## 1 Introduction
[13] performed an adiabatic CFD analysis of a sample created by replicating a small part of the channel which was derived from scanning measurements; they showed that the calculated pressure drops were matching the experimental data by employing the \(k-\varepsilon\) turbulence model with enhanced wall treatment. [PERSON] et al. [14] performed flow visualizations on samples with a dimpled structure and pressure drop measurements, compared with numerical data, highlighting the absence of an evident, fully laminar regime even at Reynolds numbers below 230. [PERSON] and [PERSON] [15]  
Figure 1: Heat plate featuring dimpled corrugation pattern [12]. carried out a series of experiments to evaluate the thermal performance of BHPEs with dimpled corrugation, in a counter-current configuration with two channels connected by a heat-conducting paste. From CFD results, they determined that the turbulent Reynolds Stress Model provided the most accurate description of heat transfer process, with the Nusselt number showing proportionality to \(Re^{0.64}\). A comparable flow pattern to that observed in BHPEs featuring a dimpled pattern is present in Pillow Plate devices (PPHEs), which are fabricated by spot-welding thin metal sheets together and subjecting them to hydroforming to achieve the desired inflation height. Indeed, the inner channel cross-section narrows to its minimum value in the vicinity of the welds, causing flow deflection and resulting in the formation of two distinct regions: the meandering core and the recirculation zone. [PERSON] et al. [16] conducted a comprehensive literature review on PPHEs, highlighting several key insights: notably, the flexibility of adjusting the outer channel depth, which offers the advantage of reducing the required heat transfer area compared to chevron PHEs, particularly in cases where flow rates on the two sides differ significantly. [PERSON] et al. [17] performed a Large Eddy Simulation of a periodic domain representing a PPHE channel supplied with water, considering both longitudinal and transverse types based on the respective pitch dimensions: they revealed differences in the vortex structures formed in the primary flow for the two cases, as well as the Nusselt number distribution, which reaches its maximum at the stagnation point in front of the welding spots for the longitudinal arrangement. [PERSON] et al. [18] carried out CFD analyses to investigate the influence of various geometric parameters on local entropy generation in a PPHE, finding that it rises with channel height and the adoption of a transverse configuration, while the diameter of the spot weld has a negligible effect. To estimate the Nusselt number in PPHEs, [PERSON] et al. [19] developed a two-zone model based on the characteristic flow pattern, as previously described, demonstrating an accuracy within \(\pm 15\%\) when compared to CFD results, under conditions of fully turbulent and a thermally developed boundary layer. Later, [PERSON] et al. [20] numerically examined the effect of dimpled surfaces on a PPHE, showing a 2.2% improvement in the heat transfer coefficient, attributed to the continuous disturbances induced in the turbulent boundary layer, which enhanced mixing; additionally, the slightly larger cross-sectional area led to a reduction in pressure losses, resulting in an 11.2% increase in thermo-hydraulic efficiency. All the cited references pertain to single-channel simulations with a fixed wall temperature, which limits the full understanding of the actual thermal interaction between the inner and outer passages. To the best of the author's knowledge, only the study by [PERSON] and [PERSON] [21] focused on conjugate heat transfer calculations for a PPHE operating in a counter current-flow arrangement.