V.._,_fiV«<93%Q'3
`
`.L!_‘:1.Voakm‘
`
`
`
`*0
`
`i ‘
`
`PUBLISHED MONTHLY BY ASME - JUNE 2006
`
`Page 1 of 16
`
`BOREALIS EXHIBIT 1088
`
`

`
`Published Monthly by ASME
`
`VOLUME 128 0 NUMBER 6 - JUNE 2006
`
`S; ACHARYA=(2006)
`. K:ANAND (2006)
`UFIMEISTER (2008)
`B5 FAROUK (2006)
`
`RESEARCH PAPERS
`
`Bubbles, Particles and Droplets
`
`8)
`(
`‘S G KANDLIKAR (2007)
`'
`KHODADADI (2007)
`"
`(2008)
`IENHARD. v (2006)
`H.
`R ‘MJLIGRANI (2006)
`MANGLIK (2008)
`CL H. OH‘(2007)
`C UMANI (200 ‘
`
`509 Simulation of Film Cooling Enhancement With Mist Injection
`Xianchang Li and Ting Wang
`
`520 Surface Deformation and Convection in ElectrostaticalIy?Positioned
`Droplets of Immiscible Liquids Under Microgravity
`Y. Huo and B. Q. Li
`
`Porous Media
`
`530 Height Effect on Heat-Transfer Characteristics of Aluminum-Foam
`Heat Sinks
`'
`w. H. Shih, w. c. Chiu, and w. H. Hsieh
`
`Modeling of Heat Transfer in Low—Density EPS Foams
`R. Coquard and D. Baillis
`
`Fully Developed Heat Transfer to Fluid Flow in Rectangular Passages
`Filled With Porous Materials
`A. Haji-Sheikh
`
`Forced Convection
`
`557 Heat Transfer in a Surfactant Drag-Reducing Solution— A Comparison
`With Predictions for Laminar Flow
`~
`'
`Paul L. Sears and Libing Yang
`
`Numerical Simulation of Flow Field and Heat Transfer of Streamlined
`Cylinders in Cross Flow
`Zhihua Li, Jane H. Davidson, and Susan C. Mantell
`
`Film-Cooling Efficiency in a Laval Nozzle Under Conditions of High
`Freestream Turbulence
`Valery P. Lebedey, Vadim V. Lemanov, and Victor I. Terekhov
`
`Natural and Mixed Convection
`
`‘ 580 Natural Convection Measurements for a Concentric Spherical Enclosure
`‘
`Peter M. Teertstra, M. Michael Yovanovich, and J. Richard Culham
`
`ll/licro/lllanoscale Heat Transfer
`588 Brownian-Motion-Based Convective-Conductive Model for the Effective
`Thermal Conductivity of Nanofluids
`Ravi Prasher, Prajesh Bhattacharya, and Patrick E. Phelan
`
`TECHNICAL BRIEFS
`
`596 Analytical Solution of Forced Convection in a Duct of Rectangular Cross
`Section Saturated by a Porous Medium
`Kamel Hooman and Ali A. Merrikh
`
`601 2 Thermal Ignition in a Reactive Viscous Flow Through a Channel Filled
`With a Porous Medium
`‘
`.
`’
`O. D. Makinde
`
`(Contents continued on inside back cover)
`
`This journal is printed on acid-free paper, vvhlch exceeds the ANSI 239.48-
`1992 specification for permanence of paper and library materials. ®""
`© 85% recycled content, including 10% post-consumer fibers.
`
`Page 2 of 16
`
`

`
`Xianchang Li
`e-mail: xli8@uno.edu -
`
`Ting Wang
`
`Energy Conversion & Conservation Center,
`University of New Orleans.
`2000 Lakeshore Dr,
`-
`New Orleans, LA 70148-2220
`
`V
`
`.
`
`
`0
`Simulation of Film Cooling ,
`Enhancement With Mist Injection
`
`_
`
`Cooling of gas turbine hot-section components, such as combustor liners, combustor
`transition pieces, and turbine vanes (nozzles) and blades (buckets), is a critical task for
`improving the life and reliability of them. Conventional cooling techniques using air—film
`cooling, impingement jet cooling, and turbulators have significantly contributed to cool-
`ing enhancements in the past. However, the increased net benefits that can be continu-
`ously harnessed by using these conventional cooling techniques seem to be incremental
`and are about to approach their limit. Therefore, new cooling techniques are essential for
`surpassing these current limits. This paper investigates the potential offilm—cooling en-
`hancement by injecting mist into the coolant. The computational results show that a small
`amount of injection (2% of the coolant flow rate) can enhance the adiabatic cooling
`efifectiveness about 30-50%. The cooling enhancement takes place more strongly in the
`downstream region, where the single-phase film cooling becomes less powerful. Three
`diflerent holes are used in this study including a two—dimensional (2D) slot, a round hole,
`and a fan—shaped dzfiusion hole. A comprehensive study is performed on the efiect offlue
`gas temperature, blowing angle, blowing ratio, mist injection rate, and droplet sizeon the
`cooling efiectiveness with 2D cases. Analysis on droplet history (trajectory and size) is
`undertaken to interpret the mechanism of droplet dynamics. [DOI: l0.1ll5/1.2171695]
`
`Keywords: film cooling, turbine-blade cooling, mist cooling
`//
`
`Introduction
`
`Cooling of gas turbine hot-section components, such as com-
`bustor liners, combustor transition pieces, turbine vanes (nozzles)
`. and blades (buckets), has always been a critical task for improving
`the life and reliability of hot-section components. Air—film cooling
`has widely been used and intensively studied as an effective
`scheme for more than half a century [1,2]. To improve the perfor-
`mance of air—film cooling, many studies have been conducted by
`examining the effect of flow and geometric parameters, including
`injection angles, injection hole configuration, density ratio, and
`blowing ratio. For example, Jia et al. [3] investigated a slot jet
`film cooling by using numerical simulations coupled with LDV
`experiments. Different jet angles from 15 deg to 60 deg with jet?
`blowing ratios ranging from 2 to 9 were studied. Their results
`showed a recirculation bubble downstream of the jet vanishes
`when the angle is 30 deg or less. They also found the blowing
`ratio has a large effect on the size of recirculation and, conse-
`quently, on film cooling. Kwak and Han [4] measured heat trans-
`fer cocfficients and film-cooling effectiveness on a gas turbine ,
`blade tip. Their results showed as the blowing ratio increased, the
`heat transfer coefficient decreased while film effectiveness in-
`creased. Heat
`transfer coefficient and film effectiveness were
`‘ found to increase with increasing tip gap clearance. Wang et al.
`[5] conducted an experimental study focusing on the flow mixing
`behavior inside the slots. Various parameters, including orienta-
`tion angle, inclination angle, slot width, effect of primaryflow,
`and slot depth, were systematically examined. The optimum slot
`depth was found to range from 2 to 2.8 times the jet diameter. The
`compound angle configuration (60 deg jet orientation and 30 deg
`slot inclination angles) was discovered to be the best choice.
`Among the typical holes are simple-angle holes with lateral or '
`forward diffusion and compound—angle holes with forward diffu-
`sion. The performance of film cooling withldifferent holes varies
`by 30-50% subject to geometric and flow conditions. Bell et al.
`[6] studied film cooling from shaped holes and measured the local
`
`Contributed by the Heat Transfer Division of ASME for publiczitioniintlie Jouizl
`NAL OF HEAT TRANSFER. Manuscript received April 5, 2005; final manuscript received!
`December 9, 2005. Review conducted by Phillip M. Ligrani,
`
`and spatially averaged adiabatic film-cooling effectiveness. They
`found laterally diffused,_ compound-angle holes and forward-
`diffused, compound-angle holes produce higher‘ effectiveness
`magnitudes over much wider ranges of blowing ratio and momen-
`tum flux ratio compared to the other three simple-angle configu-
`rations tested. All the three simple-angle hole geometries (cylin-
`drical round, simple-angle holes; laterally diffused, simple-angle
`holes; and forward-diffused, simple-angle holes) showlarger in-
`creases of spanwise-averaged adiabatic effectiveness as the den--
`sity ratio increases from 0.9 to 1.4. Brittingham and Leylek [7]
`performed numerical simulation on film cooling with’ compound-
`angle—shaped holes and concluded that superposition of individual
`effects for compound-angle cylindrical holes and streamwise—
`shaped holes do not necessarily apply to compound—angle—shaped
`holes. The compound-angle-shaped holes can be designed to n1ini—
`mize uncooled hot region between adjacent holes and, thus, some-
`'what mimic slot-jet performance. From many of the previous
`studies, the optimal blowing ratio is discovered ranging from 0.5
`to 1.0. The 35 deg injection angle and the shaped holes are found
`‘to be the most effective.
`As next—generation turbines will be required to burn alternate
`fuels with high hydrogen (H2) and carbon monoxide (CO) content
`from coal—derived syngas, cooling gas turbines becomes more dif— V
`ficult and more important. The high contents of H2 and CO will
`increase flame temperatures and flame speeds from those of '
`natural-gas combustion. Although conventional cooling tech-
`_niques using air-film cooling, impingement-jet cooling, and turbu-
`lators have significantly contributed to cooling enhancements in
`the past, the increased net benefits, which can be continuously
`harnessed by using these conventional cooling techniques, seem
`to be incremental and are about to approach their limit. /Therefore,
`new cooling techniques are essential. This paper inve‘tigates the
`potential of film-cooling enhancement by injecting mist into the
`coolant. Film cooling with mist injection can improve single-
`phase air-film cooling due to the following mechanisms: (a) the
`latent heat of evaporation’ serves as a heat» sink to absorb large
`amounts of heat; (b) direct contact of liquid droplets with the
`cooling wall can significantly increase heat transfer from wall; (c)
`steam and ‘water have higher specific heats (cp)
`than air. In 21
`
`Journal of Heat Transfer
`
`Copyright to zone by ASME
`
`JUNE 2005, Vol. 128 I 509
`
`Page 3 of 16
`
`

`
`Journal of Heat Transfer
`
`Volume 128, Number 6
`
`JUNE 2006
`
`(Contents continued)
`
`.605 Soret and Dufour Effects in a Non-Darcy Porous Medium
`M. K. Partha, P. V. S. N. Murthy, and G. P. Raja Sekhar
`
`611 Thermodynamics of Void Fraction in Saturated Flow Boiling
`Francisco J. Coliado, Carlos Monné, Antonio Pascau, Daniel Fuster, and Andrés Medrano
`
`___,§.2.1$33,"
`
`"M"*»""’45>"i,,as.~{;>»53,$co"C-r,,
`
`The ASME Journal of Heat Transfer is abstracted and indexed in the
`following:
`’
`Applied Science and Technology index, Chemical Abstracts, Chemical Engineering and
`Biotechnology Abstracts (Electronic equivalent of Process and Chemical Engineering),
`Civil Engineering Abstracts, Compendex (The electronic equivalent of Engineering
`index), Corrosion Abstracts, Current Contents, E & P Health, Safety, and Environment,
`El EncompassLit, Engineered Materials Abstracts, Engineering index, Enviroline (The
`electronic equivalent of Environment Abstracts), Environment Abstracts, Environmental
`Engineering Abstracts, Environmental Science and Pollution Management, Fluidex,
`Fuel and Energy Abstracts, lndexto Scientific Fleviews, INSPEC, International Building
`Services Abstracts, Mechanical &-Transportation Engineering Abstracts, Mechanical
`Engineering Abstracts, METADEX (The electronic equivalent of Metals Abstracts and
`Alloys Index), Petroleum Abstracts, Process and Chemical Engineering, Fteterativnyi
`Zhurnal, Science Citation index, Scisearch (The electronic equivalent of Science
`Citation Index), Theoretical Chemical Engineering
`
`Page 4 of 16
`
`

`
`R. Cuquard
`Centre Scientifique et Technique du Batiment
`(CSTB),
`24 rue Joseph Fourier,
`38400 Saint Martin d‘Heres, France
`e—maiI:
`r.coquard@csib.fr
`
`D . Ba i Ilis
`Centre Thermique de Lyon (CETHIL),
`’
`UMR CNRS 5008,
`~
`Domaine Soientifique de la Doua, INSA de Lyon,
`Batiment Saoi Carnot,
`9 rue de la physique,
`69621 Villeurbanne CEDEX, France
`e—mail: dominique.baiilis@insa-|yon.tr
`
`Modeling of Heat Transfer in
`Low-Density EPS Foams
`
`Expanded polystyrene (EPS) foams are one of the most widely used thermal insulators in
`the building industry. Owing to their very low density, both conductive and radiative heat
`transfers are significant. However, only few studies have already been conducted in the
`modeling of heat transfer in this kind of medium. This is due to their complex porous
`structure characterized by a double—scale porosity which has always been ignored by the
`previous works. In this study, we present a model of one—dimensional steady state heat
`transfer in these foams based on a numerical resolution of the radiation-conduction
`coupling. The modeling of the conductive and radiative properties of the foams takes into
`account their structural characteristics such as foam density or cell diameter and permits
`us to study the evolution of their equivalent thermal conductivity with these characteris-
`tics. The theoretical results have been compared to equivalent thermal conductivity mea-
`surements made on several EPS foams using a flux—meter apparatus and show a good
`agreement; [DOI: lO.lll5/1.2188464]
`
`Keywords: EPS foams, equivalent thermal conductivity, radiation/conduction coupling,
`radiative properties
`
`1
`
`Introduction
`
`the different porous media used for the building
`Among all
`thermal
`insulation, expanded polystyrene (EPS) foams are the
`most widely sold after glass wools. They represent approximately
`25% of the total building insulation market. Indeed, they are very
`convenient
`to manipulate due to their mechanical properties
`(lightness and stiffness), relatively cheap, and have good thermal
`performances. They are generally classified among the cellular
`materials but their porous structure is, in fact, more complex than
`a simple cellular matrix. This structure is characterized by a
`double-scale porosity as a result of their production process. Ow-
`ing to their very low density comprised between 10 and
`30 kg/m3, both conductive and radiative transfers are significant,
`leading to a coupling between these two modes of heat transfer.
`Numerous studies have dealt with radiative and conductive heat
`transfer through porous materials as reported by Baillis and Saca-
`dura [1]. Fibrous materials have been the most widely studied. We
`can cite as examples, the work of Lee [2], Jeandel et al. [3] on
`silica fibers, Milos and Marschall [4] on rigid fibrous ceramic
`insulation or Milandri [5]. Other researchers worked on glass
`foams (Fedorov and Viskanta [6]), carbon foams (Baillis [7])
`packed beds (Kaviany_ and Singh [8], Kamiuto [9], Coquard and
`Baillis [l0]), cellular foams (Glicksman et al. [1l]) or metallic
`foams (Calmidi and Mahajan [l2]), However, only few studies
`have already been conducted in the modeling of heat transfer in
`EPS foams.
`'
`
`Glicksman et al. [11] proposed a general study of radiative and
`conductive transfer through cellular foam insulation but they only
`consider materials with homogeneous cellular structure composed
`of cells struts and windows and with densities much greater than
`EPS foams. Kuhn et al. [13] studied more precisely heat transfer
`through EPS foams but they also considered that their cellular
`structure is homogeneous and thus, neglected the influence of
`macroscopic porosities. Moreover, this study was especially inter-
`ested in EPS foams with relatively high density and did not show
`good agreements with experimental measurements for foams with
`lower densities. Finally, Quenard et.al.
`[14] proposed another
`simple model which takes into account more precisely the real
`
`Contributed by the Heat Transfer Division of ASME for publication in the JOUR-
`NAL or HEAT TRANSFER. Manuscript received March 21, 2005; final manuscript re-
`L?
`ceived November 4, 2005. Review conducted by Jose L. Lage.
`'
`
`porous structure of EPS foams and, noticeably, their double-scale
`porosity for the conduction calculation. All of these studiesAne-
`glccted the coupling between radiative and conductive transfer
`and used Fourier’s law and Rosseland approximation to treat con-
`ductive and radiative transfer, respectively. Moreover, they always
`considered that the cellular material forming the foam is made of
`dodecahedric cells.
`
`In the present study, we propose a new approach to treat the
`total heat transfer in low density EPS foams based on a numerical
`resolution of the conduction-radiation coupling. The modeling of
`the radiative and conductive properties, required for the numeric
`calculation,
`take into account
`the particular structure of -EPS
`foams, notably their double scale porosity. Moreover, our model
`permits us to analyze the influence of the shape of the cells fortn-
`i11g the cellular materials contained in EPS beads, as we both take '
`into account dodecahedric and cubic shapes. First, we describe the
`particular porous structure of EPS foams. Then, we present pre-
`cisely the model proposed. Thereafter, the results of our model are
`analyzed in order to investigate the evolution of the thermal per-
`formances of the foam with its structural parameters. Finally, we
`check the validity of these theoretical results by comparing the
`equivalent thermal conductivity of several EPS foams, measured ‘
`by a guarded hot plate apparatus, with those predicted by our
`model.
`‘
`"""/
`
`’
`
`2 Description of the Structure of EP$ Foams
`As was explained before, expanded polystyrene foams are gen-
`erally classified among the cellular materials. However, contrary
`to other foams like polyurethane (PU) foams, their structure is
`more complex than a simple cellular matrix. This particular struc-
`ture is directly due to their production process.
`.
`Indeed, EPS foams are made from weakly porous cellular poly-
`styrene beads containing pentane as expanding agent.‘ At the be-
`ginning of the process, the diameter of these spherical beads is
`lower than 1 mm and they are perfectly independent from each
`other. They are expanded first (preexpansion) by simply injecting
`steam in order to noticeably increase their porosity. Thereafter,
`they are placed in a closed container and undergo a second expan-
`sion. During this second expansion, they are compressed together
`and thus. deformed, and mechanical bonds ‘are created between
`neighboring beads. At the end of the second expansion, the mate-
`rial is presented in the form of a unique rigid porous piece from
`
`538 I Vol. 128, JUNE 2006
`
`Copyright © 2006 by ASME
`
`Transactions of the ASME-
`
`K
`
`Page 5 of 16
`
`

`
`Cellular pore
`
`Contact area between
`neighboring spheres
`surface area of
`spherical Pflkticles
`
`Macroporosity
`
`Im
`a. Macroscopic structure
`
`b. Cellular structure
`
`a. 3-D view of aregular dodecahedron
`
`b.Longit11dinaI section along plan P
`of the dodecahcdmn
`
`‘ ta.
`
`Fig. 2 Illustration of the morphology of dodecahedric cells
`
`area of the windows. Finally, in lots of cellular materials, one can
`notice the presence of heaps of matter at the junction of two ’
`cellular faces like in PU foams. These so-called “cells struts” are
`thicker than the faces. Kuhn et al. [13] and Quenard et al. [14]
`took them into account for evaluating the conductive and radiative
`properties of their EPS foams. However,
`in the case of low-
`density EPS foams, as a result of the very high porosity of the
`cellular medium, they are very small and we can neglect them.
`Thus, the entire polymer is supposed to be contained in the pen-
`tagonal windows.
`To quantify the dimensions of the cells, we use the parameter
`», Dcen, called the mean cell diameter, corresponding to the distance
`between two opposite windows of the same cell (see Fig. 2): The
`volume Vdmk, of the cell is related to this diameter by the follow-
`ing relation
`
`Vdode “ 0-427 X D2211
`
`' (1)
`
`The height twin of the pentagonal windows forming the cells
`(see Fig. 2) is also related to Dee“
`
`twin =
`
`Dcell
`l16.56°)
`2 X sin<
`
`Z Dean
`1.7013
`
`(2)
`
`‘
`
`A
`2
`Assuming that the entire polymer is contained in the cell win-
`dows, it is then possible to express the thickness h of these-win-
`dows according to the mean cell diameter and the porosity seen of
`the cellular medium. Indeed, the surface area SW,“ of one window
`is: Sw,,,=O.72654>< :2
`is composed of l2 identical
`each cell
`I Moreover, given
`faces, each of these windows being shared with one neighboring
`face, there are finally six windows for one cell. And then
`
`SEM photographs representing macro and microporos-
`Fig. 1
`ity of EPS foams/
`_/
`
`which slabs can be cut up and used for the thermal insulation of
`walls. The diameter of the beads at the end of the second expan-
`sion is bctween 3 and 6 mm.
`‘
`As a result of this production process, EPS foams are charac-
`terized by a double-scale porosity. Indeed, as shown on the pho-
`tograph of Fig. l, the porous structure of EPS foams is not homo-
`geneous and is, in fact, composed of two different types of pores:
`
`-
`-
`
`cellular pores with regular shape, contained in the beads,
`pores with irregular shape, formed during the second expan-
`sion. of the cellular beads corresponding to the interparticle
`space.
`
`The order of magnitude of the size of the pore contained in the
`beads is approximately 100 ptm, whereas it is approximately 1 to
`several mm for the pores formed during the second expansion.
`Then, subsequently, we will use the terms “microporosity” and
`“macroporosity,” respectively, to make a distinction between the
`two types of pores.
`-
`The macroporosity einmbead due to the interparticle space
`formed during the second expansionvaries between 4% and 10%
`for the standard EPS foams. Generally, the lower the density of
`the foam is, the lower the macroporosity is. The diameter Dbead of
`the compressed beads is relatively homogeneous in the foam and
`we will assume afterwards that all the beads in a given foam have
`the same diameter.
`
`The characteristic macroscopic structure of the EPS foams"
`could be accurately reproduced by considering that the foam’ is an
`arrangement of overlapping spherical particles representing the
`compressed beads. The distance d between the center of two
`neighboring particles is shorter than their diameter. This distance
`depends on the expansion rate of the beads during the second
`expansion and is related to themacroporosity sinmbead. The con-
`tact area between two neighboring particles is assumed perfectly
`plane and is located on the intersection plane of the two spheres as
`shown in Fig. 1.
`'
`.—
`‘
`We can check in this figure that the morphology of the arrange-
`ment of spherical particles obtained perfectly matches the macro-
`scopic structure of real EPS foam. In particular, the shape of the
`macropores is well reproduced.
`_
`‘
`As’regards the cellular medium contained in the compressed
`beads, its porosity seen is very high and varies approximately be-
`tween 97% and 99%. The shape and the size of the cells are
`relatively homogeneous as can be seen in Fig. l(b). Therefore, we
`will assume‘ subsequently that all the cells belonging to the same
`foam have the same dimensions.
`
`Most of the previous researchers who dealt with thelinternal
`morphology of EPS foams like Quenard et al. [14] and Kuhn et al.
`[13] assumed that the cellular medium has a dodccahedral shape
`(see Fig. 2). Thus, each cell is composed of 12 pentagonal win-
`dows of polystyrene joined by their edge. The cellular material is
`then perfectly closed. Owing to the high porosity of the cellular
`medium, the thickness of the faces is negligible when compared to
`their size. We will make the same assumption as Kuhn et al. [13]
`and consider that this thickness is-constant all over the surface
`
`X h
`
`¢> h 3
`
`X
`
`— 856“) X Dee”
`
`(3)
`In real EPS foams, the cells forming the porous medium in the
`beads are not rigorously dodecamedrons but have rather irregular
`and random shapes composed of pentagonal, quadrilateral, or hex-
`agonal faces. Their shape is then more complex than the dodeca-
`hedron model and could not be perfectly described from a unique
`cell shape. However, in order to investigate the influence of the
`morphology of cellular medium on the thermal properties of EPS
`foams, we will also consider, subsequently, the case of cellular
`medium made of cubic cells. The cells are then composed of six
`square faces: The mean cell diameter is noted Dew, and the rela-
`tion expressing thetcell volume and the height, the surface area ’
`and the thickness of the windows is replaced by
`
`Vcube : D
`
`3 ‘
`cube:
`
`2
`..
`.
`_
`twin"Dcubes Swin ~ Dcube
`
`1
`
`and
`
`1_
`
`(
`
`See/ll)
`
`g
`
`3 X Vfen
`Vcube
`
`(1 ‘ Scell) X Dcube‘
`*
`‘
`<=>h= :““"m—“
`.
`3
`
`gm
`
`Journal of Heat Transfer
`
`JUNE 2006, Vol. 128 I 53§
`
`1
`
`
`I‘._‘V'"3ii'Is"_.‘4"4“'3i7§E1“9"""U‘”‘
`
`
`
`
`
`
`--W.....-.-......__.._.4_--,._.,_.._.,.:_wb"‘,~.‘\’c.°"‘5'v"l".{“"7"v.i.1.<K.'°vl\
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`
`Page 6 of 16
`
`

`
`In conclusion, our representation of the porous structure of EPS
`foams is entirely described using four different structural charac-
`teristics:
`
`-
`-
`
`the mean bead diameter Dbead of the beads
`the macroporosity eimemead or the distance :1 between the
`center of two touching beads
`the mean cell diameter Dceu
`the porosity of the cellular medium see”
`
`Generally, one prefers to use the density [JEPS of the foam in-
`stead of the porosity ace“ given that it could be measured directly
`contrary to see“ which must be determined from the measured
`density PEPS and interbead porosity a<m,,,e,,d. Indeed, we have
`
`PEPS = (1 ‘ gcell) X (1 ’ ginterbeads) X PPS -V
`PE
`@ scell: 1,_ 'pis X (1 — sinterbeads)
`PS
`
`_
`
`1
`
`3 Modeling of the One-Dimensional Coupled Ileat
`Transfer
`
`In the building insulation industry, people are usually interested
`in the total heat flux passing through a slab submitted to a one-
`dimensional steady-state heat transfer in Cartesian coordinates In-
`deed, this configuration corresponds to the thermal conditions en-
`countered by the insulator during its use. Thus, in the rest of the
`paper, we will only consider this kind of heat transfer.
`
`3.1 Numerical Resolution of the Energy Equation. As was
`previously explained, in low-density porous materials, given that
`conductive and radiative heat transfer are both dependent on the
`temperature distribution in the medium, there is a coupling be-
`tween these two modes of heat transfer. This coupling is governed
`by the energy equation explaining the thermal equilibrium in the
`material. For a one-dimensional steady-state heat transfer,
`this
`equation is
`
`Egg +— 59;
`52
`5.2
`
`= 0
`
`(6)
`
`In the model retained, we assume that the conductive heat
`transfer follows a diffusion law. Thus, the conductive flux is re-
`lated to the temperature by an analytical relation
`
`qc=—KC(T) .-—— and
`52
`5K er
`52
`5
`c
`'
`T
`qt‘
`. ~ ~ — K.
`. — —
`= —- K
`52
`5z
`La’)
`zz
`AT)
`dz
`The energy equation can be reformulated
`
`527
`. —
`522
`
`g
`
`V
`(7)
`
`(8)
`
`This equation can be solved numerically by an iterative process
`, using the control volume method. The principle of this method is
`to divide the, slab thickness in elementary control volumes. The
`temperature of the medium at the center of each volume has‘ to be
`calculated in order to know the temperature distribution in the
`medium. The iterative process goes on as follows. First, an initial
`temperature distribution in the medium is chosen, for example,
`linearly dependent on the z coordinates. Using this initial tempera-
`ture distribution, the radiative problem is solved and the radiative
`heat flux divergence 6q,/ 5z is calculated in each control Volume.
`A new temperature distribution is then calculated from this heat
`flux divergence distribution in order to satisfy the energy Eq. (8).
`From this new temperature distribution, the radiative problem is
`solved again, a new radiative heat flux divergence distribution is
`../.
`calculatecl..._and so on. The previous process is repeated until the
`
`540 I Vol. 128, JUNE 2006
`
`relative difference between the temperature distribution obtained
`for two successive iterations is lower than a very small value
`criteria equal to 10*‘, synonymous with convergence. The total
`heat flux passing through the slab could then be computed in each
`control volume by adding the conductive and radiative heat flux.
`As we can see, to solve the conduction—radiation coupling, one
`has to be able to solve the radiative heat transfer problem, that is,
`to compute the radiative transfer divergence distribution from a
`known temperature distribution.
`
`the Radiative Transfer
`3.2 Numerical Resolution of
`Equation. Radiative heat transfer in low-density porous medium
`is a relatively complex problem given that it takes into account
`not only the emission, absorption but also the scattering of radia-
`tion by the porous material. Then, contrary to the conductive heat
`transfer, the radiative heat flux could not be related to the tem-
`perature distribution by a simple analytical expression. It is thus
`necessary to solve the radiative transfer equation (RTE) governing
`the spatial and angular distribution of monochromatic radiation
`intensity I;,(z,,u.)
`in the medium. For a one-dimensional heat
`_ transfer with azimuthal symmetry, this equation is expressed by
`
`5I)\(Z»l‘)
`,,,___
`gz
`
`= - 5x[>.(ZuU~) + Kt/§(T)
`I
`
`+ Eff ¢>r(M’ —> #)Ir(z,M’)d;L’
`
`-1
`
`(9)
`
`and the boundary conditions
`
`I
`
`M0,?» > 0) =Ehot,)\Il)t(Tc) + 2-(1- Ehotlr) .f 1(0,- /~b'),U«'d,u'
`
`.
`
`0
`
`1
`
`Ix(0»7} < 0) ‘-‘Ecold,>\I())t(Tf) + 2 - (1 "Eco1d,x) - I I(l:.U«'),Uv'd/1«/
`
`0
`
`(10)
`
`0}‘, K)‘ and B(=a,\+K,\ cliaracterize the ability of the medium to
`scatter, absorb, and attenuate the radiation with wavelength A
`4>,\(,t/.,nc~> ,usc,,) characterizes the probability, for a radiation inci-
`dent in direction ,ui,,C:cos 01,13, to be scattered in an elementary
`solid angle around the direction ,u.w,=cos 03¢,
`Several numerical methods can be used to solve the RTE. For
`example, we can cite the spherical harmonics method, the zone
`method of HOTTEL or the ray—traeing methods. However,
`the
`discrete ordinates method (DOM) is the most frequently used and
`gives accurate results. In our study, we use it. The DOM is based
`on a spatial discretization of the slab,.thickness and an angular
`discretization of the space. The angular discretization allows re-
`placing the angular integrals by finite summations over fld discrete
`directions with given weighting factors wj. The spatial discretiza-
`tion must be the same as the one used for the numerical resolution
`of the energy equation. The DOM "has already been explained in
`numerous previous publications as Ref. [15] (Siegel and Howell),
`for example, and we will not describe it here more precisely. The
`accuracy of the results is entirely dependent on the angular dis-
`cretization. We then choose a very fine- discretization dividing the
`180° (Zn radians) in 180 directions of 1° and whose weighting
`factors are proportional to the solid angle they encompass.
`Theoretically, the RTE has to, be solved for the entire wave-
`lengths and the radiative heat flux divergence distribution is com-
`puted by rntegrating the contributions- of each wavelength
`co
`""4
`.
`
`'
`
`'
`
`K, 47T.’I))\(z)—21r2IM-(z),.wJ~ «ax
`ij=l
`'
`’
`
`o
`
`(11)
`
`where the subscript j refers to the j”‘ discrete direction.
`
`Transactions of the ASME
`
`Page 7 of 16
`
`

`
`In practice, to save computational time and memory usage, the
`spectral integration is made by considering that the intensity field
`in the slab is constant in N wavelength bands covering the entire
`wavelengths, Then we have
`
`where: Ik=J
`
`atk
`
`I,\.d)\
`
`where the subscript k refers to the k‘‘‘ band and AM. is the width of
`the k“‘ band.
`
`3.3 Modeling of the Radiative and Conductive Properties
`of EPS’ Foams. In order to solve the energy equation as well as
`the ‘RTE, and then evaluate the total heat transfer, the effective
`thermal conductivity KEPS and the monochromatic radiative prop-
`erties ox, Ky,
`,8,‘ and <I>)((,u,,,c——>p.m) of EPS foams have to be
`determined. In the following paragraphs we present the model
`developed in order to evaluate these properties.
`
`3.3.] Determination of the Efiective Thermal Conductivity.
`The effective thermal conductivity KEPS of porous materials
`strongly depends on the thermal conductivity of each of its con-
`stituents as well as on the morphology of the porous structure. In
`the case of EPS foams, the constituents are air (thermal conduc-
`tivity K,,;,) and polystyrene polymer (thermal conductivity Kps).
`The internal structure is characterized by two different types of
`pores which differ in size and shape (see Fig. 1). To take into
`account this specific structure, we choose the approach proposed
`by Quenard et al. [14] which divided EPS foams in three different
`phases with varying thermal conductivities. One phase corre-
`sponds to the macroporosity of air and the two other phases to the
`
`cellular materials contained in the beads. Indeed, the authors no-
`ticed a difference between the porosity and cell size of the cellular .
`medium located at the center of the beads and those located at the
`periphery. Thus, they make a distinction between these two re-
`gions of the beads. However, in the case of EPS foams with low
`density, we have analyzed several scanning electron microscopy
`(SEM) photographs of the cellular medium and we did not note a
`perceptible difference. Then, we will assume that the cellular ma-
`terial is homogeneous in the beads and then that EPS foams can
`be divided in only two phases: a gaseous phase and a cellular
`phase. The effective thermal conductivity resulting from the inter-
`action of the two phases can be estimated by the model of De
`Vries [16] which assumes that one phase constitutes a continuous
`medium (cellular phase), whereas the other phase is dispersed in
`spherical inclusions which do not interact with each other. We
`then have
`
`KEPS =
`
`(1 — Sintcrbcads) - Kcell + G“ - einterbea