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Heat transfer through source powder in sublimation growth of SiC crystal
Materials Science and Engineering B55 (1998) 174 – 183
Heat transfer through source powder in sublimation growth of SiC crystal E.L. Kitanin a, M.S. Ramm b, V.V. Ris a, A.A. Schmidt b,* a
b
State Technical Uni6ersity, 194026 St. Petersburg, Russian Federation Ioffe Physical Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russian Federation Received 27 June 1997
Abstract A problem of heat transport in a source powder is very important for sublimation crystal growth technique. This paper is devoted to the development and implementation of a model of powder heat transfer. As an example temperature field calculation in a growth chamber containing a SiC-powder layer is performed. © 1998 Published by Elsevier Science S.A. All rights reserved. Keywords: Heat transfer; SiC-powder; Sublimation growth technique
1. Introduction
2. The effective heat conductivity of granular material
Silicon carbide is a very promising material for making high temperature, high power and high frequency electronic devices [1]. The main problem of SiC electronics is fabrication of large-scale substrates. Sublimation growth is now considered to be a perspective technique which would enable one to fulfill this task. In numerous investigations devoted to SiC sublimation, growth temperature field control and optimization are the key problems [2]. At the same time heat transfer in SiC-powder, used usually as a source of vapors of Si and C in sublimation growth, is outside of the framework of most studies. Difficulties in investigating heat transfer processes in powder are caused by a variety of heat conduction mechanisms whose relative contributions depend on growth process parameters. In experimental investigations of transport processes inside the powder it is necessary to consider a wide range of parameters. Implementation of theoretical models of granular materials can reduce complicated experimental work. In the present paper a mathematical model of heat transport in SiC-powder is used for the prediction of temperature fields inside a sublimation growth chamber.
Theoretical models are mainly based on the generalized conductivity principle (GCP) [3]. The GCP is mostly effective for a cellular material when the conductivity of a continuous medium is greater than that of a dispersed one. Otherwise, calculation methods based on the GCP need corrections [4]. These corrections can be associated, in particular, with (a) non-uniform chaotic structure of the material; (b) difference of heat transfer mechanisms in gas and solid phases; (c) peculiarities of heat transfer through contact spots between particles. The problem of predicting the temperature field within a cylindrical layer of SiC-powder during sublimation is governed by Fourier equation. As the first step of the study, a starting stage of the process is considered when the powder structure is close to the initial one. Conductive and radiative mechanisms of heat transport in the powder layer are taken into account. It is assumed that the most important heat source is an external heater, while the internal source qV due to the latent heat of sublimation and chemical reaction heat release can be neglected. Under such assumptions the stationary temperature field is governed by the following non-linear energy equation:
* Corresponding author. Tel.: +7 812 2479145; fax: + 7 812 2471017.
0921-5107/98/$ - see front matter © 1998 Published by Elsevier Science S.A. All rights reserved. PII S0921-5107(98)00146-9
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183 pr leff = f(l sk eff, l eff, P),
Fig. 1. Schematic view of a granular medium with first-order structure (a) and second order structure (b).
9(leff9T)=0, where leff is the effective heat conductivity of the powder material accounting for both radiative and conductive heat transfer. Two types of multicomponent media can be distinguished: embedded media and interpenetrating media. Granular materials (powders) take intermediate place because the gas phase occupies a simply connected domain and the particles contact each other directly through contact spots. Transport properties of such media depend on transport properties of both powder material and gas in macro-pores between the particles and in micro-pores formed by roughness of the particle surfaces in contact spots. In the framework of the proposed method the effective conductivity is determined as a function of effective heat conductivity’s of powder components and the gas bulk fraction which is called porosity P. The effective conductivity can be expressed in the form
175
(1)
where P=Vg/V, V is the powder volume, Vg is the gas pr phase volume, l sk eff, l eff are the effective heat conductivities of the powder particles (a skeleton of the granular material) and of the gas pores, respectively. Geometrical parameters of the internal structure of powders can be obtained on the basis of the assumption of chaotic arrangements of particles. In accordance with [5] at low porosity (P5 0.4) it is assumed that the gas pores (first-order structure) are uniformly distributed within the powder (Fig. 1(a)) and at P\0.4 large gas pores (second order structure) appear additionally (Fig. 1(b)). According to this model the powder with the first-order structure, which is supposed to consist of monodispersed spherical particles of diameter d, is substituted by a frame with the same thermal resistance. The frame is formed by perpendicularly crossing rods of constant cross-sections. One-eighth of an elementary cell of such a frame is presented in Fig. 2(a). The frame cell is characterized by two parameters: d1 which is the characteristic size of the rod cross-section and L1 which is the characteristic rod length. These parameters are chosen to provide the porosity of a medium with the frame to be equal to P of the powder. If pores of the second order structure are present in the powder a similar model is used. In contrast to the previous one the frame now is a porous medium by itself containing the first-order pores at skeleton porossk sk ity Psk is the frame cr = V /(V −V2)= 0.4, where V volume and V2 is the volume of pores with second order structures. The bulk fraction of these pores can be determined as: P2 = V2/V=
P− Psk cr . 1− Psk cr
(2)
Fig. 2. One-eight of an elementary cell of the frame substituted into the granule medium (a) and a scheme of thermal resistances of the elementary cell (b).
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176
In this case the frame cell is characterized by two parameters d2 and L2 chosen to make the porosity of the frame (which is porous in its turn) to be equal to P2. Non-dimensional ratio Ci =di /Li satisfies the following equation: 2C 3i − 3C 2i − Pi + 1 = 0, where i=1, 2 corresponds to the first and the second order structures, respectively. Fig. 2(b) shows an equivalent scheme of thermal resistances in the elementary cell (Fig. 2(a)). In Appendix A correlation’s for the calculation of thermal resistances as functions of both geometrical parameters and effective thermal conductivities of elementary cells are presented. From Eqs. (A.1), (A.2) and (A.3) one can obtain an equation for leff,i : leff,1 = l sk eff
if
P 50.4,
leff,2 = leff,1 C 22 +n2(1 − C2)2 +
n
2n2 C2(1 − C2) , n2 C2 −C2 +1
(3)
pr,2 where n2 = l pr,2 and l sk eff /leff,1, l eff eff are effective heat conductivities for pores of the second order and for the frame skeleton. Depending on the porosity corresponding value, leff,i should be substituted into Eq. (1). In Sections 3– 5 the most important details of the model of powder heat conductivity are described.
3. Effects of geometrical and physical characteristics of the powder on heat transport through the skeleton and pores
3.1. Geometrical characteristics of the contact spot and neighboring domain In order to determine the effective conductivity of the powder skeleton, it is necessary to take into account that in a powder each particle makes contact with N neighboring particles or, in other words, has N contact spots. Thermal resistance of the contact spots is a controlling factor of heat transport through the skeleton. The number of contact spots, so called ‘the coordination number’ is a function of the bulk fraction (porosity). It can be written in the form [5] N=
P+ 3+ P2 −10P +9 , 2P
(4)
where P =P1 for P 5 0.4 and P = Psk for P \ 0.4. In a practically interesting range of porosity, N can be approximated as N =2.22P − 1.3.
Fig. 3. Schematic view of an ‘actual’ (a) and an ‘ideal’ (b) contact spot.
The contact spot and clearance around it (Fig. 3(a)) are approximated by a contact zone of radii r1, height h/2 (Fig. 3(b)) and a ring clearance around it with outer radii r2. Contact spot thickness h can be treated as an averaged height of particle surface roughness. Dimensionless values of r1, r2 and h scaled by particle radius r are z1 = r1/r, z2 = r2/r, h%= h/r. Let r3 be the radius of a part of the total particle surface which is shared by one contact spot and r4 the radius of a through pore surrounding the spherical domain of the particle surface of radius r3 (Fig. 4(a)). Dimensionless values of r3 and r4 are z3 =r3/r= 2 N − 1/N and z4 = r4/r=z3(1–P) − 1/3. In addition, two dimensionless geometrical parameters are introduced Z2 = 1 − z 22 and Z3 = 1 − z 23. Near spherical sector of radius r3 integer-averaged distance between two particles hsph is
hsph = d h%+
1 , N
where d= 2r. Size of a pore in the second order structure D can be obtained from [5]: D=3d
1 −1 . C2
3.2. Influence of a compression on the contact spot and clearance For absolutely smooth elastic spherical particles compressed by a force P, the area of contact spot Sn can be calculated using the Hertz formula. In fact the contact spot area is less because of the surface roughness effect Sreal = cSn, where c is the empirical coefficient. In the absence of external compressing forces it is possible to write expressions for non-dimensional radii z1 and z2 [5]
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177
Fig. 4. Schematic view of a conductive element with averaged parameters (a) and the connection schema (b).
z1 = z2 c, z2 = 3.3 · 10 − 3(1 − P)1/9(rsolgh1)1/3,
(5)
where rsol is the particle material density, h1 is the powder layer height, g = 9.81 m s-2. Thus the geometrical characteristics of contact spots and clearance between the powder particles depends on two empirical parameters h% and c. Regarding these characteristics all materials can roughly be divided into three groups [6,7]: rigid materials with Moose hardness numbers Mhn= 7.0–9.0 can be characterized by h% = (2.0– 6.0) · 10 − 3 and c =(3.0 – 5.0) · 10 − 4; intermediate materials Mhn =2.0 – 7.0; h% = (1.0– 8.0) · 10 − 3 and c =(5.0 – 10.0) · 10 − 4; soft materials Mhn =1.0 – 2.0; h% = (0.0 – 2.0) · 10 − 3 and c= (10.0– 50.0) · 10 − 4. Particles of MgO and SiC powders considered below (see Section 6) are close to the group of rigid materials and the following values of the parameters are used h% =2.2 · 10 − 3 and c= 5.0 · 10 − 4.
3.3. Dependence of the heat conducti6ity of gas in pores on the molecule mean free path It is known [8] that a decrease in heat conductivity of the granular materials with gas pressure in pores is greater than that due to decreasing gas heat conductivity. This phenomenon is a result of temperature jump on a solid surface at low pressure. The temperature jump depends on both gas and surface properties, namely: accommodation coefficient a, specific heat ratio g, Prandtl and Knudsen numbers [8 – 10]. The temperature jump can be neglected when the Knudsen number
Kn=L/l B B 1, where L is the molecule mean free path and l is the characteristic size of the gas pore. The characteristic size of the gas pore is : 101 –102 mm for pore side of the contact spots and :1 mm inside them. The molecular mean free path can be calculated as L=kT/ 2ps 2p, here k is the Boltzmann constant, s is the characteristic size of the gas molecule, T, p are absolute temperature and pressure of the gas. For the conditions under study (argon, T:2400 K, p:15 mmHg) L: 30 mm and, therefore, Kn:10 − 1 – 101. Effect of the temperature jump on the effective heat conductivity can be evaluated using the following correlations [10] l geff =
lg 4g 2−a L0p0 , B= · · , 1+ B/(pl) g+ 1 a Pr
(6)
where the index 0 corresponds to the normal conditions, lg = lg(T) is the gas phase molecular heat conductivity and Pr is the Prandtl number. The most complicated problem in the calculation of the temperature jump is determining the accommodation coefficient. The value of this coefficient depends on physical properties of gas and solid particle material, temperature and pressure. It can be crudely estimated from [11]: a= 1−
n
M 2g + M 2sol n , (Mg + Msol)2
where Mg, Msol are molecular masses of the phases, n= 3–4 is the number of collisions of the gas molecules with solid surface.
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178
3.4. Radiati6e heat conduction in the powder At conditions under study according to the Plank law the main portion of radiative energy transport is associated with wave lengths B 10 mm. The optical thickness for this radiation is much greater than the characteristic size of the pore. Two factors have an effect on the radiative heat flux in the powder qr: emissivity e of a surface surrounding the pore with characteristic size l and transmissivity P of the powder material surrounding the pore. The emissivity of the particle surface depends significantly on the surface condition and temperature. The radiative heat flux can be written in form qr = − l pr eff,r9T, where 3 l pr eff,r =0.227E(T/100) l
(7)
is the effective radiative heat conductivity for the pore, E(e, P) is the exchange factor. If the emissivity is constant throughout the pore surface and transmissivity can be neglected then E=
e . 2− e
(8)
In a previous study [5], the effects of various parameters, including transmission factor P, were considered on the heat conductivity of the powder. Investigations of the influence of the transmission factor P gave evidence that the effect is quite small. In the present study various forms of the transmission factor have been considered: for radiation inside a sphere, the model proposed in [7] was used (Eq. (9)) and for radiation between two parallel plates, the model proposed in [6] was used (Eq. (10)): E= E=
P +e , 1−P
Value l corresponds to the characteristic sizes of four types of pores: l =d for the pores in the first-order structures; l= D for the pores in the second order structures; l= hsph for the spherical clearance, hsph; l =h for the micropores caused by particle surface roughness.
5. The effective heat conductivity of the powder skeleton It is assumed that: heat flux can be considered as a continuous set of elementary heat flux tubes and the effective conductivity of each tube is equal to that of the whole set; the actual particle has a shape with various curvature radii but near the contact spots this radius is equal to the equivalent spherical particle one; the actual mono-particle layer of the granular material is replaced by a layer of spherical particles with the same volume. Under these assumptions the effective heat conductivity of the skeleton can be found by replacing it by a conductive element with averaged parameters (Fig. 4(a)). In this figure the longitudinal section of the heat flux tube with strong contraction at a contact spot region is shown. The heat conductivity is determined by seven thermal resistances Rj ( j= 1–7). A connection scheme equivalent to the conductive element is shown in Fig. 4(b). In Appendix B correlations for these resistances are presented. From Eqs. (B.1), (B.2), (B.3), (B.4), (B.5), (B.6), (B.7), (B.8) and (B.9)l sk eff can be obtained as follows:
(9) l sk eff =
2P + e(1−P) . 2(1− P)−e(1 − P)
(10)
+
According to Ref. [6], P is approximately equal to 0.1. + 4. The effective heat conductivity of gas in pores
(11)
Substitution of Eqs. (6) and (7) into Eq. (11) gives 3 l pr eff =0.227E(T/100) l+
lg . 1 +B/(pl)
Z3 z 22 − z 21 + 2 z3 1− 0.5h%− B/(Hh)+ 0.5h%/m4
2m3 h− Z3 Z3 − Z2 + h ln 1+ m3 h− Z2
n
+ m1(z 24 − z 23) ,
Summarizing the above results, the effective heat conductivity of the gas in the pores can be written, accounting for molecular heat transport, accommodations effect and radiative heat transport, in the form: pr g l pr eff =l eff,r + l eff.
lsol z 21 z 24 0.5h%+ (1−0.5h%)F
(12)
−1
−1
(13)
pr,3 pr,4 where m1 = l pr,1 eff /lsol, m3 = l eff /lsol, m4 = l eff /lsol and where lsol = lsol(T) is the heat conductivity of the particle material, l pr,1 eff is the effective heat conductivity of the gas in first-order structure pores, l pr,3 eff is the effective heat conductivity of the gas in the spherical clearance, l pr,4 eff is the effective heat conductivity of the gas in the contact spot micropores.
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
179
experimental reactor for SiC-bulk growth schematically, as shown in Fig. 6. The height of SiC-powder layer (vapor source) h1 is equal to 2.0 · 10 − 2 m. At the initial stage of the process the powder particle diameter is d= 100–200 mm. Tapped powder density is equal to 1850 kg m − 3 and particle material density is equal to 3210 kg m − 3. It means that the powder porosity is approximately equal to 0.42. An external source heats the reactor chamber containing the powder at constant vertical temperature gradient, the temperature varies from 2100 to 2300 K. The reactor chamber is filled with the argon at pressure p= 15 mmHg. During sublimation the following gaseous components can appear: Si, Si2C, SiC2. Porous carbon chamber walls are also sublime providing gaseous C which can condense on the particle surfaces. However at the beginning of the process the carbon film is absent and the powder structure is close to the initial one. It is supposed that the main component of the gas phase is Ar. The formulated model is used to analyze effects of particle diameter d, powder porosities P, emissivity e, transmissivity P and temperature T on the effective heat conductivity of SiC-powder at parameters characteristic for SiC-monocrystal growth reactors.
Fig. 5. Dependence of leff of MgO-powder in air on T (a) and dependence of leff of SiC-powder on p in air and in hydrogen (b).
6. Results and discussion
6.1. Test calculations To validate the proposed model, two sets of experimental data on the effective heat conductivity of powder materials have been chosen. The first set concerns the powder of MgO particles of diameter, d = 300 mm and powder porosity, P =0.52, ambient gas is air at atmospheric pressure and various temperatures. The second set concerns the powder of SiC particles, ambient gases are air and hydrogen at temperature T= 273 K and various pressures [5,12,13]. Results of the comparison of model predictions with experimental data are presented in Fig. 5(a) and (b). At high temperatures and low pressures the calculations provide good agreement with the experimental data.
6.2. Calculations of the heat conducti6ity of SiC-powder in argon Let us briefly describe the process conditions in the
Fig. 6. Schematic view of an experimental reactor for SiC bulk growth.
180
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
Fig. 8. Dependence of leff of SiC-powder on temperature for various pressures.
duction inside the reactor for sublimation crystal growth.
6.3. Predictions of temperature distribution in the growth chamber
Fig. 7. Dependence of leff of SiC-powder on temperature for various porosities (a) and particle sizes (b).
Fig. 7 illustrates dependence of leff on temperature for various P (Fig. 7(a)) and d (Fig. 7(b)). A dramatic raise in the effective heat conductivity at high porosity can be associated with an increase in the effect of radiation heat conduction due to the appearance of pores of the second order. The same reason can explain the effect of the particle diameter on the effective heat conductivity since the characteristic size of the pores is proportional to that of the particles. Fig. 8 illustrates the dependence of leff with temperature for various pressures. The noticeable difference in the effective heat conductivities at low temperatures significantly decreases at high temperatures. Nonmonotonisity of the effective heat conductivity at low temperatures is due to competition between decreasing heat conductivity of the particle material and rising radiation heat conduction. The effect of optical properties of the powder material is illustrated in Fig. 9. At the parameters under study the influence of the emissivity and transmissivity variation can be considered as negligible. Based on the above it can be mentioned that radiation plays a determining role in SiC-powder heat con-
The developed model of heat conduction in SiC-powder has been implemented for temperature field prediction in the growth chamber (Fig. 10). The axisymmetrical growth chamber consists of two co-axial graphite containers. Low pressure argon is used as an ambient gas. SiC vapor source is located at the bottom of the inner container. The global model of the growth process takes into account that both heat conductivity and radiation are the predominant heat transfer mechanisms in the equipment considered. Under the assumption of isotropy of all solid materials, the scalar heat conductivity equation is solved inside all solid and gaseous blocks. Continuity of temperature and heat fluxes is used as a threshold boundary condition at the
Fig. 9. Dependence of leff of SiC-powder on temperature for various transmissivities.
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
181
Fig. 10. Schematic view of the growth chamber and temperature contours.
internal boundaries. Strictly conservative finite volume numerical integration procedure [2] provides exact heat conservation over any control volume within the computational region. The essential role of the radiative heat exchange between all the solid surfaces being the common peculiarity of crystal growth processes requires accurate calculation of radiative heat fluxes to be entered into the flux junction conditions. A model of the grey diffusive surface radiation via the transparent gas has been adopted in the present work, which requires calculation of viewfactors for axisymmetrical configurations which account for the shadowing effects. It has been performed using the effective method proposed in [14]. To estimate the effect of specific heat conduction of powder, two types of vapor sources were considered. The ‘crystalline source’ possesses the heat conductivity of mono- or polycrystal silicon carbide [15] and the ‘powder source’ with the heat conductivity calculated on the basis of the above model. Temperature distribution in solid and powder blocks is presented in Fig. 10. It is important to note that the temperature of the surface of the seed is not influenced by the change in the source type, while the temperature distribution in the source including its surface and the bottom of the container is significantly different. In the case of the powder significantly lowering the heat conductivity of
the source results in source surface temperature almost 30° lower in comparison with the polycrystalline source. In practice, this changes the temperature difference between the source surfaces and the seed and leads to absolutely different growth conditions. Inside the powder source, there exists a temperature drop which may result in non-uniform sublimation rate across the source volume. Surface temperature distribution is significantly non-uniform with a minimum at the center which leads to a radially non-uniform species supply from the source and can also stimulate graphitization of the source.
7. Conclusion The above model of powder heat conductivity satisfactorily accounts for the main phenomena accompanying heat transfer in a wide range of powder parameters, temperature and ambient gas pressure. At high temperatures when the process is governed mainly by radiation the model provides the best agreement with experimental data at any ambient gas pressure. At relatively low temperature and pressure (T =102 K, p = 1–20 mmHg) the process is mainly limited by heat transport through contact spots. Since there is a lack of information about the structure of these spots, which
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
182
significantly depend on powder particle type and fabrication, predictions of the model are in less agreement with experiments. Nevertheless, at conditions characteristic of sublimation SiC crystal growth reactor, the model provides reliable information on SiC-powder heat conductivity. Herewith the heat conductivity of the powder appears to be two orders of magnitude lower than that for the bulk material which results in a less uniform temperature distribution in the vapor source of the reactor. This phenomenon can in its turn lead to non-uniform monocrystal parameter distribution and degradation of the source.
Acknowledgements Authors are grateful to Dr D. Hoffman and Ms B. Rexer for their support and helpful discussions.
Appendix B The thermal resistance of a solid part of the contact spot R1 is equal to that of a flat plate with thickness h/2 R1 =
It is supposed that the heat flux is directed vertically and vertical surfaces of the rods and cell are adiabatic. Under these assumptions the thermal resistances (Fig. 2(b)) can be written as follows: — for the porous medium with the first-order structure L1 1 , R 2,1 = sk , 2 l l sk d (L eff 1 eff 1 −d1) 1 L R 3,1 = pr,1 , R 4,1 = pr,1 1 , l eff d1 l eff (L1 −d1)2 R 1,1 =
(A.1)
R2 =
L2 , 2 l sk eff,1d 1 1 R 3,2 = pr,2 , l eff d2
R 1,2 =
1 , l sk (L eff,1 2 −d2) L R 4,2 = pr,2 2 . l eff (L2 −d2)2 R 2,2 =
(A.2)
1 . leff,i Li
0.5h% . pr(z − z 21)l pr eff,4
1 1 2 + + Ri= R1,i R4,i R2,i +R3,i
(A.3)
The thermal resistance R4 of the particle part between the contact spot and an isothermal surface S3 (Fig. 7(a)) R4 =
1−0.5h% −Z2 . pr(z 22 − z 21)lsol
then one can obtain leff,i (Eq. (3)).
(B.4)
The thermal resistance R5 of the spherical clearance section (Fig. 7(a)) in the form,
R5 =
1− m3
2pr Z3 − Z2 + h ln
h− Z3 pr l h− Z2 eff,3
1− m3Z3 + B/(Hhsph) . (1−m3)
.
(B.5)
(B.6)
The thermal resistance R6 of the particle part between the isothermal surface S3 and a basic surface (Fig. 7(a)) in the form, R6 =
Z3 . pr(z − z 21)lsol 2 3
R7 =
1 . pr(z 24 − z 23)l pr,1 eff
(B.7)
(A.4)
(B.8)
The equivalent thermal resistance Req is calculated in accordance with the scheme presented in Fig. 7(b). If the whole volume of the element is filled by a uniform substance with the heat conductivity l sk eff then Req =
−1
,
(B.3)
2 2
According to equivalent scheme
(B.2)
The thermal resistance R7 of the through pore surrounding the particle (Fig. 7(a)) in the form,
Equivalent thermal resistance of the cell R i can be connected with effective heat conductivity of the granular material leff,i by Ri=
1−0.5h% F, prz 21lsol
where F= 0.017+ 0.4z1 is the function approximately describing the deformation of the heat flux tube crosssection [5]. The thermal resistance of a porous part of the contact spot R3 is equal to
h= — for the porous medium with the second order structure
(B.1)
The thermal resistance of the heat flux tube contraction R2 is equal to
R3 = Appendix A
0.5h% . prz 21lsol
1 . pr4z4l sk eff
(B.9)
According to the equivalent thermal resistance scheme one can obtain:
R sk eff=
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
1 1 1 + R6+ + R1+R2 R3+R4 R5
−1 −1
n
1 R7
+
−1
,
(B.10) The expression (13) for l sk eff can be obtained from Eqs. (B.1), (B.2), (B.3), (B.4), (B.5), (B.6), (B.7), (B.8), (B.9) and (B.10).
References [1] H. Morkoc¸, S. Strite, G.B. Gao, M.E. Lin, B. Sverdlov, M. Burns, J. Appl. Phys. 76 (1994) 1363. [2] Yu.E. Egorov, A.O. Galjukov, L.A. Kadinskii, Yu.N. Makarov, A.I. Zhmakin, Modelling of flow and growth processes in epitaxial reactors using unstructured grids. Proceedings of the third ECCOMAS CFD Conference, Wiley, New York, 1996, pp. 704 – 710. [3] J.C. Maxwell, A Treatise on Electricity and Magnetism, Oxford University Press, London, 1891.
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[4] D.A. De Vries, The Thermal Conductivity of Granular Materials, Inst. Int. Froid., Paris, 1955. [5] G.N. Dulnev, Yu.P. Zarichniyak, Heat Conductivity of Mixtures and Composite Materials, Energia, Leningrad, 1974, (in Russian). [6] Z.V. Sigalova, Investigation of Heat Conductivity in Granular Materials, Ph.D Thesis, LITMO, Leningrad, 1965, (in Russian). [7] M.N. Ozisik, Complicated Heat Transfer, Mir, Moskow, 1976. [8] M.G. Kaganer, Heat-Mass Transfer in Low Temperature Installations, Energia, Leningrad, 1979, (in Russian). [9] S. Dashman, The Basis of Vacuum Technology, Mir, Moskow, 1964, (in Russian). [10] R.S. Prasolov, Heat and Mass Transfer 1, Energia, Moskow, 1968, pp. 432 – 444. [11] E.N. Frolova, Investigations of Thermal Accommodation Coefficient, Ph.D Thesis, LITMO, Leningrad, 1972, (in Russian). [12] H.W. Godbee, W.T. Siegler, J. Appl. Phys. 37 (1966) 40–65. [13] M.G. Laubitz, Can. J. Phys. 37 (1959) 798 – 808. [14] F. Dupret, P. Nicodeme, Y. Ryckmans, P. Wouters, M.J. Crochet, Int. J. Heat Mass Transfer 33 (9) (1990) 1849–1871. [15] G.G. Gnesin, Silicon Carbide Materials, Metallurgiya, Moskow, 1977, (in Russian).
.
Heat transfer through source powder in sublimation growth of SiC crystal E.L. Kitanin a, M.S. Ramm b, V.V. Ris a, A.A. Schmidt b,* a
b
State Technical Uni6ersity, 194026 St. Petersburg, Russian Federation Ioffe Physical Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russian Federation Received 27 June 1997
Abstract A problem of heat transport in a source powder is very important for sublimation crystal growth technique. This paper is devoted to the development and implementation of a model of powder heat transfer. As an example temperature field calculation in a growth chamber containing a SiC-powder layer is performed. © 1998 Published by Elsevier Science S.A. All rights reserved. Keywords: Heat transfer; SiC-powder; Sublimation growth technique
1. Introduction
2. The effective heat conductivity of granular material
Silicon carbide is a very promising material for making high temperature, high power and high frequency electronic devices [1]. The main problem of SiC electronics is fabrication of large-scale substrates. Sublimation growth is now considered to be a perspective technique which would enable one to fulfill this task. In numerous investigations devoted to SiC sublimation, growth temperature field control and optimization are the key problems [2]. At the same time heat transfer in SiC-powder, used usually as a source of vapors of Si and C in sublimation growth, is outside of the framework of most studies. Difficulties in investigating heat transfer processes in powder are caused by a variety of heat conduction mechanisms whose relative contributions depend on growth process parameters. In experimental investigations of transport processes inside the powder it is necessary to consider a wide range of parameters. Implementation of theoretical models of granular materials can reduce complicated experimental work. In the present paper a mathematical model of heat transport in SiC-powder is used for the prediction of temperature fields inside a sublimation growth chamber.
Theoretical models are mainly based on the generalized conductivity principle (GCP) [3]. The GCP is mostly effective for a cellular material when the conductivity of a continuous medium is greater than that of a dispersed one. Otherwise, calculation methods based on the GCP need corrections [4]. These corrections can be associated, in particular, with (a) non-uniform chaotic structure of the material; (b) difference of heat transfer mechanisms in gas and solid phases; (c) peculiarities of heat transfer through contact spots between particles. The problem of predicting the temperature field within a cylindrical layer of SiC-powder during sublimation is governed by Fourier equation. As the first step of the study, a starting stage of the process is considered when the powder structure is close to the initial one. Conductive and radiative mechanisms of heat transport in the powder layer are taken into account. It is assumed that the most important heat source is an external heater, while the internal source qV due to the latent heat of sublimation and chemical reaction heat release can be neglected. Under such assumptions the stationary temperature field is governed by the following non-linear energy equation:
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0921-5107/98/$ - see front matter © 1998 Published by Elsevier Science S.A. All rights reserved. PII S0921-5107(98)00146-9
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183 pr leff = f(l sk eff, l eff, P),
Fig. 1. Schematic view of a granular medium with first-order structure (a) and second order structure (b).
9(leff9T)=0, where leff is the effective heat conductivity of the powder material accounting for both radiative and conductive heat transfer. Two types of multicomponent media can be distinguished: embedded media and interpenetrating media. Granular materials (powders) take intermediate place because the gas phase occupies a simply connected domain and the particles contact each other directly through contact spots. Transport properties of such media depend on transport properties of both powder material and gas in macro-pores between the particles and in micro-pores formed by roughness of the particle surfaces in contact spots. In the framework of the proposed method the effective conductivity is determined as a function of effective heat conductivity’s of powder components and the gas bulk fraction which is called porosity P. The effective conductivity can be expressed in the form
175
(1)
where P=Vg/V, V is the powder volume, Vg is the gas pr phase volume, l sk eff, l eff are the effective heat conductivities of the powder particles (a skeleton of the granular material) and of the gas pores, respectively. Geometrical parameters of the internal structure of powders can be obtained on the basis of the assumption of chaotic arrangements of particles. In accordance with [5] at low porosity (P5 0.4) it is assumed that the gas pores (first-order structure) are uniformly distributed within the powder (Fig. 1(a)) and at P\0.4 large gas pores (second order structure) appear additionally (Fig. 1(b)). According to this model the powder with the first-order structure, which is supposed to consist of monodispersed spherical particles of diameter d, is substituted by a frame with the same thermal resistance. The frame is formed by perpendicularly crossing rods of constant cross-sections. One-eighth of an elementary cell of such a frame is presented in Fig. 2(a). The frame cell is characterized by two parameters: d1 which is the characteristic size of the rod cross-section and L1 which is the characteristic rod length. These parameters are chosen to provide the porosity of a medium with the frame to be equal to P of the powder. If pores of the second order structure are present in the powder a similar model is used. In contrast to the previous one the frame now is a porous medium by itself containing the first-order pores at skeleton porossk sk ity Psk is the frame cr = V /(V −V2)= 0.4, where V volume and V2 is the volume of pores with second order structures. The bulk fraction of these pores can be determined as: P2 = V2/V=
P− Psk cr . 1− Psk cr
(2)
Fig. 2. One-eight of an elementary cell of the frame substituted into the granule medium (a) and a scheme of thermal resistances of the elementary cell (b).
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In this case the frame cell is characterized by two parameters d2 and L2 chosen to make the porosity of the frame (which is porous in its turn) to be equal to P2. Non-dimensional ratio Ci =di /Li satisfies the following equation: 2C 3i − 3C 2i − Pi + 1 = 0, where i=1, 2 corresponds to the first and the second order structures, respectively. Fig. 2(b) shows an equivalent scheme of thermal resistances in the elementary cell (Fig. 2(a)). In Appendix A correlation’s for the calculation of thermal resistances as functions of both geometrical parameters and effective thermal conductivities of elementary cells are presented. From Eqs. (A.1), (A.2) and (A.3) one can obtain an equation for leff,i : leff,1 = l sk eff
if
P 50.4,
leff,2 = leff,1 C 22 +n2(1 − C2)2 +
n
2n2 C2(1 − C2) , n2 C2 −C2 +1
(3)
pr,2 where n2 = l pr,2 and l sk eff /leff,1, l eff eff are effective heat conductivities for pores of the second order and for the frame skeleton. Depending on the porosity corresponding value, leff,i should be substituted into Eq. (1). In Sections 3– 5 the most important details of the model of powder heat conductivity are described.
3. Effects of geometrical and physical characteristics of the powder on heat transport through the skeleton and pores
3.1. Geometrical characteristics of the contact spot and neighboring domain In order to determine the effective conductivity of the powder skeleton, it is necessary to take into account that in a powder each particle makes contact with N neighboring particles or, in other words, has N contact spots. Thermal resistance of the contact spots is a controlling factor of heat transport through the skeleton. The number of contact spots, so called ‘the coordination number’ is a function of the bulk fraction (porosity). It can be written in the form [5] N=
P+ 3+ P2 −10P +9 , 2P
(4)
where P =P1 for P 5 0.4 and P = Psk for P \ 0.4. In a practically interesting range of porosity, N can be approximated as N =2.22P − 1.3.
Fig. 3. Schematic view of an ‘actual’ (a) and an ‘ideal’ (b) contact spot.
The contact spot and clearance around it (Fig. 3(a)) are approximated by a contact zone of radii r1, height h/2 (Fig. 3(b)) and a ring clearance around it with outer radii r2. Contact spot thickness h can be treated as an averaged height of particle surface roughness. Dimensionless values of r1, r2 and h scaled by particle radius r are z1 = r1/r, z2 = r2/r, h%= h/r. Let r3 be the radius of a part of the total particle surface which is shared by one contact spot and r4 the radius of a through pore surrounding the spherical domain of the particle surface of radius r3 (Fig. 4(a)). Dimensionless values of r3 and r4 are z3 =r3/r= 2 N − 1/N and z4 = r4/r=z3(1–P) − 1/3. In addition, two dimensionless geometrical parameters are introduced Z2 = 1 − z 22 and Z3 = 1 − z 23. Near spherical sector of radius r3 integer-averaged distance between two particles hsph is
hsph = d h%+
1 , N
where d= 2r. Size of a pore in the second order structure D can be obtained from [5]: D=3d
1 −1 . C2
3.2. Influence of a compression on the contact spot and clearance For absolutely smooth elastic spherical particles compressed by a force P, the area of contact spot Sn can be calculated using the Hertz formula. In fact the contact spot area is less because of the surface roughness effect Sreal = cSn, where c is the empirical coefficient. In the absence of external compressing forces it is possible to write expressions for non-dimensional radii z1 and z2 [5]
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177
Fig. 4. Schematic view of a conductive element with averaged parameters (a) and the connection schema (b).
z1 = z2 c, z2 = 3.3 · 10 − 3(1 − P)1/9(rsolgh1)1/3,
(5)
where rsol is the particle material density, h1 is the powder layer height, g = 9.81 m s-2. Thus the geometrical characteristics of contact spots and clearance between the powder particles depends on two empirical parameters h% and c. Regarding these characteristics all materials can roughly be divided into three groups [6,7]: rigid materials with Moose hardness numbers Mhn= 7.0–9.0 can be characterized by h% = (2.0– 6.0) · 10 − 3 and c =(3.0 – 5.0) · 10 − 4; intermediate materials Mhn =2.0 – 7.0; h% = (1.0– 8.0) · 10 − 3 and c =(5.0 – 10.0) · 10 − 4; soft materials Mhn =1.0 – 2.0; h% = (0.0 – 2.0) · 10 − 3 and c= (10.0– 50.0) · 10 − 4. Particles of MgO and SiC powders considered below (see Section 6) are close to the group of rigid materials and the following values of the parameters are used h% =2.2 · 10 − 3 and c= 5.0 · 10 − 4.
3.3. Dependence of the heat conducti6ity of gas in pores on the molecule mean free path It is known [8] that a decrease in heat conductivity of the granular materials with gas pressure in pores is greater than that due to decreasing gas heat conductivity. This phenomenon is a result of temperature jump on a solid surface at low pressure. The temperature jump depends on both gas and surface properties, namely: accommodation coefficient a, specific heat ratio g, Prandtl and Knudsen numbers [8 – 10]. The temperature jump can be neglected when the Knudsen number
Kn=L/l B B 1, where L is the molecule mean free path and l is the characteristic size of the gas pore. The characteristic size of the gas pore is : 101 –102 mm for pore side of the contact spots and :1 mm inside them. The molecular mean free path can be calculated as L=kT/ 2ps 2p, here k is the Boltzmann constant, s is the characteristic size of the gas molecule, T, p are absolute temperature and pressure of the gas. For the conditions under study (argon, T:2400 K, p:15 mmHg) L: 30 mm and, therefore, Kn:10 − 1 – 101. Effect of the temperature jump on the effective heat conductivity can be evaluated using the following correlations [10] l geff =
lg 4g 2−a L0p0 , B= · · , 1+ B/(pl) g+ 1 a Pr
(6)
where the index 0 corresponds to the normal conditions, lg = lg(T) is the gas phase molecular heat conductivity and Pr is the Prandtl number. The most complicated problem in the calculation of the temperature jump is determining the accommodation coefficient. The value of this coefficient depends on physical properties of gas and solid particle material, temperature and pressure. It can be crudely estimated from [11]: a= 1−
n
M 2g + M 2sol n , (Mg + Msol)2
where Mg, Msol are molecular masses of the phases, n= 3–4 is the number of collisions of the gas molecules with solid surface.
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3.4. Radiati6e heat conduction in the powder At conditions under study according to the Plank law the main portion of radiative energy transport is associated with wave lengths B 10 mm. The optical thickness for this radiation is much greater than the characteristic size of the pore. Two factors have an effect on the radiative heat flux in the powder qr: emissivity e of a surface surrounding the pore with characteristic size l and transmissivity P of the powder material surrounding the pore. The emissivity of the particle surface depends significantly on the surface condition and temperature. The radiative heat flux can be written in form qr = − l pr eff,r9T, where 3 l pr eff,r =0.227E(T/100) l
(7)
is the effective radiative heat conductivity for the pore, E(e, P) is the exchange factor. If the emissivity is constant throughout the pore surface and transmissivity can be neglected then E=
e . 2− e
(8)
In a previous study [5], the effects of various parameters, including transmission factor P, were considered on the heat conductivity of the powder. Investigations of the influence of the transmission factor P gave evidence that the effect is quite small. In the present study various forms of the transmission factor have been considered: for radiation inside a sphere, the model proposed in [7] was used (Eq. (9)) and for radiation between two parallel plates, the model proposed in [6] was used (Eq. (10)): E= E=
P +e , 1−P
Value l corresponds to the characteristic sizes of four types of pores: l =d for the pores in the first-order structures; l= D for the pores in the second order structures; l= hsph for the spherical clearance, hsph; l =h for the micropores caused by particle surface roughness.
5. The effective heat conductivity of the powder skeleton It is assumed that: heat flux can be considered as a continuous set of elementary heat flux tubes and the effective conductivity of each tube is equal to that of the whole set; the actual particle has a shape with various curvature radii but near the contact spots this radius is equal to the equivalent spherical particle one; the actual mono-particle layer of the granular material is replaced by a layer of spherical particles with the same volume. Under these assumptions the effective heat conductivity of the skeleton can be found by replacing it by a conductive element with averaged parameters (Fig. 4(a)). In this figure the longitudinal section of the heat flux tube with strong contraction at a contact spot region is shown. The heat conductivity is determined by seven thermal resistances Rj ( j= 1–7). A connection scheme equivalent to the conductive element is shown in Fig. 4(b). In Appendix B correlations for these resistances are presented. From Eqs. (B.1), (B.2), (B.3), (B.4), (B.5), (B.6), (B.7), (B.8) and (B.9)l sk eff can be obtained as follows:
(9) l sk eff =
2P + e(1−P) . 2(1− P)−e(1 − P)
(10)
+
According to Ref. [6], P is approximately equal to 0.1. + 4. The effective heat conductivity of gas in pores
(11)
Substitution of Eqs. (6) and (7) into Eq. (11) gives 3 l pr eff =0.227E(T/100) l+
lg . 1 +B/(pl)
Z3 z 22 − z 21 + 2 z3 1− 0.5h%− B/(Hh)+ 0.5h%/m4
2m3 h− Z3 Z3 − Z2 + h ln 1+ m3 h− Z2
n
+ m1(z 24 − z 23) ,
Summarizing the above results, the effective heat conductivity of the gas in the pores can be written, accounting for molecular heat transport, accommodations effect and radiative heat transport, in the form: pr g l pr eff =l eff,r + l eff.
lsol z 21 z 24 0.5h%+ (1−0.5h%)F
(12)
−1
−1
(13)
pr,3 pr,4 where m1 = l pr,1 eff /lsol, m3 = l eff /lsol, m4 = l eff /lsol and where lsol = lsol(T) is the heat conductivity of the particle material, l pr,1 eff is the effective heat conductivity of the gas in first-order structure pores, l pr,3 eff is the effective heat conductivity of the gas in the spherical clearance, l pr,4 eff is the effective heat conductivity of the gas in the contact spot micropores.
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experimental reactor for SiC-bulk growth schematically, as shown in Fig. 6. The height of SiC-powder layer (vapor source) h1 is equal to 2.0 · 10 − 2 m. At the initial stage of the process the powder particle diameter is d= 100–200 mm. Tapped powder density is equal to 1850 kg m − 3 and particle material density is equal to 3210 kg m − 3. It means that the powder porosity is approximately equal to 0.42. An external source heats the reactor chamber containing the powder at constant vertical temperature gradient, the temperature varies from 2100 to 2300 K. The reactor chamber is filled with the argon at pressure p= 15 mmHg. During sublimation the following gaseous components can appear: Si, Si2C, SiC2. Porous carbon chamber walls are also sublime providing gaseous C which can condense on the particle surfaces. However at the beginning of the process the carbon film is absent and the powder structure is close to the initial one. It is supposed that the main component of the gas phase is Ar. The formulated model is used to analyze effects of particle diameter d, powder porosities P, emissivity e, transmissivity P and temperature T on the effective heat conductivity of SiC-powder at parameters characteristic for SiC-monocrystal growth reactors.
Fig. 5. Dependence of leff of MgO-powder in air on T (a) and dependence of leff of SiC-powder on p in air and in hydrogen (b).
6. Results and discussion
6.1. Test calculations To validate the proposed model, two sets of experimental data on the effective heat conductivity of powder materials have been chosen. The first set concerns the powder of MgO particles of diameter, d = 300 mm and powder porosity, P =0.52, ambient gas is air at atmospheric pressure and various temperatures. The second set concerns the powder of SiC particles, ambient gases are air and hydrogen at temperature T= 273 K and various pressures [5,12,13]. Results of the comparison of model predictions with experimental data are presented in Fig. 5(a) and (b). At high temperatures and low pressures the calculations provide good agreement with the experimental data.
6.2. Calculations of the heat conducti6ity of SiC-powder in argon Let us briefly describe the process conditions in the
Fig. 6. Schematic view of an experimental reactor for SiC bulk growth.
180
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
Fig. 8. Dependence of leff of SiC-powder on temperature for various pressures.
duction inside the reactor for sublimation crystal growth.
6.3. Predictions of temperature distribution in the growth chamber
Fig. 7. Dependence of leff of SiC-powder on temperature for various porosities (a) and particle sizes (b).
Fig. 7 illustrates dependence of leff on temperature for various P (Fig. 7(a)) and d (Fig. 7(b)). A dramatic raise in the effective heat conductivity at high porosity can be associated with an increase in the effect of radiation heat conduction due to the appearance of pores of the second order. The same reason can explain the effect of the particle diameter on the effective heat conductivity since the characteristic size of the pores is proportional to that of the particles. Fig. 8 illustrates the dependence of leff with temperature for various pressures. The noticeable difference in the effective heat conductivities at low temperatures significantly decreases at high temperatures. Nonmonotonisity of the effective heat conductivity at low temperatures is due to competition between decreasing heat conductivity of the particle material and rising radiation heat conduction. The effect of optical properties of the powder material is illustrated in Fig. 9. At the parameters under study the influence of the emissivity and transmissivity variation can be considered as negligible. Based on the above it can be mentioned that radiation plays a determining role in SiC-powder heat con-
The developed model of heat conduction in SiC-powder has been implemented for temperature field prediction in the growth chamber (Fig. 10). The axisymmetrical growth chamber consists of two co-axial graphite containers. Low pressure argon is used as an ambient gas. SiC vapor source is located at the bottom of the inner container. The global model of the growth process takes into account that both heat conductivity and radiation are the predominant heat transfer mechanisms in the equipment considered. Under the assumption of isotropy of all solid materials, the scalar heat conductivity equation is solved inside all solid and gaseous blocks. Continuity of temperature and heat fluxes is used as a threshold boundary condition at the
Fig. 9. Dependence of leff of SiC-powder on temperature for various transmissivities.
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
181
Fig. 10. Schematic view of the growth chamber and temperature contours.
internal boundaries. Strictly conservative finite volume numerical integration procedure [2] provides exact heat conservation over any control volume within the computational region. The essential role of the radiative heat exchange between all the solid surfaces being the common peculiarity of crystal growth processes requires accurate calculation of radiative heat fluxes to be entered into the flux junction conditions. A model of the grey diffusive surface radiation via the transparent gas has been adopted in the present work, which requires calculation of viewfactors for axisymmetrical configurations which account for the shadowing effects. It has been performed using the effective method proposed in [14]. To estimate the effect of specific heat conduction of powder, two types of vapor sources were considered. The ‘crystalline source’ possesses the heat conductivity of mono- or polycrystal silicon carbide [15] and the ‘powder source’ with the heat conductivity calculated on the basis of the above model. Temperature distribution in solid and powder blocks is presented in Fig. 10. It is important to note that the temperature of the surface of the seed is not influenced by the change in the source type, while the temperature distribution in the source including its surface and the bottom of the container is significantly different. In the case of the powder significantly lowering the heat conductivity of
the source results in source surface temperature almost 30° lower in comparison with the polycrystalline source. In practice, this changes the temperature difference between the source surfaces and the seed and leads to absolutely different growth conditions. Inside the powder source, there exists a temperature drop which may result in non-uniform sublimation rate across the source volume. Surface temperature distribution is significantly non-uniform with a minimum at the center which leads to a radially non-uniform species supply from the source and can also stimulate graphitization of the source.
7. Conclusion The above model of powder heat conductivity satisfactorily accounts for the main phenomena accompanying heat transfer in a wide range of powder parameters, temperature and ambient gas pressure. At high temperatures when the process is governed mainly by radiation the model provides the best agreement with experimental data at any ambient gas pressure. At relatively low temperature and pressure (T =102 K, p = 1–20 mmHg) the process is mainly limited by heat transport through contact spots. Since there is a lack of information about the structure of these spots, which
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182
significantly depend on powder particle type and fabrication, predictions of the model are in less agreement with experiments. Nevertheless, at conditions characteristic of sublimation SiC crystal growth reactor, the model provides reliable information on SiC-powder heat conductivity. Herewith the heat conductivity of the powder appears to be two orders of magnitude lower than that for the bulk material which results in a less uniform temperature distribution in the vapor source of the reactor. This phenomenon can in its turn lead to non-uniform monocrystal parameter distribution and degradation of the source.
Acknowledgements Authors are grateful to Dr D. Hoffman and Ms B. Rexer for their support and helpful discussions.
Appendix B The thermal resistance of a solid part of the contact spot R1 is equal to that of a flat plate with thickness h/2 R1 =
It is supposed that the heat flux is directed vertically and vertical surfaces of the rods and cell are adiabatic. Under these assumptions the thermal resistances (Fig. 2(b)) can be written as follows: — for the porous medium with the first-order structure L1 1 , R 2,1 = sk , 2 l l sk d (L eff 1 eff 1 −d1) 1 L R 3,1 = pr,1 , R 4,1 = pr,1 1 , l eff d1 l eff (L1 −d1)2 R 1,1 =
(A.1)
R2 =
L2 , 2 l sk eff,1d 1 1 R 3,2 = pr,2 , l eff d2
R 1,2 =
1 , l sk (L eff,1 2 −d2) L R 4,2 = pr,2 2 . l eff (L2 −d2)2 R 2,2 =
(A.2)
1 . leff,i Li
0.5h% . pr(z − z 21)l pr eff,4
1 1 2 + + Ri= R1,i R4,i R2,i +R3,i
(A.3)
The thermal resistance R4 of the particle part between the contact spot and an isothermal surface S3 (Fig. 7(a)) R4 =
1−0.5h% −Z2 . pr(z 22 − z 21)lsol
then one can obtain leff,i (Eq. (3)).
(B.4)
The thermal resistance R5 of the spherical clearance section (Fig. 7(a)) in the form,
R5 =
1− m3
2pr Z3 − Z2 + h ln
h− Z3 pr l h− Z2 eff,3
1− m3Z3 + B/(Hhsph) . (1−m3)
.
(B.5)
(B.6)
The thermal resistance R6 of the particle part between the isothermal surface S3 and a basic surface (Fig. 7(a)) in the form, R6 =
Z3 . pr(z − z 21)lsol 2 3
R7 =
1 . pr(z 24 − z 23)l pr,1 eff
(B.7)
(A.4)
(B.8)
The equivalent thermal resistance Req is calculated in accordance with the scheme presented in Fig. 7(b). If the whole volume of the element is filled by a uniform substance with the heat conductivity l sk eff then Req =
−1
,
(B.3)
2 2
According to equivalent scheme
(B.2)
The thermal resistance R7 of the through pore surrounding the particle (Fig. 7(a)) in the form,
Equivalent thermal resistance of the cell R i can be connected with effective heat conductivity of the granular material leff,i by Ri=
1−0.5h% F, prz 21lsol
where F= 0.017+ 0.4z1 is the function approximately describing the deformation of the heat flux tube crosssection [5]. The thermal resistance of a porous part of the contact spot R3 is equal to
h= — for the porous medium with the second order structure
(B.1)
The thermal resistance of the heat flux tube contraction R2 is equal to
R3 = Appendix A
0.5h% . prz 21lsol
1 . pr4z4l sk eff
(B.9)
According to the equivalent thermal resistance scheme one can obtain:
R sk eff=
E.L. Kitanin et al. / Materials Science and Engineering B55 (1998) 174–183
1 1 1 + R6+ + R1+R2 R3+R4 R5
−1 −1
n
1 R7
+
−1
,
(B.10) The expression (13) for l sk eff can be obtained from Eqs. (B.1), (B.2), (B.3), (B.4), (B.5), (B.6), (B.7), (B.8), (B.9) and (B.10).
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.