Modelling of thermal conductivity of porous materials:
application to thick thermal barrier coatings
F.Cernuschi a,*,S.Ahmaniemi b ,P.Vuoristo c ,T.Ma ntyla c
a
CESI S.p.a.,Via Reggio Emilia,39,20090Segrate (MI),Italy
b Metso Paper Inc,PO Box 587,FIN-40101,Jyva ¨skyla ¨,Finland
c
Tampere University of Technology,Institute of Materials Science,PO Box 589,33101Tampere,Finland
Received 27March 2003;received in revised form 2September 2003;accepted 6September 2003
Abstract
Modelling of thermal conductivity of two and three phase composite materials is used to determine the
thermal conductivity of thick porous zirconia based thermal barrier coatings for use in high temperature applications.These coatings,depending on the deposition technique and process parameters exhibit different degrees of porosity.The porosity of the coating has an affect on thermal properties in completely different ways depending on the morphology and the orientation of the pores dispersed within a continuous matrix.In this work air plasma sprayed coatings have been considered.The experimental results were successfully compared to the modelled thermal conductivities.In the model the effects of porosity were taken into account considering the shape,orientation and volumetric percentage of pores.Image analysis and mercury porosimetry was used in experimental porosity determination.
#2003Elsevier Ltd.All rights reserved.
Keywords:Functional applications;Plasma spraying;Porosity;Thermal barrier coatings;Thermal conductivity;ZrO 2
1.Introduction
Thermal conductivity plays a key role in heat transfer processes.As a matter of fact in many industrial appli-cations,materials are selected primarily by considering their mechanical and thermal properties.In
aerospace,power generation and automotive industries,materials like metal or ceramic matrix composites and porous ceramic thermal barriers are widely used for manu-facturing the most advanced components.In particular,the last generation hot path components of gas turbines (typically combustion chambers,transition pieces,tur-bine blades and vanes)are protected against hot gases (>1300 C)by ceramic thermal barrier coatings (TBCs).In practice the thickness of TBCs is in the range of 300–1000m m.
There have been high expectations of using TBCs also in diesel engines.With TBCs the mean combustion temperature could be increased in the diesel process.At
the same time the heat losses to the cooling system could be decreased.This extra heat could be recovered in a turbocharger or in a flue gas boiler in a combined cycle.Some studies have shown that TBCs can increase the coefficient of thermal efficiency of diesel process and lower the fuel consumption.1Emissions such as NO x ,SO x ,CO,CO 2,unburned hydrocarbons and particle emissions have been studied in engine tests with and without TBCs.1À3Most of these studies disclose that TBCs decrease the fuel consumption,but have a minor effect on emissions.Without question the diesel process has to be adjusted correctly to utilise the benefits of thermal barrier coating.
Porous TBC’s can drastically reduce the temperature by 100–300 C of the internally cooled metallic base materials,depending on the coating thickness and microstructure.The selection of the best insulating coating significantly increases the efficiency of the gas turbine because either the cooling flow can be reduced or a higher turbine inlet temperature (TIT)can be achieved.4Thermal properties of TBCs,apart from the intrinsic thermal properties of the coating material,depend on the deposition technique used and the
0955-2219/$-see front matter #2003Elsevier Ltd.All rights reserved.
doi:10.1016/j.jeurceramsoc.2003.09.012
Journal of the European Ceramic Society 24(2004)2657–2667
www.elsevier/locate/jeurceramsoc
*Corresponding author.Tel.:+39-22-125-8740.E-mail addresses:cernuschi@cesi.it (F.Cernuschi).
process parameters.These factors affect on the content, the shape and the orientation of porosity inside the TBC. As the high temperature performance of TBC struc-tures is strongly related to the porosity features,it is important to be able to a priori design the thermal properties of these materials as well as to theoretically interpret the experimental results.These two tasks can be faced easier if reliable modelling of thermal proper-ties of TBCs can be done.The models also should take into account the shape,orientation and the volumetric percentage of the porosity.In the following chapter a short review of modelling of thermal conductivity of two-phase systems is amic matrix
and the pores in TBC,metallic/ceramic matrix and the rein-forcement in composites,etc.).The latter part of this paper concentrates on the specific case of porous TBCs.
2.Thermal conductivity of a two-phase composites The in-serie and in-parallel models proposed by Voigt-Reuss are the simplest models which give the two extreme limits for the thermal conductivity of a two-phase composite.5This model describes a composite, constituted of an alternate sequence of layers of two phases.Depending on the disposal of layers in respect to the heatflux direction,the series or parallel scheme is obtained.
Generally,almost all the models proposed in the lit-erature could be classified into asymmetrical or symmetrical,depending on the schematisation of the roles of the different phases.A two-phase composite material could be described as constituted either by a dispersion of isolated grains embedded within a con-tinuous matrix or by a symmetrical penetration of grains of the two phases occupying the whole volume.
2.1.Asymmetrical models
Many asymmetrical models have been proposed after the pioneering studies of Maxwell5and Rayleigh.6In particular,the Maxwell model is applicable only for a very diluted(volumetric fraction below
10%)dispersion of spheres within a continuous matrix because he assumed that thefield was one-dimensional at a sufficiently long distance from each sphere.
2658 F.Cernuschi et al./Journal of the European Ceramic Society24(2004)2657–2667
If k m ,k d and f are the thermal conductivities of the matrix and of the dispersed spheres and the dilute volumetric fraction respectively,expanding the Maxwell solution in a Taylor series about =0,the thermal con-ductivity k of the composite can be written as:
k
k m
¼1À3f 1Àk d =k m ðÞ2þ
d k m
26643775ð1Þ
Subsequently the Maxwell model was also extended to consider a dilute consisting of a dispersion of ran-domly oriented ellipsoidal particles.7
A significant improvement to asymmetrical modelling was given by Bruggeman who in fact assumed th
at the sphere radius of the dilute dispersion varied within an infinite range of values and that each single sphere was embedded within the continuous matrix.8Starting from these assumptions he showed that the limitation on the volumetric fraction of the dilute dispersion can be removed (i.e 04f <1).In this case the equation derived is:9,10
k =k m À
k d
k
m
k k m 1=31À
k d
k m "#¼1Àf ð2Þ
Also in this case,analogously to the Maxwell model,it is possible to generalise the modelling to a solute dis-persion of randomly oriented ellipsoids.11À13
Of particular interest is the specific case of spheroids (ellipsoids having a revolution axis corresponding to the ellipsoid axis a and thus with the other two axes b =c )because a wide interval of real situations can be modelled by defining an orientation angle a between the field gradient (the heat flux)and the dispersed par-ticles and by varying the ratio a/c between the two axes of the spheroid.
Considering further models proposed in the literature,the Meredith and Tobias model was well suited to dilute dispersions with a particle size distribution not wide as required by the Bruggeman model.The limitation to volumetric fractions <0.6is the main drawback of this model 10
2.2.Symmetrical models
As previously described,in symmetrical models the two phases (named 1and 2)play interchangeable roles.To make more evident this symmetry,the thermal con-ductivity and the volumetric fractions of both phase 1and phase 2are indicated as k 1,k 2,f 1and f 2(where obviously f 2¼1Àf 1).In particular,if both phases con-sist of spheres of a very wide size range,the following equation could be used:14À16
f 1k 1Àk ðÞk 1þd À1ðÞk ½ þf 2k 2Àk ðÞ
k 2þd À1ðÞk ½
¼0;
ð3Þ
where d is the space dimension (typically 3).Also this model can be extended to randomly oriented ellipsoids.17
3.Modelling for porous materials
For porous materials like a TBC,the thermal con-ductivity of pores can be assumed to be negligible (i.e.k d or k i ffi0).In particular,this is true for temperatures where inside the pores the radiative contribution to the thermal conductivity of the composite can be neglected.So all the models are significantly simplified and they can be applied in semi-quantitative explanation of the experimental results related to thermophysical and microstructural characterisation 18À21as also shown in this study.With this assumption,the two extreme limits of the thermal conductivity,as determined by Voigt-Reuss,are:704k 4ð1Àf Þk m
ð4Þ
3.1.Asymmetrical models
The Maxwell and the Bruggeman models represented by Eqs.(1)and (2)respectively become:10,13,14
k k m ¼1À3
2
f
ð5Þk
m
¼1Àf ðÞ32ð6Þ
More generally,for a dispersion of spheroids,the Bruggeman model is:13k
k m
¼1Àf ðÞX ð7Þ
where X ¼1Àcos 2
1ÀF þcos 2
2F ,in which F is the shape factor of the spheroid,and a is the angle between the revolu-tion axis of the spheroid and the non perturbed heat flux.12,22Fig.1graphically represents the value of the shape factor F as a function of the axial ratio a/c.In particular,for sphere (a =c )F is 1/3while for oblate (c >a )and prolate (a >c )spheroids F values are in the range of 0–1/3and 1/3–1/2,respectively.Fig.2shows the X values for three different cases.In particular,for lamellar porosity (c >a )the X factor gets very high values when the a axis is either randomly oriented or parallel to the heat flux.On the contrary,if lamellae are oriented normal to the heat flux,X is equal to 1.For cylinders,X values tend to be 1,1.667or 2depending if
F.Cernuschi et al./Journal of the European Ceramic Society 24(2004)2657–26672659
the a axis is parallel,randomly oriented or perpen-dicular respectively.
From the previous considerations it is clear that Eq.(7)is well suited to describe a wide range of closed por-osity configurations in term of voids shape and orient-ation.Moreover Schulz showed that also randomly oriented open porosity can be accurately described by Eq.(7).13Furthermore,it is interesting to observe that for spheres,this equation reduces to Eq.(6)which expanded in Taylor series close to zer
o becomes equal to
Eq.(5).Fig.3shows the ratio k k m
as a function of the volumetric fraction of porosity as described for pores of different shape and orientation in respect to the heat flux by the Bruggeman model.In particular a small amount of lamellar (penny shaped)pores with the major axis oriented parallel to the heat flux produces a strong reduction of the thermal conductivity of the porous material.A similar result was obtained by Hasselman who described the effect of penny shaped pores on the thermal conductivity starting from the extension to revolution ellipsoids of the Maxwell model.18
The extension of the Meredith and Tobias model to the case of randomly oriented spheroidal porosity gives:10
k k m ¼2Àf 2þW À1ðÞf 21Àf ðÞ21Àf ðÞþWf ;ð8Þwhere W ¼13
12F þ21ÀF ðÞ
that is the X factor for randomly oriented spheroids.
A comparison between the models of Voigt-Reuss,Maxwell,Bruggeman and Meredith and Tobias have been considered in Fig.4for randomly oriented spher-oids with the shape factor F=0.1.As expected,the Voigt-Reuss model describes the two extreme limits of variability of the ratio k k m
while the Meredith and Tobias model represents an intermediate situation between Maxwell and Bruggeman models.As the Bruggeman model appears to be able to describe the widest range of situations,it will be applied to the selected specific experimental cases in the
following.
Fig.1.Shape factor F as a function of the axial ratio a/c of the
spheroid.
Fig.2.X factor as a function of the axial ratio a/c of the spheroid for (1)randomly oriented spheroids,(2)spheroids oriented with the revo-lution axis a parallel to the heat flux (s 2 ¼1),and (3)spheroids oriented with the revolution axis a normally to the heat flux (s 2 ¼0).The crossing point of these curves refers to the sphere (i.e.a/c=1,
X=1.5).
Fig.3.The ratio k m
vs.the volumetric fraction of porosity as esti-mated by Eq.(7)for lamellae (F =0.0369)with the revolution axis (1)parallel,(2)perpendicular and (3)randomly oriented and for cylinders (F =0.499)with the revolution axis (4)parallel,(5)perpendicular,(6)randomly oriented to the heat flux and for (7)spheres.
2660 F.Cernuschi et al./Journal of the European Ceramic Society 24(2004)2657–2667
3.2.Symmetrical models
For a symmetrical porous material (i.e.k 1ffi0)Eq.(3)simplify as follows:k k 2¼1À32
f ð9Þ
The main result is that this equation corresponds to Eq.(5)obtained for the Maxwell asymmetrical model restricted to a very diluted dispersion.This means that under peculiar conditions (very diluted not conducting sphere dispersion)the two approaches converge to the same formulas.
Furthermore,for a symmetrical dispersion of two sets of spheroids where one of them consists of pores (i.e.k 1ffi0),Eq.(3)generalises as follows:17f 1k 1ÀF 1ðÞk ½ ¼
f 2k 2Àk ðÞ
F 2k 2þ1ÀF 2ðÞk ½
ð10Þ
where F 1and F 2are the shape factors for the two spheroidal dispersions.3.3.Other models
Both asymmetrical and symmetrical models assumed well defined configurations of the porosity (i.e.disper-sion of spheroids in a continuous matrix or in spher-oids),but in some cases,the reality should be represented by an intermediate situation.McLachlan proposed a model introducing the idea of a conducting-
insulating transition.More in details,looking at the
Eqs.(6)and (9)for the three dimensional space,the conducting-insulating transition takes place for the cri-tical values f ¼1and f ¼2=3respectively.More gen-erally for a space of d dimensions,the critical value of
the volumetric fraction equal to d À1ðÞ
d .Starting from this analysis,McLachlan proposed th
e phenomenological equation:23
k k m ¼1Àf f c
df c =d À1ðÞ
ð11Þwhere f c ¼d À1ðÞD and D is the fractal dimension of the system.In particular D is a positive real number repre-senting the degree of irregularity and discontinuance of an object which has the self-similarity property.24À26In this specific contest,the two limit values of D are D=d (symmetrical model)and D=d-1(asymmetrical model)but D can assume also intermediate values depending on the specific real case.Furthermore,a possible gen-eralisation of Eq.(11)to spheroids derived from Eq.(7)is:23
k k m ¼1Àf f c
f c X
ð12Þ
Fig.5graphically represents the ratio k
k m
vs.volu-metric fraction of porosity as described by Eq.(11)for different values of the parameter D .As expected,depending to the fractal dimension D ,the transition conducting–insulating takes place at different
values.
Fig. 4.The ratio k k m
vs.the volumetric fraction of porosity as estimated by Voigt-Reuss (1)in serie and (2)in parallel models.(3)Maxwell,(4)Bruggeman and (5)Meredith and Tobias models for a randomly oriented lamellar porosity with a shape factor F=
0.1.
Fig.5.The ratio k m
vs.the volumetric fraction of porosity as esti-mated by Eq.(11)for (1)symmetrical limit D=3,(2)asymmetrical limit D=2and for fractal dimension (3)D=2.2,(4)D=2.5and (5)D=2.8.
F.Cernuschi et al./Journal of the European Ceramic Society 24(2004)2657–26672661
Another model,specifically proposed by McPherson27 for ceramic TBC,deposited by air plasma spray(APS), yields the similitude between the electrical resistance of metallic contacts and the heatflux.The model is restricted to conduction through the various points of true contact between lamellae to determine the ratio between the thermal conductivity of bulk and the porous materials.
In particular,on the basis of microstructural observa-tions with transmission electron microscopy(TEM),the porosity was assumed to consist of circular microcracks parallel to the coating surface.Microcracks were dis-tributed in planes mutually separated just by the thick-ness of the lamellae.If’is the fraction of the apparent area of the true contact on each plane,the ratio between the through-the-thickness thermal conductivity of por-ous and bulk material is:27
k k m ¼
2’
a
ð13Þ
where a is the radius of each true contact circular area. In the specific case considered by McPherson,the2a diameter and the lamellar thickness were comparable. This model has been suitably modified also for taking into consideration the gas present within the pores as well as the radiative contributions at high temperature.27
3.4.Extension to models for composites containing many porosity types
All the models previously described assume the por-osity consisting of single shape pores or cracks having also a single orientation.Nevertheless,the porosity of a real TBC(typically deposited by APS)is a mixture of different type of pores.Thus,the modelling of real cases could require the superimposition of the contributions of different porosity types to the overall thermal conductivity of the porous material.
A possible approach to extend these models to a manifold porosity system consists in applying in an iterative way a two-phase modelling.20,21,28In particular if f0is the total amount of porosity and f1and f2are the percentages of the types of porosity respectively (f0¼f1þf2),the thermal conductivity of the three-phase mixture is:
k k m ¼
1
2
F
f2
1Àf1
ðÞ
Éf1ðÞþÉ
f1
1Àf2
ðÞ
F f2ðÞ
ð14Þ
whereÉ(f)andÈ(f)indicate are the functions describ-ing the effect of porosity on the thermal conductivity of the matrix.In particular,ÉandÈcan be chosen between the different expressions previously defined depending on the morphology of the porosity which has be modelled. This iterative approach to extend models developed for a material containing a single type of porosity(for example spheres)to materials containing many types of porosity(for example spheres,cylinders and lamellae)is an hybrid between symmetrical and asymmetrical approaches.In fact,the addition of thefirst porosity to a continuous matrix requires to use an asymmetrical model but for the subsequent iterations,needed to take into account the other porosity types,the matrix could not be considered exactly continuous anymore because of the presence of thefirst dispersion.However,the result for the special case F fðÞ ÉfðÞ¼1Àf
ðÞX (corresponding to account for a given amount of por-osity of a single type using the proposed iterative approach)Eq.(16)reduces to Eq.(9)giving indications that this approach could be applied for extending the Bruggeman asymmetrical model from a single shape porosity to a spectrum of different porosity shapes if, for each porosity type,the size of the dilute dispersion varies within a very wide range of values.
Thus,when a higher number of porosity types should be simultaneously considered(for example three),it is sufficient to apply the iterative procedure ones more.In particular,if f3andÂ(f)are the volumetric fraction and the function describing the contribution of the third porosity type to the thermal conductivity respectively, the expression giving thefinal thermal conductivity k of the four phase system is:
k¼
k0
6
F
f2
1Àf1þf3
ðÞ
reactive materials studiesðÞ
É
f1
1Àf3
ðÞ
Âf3ðÞ
þÉ
f1
1Àf2þf3
ðÞ
ðÞ
F
f2
1Àf3
Âf3ðÞ
þF
f2
1Àf1þf3
ðÞ
ðÞ
Â
f3
1Àf1
Éf1ðÞ
þÉ
f1
1Àf2þf3
ðÞ
ðÞ
Â
f3
1Àf2
F f2ðÞ
þÂ
f3
1Àf1þf2
ðÞ
ðÞ
F
f2
1Àf1
Éf1ðÞ
þÂ
f3
1Àðf1þf2
ðÞ
ðÞ
É
f1
1Àf2
ðÞ
F f2ðÞ
ð15Þ
The peculiar structure of Eqs.(14)and(15)has been defined in order to obtain expressions independent from the order of mixing the different porosity types within the matrix.
4.Application to selected real cases of porous TBC As described in details elsewhere,29,30the measure-ments of thermal conductivity on some different TBC have been carried out:in particular,thick zirconia based TBC stabilised by three different oxides(8Y2O3ZrO2,22 wt.%MgO–ZrO2,and25wt.%CeO2–2.5Y2O3–ZrO2). In order to detect,if present,any densification process caused by the high temperature exposure during the measurement itself,the thermal conductivity has been
2662 F.Cernuschi et al./Journal of the European Ceramic Society24(2004)2657–2667
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