Free-molecule heat transfer in a conservative force field between parallel surfaces

The heat flux between parallel surfaces is computed analytically assuming that heat is transferred by particles moving ballistically under the influence of a conservative force field. Particle reflection at the surfaces is governed by a Maxwell-type boundary condition. It is found that the force field can give rise to a substantial reduction, but also to an enhancement of the heat flux, depending on the ratio of the temperatures at the two surfaces. The influence of the accommodation coefficients is studied. An asymmetry introduced by the force field and/or the boundary conditions at the two surfaces causes a significant heat-flux rectification, characteristic for a thermal diode.

Figure 1

Schematic showing particles moving ballistically between two surfaces. The parameters defining the boundary conditions are indicated.

Figure 2

Dimensionless heat flux as a function of the temperature ratio for the case of diffuse reflection boundary conditions at both surfaces. The curves are labeled with ${\tilde{U}}_{\mathrm{s}}$.

Figure 3

Dimensionless heat flux as a function of the temperature ratio for ${\sigma }_1=0.5,{\sigma }_2=1$. The curves are labeled with ${\tilde{U}}_{\mathrm{s}}$.

Figure 4

Dimensionless heat flux as a function of the temperature ratio for ${\sigma }_1=1,{\sigma }_2=0.5$. The curves are labeled with ${\tilde{U}}_{\mathrm{s}}$.

Figure 5

Heat-flux ratio as a function of the temperature ratio for diffuse reflection boundary conditions at both surfaces. The curves are labeled with ${\tilde{U}}_{\mathrm{s}}$.

Figure 6

Heat-flux ratio as a function of the temperature ratio for ${\sigma }_1=0.1,{\sigma }_2=1$ (top) and ${\sigma }_1=1,{\sigma }_2=0.1$ (bottom). For the latter case, only the curves corresponding to the largest and smallest value of ${\tilde{U}}_{\mathrm{s}}$ are displayed. The curves representing intermediate values of ${\tilde{U}}_{\mathrm{s}}$ fall between these two limits.