Thermal quadrupole approaches applied to improve heat transfer computations in multilayered materials with internal heat sources

• Transient temperature and heat flux calculation in multilayers containing heat sources with arbitrary profiles.
• Development of a source-position-based quadrupole transfer method, an admittance matrix method and an impedance matrix method.
• Precise computation over an arbitrary time scale, whatever the thermophysical properties.
• The source-position-based quadrupole transfer method is slightly more rapid.

An extension to the classical quadrupole method is proposed which allows computing temperature and heat fluxes anywhere inside a multilayer material containing localized and/or distributed heat sources. The distributed heat sources are not limited to being uniform in each layer. By using the superposition principle and through a treatment which depends on the relative position of the heat sources and the observation point, we get a closed-form analytical expression for the temperature/flux vector which yields stable results over an arbitrary time scale. This source-sampled quadrupole method is based on the transfer formulation related to a T-scheme two-port network representation. We propose another approach based on the impedance formulation related to the same T-scheme two-port network representation; it leads to a global impedance matrix formulation which provides first the heat flux vector and then the temperature vector. Alternatively we also propose an approach based on the admittance formulation related to a Π-scheme two-port network representation with admittances; it leads to a global admittance matrix formulation which provides first the temperature vector and then the heat flux vector. Both impedance and admittance matrix formulations are easier to program than the source-sampled quadrupole method but their computing time is slightly higher. The three proposed methods are illustrated on two four-layer slabs, one with a uniform heat source distribution in each layer and the other one with exponential heat source profiles.