**1. Introduction**

Domestic cooking appliances have significantly evolved towards a more user-friendly and efficient use during the last decades due to the importance of cooking in our daily life [1]. One of the most important elements of the cooking process is the vessel; however, its influence on the energy efficiency has not yet been widely discussed in literature, in contrast to other domestic appliances such as gas burners and induction systems [2–4].

There are few studies about the vessel influence during cooking related to thermal efficiency and bottom temperature homogenisation. Cadavid et al. [5] analysed the thermal efficiency of a pot on an electric stove using numerical simulations. Villacis et al. [4] experimentally evaluated the energy efficiency of different materials for cookware used in induction systems. Hannani et al. [6] analysed the thermal efficiency of some cookings pots using a combined experimental and neural network method. Sedighi and Dardashti [7] reported that both multilayer plates and some thermal properties, such as thermal conductivity, provide a more uniform temperature profile. Ayata et al. [8] trained a neural network to find a solution to the nonregular distribution of temperature using the most efficient thickness distribution, and Karunanithy and Shafer [9] studied the efficiency of different saucepans on various cooktops and agreed that the surface finish of the pan base significantly affects the cooking efficiency.

**Citation:** Bonet-Sánchez, B.; Cabeza-Gil, I.; Calvo, B.; Grasa, J.; Franco, C.; Llorente, S.; Martínez, M.A. A Combined Experimental-Numerical Investigation of the Thermal Efficiency of the Vessel in Domestic Induction Systems. *Mathematics* **2022**, *10*, 802. https://doi.org/10.3390/ math10050802

Academic Editor: James M. Buick

Received: 11 February 2022 Accepted: 28 February 2022 Published: 3 March 2022

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There is no study that delves into all factors involved in the properties of cooking vessels, probably due to the difficulty of conducting these tests experimentally [4]. To address this need, we have developed high-fidelity simulations with the finite element method (FEM) and designed a full factorial analysis to study the effect of the main parameters of the vessel. The finite element (FE) model was based on a previous study [10], used to analyse the thermal distribution on the bottom of a pan depending on the meat size and position in the pan.

One of the main limitations of the FE model developed at [10] was the assumption of a constant thermal conductance between the cooking vessel and the glass. The microconcavity of the vessel and the thermal-deformation of the vessel during the heating makes very complex to model accurately this thermal contact conductance. To address this issue, we explored two novel approaches: (I) including a layer of stratified air between the pan and the glass and (II) setting a variable thermal contact conductance in the interaction between the vessel and the glass along the radius, hereinafter referred to as Model I and Model II, respectively, see Figure 1a.

**Figure 1.** (**a**) Representation of the systems that form Model I (*S*<sup>1</sup> and *S*<sup>2</sup> are the same for Model II). *S*<sup>1</sup> corresponds to the system of the solid/pan, *S*<sup>2</sup> corresponds to the glass and *S*<sup>3</sup> corresponds to the air. The input heat is shown as red arrows, while outgoing heat is shown by blue arrows. *<sup>Q</sup>*˙ *pan*−*air* and *Q*˙ *air*−*glass* corresponds only to Model I. (**b**) Distribution of thermocouples in the vessel and the glass. The thermocouple (red) placed at the centre is used as input for the PI control (*Tsensor*). Blue thermocouples are placed below the glass.

The first goal of this paper was to study how the bottom of the pan affects the cooking and in detail, the influence of the contact between the vessel base and glass. For that purpose, three FE models were calibrated independently based on experimental heating tests in three different solids: two multilayer frying pans from the Würtembergische Metallwarenfabrik (WMF) and Schulte brands and a steel plate (which was used as a flat sample). Once a high-fidelity numerical model was achieved, the key parameters of the vessel were analysed through a full factorial analysis.

This paper is organised as follows. We first describe the experimental set-up in Section 2.1: the cooktop, the three vessels under investigation and the PI control algorithm used to control the temperature of the vessel. The proposed FE models and the design of the experiments (DoE) to analyse the key parameters of the vessels under heating are explained in Sections 2.2 and 2.3, respectively. This is followed by the results and discussion of the study and, finally, the main conclusions obtained are presented.
