7 Solar cooking is one possible solution to reduce the domination of fossil fuel in the domestic sector and to bene t from renewable energy. This study assesses the solar cooking potential in Switzerland. A nodal model, based on energy balance equations of a box-type solar cooker is implemented in Matlab. Model parameters that cannot be determined experimentally or analytically are evaluated through an optimization procedure based on a Genetic Algorithm (GA). The model is able to predict the temperature of the cooking vessel with an average relative error around 5%. Based on its reliability, the model is simulated over a year for di erent locations in Switzerland in order to determine the solar cooking potential. It is characterized by a metric that represents the number of days in a year the oven could be used to cook potatoes for two persons. It is found that the cooking times of potatoes can be well predicted by an Arrhenius law with an activation energy of 74.14kJ mol . The potato cooking criterion is based on the Arrhenius equation and determines if the pot simulated temperature pro le of a particular day allows to cook potatoes. The North-East of Switzerland is the least favourable area for solar cooking with theoretically around 155 cooking days per year. Around 240 days are estimated to be suitable for cooking in the cantons of Valais and Grisons, which represents a signi cant potential for solar cooking in Switzerland. Keywords: Solar cooker, Nodal modelling, Genetic algorithm, Arrhenius law, Potatoes cooking 8 Highlights 9 New method using nodal models developed for evaluating the performance of a box-type 10 solar cooker. 11 Genetic algorithms used to estimate model parameters that cannot be determined experi- 12 mentally. 13 Metric designed to estimate the cooking days potential at the national scale of Switzerland. 14 Reproducible method means it can be generalized everywhere. 15 1. Introduction 16 Solar energy is abundant on the Earth's surface but, its low density and the important 17 mass contained in the atmosphere do not allow to naturally reach temperatures high enough 18 to cook food. An early description of a solar cooker or solar oven appears in the work of the 19 Corresponding author Email address: dasaraden.mauree@gmail.com (Dasaraden Mauree) Preprint submitted to Elsevier October 24, 2019 Swiss Physicist Horace de Saussure in 1767 (Saxena et al., 2011). He built what he called an 20 \H eliothermom etre", a wood box with cork insulation and three glass layers (de Saussure, 1784). 21 Nowadays, due to the increasing pressure on fossil energy sources, there is a renewed interest 22 in solar energy exploitation and solar cookers (Schwarzer and da Silva, 2008; Geddam et al., 23 2015; Yettou et al., 2014; Esen, 2004). The rst known prototype of the kind dates from the 24 early 1950s in India (Panwar et al., 2012). In the following years, several designs and mounting 25 con gurations have been tested and their performances assessed. Enhancement of the eciency 26 has been achieved by adding sun tracking system (Al-Soud et al., 2010). Nowadays, solar ovens 27 are still built as an insulated box with transparent glass cover but in addition they integrate, 28 most of the time, re ective surfaces (booster mirrors) to increase the performance Yettou et al. 29 (2014). The energy is absorbed through the black paint of the inner wall or redirected to an 30 absorber plate at the bottom through a re ecting inner wall. Box-type solar ovens are typically 31 used to cook food around at 100C. 32 Various parameters exist to measure the performance of box-type solar cookers (Saxena et al., 33 2011). They are determined mostly by measuring the temperatures, time and solar irradiation 34 under controlled conditions. While these parameters are useful to compare solar cooker designs 35 to each other, they do not give an indication of their potential along the year for a speci c 36 the location. For this reason, a model was developed to determine a new metric providing 37 information on the number of days per year when the solar cooker can be used for each location 38 using meteorological database. 39 Most of the work done on solar cookers is through experimentation while modelling aspects 40 that would allow a yearly simulation at a national scale have not been well developed. Solar 41 ovens are modelled by separating the oven into several components and assigning a uniform 42 temperature for each component. Heat balance equations are then discretized and solved for the 43 temperatures for each time step. Terres et al. (2014) and Guidara et al. (2017) identi ed ve 44 and six components, respectively, with solar irradiation and ambient temperature as inputs to 45 the models. Maximum relative errors with the measured temperature inside the oven inferior to 46 4% are reached. It is possible to estimate several parameters of the model by minimizing the 47 di erence between the measured and the simulated temperature inside the solar oven (Saxena 48 et al., 2011). This procedure is used by Soria-Verdugo (2015) to nd the optimal values of the 49 convective heat transfer coecients which are relatively dicult to estimate otherwise. The 50 very large majority of experiments in the literature are performed in African or South European 51 countries. The reason for this is the abundant solar irradiation and relatively high ambient 52 temperature in these regions and thus a high potential for solar cooking. Currently, solar cooking 53 technology is not very popular in Switzerland. With around 1000kWh m2 per year of solar 54 irradiation, Switzerland is indeed less propitious for the development of this technology. However, 55 this amount of solar energy is already sucient to cook food during some days of the year although 56 the full potential has not yet been assessed. Switzerland o ers many isolated areas especially 57 in the mountains where fuel supply for cooking not easy and where solar oven utilization could 58 be judicious. One way of assessing this potential would be to actually try the oven at di erent 59 locations and determine its performance with a number of criteria. This is however laborious 60 and time consuming. Another more convenient way is to use a mathematical model of the oven 61 which could simulate its behaviour under di erent conditions and assess its performance. 62 The objective of the present study is to develop a nodal model of the solar oven ULOG 63 manufactured by the SOLEMYO organisation (Association Solemyo, 2019) (a Swiss organisation 64 that develops solar ovens with natural and raw materials: wood box, wool insulation). The 65 ULOG is a simple isolated wooden box with an inclined two-layers glazing. The model must be 66 able to reproduce and predict the inside oven temperature with solar irradiation and ambient 67 temperature inputs. Experiments are done to determine the thermal and physical properties 68 2 Figure 1: Oven wall of the oven. When theories are available, the experimentally determined values are compared 69 with the result of theoretical correlations. Other tests are performed by placing the oven under 70 solar exposure and by monitoring temperatures of di erent cooker components in order to obtain 71 reference data for the model validation. An optimization algorithm is used to determine some 72 parameters of the oven that are dicult to evaluate theoretically. A new metric is also developed 73 to assess the performance of the ULOG by characterizing the oven and quantifying its possible 74 utilization. This metric represents the number of days during a year when the solar oven can 75 be used to cook food in a given region. The solar oven is simulated for a typical year with solar 76 irradiation and ambient temperature of di erent locations in Switzerland. The metric is then 77 used to geographically analyse the solar cooking potential. 78 2. Method 79 2.1. Oven description 80 The ULOG is a basic solar oven made by SOLEMYO (Association Solemyo, 2019). The walls 81 are constructed with a wooden structure lled with around 5cm of bulk sheep wool (a cheap and 82 raw insulating material) as shown in Fig. 1. The inner walls are covered with a thin aluminium 83 black foil. The glazing has an angle of 30with the horizontal and has a square area of 0.5m 84 over 0.5m. The double glazing is composed of two 3mm thick clear glasses with an interior air 85 layer of 24mm. The glass layers are inserted in the wooden frame. The base of the ULOG solar 86 oven has a dimension of 66.5 cm x 61 cm. With the re ector closed, it has a maximum height 87 of 45 cm and with the re ector completely open it reaches a maximum height of 95 cm. The 88 weight without recipients or food is 9 kg. 89 This oven has two positions: one optimized for low sun positions during winter and one more 90 adapted for the solar rays coming from high in the sky. It is possible to position it as shown 91 in Fig. 2 or on the other edge. This study has been done with the oven positioned in summer 92 mode. A re ector of the same size as the glazing can also be added to increase the solar gains in 93 the oven. It is xed at the top of the glazing in the wooden frame with two hinges. 94 2.2. Nodal model 95 The basis of a nodal analysis is to subdivide a complex system into several elemental units, 96 or nodes, which represent a physical component of the initial structure like a surface or a volume 97 (Olsommer et al., 1997). It is then assumed that the physical properties, the temperature and 98 the heat ux of a node are uniform. The nodes exchange heat through convection, conduction 99 or radiation modes. The nodes are connected in a network where each connection is described 100 by a thermal resistance that characterizes the heat transfer between the nodes. 101 3 Figure 2: ULOG oven from SOLEMYO Each of the nodes has two physical properties: a temperature Tand a capacitance C(Asso- ciates, 2000). The capacitance is de ned as the product between the mass miand the speci c heat capacity cpiof the corresponding node i: Ci=micpi: (1) The thermal resistance Rijcharacterizes the heat exchange between the nodes iandj. It is also common to refer to the conductance Gij, de ned as the inverse of the thermal resistance : Gij=1 RijwithGij=Gji: (2) The governing equations of the nodal model are the heat balance equations applied for each node. The heat transfers from all the connecting nodes jmust be taken into account. For node i, it is written as : CidTi dt=X jGij(TjTi): (3) The time derivative of the temperature is discretized with Euler schemes. To avoid problems with the stability of the solution, an implicit Euler method is preferred : CiTn+1 iTn i t=X jGij(Tn+1 jTn+1 i): (4) By xing the initial conditions T0 i, Equation 4 can be solved to determine the temperature at 102 each time step. 103 The solar oven ULOG is divided into the seven following nodes: 104 Node 1: Cooking vessel 105 Node 2: All surfaces corresponding to the inner wall 106 Node 3: All surfaces corresponding to the outer wall 107 4 Figure 3: Schematic representation of the ULOG oven and the corresponding nodal network, conductances, energy inputs (red arrows) and energy loss (blue arrows) Table 1: ULOG oven dimensions. The subscript number corresponds to the identi ed numbers as referred to in Fig.3. A1A2A3A5A6 Surface area, m2 0.1767 0.4 0.7 0.2025 0.2025 Node 4: Ambient air 108 Node 5: Outer glass pane 109 Node 6: Inner glass pane 110 Node 7: Inside air 111 They are schematically represented in Fig. 3 with the conductances characterizing the heat 112 transfer between the nodes. 113 The surface area of the other components can be found in Table 1. The red arrows represent 114 the energy inputs of the system. These inputs are modelled as solar gains Gthrough the glass 115 layers. A xed fraction of the solar gains, depending on the solar oven geometry, is absorbed by 116 the inner wall and the rest by the cooking vessel. A value of 20% has been calculated for this 117 solar cooker, based on experimental data collected at the laboratory. 118 2.3. Experimental measurements 119 This study is based upon experiments conducted at the Solar Energy and Building Physics 120 Laboratory (LESO-PB) and includes work on the optical and thermal properties of the ULOG's 121 glazing and measurement of the oven air leakage. The following section details the experimental 122 measures that have been conducted. 123 2.3.1. Temperature measurements 124 In order to validate the nodal model, an experiment is set up on the laboratory roof to 125 measure the temperature inside the oven when exposed to solar irradiation (see Fig. 4). The 126 solar oven with the cooking vessel inside is placed to face the South, away from any shadow. 127 5 Figure 4: Experimental setup for temperature measurements under solar irradiation (left) and for determination of the U-value (right) K-type thermocouples are used to record the temperature inside the oven. One thermocouple 128 is xed on the back of the cooking vessel, hidden from direct solar irradiation. It corresponds to 129 the temperature T1of the pot. The second thermocouple is placed on the bottom of the oven 130 to measure T2, the temperature of the inner wall. Its location is also chosen in order to not 131 receive direct solar irradiation. The re ector is not installed on the solar oven. This reduces 132 the complexity of the model since the re ector adds modelling parameters such as the angles 133 between the re ector, the oven and the Sun. Temperatures are recorded every 30 seconds. Several 134 measurements were done starting from March 2018, during sunny days but also when there was 135 cloud cover. Experiments usually started around 10am and ended at 5pm. This experiment was 136 performed ve times with 1L of water in the cooking vessel, on the 11th, 12th, 19th, 24th and 137 26th of April 2018. 138 2.3.2. U-values 139 The experiments detailed above are performed during transient phenomena, such as the heating up of the oven. But it is also interesting to perform experiments at stationary conditions to evaluate, for instance, the U-value of the oven's walls and window. Uis the overall heat transfer coecient. This coecient is related to the heat _Qgoing through the element with surface area Aby the relation 5 : _Q=AUT; (5) where Tis the di erence between the room temperature and the air temperature inside the 140 oven. 141 Such a test is performed in the laboratory by heating up the air inside the solar oven with 142 two electric resistances that deliver 20 W of heat each (see Fig. 4). Supports are built to hold 143 the resistances vertically in order to heat up the air and not directly the inner surface of the wall. 144 Inside air, inner wall and inner glass temperatures are recorded with the same thermocouples 145 as in the experiments described above. They reach relatively quickly a stationary state when 146 the heat losses balance the heat supply of the heaters. The air temperature inside the oven is 147 measured at two di erent locations: close to the bottom and to the top of the box. The mean 148 temperature between the two is considered as the inside air temperature. A uxmeter is then 149 used to measure the heat ux through the di erent surfaces of the oven. Measurements of the 150 heat ux are made at ve di erent instants to increase the accuracy. The ambient temperature 151 in the laboratory is constant at 22C. 152 G46andG24represent the heat losses due to the thermal bridges of the oven. They are dicult to measure but they can be deduced from the U-value experiment. The heat balance at 6 steady-state is written as : _Qheater =UwallAwallT+UglassAglassT+ ( frameLframe + ovenLoven) T:(6) The heat supplied by the heaters _Qheater equals the heat losses through the walls, the window 153 and the thermal bridges of the window frame and the oven. frame and oven are the linear 154 thermal transmittances of the thermal bridges and Lframe andLovenare the length of the thermal 155 bridges. frameLframe and ovenLovenare respectively equal to G46andG24and since the other 156 terms of Equation 31 are known, the thermal bridges can be determined. The thermal losses due 157 to the thermal bridges are equally distributed among G24andG46. 158 2.3.3. In ltration rate 159 The in ltration rate is measured by arti cially increasing the concentration of carbon dioxide in the oven and monitoring its decrease over time. Both the in ltration rate and the concentration of carbon dioxide of the room are assumed constant. In that case, the concentration of CO2in the oven follows a di usion equation whose solution corresponds to an exponential decay. By tting the measurements with a function such as aet, the air leakage mass ow _ mleakcan be found through the parameter which represents the in ltration rate. The air leakage mass ow is simply given by _mleak=airVoven 3600; (7) whereairis the air density and Voven is the air volume inside the oven. The conductance G47 is therefore evaluated as the quantity G47= _mleakcp;air; (8) withcp;air being the speci c heat capacity of air. 160 2.4. Model parameters 161 Depending on the physical heat transfer mode they represent, conductances are evaluated in di erent manners. Conductive conductance is de ned as Gij=kAi L; (9) wherekis the thermal conductivity, Aithe cross-sectional area through which heat ows and L 162 is the length between the two nodes. 163 For convective heat transfers, the following formula is used Gij=hAi; (10) wherehis the convective heat transfer coecient and Aiis the area of the node iin contact 164 with the uid. 165 The heat transfer coecient is evaluated through the Nusselt number Nuwith the relation Nu=hL k; (11) whereLis a characteristic length and kis the conductivity of the uid, air in the present case. According to Churchill and Chu (1975), convective heat transfer at the surface of a vertical plate is governed by Nu=0 B@0:825 +0:387Ra1=6  1 +0:492 Pr9=168=271 CA2 ; (12) 7 with the Prandtl number Prset to 0.7 and the Rayleigh number Ra. 166 Equation 12 is valid for the exchanges between the lateral surface of the pot and the air inside 167 as well as between the vertical inner and outer oven side walls and the air. The characteristic 168 length of these surfaces is their height. Correlations that take into account the e ect of the wind 169 on the heat transfer coecient are also available and could be used. It is indeed an important 170 element which a ects the outside convective heat transfer. Its e ect is however neglected in this 171 study. 172 From Baehr and Stephan (2011), the convective heat transfer at the lid is characterized by the following equations Nu= 0:54Ra1=4;if 104Ra107(13) Nu= 0:15Ra1=3;if 107Ra1011: (14) The characteristic length is this time the ratio between the area Aand the perimeter Pof the horizontal surface. The total heat transfer coecient between the cooking vessel and the inner air,h17is evaluated as a weighted average between the lid and the lateral surfaces contributions, as h17=Acv;vhcv;v+Acv;lidhcv;lid Acv; (15) 173 withAcv;vandhcv;vrepresenting respectively the surface area and the heat transfer coecient 174 of the cooking vessel's lateral surface. Similarly Acv;lid andhcv;lid characterize the surface and 175 the heat transfer of the lid. Acvis the total surface area of the cooking vessel. 176 Equation 12 holds for the inclined glazing with the di erence that the gravity ghas to be replaced by its component parallel to the inclined glazing. This is given by the quantity gsin 6 . From Saha et al. (2007), the inclination and the space between the two layers of the glazing are too high to neglect the convection mode. The Nusselt number characterizing the convection between the two glass layers is given by the following formula from Incropera et al. (2013): Nu= 0:42Ra1 4Pr0:012H L0:3 : (16) It is a function of the aspect ratio, the height Hand widthLof the enclosure. 177 The formulation for the radiative exchanges is slightly di erent than the conduction and convection modes. The heat ux _ qis indeed dependent on the temperatures to the power four. In order to have analytical solutions available, the surfaces are assumed to be gray di use. It means that the radiative properties of the surface such as emissivity, absorptivity, re ectance are independent of the wavelength and the direction of the irradiation (Modest, 2003). For such surfaces, net radiative heat ux _ qifrom surface iamongNsurfaces is de ned as: _qi iNX j=11 j1 Fij_qj=Eb;iNX j=1FijEb;j; i= 1;::;N: (17) Eb;iis the black body emissive power of the surface i, evaluated as iT4 i.is the Boltzmann's constant equal to 5 :67108W m2K4 andiis the emissivity of the node i.Fijis the view factor between nodes iandjand only depends on the geometry. When N= 2, one equation for each _qiis derived from Equation 17. This may be solved for _ q1by eliminating _ q2 _q1=1 T4 1F12T4 2 +1(12)F12 T4 2F21T4 1 1(11)(12)F12F21: (18) 8 Table 2: Kicoecients K0K2K4K6K8 0.8214 -2.17e-5 8.05e-9 -3.28e-12 5.36e-29 Once the view factors are determined, radiative heat uxes can be calculated with Equation 178 18. 179 For simple geometries such as two parallel at plates, the view factors are equal to 1 and Equation 18 reduces to a simple form _Qij=Gij(T4 jT4 i); (19) with the conductance given by Gij=Ai 1 i+1 j1: (20) Equation 20 is valid for the radiative exchange between the two glass layers. 180 According to Mannan (2005), most monatomic and diatomic gases are non-participating 181 media. Moreover, the relatively small air volume con ned in the oven limits the interactions 182 between the radiation and the air. The inside air is thus considered as a non-participating 183 medium. Therefore, it has no in uence on the radiative heat exchanges. 184 The nodes' capacitance are other parameters needed by the model. The cooking vessel is 185 made of iron. The speci c heat capacity of iron is taken from the literature and a balance is 186 used to know the mass. The capacitance of the water is taken into account in a similar way. 187 Due to the heterogeneous structure of the oven walls, it is not straightforward to determine their 188 capacitances. The walls are made of a thin aluminium foil on the inner side, a wood structure 189 lled with wool and a thin wood panel at the exterior. The mass of every component is estimated 190 for a particular wall. Their masses are then multiplied by their speci c heat capacity and the 191 sum of these results corresponds to the capacitance of the particular wall. The same procedure 192 is applied for each wall and the sum represents the global wall capacitance of the oven. 193 The solar gains Genter the system through the glass layers. The solar radiation absorbed 194 by the exterior wooden wall of the oven is neglected. Solar gains are characterized by the solar 195 heat gain coecient g, org-value, de ned as the ratio between the radiative ux going through 196 a transparent construction element and the total incident radiative ux (Gnansounou, 2014). 197 gis usually divided in a value for direct irradiation gdirand a value for the di use irradiation 198 gdiff. The coecient for the direct irradiation is a function of the incident angle of the solar 199 beams while gdiffis de ned as the value of gdirat 60of incidence. 200 The Zenith angle of the Sun is assumed to be such that the solar beams are always perpendic- ular to the glass window. Its position in the plan perpendicular to the glazing is then determined by the incident angle .gdiris written as a polynomial function of : gdir=K0+K2 2+K4 4+K6 6+K8 8: (21) The reference = 0is determined in the normal direction of the glass window. Experiments 201 have been previously conducted in the LESO-PB to evaluate Kiand the results are presented in 202 the Table 2. 203 The solar irradiation is divided into a direct component idirand a di use component idiff. The irradiation coming from re ection from the ground is treated with an albedo coecient Abd of 0.2. Solar gains are therefore the sum of three components: the direct irradiation, the di use 9 Figure 5: Resolution algorithm irradiation from the atmosphere and the surroundings and the re ected irradiation from the ground. The solar gains are thus evaluated as: G = gdiridir A glass +A refl +gdiffidiffAglass5 6+1 6Abd ;(22) whereA glass andAglass are respectively the apparent and the real surface area of the glass 204 window, with A glass = cos  180 Aglass.A reflis the apparent area of the re ectors, if any. The 205 factors5 6and1 6take into account the fact that the window is not horizontal but has an angle of 206 30with the horizontal. 207 idirandidiffdata are available from the LESO-PB meteorological station. The meteorolog- 208 ical station of the laboratory monitors also the ambient temperature T4and it is thus available 209 as input to the model. 210 The angles de ning the Sun's position, the Zenith and the Azimuth, are taken from the cal- 211 culator (Solar Topo, 2019). They are evaluated at the point with latitude 46.51N and longitude 212 6.63E which corresponds to the city of Lausanne. 213 In order to have a rough estimation of the e ects of the re ector on the solar cooking potential, 214 it is included in the model simulation in a simpli ed way. In a similar way to Sethi et al. (2014), 215 an optimal angle of the ULOG re ector can be determined in function of the Sun's Zenith angle. 216 It would however imply to continuously adapt the re ector angle, which considerably increases 217 the complexity of the model. To avoid this problematic, the re ector is assumed to be xed and 218 aligned with the vertical. Following the method presented by Sethi et al. (2014), this position is 219 optimal for a Sun elevation of around 30. The re ector contribution is evaluated at this optimal 220 Sun angle. In that case the apparent surface of the re ector A reflis equal to Aglasscos 6 , 221 knowing the glazing and the re ector have the same surface area. 222 The solving method is summarized in Fig. 5. 223 2.5. Optimization 224 Optimization is performed on the important parameters in order to t the model output to 225 the measurements since some of the parameters cannot be determined experimentally. Genetic 226 Algorithm (GA) is used as an optimization method (Banos et al., 2011). 227 Single objective optimization is conducted. The objective function to minimize uses the least squares method, which is the square of the di erence between the measured and the simulated cooking vessel temperature, T1;mes andT1respectively. In order to increase the validity of the 10 optimization, data of all the available measurements are given to the objective function. This yields the following expression: Obj =X n Tn 1Tn 1;Mes2 |{z} Measurement 1+ Tn 1Tn 1;Mes2 |{z} Measurement 2+ Tn 1Tn 1;Mes2 |{z} Measurement 3+:::; (23) where the index nrefers to the time step. 228 To assess the robustness of the optimization procedure, the same problem is optimized with 229 AMPL (A Mathematical Programming Language), a modelling language to describe and solve 230 large-scale optimization, among other mathematical problems (Fourer et al., 2003). The MINOS 231 solver is used for the optimization procedure. 232 Five set of temperature measurements are available and are given to the objective function. 233 Upper and lower bounds are set on the important parameters determined by the sensitivity 234 analysis. The bounds are xed as plus and minus 30% of their nominal values to give exibility 235 to the optimization and take into account the uncertainties on the calculated parameters values. 236 The only constraint is that the capacitances of the two glass layers C5andC6must be equal, 237 since the two layers are supposed to be identical. To take into account the evaporation process of 238 water taking place at the constant temperature of 100C, the temperature increase in the model 239 is limited. 240 2.6. Potato cooking 241 Potato cooking experiments have been made at four di erent temperatures: 70, 80, 90 and 242 100C. The cooking times tcare evaluated for each cooking temperatures. The water is rst 243 heated up to the set point temperature. Five medium potatoes are then immersed and this instant 244 corresponds to the beginning of the cooking time calculation. Even though objective methods 245 such as the evaluation of the Young's modulus depression (Blahovec et al., 2000) have been 246 developed to determine potato cooking, the simple test of the force required to insert a pointed 247 knife in the potatoes is used to evaluate the doneness of the potatoes and the corresponding 248 cooking time tc. 249 2.6.1. Potato days metric 250 This metric corresponds to the number of days during a year where the solar oven could be 251 used to cook food in Switzerland. A day is a \potato day", when it is possible to cook ve 252 medium potatoes in 1 L of water, corresponding to food for two persons. Two criteria must 253 be satis ed for such a day: a threshold temperature, under which it is assumed the potatoes 254 would never cook, must be reached and during a sucient time to transfer enough energy to 255 the potatoes. L of (1963) showed that the energy required for physical and chemical cooking 256 processes is small compared to the energy needed to increase food temperature and balance heat 257 losses. Thus, knowing the evolution of the water temperature is the only requirement to de ne 258 if the potatoes are cooked. 259 The cooking process of the potatoes is assumed to follow an Arrhenius type behaviour: r=Be RT: (24) The exponential term depends on the activation energy of the reaction, the universal gas constantRand the temperature T. The constant Bis a characteristic of the chemical reaction. It is common to use this theory in lifetime estimation of items under speci c constraints (Sch uler et al., 2000) and in cooking processes (Petrou et al., 2002) which both involve chemical processes. In the rst case, the rate constant ris replaced by the lifetime and by the cooking time tcin the 11 second case. With potatoes, the main reaction is the \cooking" of the starch, or gelatinization, whose activation energy must be determined. Equation 24 is thus rewritten as: tc=Be RT: (25) It must be noted that in a1 Tagainst lntcplot, Equation 25 becomes a straight line whose slope is 260 equal to the ratio  R. The cooking times evaluated for four di erent cooking temperatures are 261 plotted. Then the slope of the interpolated straight line of these four points give the activation 262 energy of the potatoes cooking process, with R xed at 8.314J molK . Once the cooking times 263 are determined, the constant Bcan be determined by Equation 25. It is assumed that this 264 constant characterizes the cooking chemical reactions and thus is similar for all potato cooking 265 experiments. Its nal value is set as the average of the four results for the four experiments. 266 To ensure the potatoes are ready, a cooking criterion CCis de ned as: CC=Ztc 0Be RTdt: (26) This quantity is similar for the four cooking tests, with variations less than 5%, and is a good indicator of the doneness of the potato. Thus for a day to be cookable, it must ful ll the following requirement:Ztend t0Be RTdtCC; (27) witht0andtendbeing respectively the starting time and ending time of the cooking period. 267 Equation 27 is integrated numerically in Matlab with the temperature pro le of T1evaluated 268 with the model. 269 The model is simulated over a year with solar and ambient data for di erent locations in 270 Switzerland and the "potato days" are evaluated for each Swiss location following the procedure 271 described previously. The software Meteonorm (Remund, 2008; Remund et al., 2010) gives access 272 to a year of data for 70 meteorological stations in Switzerland. Data are interpolated in Matlab 273 in order to have a time step of 30 seconds on which the model is based. To reduce the computing 274 time, the solar oven is simulated only from 9:30am to 18:00pm each day. The results are analysed 275 with a Geographic Information System software, QGIS, in order to represent them on a map of 276 the Swiss territory. 277 2.7. Conventional thermal performance parameters 278 Performance ratings are a useful tool for the comparison of oven types. The most commonly 279 used includes the gure of merits F1andF2developed by Mullick et al. (1987) and the cooking 280 powerPas de ned by Funk (2000). F1is determined without water load in the cooking vessel 281 and based on the maximal stagnation temperature the oven reaches. It characterizes the no-load 282 conditions and is determined using Eq. 28. 283 F1=TpsTas Hs; (28) whereTpsis maximum absorber plate temperature, Tasis ambient air temperature (at stag- 284 nation) and Hsis the insolation on a horizontal surface at the stagnation time (in W m2). 285 F2andPsare obtained from measurements when the solar oven is loaded with water. F2is 286 calculated using Eq. 29. 287 F00CR=F2=F1(MC)w A ln" 11 F1(Tw1Ta H) 11 F1(Tw2Ta H)# ; (29) 12 Figure 6: Temperature measurement - Empty pot - 22.03.2018 whereF0is heat exchange eciency factor, 0is optical eciency, CRis heat capacity ratio, 288 Mis the mass of water in kg, Cis the heat capacity of water J kg1K1,Ais aperture area, 289 is time interval in seconds, Tw1is initial temperature of water, Tw2is nal temperature of water, 290 Tais average ambient air temperature and His the average solar radiation on horizontal surfaces 291 in W m2. 292 AndPsusing Eq. 30: 293 Ps=T(MC)w t; (30) where Tis a di erence of temperature of 50C and tthe time needed to obtained it in 294 seconds. 295 3. Results 296 Results of two temperature measurements on the lab roof are presented in Fig. 6 and 7. 297 Temperatures up to 120Care reached as early as March and show the good performance of 298 ULOG despite its simple construction and raw materials. Solar gains are quite volatile. The 299 temperature of the cooking pot follows the variations with a certain capacitance which smoothes 300 the curve and delays it by just a few minutes. A time lag can indeed be observed in Fig. 6 301 between the large variations in the solar gains and T1. This time lag is related to the time 302 constant of the solar oven system. The e ect of adding inertia due to the water is evident in Fig. 303 7. The blue curve is indeed much smoother than when the cooking vessel is empty. Moreover, 304 the temperature stagnates when it reaches a few degrees above 100C, caused by the water 305 evaporation phenomenon. 306 From Fig. 6, the rst gure of merit F1, which is determined using Eq. 28, can be calculated. 307 It is found to be 0.15 m2C W1(taking into account an average solar irradiation of 660 W, 308 an exterior temperature of 15C and a maximum temperature of 116C). From Fig. 7 and using 309 Eq. 29,F2was calculated to be 0.315 . From the same set of data, a cooking power Ps=30.9 310 W was determined according to the Funk procedure, Eq. 30 with a di erence of temperature of 311 50C (between 40C and 90C), which was reached after 1h53min. 312 13 Figure 7: Temperature measurement - Pot with 1L of water - 24thApril 2018 Table 3: U-values through the ve walls of the solar oven West wall Back wall East wall Front wall Bottom wall 0.31 0.33 0.39 0.38 0.35 3.1. Model parameters 313 In this section, thanks to the experiments and measurements detailed previously, the value of 314 the parameters GijandCiare determined. It represents a rst estimate of their value through 315 analytical and experimental procedures and serves as nominal value for the sensitivity and opti- 316 mization steps. 317 3.1.1. Conductances 318 The average of the ve measurements of the U-value on the di erent surfaces of the oven are 319 presented in Table 3. The U-values are given inW m2K . The di erences in the values are mainly 320 due to heterogeneity of the insulating wool and the wooden structure being slightly di erent 321 in the ve walls. The wool thickness and density are indeed quite di erent depending on the 322 surface. The heat transfer coecient of the wall Uwallof the solar oven is calculated as the 323 average of the value of the ve walls. It yields Uwall= 0:35W m2K .G23is equal to Uwalltimes 324 the walls area Awallwhich is taken as the mean between the inner and the outer surface areas. 325 ThereforeG23= 0:19W K . 326 In a similar way that for G23, an overall heat transfer coecient for the window Uglass of 327 3W m2K can be determined. With the area of the window equal to 0.2025 [ m2], it gives for the 328 conductance G56= 0:61W K . 329 G46andG24represent the heat losses due to the thermal bridges of the oven. They are dicult to measure but they can be deduced from the U-value experiment. The heat balance at steady-state is written as _Qheater =UwallAwallT+UglassAglassT+ ( frameLframe + ovenLoven) T (31) The heat supplied by the heaters _Qheater equals the heat losses through the walls, the window and the thermal bridges of the window frame and the oven. frame and oven are the linear 14 thermal transmittances of the thermal bridges and Lframe andLovenare the length of the thermal bridges. _Qheater can be written as function of the global heat transfer coecient of the oven to get UtotAtot=UwallAwall+UglassAglass + frameLframe + ovenLoven (32) By recognizing that frameLframe and ovenLovenare respectively equal to G46andG24, since the other terms of Equation 32 are known, the following result is available G24+G46= 0:03W K (33) As a rst assumption, the thermal losses due to the thermal bridges are equally distributed 330 amongG24andG46. Thus, they are both equal to 0 :015W K . These heat losses due the thermal 331 bridges represent slightly more than 5% of the total heat losses through the envelope. 332 The in ltration rate is found to be equal to 0.5vol h . Knowing the in ltration rate , the air leakage mass ow is simply given by _mleak=airVoven 3600= 4:63106kg s (34) whereairis the air density and Vovenis the air volume inside the oven. The air properties are given in the appendices and the oven volume is equal to 0.031 [ m3]. The conductance G47is therefore evaluated as the quantity G47= _mleakcp;air = 0:005W K ; (35) withcp;air being the speci c heat capacity of air. It can be noted here that the heat losses due 333 to air leakage are assumed to be very small and as such are do not play an important role in the 334 heat balance. 335 It is assumed that the heat transfer between the node 1 and 2 is largely dominated by the conduction through the bottom of the cooking pot compared to the radiative heat transfer. The conduction through the bottom of the cooking vessel Acv;bot is evaluated with the following formula G12=A11 1 k1Acv;bot+1 hcAcv;bot+2 k2Acv;bot(36) The thermal resistance due to the interface between the pot and the inner wall is taken into 336 account with the thermal contact conductance coecient hc xed to 10W m2K from Fletcher 337 (1972).  1and  2are the thickness of the nodes 1 and 2 and are xed to 0.5 [ cm] and 15 [cm]. 338 It yieldsG12= 0:36W K 339 3.1.2. Capacitances 340 The cooking vessel is made of iron. The speci c heat capacity of iron taken from the literature 341 is presented in the appendices and a balance is used to know the mass. The result is C1= 456J K . 342 When the cooking vessel is lled with 1L of water, it corresponds to an addition of 4140J K . 343 Due to the heterogeneous structure of the oven's walls, it is not straightforward to determine 344 their capacitances. The walls are made of a thin aluminium foil on the inner side, a wood 345 structure lled with wool and a thin wood panel at the exterior. The mass of every component is 346 estimated for a particular wall. Their masses are then multiplied by their speci c heat capacity 347 and the sum of these results corresponds to the capacitance of the particular wall. The same 348 procedure is applied for each wall and the sum represents the global wall capacitance of the oven. 349 15 Figure 8: Results of optimization for Genetic Algorithm and AMPL The aluminium foil and half of the inside structure are allocated to the inner wall node while 350 the other half of the wood structure and the outer wood panel are allocated to the outer wall 351 capacitance. The results are C2= 2819 and C3= 4979J K . 352 The capacitances of the glass layers have been estimated in previous work on the glazing done 353 at the laboratory and correspond to C5=C6= 1418J K . 354 The capacitance of the inside air is evaluated by the multiplication of the volume of the oven, 355 the air density (1 :08h kg m3i ) and speci c heat (1008.7h J kgKi ). It yields C7= 34J K . The ratio 356 of solar gains allocated to the cooking pot and the inner wall is determined through the resolution 357 of the integral. It gives a value for RSurf very close to 0.2. 358 3.2. Optimization 359 In order to have con dence in the optimization results, the rst test done is the comparison 360 between the two methods: GA and AMPL. Large and similar bounds are set to the variables for 361 both GA and AMPL procedures with the same objective function. The results of the optimization 362 for two di erent days are presented in Fig. 8. The similarities between the two approaches are 363 high. For the majority of the conductances and capacitances, the two methods present di erent 364 values but for the most important parameters (identi ed by a sensitivity analysis not shown 365 here), the values are similar between the two approaches. This explains the almost identical 366 curves in Fig. 8 and gives con dence in the methods. The GA procedure is preferred for the rest 367 of the study for its simple implementation in Matlab. 368 The optimization procedure has nally been implemented with the objective function de ned 369 by Equation 23. The results for the ve sets of measurements are presented in Figs. 9-13. 370 The mean relative errors between the measured and simulated cooking vessel temperature for 371 the ve measurements are shown in the following Table 4. The simulation is in good agreement 372 with the measurements, with errors either below or around 7%. Especially for the sets from 11th373 April and 19thApril with low relative errors. The uctuations of T1in Fig. 9 and 10 are well 374 reproduced by the simulated temperature. However, for the 19th, 24thand 26thApril measure- 375 ments, the heating process is not perfectly reproduced by the optimization. The curvatures of 376 16 Figure 9: Results of optimization - 11thApril 2018 Figure 10: Results of optimization - 12thApril 2018 Table 4: Relative error on T1for the ve experiments conducted in April 2018 Measurement 11th12th19th24th26th Relative error [%] 4.08 6.81 3.81 5.52 7.22 17 Figure 11: Results of optimization - 19thApril 2018 Figure 12: Results of optimization - 24thApril 2018 18 Figure 13: Results of optimization - 26thApril 2018 Conductances,W K Capacitances,J K G12 2.38 C1 5974.8 G16 11.11 C2 2792.4 G23 0.32 C6 1843.4 G45 1.13 G56 1.22 Table 5: Optimized parameters values Figure 14: Schematic representation of the nodal net- work, conductances and energy inputs (arrows) the two curves are quite di erent. This is obvious in Fig. 13 where the simulated curve heats up 377 too quickly. 378 Based on the results from the optimization, the parameter values given in Table 5 are used 379 in the model. 380 The worst t occurs for the data set from the 26thApril. It may indicate the presence of 381 errors in these measurements. The optimization is thus tried without this set of data given to the 382 objective function. Slightly better results are then obtained, especially for the 11.04 and 12.04 383 experiments shown in Figs. 15 and 16. As can be seen in Table 6, the error is almost divided by 384 two for the rst measurement. The others are also slightly smaller than in the scenario with the 385 ve measurements given to the objective function. It is interesting to note that the optimized 386 values of the parameters are similar to the previous scenario except C2whose optimized value 387 decreases to 2119.6J K . 388 All the simulated temperatures of the oven nodes are presented in Fig. 17. While the main 389 focus was on T1, it is also interesting to analyse the other results of the model simulation. As 390 expected, the temperatures of the nodes connected with the environment T3andT5are close 391 19 Figure 15: Results of optimization - 11thApril 2018 Figure 16: Results of optimization - 12thApril 2018 Table 6: Relative error on T1for the ve experiments conducted in April 2018 Measurement 11th12th19th24th26th Relative error [%] 2.26 5.61 3.72 4.98 8.89 20 Figure 17: All simulated nodes temperatures for the data set of 11thApril 2018 to the ambient temperature. The slightly higher temperature T5indicates the higher thermal 392 losses through the glass layer. T2andT7correspond to nodes with relatively low capacitances 393 and are therefore more volatile. Due to the high conductance between the nodes 5 and 6, their 394 temperatures are closely related. Even though the optimization only considered T1, it increases 395 the con dence into the validity of the model to see the others nodes' temperature values are also 396 in an acceptable range and keeping their physical meaning. 397 The model is solved once by using the correlations for convective and radiative heat exchanges 398 given in Sect. 2.4. The heat transfer coecients are plotted in Fig. 18. h56rad: is evaluated by 399 the formula 20 and h56conv: by Equation 16. Heat transfer coecients with the ambient air are 400 quite volatile due to the varying ambient temperature. Their values around 2.5W m2K are typical 401 of processes involving natural convection. The relative high value for h56rad: in comparison to 402 h56conv: indicates the importance of radiation processes between the two glass layers. The heat 403 transfer coecients for the con ned air present also typical values for such convection processes. 404 Tests have been made with data set without water load in the cooking vessel. Due to the high 405 volatility of the solar gains and the relatively large time step compared to these variations, the 406 t tends to be more dicult. An example of such results is shown in Fig. 19. The relative error 407 is also quite low and the red curve follows the general trend of the blue curve. The simulated 408 temperature however oscillates too much and reaches too high temperatures. The heat losses 409 due the thermal bridges represent slightly more than 5% of the total heat losses through the 410 envelope. The heat losses due to air leakage are very small and do not play an important role in 411 the heat balance. 412 3.3. Potato days 413 The results of the cooking experiments are presented in Fig. 20 with the tted curve. The 414 alignment of the points in the1 Tagainst lntcplot shows a very good agreement with the Arrhenius 415 theory. From these results, the following values for the activation energy , the pre exponential 416 constantBand the cooking criterion CCare found. 417 21 Figure 18: Heat transfer coecients Figure 19: Results of optimization - 14.03.2018 Table 7: Results of the cooking experiments Activation energy,kJ mol 74:14 Pre-exponential factor, [s] 9:931014 Cooking criterion, [ ] 8:08107 22 Figure 20: Cooking times versus cooking temperature Figure 21: Results of simulation for a year without re ector The threshold temperature under which it is assumed the potatoes would never cook is xed 418 to 60C. Indeed, for this temperature, the cooking time estimated with the tted curve in Fig. 419 20 is longer than 9h. It could theoretically correspond to starting cooking in the morning and 420 having the food ready in the evening. However, in practice this is not very likely to happen. As 421 a higher cooking time is not acceptable, the threshold temperature of 60C is chosen. It also 422 corresponds to the temperature above which the gelatinization of potato starch starts (Shiotsubo, 423 1984). 424 The "potato days" are calculated for the di erent meteorological stations in Switzerland. 425 The results are then integrated in a map with QGIS and interpolated to obtain values for the 426 whole Swiss territory. The triangular interpolation (TIN) method of QGIS is used. The results 427 are presented in the following map. This map gives a rst approximation of the possible use of 428 a solar oven in Switzerland. The yellow areas represent the high potential. A maximum of more 429 than 160 "potato days" is reached in Zermatt at 1640 [ m] in the South-East of the Canton of 430 Valais. Cities, especially in French speaking Switzerland, o er also a signi cant potential. By 431 contrast, the North of the territory is less favourable to solar cooking due to the smaller amount 432 23 Figure 22: Results of simulation for a year with re ector of solar irradiation. 433 The same procedure is applied to obtain the results with the re ector. They are shown in 434 Fig. 22. A signi cant increase of the potential can be observed. A maximum of more than 240 435 "potato days" is reached at Pian Rosa at 3500 m height on the South-Eastern border of the 436 Valais. 437 4. Discussion 438 The gures of merit for the ULOG solar oven were computed. Based on previous studies, F1439 should be in the range 0.12{0.16 m2C W1whereas F2 should be in the range of 0.254{0.490. 440 Therefore, the obtained values are within the recommended range (Yettou et al., 2014; Saxena 441 et al., 2011) and compare well with other solar cookers (Harmim et al., 2012; Guidara et al., 442 2017). 443 The optimization algorithm tends to increase the capacitance of the cooking vessel-water and 444 the glazing nodes. By contrast, the value for the inner wall capacitance is lower than its nominal 445 value. This could indicate that the part of the inner wall that plays a role in the thermal inertia 446 is less than half of the wall and that the important mass for heat exchanges is represented by 447 the very rst layers of the wall structure. The conductance between the cooking pot and the 448 inner wall is considerably larger than the estimated nominal value. The radiative component 449 that has been neglected in the rst approximation could be responsible for this higher value of 450 the conductance. 451 The conductance between the pot and the inner glass layer is signi cantly larger than the other 452 conductances. It represents the radiative exchanges and such a high value probably overestimates 453 the contribution of radiative phenomena. This highlights one of the drawbacks of an optimization 454 process: the loss of physical meaning of the parameters. Due mainly to modelling errors, the 455 optimization procedure nds the best value for the parameter to minimize the objective function, 456 which can greatly diverge from the physical reality. The optimized U-value however is in good 457 agreement with the experimental one. 458 24 The model seems to be in good accordance with the experimental data in particular as 459 compared to the maximum temperature obtained. This is comparable to results from Zafar 460 et al. (2019) and the error range between the simulated and measured values are also in line 461 with previous studies (Guidara et al., 2017). The focus has mainly been on the results with a 462 water load, which represents the principal situation in which the oven will be practically used. 463 This model with high capacitance is optimized more easily than sensitive models. It tends to 464 smooth the results and attenuate the e ects of solar irradiation variations. These variations are 465 low during a sunny day without clouds. During such days, the solar gains follow a smooth curve 466 which facilitates the optimization process. On cloudy days, the solar irradiation can drastically 467 change in an instant. A large amount of this information is lost due to the time step of 30 468 seconds which is much larger than the phenomena involved in the solar irradiation uctuations. 469 Tests have been performed with smaller time steps in order to capture these sharp variations. 470 However, the data set and the computing time become very large and the tting is quite dicult. 471 It can also be noted in Figs. 9-12 that there appears to be a shared feature in the computation 472 of the temperature. A slightly lower temperature is simulated in the rst hours of the day while 473 the contrary (higher temperatures simulated) are obtained in the last hours of the day. It is 474 possible that this could be due to the use of an in ltration rate that was not temperature 475 dependent. Similar ndings have also been reported by Koinakis (2005). A xed in ltration rate 476 means that the model did not account, for example, for the varying pressure inside the box Han 477 et al. (2015). 478 The very good t of the potato cooking times in Fig. 20 demonstrates that the cooking 479 process is well described by an Arrhenius law. The cooking times of potatoes double when the 480 cooking temperature decreases by 10C. In Shiotsubo (1984), an activation energy for the 481 starch gelatinization of around 99 25kJ mol is presented. The value found in the present study 482 is slightly lower but still in good agreement with the literature. The di erence could be explained 483 by the fact that the whole potatoes cooking process is characterized in this study while only the 484 starch gelatinization reaction is considered in (Shiotsubo, 1984). This chemical reaction may be 485 dominant but it is not the only chemical process involved in cooking. 486 The results of the simulation over a year show the good potential for solar oven utilization 487 in Switzerland. It is indeed possible to cook potatoes more often than every other day. The 488 calculation of the re ector contributions to the solar gains is very simpli ed since it takes into 489 account a xed and optimal value for its apparent surface. Hence this method overestimates 490 the re ector contributions. However, it gives useful information on the order of magnitude of 491 the performance enhancement due to the re ector. The e ects of the re ector remains however 492 signi cant since it increases the number of "potato days" by around 50%. It is known that 493 the South of Switzerland receives more solar radiation and is thus more propitious for solar 494 cooking. Good performances were thus observed at high altitude and encourages the solar 495 cooking utilization in isolated locations such as a mountain refuge hut. A comparative analysis 496 of the meteorological data from Jungfraujoch (located at 3580 m) and from Pully (located at 461 497 m), revealed that although the air temperature was on average 18 K lower at higher altitudes, 498 the incoming solar irradiation was on average 31% higher at the Jungfraujoch (with above 60% 499 more irradiation in December and January). This highlights the fact that more than the ambient 500 temperature, the amount of solar radiation is the main parameter for solar cooking. Cities with 501 ambient temperatures slightly higher than their surrounding countryside are also favourable to 502 solar cooking, assuming a good solar exposure. The interpolation in the GIS induces some 503 unrealistic results. Such anomaly can be observed at the far East of the Switzerland, in the 504 Grisons or around Bern with the presence of a cross. They are characterized by sharp colour 505 variation on a straight line, which represents obviously not the physical reality but only results 506 of interpolating numerical processes. In reality, such a sharp colour variation could be caused 507 25 by shadowing e ects due to the land surface and relief. Mountains or valleys for instance could 508 cause local solar irradiation variations and lead to a decrease of "potato days". This would 509 however necessitate real irradiation data for the whole Swiss territory which is quasi impossible 510 to obtain. The interpolation is useful to get an approximation of such data but its results must 511 be interpreted with care. 512 5. Conclusions 513 This study has presented an innovative nodal model for the Swiss ULOG solar oven, which 514 predicts the cooking vessel temperature's pro le along the day with solar irradiation and ambient 515 temperature as inputs. 516 The model parameters, depending on the physical properties of the oven, are determined an- 517 alytically when correlations are available, experimentally or through an optimization procedure. 518 Overall, the model is able to predict the pot temperature with average relative errors around 519 5%. In an upcoming study, the model will be improved by using an adaptive in ltration rate 520 which will be based on the formulations proposed by Malik (1978); Han et al. (2015) and by 521 incorporating new measured data that will be collected for the solar cooker. 522 Based on these reliable results and thanks to the new metric developed, the model is used to 523 assess the solar cooking potential over the Swiss territory. The North-East of Switzerland is the 524 least favourable region for solar cooking, with theoretically around 155 cooking days per year. 525 On the other hand, almost 240 cooking days are reached in the Valais and the Grisons. Similar 526 results are obtained even for locations at high altitude. This is an encouraging conclusion, which 527 supports the initial assumption that solar cooking is suitable for remote places such as high 528 altitude areas. The study highlights a signi cant potential for solar cooking in Switzerland. 529 Nowadays, solar cooking is still anecdotal in Switzerland and the potential is far from being 530 fully exploited. Obviously, solar cooking will not completely replace standard cooking, due to its 531 long cooking times, the good solar exposure needed and habits. However solar cooking interest is 532 likely to increase and the corresponding awareness might contribute to reducing the high energy 533 consumption per household in Switzerland. 534 Acknowledgements 535 The authors would like to thank the ENAC Innoseed program for its nancial support for this 536 project. This research project has been nancially supported by the Swiss Innovation Agency 537 Innosuisse through the Swiss Competence Center for Energy Research SCCER FEEB&D. The 538 authors would also like to acknowledge the work performed by Alexandre Di evart in the concep- 539 tion of the new solar cooker. 540 Nomenclature 541 T Temperature di erence between the air inside and outside of the oven, [K] 542  Thickness of the node, [ m] 543 _mleak Air leakage mass ow, [kg s] 544 _mleak Air leakage mass ow, [kg s] 545 _Q Heat transfer rate, [W] 546 26 _q Heat ux,W m2 547 _Qheater Heat supply of the heaters, [W] 548  Emissivity, [ ] 549 Solar incident angle, [] 550 Activation energy,kJ mol 551 frame Linear thermal transmittance of the frame thermal bridge, [W mK] 552 frame Linear thermal transmittance of the frame thermal bridge, [W mK] 553 oven Linear thermal transmittance of the oven thermal bridge, [W mK] 554 oven Linear thermal transmittance of the oven thermal bridge, [W mK] 555  Density,h kg m3i 556  Density,h kg m3i 557  Stefan-Boltzmann constant,W m2K4 558 A Surface area, [ m2] 559 Abd Albedo coecient, [ ] 560 Acv;bot Bottom surface area of the cooking vessel, [ m2] 561 Acv;lid Surface area of the cooking vessel lid, [ m2] 562 Acv;v Surface area of the lateral walls of the cooking vessel, [ m2] 563 Acv Total surface area of the cooking vessel, [ m2] 564 Aglass Surface area of the glass, [ m2] 565 A glass Apparent surface area of the glass, [ m2] 566 Arefl Surface area of the re ectors, [ m2] 567 A refl Apparent surface area of the re ectors, [ m2] 568 Atot Surface area of the oven's envelope, [m2] 569 Awall Surface area of the oven walls, [ m2] 570 B Pre-exponential factor 571 Ci Capacitance, [J K] 572 cp Speci c heat capacity, [J kgK] 573 Eb;i Black body emissive power,W m2 574 F1 First gure of merit, m2C W1575 27 F2 Second gure of merit, m2C W1576 Fij View factor, [ ] 577 G Solar gains, [W] 578 g Solar heat gain coecient, [ ] 579 gdif Di use solar heat gain coecient, [ ] 580 gdir Direct solar heat gain coecient, [ ] 581 Gij Conductance, [W K] 582 h Convective heat transfer coecient, [W m2K] 583 hc Thermal contact conductance coecient,W m2K 584 hcv;lid Heat transfer coecient between the air and the lid,W m2K 585 hcv;v Heat transfer coecient between the air and the lateral walls of the cooking vessel,W m2K 586 idiff Di use solar irradiation, [W m2] 587 idir Direct solar irradiation, [W m2] 588 k Thermal conductivity, [W mK] 589 L Length, [m] 590 Lframe Length of the window frame thermal bridge, [ m] 591 Lframe Length of the window frame thermal bridge, [m] 592 Loven Length of the oven thermal bridge, [ m] 593 Loven Length of the oven thermal bridge, [m] 594 m Mass, [kg] 595 Nu Nusselt number, [ ] 596 P Cooking power, W 597 Pr Prandtl number, [ ] 598 R Universal gas constant,J molK 599 r Rate constant 600 Rij Resistance, [K W] 601 Ra Rayleigh number, [ ] 602 T Temperature, [K] 603 t0 Starting time of cooking period, [ s] 604 tc Cooking time, [ s] 605 28 T1;Mes Measured cooking vessel temperature, [ K] 606 T2;Mes Measured inner wall temperature, [ K] 607 tend Ending time of the cooking period, [ s] 608 U Overall heat transfer coecient, [W m2K] 609 Uglass Overall heat transfer coecient of the oven window, [W m2K] 610 Uwall Overall heat transfer coecient of the oven walls, [W m2K] 611 Voven Volume inside the oven, [m3] 612 Voven Volume inside the oven, [ m3] 613