3.1.3. Comparative between Model I and Model II

Both Model I and Model II show a good agreement with the experimental results. Overall, the results obtained for both the WMF and Schulte pans for Model II are slightly worse than those for Model I (see Tables 4 and 5). If the concavity of the vessel is known, Model I might provide more accurate results. On the other side, Model II is indispensable for simulations where the concavity of the vessel is not known. However, it contains the tedious work of obtaining the variable thermal contact conductance by an inverse analysis.

#### 3.1.4. Heat Flux Analysis

The inbound energies, which are referred to as supplied power, and outbound energies, which are referred to as heat losses, of the WMF (using the Model II approach) are shown in Figure 8. The red dotted line represents the introduced power, the coloured areas refer to the heat losses during cooking, and the dotted and continuous green lines depict the

experimental and simulation temperatures of the sensor, respectively. At the beginning of cooking, all the power is used as heat to warm up the vessel. When the temperature of the sensor reaches the target (200 ◦C), the power decreases and maintains a constant value. The power of the steady state (approximately 400–1800 s) is converted into heat losses, mostly convection and radiation losses from the walls and the upper surface of the base (in yellow and orange).

**Figure 8.** Representation of the heat losses of computational Model II of the WMF (coloured areas), the amplitude of the power density introduced (red dotted line) and the experimental and simulation sensor temperatures (dotted and continuous green lines, respectively). The blue area corresponds to conduction losses between the base of the pan and the glass. The orange and yellow areas indicate the convection and radiation losses in the base and walls of the pan, respectively.

The inbound and outbound energies of the Schulte and steel plate models are shown in the Supplemental Data. The heat losses of the three models at their maximum power level and at the end of cooking (red and pink line markers in Figure 8) are shown in Table 6. In the transient state, conduction losses to the glass prevail above convection and radiation losses. However, in the steady state, convection losses become higher than the remaining ones. The results are consistent with Cabeza-Gil et al. [10] and Cadavid et al. [5].

**Table 6.** Conduction, radiation and convection losses in the three vessels under investigation. The Schulte pan, that has the biggest concavity, presents the lowest conductivity losses during the transient state.


The efficiency of the pans is calculated during the first 400 s as the energy used to heat up the vessel divided by the supplied energy as in Karunanithy and Shafer [9]. The efficiencies of the WMF, Schulte and steel plate are 79.95%, 82.5% and 74.18%, respectively. These values are consistent with those in Karunanithy and Shafer [9] and Villacis et al. [4], although the experimental setups were different and the experimental calculations can lead to some errors due to the approximation of the average temperature of the vessel. The energy necessary to heat up the solids during the 1800 s heating tests was 0.56 kWh, 0.38 kWh and 0.28 kWh for the WMF, Schulte and steel plate, respectively. These measurements were calculated as the temporal integration of the power supplied in the microsteel layer of the vessel.

### *3.2. Design of Experiments (DoE)*

In this section, the effects of the key parameters in the cooking and pan heating, namely, conductivity (*k*), specific heat (*ce*), emissivity (), concavity (*Con*) of the vessel, and convective coefficients of the vessel (*hconv <sup>v</sup>* ) and the glass (*hconv <sup>g</sup>* ), are presented. The main responses in the heating tests, the time to reach steady state *tst*, the introduced energy *Ein* and the homogenisation along the radius in the second 60 of the cooking *Hr*, are shown through the main effects plots (see Figure 9). All results were supported by a Pareto analysis (see Supplemental Data).

**Figure 9.** Main effects plot of conductivity (*k*), specific heat (*ce*), pan and glass convective coefficients (*hv* and (*hg*)), emissivity () and concavity (*Con*) for the responses: time to reach the steady state (**a**), supplied energy (**b**) and temperature homogenisation along the radius in the second 60 of the cooking (**c**). The units of the parameters are conductivity (W/mK); specific heat (J/kgK); convective coefficients (W/m2K); emissivity (-); and concavity (-).

Regarding the time to reach the steady state, as shown in Figure 9a, conductivity has the highest influence [20], and as it increases, the time to achieve the steady state decreases. Specific heat also influences *tst*, and as it increases, more time is needed to heat the vessel. The rest of the parameters have little influence.

The most influential parameters in the supplied energy (Figure 9b) are the conductivity, vessel convective coefficient, emissivity and concavity. These results are consistent with Villacis et al. [4], Cadavid et al. [5], Newborough et al. [21] and Karunanithy and Shafer [9], who found that the efficiency of the pan depends on the pan composition and external surface emissivity.

We performed another analysis decreasing the cooking time to 400 s (see Supplemental Data), where the influence of the specific heat was not as high as expected. The effect of the supplied energy on the variable parameters was similar to that of the total lost energy during the whole cooking period (t = 1800 s); see Supplemental Data.

Lastly, concerning the temperature homogenisation along the radius after 1 min of cooking (Figure 9c), both convective coefficients and emissivity have no influence at all. The most important parameter is the conductivity, which is directly correlated with the temperature homogenisation.
