A semi-analytical solution to estimate an effective thermal conductivity of the two-phase building materials with spherical inclusions

Abstract

In the present work, two dimensional (2D) and three dimensional (3D) semi-analytical models are proposed to estimate an effective thermal conductivity of two-phase building materials with spherical inclusions. The proposed semi-analytical approach is based on the formulation and the solution of a boundary value problem. Further, the solution is coupled with Maxwell’s methodology to estimate effective thermal conductivity. The 2D model is simple and limited in scope while the 3D model is widely applicable and is based on a multipole expansion method. The 3D model incorporates secondary parameters like size distribution, variation in thermal conductivity among the inclusions, particle interaction and statistical spatial distribution. For the 3D model, spherical representative unit cells (SRUC) based on the face-centred cubic arrangement and trimmed spheres at the boundary are proposed. Effective thermal conductivity estimated by the proposed models is compared and validated with experimental results from the literature. The 2D model predicts the thermal conductivity of conventional concrete with reasonable accuracy where the relative error is less than ±12%. The 3D model predicts accurate results for foam concrete with a relative error of less than ±15%. Predictions of thermal conductivity of lightweight concrete by the 3D model are also in good agreement with experimental results. Finally, the comparison of proposed models with models from literature has shown that the incorporation of particle interaction has improved the accuracy of the solution. Thus, the flexibility with SRUC and incorporation of particle interactions make this 3D model widely applicable for building materials with spherical inclusions.

Appendices

Appendix A - Detailed solution of 2D semi-analytical model

Boundary conditions can be given as

$$ U\left(r,0\right)=0,\kern0.75em U\left(r,\frac{\pi }{2}\right)=0,\kern0.5em U\left({R}_2,\theta \right)=\sin \theta $$

(A.1)

$$ {U}_i\left({R}_1,\theta \right)={U}_m\left({R}_1,\theta \right),\kern1em \frac{\ d{U}_i\left({R}_1,\theta \right)}{dr}=\frac{d{U}_m\left({R}_1,\theta \right)}{dr} $$

(A.2)

Using separation of variables, let

$$ U\left(r,\theta \right)=R(r)\varnothing \left(\theta \right) $$

(A.3)

Substituting Eq. (A.3) in governing equation given in Eq. (1), two separate differential equations are obtained given in Eqs. (A.4) and (A.5),

$$ {r}^2{R}^{\prime \prime (r)}+r{R}^{\prime }(r)+\lambda\ R(r)=0 $$

(A.4)

$$ {\varnothing}^{\prime \prime}\left(\theta \right)-\lambda \varnothing \left(\theta \right)=0 $$

(A.5)

Boundary conditions U(r, 0) = 0 and $U\left(r,\frac{\pi }{2}\right)=0$ can be written as,

$$ \varnothing (0)=0;\varnothing \left(\frac{\pi }{2}\right)=0 $$

(A.6)

Equations (A.5) and (A.6) together forms Sturm-Liouville eigen value problem where eigen function ∅(θ) and eigen value λ_n are given as [26, 27],

$$ \varnothing \left(\uptheta \right)=\mathrm{Sin}2 n\theta; {\lambda}_n=-4{n}^2\ where\ n=1,2,3\dots $$

(A.7)

Equation (A.4) is the Euler-Cauchy equation and solution can be given as [26, 27],

$$ {R}_n(r)=\left({A}_n{r}^{2n}+{B}_n{r}^{-2n}\right) $$

(A.8)

Thus general solution becomes,

$$ U\left(r,\theta \right)=\sum \limits_{n=1}^{n=\infty}\left({A}_n{r}^{2n}+{B}_{\mathrm{n}}{r}^{-2n}\right)\mathit{\sin}\ 2 n\theta $$

(A.9)

For the temperature field in the inclusion, B_n will become zero for the sake of boundedness of the solution. Thus, the temperature field for each component of the system i.e. matrix (U_m) and inclusion (U_i) can be given as,

$$ {U}_m\left(r,\theta \right)=\sum \limits_{n=1}^{n=\infty}\left({A}_{nm}{r}^{2n}+{B}_{nm}{r}^{-2n}\right)\mathit{\sin}\ 2 n\theta $$

(A.10)

$$ {U}_i\left(r,\theta \right)=\sum \limits_{n=1}^{n=\infty }{A}_{ni}\ {r}^{2n}\ \mathit{\sin}\ 2 n\theta $$

(A.11)

Orthogonality relations for sine and cosine functions are,

$$ {\int}_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\mathit{\sin}\left(2 n\theta \right)\ \mathit{\sin}\left(2 m\theta \right)\ d\theta =\left\{\begin{array}{c}1\kern2.25em n=m\\ {}0\kern2.25em n\ne m\end{array}\right. $$

(A.12)

Applying boundary condition given in Eq. (A.1) for temperature at boundary r = R₂,

$$ \sum \limits_{n=1}^{n=\infty}\left({A}_{nm}{r}^{2n}+{B}_{nm}{r}^{-2n}\right)\mathit{\sin}\ 2 n\theta = sin\theta $$

(A.13)

Multiplying both the sides by Sin(2mθ) and integrating from $-\frac{\pi }{2}$ to $\frac{\pi }{2}$,

$$ \left({A}_{nm}{R}_2^{2n}+{B}_{nm}{R}_2^{-2n}\right)=\frac{\int_0^{\frac{\pi }{2}}\sin\ \uptheta \ast \sin 2 n\theta d\theta}{\int_0^{\frac{\pi }{2}}{\left(\sin 2 n\theta \right)}^2\ d\theta}=C $$

(A.14)

Applying boundary conditions given in Eq. (A.2) for temperature and orthogonality property,

$$ \left({A}_{nm}{R}_1^{2n}+{B}_{nm}{R}_1^{-2n}\right)={A}_{ni}\ {R_1}^{2n} $$

(A.15)

Applying boundary condition given in Eq. (A.2) for heat flux and using orthogonality property for sine,

$$ {k}_m\left({A}_{nm}{R}_1^{2n}-{B}_{nm}{R}_1^{-2n}\right)={k}_i{A}_n\ {R_1}^{2n} $$

(A.16)

Appendix B - Regular and irregular solid spherical harmonics respectively and can be given by [22]

$$ {y}_t^s\left(\boldsymbol{r}\right)=\frac{r^t}{\left(t+s\right)!}{P}_t^s\left(\cos \theta \right)\exp \left( is\phi \right);\kern0.5em {Y}_t^s\left(\boldsymbol{r}\right)=\frac{\left(t-s\right)!}{r^{t+1}}\ {P}_t^s\left(\cos \theta \right)\exp \left( is\phi \right)\kern0.75em $$

(B.1)

Where ${P}_t^s$are the associated Legendre functions of the first kind of order ‘s’ and degree ‘t’.

Temperature disturbance in terms of solid harmonics for q^th inclusion where origin is at the centre of q^th inclusion is given as _,

$$ {T}_{dis}^q\left({\boldsymbol{r}}_{\boldsymbol{q}}\right)=\sum \limits_{t=1}^{\infty}\sum \limits_{s=-t}^t{A}_{ts}^{(q)}{Y}_t^s\left({\boldsymbol{r}}_{\boldsymbol{q}}\right) $$

(B.2)

Superposition of temperature disturbance due to all the inclusions with origin at the centre of the q^th inclusion is as [22],

$$ \sum \limits_{p\ne q}^N{T}_{dis}^p\left({\boldsymbol{r}}_{\boldsymbol{p}}\right)=\sum \limits_{t=0}^{\infty}\sum \limits_{s=-t}^t{b}_{ts}^{(q)}{y}_t^s\left({\boldsymbol{r}}_{\boldsymbol{q}}\right) $$

(B.3)

Where ${b}_{ts}^{(q)}$ is given as [22],

$$ {b}_{ts}^{(q)}={\left(-1\right)}^{t+s}\sum \limits_{p\ne q}^N\sum \limits_{k=1}^{\infty}\sum \limits_{l=-k}^k{A}_{kl}^{(p)}{Y}_{k+t}^{l-s}\left({\boldsymbol{R}}_{pq}\right),\kern1.5em {\boldsymbol{R}}_{pq}={\boldsymbol{r}}_{\boldsymbol{p}}-{\boldsymbol{r}}_{\boldsymbol{q}}\kern0.5em $$

(B.4)

Temperature far field T_far(r) can be expressed in terms of solid harmonics as [22]

$$ {T}_{far}\left(\boldsymbol{r}\right)=\sum \limits_{t=1}^{\infty}\sum \limits_{s=-t}^t{c}_{ts}{y}_t^s\left({\boldsymbol{r}}_{\boldsymbol{q}}\right),\kern0.5em {c}_{ts}=\frac{\left(t+s\right)!}{4\pi {R}^{t+2}{\alpha}_{ts}}\int {T}_{far}\overline{\chi_t^s} dS $$

(B.5)

Appendix C - Relation for effective thermal conductivity in 3D model

According to the boundary condition, far field temperature is linearly varying in z direction. For equivalent inclusion, writing far field temperature in the form of Eq. (B.5),

$$ {T}_{far}=z={c}_{10}{y}_1^0(r), thus\ {c}_{10}=1; $$

(C.1)

Also, the disturbance field for equivalent inclusion which has similar form as given in Eq. (B.2) becomes,

$$ {T}_{dis\_ EI}={A}_{10}^{\ast }{Y}_1^0(r)\kern0.5em $$

(C.2)

Now ${A}_{10}^{\ast }$ can be obtained by equating temperature disturbance due to cluster of inclusions given in Eq. (16) with temperature disturbance due to equivalent inclusion given in Eq. (C.2)

$$ {A}_{10}^{\ast }=\sum \limits_{n=1}^N{A}_{10}^{(n)} $$

(C.3)

Now for single inclusion problem in case of equivalent inclusion, considering only the terms of the order 1 (t = 1) and degree 0(s = 0), Eq. (17) boils down to following relation, where second term is eliminated,

$$ \frac{\left({\alpha}^{\ast }+2\right)}{\left({\alpha}^{\ast }-1\right)\ast {\left({R}_{eff}\right)}^3}\ {A}_{10}^{\ast }=-{c}_{10},\kern1.25em where\ {\alpha}^{\ast }=\raisebox{1ex}{${k}_e$}\!\left/ \!\raisebox{-1ex}{${k}_m$}\right. $$

(C.4)

Thus, relation for effective thermal conductivity can be given from Eqs. (13) and (14) is given as,

$$ {\alpha}^{\ast }=\frac{1-2\mu }{1+\mu },\kern0.75em where\ \mu =\frac{A_{10}^{\ast }}{R_{eff}^3}, $$

(C.5)