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TECHNICAL PAPER

connected through regions of varying thickness. In the first the elementary cells is likely to take place in all three orthogonal
pore-solid configuration (enclosed pores), heat transfer occurs dimensions even under imposed one-dimensional temperature
predominantly through the solid skeleton, with the primary gradient. The heat conduction at the boundaries of elementary
role of the pores being the reduction of the effective cross- cells is negligible in the direction other than the direction of
sectional area available for heat flow. In contrast, in the second heat flow and therefore ignored. The general time dependent
configuration (enclosing pores), the pore phase interrupts the one-dimensional equation of heat transfer is,
solid heat-flow path, and heat transfer must occur primarily
through the pore fluid. This results in a substantial reduction (14)
in the overall heat-transfer rate. Assumption of these two where, ρ, C and k are the density, specific heat, and thermal
types of pore geometry configuration provides a more realistic conductivity of the medium respectively. T is the temperature.
representation of heat transfer pathways and allows for an The k is assumed to be independent of temperature but may
extension of Ohm’s Law model concept to three dimensions. vary in space and is isotropic at any location. It is convenient

Within this framework, each pore type is idealized within to express the spatial and temporal co-ordinates and thermal
a corresponding unit cell each containing both solid and conductivity in a non-dimensional form by using the following
[41]
2
pore phases, as shown in Figures 14 and 15. Each unit cell is relationships as, X = x/L x ; Y = y/L y ; Z = z/L z ; l = k/k s ; τ = t(L ρ s
assumed to be cubic so that, when packed together, the cells C s )/k s , and θ = (T – T L )/(T 0 – T L ). The x, y, z are actual spatial co-
fill space without leaving additional voids. The cubic form ordinates and X, Y, Z are the corresponding non-dimensional
of the solid core within each elementary cell also facilitates spatial co-ordinates. L x , L y , L z are the dimensions of elementary
straightforward computation of porosity, assumed to be equal cells in x, y, z directions respectively. Further, as the elementary
to the porosity of the bulk material. The dimensionless length cells are cubes; L x = L y = L z = L; and k s is the conductivity, ρ s is
of each unit cell is taken as unity. The model assumes that the the density, and C s is the specific heat of the void free solid
bulk material is composed of a random assembly of such unit frame and λ is non dimensional thermal conductivity. τ is the
cells. Superposition of effects of all the modes of heat transfer non-dimensional time. T 0 and T L are the boundary temperatures
allows the effective thermal conductivity estimation of a cell along the direction of imposed temperature gradient at x = 0
under steady temperature gradient in one-dimension. Further, and x = L, respectively. θ is the non-dimensional temperature
the effects of both types of cells, when combined, provide the field. Thus, Equation 14 in non-dimensional form is,
thermal conductivity of the material. (15)
The following assumptions are adopted regarding the In steady state, Equation 15 reduces to,
homogeneity and isotropy of the medium. Although, the
elementary cells themselves are not homogeneous, overall (16)
medium is macroscopically homogeneous, as the elementary
cell, distribute themselves uniformly and randomly in the Following are the relevant boundary conditions for solving
material medium. The overall medium is isotropic and the Equation 16 in 3-D. space,
overall effective conductivity in any orthogonal direction is for X = 0, θ = 1 and for X = 1, θ = 0,
identical to that in any other direction. The elementary cells are
symmetrical about all the axes. Conduction heat transfer within for Y = 0 and Y = 1, (dθ/dY) = 0, and for Z = 0 and Z = 1, (dθ/dZ) = 0.

Figure 14: Enclosing pore representation Figure 15: Enclosed pore representations

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