Generalized Theory of Thermal Conductivity for Different Media: Solids

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C: Physical Processes in Nanomaterials and Nanostructures

Generalized Theory of Thermal Conductivity for Different Media: Solids to Nanofluids Swapna Mohanachandran Nair Sindhu, and Sankararaman Sankaranarayana Iyer J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b07406 • Publication Date (Web): 28 Aug 2019 Downloaded from pubs.acs.org on August 28, 2019

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The Journal of Physical Chemistry

Generalized Theory of Thermal Conductivity for Different Media: Solids to Nanofluids Swapna Mohanachandran Nair Sindhu and Sankararaman Sankaranarayana Iyer * Department of Optoelectronics and Department of Nanoscience and Nanotechnology, University of Kerala, Trivandrum, Kerala, India- 695581. *Corresponding author email: [email protected]

ABSTRACT: The advent of nanotechnology in the twenty first century opened a new branch of nanoscience known as nanofluids finding wide range of industrial applications especially in heat transfer. Though the theory of thermal conductivity of solids is well established, there is no such conclusive model to explain the thermal conductivity of nanofluids. In the present work we propose a generalized theory for thermal conductivity applicable to materials ranging from heterogeneous solids, porous materials, nanofluids, and ferrofluids. The model could explain the effective thermal conductivity of not only the combination of solids but also solid-fluid mixtures. The proposed theory could successfully link the existing models for porous solid materials and nanofluid as its special cases. The proposed model is verified against the experimental data by simulating the theoretical equations.

INTRODUCTION Studies on the thermal properties of materials have always been the focus of interest of mankind as the inventions and discoveries in the field have revolutionized our life. The incessant search for better materials with improved thermal properties resulted in the development of nanofluids and nanofluidics during the last two decades.1–8 If the 20th century witnessed the development of

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several models and theories9–13 to explain the thermal properties of matter and thermal energy transport, the 21st century is witnessing the development of theories explaining the conductivity of nanofluids,1,13–20 though no one has succeeded in arriving at a conclusive model. Unlike theoretical support for thermal conductivity of solids, that of nanofluids cannot be predicted by any of the conventional multiphase conductivity models given in literature.14,15,18,21–26 For solids there exist lot of models relating thermal conductivity and porosity that include Eucken, Russell, Loeb, Radiation (Russell) equations9–13,27 derived from Maxwell’s relation for conduction on isometric pore assumption. Among these, the Loeb’s model27 developed in 1954 for the thermal conductivity of porous material is well noticed and used by several researchers because of its simplicity.11,28,29 The possible hurdle for the generalized theoretical modeling may be the micro convective heat transfer and Brownian motion that are predominant in nanofluids. Patel et al. (2008)16,26 have proposed a model for thermal conductivity of nanofluids on the assumption that two continuous media (solid and liquid) are considered participating in the conductive heat transfer and the total heat transfer through the nanofluid can be expressed as the sum of the heat transferred by the solid and the liquid parts. The major limitation of this model is in its basic assumption that the temperature gradient across the ‘particles’ of solid and liquid are the same, which is possible only if these two types of ‘particles’ are imagined to be arranged perpendicular to the direction of heat flow and is far from reality. Another limitation with this model is in its second assumption that this temperature gradient remains the same irrespective of its distance from the hot end. The real nanofluid resembles more to the Lobe’s model of laminar heat flow through a medium containing solid particles distributed in a liquid along and perpendicular to the direction of heat flow. Hence, a complete analysis leading to a better


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understanding of the thermal energy transport in nanofluid is possible if and only if the distribution of solid particles in both the directions are considered with liquid in between. In the present work an attempt has been made in developing a generalized theory of heat conduction, modifying the Loeb’s model (which we propose to name as Sankar-Loeb’s model), which is applicable to a wide class of materials like solids, liquids, homogenous and heterogeneous mixtures, porous materials, nanofluids, ferrofluids, fluid containing micro/meso/macro porous particles and core-shells. The expressions for thermal conductivity of heterogeneous solids, porous materials (Loeb’s model27), and nanofluid (proposed by Patel et al.16,24,26) appears as special cases of our proposed generalized theoretical model for the thermal conductivity.

THEORY AND DISCUSSIONS One Dimensional Model

Figure 1. (a) Laminar view of the system with hollow solid particles in liquid (b) one-dimensional array of hollow solid particles with the heat flow direction. In the proposed model we have assumed the nanoparticle dispersed in a liquid medium and the laminar flow of heat as shown in Figure 1. As a general case let us consider one such onedimensional array (shown separated in Figure 1a), of length L and unit area of cross-section, containing the hollow solid and the liquid portions. Figure 1b shows such one dimensional parallel


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arrays in a plane. Let l, ls, and lp be the total length of the liquid, solid, and pore region in the direction of heat propagation as marked in Figure 2. The temperature gradient across the liquid, solid, and pore region along the direction of heat propagation is not equal, as their conductivities differ.

(1)

But the temperature gradient perpendicular to the direction of heat propagation can be considered the same provided the layers are identical. If W is the amount of heat energy flowing through a layer of unit cross-sectional area the total amount of heat energy getting transferred across the sample will be the sum of heat energy flowing through all layers. We know that W is related to the thermal conductivity (k) and the negative temperature gradient

as follows30 (2)

W = -k

The temperature drop (ΔT) across the sample of length L and effective thermal conductivity conducting the heat energy (W) is given by ΔT = -W If

,

(3)

, and

are the thermal conductivities of the liquid, solid, and porous region the total

temperature drop across the sample can be written as ΔT = -W ∑

∆

where ls = ∑ ∆ lp = ∑ ∆

∑

∆

∑

∆

= -W

– combined length of solid section; l = ∑ ∆

(4) – combined length of liquid section;

– combined length of the porous section, and np – the number of pores. The thermal

conductivity of porous region is related to its shape (γ- geometrical factor), size (∆

, temperature

(T), emissivity (e) of the radiating surface and Stefan’s radiation constant (σ) through the relation27


4

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kp = 4∆

γeσT3

(5)

From eqs (3) and (4)

-W

-W

=

(6)

Figure 2. (a) The hollow solid particles dispersed in the liquid medium (b) the hollow solid particle. The length of the sample L = ∑ ∆ or

l=L-

∑∆

∑∆

= l + +

(7)

-

Let us define porosity as the fraction of total length occupied by the pore as =P

(8)

 Eq 6 becomes =

1

1

Using eq 8, we get = (1-P)

(9)

Neglecting the second and third terms being small (1-P) or

= (1-P) 1

(10a)

(10b)


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Equation 10 shows that for a given solid volume fraction, as the porosity P increases the ratio of thermal conductivity decreases. Here we have considered only the pores enclosed by a solid within the liquid. Hence, the increase of porosity (P = ) (Figure 3) suggests the increase in , or the size of the hollow solid particle. The decrease in the ratio of thermal conductivity suggests an increase of

. This result can considered to be in agreement with the literature report31 of an

increase of thermal conductivity with particle size in the liquid medium. These results show that the variation of thermal conductivity depends only on the particle size along the direction of propagation of heat and not on its pore size when it is within a liquid. As the particle size along the direction of propagation of heat increases, the effective path length of propagation of heat energy through the solid increases which in turn increases the conductivity. Here, the open pores that are also commonly treated as porous need only be considered as particles of irregular shape.

Figure 3. Hollow solid particles of different pore size. Case 1: For the solid particles in liquid medium

Figure 4. Solid particles in liquid medium. Consider the solid particles (P = 0) dispersed in a liquid medium as shown in Figure 4. Hence eq 10 can be written as


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(11)

To be more precise, consider eq 9 with =

= 0, then eq 9 can be written as

(12)

From this equation, it is clear that as the total particle size (ls = ∑ ∆ ) increases k

also increases.

In one of our earlier work in titanium dioxide (TiO2) nanofluid, we have shown that the thermal conductivity increases with particle size.31 The variation of effective thermal conductivity of the nanofluid calculated using eq 12 with the particle size is shown in Figure 5. From literature,31–33 we can see several reports of increase of thermal conductivity with particle size which can be explained on the basis of this equation.

Figure 5. Variation of effective thermal conductivity with particle size for TiO2 nanofluid. Case 2: For two solids with no pores

Figure 6. Two solids with no pores. When two solids of conductivities

and

are joined in series as shown in Figure 6, the effective

conductivity calculated using eq 10 is


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with

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(13) and

.

Equations (11) and (13) are in good agreement with the literature and are discussed in all standard books on Heat.30 From eq 13 it is clear that when different materials of the same cross-sectional area are joined together the effective thermal conductivity will be smaller than the smallest. As an example, the effective thermal conductivity of an aluminium rod joined to different materials is shown in Figure 7.

Figure 7. Variation of the effective thermal conductivity of Al-X combinations, (X = Platinum (Pt), Iron (Fe), Tungsten (W), Gold (Au), Silver (Ag)).34

Case 3: For porous solid with no liquid Here the length of liquid region l = 0, hence the term corresponding to the liquid vanishes in eq 4 and we get eq 10 modified as (1-P

+

(14)

which is same as the expression for the effective thermal conductivity derived by Loeb’s.27 Case 4: Incorporating solid volume fraction


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If

and

are the cross sectional area of heat transfer for the liquid and solid portion such that

the total area A =

. Assuming the solid and liquid media as spheres of radii

and

respectively (as assumed by Patel et al.16), the volume fractions of solid (ξ) and liquid (1- ξ) are related to the respective number of particles through the eq 15

;

(15)

The areas of cross section

4

;

=

4

(16)

Substituting eq 14 in 15 we get =

(17)

Incorporating the cross sectional areas of solid and liquid in eq 10 we get = (1-P)

1

(18)

Using eq 14 

= (1-P) 1

(19)

Assuming radius of hollow alumina nanoparticles glycol

= 20 nm35, the molecular size of ethylene

= 0.23 nm and the porosity P = 0.2, the variation of

with solid volume fraction

calculated using eq 19 is shown in Figure 8. The experimental results of Beck et al.36 shown in Fig .8b is also in good agreement with eq 19 and the proposed model (Figure 8a). The difference between the theoretical and experimental values may be due to the input parameters selected for evaluating eq 19. As the volume fraction or weight fraction of nanoparticle in the nanofluid increases, the number of particles along the direction of propagation of heat also increases. Thus


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it can be seen that the effective path length of propagation of heat energy through the solid increases, which in turn enhances the

Figure 8. The variation of

.

with solid volume fraction (a) the theoretical plot for

= 20 nm,

= 0.23 nm, and P = 0.2 (b) from the experimental data.35

Case 5: Considering particle movement When the particle size is very small, the temperature-dependent motion of particles cannot be ignored. The thermal conductivity of solid particle is proportional to its average velocity due to Brownian motion and is given by24,37

(20)

where ‘c’ a constant and the average velocity (

obtained from Stokes-Einstein’s formula24,38 is

Here,

(21)

, T, and

represent the Boltzmann constant, temperature, and the dynamic viscosity

respectively. With P = 0, eq 10 becomes = 1

(22)


10

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or eq 19 becomes = 1

(23)

Case 6: For Core-shell particles dispersed in liquid -

Figure 9. Core-shell model. In the case of core-shell nanoparticles as shown in Figure 9, the core and shell materials are different and let

and

be their respective thermal conductivities. The effective conductivity

of the core-shell nanofluid can be written from eq 10 as

(24)

For the hollow core shell nanoparticles, eq 10 becomes

(25)

Two Dimensional Model In the previous section, we have considered an array of hollow solid spheres dispersed in liquid medium in one dimension and discussed the various possibilities of applying eq 10 for solids/porous particles/liquids and a combination of one or more of these. In this section, a twodimensional model is considered. A plane parallel to the direction of heat flow can be imagined to consist of a number of such one-dimensional arrays as displayed in Figure 10.


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Figure 10. Two-dimensional array of hollow solid spheres dispersed in the liquid medium. Let

,

,

be the effective thermal conductivity of first, second, third,…and nth array

… and

respectively. These parallel arrays are imagined to form a layer parallel to the direction of heat flow. Hence, the total amount of heat energy (W) entering the layer can be written as W= ∑ The temperature gradient W=-

⋯

being the same across the plane eq 2 can be written as

(26)

or, the effective thermal conductivity of these parallel arrays can be written as ⋯ where, from eq 10,

=

∑ (1-P)

(27)

Case 1: For the parallel array of solid and liquid particles


12

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Figure 11. Parallel array of solid and liquid particles. When the solid and liquid particles are parallel to each other and perpendicular to the direction of heat flow as shown in Figure 11, using eq 10 with P = 0, the thermal conductivity of the solid and liquid layer can be written as

or

or

The effective thermal conductivity for the model shown in Figure 8 is given by eq 27 as =

1

=

(28)

Assuming the solid and liquid media as spheres of radii

and

respectively, the ratio of

is

given by eq 17

=

Incorporating the cross-sectional areas of solid (

and liquid (

and their volume fractions in

eq 28 we get =

1

(29)


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which is the expression derived by Patel et al.16 for the effective thermal conductivity of nanofluids. Figure 12 shows the experimental and theoretical curve for alumina and TiO2 nanofluids.31,36 Here, one can see that the experimental results reported by several researchers do not agree with the theoretical curve drawn using eq 29 which is same as the model proposed by Patel et al.26 However the experimental data matches well with the Figure 12 discussed as case 1 in the one-dimensional model.


with particle size - (a & b) the theoretical plot and (c & d) the

experimental plot - for alumina and TiO2 nanofluid.31,36

Case 2: For the parallel array of solids When two solid particles are parallel to each other and perpendicular to the direction of heat flow as shown in Figure 13, using eq 10 with P = 0, the thermal conductivity of the solid layers can be written as


14

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For the first solid layer

or

For the second solid layer

or

Figure 13. Parallel array of solids

The effective thermal conductivity for the model shown in Figure 9 is given by =

(30)

which is in agreement with literature. Figure 14 shows the effective thermal conductivity for the system containing aluminium rod connected parallel to material X (X = Pt, Fe, W, Au, Ag).

Figure 14. Variation of the effective thermal conductivity of Al-X combinations (X = Pt, Fe, W, Au, Ag).34


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Case 3: For a material with pores within solid (Loeb’s model) In the model proposed by Loeb’s the porous layer is in between solid layers as shown in Figure 15. The effective thermal conductivity for the system can be obtained from eq 10 which is same as the one derived by Loeb’s. By eq 14, For the solid layer

or

For the porous layer

or

Figure 15. Material with pores within solid. If A is the cross-sectional area of the sample, ПA is the fraction of area occupied by the porous region, and (1- П) A that by the solid region, then

∆

or

= (ПA =П

1

П

∆

)

+ (1- П)

(31)

П gives the porosity transverse to the direction of propagation. From eq 31 it is clear that as П increases

decreases. Figure 16 shows the experimental and theoretical variation of

with


16

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porosity for porous alumina.35 The small difference between the theoretical and experimental data may be due to the approximations.


with porosity for porous alumina.

Case 4: Orientation dependence From eq 10 it is clear that as the longitudinal porosity (parallel to the direction of heat propagation - Figure 17a) increases

also increases. But eq 31 suggests that with the increase

of transverse porosity (perpendicular to the direction of heat propagation - Figure 17b) decreases. This shows that the orientation of the particle along the direction of heat propagation is highly significant in the thermal conductivity of its nanofluid. The longitudinal and transverse porosity can generally be expressed as Porosity ρ =

sin

(32)

where a and b are the length and thickness of the solid/hollow nanoparticle along the direction of heat propagation and θ is the orientation of the particle with respect to the direction of heat propagation (Figure 17c). Equation 32 can be used for finding the effective thermal conductivity for ferrofluids exhibiting strong magnetic field dependence on the orientation of the suspended nanoparticles in the nanofluid. This suggests the possibility of tuning the thermal conductivity of ferrofluids by varying the applied magnetic field.


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Figure 17. Orientation of particles - (a) parallel (b) perpendicular (c) at an angle θ - to the direction of heat propagation.

CONCLUSIONS Unlike the strong theoretical backing for the thermal conductivity of solids, nanofluids do not have a generalized theory in spite of several models proposed so far. The present paper proposes a generalized model for thermal conductivity applicable for materials ranging from solids to nanofluids. We could successfully bring the Loeb’s model for porous nanomaterials and the present nanofluid models as special cases of our proposed model which we would like to name as Sankar-Loeb’s model. The proposed one-dimensional model is also extended to two-dimension to obtain the expressions for effective thermal conductivity. Through this model, we could obtain the equations for thermal conductivity for solids, liquids, homogenous and heterogeneous mixtures, porous materials, nanofluids, ferrofluids, fluid containing micro/meso/macro porous particles and core-shells from our generalized equation (eq 10 – the Sankar- Loeb’s equation). The dependence of thermal conductivity of nanofluid on the size, orientation, and porosity of the nanoparticle are also proved and verified with the experimental data available in the literature.


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ACKNOWLEDGEMENT The authors are thankful to Dr. K. V. Dominic, Professor of English (Retired) and Editor-in-Chief, Writers Editors Critics (WEC) for the support given in English language editing. This study was not funded by any agencies in the public, commercial, or not-for-profit sectors. The authors declare no competing financial interest.

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(36) Serebryakova, M. A.; Zaikovskii, A. V; Sakhapov, S. Z.; Smovzh, D. V; Sukhinin, G. I.; Novopashin, S. A. Thermal Conductivity of Nanofluids Based on Hollow γ-Al2O3 Nanoparticles, and the Influence of Interfacial Thermal Resistance. Int. J. Heat Mass Transf. 2017, 108, 1314–1319. (37) Tien, C. L.; Lienhard, J. H. Statistical Thermodynamics/Revised Printing. Washington, DC, Hemisph. Publ. Corp., 1979; 413. (38) Kalteh, M.; Abbassi, A.; Saffar-Avval, M.; Frijns, A.; Darhuber, A.; Harting, J. Experimental and Numerical Investigation of Nanofluid Forced Convection inside a Wide Microchannel Heat Sink. Appl. Therm. Eng. 2012, 36, 260–268.


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TOC GRAPHIC


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Generalized Theory of Thermal Conductivity for Different Media: Solids

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