has
shown that independent of the fuel and experimental conditions the linear dependency
of
with
prevails for a wide range of conditions [9, 13-15]. So far there is no
complete analysis that shows why this is the case when variables other that the simple
thermal model are incorporated.
If it is assumed that this functional dependency between external heat flux and time is
valid, integration over time shows that time scales with
.
This dependency can be readily demonstrated if
is constant, but showing its validity
for a time evolving incident heat flux will be part of the present study. If time can be
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scaled by the squared integral of the incident heat flux, then surface temperature and
ignition delay time can be presented as a function of the integral of the heat insult in a
single curve that can be used to completely de-couple solid and gas phases in the
numerical modelling of the ignition process.
The methodology used to demonstrate the dependency of the ignition delay time with
the squared integral of the incident heat flux will now be shown. If both sides of
Equation (3) are integrated over time, setting the upper integration limit as the time to
ignition,
(15)
the equation becomes
(16)
Finally, the dependence of the ignition time with the squared integral of the incident
energy can be established:
(17)
In this case, the integral of the incident heat flux is trivial. The following figure shows
the scaling of experimental ignition data for PMMA taken from the work of Dakka et al.
[14]. Scattering is greater for the lower heat fluxes (i.e. higher ignition times), showing
that the magnitude of the surface heat losses compared to the incident heat flux has an
effect on the scaling. This will be discussed in more detail in the following sections.
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350
300
Time (s)
250
200
150
100
50
0
0.0E+00
5.0E+06
1.0E+07
1.5E+07
2.0E+07
2.5E+07
3.0E+07
(Integral q_e)^2 (kJ/m2)^2
Figure Error! No text of specified style in document..4. Scaling of experimental ignition
data for PMMA [14]. The scaling was done using Equation (17).
In a similar way, the two other analytical solutions are integrated. In both cases the
term mt has been taken as
. So, for Equation (5) the integral is
(18)
Then, the time to ignition is expressed as
(19)
And for Equation (14) the solution is equivalent (following the definition in Equation
(10)):
(20)
For the two previous equations, the heat fluxes are more complex functions of time, but
the solutions for both integrals are nevertheless simple. The following table presents the
solutions to the integral of the incident heat flux for three cases.
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Table Error! No text of specified style in document.-2. Solution to the time
integral of the incident heat flux for the three cases discussed in this
report.
Thus, if the net heat flux on the surface of the solid as a function of time is known, the
integral can be calculated for the elapsed time (i.e. replacing tig by telapsed), and by
using the scaling results for the particular material and heating regime (i.e. Figure Error!
No text of specified style in document..4 or equivalent), a time to ignition can be calculated.
Ignition will be attained when the calculated time to ignition is equal or greater than the
elapsed time. An example is provided in Section 1.7 for illustrative purposes.
In order to validate the conclusions of this analysis, tests were conducted using ramping
heat fluxes. The aim was to compare the different scaling methods applying real cases.
The next sections will discuss the experimental procedure and results, to later examine
the de-coupling methodologies.
1.4.
Experimental Description
Three materials, Nylon 6 (PA6), PA6 with ceramic nano fillers, and
polymethylmethacrylate (PMMA) were chosen to illustrate the methodology. The
materials were tested using the FM Global Fire Propagation Apparatus (FPA) [16].
Piloted ignition tests with in-depth temperature measurements have been carried out on
110 x 110 x 12 mm samples of pure PA6, and of PA6 with nano-composites (Table
Error! No text of specified style in document.-3). A sample consists of two 110 x 110 x 6
mm blocks of material bonded together using super glue, with four 2 mm holes drilled
from the side at varying depths throughout the sample (Figure Error! No text of specified
style in document..5). The specimen is attached to an aluminium block, with heat contact
paste applied between the two surfaces to reduce the thermal contact resistance. Type
K thermocouples are inserted horizontally through the drilled holes at four depths within
the sample and vertically at three depths within the aluminium block (see Figure Error!
No text of specified style in document..5). These are connected to a data logger, and
temperature readings are taken every second. The specimen is then wrapped with
insulation to reduce any effect on the thermocouples of radiation from the sides. Time to
ignition is also recorded.
Piloted ignition tests were also performed on PMMA samples of dimensions 100 x 100 x
4.9 mm, using a similar methodology. No temperature measurements were carried out.
The heat flux ramps ranged from 0.01 to 5.0 kW·m−2·s−1. 36 samples were tested.
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Figure Error! No text of specified style in document..5. Side schematic of the specimen
arrangement. The depths of Thermocouples (TCx) for each sample can be found in
Table Error! No text of specified style in document.-3.
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Test
No.
1
Heat Flux Ramp
Thermocouple Depths
[kW/m2s]
[mm]
Material
PA6
TC1 = 1.5
TC3 = 6.5
TC2 = 3.1
TC4 = 8.9
TC1 = 1.6
TC3 = 6.0
TC2 = 3.3
TC4 = 9.3
TC1 = 1.9
TC3 = 6.0
TC2 = 3.1
TC4 = 9.0
TC1 = 1.5
TC3 = 6.9
TC2 = 3.2
TC4 = 9.2
TC1 = 2.2
TC3 = 6.9
TC2 = 3.3
TC4 = 9.3
-
-
0.1
2
PA6+NC
0.1
3
PA6+NC
0.1
4
PA6+NC
0.2
5
PA6+NC
0.2
6 – 41
PMMA
0.01 – 5.0
Table Error! No text of specified style in document.-3. Experimental
conditions.
In standard operation, samples tested in the FPA are exposed to constant heat fluxes.
Since this study required the samples to be subjected to time-dependent heat fluxes, a
MATLAB script was developed to linearly increase the heat flux over time. Figure Error!
No text of specified style in document..6 shows ramps that correspond to typical wildland
fire heating regimes as the fires approach a target. In all tests, calorimetry was
performed; however, for the purpose of this study, only ignition data and in-depth
temperature measurements will be presented.
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Figure Error! No text of specified style in document..6. Typical heat flux ramps used
during the testing campaign, which are representative of wildland fires.
1.5.
Experimental Results and Discussion
1.5.1. Validity of Semi-Infinite Solid Assumption
The analytical solutions presented in Section 1.2 are all based on the assumption of a
semi-infinite solid behaviour. The in-depth temperature measurements have been used
to test the validity of this assumption, thus assessing the applicability of the solutions to
the present experimental results.
Under ramping incident heat flux conditions, the semi-infinite solid assumption is no
longer valid when ignition is attained. The in-depth temperature heating of one sample
is presented in Figure Error! No text of specified style in document..7 and Figure Error! No
text of specified style in document..8 for two sets of times. It will nevertheless be seen in
the following sections that the proposed scaling gives good results, thus it can be
inferred that the departure from the semi-infinite solid behaviour is not as considerable.
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Figure Error! No text of specified style in document..7. Initial heating of the sample for
PA6.
Figure Error! No text of specified style in document..8. Final heating of the sample for PA6,
showing the onset of pyrolysis.
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1.5.2. Benchmark Tests for Constant Incident Heat Flux
A series of benchmark tests were conducted for a constant heat flux for PA6. The data is
presented in Figure Error! No text of specified style in document..9.
0.2
0.18
t_ig^-(1/2) (s^-(1/2))
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
10
20
30
40
50
60
Incident Heat Flux (kW/m2)
Figure Error! No text of specified style in document..9. Ignition delay times as a
function of the incident heat flux for PA6. For ease of comparison, the scales are
identical to the ones used in Figure Error! No text of specified style in document..10.
1.5.3. Ramping Heat Fluxes
Figure Error! No text of specified style in document..10 shows the ignition delay times for
the PMMA and PA6 ramping heat flux tests. The heat flux on the abscissa corresponds to
the average incident heat flux up to the time of ignition. The observed behaviour is
similar to the constant incident heat flux tests.
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0.2
0.18
t_ig^-(1/2) (s^-(1/2))
0.16
0.14
0.12
0.1
0.08
0.06
0.04
PMMA
0.02
PA6
0
0
10
20
30
40
50
60
Average Incident Heat Flux (kW/m2)
Figure Error! No text of specified style in document..10. Ignition delay times as a
function of the average incident heat flux (ramping) for PMMA and PA6.
However, if the times to ignition are plotted against the total incident energy for both
the constant incident heat flux and the ramping incident heat flux cases, it can be seen
that both cases follow a trend, as shown in Figure Error! No text of specified style in
document..11. This figure shows that for higher heat fluxes and steeper ramps the total
energy received by the sample at ignition is less than for lower heat fluxes. Thus, it is
not the incident energy but rather the relationship between the net heat flux on the
sample and the rate at which that energy is conducted into the sample which controls
ignition. In other words, assuming that ignition occurs at a constant temperature, it is
the speed at which this temperature is attained that will control the process.
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500
450
400
t_ig (s)
350
300
250
200
150
Ramping
100
Constant
50
0
0
2000
4000
6000
8000
10000
12000
Total Incident Energy (kJ/m2)
Figure Error! No text of specified style in document..11. Ignition delay times as a
function of the total incident energy for PA6. The results for constant and ramping
incident heat fluxes are presented.
Because the ramping heat flux tests done for PMMA covered a much wider range than
those done for PA6, only the former results will be presented and discussed from this
point onwards. Ignition times were plotted against the squared integral of the incident
heat flux, for the different analytical solutions presented in this work. Results are shown
in Error! Reference source not found. below.
The decoupling methodology involves the calculation of the time integral of the net heat
flux according to the different scalings discussed previously (Table Error! No text of
specified style in document.-2), and the calculated times to ignition will be dependent
on the scaling applied. This is demonstrated in Error! Reference source not found.,
where for a given value of the squared integral of the net heat flux, ignition times
present important differences depending on the type of scaling utilized. These
differences become less important for the higher values of the ramping heat fluxes. In
particular, it is observed that there is no significant difference in the predicted ignition
times obtained by the scalings for ramping incident heat fluxes with and without heat
losses, which means that for higher heat fluxes, the influence of the surface losses is not
important. At less steeper ramps surface heat losses become significant, and the
calculated ignition times can present considerable differences.
The constant heat flux scaling was carried out for incident heat fluxes corresponding to
those at 40% and 60% of the observed ignition time. In the former case, calculated
times to ignition will be overestimated. If the constant incident heat flux is set at values
above 50% (the latter case), then the ignition times obtained using this method will be
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shorter. For constant incident heat fluxes corresponding to the ramping heat fluxes at
50% of the ignition time, the scaling results match those corresponding to ramping
incident heat flux and no losses. This behaviour is easily explained by the fact that for
an increase in the incident heat flux, the total amount of incident energy on the surface
at the time of ignition decreases (see Figure Error! No text of specified style in
document..11).
700
Ignition Time (s)
600
500
400
300
200
Ramping, Surface Losses
Ramping, No Surface Losses
Constant Heat Flux (40%)
Constant Heat Flux (60%)
100
0
0
5
10
15
20
25
Squared Integral of Net Heat Flux (GJ2/m4)
30
35
Figure 1.12. Time to ignition scaling by the squared integral of the incident heat
flux, for the different analytical solutions. Results are done for PMMA. The
constant heat flux scaling results were carried out for the heat fluxes values
corresponding to the irradiance levels of the ramping heat flux at 40% and 60%
of the ignition time. The convective coefficient used in the scaling is 25 W·m-2·K-1.
1.6.
Effect of Material Thermophysical Properties
The effect of material thermophysical properties on the relative magnitude of the
incident heat flux and the surface heat losses terms in Equation (9) has been studied.
The ratio of the heat losses term to the incident heat flux term was plotted against the
surface temperature given by Equation (9) for a ramp of 50 W·m-2·s-1, using properties
for several standard building materials [17-20]. Note that the expression in Equation (9)
corresponds to an approximation of the analytical expression carried out for small times.
Its range of applicability is therefore restricted; see Figure Error! No text of specified style
in document..3 for a comparison between this approximation and the exact solution. The
results are presented in Figure Error! No text of specified style in document..12.
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0.7
Foam
PA6
PE
0.6
Wood
PMMA
Hardboard
Ratio of terms
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
300
Surf. Temp. (°C)
Figure Error! No text of specified style in document..12. Ratio of the surface heat losses
term and the incident heat flux term in Equation (9) as a function of the
calculated surface temperature for a ramp of 50 W·m-2·s-1. To calculate the
surface temperatures, the same range of times was used in each case, from 10 –
300 s. In the case of Foam, where the value for thermal inertia used was an order
of magnitude smaller than the rest of the values, for the given conditions surface
temperatures actually started to decrease after 70 s; therefore, the times used
ranged from 5 – 60 s. The data was obtained from the following references: Foam
[17]; Wood [18]; PA6 [19]; PMMA [20]; PE [20]; Hardboard [17].
This analysis permits testing the validity of performing the scaling using Equation (5),
that is without considering the heat losses. For cases of low thermal inertia, as can be
seen in Figure Error! No text of specified style in document..12, the term corresponding to
surface losses attains significant values compared to the incident heat flux term, thus
rendering the assumption of no surface losses invalid. For materials with higher values
of thermal inertia this assumption becomes valid, and it will be shown in the next section
that for a material with an average thermal inertia, the errors incurred when using this
assumption are negligible.
1.7.
Example of the use of the decoupling methodology
The following example is provided in order to give a more detailed description of the
methodology that is proposed in this work. The example is done for PMMA subjected to
a ramping incident heat flux. Using experimental ignition data, the scaling is carried out
in order to obtain a graph relating the time to ignition to the square of the integral of
the incident heat flux. This graph is shown in Error! Reference source not found.. A
curve fit is then performed on the data to obtain an expression for that relationship
(depending on the scaling employed, the relationship can be linear or of a higher order;
see Error! Reference source not found.).
DX.Y-Z-MM-0100-1
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The time to ignition is calculated by computing the squared integral of the incident heat
flux up to the elapsed time, and using the aforementioned expression to obtain a value
for an ignition time. The proposed scaling implies that the ignition time at a given
incident heat flux (be it ramping or constant) is proportional to the square of the integral
of the incident heat flux evaluated between time zero and the time to ignition. This
means that ignition will be attained once the calculated time is equal to the elapsed
time.
Error! Reference source not found. displays the evolution of the calculated times to
ignition to the elapsed time. Thus, every ramp will have its own distinct curve for the
calculated ignition time. The computed times to ignition compare well with the
experimental times to ignition.
As was stated before, the difference in the predicted ignition times for the scaling
considering surface heat losses as opposed to the scaling without surface losses will
depend on the material thermophysical properties. Table Error! No text of specified style
in document.-4 shows the differences in ignition times obtained for both cases. Only at
very low ramp values the difference becomes appreciable, but these cases can be
considered as impractical for fire applications. Figure Error! No text of specified style in
document..13 presents the comparison of the ignition times obtained using both scalings
for a ramp of 15 W·m-2·s-1.
Calculated Ignition Time (s)
700
600
500
400
30.00 (W/m2s)
50.00 (W/m2s)
300
150.00 (W/m2s)
600.00 (W/m2s)
200
100
0
0
100
200
300
400
500
600
700
Elapsed Time (s)
Figure 1.14. Calculated ignition time as a function of the elapsed time, for
different values of the heat flux ramp. The blue line corresponds to ignition times
equal to elapsed times, therefore the intersect of the calculated ignition times for
each particular ramp with the blue line will determine the time to ignition.
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Calculated Ignition Time (s)
1200
1000
800
Surface Losses
600
No Losses
400
200
0
0
200
400
600
800
1000
1200
Elapsed Time (s)
Figure Error! No text of specified style in document..13. Comparison of the results
obtained for the scalings with and without considering surface heat losses. The
ramp used in this example is of 15 W·m-2 ·s-1. The blue line corresponds to ignition
times equal to elapsed times, therefore the intersect of the calculated ignition
times for each particular ramp with the blue line will determine the time to
ignition.
Table Error! No text of specified style in document.-4. Calculated ignition times
for both scalings. The error was calculated using the surface losses case as
the base value.
1.8.
Concluding Remarks
Forest fire models must resolve different phenomena occurring at a wide range of time
and length scales, and the application of simplifications to certain events can greatly
improve their computational efficiency. The decoupling of the solid and gas phases
greatly reduces the computational effort in these models. This decoupling can be readily
accomplished using standard analytical solutions for the ignition time, but in some cases
like approaching fires in the WUI these solutions are not realistic in terms of boundary
conditions. The proposed decoupling methodology aims at providing a more realistic
DX.Y-Z-MM-0100-1
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approach by considering an incident heat flux linearly growing with time. This
methodology incorporates a novel analytical solution for the ignition of solid samples
exposed to ramping incident heat fluxes and convective cooling at the exposed surface.
This method has been validated for constant and time-varying incident heat fluxes, and
has been shown to be accurate. It is recommended that materials with low thermal
inertia incorporate the scaling that considers surface heat losses, while it has been
shown that materials with higher values of thermal inertia can use the scaling without
surface losses.
The authors believe this method will prove to be a useful tool in fire modelling, where
there is a growing need to model more complex situations, such as those that can be
encountered in the wildland-urban interfaces. This contribution will be particularly useful
in the scope of the Fire Paradox project, where numerous attempts in forest fire
modelling have been proposed. In particular, it is expected to incorporate these results
in the work carried out by VTT on W-FDS.
In terms of future work, in order for this methodology to be used in real applications, a
database of ignition tests of different materials should be compiled. An obvious first
choice would be doing the testing on wood, and perhaps study the applicability of the
model to predict the ignition of live trees.
1.9.
References
1. Torero, J.L., “Flaming Ignition of Solid Fuels”, in: SFPE Handbook of Fire Protection
Engineering, 4th Edition, Society of Fire Protection Engineers, 2009.
2. Sullivan, A., “A Review of Wildland Fire Spread Modelling, 1990-Present, 1: Physical
and Quasi-Physical Models”, arXiv:0706.3074v1[physics.geo-ph] (2007).
3. Novozhilov, V., “Computational Fluid Dynamics Modeling of Compartment Fires”,
Progress in Energy and Combustion Science 27:611–666 (2001).
4. Manzello, S. L., Park, S. H., Cleary, T. G., Shields, J. R, “Developing Rapid Response
Instrumentation Packages to Quantify Structure Ignition Mechanisms in WildlandUrban Interface (WUI) Fires”, Fire and Materials 2009. 11th International Conference
Proceedings, San Francisco, CA, 2009, pp. 215–224.
5. Rehm, R., “Effects of Winds from Burning Structures on Ground-Fire Propagation at
the Wildland-Urban Interface. Final Report”, NIST GCR 06-892, National Institute of
Standards and Technology (2006).
6. Cohen, J.D., “Relating flame radiation to home ignition using modeling and
experimental crown fires”, Can. J. For. Res. 34:1616–1626 (2004).
7. Quintiere, J., “A Simplified Theory for Generalizing Results from a Radiant Panel Rate
of Flame Spread Apparatus”, Fire and Materials 5:52–60 (1981).
8. Lawson, D.I. and Simms D.L., “Reply to Ignition of Wood by Radiation”, British
Journal of Applied Physics, 3:394–396 (1952).
9. Quintiere, J., Fundamentals of Fire Phenomena, John Wiley & Sons, Chichester,
2006.
10. Long, R., Torero, J.L., Quintiere, J., Fernandez-Pello, C., “Scale and Transport
Considerations on Piloted Ignition of PMMA”, Fire Safety Science, Proceedings 6th
International Symposium, Poitiers, France, 2000, pp. 567–578.
DX.Y-Z-MM-0100-1
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11. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, Oxford University Press,
London, 1959.
12. Standard test Method for Determining Material Ignition and Flame Spread Properties,
ASTM-E-1321-90, ASTM, Philadelphia, 1990
13. Atreya, A., “Ignition of Fires”, Philosophical Transactions Royal Society London A
356:2787–2813 (1998).
14. Dakka, S., Jackson, G. and Torero, J.L., “Mechanisms Controlling the Degradation of
Poly(Methylmethacrylate) Prior to Piloted Ignition”, Proceedings of the Combustion
Institute 29: 281–287 (2002).
15. Boonmee, N. and Quintiere, J.G., “Glowing and flaming autoignition of wood”,
Proceedings of the Combustion Institute 29: 289–296 (2002).
16. “ASTM E2058-03 Standard Test Method for Measurement of Synthetic Polymer
Material Flammability Using a Fire Propagation Apparatus”, ASTM International, West
Conshohocken (2003).
17. Quintiere, J., “Surface Flame Spread”, in: SFPE Handbook of Fire Protection
Engineering, 3rd Edition, Society of Fire Protection Engineers, 2002.
18. Reszka, P., “In-Depth Temperature Profiles in Pyrolyzing Wood”, PhD Thesis, The
University of Edinburgh, 2008. http://hdl.handle.net/1842/2602
19. 19. Steinhaus, T., “Determination of Intrinsic Material Flammability Properties from
Material Tests Assisted by Numerical Modelling”, PhD Thesis, The University of
Edinburgh, 2009.
20. Tewarson, A., “Generation of Heat and Chemical Compounds in Fires”, in: SFPE
Handbook of Fire Protection Engineering, 3rd Edition, Society of Fire Protection
Engineers, 2002.
DX.Y-Z-MM-0100-1
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2. Criteria for the assessment of wildland fire effect on burn injuries:
Model description and experimental results.
2.1. Introduction
This report is concerned with the development of a numerical model for the assessment
of burn injuries affecting fire fighters. For a more detailed discussion on the mechanisms
of burn injuries and the state of the art of burn injury prediction and modelling of skin
when subjected to heat, please refer to the literature review in IR 2.5-3. A description of
the model developed to assess the influence of an impermeable membrane in the
degree of burning and a comparison with experimental results is also presented here.
Burn injuries when fighting fires are common [1]. Protective fireman clothing is indeed
effective in the prevention of burns, but the design requirements and testing procedures
do not address all the potential modes of occurrence of burn injuries, and thus
protective garments fail in some scenarios. Protective gear can normally fail in three
modes during a fire. The first mode is direct impingement from flames or exposure to
high heat fluxes for times longer than a critical period of time; a second type of failure
occurs when the clothing is compressed, because the fabric significantly reduces its
thermal resistance [2], thus increasing the heat flowing from the hot exterior to the skin.
This type of injury is caused by direct contact with hot solids or when the fireman kneels
or crawls. Finally, steam passing through the clothes and getting into contact with the
skin can also cause scalding. For this reason, moisture barriers are included in the
protective fabrics [3].
In summary, studies on fire fighter gear have shown that the performance of protective
clothing is affected dramatically by three different variables, the structure of the material
(thermal properties), the compression level and the water content. The first variable is
a design parameter but the other two cannot be specified since they are related to
usage.
The models developed to predict burn injuries have dealt with the protective clothing as
an external boundary condition [4]. The boundary condition imposed on the skin model
is normally a heat flux at the inner side of the layer. Although there is general
recognition that water transport and latent heat play a significant role in the
establishment of a burn injury and in the properties of the protective gear, currently no
model exists that incorporates the interaction of the skin with the protective barrier
through the transport of water. Water will not only be present as the steam coming from
the exterior of the protective barrier, it will also be present in condensed and gaseous
phase within the barrier, as a result of perspiration. Water migration from the interior of
the barrier can constitute an important mechanism through which heat can be
transferred away from the skin, thus being beneficial in terms of the thermal damage to
the skin. Thus, the presence of a moisture barrier in the protective layer will prevent this
migration.
The aim of this study is to analyze the effect of the moisture barrier in the heat insult to
the skin. For this reason, a series of simple tests have been performed on water- soaked
polyurethane foam exposed to different heat fluxes with and without the presence of an
DX.Y-Z-MM-0100-1
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impermeable layer. The experimental results were compared to the results of a onedimensional heat transfer model, which includes the migration of water and the
presence of the moisture barrier.
2.2.
Experimental Procedure
Polyurethane foam samples were exposed to a range of incident heat fluxes with two
different vapour transfer boundary conditions: an open and impermeable. The test
results provided temperature histories at various depths in the simulated skin samples.
These temperature histories allow for simple comparison with modelling results.
The samples were 100 x 100 x 40 mm blocks of open cell, high porous texture, flexible,
non- flame retarded polyurethane foam. Each specimen was placed in a Cone
Calorimeter-type sample holder [5], sealed with an unbroken layer of aluminium foil to
prevent water escaping through the joint of the base and cover of the holder. The
moisture barrier used in the experiments to generate an impermeable moisture
boundary condition was a copper plate. The plate was painted black in order to create a
definite thermal boundary condition minimizing radiant reflection. The samples were
heated using a cone-heating element. The imposed heat flux on the samples ranged
from 0 – 20 kW·m-2.
Type K thermocouples with 0.25 mm wire diameter were used. Six thermocouples were
used in each test, one located 10mm above the surface of the sample and the remaining
five were inserted at 5 and 10 mm intervals along the centreline of the sample, starting
at 5mm down into sample. The thermocouples were labelled 1 to 6, with thermocouple
no. 6 being the one at the surface of the sample and thermocouple no. 1 at the lowest
point in the sample. The thermocouple layout is shown in detail on Figure Error! No text
of specified style in document..14.
6
5
4
3
1
2
Figure Error! No text of specified style in document..14 Sample thermocouple layout
Each hole in the foam specimen was carefully made with a 0.25 mm hole punch. The
thermocouples were the inserted to a depth of 50 mm, the centreline of the sample.
The sample was inserted into the sample holder and then saturated with approximately
DX.Y-Z-MM-0100-1
Page 30 of 43
200 ml of cold water. The initial specimen mass was recorded using an electronic scale.
The moisture barrier was put in place before the initial mass was recorded.
The selected incident heat fluxes were 3.0, 5.8, 9.8 and 17 kW·m-2, which corresponded
to cone heater temperatures of 400, 500, 600 and 700° C respectively. Each
experimental condition – incident heat flux and water migration boundary condition –
was repeated four times.
Temperature
(°C)
Heat Flux
(kW · m-2)
400
3.0
500
5.8
600
9.8
700
17.0
Table Error! No text of specified style in document.-5 Correlation between
heater temperature and incident heat flux on the sample.
2.3.
Experimental Results
The experiments showed good repeatability. As a sample, the following figure presents
the results for all four repetitions for thermocouple 5 (closest to the surface) for an
external heat flux of 5.8 kW·m-2, with no barrier.
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Comparison of Thermocouple 5 For Each Experiment
120
110
100
90
80
500-1
Temp
70
500-2
60
500-3
500-4
50
Average
40
30
20
10
0
0
500
1000
1500
2000
2500
Time
Figure Error! No text of specified style in document..15 Temperature histories for
thermocouple #5 (closest to the surface), for an incident heat flux of 5.8 kW·m-2,
with no moisture barrier. The average curve is also shown, in red.
The first and most noticeable stage corresponds to the preheating stage and took place
during the first 500 seconds of the experiments. As the water-saturated sample
continued to be heated, water began to migrate freely through the sample. At
approximately 1600 seconds into the experiment it was clear that the upper part of the
foam had become distinctively dryer than the rest of the sample as the surface began to
char slightly, as can be seen in the following figure.
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Figure Error! No text of specified style in document..16 Surface of a foam specimen
after 1600 s for an incident heat flux of 5.8 kW·m-2, with no moisture barrier.
The effect of the boundary conditions on the temperature histories is clear. Precluding
moisture migration from the front face of the specimen will yield higher temperatures in
the interior of the foam sample. Moisture migration represents an important source of
heat loss from the sample, which helps lower the in-depth temperatures. The
comparison of the temperature histories for thermocouple #5, for experiments with and
without the presence of the moisture barrier is presented in Figure 1.4. In terms of mass
loss, specimens with free moisture migration lose water at higher rates than those with
an impermeable barrier.
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Thermocouple No. 5
120
110
100
90
o
Temperature ( C)
80
70
500 - No Plate
60
500 - Plate
50
40
30
20
10
0
0
500
1000
1500
2000
2500
Time (seconds)
Figure Error! No text of specified style in document..17 Comparison of averaged
temperature histories for thermocouple # 5 (closest to the surface), for an
incident heat flux of 5.8 kW·m-2;the red curve corresponds to the experiments
with a moisture barrier and the blue curve to experiments without any barrier.
Figure 1.5 shows the results for thermocouple #5 at various incident heat fluxes and
with a permeable boundary condition. Temperature histories vary as expected with the
varying heat fluxes. Only in the higher heat fluxes temperatures get close to the water
boiling point, but a clear temperature plateau indicating the change of phase is not
observed, which indicates that the water mass loss process is mainly by migration and
evaporation at the surface. Temperature histories for tests with the moisture barrier in
place show faster heating regimes, and an indication that, for the higher heat fluxes, a
change of phase is attained, as presented in Figure 1.6.
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TC5
120
400 - No Plate
110
500 - No Plate Benchmark
600 - No Plate
100
700 - No Plate
90
o
Temperature C
80
70
60
50
40
30
20
10
0
0
500
1000
1500
2000
2500
Time (sec)
Figure Error! No text of specified style in document..18 Comparison of averaged
temperature histories for thermocouple # 5 (closest to the surface), for incident
heat fluxes of 3.0, 5.8, 9.8 and 17.0 kW·m-2; all the experiments were carried out
without a moisture barrier.
TC5
120
400 - Plate
110
500 - Plate Benchmark
600 - Plate
100
700 - Plate
90
o
Temperature C
80
70
60
50
40
30
20
10
0
0
500
1000
1500
2000
2500
Time (seconds)
Figure Error! No text of specified style in document..19 Comparison of averaged
temperature histories for thermocouple # 5 (closest to the surface), for incident
DX.Y-Z-MM-0100-1
Page 35 of 43
heat fluxes of 3.0, 5.8, 9.8 and 17.0 kW·m-2; all the experiments were carried out
with a moisture barrier.
The thermal penetration depth is greater for the tests with the moisture barrier, as
shown in Figure 1.7 and 1.8. This implies that the thermal damage to the skin will be
deeper when the boundary condition at the skin surface prevents an adequate moisture
migration.
70
60
Temperature (°C)
50
TC 1
40
TC 2
TC 3
TC 4
30
TC 5
20
10
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Time (s)
Figure Error! No text of specified style in document..20 Comparison of averaged
temperature histories at various depths for an incident heat flux of 5.8 kW·m-2,
and no moisture barrier.
120
100
Temperature (°C)
80
TC 1
TC 2
60
TC 3
TC 4
,
TC 5
40
20
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Time (s)
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Figure Error! No text of specified style in document..21 Comparison of averaged
temperature histories at various depths for an incident heat flux of 5.8 kW·m-2,
and with moisture barrier.
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2.4.
Modelling
It has been recognized that the effect of moisture migration on the severity of skin
burns plays an important role, and therefore it becomes important to model this process
in an accurate way. Modelling the behaviour of the foam samples subjected to heat
fluxes representative of those encountered in skin burn accidents with and without the
presence of a moisture barrier is a first step in the development of a mathematical
model of human skin. As an initial approximation, the model has only included heat and
moisture transfer, but has not taken into account any chemical or physical changes on
the foam matrix. The model treats the foam specimen as one-dimensional.
The model considers three species, foam, liquid water and vapour. The physical
phenomena included are rise in control volume sensible heat, conduction, gas
convection, and water evaporation. No movement of liquid water or diffusive mass
transfer was modelled. These effects are however deemed to be of little importance to
the results [6, 7]. The porosity of the foam matrix is given by
, and water is
considered to be located inside the pores. The energy equation is presented below.
(1)
Mass conservation is done for water and vapour. Moisture evaporation was modelled as
a first order Arrhenius expression.
(2)
Initial conditions are as follows:
(3)
At the boundaries, radiative and convective losses from the surface are considered.
DX.Y-Z-MM-0100-1
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(4)
Pressure is calculated by doing a mass conservation equation for vapour and, by
assuming ideal gas behaviour, getting an expression for the pressure in terms of the
total vapour mass concentration, as shown in Equation (5). The flow of vapour was then
calculated using Darcy’s law.
(5)
The model was solved using the method developed by Crank, Nicolson, Bamford and
Malan [8, 9], as formulated by Kung [10]. The temperature for the next time step is
assumed to be equal to the temperature at the present time, and with that, new values
for densities, solid and gas phase volumes, porosities, permeability, pressure, velocity,
effective thermal conductivity, mass fluxes, enthalpies and boundary conditions are
calculated. The energy equation is then solved using the updated parameters. With the
new temperatures, the previous parameters are calculated again, and after that the
energy equation is solved. This loop is repeated until the temperatures converge to a
given tolerance value. A Matlab code was developed for the solution of the set of
equations.
2.5.
Numerical Results
The model was compared with two experimental results, namely those for incident heat
fluxes of 5.8 and 9.8 kW·m-2. The results are satisfactory, showing close agreement
between the measured and predicted results, especially for the time scale of skin burns,
which is of the order of minutes.
Some numerical instabilities were observed for the case with the presence of the
moisture barrier, which is believed to be caused by the solution algorithm. Specifically,
the way the vapour velocities are calculated should be reviewed (see Equation (5)
above).
DX.Y-Z-MM-0100-1
Page 39 of 43
350
340
Temp (K)
330
Model 5 mm
320
Thermocouple 5
Thermocouple 4
310
Model 10 mm
300
290
0
100
200
300
400
500
600
Time (s)
Figure Error! No text of specified style in document..22 Comparison of numerical and
experimental results at depths of 5 and 10 mm from the exposed surface for an
incident heat flux of 5.8 kW·m-2, without moisture barrier.
350
340
Temp (K)
330
Model 5 mm
Model 10 mm
320
Thermocouple 5
Thermocouple 4
310
300
290
0
100
200
300
400
500
600
Time (s)
Figure Error! No text of specified style in document..23 Comparison of numerical and
experimental results at depths of 5 and 10 mm from the exposed surface for an
incident heat flux of 5.8 kW·m-2, with moisture barrier. Note that the
thermocouple reading for a 10 mm depth shows higher temperatures than the
thermocouple located at a 5 mm depth, which can be attributed to experimental
error.
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370
360
350
340
Model 5 mm
Model 10 mm
330
Thermocouple 5
320
Thermocouple 4
310
300
290
0
100
200
300
400
500
600
Figure Error! No text of specified style in document..24 Comparison of numerical and
experimental results at depths of 5 and 10 mm from the exposed surface for an
incident heat flux of 9.8 kW·m-2, without moisture barrier. The differences
between the model predictions and the experimental results can be due to
misplacement of the thermocouple.
370
360
350
340
Model 5 mm
Model 10 mm
330
Thermocouple 5
320
Thermocouple 4
310
300
290
0
100
200
300
400
500
600
Figure Error! No text of specified style in document..25 Comparison of numerical and
experimental results at depths of 5 and 10 mm from the exposed surface for an
incident heat flux of 9.8 kW·m-2, with moisture barrier. Note that the differences
for the 10 mm depth thermocouple can be attributed to experimental error.
DX.Y-Z-MM-0100-1
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2.6.
Conclusions
It has been shown that the transport of moisture can be a significant factor affecting the
severity of skin burns on fire fighters. Thus, if water is allowed to freely exit the skin, the
penetration of the heat wave is going to be less important, and the consequent burn
less severe. This information can serve as the basis for research on the development of
new protective clothes and on the physiology of skin burns. On a shorter timescale, this
information can be passed on to fire fighters for them to apply the conclusions and
protect the men on the field.
2.7.

hfg



Nomenclature
2.8.
References
surface absorptivity
latent heat of vaporization (J·kg−1)
surface emissivity
reaction rate (kg·m−3·s−1)
porosity
h
specific enthalpy (J·kg−1)
hc
convective heat transfer coefficient (W·m−2·K−1)
k
thermal conductivity (W·m−1·K−1)
K
reaction rate coefficient (s−1)
L
total depth of sample (m)
p
pressure (N·m−2)

density (kg·m−3)
R
Universal Gas Constant (J·mol−1·K−1)
T
time (s)
T
temperature (K)

Stefan-Boltzmann constant (5,67x10−8 W·m−2·K−4)
u
gas velocity (m·s−1)
x
depth dimension (m)
Subscripts
f
foam
vap
vapour
wat
water
1.
2.
3.
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