Prediction and measurement of effective thermal conductivity of three-phase systems

J. Phys. D: Appl. Phys. 24 (1991) 151~1526. Printed in the UK

I Prediction and measurement of 1 effective thermal conductivity of three- 1 1 phase systems I L S Vermats, A K Shrotriyat, Ramvir Singht and D R ChaudharyS t MSJ College, Bharatpur 321 001, India + Physics Department. University of Rajasthan, Jaipur 302 004, India

Received 9 August 1990, in final form 18 March 1991

Abstract. Effective thermal conductivities of three-phase systems have been experimentally determined by a differential temperature sensors method. The results are in good agreement with the values predicted by the geometry- dependent resistor model developed for three-phase systems. Reported values of ETC in t h e literature are also compared with those predicted by the model which show good agreement.

1. Introduction

The thermophysical study of granular substances in both consolidated and unconsolidated forms has gained much importance recently. Their use as insulating envelopes in solar ponds, refrigerators and air con- ditioners is of great importance to engineers and physicists. Study of their thermal parameters is also valuable for the explosive material industry, the ceramic industry, nuclear reactors and oil exploration. Granular suh- stances of practical importance range from simple two- phase systems of spherical particles to complex systems of irregular particles with more than two constituent phases. Three-phase systems with porous cementing materials are yet more complex. The theoretical modelling of such systems with complex parameters is a challenging task for physicists, engineers and geol- ogists.

Lichtnecker (1926), Brailsford and Major (1964), Cheng and Vachon (1969), Huetz (1970), Kumar and Chaudhary (1980), Pande er a1 (1984), Hadley (1986) and others have proposed models for predicting effective thermal conductivities (ETCS) of three-phase systems. These models have a limited applicability and cannot be used for all types of systems. A general expression which can predict the ETC of a complex multiphase system is still lacking. This paper aims to find a suitable expression to predict ETCs of granular substances in consolidated, unconsolidated and packed structures. Here we limit ourselves to deal with complex three-phase systems.

8 Address for correspondence: Thermal Physics Laboratory, Department of Physics, University of Rajasthan, Jaipur 302 004, India.

Effective thermal conductivities of various granular systems have also been determined experimentally. A differential temperature sensor probe developed by the authors (Verma et a1 1990, 1991) has been used for the measurement of the ETC. This technique has the advantage that it measures all the thermophysical parameters of granular substances simultaneously and more accurately in comparison with other transient methods.

This paper also paves a way for predicting the ETC of a still more complex system having more than three phases.

2. Theoretical formulation

Hadley (1986) has given the following closure equations to predict the ETC of a multiphase system having n components.

V ( T ) = E V,(VT,)’ (1)

Ae ’ V(T) = A,V,PT])’ (2)

,=1 n

] = I

where A, is the ETC of the mixture; p, and A, are the volume fraction and thermal conductivity of the jth phase respectively, V(T) represents the gradient of average temperature and (VT,)’ gives the average of temperature gradient in the jth phase. For a two-phase system these equations can be written as

V ( T ) = dVT1)’ + (1 - V)(VT2)2

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L S Verma et a/

101 l b l l C 1

Figure 1 . Configuration of slabs with respect to heat flux

where (VT, ) ' and (VT2)2 are the average temperaturc gradients in the continuous phase and dispersed phase respectively, and rp is the porosity (volume fraction of the continuous phase). These two equations contain three constants V ( T ) , (VT , ) ' , and (VT2)2-and hence cannot be solved unless some relation connecting the constants is assumed.

If (VT,) ' = (VT,)' then we obtain the expression for parallel resistors given by (figure l(a))

All = [ @ 1 + (1 - r p ) * 2 1 ( 5 ) and the assumption (VT, ) ' = (A2/A,)(VT2)2 leads to an expression for perpendicular resistnrs (figure l(b)):

_ _ tiere All and ,IL reprcscnt the upper and lowei bounds of thermal conductivity for a porous system. Any model for a two-phase system which depends upon rp and A z / A l can be represented by a general equation

(VT1)l = [ F + (A2/* ' ) (1 - F)1 ' W*Y (7)

where F is a parametcr lying between U and 1 . At this juncture, if we assume

* (VT,)1 = 2 ' (VT?)?

*/I ( 8 )

then equations (5) and (6) can be obtained as special cases of equation (8)? i.e. Al = ,IIl leading to equation ( 5 ) and A, = (J.2/A,) 2.i giving equation (6). For nther situations equation (8) describes a resistor model for slabs inclined to the direction of heat How. The ratio AL/All represents the tangent of the angle of inclination 8. As is always greater than Al our assumption given by equation (8) is in accordance with the condition imposed by equation (7). Thus, by varying the angle of the slabs according to the geometry of the components, the ETC of different systems can be predicted.

Therefore, for inclined slabs as shown in figure I(c), we have

= [ r p ~ ' + (1 - ~ ) A ~ I cos a (9)

and

The effective thermal conductivity of the system will

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be given by

Ae = [*f +A:]'". (11) Equations (9) and (10) suggest that an increase in

the angle 0 will increase Al and decrease All components. The net result will be a decrease in Ac. Similarly, a decrease in 0 will have the reverse effect and Ac will increase.

3. Mathematical formulation for angle 0

To determine the angle 0 in terms of structural parameters, we first consider the geometry of the particles which can bc convcnicntly dcfined in terms of sphericity.

3.1. Sphericity of particles (+)

Angularities of the grains in a porous system greatly alter its thermophysical properties. Experiments show that the behaviour of a system packed with non-spherical particles is radically different from that involving spherical particles. Attempts have been made to des- cribe the deviation of particle shape from a sphere by a factor called sphericity which is a measure of the roundness of the particles. It is defined as $ = s / S where s is the surface area of the sphere having the same vo!~me as !ha.! of the particle and ,? is its actual surface area. For spherical particles v = 1 and for particles of any other shape @ < 1. Wadell (1935) has described a method for the experimental determination of v . The value of the resistivity formation factor has been found to depend upon sphericity. The resistivity formation factor for a system with angular particles will be greater than one. An increase in the value of I) means a decrease in grain angularities, which brings the particles closer.

A good account of the dependence of physical properties on particle angularities is given by Haughey and Baveridge (1969). To consider the effect of this parameter let us consider heat conduction in the system at microscopic level. In a porous system the conduction of heat takes place by the following processes.

3.2. Particle to particle conduction: resistivity formation factor F

Alberts er a/ (1966) have explained this mode of heat conduction in a linear array of spheres. In this analogy a simplified approach to heat conduction in a two- dimensional arrangement of spherical particles is shown in figure 2(a). Such a state of affairs can be interpreted in terms of the resistivity formation factor. This factor was first introduced by Archie (1942) to account for the How of electrical energy in a porous medium. Wyllie (1963) has defined it as the ratio of resistivity of a porous medium fully saturated with a conductive fluid to the resistivity of the Huid itself. In the same analogy it can also be defined for the How of thermal energy in a porous medium. Thus, F is the

ETC of three-phase systems

E

(a1 (b) (e1 Figure 2. Heat flux through randomly dispersed particles.

thermal resistance with inclusion of particles in the fluid divided by the thermal resistance of thc fluid itself. It is a measure of the average path traversed by the lines of flux in the porous medium. To evaluate its value let us consider a portion of the medium (fluid) placed in a vessel of volume V and cross sectional area A,. If I is the length of the vessel then we have V = AJ. In an electrical analogy its thermal resistance will be

1 r = - AIAC'

~

Let us now disperse spherical particles such that there is perfect particle-to-particle contact. If the volume of the solid particles is U and that of fluid particles U then we have V = ( U + U ) . If L,, is the average length of the path traversed by the heat flux in the fluid and A,, is the average of the transverse cross sections of fluid, then the thermal resistance offered by the column of fluid will be R = L J ( A l Aav). The formation factor would then be

F = - = - . > = - - . R La, A V (Lyj2 - r 1 A,, U

or

F = V2/V (12) where q = is a measure of the average deviation of heat flux due to inclusion of particles in the fluid. Also = u/V is the fractional volume of fluid (porosity). The deviation q depends upon the angularities of the particles and the mode of packing. The path of flux lines with such particles is shown in figure

To derive the expression r) for non-spherical particles, let us take the simplest case of ellipsoidal particles with semi-axes ( o # 6 = c ) dispersed in the fluid. The fluid being contained in a cylinder of dia- meter 2a and length 26. Its section in the X Y plane is shown in figure 2(c). Let the heat flux pass along the Y direction as shown in the figure. After meeting the ellipsoid, the flux lines may follow a circular or ellip- tical path. The average of the path can be obtained by averaging the paths such as ABCDE which is equal to 2(AB + BC). The average of paths such as AB will be

W ) .

(AB),, = 1' (6 - y) d x / j n dx = b (1 - :) U 0

The average of paths like BC can be determined by dividing OC into n equal parts. The length of the rth part (arc BC) will be

BC= [I + ( d y / d ~ ) ~ ] " ~ dx. I":," The equation of the boundary of the particle will be

x2 y ? - + - - 1 a' b 2 -

hence,

(13)

where /3 = b/u < 1 and terms having higher powers of (1 - p2) have been neglected. The right-hand side of equation (13) is a function of ( r / n ) . Its average value will be given by

If n is large, then in the limit n- CO, the sum can be expressed as

where ( r / n ) = x . Hence the average of paths will be given by

(BC), , = a I {cost' x - [(I - P')/41 I

(I

X [COS-' x + ~ ( l - x2)1/2] - $[(I - p2) /4 ] * x [I + x ( l - x 2 ) " ' + $ x 3 ( l - ~ ' ) ' ' ~ ] } d x

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or

(BC), ,= ~ [ l - f(1 - p’) - h(1 - p2)’]. (14) Thus La, = 2{b[l - ( ~ / 4 ) ] + ~ [ l - j ( l - p’) - & ( l - /32)2]}

As we have I = 2b, the average deviation in path will be

q = - = [ I - - ; + i(l - f(1 - p’) - &(l - P2)2)]. 1

This is the value of the average deviation when the flux is travelling along the Y axis. Let us denote it by qy. So we have

- *(l - 82)’)l. (15) As the axes of the ellipsoid are equal along Y and Z , we have q y = qz.

When the particles are spherical p = 1, hence the deviation will be

q = [2 - ( ~ / 4 ) ] 1.2146 (16) which is independent of the radius of the circle.

Similarly for heat flowing along the X axis, the same procedure can be adopted to determine the deviation. It can be shown that

V x = 1 - - + ( 1 - 6 ( 1 - p 2 ) - & ( l - p 2 ) 2 ) ] . (17) [ : When heat flux is randomly directcd the cffcctive

value of the deviation will be the geometric mean of the deviation with heat flow along the three axes.

The values of q given by the above equations can be fitted into the equation

where 6 is a constant for a given value of p . To gen- eralize this equation for angular particles we consider the geometry of the particles in terms of sphericity defined earlier. We have

q = 1 + 6(1- q) (18)

6 = p. (19) The relation between 6 and 9 consistent with equations (18) and (19) is

6 = 0.3219/* so, from equation (18), we get

0.3219(1 - p)

lil q = l +

Putting the value of 0 from equation (20) into equation (12) gives the average deviation of path as

(21) {l + [0.3219(1 - cp)]/~)}’

P F =

This equation represents the formation factor for non- spherical particles, It has no dimensions and its value is always greater than one.

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Agrawal and Bhandari (1969) have calculated its value for electrical flux lines in a system. This factor is widely used in the petroleum industry to characterize the pore structure of sedimentary rocks and shales. Maxwell (1904) derived the formation factor for a system of dispersed spheres:

F = - 3 - P 2 P ’

Fricke (1931) generalized Maxwell’s expression for oblate and prolate spheroids. For a system of touching spheres equation (21) reduces to that given by Swa- linski (1926). A good account of this factor is given by Wyllie and Gregory (1953).

Woodsidc and Messmer (1961) and Chaudhary and Bhandari (1968) have shown that the thermal conductivity of sand and sandstones depends upon the formation factor. We feel that the thermal conductivity of a porous medium should depend upon the average deviation of the flux lines in the medium. The greater the deviation, the higher should be the thermal resistivity and the lower should be the thermal conductivity. For no dispersion (p’ = l ) , the flux lines will remain parallel and we have F = 1.

3.3. Fluid to particle conduction and vice versa

Tarcev (1975) has sh~w:: tha: dcA-;-g the f i ~ v ; ~f electrical energy in a dielectric medium when the flux passes from one dielectric to another, the deviation of flux lines depends upon the ratio of the dielectric constants of the two media. In the same analogy when the heat flux passes from one medium to another the deviation should depend upon the ratio of their thermal conductivities (&/Al), In a real situation where the particles are of irregular shape the deviation would depend upon [w(A2/A,)]. This deviation causes a part of the flux lines to travel through the pores within the system. As the particles are randomly distributed, the direction of phonons (flux lines) is changed on each encounter with the particles. One can see that the number of such encounters will be proportional to [q(A2/h1)] . This deviation causes flux lines to follow a zig-zag path within the medium. To find the net effect of such conduction a suitable average of the deviation should be taken. Everret and Stone (1958) have mentioned that such a situation can be thought of as a problem of random walk. In this analogy the average deviation should be proportional to [V(AJAd]’”.

3.4. Dependence on porosity 9

The porosity has a twofold effect on the thermal conductivity of a porous medium. Firstly, it is a measure of the concentration of the continuous phase. Gen- erally the dispersed phase has a higher thermal conductivity, therefore an increase in q will decrease the ETC of the system by virtue of a lowering in concentration of the dispersed phase. Secondly, an increase in rp will increase the distance between the


particles, SO decreasing particle-to-particle conduction which again would lead to a decrease in the ETC. Hence the angle of inclination of the proposed slabs should depend upon q2.

In the real picture of a porous system the particles are randomly dispersed. As the flux lines have been assumed to have a fixed direction, all the above factors ( F , ‘p2, and [$(A2/AI)]’”) can be taken as a measure of the probability of orientation of the phases (assumed in the form of slabs). When all the factors are taken into account, the composite probability of orientation will be proportional to the product of all the above factors. Hence we should have

where Bo is the constant of proportionality whose value depends upon the mode of packing of the particles and the nature of the system. For two-phase systems B,, has been evaluated numerically. For two-phase granular systems it has been found to be 1.15, while its values for two-phase suspensions, emulsions and solid-solid mixtures are 1.7, 0.65 and 1.05 respectively.

3.5. Cross section ratio

The concept of parallel slabs for different phases is a way of describing the real state of affairs of a porous system. If a system actually consists of different slabs, the theory should also give the resultant thermal conductivity of the composite structure. To incorporate this view, the factor cross section ratio is also introduced in the expression for 8. If A, and All are respective areas of the slab perpendicular and parallel to the direction of heat Row, then its value is given by (A&; for spherical particles it is equal to one. For a dispersed system where particles are oriented in all possible directions, its average value is also one. Its usefulness lies in predicting the ETC of fibre reinforced composites.

So, considering the cross section ratio as well, the expression for angle 8 should be

1

or

8 = tan-’ [Bo e) Fm2 ( V ? ) ’ I 2 ] . (23)

4. Composition of a three-phase system into a two-phase system

For a three phase system n = 3, so we have

V(T) = P,(VT,)~ + P~(VT,)* + P,(VT,)~ (24)

A,V(T) = A1q1(VT1)I + A2P2(VT2)Z + A3q3(VT3)3.

and

(25)

If we put

PI(VTl?’ + P2(VT2?2 = ( V I + P1)(VTi)‘

A‘PI(VTl)’ + A,qdVT2)2 = Ai(P1 + PZ)V(Ti)’

and

in equations (7) and (8), we obtain

V(T) = (Pi + qz)(VTi)’ + ~3(vT3)’ (26)

L v ( T ) = A i ( ’ P 1 + P~)V(T;)’ + A3V3(VT3)3. (27) Equations (26) and (27) show that a three-phase

problem has been reduced to a two-phase problem, one phase being the intermediate phase having an average temperature gradient (VT,)’and the second phase being the remainder with an average temperature gradient (VTJ3. If we compare equation (26) with (3) and equation (27) with (4) we see that the volume fraction and thermal conductivity of the intermediate phase should be given by

Pi = ( W l + v2) (28) and

The usefulness of this method depends upon the proper pairing of different phases. No selection rules for the pairing of different phases have been given by Hadley, but we suggest the following guidelines for composing the phases.

(i) When the amount of dispersed phase is small, the continuous phase should he treated as independent and the rest should be combined to form an intermediate phase.

(ii) If the fractional volume of phases are comparable, the intermediate phase should be formed in such a manner as to reduce the ratio of thermal conductivities of intermediate and independent phases.

(iii) To define the intermediate thermal conductivity, the constituent of the pair playing a greater role in conduction should be considered. This has to be decided judiciously by considering the concentration and thermal conductivity of the constituents within the pair itself. It can he judged by comparing the products of their concentration and respective thermal conductivity. Thus the intermediate thermal conductivity can also be defined by

5. Experimental arrangement and measurement technique

A differential temperature sensor method was used to determine the thermal conductivity.

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Table 1. Comparison of predicted and experimental values for different three-phase systems. Values of thermal conductivity are in W m-' K-I. System I, iron powderialuminium powderiVaseline system: A,(Fe) = 65.402. A,(Al) = 204.24, ),,(Vas) = 0.185, 8, = 0.55. 11, aluminaizirconiaiVaseline system; A,(AI,O,) = 24.416, A2(Zr02) = 1.998, A,(Vas) = 0.185, 6, = 0.55. Ill, cupric oxideialuminiumib'aseline system: A,(CuO) = 13.490, &(AI) = 204.24, &,(Vas) = 0.185, E,, = 0.55. A and ry are the thermal conductivities and volume fractions of the phases respectively. Suffix 0 represents the continuous phase. Suflixes 1 and 2 represent the first and second dispersed phases respectively.

Volume fraction Proposed Brailsford Lichtnecker Hadley

First Second model model model model phase phase ETC

System ry, ry2 exp. ETC %Error ETC %Error ETC %Error ETC %Error

I 0.008 0.003 0.176 i 0.006 0.187 6.3 0.191 8.6 0.198 12.5 0.161 8.4 0.016 0.006 0.182 i 0.007 0.191 4.8 0.197 8.4 0.212 16.4 0.162 11.0 0.023 0.009 0.199 t 0.011 0.198 0.6 0.203 2.1 0.226 13.3 0.162 18.4 0.031 0.012 0.242 f 0,009 0.214 11.4 0.210 13.4 0.241 0.3 0.163 32.6 Average deviation 5.8 8.1 10.6 17.6

II 0.012 0.003 0.182 i 0.007 0.188 3.2 0.193 6.2 0.198 8.6 0.163 10.4 0.024 0.006 0.178 i 0.009 0.192 7.8 0.202 13.3 0.211 18.5 0.164 8.0 0.035 0.010 0.192 i 0.008 0.199 3.8 0.210 9.6 0.225 17.1 0.164 14.5 0.047 0.013 0.205 t 0.011 0.214 4.3 0.219 7.0 0.240 17.1 0.165 19.5 Average deviation 4.8 9.0 10.6 13.1

I l l 0.010 0.006 0.178 + 0.008 0.188 5.6 0.193 8.7 0.201 13.1 0.159 10.8 0,021 0.013 0.193 i- 0.009 0.192 0.6 0.203 5.3 0.222 14.9 0.160 16.9 0.031 0.019 0.209 i 0.013 0.196 6.1 0.212 1.6 0.241 15.5 0.162 22.4 0.041 0.025 0.228 2 0.009 0.203 11.1 0.222 2.8 0.263 15.3 0.164 28.1 Average deviation 5.9 4.6 14.7 19.6

A differential temperature sensor probe consists of a transient probe with two parallel steelkonstantan thermocouples mounted at different distances from the probe. One thermocouple is mounted close to the needle at a distance of nearly 0.2cm. Another thermocouple is mounted diametrically opposite the needle at about 0.6 cm. The two constantan tcrminals of thc thermocouple are joined together so that the free steel terminals measure the difference in thermo-EMF generated by the thermocouples. In this respect the device is a modified form of a parallel wire arrangement for thermal diffusivity determination. This modification eliminates the errors associated with the transient probes, so the performance is improved. i t aiso makes it a device for the simultaneous determination of all the thermophysical parameters.

5.1. Theory of the method

If the rise in temperature at two near points, distant r , and r2, be 0, and O2 respectively, then from the theory of line heat source we have

- -~ 1 r: + "1 4 ( 4 ~ ~ t ) ~ bH

and

where y is the power generated per unit length in the source and CY is the thermal diffusivity.

The term (2AlbH) represents an initial lag error which arises due to the thermal mass of the probe and contact resistance between the probe and the sample. Subtracting (32) from (31) we get

Therefore, the lag correction term has been eliminated from the final expression (33).

To account for time lag error a correction term t, has to be subtracted from the observed time t . Hence equation (33) becomes

(1 r /r , \2 r: - r: 1 .-__

i 1"' \ I I / 4 a ( I - I ~ )

64n2 (t - t,)2 3 ' ri - r;' 1 +-- As t * I, we have

_ _ _ - 4 n t 4nA

Expanding and neglecting second and higher powers

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Table 2. Comparison of predicted and experimental values for three-phase granular systems. Values of thermal conductivity are in W m-l K-'. System I, iron powderlaluminium powder and air: ),,(Fe) = 65.402, &(AI) = 204.24, &(air) = 0.0274; 6, = 3.8. 11, aluminaizirconia and air: Aj(A1203) = 24.416. .12(Zr02) = 1.998, &(air) = 0.0274, 6, = 6.8. Ill, cupric oxideialuminium powder and air: A,(CuO) = 13.490. A2(Al) = 204.24, &,(air) = 0.0274, 6, = 6.8. A and q are the thermal conductivities and volume fractions of the phases respectively. Suffix 0 represents the continuous phase. Suffixes 1 and 2 represent the first and second dispersed phases respectively.

Volume fraction

First Second phase phase

System 'pl W2

I 0.537 0.049 0.516 0.094 0.438 0.120 0.427 0.158 Average deviation

II 0.387 0.027 0.368 0.049 0.365 0.074 0.328 0.089 Average deviation

111 0.285 0.042 0.286 0.086 0.266 0.121 0.248 0.154 Average deviation

ETC exp

Proposed model

Brailsford model

Hadley model

ETC % Error ETC %Error ETC %Error

0.349 2 0.008 0.226 * 0.007 0.218 * 0.013 0.293 t- 0.012

0.142 * 0.007 0.163 C 0.012 0.185 * 0.013 0.170 f_ 0.008

0.183 * 0.011 ~ ~~

0.222 * 0.009 0.228 C 0.012 0.211 * 0.007

0.215 38.3 0.223 1.4 0.310 42.4 0.313 6.8

22.2

0.154 8.2 0.164 0.5 0.162 12.3 0.191 12.6

8.4

0.187 2.0 0.179 19.2 0.194 14.7 0.210 0.5

9.1

0.026 92.5 0.025 89.1 0.624 89.0 0.023 92.2

90.7

0.027 81.2 0.026 84.0 0.025 86.3 0.025 85.3

84.2

0.026 85.7 0.025 88.7 0.024 89.5 0.023 89.0

88.2

0.211 39.5 0.230 1 .E 0.314 44.1 0.329 12.2

24.4

0.381 168 0.413 153 0.418 126 0.491 189

159

0.581 217 0.581 162 0.641 181 0.703 233

198

Table 3. (Sugawara 1961). Predicted and experimental values of thermal conductivity for calcareous sandstone, air and water. Values of thermal conductivity are in W m-' K-'. A, (stone) = 1.856, A,(air) = 0.0275, &,(water) = 0.6322, E, = 11; other symbols as in table 2.

___


First Second model model model model System phase phase ETC No Vl W2 exp. ETC % Error ETC % Error ETC % Error ETC % Error

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

0.120 0.025 0.157 0.025 0.301 0.025 0.227 0.025 0.095 0.050 0.132 0.050 0.202 0.050 0.276 0.050 ~~ .~ 0.070 0.075 0.107 0.075 0.177 0.075 0.231 0.075 0.045 0.100 0.082 0.100 0.152 0.100 0.226 0.100 0.127 0.125 0.201 0.125 0.176 0.150 0.151 0.175 Average deviation

1.293 1.172 0.742 0.974 1.346 1.21 8 1.032 0.789 1.380 1.253 1.067 0.827 1.41 5 1.291 1.090 0.847 1.117 0.858 0.885 0.899

1.389 1.282 0.966 1.114 1.346 1.204 0.999 0.841 1.373 1.209 0.968 0.834 1.424 1.251 0.984 0.785 1.023

0.839 0.883

0.805

7.4 9.4

30.1 14.3 0.0 1.2 3.2 6.6 0.5 3.5 9.2 0.8 0.7 3.1 9.8 7.3 8.4 6.2 5.2 1 .E 6.4

1.510 1.421 1.102 1.261 1.539 1.449 1.288 1.127 1.566 1.477 1.314 1.195 1.593 1.503 1.340 1.178 1.366 1.202 1.227 1.250

16.8 21.2 48.5 29.4 14.3 19.0 24.8 42.9 13.5 17.9 23.2 44.5 12.6 16.4 23.0 39.0 22.3 40.1 38.6 39.1 27.4

1.090 0.933 0.508 0.694 1.179 1.009 0.751 0.550 1.275 1.091 0.812 0.647 1.379 1.180 0.879 0.643 0.950 0.696 0.752 0.814

15.7 1.569 21.3 20.4 1.497 27.8 31.5 1.220 64.4 28.7 1.362 39.8 12.4 1.576 17.1 17.2 1.507 23.7 27.2 1.374 33.1 30.3 1.233 56.3 7.6 1.582 14.7

12.9 1.516 21.0 23.9 1.385 29.8 21 .E 1.284 55.2 2.6 1.589 12.3 8.6 1.525 18.1

19.4 1.397 28.1 24.1 1.259 48.6 14.9 1.408 26.1 18.9 1.271 48.2 15.0 1.284 45.1 9.5 1.297 44.3

18.1 33.7

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Table 4. (Cheng et a/ 1972). Comparison of predicted and experimental values of effective thermal conductivity for different three-phase systems. Values of thermal conductivity are in W m-l K-'. System I. aluminium and nickel powder in silicon rubber: Al(Al) = 204.258. A,(Ni) = 90,012, A,(rub) = 0.384. = 5.2. (I, aluminium and bismuth in silicon rubber: &(AI) = 204.258, A@) = 8.039. A,(rub) = 0.384, 6, = 2.0. 111, lead and bismuth in silicon rubber: Al(Pb) = 34.62, A2(Bi) = 8.039. M u b ) = 0.384, 6, = 2.0. Other symbols as in tables 1 and 2.



System 'pl M exp. ETC % Error ETC %Error ETC %Error ETC % Error

I 0.06 0.04 0.559 0.631 12.8 0.508 9.2 0.696 24.5 0.380 32.0 0.13 0.09 0.805 0.685 14.9 0.696 13.5 1.419 76.3 0.900 11.8 ~~~

Average deviation 13.8 11.4 50.4 21.9

II 0.15 0.03 0.770 0.501 34.9 0.489 36.5 1.080 40.2 0.418 45.7 0.04 0.08 0.637 0.662 3.9 0.509 20.1 0.631 0.9 0.414 35.1 Average deviation 19.4 28.6 20.6 40.4

111 0.08 0.10 0.692 0.658 4.9 0.574 17.1 0.749 8.2 0.542 21.7 0.08 0.04 0.592 0.555 6.2 0.485 18.0 0.623 5.2 0.390 34.1 Average deviation 5.6 17.5 6.7 27.9

of (tJr) we get

or

Here we have retained terms up t o the second power of (l/t). Equation (34) shows that the correction term appears with the coefficient of (l/r2). This equation between (0, - 0,) and (l/r) represents a parabola of the form

y = A ~ + A , X + A , x ~ (35) where An, A , , and A, are constants called coefficients of the curve. The values of these coefficients can be determined by a data fitting technique using a standard computer. For larger values of time (I > 100s) equation (35) almost approaches a straight line. The values of the coefficients will be

A " -"in(') - 2nA (36)

(37)

and

Knowing the value of A,, the thermal conductivity of the specimen can be determined by equation (36).

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So we have

A=-ln(') 4 2nA (39)

Detailed theory and the experimental arrangement have been given by Verma et a1 (1YY0, 1991).

5.2. Procedure

After switching on the current in the heater wire, the temperature difference in the sample at the midpoints of the sensors is noted with respect to time. A plot between the reciprocal of time (1/1) and the temperature difference (0, - 0,) will be a parabola. Its coefficients are determined by a data fitting technique using a personal computer PC-XT. The thermal conductivity can be determined by equation (39) as discussed above. The computer program is designed to display the values of various coefficients and thermal conductivity directly.

5.3. Sample preparation

Different three-phase systems were formed by mixing known proportions of the constituents. Fine powders of iron and aluminium were used to form an iron/ aluminium/air system. In a similar manner cupric oxide(CuO)/aluminium/air and alumina(A1203)/ zirconia(ZrO,)/air systems were also formed and their thermal conductivity was determined. Known amounts of these powders were also mixed in a known volume of Vaseline to form iron/aluminium/Vaseline, cupric oxide/aluminium/Vaseline and alumina/zirconia/ Vaseline systems. All the powders used were dried in an oven at l l0"C for about 24h to remove possible moisture content in the sample. Three-phase mixtures formed by building material powders were also studied.


Table 5. (Hadley 1986). Comparison of predicted and experimental values for packed three-phase metallic systems. Values of thermal conductivity are in W m-l K-' . Other svmbols as in tables 1 and 2.



No VI P2 exp. ETC % Error ETC % Error ETC % Error ETC %Error

I , brassisteeliair system: &(Brass) = 113, &(steel) = 12.4, &(air) = 0.0274. S, = 15.6 1 0.08 0.784 23.20 31.71 36.7 0.96 95.9 37.42 61.3 25.44 10 2 0.14 0.784 17.22 18.39 6.8 0.53 96.9 23.37 35.7 15.97 7 3 0.19 0.784 14.71 13.26 9.9 0.38 97.4 15.79 7.3 10.83 26 4 0.24 0.784 8.15 10.16 24.7 0.29 96.5 10.66 30.8 7.12 13 5 0.10 0.577 13.92 16.49 18.5 0.76 94.6 21.19 52.3 14.56 5 6 0.15 0.577 11.28 10.88 3.6 0.49 95.7 14.65 29.8 9 ~ 5 16 7 0.20 0.577 8 0.26 0.577 9 0.11 0.380

10 0.17 0.380 11 0.22 0.380 12 0.27 0.380 13 0.12 0.187 ~~

14 0.18 0.187 15 0.24 0.187 16 0.29 0.187

Average deviation

8.41 7.94 5.6 0.35 5.92 5.85 1.1 0.26 8.00 8.11 1.4 0.68 6.90 5.11 26.0 0.42 5.71 3.82 33.1 0.32 4.27 3.00 29.8 0.25 5.80 2.59 55.3 0.61 4.52 1.67 63.1 0.39 3.54 1.20 66.1 . 0.28 2.82 0.95 68.2 0.23

28.0

~~ ~. . 95.8 10.12 20.4 7:37 12 95.6 8.50 9.7 4.26 28 91.5 13.36 67.0 9.a3 13 93.9 8.80 27.6 8.47 7 94.5 6.22 8.9 4.29 25 94.2 4.39 2.8 2.98 30 89.5 8.56 47.7 5.78 0 91.3 5.79 28.1 3.80 18 92.0 3.91 10.5 2.52 29 92.0 2.82 0.1 1.65 41 94.1 27.5 17

11, brassisteelivacuum system: &(brass) = 113, &(steel) = 12.4, h,(vac) = 0.001, €3, = 15.6 1 0.08 0.784 20.76 31.71 52.7 0.04 99.8 28.72 38.3 - - 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

0.14 0.784 14.88 0.19 0.784 12.78 0.24 0.784 6.07 0.10 0.577 12.95 0.15 0.577 9.1 1 0.20 0.577 6.65 0.26 0.577 4.30 0.11 0.380 6.00 0.17 0.380 4.90 0.22 0.380 3.89 0.27 0.380 2.63 0.12 0.187 3.78 0.18 0.187 2.81 0.24 0.187 1.91 0.29 0.187 1.39 Average deviation

18.39 23.6 0.02 99.9 13.25 3.7 0.01 99.9 10.16 87.4 0.01 99.8 16.49 27.3 0.03 99.8 10.88 19.4 0.02 99.8 7.94 19.4 0.01 99.8 5.85 36.1 0.01 99.8 8.11 35.1 0.03 99.6 5.11 4.2 0.02 99.7 3.82 1.8 0.01 99.7 2.99 13.8 0.01 99.7 2.58 31.7 0.02 99.4 1.66 36.3 0.01 99.4 1.19 37.5 0.01 99.5 0.95 31.8 0.01 99.4

27.6 99.7

14.70 1.2 - - 8.42 34.1 - - 4.82 20.6 - -

15.22 17.5 - - 8.91 2.2 - - 5.22 21.5 - - 2.75 36.1 - - 9.28 54.7 - - 5.01 2.3 - - 3.00 22.8 - - 1.80 31.7 - - 5.76 52.3 - - 3.19 22.2 - - 1.77 7.5 1.08 22.2 - -

24.2

- -

6. Results and discussion

Results of the study of different three-phase systems are listed in tables 1 to 7. The effective thermal conductivity of the samples has also been determined by other models (Brailsford and Major 1964, Lichtnecker 1926, Hadley 1986). Values of the thermal conductivity of different constituent phases cited in the literature have been used for calculations. The values of ETC predicted by these models and the percentage deviations from the exper- iment are shown in the tables. Expressions used in the calculations are given in the appendix.

Table 1 gives the values for systems formed by suspending powders in Vaseline. The composition is given in the table. Here the fractional volume of dispersed oxide phase mixture is small, so the continuous phase (Vaseline) has been taken as an independent phase and granular phases have been composed to give the intermediate phase. We see that the proposed model

predicts values of ETC quite close to the experimental values, whereas other models give a higher deviation. Table 2 represents the study of powder-air systems. Here the concentration of the powder phase is comparable with that of air. Therefore the second phase having thermal conductivity A 2 has been combined with air to form the intermediate phase. This combination lowers the intermediate thermal conductivity. In a similar manner thermal conductivities of air and water have been combined to form an intermediate phase in a calcareous stone/water/air system (stone, as the majority phase, has been taken as an independent phase). Within the paired phases, second phase water has a greater contribution towards heat conduction; this can be readily seen by comparing the product of the fractional volume of water and its thermal conductivity with respect to that for air. Here the former has a greater value, hence relation (30) has been used to define the intermediate thermal conductivity. The results of the

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L S Verma et al

Table 6. (Hadley 1986). Comparison of calculated and experimental values of three-phase granular systems. Values of thermal conductivity are in W m-’ K-’. System I, brassisteeliwater system: h,(brass) = 113, A,(steel) = 12.4, A,(water) = 0.6, Bo = 0.9. 11, brassisteeliair system: &(brass) = 113, &(steel) = 12.4. &,(air) = 0.0274, 6, = 2.8. 111, brassisteeli vacuum system: &(brass) = 113, &(steel) = 12.4, &(vac) = 0.001, 8, = 2.8. Other symbols as in tables 1 and 2.


First Second model model model model phase phase ETC -.

System ql P2 exp. ETC % Error ETC % Error ETC % Error ETC % Error

I 0.14 0.784 29.15 28.62 1 .E 10.25 64.8 36.00 23.5 30.5 5 18.0 10 0.24 0.784 16.38 16.29 0.6 5.85 64.3 22.36 36.5

0.29 0.187 6.36 6.23 2.1 3.98 37.4 6.91 8.6 7.01 10 Average deviation 1.5 55.5 22.9 8

0.248 16 II 0.56 0.784 0.296 0.339 14.6 - - - - 0.252 20 0.316 0.324 2.4 - - - - 0.53 0.577 0.271 8 0.294 0.303 3.0 - - - - 0.49 0.380 0.292 12 0.45 0.187 0.332 0.248 25.4 - - - -

Average deviation 11.4 14 111 0.56 0.784 0.0431 0.0473 9.6 - - - -

0.577 0.0473 0.0450 4.9 - - - - 0.53 0.49 0.380 0.0424 0.0419 1.1 - - - - 0.45 0.187 0.0666 0.0338 49.2 - - - - - - Average deviation 16.2

- - - - - -

Table 7. Comparison of predicted and experimental values for some building materials granular systems. Values of thermal conductivity are in W m-’ K-’. System I, marbleiairiwater system: ,%,(marble) = 3.3, kl(air) = 0.0274, ,%,(water) = 0.6, B6 = 1.15. !I, Knta stoneiairlwater system: &(stone) = 3.1, &(air) = 0.0274, Adwater) = 0.6, S, = 1.15. Other symbols as in tables 1 and 2.

Volume fraction Proposed Brailsford Lichinecker Hadley


System v1 P2 exp. ETC % Error ETC % Error ETC YO Error ETC % Error

I 0.307 0.083 1.192 i 0.018 1.757 47.4 0.200 0,190 1.795 i 0.013 1.834 2.2 0.105 0.285 1.923 ? 0.022 1.913 0.5 0.007 0.383 2.310 i 0.025 2.006 13.2 Average deviation 15.8

II 0.399 0.115 1.180+0.017 1.245 5.5 0.317 0.197 1.573 t- 0.021 1.287 18.2 0.231 0.283 1.621 t 0.018 1.337 17.5 0.044 0,470 1.982 +_ 0.028 1.469 25.9 Average deviation 16.8

1.698 42.4 0.658 1.693 5.7 0.916 1.689 12.2 1.226 1.685 27.1 1.661

1.214 2.9 0.389 21 .E

1.211 23.0 0.501 i.206 25.5 0.653 i.200 39.4 1.164

22.7

44.8 49.0 36.2 28.1 39.5 67.0 68.1 59.7 41.3 59.0

2.759 131 2.479 38 2.223 16 1.949 16

50 2.496 1 1 1 2.283 45 2.055 27 1.547 22

51

study are shown in table 3. In the majority of cases the predicted values agree within 7%. The average deviation is6.4whichisfarsmallerthan thatgiven by other models. Table 4 represents the study of powders embedded in silicon rubber. Here aluminium and nickel, aluminium and bismuth, lead and bismuth have been combined to form the intermediate phase for the first, second, and third sample respectively. In the first sample aluminium and nickel have higher thermal conductivities. Therefore the parameter Bo has a higher value, whereas in the second and third samples the contribution of bismuth is smaller so the perpendicular component of ETC is small. ThiscausesB,tohaveasmallervalue. Here the predicted

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values agree within 15%, whereas the other modelsgive larger deviations.

In tables 5 and 6 Hadley’s experimental results have been discussed. He obtained these values for a packed system of metallic powders. The results have been classi- fied into various three-phase systems as mentioned in the tables. Here the metallic constituents brass and stainless steel have a higher fractional volume. So these phases have been combined to form an intermediate phase. For a brass/steel/air system the deviation for values predicted by proposed model is slightly on the high side. The reason for this is obvious: Hadley used highly angular (sharp corners and projections) brass

Erc of three-phase systems

particles to compose the disc which are bound to have a certain value of sphericity v . This value is unknown to us, so we have assumed its value to be unity (therefore Some error is liable to occur in the predicted values). Moreover he used two adjustable parameters (fo and a) toexplain the results. For the brass/steel/vacuum system Hadley has not mentioned the theoretical values. How- ever, we have made an attempt to predict the ETC of the system. Vacuum has been treated as one of the constituents and has been assigned an arbitrary thermal conductivity close to zero (=0.001). Theoretically, it should have zero thermal conductivity but in practical situations the vacuum is not perfect and some conduction is always present due to residual molecules. Moreover, heat transfer by radiation also takes place within the system. The pores play an important role in this kind of heat transfer. Looking at the situation assigning a low thermal conductivity to vacuum is justified. The model of BrailsfordandLichtnecker gave amuchhigher deviation for the packed metallic systems studied by Hadley. Wherever these models gave more than 100% deviation, the values are not shown in the table. Hadley's model gave more than 100% deviation for alumina/zirconia/ air and cupric oxide/aluminium powder/air systems.

Theexperimental study of building materials is shown in table 7. Here the fractional volume of solid is nearly 50%. Therefore, on the same lines as discussed above, the granular phase of higher fractional volume has been treated as independent and the others comprise the intermediate phase. The predicted values agree within 20% and are comparable to those given by other models. Hadley's model is unable to predict ETC and gives con- siderably higher deviations.

The term [@(A$. l)]'/2andconstant &play an important role in predicting the ETC. They determine the component of thermal conductivity perpendicular to the direction of heat flow. In the metallic powder systems this part of the conductivity has a higher value. Hadley used two parameters: f o to account for the parallel component, and a to determine the perpendicular component. The proposed model has onlyone such constant to decide all these. Values of the constants for the different systems are given in the tables.

7. Conclusions

From a close study of the various tables one can see that different models have limited applicability. When one model developed for a particular type of system is tested on the other, it fails to predict the effective thermal conductivity. However, the proposed model successfully predicts the ETC of different three-phase systems. When the dispersion is small and thermal conductivities of the phases are low, the predicted values agree within 7%. For intermediate dispersions the agreement is within 15%. For higher dispersions with metallic phases the predicted values are found to agree within 20%. The proposed model is equally good for granular, consohdated and packed systems. Moreover,

it has potential for development to predict ETCS of composites and other complex systems.

Acknowledgments

Financial assistance from DNES, New Delhi and ICAR, New Delhi, and equipment support from TWAS. Trieste, Italy is gratefully acknowledged.

Appendix

Expressions used in the calculations are as follows

systems) (i) Brailsford and Major model (two-phase

(ii) Lichtnecker model (three-phase systems)

he = (A,,).~l ' ( A l ) " l ' (A2)"? .

(iii) Hadley model (two-phase systems)

A and cp are the thermal conductivities and volume fractions of the phases respectively. Suffix 0 represents the continuous phase. Suffixes 1 and 2 represent the first and second dispersed phases respectively.

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Prediction and measurement of effective thermal conductivity of three-phase systems

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