materials selection for optimal design of a porous radiant burner for environmentally driven...

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DOI: 10.1002/adem.200900089
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Materials Selection for Optimal Design of a Porous RadiantBurner for Environmentally Driven Requirements**

CATIO
By Jaona Randrianalisoa*, Yves Brechet and Dominique Baillis
[*] Dr. J. Randrianalisoa, Prof. D. BaillisCETHIL, UMR5008, CNRS, INSA-Lyon Universite Lyon1,F-69621 Villeurbanne, FranceE-mail: [email protected]

Prof. Y. BrechetSIMAP, CNRS, Grenoble INP, Univesrite Joseph FourierF-38402, Saint Martin d’Heres, France

[**] This work was supported by the Rhone-Alpes Region Energycluster. The authors are grateful to the French PetroleumInstitute (IFP) of Solaize for initiating this work and to S.Gauthier, E. Lebas, J. Rosler, and A. Nicolle for their helpfuldiscussions.

ADVANCED ENGINEERING MATERIALS 2009, 11, No. 12 � 2009 WILEY-VCH Ve

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Radiant porous burners are promising technological

solutions to provide radiative and uniform heating without

gas convection, possible directional heat flux, and efficient

combustion of the gas. In addition, the efficiency of

combustion allows fulfilling requirements on pollution

(NOx, CO, etc.).[1,2]

In order to produce such burners, materials issues are a key

problem. Some of the requirements are straightforward to

identify, such as operating at high temperature (around

1000–1500 K), in a chemically aggressive environment. Some

other requirements on materials are more convoluted: the

operating conditions of the burner (such as the flame position

and the heat generated) are the keys to optimize the irradiated

power, and are also related to the materials used in its

fabrication. These operating conditions are function of

the radiative and thermophysical properties of the constitu-

tive material, on the permeability of the porous component.

The materials used also have a variety of degrees of freedom:

they are porous which means that both the constitutive

materials and the inner architecture are variables that can be

used to optimize the burner.

The current study focuses on open cell foam materials. As

such the optimization of materials for radiant porous burners

can be seen as a paradigm of a ‘‘materials by design’’

approach[3] which requires a rather advanced modeling of the

fluid flow, heat generation from chemical reactions, and

coupled heat transfer (radiation, convection, and conduction

transfer) inside the burner.

In Basic Physical Phenomena Section, we will present the

governing equations for the thermophysical problem, which

will enter the modeling. This will allow us to identify the

relevant materials properties. In Preselection Section, the

detailed set of requirements in terms of constraints and

objectives will be outlined. The Optimal Choice Section will

propose a first material selection in terms of screening via the

structure of the material (open porosity), the operating

temperature, and the oxidation environment. In Conclusion,

this preselection will serve as a basis in which an optimization

procedure will be proposed both for emitting radiative power

and for pollution control.

Basic Physical Phenomena

The basic idea of a porous burner is to perform the

combustion of a premixed gas (combustible and air) inside a

porous support (see Fig. 1).[4] The burner can be conveniently

subdivided into three parts, namely the entrance zone, the

porous combustion support, and the oxit zone. First, the gas

combustible is injected in the porous support from the

upstream (or entrance) zone while the burner ignition is

performed at the porous support boundary in downstream (or

exit) zone. As a consequence, the flame propagates from the

exit zone to the entrance zone. During the combustion, a part

of the energy of chemical reactions is transferred to the porous

support by convection heat transfer mode. Amount of this

thermal energy propagates through the porous support by

conduction and radiation. Thus, the porous support heats the

combustible gas while limiting the excess of temperature

reaching by the combustion through the thermal dissipation.

The thermal energy of the porous support is transferred to the

surrounding medium by convection at the boundaries of the

porous support and by radiative emission. It can be noted

that through this combustion mode, the heat produced by

radiative emission is much significant than that transferred to

the outside medium by convection.

To model the porous burner, a one-dimensional config-

uration (illustrated in the Fig. 2) is considered. From bottom to

top are the upstream or entrance zone, porous support, and

the exit or downstream zone. In addition, it is assumed that:

(1) T

rlag

he flow entering the porous zone is assumed perfectly

mixed and has a uniform velocity.

(2) T
he flow is a plug flow and the combustion front is one
dimensional. Neither turbulence nor stretch is induced by

the flow through the foam. The thickness of the foam is

much smaller than its diameter, thus, the heat and gas

losses from the edges can be neglected.

(3) T
he foam is sufficiently porous and the flow velocity is
sufficiently low that the process is isobaric.

(4) T
he porous medium emits, absorbs, and scatters thermal
radiation as a gray homogeneous medium, and gaseous

radiation is negligible compared to the solid radiation.

GmbH & Co. KGaA, Weinheim 1049

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J. Randrianalisoa et al./Materials Selection for Optimal Design of a . . .

Fig. 1. Photography of a porous burner. The porous piece is turned red by the heatresulting from combustion.

Fig. 2. One-dimensional porous burner configuration.

(5) N

105

o catalytic effect is induced by the solid. Thermal equi-

librium between the gas phase and the solid phasewas not

imposed; heat exchange between them is calculated using

a volumetric heat transfer coefficient.

Note that in this model the position of the combustion front

was not fixed but results from the equilibrium between the

flame speed and the in-flux velocity.

The numerical model of the burner accounts for the heat

transfer (by conduction, convection, and radiation), the mass

transfer (by convection and diffusion), and the combustion. As

a consequence, the governing equations, in absence of

catalytic reactions, are constituted of the following:[5,6]

An Energy Equation in the Solid Phase

The enthalpy balance in the solid phase having tempera-

ture Ts at the abscise x can be written as[7]

ð1� fÞrsCps@Ts

@t¼ @

@xls

@Ts

@x

� �� hðTs � TgÞ �

@qr

@x(1)

with f porosity, rs Cps the volumetric specific heat of the

constitutive material, Tg the gas phase temperature at the

abscise x. The terms in the right-hand side of Equation 1

correspond to the heat exchanges by conduction (with an

effective thermal conductivity ls), convection between gas and

solid (with exchange coefficient h), and radiation (with a

radiation flux qr). In the gray medium approximation, the

0 http://www.aem-journal.com � 2009 WILEY-VCH Verlag GmbH & C

divergence of the radiation flux in Equation 1 can be expressed

as[7]

@qr

@x¼ sa 4pI0ðx;TsÞ � 2p

Zþ1

�1

Iðx;mÞ dm

24

35 (2)

The right-hand side of Equation 2 corresponds to the

difference between the radiation power emitted per volume

unit at temperature Ts and the radiation power per unit

volume coming from all directions. sa is the absorption

coefficient of the porous material, I(x,m) is the radiation

intensity at the spatial coordinate x (along the sample

thickness) and propagating along the direction of cosine m

with respect to the x axis. I0(x,Ts) is the equilibrium intensity at

abscise x and temperature Ts.[7]

The Radiative Transfer Equation

The determination of I(x,m) requires solving the radiative

transfer equation (RTE). Assuming that there is an azimuthal

symmetry around the x axis, the usual one-dimensional RTE is

written as follows:[8]

m@Iðx;mÞ

@x¼ �ðss þ saÞIðx;mÞ þ saI

0ðxÞ

þ ss

2

Zþ1

�1

Iðx;m0ÞFðm0;mÞ dm0 (3)

The first term in the right-hand side of Equation 3

corresponds to the attenuation of the radiation due to

scattering (with scattering coefficient ss) and absorption while

remaining terms are, respectively, the radiation reinforcement

due to emission and incoming scattering (with a phase

function F(m0,m) denoting the transition probability from the

direction of cosine m0 to m).

Transport Equation of Chemical Species

The following equation written in terms of molar fraction X

models the transport of each species[6] and has to be solved

f@

@tðCXÞ þ f

@

@xðCXÞu

¼ f@

@xCD

@X

@x

� �þ f

@

@xC

DQ

Tg

@Tg

@x

� �þ f

Xnr (4)

with C the molar concentration of gas phase and m its molar

average velocity. Assuming a perfect gas of constant R and

pressure P, C is given by P/RTg. n and r are the Stoichiometric

coefficient and reaction rate of the considered reaction,

respectively. The sum is performed over all species.

In the right-hand side of Equation 4, the first term in

bracket corresponds to the diffusion of chemical species due to

concentration gradient (with a diffusion coefficient D); the

second term in bracket is the diffusion due to gradient

temperature known as Soret effect (with a thermal diffusion

factor Q). The last term accounts for the annihilation or the

creation of species during combustion.

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An Energy Equation in the Gas Phase

The enthalpy balance in the gas phase can be written as

follows:[6]

f@

@t

XðCXCpgTgÞ þ

@

@x

XðCXCpgTgÞu

¼ fX

D C@X

@xCpg

@Tg

@x

� �þ f

XDQ

Tg

CCpg

@2Tg

@x2

� �

þ f@

@xlg

@Tg

@x

� �� hðTg � TsÞ � f

XXnrHðTgÞ

(5)

with H(Tg) the enthalpy of species at temperature Tg and Cpgthe heat capacity of species. As in Equation 4, the sums are

performed over all species.

The first three terms in right-hand side of Equation 5

account for the enthalpy reinforcement due to (i) concentra-

tion gradient of species, (ii) thermal gradient, and (iii) thermal

conduction while the two last terms are the enthalpy

attenuation due to gas–solid convection exchange and

combustion, respectively.

Continuity Equation

According to the plug and isobaric flow assumptions, the

gas phase velocity can be calculated from the continuity

equation written, hereafter, in terms of molar concentration of

gas phase C [6]

f@

@tC þ f

@

@xCu ¼ f

X @

@xCD

@X

@x

� �

þ fX @

@xC

DQ

Tg

@Tg

@x

� �þ f

Xnr

(6)

The sums in Equation 6 are performed over chemical

species.

The Boundary conditions associated to Equations 1, 3–5 can

be found in ref. [5] However, it is interesting to note that these

boundary conditions are dependent on the properties of the

medium surrounding the burner (such as its emissivity

denoted by eout, and thermal conductivity lout) and the

properties of the porous material (such as the specific area

of pores, namely Sc, porosity f, and strut emissivity, namely

by es).There are three types of unknown parameters involved in

Equations 1–6 and in the boundary conditions. First, the

thermodynamic and transport properties of gases (Cpg,D,Q, r,

n, lg) are evaluated using the Chemkin and Transfit codes.[9,10]

Then, the surrounding medium is assumed as a blackbody at

room temperature, i.e., eout¼ 1, and the thermal conductivity

of air lout at ambient pressure is used. Finally, the functioning

of the burner imposes a gas flow, i.e., open porosity. The

effective material properties (ls, h, sa, ss, F, e, Sc) can

be obtained playing both of the constitutive material of which

the porous is made, and on the inner architecture of this

cellular solid. Therefore, in the strategy we have adopted here,

‘‘material by design,’’ we need to establish explicit relations

between constitutive materials and architecture, and the

macroscopic effective properties. Tables 1 and 2 summarize

ADVANCED ENGINEERING MATERIALS 2009, 11, No. 12 � 2009 WILEY-VCH Verl

the effective properties involved in the above numerical

model (i.e., in Eqs. 1–6) and their relationship with the

constitutive material and architecture.[11–17]

The current numerical model is constituted of 36 coupled

equations (27 equations for species transport according to the

famous Mech 1 chemistry mechanisms;[18] 3 equations for

energy transfer and gas phase velocity (i.e., Eqs. 1, 5, and 6);

and 6 RTEs corresponding to 6 Gaussian angular discretiza-

tion[19] between 0 and p). They are partial differential equation

(PDE) type, so the Comsol[20] code based on the finite element

solver, especially appropriate for PDEs, is used.Moreover, the

Premix code is combinedwith the Comsol code to improve the

treatment of chemical phenomena. Due to the high number of

equations to be solved simultaneously, a solution strategy

needs to be adopted as detailed in ref. [5].

Set of Requirements

The first step in any material selection process is a clear

definition of the set of requirements.[3] In particular, it is

essential to distinguish between objectives (What has to be

minimized or maximized?) and constraints (the threshold

level which has to be fulfilled). It is also very important to have

a clear understanding on the variables on which the

optimization relies.

Constraints

The combustion support, i.e., the porous material, must

satisfy several requirements. It needs to resist high tempera-

ture, oxidative atmospheres, and thermal choc. It must be

cheap. Moreover, the radiant mode of the burner imposes that

the combustion front takes place inside the porous zone.

Objectives

An optimized burner is an excellent radiant heater and, at

the same time, produces low concentration of pollutants. For a

given in-flux, the porous support is chosen so that the emitted

radiation by the burner is maximal while the pollutant rates

are minimum, at least smaller than standard values. Finally, a

large functioning range in terms of in-flux is desired.

Free Variables

To design an optimal burner fulfilling the above con-

straints, one can play on a number of free variables. The

constitutive material of the porous burner is one of them. But

operating conditions such as in-flux and the gas composition

can also be optimized for maximum burner efficiency. In the

following paper, the operating conditions will be considered

as given and the optimization will be carried out on the

material choice.

Table 3 summarizes the set of requirements considered in

the current porous radiant burner optimization.

Preselection

In the above set of requirements, a first group of criteria can

be treated as a ‘‘filtering step.’’[21] For instance, gas must flow

inside the burner, and therefore porousmaterials are required.

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Table 3. Set of requirements.

Constraints Objectives Free variables

Materials: resisting at high temperature, in oxidative atmospheres, and to thermal choice. High radiation emission Architectural properties

Functioning: the flame front inside the porous zone. Low pollutant emission Constitutive materials

Wide range of in-flux

Table 1. Radiative properties of porous materials.

Properties Formula Assumptions and references

Scattering phase function, F(u)

Metallic-based materials 8

3pðsin u � u cos uÞ (7) A metallic strut absorbs or scatters radiation beam. The local reflection

is almost diffuse since the strut roughness and the radiation wavelengths

are comparable. Hence, the phase function of an opaque diffuse

particle prevails.[11]

Ceramic-based materials 1 (8) The radiation interaction with ceramic struts is still not well known.

In a first approximation, an isotropic scattering is usually adopted.[12,13]

Extinction coefficient, se¼ saþ ssMetallic-based materials

2:656

ffiffiffiffiffiffiffiffiffiffiffi1� f

p

dc

(9) Relation derived for foam materials with tetracaedecaedric elementary

cells.[14]

Ceramic-based materials2:656

ffiffiffiffiffiffiffiffiffiffiffi1� f

p

dc1� hmið Þ

(10) mh i is a correction accounting for the anisotropy of scattering.

Comparison of se from X-ray tomography image analysis and that

from spectrometric measurements considering isotropic scattering

(data from ref. [15]) shows that mh i is about 0.2 for Mullite and

Zirconia foams.

Scattering albedo v¼ ss/(saþ ss)

Metallic-based materials g (11) For opaque struts, the scattering albedo reduces to the strut

reflectivity g .[7]

Ceramic-based materials 0:7 forTs > 600K

0:3 forTs � 600K

(12) The values of 0.7 and 0.3 correspond to the scattering albedo of

Mullite or PSZ foams calculated by using the Planck mean[7] at

800 and 450 K, respectively.[16]

Emissivity, es 1 – v (13) Equation 13 is exact for opaque struts. For semitransparent struts, it

is assumed that most part of the radiation energy is either reflected

or absorbed by a strut.

Table 2. Conduction and convection properties, and specific area of porous materials.

Properties Formula Assumptions and references

Thermal conductivity, ls 13(1–f) lbulk (14) The ls value is either issued from literature or estimated from this relation using

the measured bulk conductivity lbulk in literature.

Convection coefficient, hlg

1:2Re0:43Pr1=3

b

(15) Model previously suggested by Giani et al.[17] and commonly adopted for porous

burner modeling.[5] The strut size b is calculated knowing the cell size or volume

and porosity. Re and Pr the famous Reynold’s and Prandtl’s numbers, respectively.

Specific area of pores, Sc 36ab

Vc

(16) Assuming foam materials constituted of tetracaedecaedric cells having triangular

struts of length a¼ dc/2.995 and size b. The cell volume Vc is connected to a

parameter by Vc¼ 8(2)1/2a3.

The operating temperature and the oxidizing atmosphere will

screen out a number of constitutive materials. The operating

range in terms of flux and the required Reynold’s numbers

imposes some restrictions on the pore size.

In a first step, imposing a maximum service temperature

larger than 1000 K and a very good oxidation resistance, and

limiting the selection to open cell foams, the Cambridge

Engineering Selector (CES) software[22] was used to provide a

list of candidate materials. This list is given in Table 4. Four


additional materials currently studied in our laboratory were

added to this list.

In addition, fluid mechanics imposes extra conditions on

the material. An excessive Reynold’s number would lead to a

turbulent flow, and to a combustion taking place outside the

burner. A low porosity and small pore size induce significant

pressure drop, as a consequence they may cause a flow with

position dependent pressure. Therefore, the pore size must be

larger than a minimum value (about 100mm).


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Table 4. Candidate open cell materials with the appropriate temperature and oxidation resistance.

Base materials (purity ratio) (bulk density) Current acronym Cell number per unit volume [mm-3] Relative density

Alumina (99.8%) (1.2) Alumina 300–6� 10þ4 0.297–0.328



Alumina (99.5%) (0.745) Alumina 0.20–88 0.17–0.22

Alumina (92%) (0.61) Alumina 0.28–110 0.13–0.18

Alumina (99%) (0.825) Alumina 0.10–15 0.205–0.215

Cordierite (0.5) Cordierite 0.10–15 0.16–0.18

Mullite (0.70) Mullite 0.10–15 0.23–0.24

Mullite (0.65) Mullite 0.10–15 0.215–0.225

Mullite (NCL) (0.46) Mullite-NCL 0.20–42 0.15–0.16

Silicon carbide (0.5) SiC 0.10–15 0.15–0.16

Zirconia (partly stabilized) (1.28) PSZ 0.10–15 0.205–0.215



Zirconia Mullite Alumina (0.63) ZrO2�Al2O3�SiO2 0.33–57 0.15–0.19

Zirconia with calcia (fully stabilized) (0.74) ZrO2�CaO 2.30–38 2.24

Zirconia with magnesia (partly stabilized) (0.81) ZrO2�MgO 0.50–60 0.123–0.147

Additional materials Porosity Cell size-Strut size [mm] Specific area [mm�1]

FeCrAlY 0.951 1.27–0.23 2.00

NiCrAl 0.928 1.30–0.26 2.45

Mullite 0.809 1.25–0.19 3.60

PSZ 0.832 0.83–0.19 4.43

Optimal Choice

The materials preselected in the previous section have to be

evaluated, using the models presented in Basic Physical

Phenomena Section, in order to calculate their efficiency in

terms of radiated power and pollution rate. In order to do so,

we need the effective properties of these materials. Some of

these properties (such as thermal conductivity, specific heat,

porosity, and cell size) are referenced in (or inferred from)

the CES database. Other properties (such as extinction

coefficient, scattering albedo, exchange coefficient, and

Table 5. Effective materials properties.

Material acronyms Porosity Specificarea [mm�1]

Thermal conduc[W m�1 K�1

Alumina 0.67 26.60� 3.10

0.90 0.16� 0.72

Cordierite 0.82 6.80� 0.25

0.84 1.19� 0.41

Mullite 0.77 1.47� 0.33

Mullite-NCL 0.84 1.51� 0.31

SiC 0.85 6.36� 0.55

0.84 1.15� 0.71

PSZ 0.79 7.54� 0.26

ZrO2-Al2O3-SiO2 0.85 1.72� 0.50

0.81 10.95� 0.33

ZrO2–MgO 0.87 1.77� 0.26

0.85 9.63� 0.24

FeCrAlY 0.951 2.00 2.67� 10�4 TsþNiCrAl 0.928 2.45 6.18� 10�4 TsþMullite 0.809 3.60 0.26

PSZ 0.832 4.43 0.27

The properties which are calculated using Equations 7–16 are indicat


specific area) have to be estimated using Equations 7–16.

The values are given in Table 5 for the porous supports

representing the preselected materials. The comparison

between experimental results and modeling concerning the

NiCrAl foam recently carried out by Gauthier et al.[5] gives a

feeling for the accuracy of the model.

In the current study, combustion of premixed gas

constituted of air and natural gas is studied. The gas

combustible is injected at room temperature with an

equivalence ratio of 0.77. Two different specific powers (or

tivity]

Extinctioncoefficient [mm�1]

Scattering albedo

9.36� 0.3 for Ts< 600 K�; 0.7 for Ts> 600 K�

0.058�

2.39�

0.42�

0.52�

0.53�

0.40�

2.23�

2.58�

0.64�

3.85�

0.62�

3.38�

0.09 0.41 0.68

0.07 0.59 0.63

0.26 Tsþ 609 7� 10�4 Tsþ 0.069

0.267 Tsþ 846 5� 10�4 Tsþ 0.193

ed by an asterisk (�)


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in-flux) of 100 and 300 kW m�2 are considered. According to

the previous experiment carried out byGauthier et al.,[23] these

operating conditions allow to operate the burner in the radiant

mode with different types of porous supports. Moreover, they

approximate the operating conditions used in practice. In the

following figures, the numbers associated to the material

name refer to the porosity and cell size (in millimeter).

Some of the preselected materials such asMullite-NCL and

PSZ foams would lead to a combustion front outside the

burner, as the full numerical calculationwould show it. Others

such as Alumina(0.90,15.27) presents a combustion front close to

the entrance of the porous region. This is unwanted since it

would lead to a heating of the gas combustible diffuser

system. The combination of properties ruling the front

position is not straightforward and only a full numerical

calculation[16] can allow such an evaluation. In the following

paper, we will consider only the materials for which the

combustion front is well inside the porous region.

Fig. 3. (a) CO concentration versus NOx concentration of preselected materials ful-filling the set of requirements. Case of in-flux of 100 kW m�2. Open gray circles:additional materials; open circles: materials from CES. (b) CO concentration versusNOx concentration of preselected materials fulfilling the set of requirements. Case ofin-flux of 300 kW m�2. Open gray circles: additional materials; open circles: materialsfrom CES.

Pollution Rate

The pollution rate as well as the radiated power are

functions of the injected specific power, denoted by P.

Classically, the emission rate is measured and computed at

a distance of 0.5–2 cm from the burner’s exit side.[5] In the

current study, the calculation results are evaluated at 2 cm

from burner exit. We have investigated the burner’s behavior

for the maximum and minimum values of P in a standard

range of operation (100–300 kWm�2). The classification of the

different possible materials listed in Table 4, for minimal NOx

and CO concentration is shown in Figure 3(a) and 3(b).

For the two considered in-fluxes, the ceramic porous

materials lead to high pollution rates. For instance, with a

specific power of 100 kW m�2, the PSZ that can be

commercially available gives a pollution rate about 12–20%

higher than the FeCrAlY. The other porous ceramics selected

from CES database lead to pollution rates in the same range.

Indeed, most of these materials have a porosity around 0.8.

Metallic foams such as FeCrAlY, with a porosity close to 0.95,

for the same specific power lead to lower pollution rates in

terms of CO and NOx.

Radiation Power

As far as the radiation power is concerned, the porous

alumina is less efficient with an emitted flux of 15 kW m�2

(resp. 19 kW m�2) only for an in-flux of 100 kW m�2 (resp.

300 kW m�2). Other materials are significantly better but they

lie in a range of about 21–24 kW m�2 for 100 kW m�2 in-flux

and of about 25–28 kW m�2 for 300 kW m�2 in-flux. The best

one is Mullite foam for an in-flux of 100 kWm�2, whereas it is

FeCrAlY when the in-flux is 300 kW m�2 (see Fig. 4(a)

and 4(b)).

Optimal Material Choice

The optimal material choice depends on the relative

importance of the thermal efficiency objective and of the

low pollution objective. From the viewpoint of pollution, the


FeCrAlY foam solution is always preferable. From the

viewpoint of thermal efficiency, the situation ismore complex:

at high in-flux, metallic foams are better while at low in-flux,

Mullite foam takes over.

The problem of durability of the burner may be an issue. It

may be that the gas contains aggressive elements which are

more lightly to damage metallic foams rather than ceramic

foams. In this respect, it is worth considering the porous

ceramics, which provide the best compromise both for the

radiant efficiency and for pollution rates. From the materials

investigated above, the Zirconia Magnesium Oxide

(ZrO2�MgO) with a porosity of 0.87 gives the best

compromise for both objectives and high and low in-flux

pressures.


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Fig. 4. (a) Radiation emitted flux corresponding to preselected materials fulfilling theset of requirements. Case of in-flux of 100 kW m�2. (b) Radiation emitted fluxcorresponding to preselected materials fulfilling the set of requirements. Case of in-fluxof 300 kW m�2.

It is worth noticing that the best materials have the highest

porosities and a cell size of about 1.2–1.3mm.

Conclusions

The radiant burner optimization involves both energy

efficiency criteria and pollutant rate limitations. These two

criteria put very demanding requests on materials selection.

We have shown in this contribution that the material selection

depends both on operating conditions, and on the relative

importance of the two above criteria. We have outlined a

selection procedure, relying on a physically based model for

thermochemical processes, which allows not only to select

from an existingmaterial database, but also to guide amaterial

by design strategy.

Further improvement will have to relax the underlying

hypothesis of the present paper: the best solution may not be a

singlematerial. Possibilities of multilayers, gradient porosities


open a whole range of new options which will have to be

investigated using the same modeling tools and selection

tools[24] as the ones illustrated in this contribution.

List of symbols

a strut length (m)

b strut size (m)

C molar concentration (mol m�3)

Cp specific heat (J mol�1 K�1)

d cell size (m)

D diffusion coefficient (m2 s�1)

I, I0 radiation intensity and equilibrium or Planck

intensity, respectively (W m�2 sr�1)

h convection exchange coefficient (W m�2 K�1)

H molar enthalpy (J mol�1)

L material thickness (m)

P in-flux or injected specific power (W m�2)

Pr Prandtl’s number

qr radiation flux (W m�2)

r reaction rate (mol m�3 s�1)

R perfect gas constant (J kg�1 mol�1)

Re Reynold’s number

Sc specific area of pores (1 m�1)

t time (s)

T temperature (K)

� molar velocity (m s�1 mol�1)

V cell volume (m3)

x axis of a Cartesian reference

X mole fraction

e emissivity

f porosity

F scattering phase function

g reflectivity

l thermal conductivity (W m�1 K�1)

m, m0 cosines of radiation direction

hmi extinction coefficient correction accounting for the

anisotropy of scattering

n Stoichiometric coefficient

u angle between the radiation incident direction of

cosine m0 and the scattering direction of cosine m

(rad)

Q thermal diffusion coefficient (m2 s�1 K�1)

r density (kg m�3)

s absorption, scattering, or extinction coefficient

(1 m�1)

v scattering albedo

Subscripts

a refers to absorption coefficient

bulk refers to the constitutive material

e refers to absorption coefficient

c refers to cell

g refers to the gas phase

out refers to the medium surrounding the burner

s refers to scattering coefficient or to solid phase


COM

MUNIC

ATIO

N


Received: March 16, 2009

Final Version: April 20, 2009

Published online: July 1, 2009

[1] M. M. Kamal, A. A. Mohamad, Proc. Inst. Mech. Eng.

2006, 220, 487.

[2] R. Viskanta, in Handbook of Porous Media, (Ed: K. Vafai),

2nd edn., Taylor & Francis, New York 2005, 607.

[3] M. Ashby, Y. Brechet, D. Cebon, L. Salvo, Mater. Des.

2004, 25, 51.

[4] S. B. Sathe, M. Kulkarni, R. E. Peck, T. W. Tong, Proc.

Combust. Inst. 1991, 23, 1011.

[5] S. Gauthier, A. Nicolle, D. Baillis, Int. J. Hydrogen Energy

2008, 33, 4893.

[6] T. Poinsot, D. Veynante, in Theoretical and Numerical

Combustion, (Ed: R. T. Edwards), 2nd edn., Edwards,

Philadelphia 2005.

[7] R. Siegel, J. Howell, in Thermal Radiation Transfer, (Ed: R.

H. Bedford), 4th edn., Taylor & Francis, New York

2002.

[8] M. N. Ozisik, Radiative Transfer and Interaction With

Conduction and Convection, John Wiley & Sons, New

York 1973.

[9] R. J. Kee, J. A. Miller, T. H. Jefferson, in Sandia National

Laboratories Report 1980, SAND80-8003.

[10] R. J. Kee, G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, J. A.

Miller, in Sandia National Laboratories Report 1986, SA-

ND86-8246.


[11] M. Loretz, R. Coquard, D. Baillis, E. Maire, J. Quant.

Spectrosc. Radiat. Transfer 2008, 109, 16.

[12] A. Drift, G. J. J. Van der Beckers, K. Smit, J. Beesteheerde,

in 37th Eurotherm Seminar Proceeding (Ed: M. C. Rana),

Sallugia, Italy 1994, p. 275.

[13] I.-G. Lim, in 4th International Conference on Technologies

and Combustion for a Clean Environment Proceedings (Ed: S.

Samuelson), Lisbon, Portugal 1997, p. 28.

[14] M. Loretz, E. Maire, D. Baillis, Adv. Eng. Mater. 2008, 10,

352.

[15] M. Loretz, Characterization of Thermal Properties of

Ceramic and Metallic Foams From X-Ray Tomography

Analysis, Ph.D. Dissertation, National Institute of

Applied Sciences, Lyon 2008.

[16] S. Gauthier, Contribution to the Study of Combustion of

Natural Gas and Hydrogen Mixture in Porous Catalytic

Media, Ph.D. Dissertation, National Institute of Applied

Sciences, Lyon 2008.

[17] L. Giani, G. Groppi, E. Tronconi, Ind. Eng. Chem. Res.

2005, 44, 9078.

[18] A. Nicolle, P. Dagaut, Fuel 2006, 85, 2469.

[19] W. A. Fiveland, J. Thermophys. Heat Transfer 1988, 2, 309.

[20] Comsol MultiphysicsTM software. Comsol Inc., 2008.

[21] M. Ashby, in Materials Selection in Mechanical Design, 3rd

edn., Butterworth Heinemann, Oxford 2005.

[22] CES 4, The Cambridge Engineering Selector, Granta Design

Ltd, 2002.

[23] S. Gauthier, E. Lebas, D. Baillis, Int. J. Chem. Reactor Eng.

2007, 5, 114.

[24] M. Ashby, Y. Brechet, Acta Mater. 2003, 51, 5801.


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