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### Solid with Spherical Inclusions Effective Thermal Conductivity Equation and Calculator

Solid with Spherical Inclusions Effective Thermal Conductivity Equation and Calculator

Effective thermal conductivity of a two-phase system

Construction of two systems that both contain within a spherical region of radius R' (a) the "true" system, a medium with thermal conductivity ko, in which there are embedded n tiny spheres of thermal conductivity k, and radius R; and (b) an "equivalent" system, which is a continuum, with an effective thermal conductivity keff. Both of these systems are placed in a temperature gradient A, and both are surrounded by a medium with thermal conductivity ko.

Eq. 1
Tr = T + [ 1 - ( ( keff - ko ) / ( keff + 2 · ko ) · ( R / r ) 3 ] · A · r · cos (Θ)

Where:

T = Initial Temperature (K)
keff = Effective Thermal Conductivity (J/ft s K)
ko
= Thermal Conductivity of Solid (J/ft s K)
R =
r
= Distance from Center (ft)
A
Θ
= Position of Substance from Center (rad)
Tr
= Temperature at R (K)

Figure 1a

Figure 2a

Figures 1a and 1b thought experiment used by Maxwell to get the thermal conductivity of a composite solid: (a) the "true" discrete system, and (b) the "equivalent" continuum system.

Source:

1. Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (2002). Transport Phenomena (Second Ed.). John Wiley & Sons, Chapter: 11, Page: 371.

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