Interaction of a sparce particle stratum with a constantly heated plane in presence of a transverse temperature gradient

While modelling dispersions in containers or tubes it may be necessary to find distortions brought by the suspended particles in the temperature distribution in a vessel. Essential step of such a calculation is to determine of temperature field emerging when the particles are placed near the plane wall of the vessel. For simplicity one may suppose additionally that the carrying medium is stationary and that the particles are spherical. Solving this problem, the authors replace the plane by a fictitious particle that is mirror-positioned with respect to a given one. This allows to use multipole expansion for representation of the temperature that is a harmonic function in the case discussed. The obtained solution is used to find effective heat conduction coefficient of particles’ stratum placed in a half-space bounded by a plane with constant temperature. To do this, the authors average the temperature in the medium by the particles’ positions and compare the result with the solution of reference problem about temperature distribution in a half-space with a uniform stratum of other thermal conductance. The calculation is provided under the assumption that suspended spheres are placed rarely and therefore interact only with the plane but not with each other. A correction term is obtained that must be included in the expression for heat conduction coefficient if the total medium longitude in the direction orthogonal to the plane is finite.