A new formulation for convection problems entailing multiple isothermal boundaries

Many convection problems entail more than two isothermal boundaries. In previous work, a resistor-network model was proposed for this class, multi-temperature convection problems. A technique dubbed dQdT was also developed to obtain the paired convective resistances that characterize the thermal network of a multi-temperature convection problem. In the present paper an extension of the Newton law of cooling is proposed as a general formulation of multi-temperature convection in terms of multiple driving temperature differences. Most notably, the proposed formulation eliminates the need for an effective temperature difference. The formulation is characterized by functionality coefficients which give the relation between a heat transfer rate and one of the temperature differences. These coefficients can be obtained using the dQdT technique. The new formulation and the application of the dQdT technique are demonstrated for classical three-temperature convection problems. The connection between the extended Newton formulation and the resistor-network model of multi-temperature convection is also discussed. It is shown that dQdT can be used to determine the applicability of the resistor-network model of convection.