Chemical Engineering Science 71 (2012) 239–251
Contents lists available at SciVerse ScienceDirect
Chemical Engineering Science
0009-25
doi:10.1
Abbre
coefficie
Differen
Dry-filmn Corr
E-m
rjcrewe1 Te2 Te
journal homepage: www.elsevier.com/locate/ces
A theoretical and experimental investigation of intumescent behaviour inprotective coatings for structural steel
J.E.J. Staggs a,n, R.J. Crewe b,1, R. Butler c,2
a Energy Research Institute, University of Leeds, Leeds, LS2 9JT, United Kingdomb Department of Forensic and Investigative Sciences, University of Central Lancashire, Preston, PR1 2HE, United Kingdomc International Paint Ltd., Gateshead, NE10 0JY, United Kingdom
a r t i c l e i n f o
Article history:
Received 27 August 2011
Received in revised form
5 December 2011
Accepted 9 December 2011Available online 17 December 2011
Keywords:
Heat transfer
Intumescent coating
Mathematical modelling
Porous media
Fire resistance
Steel protection
09/$ - see front matter & 2011 Elsevier Ltd. A
016/j.ces.2011.12.010
viations: CHTC, Convection heat transfer co
nt; MLC, Mass loss calorimeter; TGA, Therm
tial thermal analysis; DSC, Differential scann
thickness
esponding author. Tel.: þ44 133 343 2495.
ail addresses: [email protected] (J.E.J. Sta
@uclan.ac.uk (R.J. Crewe), rachel.butler@akzo
l.: þ44 1772 89 3578.
l.: þ44 191 401 2432.
a b s t r a c t
A mathematical model describing heat transfer and expansion processes within an experimental
intumescent coating is described. The model has been developed alongside a relatively comprehensive
experimental programme involving analytical methods, standard and non-standard furnace tests and
mass-loss calorimeter (MLC) tests. The model is fully continuous (rather than semi-discrete as in other
approaches) and uses a simple competitive reaction scheme to describe the kinetics of the initial gas-
forming step of the coating degradation reaction. The degradation mechanism is coupled with a char
expansion sub-model, where a fraction of the evolved gas is trapped causing expansion. This scheme
incorporates endothermic and exothermic reactions, the heats of which have been estimated from DTA.
Much effort has been expended on a realistic description of the heat transfer processes within the
expanding char and a detailed composite thermal conductivity model including radiation transfer
across pores is included. This has been calibrated for fully expanded chars using empirical temperature
dependent thermal conductivity data. Model results compare well with furnace test results. However,
results from MLC experiments demonstrate a larger than expected range in coating expansion than
predicted by the model. These observations emphasise the importance of the basic expansion
mechanism and demonstrate that this critical area requires more research.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The practice of protecting structural steel members inbuildings with intumescent coatings is now well established.In the event of a fire, the coatings are designed to expand oncontact with heat to provide a thermally insulating char thatdelays diffusion of heat to the steel. The coatings are typicallyapplied at a dry film thickness (DFT) of a few mm and so donot interfere with the architectural aesthetics of the steelmember. The expansion ratios are typically high – of the orderof 10 or more – and the resulting chars are therefore highlyporous, with low effective thermal conductivities at room
ll rights reserved.
efficient; HTC, Heat transfer
ogravimetric analysis; DTA,
ing calorimeter; DFT,
ggs),
nobel.com (R. Butler).
temperature (at elevated temperature thermal conductivity isaugmented by radiation heat transfer across the char pores).Provided that the expanded char remains in place during thefire, the enhanced thermal resistance imparted to the steelimplies that there is greater time available for evacuation orfire-fighting before the structural strength of the member iscompromised.
The process of intumescence is complex and despite the factthat it has been exploited in commercial coatings, remains poorlyunderstood. It involves an interplay of physical and chemicalprocesses that must occur in correct sequence in order to producethe insulating char. Furthermore, since the char must be highlyporous to provide thermal insulation, the average wall thicknessof the solid matrix must necessarily be low. This in turn presentsdifficulties in maintaining sufficient strength so that the charremains in place during a fire.
The fire resistance of a commercial coating is tested usingexpensive, large-scale methods where a coated beam section issubjected to a prescribed temperature regime such as describedin ISO 834-1 or BS EN 1991-1-2:2002. These tests can beconducted on free-standing beam sections (when the time takento reach a specified temperature is the test criterion) or on
Fig. 1. Simple competitive mechanism for char formation.
Virgin Coating
Δmp
Gas
Char
Δmg
Δmc
Gas
Char
Δm
Δm
ΔtΔt
Δmc + rg Δmg
Expanded char
(1 − rg) Δmg Gas escapes
Fig. 2. Simplification of the expanded char forming process.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251240
horizontal loaded sections, where the deflection of the beam isalso part of the test criteria. The need for a detailed understandingof the transient heat transfer properties of an expanding char isclear and a predictive tool to aid product development byreducing the number of experimental tests is therefore important.Whereas the performance of an inert insulating coating is rela-tively well understood and quantifiable (Staggs, 2011a), a dyna-mically expanding char, where the ultimate insulation propertiesdepend on the heating regime, is not well understood. There havebeen several attempts to model heat transfer in intumescentsystems and these are neatly reviewed by Griffin (2010). Most ofthe more complex models use a kinetic scheme of parallelreactions to model char degradation (Di Blasi and Branca 2001;Di Blasi, 2004; Griffin, 2010) where the Arrhenius parameters areestimated from TGA data. In this approach it is argued that themost important step in the degradation process is the gas-producing step and rather than adopt an involved description ofthe reaction kinetics across the whole temperature spectrum, acompetitive scheme is used to describe the behaviour in thevicinity of this region. The model is 1-D in space, assuming thatthe char expansion is normal to the coated substrate. Althoughexpansion is included, allowing for change of volume with time,the equations are re-cast using a front-fixing transformation forconvenience in the numerical solution. Standard finite differencemethods are used with adaptive time-stepping to solve thetransformed equations in the constant volume domain. Thetime-stepping method is fully implicit and the discrete nonlinearequations are solved at each time level using an iterative relaxa-tion method.
This paper describes the results of a two-year collaborationbetween the University of Leeds and International Paint Ltd. (acompany that is part of the Akzo Nobel group). The project waspart-sponsored by the Technology Strategy Board of the UKgovernment (the remainder of the funding coming from theindustrial partner) and its aim was to investigate the behaviourof an experimental intumescent coating in a variety of testconditions. The main goal of the work was to develop amathematical model of intumescence, guided by experimentalobservation and analytical data, whose effectiveness could beevaluated directly against high-quality test data. Although ana-lytical methods such as TGA, DTA, He pycnometry and tempera-ture-dependent thermal conductivity measurements have beenused in the work, the bulk of the experimental effort wasdirected towards obtaining data from standard and non-stan-dard furnace tests (utilising steel plates rather than beam orcolumn sections) and also from bench-scale mass-loss calori-meter (MLC) tests.
2. Mathematical model
2.1. Char formation sub-model
A detailed description of the chemistry of the degradationprocesses involved in forming the intumescent char from thevirgin coating is beyond the scope of this work and will beinvestigated in a separate paper. Instead, the focus is on obtainingthe simplest description of the reaction kinetics that adequatelydescribes the main mass loss step observed in TG experiments.This step is associated with the formation of the blowing agentand hence char expansion. It transpires that a simple competitivereaction pair as shown in Fig. 1 is sufficient to crudely approx-imate the mass loss process. Naturally this mechanism is far toosimplistic to describe any of the fine details of the char-formingprocess.
Thus if a small enough section of coating is considered, suchthat temperature gradients are negligible, it is assumed that the
equations defining the amounts of unreacted coating, gas andsolid char are given by
dmp
dt¼�ðkgþkcÞmp,
dmg
dt¼ kgmp,
dmc
dt¼ kcmp, ð1Þ
where kj ¼ Ajexpð�Tj=TÞ, j¼c, g. Here Aj and Tj represent pre-exponential factors and activation temperatures respectively.
The mechanism of char expansion is problematic in anyintumescent model. In this approach the char mc in the compe-titive reaction is initially viewed as an expanded char precursor.Gas is then trapped within the char precursor, generating theexpanded char. It is assumed that the expansion occurs as soon asgas and char precursor are formed together. The amount ofexpanded char produced in a given time step depends on thelocal amounts of char precursor and blowing agent formed. Indetail, consider a small control volume V of reacting coating. Thevolume V contains up to three species: un-reacted coating ofdensity rp, solid char of density rc,s and trapped gas of density rg.Let m be the mass of V and let r be the overall bulk density.Suppose that in time step Dt an amount of gas Dmg and anamount of char precursor Dmc is generated. We assume that afraction rg51 of gas is trapped within Dmc, forming expandedchar (Fig. 2).
Thus, in a given time step Dt, the change of volume will begiven by
DV ¼�Dmp
rp
þDmc
rc,s
þrgDmg
rg
� rgDmg
rg
: ð2Þ
Here the first term corresponds to the loss of volume fromconsumption of un-reacted coating, the second term correspondsto the change in volume of char precursor and the final termcorresponds to the volume of trapped gas, which will be muchgreater than the other two volume changes for an intumescentsystem. The change in mass will be Dm¼�(1�rg)DmgE�Dmg.Then since r�1dr/dt¼m�1dm/dt�V�1dV/dt, it follows that, to agood degree of approximation, the equation defining the evolu-tion of bulk density may be written in terms of the local massfraction of un-reacted coating m¼mp/m as
1
rdrdt¼� 1þ
rgrrg
!kgm, rð0Þ ¼ rp: ð3Þ
Since mp¼mm, it also follows from the kinetic equations that
1
mdmdt¼�ðkgð1�mÞþkcÞ, mð0Þ ¼ 1: ð4Þ
These last two equations are sufficient to track the state of theintumescent reaction throughout the total volume of the coating
Fig. 4. Section through a typical expanded char.
Fig. 5. SEM image of a typical expanded char section.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251 241
(see below). Also if j is the local porosity and since (1�j)Vcorresponds to the volume of solids in V, it follows that the localporosity may be written as
f¼ 1�m rrp
�ð1�mÞ rrc,s
, ð5Þ
If we consider the expanding coating as being comprised of alarge number of thin layers (Fig. 3), in a given time step Dt layer i
will expand an amount DVi¼rgDmg,i/rg¼rgkg(Ti)miriViDt/rg.Therefore the total movement at layer j will be given by thesum of expansions of every layer below j, i.e.
dyj ¼Dt
S
Xj
i ¼ 1
rgkgðTiÞmiri
rg
Vi, ð6Þ
where S is the cross-sectional area (assumed to be constantthrough the thickness of the expanding char). In the limit as eachlayer’s thickness approaches 0, we see that the rate of displace-ment d _y due to expansion, at distance y above the base of thecoating, is given by
d _y ¼Z y
0
rgrrg
kgðTÞm dy: ð7Þ
2.2. Effective thermal conductivity
As the expanded char forms, the visible pores aremostly oblate spheroids with a wide range of sizes as Fig. 4confirms. An SEM (Scanning Electron Microscope) image of atypical expanded char section, obtained using a FEI Quanta 600machine, is shown in Fig. 5 and this reveals that there is muchfine detail to the pore structure that is not apparent to thenaked eye.
The effective thermal conductivity (ETC) l of a porous solid liesbetween two bounds (Kantorovich and Bar-Ziv, 1999; Staggs,2008, 2010):
l0=l1
l0=l1þjð1�l0=l1Þr
ll1
r1�j 1�l0
l1
� �: ð8Þ
here l0 represents the thermal conductivity of a pore and l1 thethermal conductivity of the solid matrix. The actual value of ETCdepends mainly on the shape and distribution of pores and to alesser extent on their size. In this model, Bruggeman’s model forrandomly distributed spheres with randomly distributed radii isemployed (Kantorovich and Bar-Ziv, 1999; Bakker, 1997). Here
Laye
Laye
Laye
Laye
Lay
Time ty
Steel
Heat
Fig. 3. Illustration of
the ETC is given implicitly by the equation
L�L0
L1=3ð1�L0Þ
¼ 1�j, ð9Þ
r 1
r 2
r 3
r 4
er j
Time t + Δt
Steel
char expansion.
3 In fact these forms of the equations are more convenient for numerical
solutions.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251242
where L¼l/l1 and L0¼l0/l1. Note that in order to use thisequation in the heat transfer model, it must be solved for L. Thepore thermal conductivity is assumed to be the same as that ofair, i.e. 0.026 W m�1 K�1. In reality the pore thermal conductivitywill vary with temperature (the thermal conductivity of air variesfrom 0.026 W m�1 K�1 at ambient temperature to 0.0657W m�1 K�1 at 973 K). However it is likely that this variation willbe swamped by the radiation enhancement effect arising fromradiation heat transfer across pores. This point is revisited andventilated further below.
There is a complication in applying this (or any) thermalconductivity model to the present case because as the char-forming reaction proceeds, the solid consists of three componentsas noted above. In order to work around this issue, it is assumedthat the solid comprises a uniformly distributed combinationof expanded char, consisting of solid matrix plus air-filledpores, together with un-reacted coating. Now, let the ETC of atwo-component system be given by leff¼K(ldisp,lsolid,jdisp), whereldisp is the thermal conductivity of the dispersed phase, lsolid is thethermal conductivity of the solid phase and jdisp is the volumefraction of the dispersed phase. The ETC of the expanded char willbe given by lc¼K(l0,lc,s,jc), where lc,s is the skeletal thermalconductivity of the expanded char and jc is the volume fractionof pores in the expanded char (not to be confused with the overallporosity given by Eq. (5), which takes into account the volume ofun-reacted coating). It may easily be shown that jc is related to jby jc¼j(1�mr/rp)�1. The overall ETC of the solid will then begiven by l¼K(lc,lp,1�Z), where lp is the thermal conductivity ofthe un-reacted coating and Z¼mr/rp is the volume fraction of un-reacted coating.
At high temperature, thermal conduction will be enhanced byradiation heat transfer across pores. If the pore size is small, thenit may be shown that this results in a temperature-dependentaugmentation to the basic ETC, which may be written in the form(Staggs, 2010)
ln¼ lþkRj
T3
T3a
�1
!: ð10Þ
This expression is obtained by considering the extra heat fluxresulting from radiation across small pores. Here kR is a parameterthat will depend on the properties of the pores including theirgeometrical shape and internal emissivity. It is also possible thatkR may depend on porosity itself if the pore sizes are not small.Thus we see that it is perfectly possible for a fixed coating subjectto different thermal histories to produce different pore structuresand hence different ETCs. In particular for a coating expandingvertically downwards, we would expect the added effect ofgravity to produce larger pores of different aspect ratio than fora coating expanding vertically upwards, where gravitationalresistance would cause more oblate pores to form. The estimationof kR is described in y3 below, where samples of expanded charsfrom furnace tests were used to obtain empirical measurementsof ETC across a temperature range up to 973 K.
2.3. Heat and mass transfer sub-model
Conservation of mass applied to the unreacted coating andsolid char respectively implies that
@ðmrÞ@tþ@ðmd _yÞ@y
¼�ðkgþkcÞmr, ð11Þ
@ðð1�mÞrÞ@t
þ@ðð1�mÞrd _yÞ
@y¼ kcmr: ð12Þ
As expected, adding these two equations recovers Eq. (3) withd/dt replaced by the advective derivative @=@tþd _y@=@y and using
this relation in the first Eq. (11) recovers Eq. (4) again with d/dt
replaced with the advective derivative.3
Conservation of energy in the coating (neglecting any workdone during expansion) implies that the temperature satisfies theequation
@ðrcTÞ
@tþ@ðrcTd _yÞ
@y¼�rm ðDHgþcgTÞkgþDHckc
� �þ@
@yln @T
@y
� �:
ð13Þ
here c is the net solid specific heat capacity given byc¼mcpþ(1�m)cc (the subscript p refers to a virgin coatingproperty and the subscript c to a char property) DHg is the heatchange as unit mass of coating is converted into gas, DHc is theheat change as unit mass of coating is converted into solid charand the ETC ln is given by Eq. (10). The local rate of displacementdue to expansion d _y is given by Eq. (7). In order to evaluate l* thelocal porosity j is required and this is given by Eq. (5). Again,using the relations above for dr/dt and dm/dt it is possible to showthat the temperature equation may be written in the moreconvenient form
rcdT
dt¼�rmD _Hþ @
@yln @T
@y
� �, ð14Þ
D _H ¼ kgfDHgþðcg�cpÞTgþkcfDHcþðcc�cpÞTg: ð15Þ
The location of the exposed surface y¼s(t) is given by thesolution of the integro-differential equation
ds
dt¼
Z s
0
rgrrg
kgðTÞm dy: ð16Þ
Note that the solution of this equation allows the model topredict the char expansion as a function of time: the predictedexpansion ratio being s(t)/l.
The situation for the substrate is straightforward and wesimply solve
rscs@T
@t¼
@
@yls@T
@y
� �ð17Þ
throughout the thickness, where the subscript s refers to asubstrate property.
The initial conditions are
Tðy,0Þ ¼ Ta, mðy,0Þ ¼ 1, rðy,0Þ ¼ rp, sð0Þ ¼ l, ð18Þ
where Ta is ambient temperature and l is the initial thickness ofcoating. The boundary conditions depend on the particularscenario being modelled. For the furnace test on the unexposedsurface y¼0 we have
ls@T
@y¼ haðT�TaÞ ð19Þ
and on the exposed face y¼s(t) following Staggs and Phylaktou(2008), Staggs (2011a):
ln @T
@y¼ hf ðTf�TÞþefsðT4
f �T4Þ: ð20Þ
here ha is the convection heat transfer coefficient (CHTC) for heattransfer from the unexposed surface to the ambient environmentoutside the furnace, hf is the CHTC for convection inside thefurnace, Tf is the temperature of the furnace gases and ef is theeffective emissivity of the furnace interior. The effective emissiv-ity approximately accounts for radiation both from the furnacegas and also from the interior surfaces of the furnace walls. The
300
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mass of gas
Mass of char
Experimental Competitive Scheme
Mas
s / I
nitia
l Mas
s
Temperature / K
Total Mass
400 500 600 700 800 900
Fig. 7. Comparison of competitive reaction scheme with constant heating rate TG
data at 10 K/min. The char and gas masses are calculated from the reaction
scheme.
4
)
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251 243
reader should refer to Staggs and Phylaktou (2008) for furtherdetails.
For the MLC, it is assumed that the sample is well insulated onits unexposed surface, so qT/qy¼0 on y¼0. On the exposedsurface y¼s(t) we have (Staggs, 2011b)
ln @T
@y¼ es FðT4
c�T4a ÞþðT
4a�T4
Þ
n oþhcðTa�TÞ: ð21Þ
here e is the emissivity of the exposed surface, Tc is the absolutetemperature of the heating element, hc is the CHTC for convectionunderneath the MLC heating element and F is the view factorfrom the cone heater to the exposed surface. Note that as theintumescent coating expands, its top surface will move uptowards the cone heating element, which has the effect ofchanging F. If FS�dA is the view factor from the interior surfaceof the cone heating element S to a differential target dA under-neath, then for a target of area A underneath the heating element,F¼ ðS=AÞ
R RAFS�dAdA. It may be shown that (Staggs, 2011b)
FS�dA ¼2dA
pS
Z z2
z ¼ z1
Z p
y ¼ 0
Rzðr1þtz1�acosyÞðR2þa2þz2�2RacosyÞ2
dydz, ð22Þ
where t¼(r1�r2)(z2�z1)�1, R¼r1�t(z�z1) and rj, zj are as shownin Fig. 6.
For typical MLC dimensions, the calculation above shows thatF ranges from 0.74 when the gap between the cone heater andsample (z1 in Fig. 6) is 25 mm to 0.84 when z1¼0.
0
-2
-1
0
1
2
3
Gas-forming reaction Char-forming reaction
ln (P
re-E
xpon
entia
l Fac
tor /
s-1
Heating Rate / Kmin-1
50 100 150 200
Fig. 8. Variation of pre-exponential factors with heating rate.
3. Model parameter values
When compared with constant heating rate TG experiments,the simple competitive mechanism adequately describes themain mass loss step, but the high-temperature thermal degrada-tion (which affects the final char yield) is not well reproduced.The graph in Fig. 7 illustrates this for a heating rate of 10 K/min.
TG experiments were conducted over a range of heating ratesusing a Mettler Toledo TGA/DSC 1, and Arrhenius parametersobtained for the competitive reaction. It was found that theactivation temperatures for each reaction were approximatelyconstant (�5065 K for the gas-forming reaction and �2639 K forthe char-forming reaction), but the pre-exponential factors variedstrongly for heating rates less than 50 K/min and mildly forheating rates greater than 50 K min�1, as shown in Fig. 8. Thissuggests that at heating rates below 50 K min�1 the assumeddegradation scheme is invalid and is approximately valid at highheating rates.
As with any reaction scheme incorporating Arrhenius kinetics,the cold boundary difficulty arises. This effect stems from the factthat terms like expð�TA=TÞ are non-zero at room temperature,implying finite reaction rate. In the present case, this problem iscircumvented by the usual device of a limiter, where the reaction
dA
a
z 2
z 1
r1
r2
MLC Heating Element
Differential Target
Fig. 6. Definition of geometrical parameters for MLC view factor.
rate is taken to be yexpð�TA=TÞ. Here y is assumed to be 0 attemperatures below a pre-set threshold (taken as 300 K) and1 when the temperature exceeds the threshold.
We can estimate the heating rate at the onset of intumescencein either the MLC or test furnace environment. The graphs in Fig. 9show both temperature and heating rate for blank steel plates of5 mm thickness in the MLC and also the test furnace (subject tothe standard hydrocarbon heating curve BS EN 1991-1-2:2002). Inthe case of the MLC test, the steel plate was coated with an inerthigh-emissivity paint of similar emissivity to the intumescentcoating. If 500 K is taken as a nominal temperature for the start ofthe intumescent reaction (a reasonable estimate from the data inFig. 7), we see that the heating rate at this temperature is high inboth cases (�80 K min�1). This observation is serendipitous sinceat heating rates in excess of 50 K min�1, the variation in derivedkinetic parameters from the TG experiments is small. In the lightof these results it was decided to take values for the pre-exponential factors in the middle of the slowly-varying regionat 100 K min�1, which gave 24.3 s�1 for the gas-forming reactionand 0.3 s�1 for the char-forming reaction. Although this is likelyto represent the initial char expansion reaction at the exposed
0
300
400
500
600
700
800
900
Time / s
Tem
pera
ture
/ K
Plate at 40 kWm-2
0
20
40
60
80
100
120
140
dT/d
t / K
min
-1
0200
400
600
800
1000
1200
1400
Time / s
Tem
pera
ture
/ K
Furnace
Plate
0
20
40
60
80
100
dT/d
t / K
min
-1
400 800 1200 1600 2000
400 800 1200 1600 2000
Fig. 9. Heating rates for blank steel plates in the MLC at 40 kW m�2 (top) and test
furnace (bottom).
Table 1Experimental values of expanded char thermal conductivity.
T (K) x¼(T/Ta)3�1 Thermal conductivity
(W m�1 K�1)
298 0.052 0.08
373 1.063 0.102
523 4.687 0.154
673 11.118 0.184
823 21.161 0.38
973 35.622 0.36
00.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Experimental Data Linear Fit 95 % Confidence Intervals
Ther
mal
Con
duct
ivity
/ W
m-1
K-1
x5 10 15 20 25 30 35 40
Fig. 10. Variation of measured ETC with x.
4 The main uncertainty for this value comes from the estimate of bulk density
and the main uncertainty for that comes from the determination of sample
volume, which is approximately 75%. If the uncertainty in the sample volume is
7e, then the uncertainty in j is approximately 7(1�j)e, i.e. about 0.2%.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251244
surface reasonably, as the intumescent reaction proceeds the localheating rate below the exposed surface will be much lower thanthe estimate above and so we should not expect the model topredict char formation well in these regions. It is tempting tosuggest that the kinetic parameters should be dependent onheating rate in order to circumvent this difficulty. However, inthe authors’ opinion this approach is unphysical and a bettersolution would be to implement a more detailed degradationmechanism.
The ETC model implies that a plot of ln against x¼(T/Ta)3�1
should be linear. Thermal conductivity measurements of anexpanded char section using a Netzsch model 457 MicroFlashTMlaser flash diffusivity apparatus were conducted with the resultsshown in Table 1 and a plot of measured ETC against x is shown inFig. 10. The dashed curves in this plot correspond to 95%confidence intervals for the linear fit, suggesting that the 823 Kdata point may be an outlier. These data confirm that themeasured variation in ETC is an order of magnitude greater than
the expected variation of the thermal conductivity of air over thegiven temperature range.
The skeletal density of the expanded char section was found byhelium pycnometry using a Micromeritics AccuPyc 1330 machineand this, combined with an estimate of the bulk density of theexpanded char, was used to estimate its porosity; giving a value ofjE0.96.4 The gradient of the line of best fit in Fig. 10 is 0.0086,with standard error 0.0018. Using this observation and the factthat Eq. (10) suggests that the gradient of a plot of ETC against x
should be jkR, we find that kRE0.009070.0019 W m�1 K�1.Unfortunately it was not possible to obtain other expanded charsamples with a sufficiently wide range of porosity to investigatethe suggested porosity dependence implied by Eq. (10). Therelative size of the radiation enhancement to conductivity com-pared with the room-temperature thermal conductivity is givenby the ratio R¼kRjx/l. When R51 radiation enhancement isnegligible, but when R41 it represents a significant contributionto the overall thermal conductivity. Across the temperature rangeof the data in Table 1, the derived parameters give a maximumvalue for R of 3.1, indicating the importance of radiation.
The trapped gas fraction was estimated from the bulk densityof expanded char samples obtained from furnace tests. If m0 is theinitial coating mass and mN is the final char mass, then the massof evolved gas is mg¼m0�mN. If rN is the char bulk density, thenneglecting the skeletal volume of the char (which is negligible forthe char samples considered here when compared with the voidvolume) the volume of trapped gas is approximately mN/rN,corresponding to a mass of mNrg/rN, where rg is the gas density.This implies that an estimate of the fraction of trapped gas isgiven by the ratio of the mass of trapped gas to the mass of gasproduced, i.e. mNrg/(mgrN). Now if rc¼mN/m0 is the char yield,then mg¼m0(1�rc) and so mN/mg¼rc/(1�rc) and it follows that
rg �rcrg
r1ð1�rcÞ: ð23Þ
The skeletal thermal conductivity of the expanded char wasestimated by compressing a sample of char under high pressurein an Instron compression tester and measuring the thermalconductivity of the compressed sample using the same techniqueas described above. The thermal conductivity of the virgin coatingwas determined using the same technique and the specific heat
800
1000
1200
1400m
pera
ture
/ K
Furnace
Low emissivity
High emissivity
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251 245
capacity obtained from DSC experiments. A sensitivity analysis ofthe model parameters (not included for brevity) showed that theskeletal specific heat capacity of the coating had virtually noeffect on model predictions (since the mass of coating and hencethe char is so low compared to the mass of the substrate).Therefore no separate measurement of this parameter was con-ducted and it was assumed to be the same as the virgin coating.The density of the virgin coating was found from measurementsof the mass of applied coating combined with a large number ofspot measurements of thickness using a micrometre to determinethe volume. The thermal properties of the evolved gas were takento be identical to air (Incropera and DeWitt, 1996).
The CHTC in the MLC hc was determined by a methodpresented in earlier papers (Staggs, 2009, 2011b). The resultsare shown in Fig. 11. Although there is no particular difficulty inincorporating a temperature-dependent correlation for the CHTC,it was decided to simply employ a temperature-averaged value.The data in Fig. 11 suggest that a reasonable correlation for hc ishc(T)E20.8þ0.0093T, where T is in K. Therefore if the cone heatflux is _q 00c , the steady surface temperature TN will be given by thesolution of e _q00cþhcðT1ÞðTa�T1ÞþesðT4
a�T41Þ¼ 0.
The average CHTC hc over the range Ta to TN is given by thecorrelation for hcevaluated at T ¼ ðTaþT1Þ=2. Taking an emissiv-ity of 0.95, consistent with the samples used to obtain the datashown in Fig. 11, the values for average CHTC are shown inTable 2 for a range of representative heat fluxes. In the light ofthese data it was decided to take the value 26.3 W m�2 K�1 forthe CHTC.
The test furnace heat transfer properties were found using amethod again detailed in earlier papers (Staggs and Phylaktou,2008; Staggs, 2011a). The method involves heating steel panelscoated with inert high temperature paint of known emissivityaccording to a standard fire test regime. The furnace CHTC hf andeffective emissivity are found by fitting an appropriate equation
50010
12
14
16
18
20
22
24
26
28
30
32
Con
vect
ion
Hea
t Tra
nsfe
r Coe
ffici
ent /
Wm
-2K
-1
Temperature / K600 700 800 900 1000
Fig. 11. Measured convection heat transfer coefficients in the MLC.
Table 2Average CHTC values.
_q 00c (kW m�2) TN (K) hc (W m�2 K�1)
30 751.1 25.7
40 822.4 26.0
50 881.0 26.3
60 931.1 26.5
70 975.1 26.7
for the steel temperature to the test data. Two furnace tests usingthe standard hydrocarbon curve, each with four panels coatedwith either high emissivity paint or low emissivity paint wereconducted and the averaged results are shown in Fig. 12.
The optimised fits to the experimental data were obtainedwith the following furnace heat transfer parameters:hf¼33.4 W m�2 K�1, ha¼3.0 W m�2 K�1, ef¼0.07.
The individual heats of reaction are difficult to determinewithout careful and detailed analytical techniques. Although adifferent reaction scheme was employed, Griffin (2010) used DSCto estimate heats of reaction. He found that for his particularcoatings, the main endothermic reaction was the first step, whichhe referred to as the ‘‘melting’’ reaction. He reported values of theorder 0.65–1.0 MJ kg�1, depending on the particular coating. Inher models, Di Blasi (2001, 2004) has used either estimated valuesor values obtained from Cagliostro et al. (1975).
In the present study DTA at 10 K/min under nitrogen was usedto estimate the heats of reaction for the simplified competitivescheme. Fig. 13 shows the DTA curve, corrected for baseline drift,together with sample mass and rate of mass loss from a TAInstruments Q600 Thermal Analyser. The DT curve clearly indi-cates that there is a mixture of endothermic and exothermicreactions occurring during the initial mass loss phase. Inspection
0200
400
600Te
Time / s1000 2000 3000 4000 5000 6000 7000 8000
Fig. 12. Results of test furnace calibration. The symbols correspond to test furnace
results (apart from the furnace temperature curves) and the curves correspond to
optimised fits.
3007
8
9
10
11
12
Temperature / K
Sam
ple
Mas
s / m
g
0.000
0.002
0.004
0.006
0.008
0.010
R2
-dm/dt / mgs ΔT / K
R1
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
350 400 450 500 550 600
Fig. 13. DTA results.
8
10
12
14
16MLC
-2 Dev
iatio
n / K
75 kWm-2
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251246
of the mass derivative �dm/dt (and also the presence of twoshoulders in the mass curve occurring after the main mass loss hasstarted) also shows that there are at least two further reactions(labelled R1 and R2 in Fig. 13) that are not included in the simplemodel presented above. Furthermore, since R1 and R2 coincidewith peaks in the DT curve, it may be inferred that these reactionsare both exothermic. A complete analysis of Fig. 13 together withan estimation of the heats of reaction for the competitive schemeemployed in the model is probably best deferred to a separatepaper. However a theoretical DT curve based on the competitivescheme with best estimates of the reaction heats is also shown inthe figure (dashed curve) for comparison.
Unless otherwise stated below, the model parameters (includ-ing those not referred to above) are summarised in Table 3. TheDSC measurements were obtained using a Netzch DSC 200F3Maia instrument.
00
2
4
6
00
5
10
15
20
25
30
35 kWm-250 kWm
Sta
ndar
dTime / s
4mm DFT
6mm DFT
Sta
ndar
d D
evia
tion
/ K
Time / s
10mm DFT
Furnace
200 400 600 800
500 1000 1500 2000 2500 3000 3500
Fig. 14. Variation in observed temperature for MLC (top) and hydrocarbon furnace
test (bottom). The MLC data are for 1 mm DFT.
4. Comparison with experiment
Experiments were conducted in both the MLC and test furnaceenvironments. The MLC study involved testing a variety of dry-film thicknesses (DFTs) of coating at three heat fluxes:35 kW m�2, 50 kW m�2 and 75 kW m�2. The bulk of the furnacetests used the standard hydrocarbon heating curve (BS EN 1991-1-2:2002) where the furnace temperature is given as a function oftime (in seconds) by
Tf ðtÞ ¼ 293þ1080ð1�0:325e�0:167t=60�0:675e�2:5t=60Þ: ð24Þ
A range of coating DFTs were again used in the tests. However,a non-standard heating curve was also used for completeness. Inboth test regimes, coatings were applied to insulated 5 mm thicksteel panels (10 cm�10 cm in the MLC and 30 cm�30 cm in thefurnace). However, an additional thicker substrate was also testedin the furnace environment.
A modestly large variation was observed in both furnace andMLC test results. The graphs in Fig. 14 show the standarddeviations of substrate temperature for 10 repeats at each heatflux in the MLC and 4 repeats at each DFT in the furnace. Ingeneral the variation of substrate temperature was larger for thefurnace than the MLC.
Table 3Summary of model parameters.
Kinetic parameters (all estimated from TGA)
Activation temperature (K)
Gas-forming 5065
Char-forming 2639
Thermal conductivity parameters
Void (l0) (W m�1 K�1) Skeletal (l1) (W m�1 K�1)
0.026 (taken as air) 0.36 (direct experimental
measurement)
Density parametersVirgin coating (rp)/kg m�3 Gas (rg) (kg m�3)
1200.0 (product data sheet) 1.0 (taken as air)
Specific heat capacitiesVirgin coating (cp) (J kg�1 K�1) Gas (cg) (J kg�1 K�1)
1498 (DSC measurment) 2170.0 (Taken as air)
Heat transfer coefficients (obtained from calibration experiments as detailed in Sta
MLC (hc)/W m�2K�1 Furnace interior (hf) (W m�2K�1)
26.3 33.4
Substrate properties (assumed to be steel)
Density (rs) (kg m�3) Specific heat capacity (cs)
(J kg�1 K�1)
7500.0 434.0
4.1. Furnace experiments
The photograph in Fig. 15 shows the 1.5 m3 furnace used atInternational Paint for the tests. The interior temperature ismonitored at a number of locations and the fuel flow rate to theburners is adjusted to reproduce a particular heating regime. Foursamples are usually tested together, mounted in a frame that is
Pre-exponential factor (s�1) Heat of reaction (MJ kg�1)
24.3 0.757
0.3 �1.550
Radiation (kR) (W m�1 K�1)
0.009 (inferred from direct
measurements)
Char skeletal (rc,s) (kg m�3) Trapped gas fraction (rg)
1886.0 (helium pycnometry) 0.017 (estimated from expanded
furnace chars)
Char (cc) (J kg�1 K�1)
663.0 (DSC measurment)
ggs and Phylaktou (2008) and Staggs (2009)
Furnace exterior (hf) (W m�2K�1) Furnace effective emissivity
3.0 0.07
Thermal conductivity (ks)
(W m�1 K�1)
52.0
0250
300
350
400
450
500
550
600
650
700
10mm DFT
4mm DFT
6mm DFT
Ste
el T
empe
ratu
re /
K
Time / s500 1000 1500 2000 2500 3000
Fig. 17. Comparison between model and furnace test results for the hydrocarbon
heating curve.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251 247
insulated on the unexposed face and mounted vertically in theopening shown in the photograph (Fig. 16, left). Five thermo-couples are attached to each steel panel as shown in Fig. 16, right.
The graph in Fig. 17 compares model results with furnace testresults for the standard hydrocarbon test curve for a range ofdifferent DFTs (the variation in experimental data is given in thebottom graph of Fig. 14). The model predictions are shown by thesolid curves and the experimental results by the symbols. Esti-mates of the final expansion ratios for the 4 mm and 6 mm DFTcoatings were obtained from the final volumes of expanded chars(3 estimates for the 4 mm DFT and 8 for the 6 mm DFT). Thesedata showed large variation in the final expansion for the thickercoating and yielded average expansion ratios of 10.4 for thethinner coating and 10.1 for the thicker coating (with a standarddeviations of 0.7 and 2.1, respectively). Model predictions for theexpansion ratios of 4 mm and 6 mm DFT coatings as functions oftime are shown in Fig. 18. Although ultimate expansion is over-predicted when compared to the experimentally observedaverages (obtained from fully expanded chars), the estimates fallswell within the observed ranges.
In order to investigate the effect of substrate thickness, 4 testsamples were prepared by applying 4 mm DFT of coating to10 mm thick steel plates. These were subject to the standardhydrocarbon heating regime in the test furnace and the resultsare compared with model predictions in Fig. 19. The average ofthe four test results are shown by symbols and model results by
Fig. 15. The test furnace.
Fig. 16. Configuration of test panels
the solid curve. The dashed curves are the experimental average7 one standard deviation. Again the agreement between modeland experiment is good.
and thermocouple attachment.
00
2
4
6
8
10
12
4 mm DFT Obsereved range of finalexpansion for 4 mm DFT(average +/- 1 std. dev.)
Exp
ansi
on R
atio
Time / s
Obsereved range of finalexpansion for 6 mm DFT(average +/- 1 std. dev.)
6 mm DFT
500 1000 1500 2000 2500 3000
Fig. 18. Computed expansion ratios compared with final measured values.
0
300
400
500
600
700 Experimental Average Average +/- 1 Std. Dev. Model
Tem
pera
ture
/ K
Time / s1000 2000 3000 4000
Fig. 19. Comparison with test results for different substrate thickness.
0
300
350
400
450
500
550
600
650
700
Time / s
Ste
el T
empe
ratu
re /
K
250
500
750
1000
1250
1500Fu
rnac
e Te
mpe
ratu
re /
K
500 1000 1500 2000 2500
Fig. 20. Comparison of furnace results with model results for a non-standard fire
test curve.
4
10
20
30
40
50
60
70
80
90
MLC Results ModelB
ulk
Den
sity
/ kg
m-3
Expansion Ratio6 8 10 12 14 16 18
Fig. 21. Ultimate expansion and bulk density for a range of DFTs.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251248
Finally, the effect of a different heating regime in the testfurnace was investigated by subjecting two test specimens (withcoating applied at 4 mm DFT and 6 mm DFT) to a non-standardtest curve. It was not possible to provide experimental variationfor this test because of limited availability of the test furnace. Thetest curve has a lower initial heating rate and a lower ultimatetemperature. The furnace test results are compared with modelcalculations in Fig. 20. Also shown in this figure is a typicalpractical interpretation of the standard hydrocarbon curve. Againthe agreement between the model and fire test results is good forboth test specimens.
4.2. MLC experiments
Given the nature of bench-scale tests in the MLC, they aremuch cheaper to carry out than furnace tests. Consequently it waspossible to conduct a more thorough experimental programmeusing the MLC than the test furnace (within the limits of theapparatus5). In particular, char expansion and bulk density weredetermined for a useful number of tests. Analysis of expanded
5 For example, the maximum coating thickness was limited to approxi-
mately 2 mm to prevent problems with the expanding char making contact with
the cone heater.
char residues after heating in the MLC was carried out todetermine ultimate bulk density and expansion ratio and theresults are shown in Fig. 21 (the symbols are coloured accordingto the heat flux used in the experiment). It is immediatelyapparent from this figure that the variation of sample expansionand bulk density is much greater in reality than that predicted bythe model. These data suggest that the amount of gas trappedduring expansion is related to heat flux (and therefore heatingrate) in a manner that has not been accounted for. After theexcellent agreement reported above for the furnace tests, thisresult came as something as a surprise. It also shows that, despitethe limited investigation in the furnace tests, the heating envir-onment can have a major effect on the resulting expansion of anintumescent coating.
In order to improve the agreement, without re-formulating theexpansion sub-model (which would be the best procedure in thelight of Fig. 21, but beyond the scope of this paper) a parametersensitivity analysis suggested that adjusting the fraction of gastrapped as a function of heat flux would produce better agree-ment. From a scientific (rather than an engineering) viewpoint weaccept that this deficiency in the model points to a deeperproblem in the expansion sub-model and this is an area of on-going research.
In order to improve the model predictions, the MLC results at35, 50 and 75 kW m�2 were analysed for coatings of 1 mm DFTand the optimum trapped gas fractions obtained. These are shownin Fig. 22. These data suggest that for coatings of 1 mm DFT, thetrapped gas fraction is an increasing function of heat flux andappears to approach a constant value for high heat fluxes. If weconsider the wider implications of this observation, then we areagain led back to the effect of heating rate on char expansion. Theeffect of heating rate on degradation kinetics was briefly dis-cussed in y3, where it was found that the derived Arrheniusparameters became quasi-constant at high heating rates. It isprobable that the range of heating rates that an expanding charexperiences in the MLC is greater than in the furnace test. If this iscorrect, then it would explain why the model performed better inthe furnace test scenario, where the heating rates are probablyhigher overall.
If the trapped gas fraction is adjusted for heat flux according toFig. 22, then the agreement across the range of applied DFTs ismuch improved, as Figs. 23 and 24 confirm.
The observed substrate temperatures are compared withpredictions from the modified model in Fig. 24 for a range of
4
10
20
30
40
50
60
70
80
90 MLC Modified Model
Bul
k D
ensi
ty /
kgm
-3
Expansion Ratio
Heat Flux / kWm-2
6 8 10 12 14 16 18
Fig. 23. Effect of incorporating variable trapped gas fraction on predicted bulk
density and expansion ratio.
0
300350400450500550600
300400500600700800900
300400500600700800900
2.0 mm DFT
1.67 mm DFT35 kWm-2
50 kWm-2
1.83 mm DFT
1.00 mm DFT
0.22 mm DFT
Sub
stra
te T
empe
ratu
re /
K
1.82 mm DFT
0.98 mm DFT
0.38 mm DFT75 kWm-2
Time / s
200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Fig. 24. Substrate temperatures for MLC study. The symbols are experimental
results and the curves are model predictions.
300.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
Frac
tion
of G
as T
rapp
ed
Heat Flux / kWm-2
40 50 60 70 80
Fig. 22. Trapped gas fraction relationship with heat flux.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251 249
coating DFTs at heat fluxes of 35 kW m�2, 50 kW m�2 and75 kW m�2. The variation in experimental data for 1 mm DFT isgiven in the top graph of Fig. 14. It was not possible to provideexperimental variation for coating thicknesses other than 1 mmbecause of time and lab equipment constraints. Experimentalresults are sensitive to coating DFT and it requires a great deal oftime and expertise to prepare test samples at specific DFTs.
5. Conclusion
A mathematical model of intumescent behaviour has beendeveloped and comprehensively investigated by a variety ofexperimental methods, including furnace and MLC testing of anexperimental intumescent coating. Most of the important modelparameters were determined by a range of analytical techniques,with the exception of the trapped gas fraction, which wasestimated from the bulk density of expanded char samples fromfurnace tests. The model was found to perform well whenpredicting intumescent behaviour in the test furnace. A numberof tests were conducted at different DFTs, substrate thicknesses
and heating regimes and in all cases the substrate temperaturescompared well with experimental observations.
The agreement with MLC experiments was less good and itappears that this is attributable to an insufficiently detailedrepresentation of the expansion process in the mathematicalmodel—the Achilles heel of most intumescent models publishedto date. In particular it was found that the range of char expansionwas much greater in the MLC experiments than the model wascapable of predicting using the same parameter set as the furnacetests unless the trapped gas fraction was adjusted. As heat fluxincreased, it was found that char expansion was much greaterthan expected and in order to obtain agreement between modeland MLC results it was necessary to increase the trapped gasfraction as a function of heat flux. This observation is mostirksome in the light of the success of the model in predicting awide variety of behaviour in the furnace environment, but alsomost enlightening. Firstly it demonstrates the large effect that aparticular heating regime can have on the performance of aparticular coating. This clearly has ramifications in the widerworld of fire safety, where performance in furnace tests is reliedon to assess a coating’s performance in a real fire. Secondly it alsodemonstrates that a detailed representation of the expansionprocess is a critically important component of a mathematicalmodel of intumescence and the somewhat simplified viewadopted here is insufficiently detailed to explain experimentalobservation. The TG results used to estimate the Arrheniusparameters for the simple degradation mechanism in the charformation model are only valid for moderately high heating rates.At lower heating rates the degradation mechanism is almostcertainly insufficiently complex to explain the observed mass loss.Consequently it follows that the gasification rate will be poorlypredicted. The MLC results demonstrate that char expansiondepends on external heat flux, which suggests a dependency onlocal heating rate. However, overall char expansion will be con-trolled by local gasification rate and also by how much gas istrapped locally throughout the char. It is relatively easy to modifythe degradation mechanism; however, it is not at all easy to
Sample from a Furnace Test
Sample from a MLC Test
Fig. 25. Typical expanded chars from the furnace test (top) and MLC test (bottom).
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251250
model the gas trapping process. Further work is presently focussedon attempting to develop more realistic models in both of theseareas.
The intumescent model has the restriction of being one-dimensional in space and it is assumed that char expansionis normal to the exposed surface. Given the severity of thisassumption, it is remarkable that anything approachingthe agreement with experiment reported above is achievable.In both furnace tests and MLC tests, char expansion is notone-dimensional, as shown in Fig. 25. Unfortunately, since thephysics of the char expansion process are so poorly understoodat present, a fully three-dimensional model seems a ratherdistant goal.
The effect of local oxygen concentration on the degradationkinetics and hence char expansion was also briefly considered,although not reported above for reasons of brevity. Typically,ambient oxygen concentration is lower in furnace tests (approxi-mately 5%) than tests conducted in the MLC at ambient atmo-spheric conditions. Constant heating rate TGA was conducted inboth air and nitrogen. The results show that oxygen concentrationdoes indeed have an effect on the degradation kinetics and that ingeneral, the main mass loss step occurs more slowly in air thannitrogen (a seemingly contradictive result in the light of thehigher than expected expansion ratios observed in the MLC).However, it is not clear if this observation is relevant for MLCexperiments. When gas is produced and vented from the coating,oxygen concentration in the locality of the reactive coating may inreality be low: only the earliest stages of the reaction or ultimatechar degradation will be susceptible to the effects of oxygenconcentration.
At present, it is difficult to see how a comprehensive descrip-tion of the expansion process will be achieved and also how it willbe translated into a useable mathematical model. As gas isproduced, the amount trapped (and therefore contributingtowards expansion) depends on the viscoelastic properties ofthe surrounding matrix, the gas pressure and the porosity. Evenif a sufficiently detailed model could be formulated, it is dubiousthat the rheological properties could be obtained. However, theresults discussed above have prompted a revised view of theexpansion sub-model incorporating a more realistic char forma-tion mechanism. This sub-model is currently under development
and will be incorporated into a revised version of the intumescentmodel in due course.
Nomenclature
A Pre-exponential factor, s�1
c Specific heat capacity, J kg�1 K�1
FS�dA View factor from surface of MLC heater to differentialelement dA
h Convection heat transfer coefficient, W m�2 K�1
k Reaction rate, s�1
kR Radiation enhancement coefficient in effective thermalconductivity, W m�1 K�1
l Initial coating thickness (DFT), mm Mass, kgrg Trapped gas fractions Location of exposed surface of coating, mt Time, sT Temperature, KV Volume, m3
y Distance through thickness of coating, mdy Local change of thickness of coating, md _y Local rate of change of coating thickness, m s�1
DH Heat of reaction, J kg�1
D _H Net rate of heat release per unit mass for degradationreactions
Dm Infinitesimal mass, kgDt Time step, se EmissivityZ Volume fractionj PorosityF View factor for MLCl Thermal conductivity, W m�1 K�1
L Thermal conductivity ratiom Mass fractionr Density, kg m�3
s Stefan–Boltzmann constant, W m�2 K�4
Main subscripts other than used above
a ambient conditionsc char bulk propertyc,s skeletal char propertyf furnace interior conditionsg gas propertyp unreacted coating propertys substrate property
References
Bakker, K., 1997. Using the finite element method to compute the influence ofcomplex porosity and inclusion structures on the thermal and electricalconductivity. Int. J. Heat Mass Transfer 40, pp3503–3511.
BS EN 1991-1-2, 2002. Eurocode 1: Actions on Structures.Cagliostro, D.E., Riccitello, S.R., Clark, K.L., Shimizu., A.B., 1975. Intumescent
coating modelling. J. Fire Flammability 6, pp205–221.Di Blasi, C., Branca, C., 2001. Mathematical model for the nonsteady decomposition
of intumescent coatings. AIChE J. 47, pp2359–2370.Di Blasi, C., 2004. Modeling the effects of high radiative heat fluxes on intumescent
material decomposition. J. Anal. Appl. Pyrolysis 71, 721–737.Griffin, G.J., 2010. The modeling of heat transfer across intumescent polymer
coatings. J. Fire Sci. 28, pp249–277.Incropera, F.P., DeWitt, D.P., 1996. Introduction to Heat Transfer, third edition John
Wiley (Appendix A).Kantorovich, I.I., Bar-Ziv, E., 1999. Heat transfer within highly porous chars: a
review. Fuel 78, 279–299.
J.E.J. Staggs et al. / Chemical Engineering Science 71 (2012) 239–251 251
Staggs, J.E.J., 2008. Image Analysis of 2D Intumescent Char Sections to EstimatePorosity. Fire Retardancy of Polymers: New Strategies and Mechanisms. RoyalSociety of Chemistry.
Staggs, J.E.J., Phylaktou, H.N., 2008. The effects of emissivity on the performance ofsteel in furnace tests. Fire Saf. J. 43, 1–10.
Staggs, J.E.J., 2009. Convection heat transfer in the cone calorimeter. Fire Saf. J. 44,pp469–474.
Staggs, J.E.J., 2010. Thermal conductivity estimates of intumescent chars by directnumerical simulation. Fire Saf. J. 45, pp238–247.
Staggs, J.E.J., 2011a. Numerical characterisation of the thermal performance ofstatic porous insulation layers on steel substrates in furnace tests. J. Fire Sci.29, 177–192.
Staggs, J.E.J., 2011b. A reappraisal of convection heat transfer in the conecalorimeter. Fire Saf. J. 46, pp125–131.
Recommended
