Numerical Heat Transfer, Part B: Fundamentals ...

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Numerical Heat Transfer, Part B: Fundamentals

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Thermal and Stress Analysis of Glazing in Fires and Glass Fracture Modeling with a Probabilistic Approach

S. Dembelea; R. A. F. Rosarioa; Q. S. Wanga; P. D. Warrenb; J. X. Wena a Faculty of Engineering, Kingston University, London, United Kingdom b Pilkington European Technical Centre, Lathom Ormskirk, Lancashire, United Kingdom Online publication date: 20 December 2010

To cite this Article Dembele, S. , Rosario, R. A. F. , Wang, Q. S. , Warren, P. D. and Wen, J. X.(2010) 'Thermal and Stress

Analysis of Glazing in Fires and Glass Fracture Modeling with a Probabilistic Approach', Numerical Heat Transfer, Part B: Fundamentals, 58: 6, 419 — 439 To link to this Article: DOI: 10.1080/10407790.2011.540953 URL: http://dx.doi.org/10.1080/10407790.2011.540953

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Numerical Heat Transfer, Part B, 58: 419–439, 2010 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7790 print=1521-0626 online DOI: 10.1080/10407790.2011.540953

THERMAL AND STRESS ANALYSIS OF GLAZING IN FIRES AND GLASS FRACTURE MODELING WITH A PROBABILISTIC APPROACH

Downloaded By: [China Science & Technology University] At: 14:32 21 December 2010

S. Dembele1, R. A. F. Rosario1, Q. S. Wang1, P. D. Warren2, and J. X. Wen1 1

Faculty of Engineering, Kingston University, London, United Kingdom Pilkington European Technical Centre, Lathom Ormskirk, Lancashire, United Kingdom

2

The aim of the study is to predict the thermal and stress behavior of a framed glass subjected to typical fire conditions, and the initial glass fracture time and locations using a probabilistic approach as an alternative to Pagni’s deterministic criterion. Thermal stresses in glass have been little researched. The probabilistic approach has the advantage of taking into account some uncertainties such as the edge conditions. The model employed is based on stress and conduction heat transfer models, a spectral discrete ordinates radiation model, and a failure probability model. Some results of its verification and applications are reported here.

1. INTRODUCTION The fracture and subsequent partial or complete fallout of a window or door glass pane during fires significantly affects the dynamics of fires in compartments or enclosures. The sudden venting resulting from the fallout of the glass provides an inlet for fresh air, which may result in flashover or backdraft depending on the development stage of the fire. Emmons [1] underlined the need for research on glass behavior in fires, and suggested that glass fracture is caused by thermally induced tensile stresses. In a typical compartment fire, there are two main physical processes involved when a glass pane is subjected to heat from fire. The first is the heat transferred by radiation and convection from the fire source and the hot combustion products to the glass. Radiation remains the dominant mode of heat transfer in the fire environment. The second process is the mechanical stress distribution and glass fracture [2]. The initial cracks in a framed window glass are generally initiated at edge defects, and cracks propagate away from the glass edge. The glass pane at this stage could still be in place in the frame, with no vent opened. When multiple cracks join, partial or complete fallout could occur, then affecting the fire ventilation conditions in the enclosure. Received 28 July 2010; accepted 26 October 2010. The authors gratefully acknowledge the financial support of Pilkington plc for this work. Address correspondence to Siaka Dembele, Faculty of Engineering, Kingston University, Roehampton Vale, Friars Avenue, London SW15 3DW, United Kingdom. E-mail: S.Dembele@ kingston.ac.uk

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NOMENCLATURE A RA c E h H Ik


k Kgk L m Nd

q Q rqrad s t T wm x, y, z a

area of material in tension, m2 stress-area integral specific heat of glass, J=kg K Young’s modulus, N=m2 convection heat transfer coefficient, W=m2 K glass pane half-height, m spectral radiation intensity, W=m2 sr mm glass thermal conductivity, W=m K glass spectral absorption coefficient, 1=m thickness of the glass pane, m Weibull modulus number of discrete directions in spectral discrete ordinates method (SDOM) element nodal displacement vector or heat flux, W=m2 global displacement vector radiative source term, W=m3 width of glass shaded edge, m time, s temperature, K weight of discrete direction m in SDOM coordinate variable, m absorptivity of glass or coefficient of linear expansion, 1=K

b c D e k m, n, g q r rmax rnom u r rVM r1,2,3 s Subscripts b f g gf g1 x, y, z 1 k

coefficient of thermal expansion, 1=K shear strain difference emissivity of glass or strain wavelength of radiation, mm direction cosines in SDOM reflectivity or density, kg=m3 stress, N=m2 maximum tensile stress, N=m2 nominal stress, N=m2 unit of area strength, N=m2 equivalent stress, N=m2 principal stresses, N=m2 transmissivity of glass or shear stress, N=m2 blackbody relative to fire side of the glass pane relative to the glass glass surface on fire side glass surface on ambient side relative to coordinates x, y, z relative to ambient side of the glass pane spectral

Superscripts m discrete direction in SDOM

Following Emmons’s studies, Pagni [3] and Keski-Rahkonen [4] carried out valuable research on the heat transfer modeling of glass in compartment fires. Pagni [3] suggested the most widely used glass fracture criterion based on the glass temperature increase, and subsequently developed a theoretical model for heat transfer and glass fracture which was implemented in the Break1 code [5]. Cuzzillo and Pagni [6] carried out a good review of existing heat transfer models for glazing in a fire environment. The simplest heat transfer model treats the glass as a lumped mass and uses a constant heat transfer coefficient. Another approach treats the glass as a distributed mass that absorbs radiation through its thickness with nonlinear radiative boundary conditions [6]. Sincaglia and Barnett [7] developed a glass fracture model with emphasis on radiation wavelength dependence which was implemented in the zone-type computer code Branzfire by Parry et al. [2]. More recently, the present authors have developed an advanced radiation heat transfer model, based on a spectral discrete ordinates method (SDOM), which addresses some limitations of literature models [8–10]. The model is spectral (glass is a selective material that absorbs, reflects, and transmits radiation within specific wavelengths), accounts for the diffuse nature of the radiation incident on the glass pane, and provides better handling of the boundary conditions. Validation studies of the one-dimensional


THERMAL AND STRESS ANALYSIS OF GLASS IN FIRES

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radiation model (1D-SDOM) have been presented in terms of temperature distribution in the glass and the initial fracture time using Pagni’s criterion [8, 9]. In order to simulate the dynamic interaction between the fire and the glazing system in a compartment, the spectral radiation model was coupled with the computational fluid dynamics (CFD) code Fire Dynamics Simulator (FDS 5.0) developed at the National Institute of Standards and Technology (NIST) [11]. Successful validation and application results of the coupled tool, referred to as FDS-SDOM, were presented in [10]. All the aforementioned theoretical studies were mainly concerned with heat transfer modeling and thermal analysis, the calculations of temperature distribution in the glass pane, and the prediction of the time to first crack occurrence (fracture initiation), with no attempt to predict partial or complete fallout. However, they did not investigate the detailed stress distribution in the glass, which is important for the location of the initial fracture zones and also for better understanding of the glass fracture and fallout mechanism. The theoretical study of glass thermal stress in fire conditions, which has been little considered in the literature, is undertaken in the present study. Some experimental studies have been conducted to understand the behavior of glazing in fires. Among others, Joshi and Pagni [12] carried out some tests to determine the statistical fracture stress distribution of glass samples for input in the Break1 code. Skelly et al. [13] tested the behavior of framed and unframed glass panes exposed to compartment fires by measuring temperatures and the time of first crack occurrence (fracture initiation). Mowrer [14] studied the effect on glazing of external fires typical of earthquake-induced and wildland–urban interface fires. Harada et al. [15] also measured the time of initial fracture of glass panes subjected to different radiant heat fluxes. Experiments were conducted on large glass panes exposed to wood crib fires in a compartment by Shields et al. [16] to quantify both fracture and fallout. These experimental data have provided valuable information for model verifications and validations. It is worth mentioning that in some recent studies, attempts have been made to predict the window glass fallout in compartment fires. To account for the probabilistic nature of the glass fracture temperature, which depends on many parameters (type of glass, manufacturing conditions etc.), Pope and Bailey [17] developed a Gaussian glass breakage model that was coupled with FDS [11]. In simulations, the fracture temperature of a glass pane is selected from the results of a Gaussian distribution in which the average fracture temperature and standard deviation are user-defined quantities. Once any point on the pane reaches its predefined fracture temperature, the pane is removed (fallout) [17]. Hietaniemi [18] proposed a probabilistic approach to predict glass fallout. The time and glass temperature at the first occurrence of a crack (initial fracture) was modeled using Break1 [5] and Pagni’s criterion, together with a Monte Carlo method to account for the probabilistic nature of parameters such as the thermal and mechanical properties of the glass. The glass fallout criterion is based in [18] on a prescribed number of successive multiple cracks obtained from literature experimental data. Hietaniemi’s approach for fallout prediction was employed by Kang [19] using FDS, Break1, and Pagni’s breakage criterion. The entire glass pane is removed (fallout) when the number of glass cracks exceeds typically three [19].


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Although these recent studies [17–19] to predict glass fallout have their own merits, they seem to be more computer-based and too simplistic, more research is needed to further validate them and provide a more robust theoretical justification of the physics of fallout employed. Glass fallout prediction is a very complex mechanism that requires detailed thermal and stress modeling, crack propagation modeling etc. Fallout prediction is beyond the scope of the present study. However, the probabilistic nature of the glass initial fracture mentioned in [17, 18] that has been little researched should clearly be accounted for in models for initial cracks and fracture predictions. Glass panes produced from the same batch may have different mechanical properties because the edge conditions after cutting will not be exactly the same. These differences in the properties of the glass, resulting from the manufacturing process, which are of statistical or probabilistic nature, will be reflected in the actual fracture and subsequent fallout characteristic of the glass in fires. It is worth noting that the glass fracture criterion employed in nearly all studies for framed glass in a fire environment is based on Pagni’s criterion [3], which states that glass fracture occurs when the temperature increase in an exposed portion of the glass is sufficient to induce a predetermined fracture (failure) stress in the shaded framed edge. However, this widely used criterion has two major limitations. First, it was developed for a uniform exposure of the glass pane to fire, which is not the case for many real compartment fire scenarios, where the glass may be subjected to heat flux from a hot upper layer and a cooler lower layer (nonuniform heating) [16]. Second, Pagni’s criterion does not account for some real physical parameters of the glass, such as edge conditions when the glass, is manufactured or cut to size. Those parameters, which are of a statistical or probabilistic nature, have a strong influence on the glass fracture. The objectives of the present study are twofold: (1) to model the thermal stress distribution in glass for a better understanding of the fracture phenomenon and to predict the location of initial fracture zones; and (2) to investigate a probabilistic glass fracture criterion that accounts for some of the variability of the problem, such as the edge condition, as an alternative to Pagni’s deterministic approach. The overall aim is to analyze the thermal and stress behavior of a glazing assembly under fire conditions, and also predict the locations and time of initial fracture in the glass pane based on a probabilistic approach, which is the originality of the study. Three-dimensional stress and conduction models and a failure probability model have been implemented by the present authors and are coupled with the model SDOM previously developed by the same authors [8–10]. The model is applied to some verification scenarios from the literature, and further analysis of the results is carried out. 2. MATHEMATICAL MODELING The modeling approach employed for thermal and stress analysis in the glass, and the prediction of initial fracture occurrence in the glass pane, is comprised of the following 3-D submodels: conduction heat transfer and spectral radiation heat transfer models, a thermal stress model, and a probability of failure model for glass fracture. For simulations of a typical glazing assembly in a compartment fire, the fire dynamics (turbulence, combustion, fluid flow) should be dynamically coupled with


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the glazing response. This is achieved by coupling the developed submodels with the CFD code FDS. Only the main equations important for understanding of the submodels are provided here for clarity.

2.1. Heat Transfer Model


2.1.1. Energy equation and boundary conditions. In a compartment fire, the temperature distribution in a framed glass (Figure 1) is determined from the transient energy equation for a differential element of glass. In 3-D Cartesian geometry, this equation is expressed as qTðx; y; z; tÞ q2 T q2 T q2 T qc ¼k þ þ 2 qt qx2 qy2 qz

! rqrad ðx; y; zÞ

ð1Þ

The first terms in parentheses on the right-hand side (RHS) of Eq. (1) represent the conductive heat flux in the glass determined from Fourier’s law. The second term, rqrad, is the local total radiative source term, also called the divergence of the radiative heat flux, arising from the absorption and emission of thermal radiation by the glass. This source term is calculated in the present study from the spectral discrete ordinates method (SDOM) described below. It is worth noting that absorption and emission by glass is a spectral phenomenon that depends on the wavelength of thermal radiation. Boundary conditions are needed to solve Eq. (1). On the surface of a glass pane exposed to fire or heat (x ¼ 0), Z 1 Z 1 qT k ¼ hf Tf Tgf þ ak qkf dk ekg Ibk ðTgf Þp dk qx x¼0 0 0 On the surface of a glass pane exposed to ambient (x ¼ L),

Figure 1. Window glass geometry.

ð2Þ

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S. DEMBELE ET AL.

k

Z 1 Z 1 qT ¼ h T T e I ðT Þp dk ak qk1 dk þ 1 g1 1 kg bk g1 qx x¼L 0 0

ð3Þ


Similar boundary conditions are also derived for the framed edge. To determine the 3-D transient temperature distribution, T(x, y, z, t), in the glass, Eq. (1) is integrated over time using the finite-difference method and over the volume with the finite-element approximation. 2.1.2. The spectral discrete ordinates radiation model (SDOM). The absorption, emission, and transmission of radiation in glass are spectral phenomena which should be accounted for in calculations. The 3-D spectral radiative transfer equation (SRTE) in an absorbing and emitting, nonscattering medium such as glass can be written as [20] m

qIk qIk qIk þn þg ¼ Kgk ½Ibk Ik qx qy qz

ð4Þ

In Eq. (4), Ik is the spectral radiative intensity, which is a function of the spatial ^ (m, n, k). The discrete ordinates method location (x, y, z) and the angular direction X [21] is based on the separation of the angular dependence from the spatial dependence of the intensity in the SRTE. This is achieved by choosing a set of discrete directions spanning the angular range 4p. For each discrete direction, m, Eq. (4) becomes [21] mm

qIkm qI m qI m þ nm k þ gm k ¼ Kgk Ibk Ikm qx qy qz

ð5Þ

The following radiative boundary conditions are used to solve Eq. (5): At x ¼ 0;

Nd X Ikm ¼ ekg Ibk ðTgf Þ þ 2 1 ekg skg mm jnm jIkm ðx ¼ 0Þ

ð6Þ

m¼1;n0

The radiative boundary conditions at the others boundaries in the y and z directions take a similar form. Equation (5) is integrated over each control volume to determine the local intensity field Ikm . Results presented in the present study are based on an S6 quadrature scheme [21], which provides solutions that do not vary significantly when higher orders (e.g., S8, S12) are used at the expense of higher computing times. Once the local intensity field is determined, the spectral and total net radiative heat fluxes at any location in the glass pane are respectively given by qk ðxÞ ¼

Nd X m¼1

wm mm Ikm ðxÞ

ð8Þ


qðxÞ ¼

X

qk ðxÞ Dk

425

ð9Þ

all Dk

The total radiative source term, rqrad, used in Eq. (1) is calculated as

rqrad ¼

X

" Kgk 4pIbk

# m m

w m jn

m

jIkm

ð10Þ

m¼1

all k


Nd X

The spectral range 0.1 mm to 100 mm is used in the integration for total quantities in the present study. A comparative study between the spectral discrete ordinates method (SDOM) and simplified radiation models such as the two-flux model, carried out by the present authors [10], showed the major improvements in terms of accuracy of the SDOM approach. The justification of the choice of SDOM model for fire=glass problems are discussed in [8, 9]. The model was investigated independently and some sensitivity studies carried out in previous studies [8, 9] before being coupled with other models.

2.2. Thermal Stress Model The thermal stress model employed is an adaptation of the model presented in [22], which was modified by the present authors, adapted, and validated for thermal loads relevant to glass. The reader may refer to [22] for more details. The methods adopted to calculate the thermal stresses are the potential energy and the Galerkin finite-element approaches. The potential energy and Galerkin approaches yield the set of equations KQ ¼ F

ð11Þ

where Q is the global displacement vector, K is the stiffness matrix, and F is a global matrix. The calculation of the stress and strain values can be performed once Eq. (11) is solved and the results of the global displacement (Q) and element nodal displacements (q) are obtained. The elements strains, e ¼ [ex, ey, ez, cyz, cxz, cxy]T, and stresses r ¼ [rx, ry, rz, syz, sxz, sxy]T vectors in the glass can be obtained from e ¼ Bq

and

r ¼ De ¼ DBq

ð12Þ

where D is the material matrix and B is a (6 12) matrix [22]. Once the temperature field in the glass is calculated from Eq. (1), if the glass temperature changes by DT(x, y, z) with respect to the initial state, the thermal stresses can be calculated by

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S. DEMBELE ET AL.

r ¼ Dðe e0 Þ

ð13Þ

where e0 ¼ [a DT, a DT, a DT, 0, 0, 0]T. Once the normal and shear stresses and strains are obtained, the principal stresses r1, r2, and r3 are calculated as I1 þ c cosðhÞ 3 I1 2p r2 ¼ þ c cos h þ 3 3 Downloaded By: [China Science & Technology University] At: 14:32 21 December 2010

r1 ¼

I1 4p r3 ¼ þ c cos h þ 3 3

ð14Þ ð15Þ

ð16Þ

where rffiffiffi a c¼2 3

1 3b 1 h ¼ cos 3 ac

I1 ¼ rx þ ry þ rz ;

I2 a ¼ 1 I2 3

3 I1 I1 I2 I3 b ¼ 2 þ 3 3

I2 ¼ rx ry þ ry rz þ rx rz s2yz s2xz s2xy

I3 ¼ rx ry rz þ 2syz sxz sxy rx s2yz ry s2xz rz s2xy The principal strains (e1, e2, e3) are calculated using similar relations, as reported elsewhere in the literature. A quantity called the equivalent stress or Von Mises stress, rVM, is commonly used in solid mechanics and is strictly applicable to ductile material failure: rVM

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iffi 1h 2 2 2 ðr1 r2 Þ þ ðr2 r3 Þ þ ðr3 r1 Þ ¼ 2

ð17Þ

For the fracture or failure of brittle materials such as glass considered in the present study, the principal stresses are more relevant and are used in our calculations (see Section 2.3.2). However, rVM still provides a good indication of the equivalent tensile stress to which the material is subjected. 2.3. Glass Fracture Criteria 2.3.1. Pagni’s glass fracture criterion. The fracture of a framed glass exposed to heat is generally initiated at an edge defect resulting from the manufacturing process or when the glass is cut to size. When the framed glass pane is exposed to fire, fracture occurs when the thermally induced tensile stresses in the shaded area reach the tensile defect strength. The most widely used glass fracture criterion for fire exposure is based on Pagni’s criterion [3], which states that glass fracture occurs when the temperature increase in an exposed portion of the glass is sufficient to induce a predetermined fracture (failure) stress in the framed shaded edge (Figure 1):



s rmax DT ¼ T exposed Tframed edge 1 þ H Eb

427

ð18Þ

The advantage of this criterion is its simplicity, and it yields relatively good initial fracture time under some conditions. However, the criterion in Eq. (18) does have some limitations. First it was developed for a uniform exposure of the glass pane to fire heat. As discussed by Parry et al. [2], experimental studies such as [16] suggest that for some compartment fires, the glass heating is not uniform, as it is exposed to both the upper hot combustion gases and the lower cold gases. Only approaches that can predict the actual stress distributions, as undertaken in the present work, can handle fracture resulting from nonuniform exposure. Second, another limitation of Pagni’s criterion arises from the fact that glass fracture, which initiates at edge defects, is strongly dependent on the edge conditions for a given type of glass. The edge conditions are of a statistical or probabilistic nature, as they dependent on many variable parameters. Pagni’s criterion does not take into account the variability of some key parameters such as the edge conditions in fracture calculations. The glass fracture model employed here can be an alternative to Pagni’s criterion. It involves a full stress analysis and a probability-of-failure model based on Weibull statistics. 2.3.2. Probability of failure model. Brittle materials such as glass show variability in strength because of the presence of a distribution of flaws or cracks, both in size and position [23]. Glass strength depends on the treatment and handling of its surface. The Weibull probability distribution [24] was shown to successfully quantify statistically the uncertainties in the breaking strength of glass. The probability-of-failure model based on the Weibull distribution employed in the present work for glass fracture was developed and validated at Pilkington [23, 25]. It quantifies the probability of the glass material failure due to thermal and load stress. The general form of the failure probability (F) of the glass is given by [23, 26]

1 m rnom m F ¼ 1 exp C 1 þ A RA u m r

ð19Þ

u is the unit of area where m is the Weibull modulus, C is the gamma function, r strength, and RA is the stress-area integral. The maximum principal stress value in the glass pane is assigned to the nominal stress rnom. The stress-area integral is expressed as N 1X RA ¼ N 1

r1 rnom

m r2 m r3 m þ þ rnom rnom

ð20Þ

where N is the number of nodes, and r1, r2, r3 are the principal stresses calculated by the thermal stress model in Section 2.2. Equations (19)–(20) are used to determine the probability of failure of the glass surface and the edge elements. The overall failure probability (OFP) of a glass surface consisting of k elements is calculated as

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S. DEMBELE ET AL.

OFP ¼ ð1 F1 Þð1 F2 Þ . . . ð1 Fk Þ

ð21Þ

u , are different for The values of the Weibull modulus, m, and unit of area strength, r each type of glass and edge conditions. The values employed in the present study were obtained from a compilation of fire testing experimental data by Pilkington [25] for different type of glass (ordinary, safety glass) and edge conditions (as-cut, grounded, and polish edge).


3. RESULTS AND DISCUSSION 3.1. Investigation of Glass Panes Subjected to Heat Fluxes To verify the approach developed, the experiments of Harada et al. [15] are considered. Several experiments were conducted on ordinary framed glass to predict the glass breakage by imposing different heat fluxes between 2.7 and 9.7 kW=m2 [15]. This was achieved by changing the distance between the propane-fired radiant panel and the glass pane as shown in Figure 2. Only the time of first crack occurrence (initial fracture time) was measured in these experiments. No temperature, stress, or strain measurements were carried out by Harada et al. [15]. The code ANSYS [27] is employed for temperature and stress comparisons. The glass dimensions are thickness L ¼ 3 mm, 200 mm width, 200 mm height, and s ¼ 15 mm (framed edge width). For simulations with the authors’ model, the following data are used: E ¼ 73 GPa, b ¼ 8.75 106 K1, a ¼ 4.6 107, and k ¼ 0.95 W=m K. The SDOM is based on an S6 quadrature scheme (48 angular directions in total). Meshes of 2,312 elements [34(Y) * 34(Z) * 2(X)] were generated for both codes (ANSYS and authors’), as they provide a good compromise between accuracy and computing effort. A preliminary grid sensitivity analysis shows that the results obtained do not vary significantly with finer meshes. For this particular scenario, the glass is assumed to be uniformly exposed to the radiant heat. This scenario allows a verification of the model for a uniform heat exposure. Two variants of the authors’ model are used in the simulations: one

Figure 2. Experiment of Harada et al. [15].



429

accounting for the radiative source rqrad in Eq. (1) (referred to as ‘‘Model’’) and another neglecting this source term (referred to as ‘‘Model-No radiation’’). Comparisons are made between the authors’ model and ANSYS [27] for the temperature, stress, and strain distributions predicted for this experimental scenario. No coupling of the model with the CFD code FDS is done for this heat exposure scenario, for a better assessment of the model independent of the fluid dynamics and turbulence, which introduce more uncertainties. A total of 2,312 meshes was employed in simulations. ‘‘Model-No radiation,’’ as ANSYS, does not account for the radiation source term in Eq. (1). Figure 3 presents the temperatures for Model, Model-No radiation, and ANSYS for an incident uniform heat flux of 6.69 kW=m2. This value of the heat flux provides a good representation of the results obtained for other fluxes values. Relatively good agreement is obtained between the temperature predicted by Model-No radiation and ANSYS, with a maximum difference of 2 C over 140 s for the exposed surface of the glass (surface facing the propane burner) and a 1 C difference on the shaded area of the glass (edge side). By accounting for radiation as in ‘‘Model,’’ the temperatures on the exposed and unexposed surfaces of the glass decrease by up to 5 C and increase by up to 7 C on the shaded edge of the glass in comparison with ANSYS. This is due to the fact that ANSYS does not account for the radiation source term. The glass simultaneously absorbs and emits thermal radiation, and its temperature can increase if the material is a net absorber or decrease if

Figure 3. Temperature distribution at the center and edge of the glass for Model, Model-No radiation, and ANSYS for an incident heat flux of 6.69 kW=m2.


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it is a net emitter. In the exposed area, with increasing exposure time, the glass gradually becomes a net emitter for this particular scenario, which results in relatively lower temperatures compared to ANSYS calculations. In the shaded edge area, where temperature is relatively lower compared to the center of the pane, radiation absorption becomes important, and its inclusion in Model calculations yields to relatively higher temperature in the shaded area compared to ANSYS, which neglects this radiation absorption. The temperature predictions with the model in the Y and Z directions (along the width and height of glass pane, respectively) for different times are presented in Figure 4 for a uniform incident heat flux of 6.69 kW=m2. The temperature on the exposed area of the glass increases rapidly from 300 to 405 K after 140 s in the exposed (unframed) area of the glass. In comparison, the temperature in the framed shaded area increases only by 20 C after 140 s. The large temperature gradients between the hottest points in the glass and the coldest ones induce a high stress in the glass pane. The central area of the glass tries to expand due to the increase of the temperature and is constrained by the shaded edge of the glass at lower temperature, resulting in thermal stresses in the glass. Figures 5a–5c show the temperature distributions in the glass pane subjected to an incident heat flux of 6.69 kW=m2 after 2 min of exposure, respectively, for Model,

Figure 4. Temperature predictions with Model in Y and Z directions at different times for an incident heat flux of 6.69 kW=m2.



431

Figure 5. Temperature distribution (K) in the glass for an incident heat flux of 6.69 kW=m2: (a) Model; (b) Model-No radiation; (c) ANSYS.

Model-No radiation, and ANSYS. In Figure 5a, the temperature in the exposed area is about 400 K. The temperature distribution in the shaded area of the glass panel is more gradual with the Model (Figure 5a) than with ANSYS (Figure 5c) and Model-No radiation (Figure 5b), both not accounting for the radiation source in the glass. This behavior has a direct effect on the stress distribution in the glass pane, as shown in Figures 6a–6c for the principal stress r1 (which is the maximum of the three principal stresses r1, r2, r3). For r1 the highest tensile stress values (50 MPa) are located at the edge of the shaded area of the glass with Model (Figure 6a), whereas with Model-No radiation (Figure 6b) and ANSYS (Figure 6c), the highest values are located near the transition area between the exposed area of the glass and the shaded area. A negative stress (compression) is obtained in the central exposed area with all models, which is consistent with the framed glass behavior. In terms of verification, Model-No radiation and ANSYS, which both neglect radiation source terms, provide results that agree relatively well. However, the most accurate


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Figure 6. Principal stress r1 (N=m2) in the glass for an incident heat flux of 6.69 kW=m2: (a) Model; (b) Model-No radiation; (c) ANSYS.

predictions are given by Model, and results are therefore only shown for Model hereafter. Figure 7 presents the principal strain e1 (maximum of the three principal strains, e1, e2, e3) for Model. As expected, the highest strains (5.8 * 104) are located in the areas with higher values of tensile stresses. In order to assess the probabilistic glass fracture model presented in Section 2.3.2, the overall failure probabilities (OFP) of the glass have been calculated. In the calculations the glass is considered as ordinary glass and the edge condition ‘‘as-cut.’’ Figure 8 presents the OFP for the different incident heat fluxes on the glass pane. When the incident heat flux is increased from 5.48 to 9.11 kW=m2, the OFP increases rapidly from 10% to 90% in 30 s as a result of higher radiant heat absorption. With a lower incident heat flux (5.48 kW=m2), it takes almost 200 s to increase the probability of failure from 10% to 90%. The results show the strong correlation between the amount of radiant heat and the glass fracture probability.



433

Figure 7. Principal strain e1 predicted by Model for an incident heat flux of 6.69 kW=m2.

The experimental and predicted times of the first crack occurrence (initial fracture) considering an OFP of 50% are presented in Table 1. Experimental data are averaged over a series of tests. For the heat flux of 5.48 kW=m2, the experimental

Figure 8. Overall failure probability (OFP) of the glass pane for different heat fluxes.

434

S. DEMBELE ET AL. Table 1. Overall failure probability (OFP) predicted with Model-FDS and experimental values of time to first crack for different incident heat fluxes

Incident heat flux 2


5.840 kW=m 6.690 kW=m2 9.110 kW=m2

Experimental data (time of first crack) [15]

50% OFP (Model)

207 s 144 s 90 s

203 s 150 s 98 s

and model predicted times are respectively 207 and 203 s (1.9% difference), 144 s and 150 s (4.2% difference) for 6.690 kW=m2, and 90 s and 98 s (8.8% difference) for a heat flux of 9.11 kW=m2. These results confirm the relatively good agreement between Model predictions and experiments for a range of incident uniform heat flux exposure typical of fire scenarios. They also show the good potential of the probabilistic approach to predict the initial fracture times using detailed stress calculations.

3.2. Simulation of a Glazing Assembly in a Compartment Fire The scenario considered in this section is typical of framed glass window behavior in a compartment fire. To account for the dynamic interaction between the glass behavior and the fire dynamics, the model developed by the authors is coupled with the CFD code FDS [11]. FDS solves the Navier-Stokes equations for low Mach number (Ma < 0.3). The numerical algorithm employed is an explicit predictor=corrector scheme, second-order-accurate in space and time, using a direct Poisson solver. Turbulence is treated by means of large eddy simulation (LES), via the Smagorinski subgrid scale model. For combustion, a mixture fraction model is used. The coupled tool is referred to as ‘‘Model-FDS’’. FDS predicts the fire dynamics (turbulence, combustion, fluid dynamics) and the heat flux incident on the glass pane, which is not uniform for this scenario [13]. The authors’ model then calculates the temperature distributions and thermal stresses in the glass, as well as the probability of failure. The glazing=compartment fire experiments of Skelly et al. [13] are simulated in this section. The experiments were carried out in a rectangular compartment (1.0 m 1.2 m 1.5 m high) constructed of iron slats and ceramic fire board. There are three vents in the compartment: an open upper-layer exhaust vent, an open vent near the base of the compartment for fire ventilation, and a glass window assembly. The glass window assembly measures 0.50 m 0.28 m. The tested ordinary glass is 2.4 mm thick. The fire considered here consists of a burning hexane liquid in a 20 cm 20 cm tray. The temperatures denoted by ‘‘tests 4, 5, 6’’ in Figure 9 represent the temperature of the exposed side surface and shaded edge of the glass pane measured in three repeated tests [13]. The exposed glass surface temperature is measured with a thermocouple in the center of the glass pane. The temperature of the shaded area of the glass is measured at the center point of the bottom edge of the glass pane. Model-FDS predicted surface temperatures at the exposed side, shaded edge, and unexposed side are also presented in Figure 9. Relatively good agreement is obtained between Model-FDS predictions and the experimental data. For the first 30 s the glass exposed surface temperature is underestimated in



435

Figure 9. Glazing in compartment fire—pool fire 20 cm 20 cm—temperature predicted by Model-FDS and experimental data.

comparison with the experimental data of the three experiments (the measurements errors in the experiments were not provided in [13]). However, good agreement is obtained for the rest of the simulation time. Model-FDS predicts a temperature increase of 10 K in the shaded area; the same temperature difference was measured in the experiments. The temperature distribution on the fire side of glass is presented in Figure 10a. The temperature of the glass surface is higher on the left side of the glass pane (394 K), due to the proximity of the left side of the glass pane to the pool fire. This nonuniform exposure of the glass can clearly be seen in this real fire scenario. Pagni’s criterion, as mentioned before, does assume uniform heating. The temperatures in corners remain at their initial values (294 K), due to the low conductivity of the glass and frame. Along the edges, the glass presents a smoother temperature differential from the exposed area to the shaded area. The equivalent tensile stress rVM to which the glass pane is subjected due to the temperature gradient is presented in Figure 10b. rVM represents the equivalent tensile stress to which the material is subjected (by definition, it is a positive quantity, i.e., tensile stress); this stress is not employed in the glass fracture prediction in the present work, but it provides a good indication of the overall stress and the locations of the maximum equivalent tensile stresses in the framed edges. The stresses in the four framed corners are around 1 107 Pa. The higher equivalent tensile stresses


436

S. DEMBELE ET AL.

Figure 10. Glazing in compartment fire—pool fire 20 cm 20 cm: (a) temperature distribution in the glass pane (K); (b) equivalent stress in the glass pane (N=m2).

(4.5 107 Pa) are located in the left, top left, and bottom left glass edge surfaces, where the temperature gradient is higher. In this particular case the temperature distribution in the glass pane is not uniform; consequently, the higher tensile stresses are not uniformly and symmetrically distributed in the edges of the glass pane, as in Figure 6a for the uniform exposure. The highest tensile stresses (4.58 107 Pa) are located in the left-hand framed edge surface of the glass pane, where the temperature gradients are the highest. Consequently, the probability of the glass failure in this region is higher than any other, and the first crack will occur in this zone, as shown in Figure 11a for the failure probability of the glass surface elements (PFE is 1.2% in left edge surface). The glass is assumed ordinary and edge conditions ‘‘as cut’’ in model calculations. The edge surfaces have a higher probability of failure than the exposed and unexposed surfaces of the glass pane, as can be seen in Figure 11a, which is well consistent with the theory because fracture initiates at an edge defect.



437

Figure 11. Glazing in compartment fire—pool fire 20 cm 20 cm: (a) probability of failure of elements (PFE) in the glass; (b) overall failure probability (OFP), of the glass pane.

The overall failure probability (OFP) of the glass pane is presented in Figure 11b. The OFP increases rapidly in the last 40 s of the simulation, rising from 50% to 100% in 20 s. Therefore the glass pane has a high probability of initial cracks or fracture occurrence between 85 s (50% OFP) and 110 s (100% OFP) from predictions (98 s on average). The averaged experimental time for first crack occurrence in the three tests is 107 s [13], which is well within range of the time predicted with Model-FDS. Simulations were also carried out for the 30 cm 20 cm tray pool fire [13]. Although the detailed results are not presented here for lack of space, the averaged experimental time for the first crack occurrence is 53 s [13], and the predictions by Model-FDS are between 50 s (50% OFP) and 65 s (100% OFP)—57 s on average— showing good agreement. The simulations show the capability of the model to predict the time of initial cracks or fracture occurrence as well as the location of these cracks. It is worth noting that for the simulations specific to this compartment fire scenario (as well as in Section 3.1), there is a temperature difference between the exposed and unexposed surfaces of the glass, and hence a temperature gradient exists through the thickness of the glass. As a result, an analysis of the glass displacement plots, not presented here, shows that the expansion of the exposed glass surface leads to thermal bending along the direction of the glass thickness. This shows the limitation of some literature approximations that neglect such through-thickness temperature gradients in the glass. Although the simulations carried out and presented in this article were based on ordinary glass, for tempered (toughened) glass, which is stronger than ordinary glass, one would expect longer times for initial break occurrence, since the thermally induced tension must first overcome the compression manufactured into toughened glass, as noted by Pagni [3]. 4. CONCLUSIONS A model based on spectral radiation and conductive heat transfer, detailed thermal stress, and a probability of failure is employed to analyze glass behavior


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in fires. The approach can represent an alternative to the deterministic Pagni’s criterion, as it takes into account nonuniform exposure of the glass pane to heat as well as the edge conditions. The approach also provides a better understanding of the stress distribution prior to glass fracture. Numerical simulations have been carried out to verify the accuracy of the model. In the first set of simulations, for uniform heating of a framed glass pane, the temperature, stress, and strain predicted by the model agree reasonably well with the predictions of ANSYS. Relatively good agreement is also obtained between the model and experimental data for the time of first crack occurrence in the glass. The simulations show the importance of accounting for the radiative source term and the probabilistic nature of glass fracture. In the second set of simulations, the model is coupled with the CFD code FDS to predict the dynamic interaction between the framed glass and the fire in a compartment, where nonuniform exposure to heat is generally encountered. The predicted overall probability of failure for the initial fracture of the glass agrees relatively well with the experimental time of first crack. The model also has the potential of providing the locations of areas with highest tensile stresses and higher probability of fracture. The approach could be employed to predict initial fracture times and the locations of the first cracks. It could also serve as the foundation to develop a more rigorous approach for glass fallout prediction, which could be undertaken as further work. More verification studies will be undertaken upon data availability in the literature to further validate the presented model and its submodels. There is clearly a need in the fire research community for more experimental data, in particular on through-thickness temperature data, stresses, and strains in framed glass subjected to fires. Properties such as the spectral emissivity and absorption coefficient of glass are important inputs to the model reported here. Although these data are available for ordinary glass, they are lacking for other types of glass on the market (e.g., toughened, etc.).

REFERENCES 1. H. W. Emmons, The Needed Fire Science, Fire Safety Science: Proc. 1st Int. Symp., pp. 33–53, Hemisphere, Washington, DC, 1986. 2. R. Parry, C. A. Wade, and M. Spearpoint, Implementing a Glass Fracture Module in the Branzfire Zone Model, J. Fire Protect. Eng., vol. 13, pp. 157–183, 2003. 3. P. J. Pagni, Fire Physics—Promises, Problems and Progress, Fire Safety Science: Proc. 2nd Int. Symp., pp. 49–66, Hemisphere, Washington, DC, 1989. 4. O. Keski-Rahkonen, Breaking of Window Glass Close to Fire, Fire Mater., vol. 12, pp. 61–69, 1988. 5. P. J. Pagni and A. A. Joshi, User’s Guide to Break1, The Berkeley Algorithm for Glass Breaking in Compartment Fires, Rep. NIST-GCR-91, Gaithersburg, MD, 1991. 6. B. R. Cuzzillo and P. J. Pagni, Thermal Breakage of Double-Pane Glazing by Fire, J. Fire Protect. Eng., vol. 9, pp. 1–11, 1998. 7. P. E. Sincaglia and J. R. Barnett, Development of a Glass Window Fracture Model for Zone-Type Computer Fire Codes, J. Fire Protect. Eng., vol. 8, pp. 101–118, 1997. 8. S. Dembele, R. A. F. Rosario, J. X. Wen, K. S. Varma, S. Dale, M. Wood, and P. D. Warren, Study of Glazing Behaviour in Fire Conditions Using Advanced Radiation Heat


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