## 1 Introduction  
During the last twenty years, several studies were carried out on the application of open cell foams as heat-exchange devices, catalytic reactors, or substitutes for traditional after-treatment honeycombs. [PERSON] et al. [1] performed the characterization of the mass-transfer properties of metallic foams for structured catalysis. For fast-diffusion limited processes, such structures were found to provide conversion efficiencies equivalent to those of traditional honeycombs, with the benefits of a reduction in the required reactor size. [PERSON] et al. [2] demonstrated that, as a function of the open cell foams geometrical features, higher momentum and mass-transfer properties could be achieved with respect to traditional honeycombs. A further development of such applications relies on the advancements of the additive layer manufacturing technology, which offer the possibility for investigating and evolving the open-cell foams to periodic open cell structures (POCS). The resulting higher flexibility in the production process leads to an optimization of the structures' shapes and to the capability of achieving peculiar features, as a function of the obtained porosity, surface area-to-volume ratio, and permeability. [PERSON] et al. [3] studied the influence of the main geometrical features of ideal cubic cell POCS, namely porosity and cell orientation, on the induced pressure drop. [PERSON] et al. [4] carried out an experimental campaign on eight different cubic cells to study the effects of different design parameters, such as the struts and cell dimensions or the angular orientation, on the pressure drop and heat transfer properties. The cells' shapes and geometrical features strongly influenced the behavior of the working fluid and the spatial distribution of the different flow fields. [PERSON] and [PERSON] [5] performed a characterization of the pressure drop, interstitial heat transfer and flow dynamics properties of four different unit cells. In particular, the octet topology resulted in the highest heat dissipation, weighted by the increment in the required pumping power. In the field of after-treatment systems (ATS) for internal combustion engines (ICEs), the open-cell foams' peculiar features and the enhancements that derive from the AM application are such that their equipment is significantly appealing for a further improvement in the after-treatment performances. In this context, [PERSON] et al. [6] performed a dedicated study on open-cell foams equipped on a small heavy-duty truck. Such an application was tested on a chassis dynamometer and the experimental results showed that comparable pollutants' conversion efficiencies, with respect to traditional honeycombs, could be obtained with a lower amount of catalyst loading. [PERSON] et al. [7] conducted a preliminary numerical investigation on the application of a cubic POCS as an innovative mixing device for selective catalytic reduction (SCR) systems. [PERSON] et al. [8] performed a comprehensive characterization of POCS structures as automotive catalysts. The observed mass transfer enhancement with respect to traditional honeycombs allows to achieve the same target pollutants' conversion with a lower wetted surface area and consequently a lower quantity of precious metals. However, as a response to the tightening emission legislations, the specific field of application that the after-treatment systems have to address is that of the pollutants' abatement efficiencies for some critical conditions. The pollutants' conversion is strongly dependent on the temperature of the after-treatment system components, commonly the three-way catalyst (TWC), diesel oxidation catalyst (DOC), or SCR devices. In the early 1990s, [PERSON] et al. [9] pointed out that the typical characteristic time of the catalyst warm-up operations is such that the registered cumulative emissions before the light-off event constitute over 70-80% of the total cumulative emissions in a test cycle.