The effect of partial loads on the performance of a funnel solar cooker

Several solar cooker designs have been proposed over the last decades. The funnel cooker is a well-known model, representative of the category of panel-type solar cookers. From the empirical experience of real practice, it is known that the use of partial loads has an impact on the performance of the cooker. However, this effect has not yet been investigated rigorously. This work aims to fill this lack. Extensive experimental work was performed to determine the effect of partial loads on a funnel cooker thermal performance. Tests were conducted on two identical funnel cookers, in Malaga, Spain, with low sun elevation. Cookers were tracked only azimuthally. Experimental protocol was based on ASAE 580.1 Standard for better replicability. Results showed that the standardised power drops by about 15% of the original value when the water volumetric load fraction drops by 25%, for both cooker operations, with and without glass enclosure. This important reduction is explained on how the fill level of the cooking vessel affects its function as a thermal radiation receiver. Results from experimental tests were correlated into a simple formula of practical interest. Finally, a new cooking vessel design, that improves performance at partial loads, i.e., a 25.4% increase in cooker power, was proposed and tested.


Introduction
Solar cookers are devices that use solar energy to heat, cook, or pasteurise food or drink.Solar cookers are useful tools in mitigating situations of energy poverty, which affects millions of human beings [1].In addition, in industrialised countries, a larger-scale use of solar cooking would have a positive impact in reducing the carbon footprint [2].Solar cookers are also excellent educational tools [3,4] and a valuable cooking option in some emergency situations such as natural disasters or power outages [5,6].
Over decades, many different solar cookers designs have evolved.Some comprehensive reviews can be found in literature [7][8][9]10].Saxena et al. [10] have discussed on thermal and photovoltaic solar cooking systems giving attention to the associated social, cultural, and economic aspects.Most of today's solar cookers can be classified into four main categories: parabolic, panel, box, and tube cookers [11][12][13].This work focuses on the performance of panel solar cookers, which are usually low-cost devices, without heat storage media, easy to build and often portable.They use various reflective panels to direct sunlight onto a cooking vessel, usually contained in an enclosure, made of clear plastic or glass.Solar panel cookers require little tracking of the sun and allow unattended cooking.More than 40 designs of panel solar cookers are available in open domain [14].Some popular designs are the Cookit [15], HotPot [16], Haines [17,18], Copenhagen [19], and Funnel [20][21][22][23].
Solar cookers are essentially thermal machines, so their thermal performance is a key factor.It can be determined by analysis of optical and thermal characteristics of the cooker design or by experimental testing.There are several parameters commonly used for performance investigation of solar cooking systems [9], such as the figures of merit F1 and F2 [24], the opto-thermal ratio (COR) [25] or the standardised cooker power [26,27].
Several papers have recently appeared investigating different aspects that influence the thermal performance of solar panel cookers.In case of funnel type cookers, Ruivo et al. [20] experimentally found that adopting a glass lid for the cooking vessel increased the standardised cooker power [27] by 46% compared to the value when using an opaque black metal lid.Apaolaza-Pagoaga et al. [22] investigated the influence of changing the vertical position of the cooking vessel and determined the optimal trivet height to be 25 mm.Sagade et al. [21] studied the performance of funnel solar cookers at intermediate temperatures, i.e., for a range of temperature higher than water boiling point at sea level.They investigated the influence of using the glass cover on the thermal performance.Carrillo-Andrés et al. [23] used a raytracing computational model to study the optical performance and the influence of misalignments on ideal sun tracking.All these works have contributed to a better understanding of the performance of funnel-type solar cookers.
Kumar et al. [28] tested a box solar cooker with a common aluminium black plate placed at the bottom of the cooking chamber and the same box cooker design with a layer of hollow capsules distributed over the aluminium plate.The capsules were filled with a phase change material with a melting temperature of 64 • C. Experiments of testing two box cookers side by side were conducted for investigating the performance of each configuration with load distributed in four pots and also without pots.The reported results indicated that the difference between the stagnation temperatures of the aluminium plate for the two configurations of the cooker tested without pots is small.The use of a phase change material with a low melting point should be had well justified because, at a first view, the use of a phase change material with a melting point over 100 • C would make more sense to be investigated in the context of a cooking process.
Ghosh et al. [29] investigated on the impact of different glazing on the performance of a box solar cooker without load.They found that the use of a cover of single glazed low-e antimony indium oxide coating is promising for domestic usage.
There is an aspect that has not been studied so far, the influence of the load level of the cooking vessel on the cooker performance.In the case of funnel solar cookers, it is quite common to process a small amount of food or water occupying a volume smaller than the full capacity of the cooking vessel (i.e., partial load).From empirical experience of the intensive solar cooking user and co-author of current paper QS,CI95% 95% confidence interval for the standardised power (W) QS,PI95% 95% prediction interval for the standardised power (W) qS,m,50 Standardised power per unit of mass load for ΔT w,a = 50 Wind velocity (m s  Celestino Ruivo, using a partial load influences the cooker thermal performance.This effect has not yet been rigorously investigated.This work aims to fill this lack, through extensive experimental tests of two funnel solar cookers using pots with different load levels.The experimental adopted protocol is based on ASAE S580-1 Standard for a better replicability.Results from experimental tests are correlated into a simple formula of interest to the practical user.A physical interpretation of the results is made, based on how the fill level of the cooking vessel affects its function as a thermal radiation receiver.Finally, a new vessel design that improves performance at partial loads is proposed and tested.

Description of the funnel solar cooker
The funnel solar cooker uses a multifaceted reflector to concentrate Fig. 2. Assembled funnel reflector.

Table 1
Funnel cooker geometrical data summary.solar radiation on a cooking vessel that is placed inside a transparent enclosure to create a greenhouse effect.The model studied in this work is made from two identical rectangular composite panels, as shown in Fig. 1, with aluminium mirror finish and a solar reflectance of 0.85 [30].Panels are cut and assembled to form the 3D funnel shape reflector as illustrated in Fig. 2.More details on the construction and assembling of each reflector can be found in Ruivo et al. [20].Detailed geometrical data can be found in Table 1.A central axis of the funnel reflector can be defined as the intersection of the horizontal and vertical bisector planes.
The resulting angle ψ axis shown in Fig. 2 is 38 • .The intercept area is the funnel aperture area projected onto a perpendicular plane to direct beam radiation.In this case, the maximum possible intercept area, A c = 0.4995 m 2 , occurs when the sun's rays are parallel to the central axis of the reflector.Two identical funnel reflectors, named CSR1 and CSR2, were constructed and tested side by side.The two complete devices are depicted in Fig. 3.
Regarding the cooking set, the load is contained in a cylindrical, black-enamelled steel cooking pot with a glass lid, 200 mm diameter and 100 mm height, see Fig. 4a).The maximum capacity of the pot is 3 L.The mass of the pot and glass lid are 540 g and 260 g, respectively.Fig. 4b) depicts the pot inside a glass enclosure, which acts as a heat trap.The mass values of the two enclosures are 2250 g and 2207 g, respectively, for CSR1 and CSR2.These massive glass enclosures are made of recycled windows from discarded washing machines.They are 'as new' when properly cleaned.Table 1 lists some other data about these two enclosures.The differences between the mass values and shapes of these two enclosures are negligible.Even though they are not strictly identical, they were used previously in side by side experiments for testing more than one solar cooker [19][20][21][22]31].The results show that the performance of funnel solar cooker tested with these two transparent enclosures can be considered identical.As an example, in Ruivo et al. [20], experiment no.63, the difference between the measured standardised power of two identical cookers was less than 0.8%.It is important to note that using a massive glass enclosure, 5 mm thick, has a considerable impact on the total mass of the cooking set.Thus, a large amount of the solar energy captured by the funnel reflector is stored in the glass enclosure.As expected, if a greenhouse with negligible mass, such as a thin-walled glass enclosure or a plastic bag, were used instead, the cooker power would be higher.However, these alternatives have disadvantages.There is a risk of breaking a thin glass enclosure when washing it, and plastic bags can only be used a few times and may melt if they come into contact with the cooking pot.This cooker design does not require the user to be close to the cooker during whole cooking period, because it can successfully cook the food in periods of about 2 h without need of tracking.Moreover, the risk of burning is very small.Cleaning of the funnel reflector is very easy.Simple cleaning operation with water and a soft cloth is enough.

Parameters of performance
Next, some theoretical background will be presented to support discussion on the influence of the load level on the performance of the cooker.Most of the performance parameters commonly used to rate solar cookers are based on the well-known Hottel-Whillier-Bliss (HWB) [32] formulation: where Q is the rate of useful heat gain by the water, A c is the collecting area of the funnel reflector, η 0 is the optical efficiency, defined as the fraction of incident solar radiation on the collecting area that is absorbed at the pot surfaces, I is the total solar irradiance on the collecting area, and A r is the receiver area, i.e., the surfaces of the pot that absorb solar radiation.The parameter U L represents the heat loss factor, T w the water temperature, T a the ambient temperature and F ′ the heat exchange efficiency factor.The parameter F ′ represents the ratio of actual useful energy gain to the useful gain that would result if the receiver absorbing surface had been at the water temperature [32].F'multiplies both the heat loss factor U L and the optical efficiency η o .The efficiency of the cooker can be defined as the ratio of useful heat gain to the solar energy intercepted by the funnel reflector, Eq. ( 2): Combining Eqs. ( 1) and ( 2), a common form of HBW formulation is obtained: where C g = A c /A r is the cooker geometric concentration ratio.In the case of this cooker C g = 3.97.More details can be found in the work of Carrillo-Andrés et al. [23].
The pair of parameters Cg is usually determined experimental.In a typical test, a certain load inside a pot, water or other suitable fluid, is placed in the solar cooker and heats up in the sun.Solar irradiance, ambient temperature and water temperature are recorded during the experiment.Useful heat gain rate is calculated from the rate of increase in water temperature by: where m w and c w are the mass and the specific heat of water, respectively.The experimental data is used to plot Q AcI vs (Tw− Ta) I , and a linear fit gives the parameters F ′ η 0 and F ′ UL Cg .This linear formulation has some limitations [33], but it is the most frequently used to date and is the basis for some of the most widely used methods for evaluating the performance of solar cookers.For example, the recently proposed optothermal ratio performance parameter (COR) is simply related by [23,25,34]: The widely accepted ASAE S580.1 standard [26,27] protocol for testing and reporting solar cooker thermal performance also proposes a linear formulation.More details can be found in Section 2.6.

Influence of partial load on the cooker performance
In the case of funnel solar cookers, it is quite common to cook with a partial load, i.e., to process a small amount of food or water occupying a volume smaller than the full capacity of the cooking vessel.In fact, ASAE S580.1 Standard indicates a specific load ratio to be used in the tests, defined as the ratio between the water load and the intercept area of the cooker: "cookers shall have 7,000 g potable water per square meter intercept area distributed evenly between the cooking vessels supplied with the cooker.If no cooking vessels are provided, inexpensive aluminium cooking vessels painted black shall be used".In this work, this parameter is denoted as "load-to-area ratio", LAR: Funk [26] mentioned that water was chosen as working fluid in ASAE S580.1 because it resembles food in density and specific heat.He declared that "thermal performance is sensitive to loading rate", but he did not offer more details on this aspect.
In an early work by Mullick et al. [35], it was observed, for a box type cooker, that cooker performance increased with increase of water load.This result was attributed to an improvement in the ratio between the capacity of the water load and the added capacity of water, vessels, and a certain portion of the interior of the cooker.Mullick et al. [35] also investigated the effect of number of vessels, keeping the total water load constant.They found that performance increased with the number of vessels, which was attributed to an improvement in the heat-exchange efficiency factor F ′ , which indicates the effectiveness of the heat transfer from the plate or the vessel top to the contained load.As the latter results are obtained with a constant LAR, it suggests that the geometric configuration of the space occupied by the load may influence the thermal performance of the cooker.In the case of a funnel solar cooker, solar radiation hits mainly on the lid and side of the pot [23].Consequently, the absorbing surfaces increase their temperature, driving the heat to the water load.A pot full of water is much more efficient as a receiver because the temperature of irradiated surfaces in contact with water tends to be closer to that of water, as the convection heat transfer between pot walls and water is very effective.Heat transfer to the water is very effective when solar radiation impinges the outside face of the pot side below the water line, or when radiation passes through a glass lid and the water itself, and then is absorbed at the inside face of the pot side, below the water line, or at the bottom of the pot.On the opposite side, when the solar radiation impinges the side of the pot at its outside face above the water line, the heat transfer is not so effective.The metal of the pot wall acts as an extended surface like a fin, but its temperature will inevitably rises above the water temperature, causing more heat losses to the external environment.This effect is depicted in Fig. 5, that shows an infrared image of the pot made on December 17, 2021, with 1.5 kg partial water load, glass lid.The image was made 9 min after starting the a heating test of load using cooking vessel, without enclosure.It can be noticed how the side wall of the pot has a much higher temperature in the part that is above the water line.
The increase in temperature of the absorbing surface over that of water is very noticeable in the case of using an opaque lid.In the work of Ruivo et al. [20], the temperature of a black metal lid was measured, being much higher than that of the water load and easily reaching temperature differences of around 100 ℃.The associated increase in heat losses to the environment can be significant.In fact, the use of a glass lid increased standardised cooker power [27] by 46% compared to the use of a metal lid [20].
From the point of view of the HWB model, a reduced water load leads to a decrease of the heat exchange factor F ′ , but also changes in parameter U L may be possible, since natural convection and long wave radiation exchange are dependent on the surface temperature.In view of the above, it follows that predicting the influence of the load level on the cooker performance is a complex task, since very detailed optical and thermal models are necessary.An alternative approach is experimentation, then correlating the results with some significant parameter.Since the filling level of the vessel is important, it is convenient to introduce a new parameter here named "volumetric load fraction".It is defined as is the ratio of the volume occupied by the water load (V w ) to the total volume of the cooking vessel (V pot ): For a given intercept area and pot volume, water load is related with LAR and VLF, but they focus on different concepts.

Experimental set-up
Two identical funnel cooker reflectors, CSR1 and CSR2, were tested side by side as shown in Fig. 3.The experimental set-up was installed outdoors, on the roof of the School of Industrial Engineering of the University of Malaga, Spain, at a latitude of 36.9 • N. Tests were conducted from November 2019 to February 2020 and in January and February 2021.Only azimuthal sun tracking was applied, i.e., each prototype was tested without tilting adjustment.Five T-type thermocouples having and uncertainty of ± 1 • C were positioned 10 mm from the bottom of the pot to measure the temperature of the water.One sensor is in the centre of the pot and the positions of the other four sensors form a square whose vertices are all approximately 10 mm from the pot wall.The average temperature value, T w , provides a more robust measure of the water temperature than using the value of single sensor.Global solar irradiance was measured by two Hukseflux LP02 pyranometers near the cookers.One was placed horizontally, and the other was placed on a tilted plane at an angle of 40 • to the horizontal plane.Global normal solar irradiance I n and direct normal solar irradiance I bn were computed from measurements of the two pyranometers measurements, using the Liu Jordan isotropic sky model [32].The albedo was assumed to be 0.2 and the diffuse fraction of global radiation was taken from data measured in a nearby meteorological station.A Campbell Scientific CR1000 data logger recorded both global solar irradiance and temperature every minute.Wind speed and ambient temperature were measured by a dedicated Onset weather station located near the cookers.Measurement ranges and technical specifications of the used instruments are summarized in Table 2.

Test cases
The LAR value recommended by ASAE S580.1 Standard is 7 kg m − 2 .

Table 2
Instruments used to measure the solar irradiance, air speed and temperature.3. The lower ratios correspond, for example, to the amount of water needed for preparing tea or coffee for two or three people.Each water load is also related to the portion of absorbing surfaces below or above the waterline, see Fig. 6.A total of 46 individual tests have been carried out.Data for each individual test can be found in Tables A1 and A7 included in Appendix A. Part of the tests were carried out without glass enclosure to determine if the presence of this element influences the impact of partial loads on performance.As shown in Table 3, there are a total of eight test cases.Cases with glass enclosure are indicated with the letter G, whereas cases without glass enclosure are indicated with the letter N.

Data processing
The standardised power of cookers CSR1 and CSR2 for the two configurations of cooking set and for different mass values of the water load was evaluated by following mainly the protocol of ASAE S580.1 Standard [27].Water was heated from ambient temperature until a point close to the boiling point, typically around 95 • C. When starting each experiment, the pot, the lid, and the glass enclosure were at a temperature close to the ambient temperature because they had not yet been exposed to the sun.The measured values of all variables are clustered in time intervals.For each time interval Δt i , the water temperature increases by ΔT w,i and the cooker power Qi is computed using: where m w and c w are the mass and the specific heat of water, respectively.The mass of water inside the pot is assumed to be constant because, as boiling point is not achieved, the mass of water changing to vapour is negligible during a test.
As recommendation of ASAE S580.1 Standard [27], the cooker power at each time interval is corrected to a reference value for a global normal solar irradiance of 700 W m − 2 , using: The standardised cooker power of all valid observation points is then plotted against the difference between the water temperature and the ambient temperature and the following linear regression linear, relating the standardised cooker power QS and the temperature difference, is determined: The ASAE S580.1 Standard promotes the use of a single measure of performance QS,50 that is the standardised cooker power for ΔT w,a = 50 • C and a standard value solar irradiance I = 700 W m − 2 , predicted by the linear regression of Eq. ( 10).This value is deemed to be representative of the cooker power throughout the process of heating water from ambient temperature to the boiling point.The global normal solar irradiance was adopted because it leads to a fairer comparison between cooker designs, as many solar cookers can collect diffuse solar radiation, at least in some extension.Besides it is a recommendation of ASAE   9), considering 90% of 700 W m − 2 as the corresponding standard direct solar irradiance value, the obtained standardised power values, in the case of present study, would not change strongly because, according to the data published in Tables A1 to A7, the average fraction of direct solar irradiance is 0.902, being the maximum and the minimum fraction values 0.942 and 0.821.As water load changes, it is convenient to introduce a standardised power per unit of mass load, to provide a better description of the cooker performance at partial loads: qS,m,50 = QS,50 where m w is the mass of water.In this work, the 10-minute clustering time interval specified in the ASAE S580.1 Standard was found to be too large to adequately sample short duration tests.As an example, Fig. 7 shows the time series for some relevant variables measured during the experiment 109B, conducted on 29th January 2021, using funnel reflectors, CSR1 without the glass enclosure and CSR2 with the glass enclosure, with just 0.5 kg of water.This experiment started at 11:54 am (solar time) and took just about 40 min to finish.Once the data is processed, Fig. 8a) shows the results of standardised power values for the experiment 109B.Only three valid points were obtained to describe the whole heating process when using the Standard clustering time interval of 10 min.This low sampling frequency may cause a loss of information when deriving the linear regression used for estimation of the standardised power.Using a reduced interval of 5 min seems to be more adequate, Fig. 8b).However, to justify the decision on which time interval to adopt, the entire set of experiments with 0.5 kg of water load was analysed in time intervals of 3, 5 and 10 min.
Table 4 summarises the data and results of the calculations of some parameters, operating with 0.5 kg of water, for time intervals of 10, 5 and 3 min.It was found that some regression parameters have differences up to 18% when comparing the results obtained with time intervals of 10 min and 5 min.The largest difference, 18%, is observed on the slope (a) of the linear regression for the cooking set configuration not using the glass enclosure.For the same configuration, the difference on QS,0 and QS,50 are 12% and 5%, respectively.These findings suggest that information was lost when using a time interval of 10 min.When comparing the results obtained with time intervals of 5 and 3 min, those differences are practically negligible, so it is concluded that a time interval of 5 min is adequate to sample short-duration tests.
On the other hand, to validate the adequacy of a 5 min clustering time for slow tests, the same analysis is applied to the set of experiments using 2 kg water load, i.e., for a LAR of 4 kg m − 2 .Table 5 lists the calculated parameters.It can be observed that regression parameters are practically the same for time intervals of 10 and 5 min.Therefore, a time interval of 5 min is an adequate clustering time interval to process the entire range of slow and fast tests conducted in this work.Finally, confidence and prediction intervals were computed for the standardised cooker power.A confidence interval reflects the uncertainty around the mean prediction values, while a prediction interval reflects the uncertainty around a single value [36].The use of confidence and prediction intervals is considered an improvement on the ASAE S580.1 Standard data processing protocol.The R language [37] was used for the data   processing, linear model fitting, and graph production.

Results and discussion
Table 6 summarizes results for the four tested configurations N1 to N4, without glass enclosure.The total number of tests carried out is denoted by n t , being n t1 and n t2 the number of tests perfomed with devices CSR1 and CSR2, respectively.Similarly, Table 7 shows results for the four tested configurations, G1 to G4, with glass enclosure.
Figs. 9 to 12 show the regression analysis for LAR values of 1, 2, 3, and 4 kg m − 2 , respectively.A visual inspection suggests there may be a non-negligible dependency of power of the cooker on the sun elevation angle.Tests conducted with a sun elevation angle close to the central axis of the cooker (~38 • ) tends to deliver more power as can be seen, for example, in the results of the experiments 60A with CSR1 and 60A with CSR2, compared to the other tests plotted in the same Fig.11.Future works should investigate, in more detail, the influence of the sun elevation angle on the cooker performance.However, in this work, all tests were conducted within a limited range of average values of sun elevation angle, i.e., in the range of 28 to 41 • .Thus, the impact of the sun elevation angle on the performance of the cooker, for the experiments carried out in present work, is also limited.Moreover, the dispersion of data seems to be acceptable because the R 2 values comply with the ASAE S580.1 Standard, being relatively high in all tests, except for the configuration G3.In this set of tests, shown in Fig. 6 in blue colour, the R 2 value is at the limit of the Standard.
The results summarized in Table 8 show clearly that lower load ratios lead to lower cooker power values QS,50 , and QS,0 .This phenomenon has already been verified by the authors in other recent published works [17,19], as it can be seen also in Table 9.The relationship between mass load and the QS,50 and QS,0 parameters was not investigated in these two previous works [17,19].According to the plots depicted in Fig. 13, a linear trend is observed.The dotted lines shown in Fig. 13 represent the extrapolation of the linear correlations.This linear behaviour is in accordance with what was stated previously in Section 2.3.

Table 6
Summary of regression analysis of tests with different load ratios by using cooking sets without glass enclosure.The parameter QS,0 corresponds to the power when difference between water load temperature and ambient is null, ΔT w,a = 0 • C. Thus, the power QS,0 is mainly related to the optical efficiency, but heat losses have also influence on it, as explained below.When ΔT w,a = 0 • C, thermal losses from the cooking set to the environment are reduced, but

Table 10
Proposed formulas for predicting QS,0 and QS,50 as a function of VLF. they are not zero.There are pot surfaces that receive solar radiation and achieve higher temperature than that of the water.Those parts of the side wall of the pot that are above the waterline tend to reach a much higher temperature than those below the waterline, see Fig. 5.The same applies to an opaque lid.These high surface temperatures increase the rate of heat losses to the environment.Therefore, an under-filled pot is a low-efficiency radiation receiver.This explains the strong influence of load on QS,0 .In the case of QS,50 , the relationship with load is similar to that of QS,0 .This is expected, as load affects the heat exchange parameter F ′ , that multiplies both the optical and thermal losses terms in the linear model, as was explained in Section 2.3.Since the influence of partial load on the cooker performance is very strong, it is of interest to the practical user to have a simple linear formula, see Table 10, to estimate the change in performance parameters when using partial loads.
Hence, as a general conclusion, it is desirable to work with a cooking vessel as full as possible, to get the most power from the cooker.However, the standardised power qm,S,50 per mass load unit (W kg − 1 ) is still higher at partial loads, see Tables 6 and 7. So, a lower load will achieve boiling temperature faster.
Secondarily, it is interesting to analyse the impact of the glass cover on cooker performance.Results show that differences in QS,0 between cases with or without glass enclosure are small and within the uncertainty bands.So, results are not conclusive, see Fig. 13.It seems that, in cases without glass enclosure, the higher thermal losses are compensated with an increment of optical efficiency (no glass transmission losses), and a lower thermal inertia (no massive glass to be heated).On the contrary, in the case of QS,50 there is a significant influence of the glass cover.The increment in standardised power QS,50 is around 30 to 40%.Results for 1 kg water load are a bit off the mark.A possible reason is that, for that water load value, tests performed without glass enclosure were conducted with a noticeably higher sun elevation angle than those tests with glass enclosure, see Fig. 1.For the practical user, a rule of thumb can be stated: when the volumetric load fraction drops by 25%, the standardised power will drop by about 15% of the original value.This rule applies to both operations with and without glass enclosure.

Improved cooking vessel design for partial loads
A device is proposed to increase the volumetric load fraction (VLF) when using partial loads.This device is a hollow cylinder made of black metal, like a steel can, weighing 57 g.It was inserted in the centre of the cooking vessel, as depicted in Fig. 14.In the scope of the conducted experiments, it was attached to the bottom of the pot using high temperature resistant aluminium foil tape.All added elements were black painted.This is only a concept prototype.It is possible to develop a more practical design of this concept, but it is out of the scope of this work.Side-by-side tests using 1.5 kg water load, and adopting a glass enclosure, were conducted to evaluate the cooker performance with a standard vessel and with an improved vessel.The obtained results are depicted in Fig. 15.The use of this cylindrical device causes an increase of VLF from 0.50 to 0.64.Standardized power of the cooker with standard vessel was QS,50 = 54.2W. This result agrees with the result of other tests made using the same load, with similar sun elevation angle α s ≈ 30 • (experiments 56A with CSR2, 57A with CSR1 and 105B with CSR2, see Table A3).The standardized power Qs,50 increases to 68 W. This result outperforms the conventional cooking vessel by 25.4%.This achievement clearly demonstrates that the main factor influencing the performance at partial loads is the geometrical configuration of the vessel, that acts as a radiation receiver, as was explained in Section 2.3.

Conclusions and future research
This work analyses the influence of using a cooking vessel with partial loads on the performance of a funnel solar cooker.The main conclusions are: 1) Results from a total number of 48 experimental tests showed clearly that partial loads lead to significant reductions in cooker performance parameters in terms of both powers QS,50 and QS,0 , because an under filled vessel receives proportionally more radiation on surfaces above the waterline (Section 3).2) Simple correlations between the cooker powers QS,50 and QS,0 and the volumetric load fraction were derived, which are of interest to the practical user (Section 3). 3) Results also showed that the adoption of a glass enclosure increases the standardised power QS,50 by about 30 to 40%, compared to the case without glass cover (Section 3).4) A new design of cooking vessel, that improves cooker performance at partial loads was tested (Section 4).The standardized power obtained with the novel design outperform that obtained with the standard vessel by 25.4%, even if the same mass of water is adopted in both cooking vessel configuration.5) ASAE S580.1 clustering time interval size of 10 min is not adequate for sampling fast tests (~40 min duration).A smaller 5 min interval has been found to be more suitable for both slow and fast tests, see Section 2.6.6) It is recommended that a future version of the Standard should include testing at two different load levels, for example, at full load and at half load, in order to provide a more complete characterization of cooker performance under real-life operating conditions.
This work contributes to a better understanding of the factors that affect the thermal performance of solar cookers, investigating an aspect that has not been rigorously investigated before.Results are useful for practical users and also for developers of solar cookers that are looking to improve the vessel cooking design.
Intercept area of the solar cooker (m 2 ) A r Receiver area (m 2 ) a Slope of the linear regression of standardised power (W • C − 1 ) C g Cooker geometric concentration ratio (-) COR Opto-thermal ratio (m 2 • C W − 1 ) c w Specific heat of water (J • C − 1 kg − 1 ) F ′ Heat exchange efficiency factor (-) F1 First figure of merit (W m − 2 • C -1 ) F2 Second figure of merit (-) h w Water line height (m) I Solar irradiance (W m − 2 ) I bn Beam normal solar irradiance (W m − 2 ) I bn Average beam normal solar irradiance during a test (W m − 2 ) I n Global normal solar irradiance, i.e., global solar irradiance on plane perpendicular to beam radiation (W m − 2 ) I n Average global normal solar irradiance during a test (W m − 2 ) LAR Load-to-area ratio (kg m − 2 ) m w Mass of water (kg) n p Number of valid observation points for deriving the linear regression n t Number of tests n t1 Number of tests performed with cooker CSR1 n t2 Number of tests performed with cooker CSR2 Q Cooker power (W) QS Standardised cooker power (W) QS,0 Standardised cooker power for ΔT w,a = 0 • C (W) QS,50 Standardised cooker power for ΔT w,a = 50 • C (W)

Fig. 4 .
Fig. 4. Cooking set: a) without glass enclosure and b) with glass enclosure.

Fig. 5 .
Fig. 5. Thermal image of a pot with partial load during a heating test inside the funnel reflector.
-Pagoaga et al.This value does not correspond to the most common solar cooking operations in real daily practice.Moreover, the dimensions of the pot and the amount of food load adopted in practice with the funnel cooker tested are not compatible with the standard load ratio.Thus, different amounts smaller than the value recommended by the ASAE S580.1 Standard have been tested.Each test with different amount of water corresponds to a different load to area ratio (LAR) and also to a different level of load inside the cooking vessel when the same pot is used.The LAR values tested in this work are 4, 3, 2 and 1 kg m − 2 , corresponding to 2.0, 1.5, 1.0 and 0.5 kg water load, as indicated in Table

Fig. 7 .
Fig. 7. Time series of several variables during test of cookers CSR1 and CSR2 in experiment 109B by using 0.5 kg of water.

Fig. 8 .
Fig. 8. Plots of standardised power of cookers CSR1 and CSR2 for the experiment 109B with clustering time interval of a) 10 min and b) 5 min.

Table 3
Test cases considered.
Fig. 6.Water line height h w for different load levels.

Table 4
Summary of regression analysis of tests with 0.5 kg water load, for clustering time intervals of 10, 5 and 3 min.

Table 5
Summary of regression analysis of tests with 2 kg water load, for different clustering time interval of 10 and 3 min.

Table 7
Summary of regression analysis of tests with different load ratios by using cooking sets with glass enclosure.

Table 8
Summary of cooker power for the eight tested configurations.

Table 9
Performance results summary of some solar cookers tested with different water loads.

Table A1
Data of a first set of experiments carried out with a glass enclosure.

Table A4
Data of a fourth set of experiments carried out with a glass enclosure.

Table A5
Data of a first set of experiments carried out without a glass enclosure.

Table A6
Data of a second set of experiments carried out without a glass enclosure.

Table A7
Data of a recent set of experiments carried out with the standard pot and the modified pot.