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W. F. Huebner, D. C. Boice (SwRI), J. R. Green (JPL)
We present a mathematical model using electric analogs to simulate vertical and lateral conductive heat flow in surface layers of planetary bodies with topography. The model can also be used to determine average electric and thermal conductivities of small-scale granular mixtures (as opposed to molecular mixtures). The algorithm is general and applicable to complex compositions. Analogies between thermal and electric conductivities are basic and well known. The model uses Kirchhoff's rules for electric networks.
If a temperature difference is maintained across a solid body, the thermal energy transported per unit time and unit area, (the vector heat current per unit area, {\bf Q}), is proportional to the negative temperature gradient, such that {\bf Q} = - \kappa \nabla T. Here \kappa is the heat (or thermal) conductivity of the material. For the electric analogy we use Ohm's law. If a potential difference is maintained in a resistive (ohmic) body, the electric charge transported per unit time and unit area (the vector current density, {\bf i}) is proportional to the electric field, such that {\bf i} = \sigma {\bf E} = - \sigma \nabla V. Here \sigma is the electric conductivity (or specific conductance) of the material and V is the electric potential. With {\bf i} replacing {\bf Q} and V replacing T the parallel nature of thermal and electric conductivity is established. The thermal conductivity, \kappa, is a direct analog to the electric conductivity, \sigma.
The model will be used to verify heat flow measured through porous mixtures of ice and dust as an analog of comet matter in the laboratory. Heat flow is simulated by electric currents through a three-dimensional network of resistors with emfs representing temperatures at boundaries. We illustrate our model, for simplicity, with a two-dimensional network. Each type of material with given conductivity is represented by a corresponding value for the electric resistance. The {\it number} of each type of resistor is proportional to the relative abundance of each material type. For mixtures, resistors are selected randomly.