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

Heat Transfer Engineering

Thermodynamics

Solid with Spherical Inclusions Effective Thermal Conductivity Equation and Calculator

Effective thermal conductivity of a two-phase system

Preview: Solid with Spherical Inclusions Effective Thermal Conductivity Calculator

Construction of two systems that both contain within a spherical region of radius
R' (a) the "true" system, a medium with thermal conductivity *k _{o}*, 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

*k*. Both of these systems are placed in a temperature gradient A, and both are surrounded by a medium with thermal conductivity

_{eff}*k*.

_{o}Eq. 1

*T _{r} = T + [ 1 - ( ( k_{eff} - k_{o} ) / ( k_{eff} + 2 · k_{o} ) · ( R / r ) ^{3} ] · A · r · cos (Θ)*

Where:

T = Initial Temperature (K)

k_{eff} = Effective Thermal Conductivity (J/ft s K)*
k _{o}* = Thermal Conductivity of Solid (J/ft s K)

*Effective Radius (ft)*

R =

R =

*= Distance from Center (ft)*

r

r

*= Temperature Gradient (K/ft)*

A

A

*= Position of Substance from Center (rad)*

Θ

Θ

*= Temperature at R (K)*

T

T

_{r}

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:

- 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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