This paper is the first of set of two dealing with a multidimensional, multiphase and multiphysics model of conjugate and coupled transport phenomena during electromagnetic treatments to moist substrates. Exposure to electromagnetic energy can be controlled and optimized by providing localized convection heat and mass transfer. The model features the stationary Maxwell's equation, coupled to the transient equations of heat conduction and mass diffusion. Moreover, the stationary Navier-Stokes equations are devised, in conjunction with the energy and mass convective equations, as the dependence on the localized heat and mass convection is accounted for. Energy and vapor transport are applied regardless of the substrate interface, to exploit the advantages of a conjugate formulation. An optimized kinetic formulation is employed to deal with water phase change. Due to the complex interdependence of the various transport phenomena, a computational strategy is. set up to solve the model, by means of a finite element code which is run in a cycle of 2 consecutive sweeps, depending on the assumption of the dielectric properties of the substrate. Grid independence tests gave acceptable results for more than 0.5 M elements and more than 5 M degrees of freedom. This paper sets the grounds for the presentation of selected results, reported in Part II. (C) 2015 Elsevier Ltd. All rights reserved.