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.