Applied Energy 111 (2013) 1136–1146

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Applied Energy
journal homepage: www.elsevier.com/locate/apenergy

Solar cooker of the portable parabolic type incorporating heat storage

based on PCM
Antonio Lecuona a,⇑, José-Ignacio Nogueira a, Rubén Ventas a, María-del-Carmen Rodríguez-Hidalgo b,
Mathieu Legrand a
a
Universidad Carlos III de Madrid, Grupo ITEA, Dpto. de Ingeniería Térmica y de Fluidos, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain
b
Universidad Politécnica de Madrid, ETSI Navales, Dpto. de Sistemas Oceánicos y Navales, Avda. Arco de la Victoria 4, 28040 Madrid, Spain

h i g h l i g h t s

" A portable utensil for commercial paraboloid type solar cookers is proposed.
" It includes heat storage with phase change materials (PCMs).
" The utensil is stored indoors in a thermally insulating box after charging.
" A thermal 1-D model predicts its performance in sunny days.
" The set allows cooking lunch, dinner and next day the breakfast for a family.

a r t i c l e i n f o a b s t r a c t

Article history: This paper reviews relevant issues on solar cooking in order to define and evaluate an innovative layout of
Received 6 August 2012 a portable solar cooker of the standard concentrating parabolic type that incorporates a daily thermal
Received in revised form 26 January 2013 storage utensil. This utensil is formed by two conventional coaxial cylindrical cooking pots, an internal
Accepted 30 January 2013
one and a larger external one. The void space between the two coaxial pots is filled with a phase change
Available online 13 March 2013
material (PCM) forming an intermediate jacket. The ensemble is thermally simulated using 1-D finite dif-
ferences. A lumped elements model with convective heat transfer correlations is used for the internal
Keywords:
behavior of the utensil, subjected to external radiation. This numerical model is used to study its tran-
Solar cooking
Portable parabolic solar cooker
sient behavior for the climatic conditions of Madrid, and validated with experimental data. Two options
PCM have been checked as possible PCMs: technical grade paraffin and erythritol. The results indicate that
Heat storage cooking the lunch for a family is possible simultaneously with heat storage along the day. Keeping after-
LHTES wards the utensil inside an insulating box indoors allows cooking the dinner with the retained heat and
Numerical model also the next day breakfast. This expands the applicability of solar cooking and sustains the possibility of
all the day around cooking using solar energy with a low inventory cost.
Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction often imply a relevant amount of work and health hazards as pop-
ulation lacks the access to modern forms of energy or they are too
The energy needs for cooking in developed countries are gener- expensive for their incomes [1,2]. The world population involved in
ally of minor importance compared with other domestic needs, this later case is huge, in the order of 2000 million people [3,4].
such as electronic appliances, lighting, washing, heat, and cold pro- This population mainly cooks with wood or barely processed bio-
duction for space acclimatization. In contrast, for many less devel- mass, implying this large times for collection and preparation,
oped countries cooking becomes the main energy consumption as mainly in extra-urban environments.
covering primary needs is the focus of a large percentage of the Burning wood or dung indoors by this population generates
population. In addition, the energy sources used in these countries much smoke. Particles and toxic compounds are contained in the
fumes, generating damages and illnesses on eyes and on the respi-
⇑ Corresponding author. Tel.: +34 916249475; fax: +34 916249430. ratory track [5]. According to some authors they are the origin in
E-mail addresses: lecuona@ing.uc3m.es (A. Lecuona), goriba@ing.uc3m.es (J.-I. the order of 1.5–3.1 million premature deaths worldwide per year
Nogueira), rventas@ing.uc3m.es (R. Ventas), mariadelcarmen.rodriguez.hidal- [2,4,5] and even larger figures [1]. Accidental fires and burns are
go@upm.es (María-del-Carmen Rodríguez-Hidalgo), mlegrand@ing.uc3m.es (M. also an additional risk.
Legrand).

0306-2619/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.apenergy.2013.01.083
 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146 1137

Glossary

A transversal area for heat transfer (m2) Greek

Aap aperture area of the reflector (m2) a absorptivity of utensil surface (–)
c specific heat (J kg1 K1) b volumetric expansion coefficient (K1)
Dap diameter of the aperture area of the reflector (m) DTm temperature interval for phase change of PCM (K)
F0 modifier for overall heat transfer coefficient (–) e external surface emissivity (–)
f focal distance (m) go optical efficiency = rsca (–)
Gb beam solar irradiance (W m2) c fraction of transmitted rays captured by utensil (–)
g acceleration of gravity (m s2) l dynamic viscosity (Pa s)
h heat transfer coefficient (W m2 K1) q density (kg/m3)
ie external element index, 6 outdoors, 11 indoors (–) r Stefan–Boltzmann constant (W m2 K4)
Kv evaporation scale factor (–) s transmissivity from reflector to utensil surface (–)
k heat conductivity (W m1 K1)
keq equivalent heat conductivity (W m1 K1) Subscript
Lm specific melting heat of PCM > 0 (J kg1) amb ambient
l thickness of element (m) box for insulating box
M mass (kg) co food cooking
Nu Nusselt number (–) coo cooling test
Qc heat power lost by convection (W) er erythritol
Qco food cooking heat power (W) e external, either of the utensil or box
Qr heat power lost by radiation by the external surface (W) f melted PCM
Qs solar heat power received on utensil external surface fo food
(W) for forced convection
Qv cooking medium evaporation heat power (W) free free convection
r reflectivity (–) i element index, =1 for inner most
s exponent for evaporation rate estimation (–) l liquid PCM
T temperature (K) m melting
Tend,1 end of breakfast cooking temperature, i = 1 (K) p paraffin
Tm PCM melting temperature (K) ou outer pot
Tsat atmospheric saturation temperature (K) pot pot
Tsat,c empirical correction of Tsat (K) s solid PCM
t true solar time (s) sc start of cooking
to time for outdoors to indoors move (s) w water
tcoo e1 over-temperature reduction time (s)
U overall heat transfer coefficient (W m2 K1)

In wide areas of most of the economically developing coun- 2. Technology overview and need of a novel storage solar
tries there is plenty of solar energy and incidentally they are cooker
of dry weather, so that wood is scarce. Solar cookers can be
used in these areas for water sterilization, for cooking and 2.1. Solar cooking
even for preserving and drying food, not neglecting industrial
and medical uses [3]. These devices can alleviate families from The possibility of solar cooking is primarily conditioned by the
the cooking related problems associated with wood and bio- required amount of energy. 1.4 MJ of final energy has been consid-
mass in a sustainable way. The potential of use is very large; ered as the representative family meal cooking needs in India using
in India Purohit et al. [6] estimate it as 75 million units of so- a solar box cooker [9] and similar values are given for more general
lar box cookers despite their use is only supplementary to cases in [10,11].
other cooking options because of the intermittency of sun. So- The heat required for cooking is the addition of the sensible heat
lar cooking is also a good resource for refugees’ camps and for to reach the desired temperature inside the food, a minor heat frac-
near zero energy advanced buildings. There have been many tion required for the culinary transformation (simmering) and the
experiences on its use, e.g. [1–3,7]; the research open literature heat losses during the time required for both processes. Boiling
about the subject is ample, but still social and technical barri- sometimes helps in agitating the food and eliminates extra water
ers limit their spread, e.g. [8]. of some recipes, but it constitutes a large latent heat loss. The solar
This enormous potential of use means that any development cooker must be designed accordingly if this last process is required.
that results in an easier, safer or more economical practice is of One solution to this loss is pressure cooking, at the expense of an
paramount relevance. The device proposed in this paper take steps increased cost [12,13].
in these three lines. It is economically feasible because its con-
struction is almost straight ahead with off the shelf materials in
hardware stores. The heat accumulation makes easier to cook with 2.2. The parabolic solar cooker
some solar intermittence and even allows for extended indoor
operation. The type of solar collection used, combined with the There are many varieties of solar cookers [14–18], from the
heat accumulation capability, make possible that cooks are more very cheap to sophisticate ones and consequently non suitable
inclined to briefly cut the solar irradiation over the utensil to stir for low income economies. Although the ‘‘parabolic cooker’’
or inspect the cooking. This reduces the hazards due to unwanted [19] is not one of the most affordable models it can be used
eye or hands exposition to concentrated insolation. in less developed regions owing to its reasonable cost and high
1138 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146

throughput. This cooker type, Fig. 1a, was developed between surface. Thus it is in direct contact with the atmosphere. This
1950 and 1960, e.g. [18,20,21] although its improvement and implies much higher infrared radiation and convective thermal
characterization still continues today [22,23]. Some varieties of losses than those other types of solar cookers that take advan-
its basic design allow local production using also locally avail- tage of the greenhouse effect, such as the solar box cooker
able materials and tools in a participative way, which promotes [14,15,17,18]. These higher losses are compensated by the larger
its enduring use, e.g. [24,25]. It has demonstrated ability to boil sun colleting aperture [22]. One current design is the deep
and cook fast. These characteristics are appropriate for some tra- paraboloid short focused (f < Dap/2) SK-14 model and evolutions,
ditional recipes, ease its implementation and also facilitate its e.g. [26], which has an aperture diameter Dap = 1.4 m [16]. This
implementation in urban areas where time is scarce and where concept implies a large heating power and a higher cooking tem-
cooking at short notice is important [8]. It is robust and durable. perature, allowing baking, roasting and frying in sunny days
One disadvantage is that it needs to be aligned with the sun [19].
every 15–20 min, although this is compatible with food care, Portable solar cookers are an option for the low income commu-
stirring and ingredient addition. The parabolic cooker is based nities and they are compatible with the local tradition of cooking
on the concentration of only the beam sun radiation on the outdoors. They are also convenient for implementing solar cooking
external surface of a cooking utensil located in the focal point, in high-rise buildings as it is possible to install them in flat roofs or
smaller in size (in the order of 1/10) than the aperture area of balconies. Wind, privacy and sometimes hygiene [12] are not com-
an axis-symmetric reflecting paraboloid. The utensil, pot or patible with this, so that cooking indoors can be an advantage
pan, is generally a conventional cooking utensil with a blackened worth considering [7,17].

Fig. 1. (a) Cooker A in operation. (b) Focused sun on utensil. (c) Inner pot view with water.
 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146 1139

2.3. The storage option for a solar cooker ough for immediate use under normal conditions, as well as water,
e.g. [31,46]. Accordingly, the objective established in Section 5 of
Portable heat storage during the day would allow cooking out- this paper is that this temperature is reached after cooking a rep-
doors simultaneously to heat storage and later cook indoors in the resentative amount of food for three consecutive family hot meals.
late afternoon with no sun. Indoors cooking early next morning The food representative load selected for each meal is Mco = 1 kg of
would be also possible if the heat retention during the night is suf- cold water, coherent with the work of other researchers [23,47]. An
ficient for the heat storage to lasts enough. The heat delivery for additional load, in the order of four times larger, has to be added to
the breakfast must be fast enough for cooking successfully. This account for the usual practice of surrounding the food with a cook-
work proposes a way of integrating this concept in a compatible ing medium. This cooking medium is mostly a water based fluid, so
way with normal the advantages of fast non-storage solar cooking, that for modeling purposes a fixed load of water is considered as
resulting in an enhancement of the versatility of the parabolic solar representative inside the cooking utensil during sun exposure
cooker. In addition, the simultaneous heat storage eliminates fail- and storage. According to the well-known performances of the par-
ures in cooking during cloudy intervals. abolic cooker this water would reach boiling conditions on sunny
The proposal in this work modifies a pot to include a small tank days in less than 1 h implying the vapor generated afterwards an
for sensible and even latent heat storage as a possibility of produc- energy loss when cooking. Because of that and avoiding the extra
ing affordable heat storage and retention for later use. This concept expenditure that a pressure vessel would imply, there is literature
offers a lower cost and simplicity than other proposals that involve where edible oil is considered as alternative cooking medium, such
indirect cooking, which on the other hand offer a very convenient as in [48]. In Section 5 another alternative is proposed, keeping the
use when conditions are appropriate for such installations [17,27– inner pot empty after lunch and dinner cooking, until next day
31]. breakfast cooking is performed. Elsewhere the cooking time is con-
Due to the high thermal losses of a pot exposed to air draughts, sidered the time for sensible heating plus half an hour for simmer-
thermal insulation has to be used if the objective is to retain stored ing, according to [9], thus in this work 1 h will be conservatively
heat. Baskets filled with cloths, hay or wool, are currently used in considered as the time for cooking.
some communities for covering the pots to prolong cooking in-
doors, e.g. [32] but also to keep heat. In this work, an inexpensive 2.6. A proposal from the knowledge basis
insulating box is proposed for heat retention during the late after-
noon and night. Up to the knowledge of the authors of this paper, the herewith
proposed storing utensil has not been considered in the open liter-
2.4. The PCM alternatives ature, neither characterized. Even in recent reviews no similar de-
vice can be found, e.g. [17,18]. In previous studies some different
Filling the small tank in the modified pot with a phase change designs for storage have been studied, which have been reviewed
materials (PCMs) completes the proposal of this work. These mate- recently in these two papers and by Sharma et al. [49]. Those stud-
rials are now subject of intense research owing to the high density ies offer an essential basis of knowledge to further advances. For
of the stored energy, e.g. [17,18] and more specifically [33–36]. this reason, the most relevant for this study are briefly discussed
Among them paraffins from the oil industry refinery are safe to in what follows with the aim to support the thermal study pre-
use, non-toxic, stable [37], non-corrosive, hydrophobic, widely sented in this paper.
available, and cheap. Thus, they are suitable for our aim. They Domanski et al. [50] studied a box-type cooker with an inner
can be tailored to melt up to about 110 °C, so that heat transfer storage of 2.0 kg of magnesium nitrate hexahydrate (melting tem-
to the cooking food is fast when solidifying, even for boiling but perature Tm = 89 °C) as PCM, allowing cooking during 1 h off-sun-
not for frying. Their melting heat is modest but valuable, shine with Mfo = 1 kg of water as load. Tm higher than 100 °C for
Lm  120–170 kJ/kg, losing about 20 kJ/kg after thermal cycling, the PCM is suggested. In [51] the same issue is raised, but 2 kg of
according to [38]. These values are approximate and may be differ- a lower melting temperature PCM (acetamide Tm = 82 °C) are used
ent as the nature of paraffins is variable. Organic materials from a in a box-type solar cooker to successfully cook in the evening. The
biological origin are offering competitive performances [39], seem- PCM cylindrical tank includes eight fins to enhance heat transfer; it
ingly already commercial and of renewable origin. The low heat surrounds the cooking vessel which contains Mfo = 0.75 kg of
conductivity of paraffins and the alike can be enhanced by adding water. The authors suggested that a PCM with Tm of 105 °C to
heat conducting particles [40], such as graphite, adding a conduc- 110 °C would allow cooking during the night with their apparatus.
tive solid matrix [41,42], and even carbon nanotubes [43], although Budhi et al. [52] responded to these suggestions and tested a box-
convection can be negatively affected. Alternatively, metallic fins type solar cooker of 50 cm  50 cm solar aperture during the win-
or rings can be installed inside the tank with different configura- ter season in India, with 4 kg of acetanilide as PCM (Tm = 118 °C,
tions, e.g. [33,36,44,45]; even steel wool can be added. Technical melting heat Lm = 222 kJ/kg) and boosting solar heat with three
paraffins do not suffer from the super-cooling (also called sub- external plane reflectors. In summer a single reflector and
cooling) effect to the same extent that some other PCMs do, e.g. 2.25 kg of PCM was suitable. The results showed that Mfo = 1 kg
[38]. of food could be cooked as late as 8:00 pm in winter with no neg-
Paraffins have been recommended for PCM storage jointly with ative effect of storage on simultaneous cooking of the lunch. All
erythritol by Shukla et al. [38]. PCMs of higher melting tempera- these experiments were performed outdoors. We arrive as a preli-
tures allow faster cooking, but solar heat collection would be of minary conclusion that Tm higher than 100 °C needs to be explored
lower efficiency due to losses; moreover, melting during winter and also a different layout in winter than in summer seems neces-
would be difficult. sary for those regions where winter is characterized by both a low-
er solar irradiation an lower temperatures.
2.5. Design variables Although other tests are possible, in this paper the capacity of
the proposed layout of the parabolic cooker with thermal storage
One of the main limitations of solar cooking is the small heating for cooking three meals is evaluated by means of a calibrated
capability. Taking this into account, the minimum requirement numerical model. Initially, the member of the family in charge of
that has to be pursued is to reach 70 °C in the cooked food, as it cooking can store heat along the day, including lunch cooking. La-
is widely recognized that when reaching 70 °C food is sterilized en- ter, the dinner can be cooked with the stored heat indoors, retain-
1140 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146

ing the heat with an insulation box. This would allow cooking the
breakfast next morning.
In addition to paraffin other PCMs that seem interesting for the
herewith purpose are sugar alcohols due to their food compatibil-
ity for the case of leaking. A possible option is erythritol that offers
Tm = 118 °C [38,53]. It shows a super-cooling effect (decrease of
solidifying temperature) of 14 °C, according to [38] and a loss of
35–40 kJ/kg in Lm and a loss of 9–10 °C in Tm after thermal cycling.
Although it is notorious for its non-predictable degree of super-
cooling its properties can be considered as representatives of high-
er melting temperature PCMs for the purpose of this study. The
reduction of the super-cooling effect of this particular PCM by
materials technology or a thermally equivalent alternative has to
be confirmed. It must be taken into account that it is more expen-
sive than paraffins, but its mass production as food sweetener is
reducing its price substantially. In present times it can be acquired
in small quantities for less than 10 €/kg, food grade. Sharma et al.
[54] stored heat in 45 kg of commercial grade erythritol. The of-
fered enthalpy curve for his PCM seems to extend complete melt-
ing up to 130 °C.
The valuable experience of the references above cited, among
others, allows defining the rationale of the scheme proposed in
Section 3.
Within the described scenario, the main objective of this work
is to offer a concept: a simple portable utensil made with commer-
cial pots for its use with commercial parabolic solar cookers aiming
at family use. It includes heat storage in a PCM, but it is able to
simultaneously cook outdoors. Storing the pot in a heat retention
box in the afternoon could allow cooking the dinner several hours
later and the next day breakfast in due time, about nine further
hours later. In order to ascertain these capabilities a numerical 1-
D model is developed to characterize its performances and allow-
ing for optimization. Some parameters are experimentally cali-
brated and validated to increase the model accuracy and reliability.

3. Experimental setup

Two commercial paraboloid based solar cookers have been se-

lected for the present study as representative. Cooker A is of Chi-
nese manufacturing [55] with f = 0.52 m, Dap = 1.4 m, Fig. 1. The
reflector is constructed from six sectors, formed by identical
stamped thin steel sheets, apparently as a byproduct of satellite
antennas manufacturing. They are covered by highly reflecting
adhesive plastic tape. It was modified to allow a repetitive test
campaign by substituting the supporting platform by a more rigid
one, including a hanging fixture for the cooking utensil, Fig. 1. It
Fig. 2. (a) Cooker B in operation showing the focused sun on the utensil. (b)
concentrates most of the sun rays on a spot of a diameter smaller Insulating box closed and instrumentation.
than 0.25 m. On the other hand, Cooker B is a variety of the short
focused SK-14 [26] with identical aperture diameter and
f = 0.36 m, so that the focus point is inside the paraboloid, Fig. 2. using their center handles to form a double skin removable cover
The reflector is constructed from 24 anodized reflecting aluminum with a gap between them. This is helpful for both thermal insula-
strips getting an unsupported flexed shape. The reflector spreads tion and steam retention.
rays around the pot in what is called ‘‘solar flames’’. Actually for Temperatures were measured using handheld indicators of
low solar altitude angle the solar rays illuminate the pot lid, thus 0.1 °C resolution for type T thermocouples, which are of £
spreading heat input. It was tested without modifications. Table 1 0.5 mm bead. The measuring chain was calibrated to about
summarizes the basic parameters of both cookers. ±0.7 °C uncertainty at 95% probability confidence interval.
A prototype of the proposed utensil has been built. It is formed The sun irradiation was measured using a Mac-SolarÒ handheld
on the basis of two stainless steel commercial pots, Fig. 1. The open apparatus with 2% nominal uncertainty. The diffuse component
sides of either pot are soldered to two thin flanges that are bolted was measured locally using a small screen to shade the sun just
together with a gasket to close the peripheral PCM tank, Fig. 1c. It over the sensor and afterwards it was subtracted from the global
was externally blackened with technical paint. The upper flange measurement. The diffuse component was checked against two
incorporates two diametrically opposed vertical probes for mea- pyranometers measurements (one fully open and the other with
suring the PCM temperature with thermocouples. It incorporates a band to stop direct radiation) and a sky model, giving a ±20%
also a safety pressure relief valve meant for thermal storage mate- uncertainty at 95% probability confidence interval. Only clear days
rials of high vapor pressure. Both pot lids were bolted together without clouds were selected for measurements.
 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146 1141

Table 1
Parameters for both solar cookers.

Pots Reflector
Inner pot inner height, 14 cm Aperture, Aap = 1.5 m2
Outer pot inner height, 17 cm Optical eff. ICO-GEN (Cooker A) go = 0.5 (experimental)a
Inner pot inner diam., 22 cm Optical eff. SK-14 (Cooker B), go = 0.45 (experimental)a
Outer pot inner diam., 28.5 cm Paraffin
Filling fraction of cooking medium, 75% PC, Tm,p = 100 °C
Filling fraction of PCM, 80% PC heat, Lm,p = 140 kJ kg1
Water representing cooking medium, 4.0 dm3 Heat conductivities ks,p = 0.21; kf,p = 0.2 W m2 K1
PCM volume, CASE I 3.41 dm3 CASE II (synthetic) 6.9 dm3 Densities qs,p = 880; qf,p = 770 kg m3
PCM symmetric melting interval DTm = 20 °C Heat capacity cs,p = 1.8; cf,p = 2.4 kJ kg1 K1
PCM element thickness l2–6 = 6.88 mm Liquid thermal expansion bp = 0.76  103 K1 (aprox.)
Pot external emissivity, epot = 0.95 Liquid viscosity lp = 4.90  102 Pa s
Pot external surface Ae,pot = 0.31 m2 Erythritol
Mass of empty utensil, M = 5.2 kg PC, Tm,er = 118 °C
Cooking starting times: 05:00, 12:00, 20:00 h PC heat, Lm,er = 340 kJ kg1
Box Heat conductivities ks,er = 0.733; kf,er = 0.326 W m2 K1
Element thickness, l7–11 = 3.2 cm Heat capacities cs,er = 1.38; cf,er = 2.76 kJ kg1 K1
Insulation conductivity, kbox = 0.08 W K1 m1 Densities qs,er = 1480; qf,er = 1300 kg m3
External surface, Ae,box = 1.26 m2 Liquid thermal expansion ber = 0.67  103 K1 (estimated)
External emissivity, ebox = 0.2 Liquid viscosity ler = 102 Pa s
Mass, 2.9 kg Wind velocity (only outdoors): 1 m/s
*
Whole day conservative value used for both solar cookers calculations: go = 0.40.

The insulating box, Fig. 2b, was constructed from a commercial of the box. Fig. 4 shows the initial and boundary conditions
corrugated cardboard box that was filled with polyurethane spray- applicable.
can hardening foam keeping a tight space for the utensil. A lid was The high longitudinal heat conduction along the metal pot walls
cut from the upper part of the box. All the surfaces were covered by allows accepting the approximation of external uniform tempera-
MylarÒ aluminized foils, inside and outside, in order to minimize ture for convection and radiation and using the full heat capacity
infrared radiation losses and to reduce humidity absorption. The of both pots in Eq. (1). This approximation circumvents the diffi-
characteristics of pots, box, reflectors and both PCM’s, paraffin culty in assessing the spatial distribution of incident sun radiation
and erythritol, used in this work are summarized in Table 1. on the external pot surface. This distribution is different in the two
paraboloids A and B, and also changes along the day; moreover, sun
4. Numerical model tracking is imperfect when done manually, so it would be much
complicate to take into account all these details. The temperature
The utensil, including the insulating box is divided into isother- differences found experimentally, especially inside the PCM, some-
mal coaxial elements following the two coaxial pots geometry, times reaching 20 °C, were reduced by periodically rotating the
shown in Fig. 1, and approximating the box as a cylinder of equal utensil.
volume. Fig. 3 shows a conceptual scheme. On each of the elements The general equation for the transient heat balance for element
the heat balance is performed assuming either 1-D heat conduc- i can be written as:
tion, convection and radiation, where applicable. The whole set dT i keq;i
comprises: ci M i ¼ Q s  ðQ r þ Q c Þ þ Ai ðT i1 þ T iþ1  2T i Þ
dt |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} li
Only for the external element ie
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
keq;i
 1 cylindrical lumped element representing the contents of the ¼0 for i¼1¼Ae li
e ðT
ie 1 T ie Þ for i¼ie
e
inner pot, i = 1. It includes water as the cooking medium and
 ðQ co þ Q v Þ ð1Þ
the food load, assumed at uniform temperature because of the |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
strong free convection effect and eventual stirring. It exchanges Only for the inner pot element; i¼1

heat with the wet inner pot wall and loses vapor when The terms in the right hand side are assumed as quasi-steady.
approaching the boiling temperature. The solar heat power is modeled as:
 5 layer elements of the PCM, which are of uniform thickness and
of linear variation inner temperature, each one coating its inner Q s ¼ Aap go Gb ð2Þ
neighbor and cup shaped, i = 2–6. Each of them incorporates a go includes all the optical losses, indicated in the Glossary. The solar
horizontal circular portion and a cylindrical layer vertical por- intensity Gb is modeled for sunny days according to a Hottel corre-
tion, according to Fig. 3. The first and last elements respectively lation, assuming perfect sun tracking [56]. Madrid, Spain (40°230 N
include the inner and outer pot wall heat capacity at uniform 3°430 W, 670 m altitude, 2700 yearly sun hours), with a dry conti-
temperature and with both negligible heat resistance and nental Mediterranean climate, is chosen as a location where the
thickness. authors can experimentally check the numerical results and where
 5 layer closed elements of uniform thickness and linear varia- the sun irradiation is similar or smaller than that in many develop-
tion inner temperature, accounting for the whole uniform mate- ing countries.
rial of the insulating box, i = 7–11. The re-radiated heat power from the external surface, either of
pot or box is formulated as:
Two states arise: 1. When the utensil is outdoors before moving  
indoors, time t 6 to; the model is run without the insulating box; Q r ¼ Ae er T 4e  T 4amb ð3Þ
the external element corresponds to ie = 6. 2. When the utensil is
indoors, inside the insolating box t > to, ie = 11, according to We use Te = T6, Ae = Ae,pot, e = epot outdoors when t 6 to, and
Fig. 4. Solar heat power Qs = 0 in Eq. (1) and both heat power losses Te = T11, Ae = Ae,box, e = ebox indoors when t > to, as Fig. 4 depicts. Tamb
Qr and Qc switch to the ones corresponding to the external surface is the corresponding outdoor or indoor temperature. The outdoor
1142 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146

are experienced. But more importantly, the formulation in those

experiments yields experimental values for an overall heat transfer
coefficient U modified by a non-separable factor F0 devised for
using the cooking temperature T1 instead of the external surface
temperature Te, which is evident comparing Eq. (4) with Eq. (5).
Moreover, in the referred cases U also incorporates the thermal
resistance by convection inside the pot and the conduction resis-
tance of the pot wall. This means that for those experiments the
summation in the left hand side of Eq. (5) is non-separable:

Q c þ Q r þ Q v ¼ Ae F 0 UðT 1  T amb Þ ð5Þ

0
Comparison of the F U values obtained through the modeling
performed in this work with those experimental values yield val-
ues in the range. Owing to the above mentioned limitations and
the fact that in those studies wind speed has been found to have
a substantial effect and its measurement has been performed un-
der different conditions, a deeper study is needed to offer a de-
tailed comparison, which is out of the scope of this paper.
Dealing with heat transfer inside the PCM, in respect to Eq. (1),
when an element of the PCM is in solid state we take keq = ks, the
molecular solid heat conductivity, thus considering that pure con-
Fig. 3. Scheme of the discretization for modeling of the utensil for cooking and heat duction is occurring. Alternatively, the correlation proposed in [63]
storage. The dash lines indicate the boundaries of the elements. is used in our model for the melted PCM elements with keq = kfNu,
and using the Nusselt number given by Eq. (6). Both horizontal and
temperature has been modeled using monthly averages of daily vertical portions of the elements, as Fig. 3 shows, have been com-
maximum and minimum temperatures and locating them at bined in each element. Fortunately this correlation, for heat trans-
14:00 and sunrise, respectively, from [57]. A sinusoidal fit defines fer close to the melting point, does not include any characteristic
the temperature between them (all the times are expressed as true length so that no modification is needed when several neighboring
solar time). The indoor space is a comfort zone, with a temperature elements are melted:
of 22 °C plus half the difference with the outdoors temperature if ) 2 T T  2
31=3
higher, or minus one quarter of this difference if lower.
h l
Nu ¼ kf i > 1 g i 2 m q2f cf kf bf
f hf ¼ 0:072 4 5
Convective heat transfer to atmosphere is modeled adding the i ¼ melting elements in the range 1 to 6 lf
free convection with the forced convection coefficients using a cu-
ð6Þ
bic algorithm, Eq. (4) according to Incropera and De Witt [58]:
 1=3 The specific heats of the PCM have been considered as constant
3 3 for the solid state cs and for the liquid state cf. During melting of an
Q c ¼ Ae hfree þ hfor ðT e  T amb Þ ð4Þ
element two issues have to be considered: (i) some PCMs are mix-
For both, the utensil and the insulating box, the forced convec- tures so that they show an interval for phase change DTm = Tl  Ts.
tion coefficient for spheres from Whitaker and the free convection (ii) Discretization of the PCM into elements need to consider some
from Churchill are used, both referenced in [58]. This is so because virtual temperature interval for the complete phase change within
the differences for compact bodies such as spheres, cylinders and the element. During phase change of an element a uniform heat
cubes are considered of minor relevance for the purpose herewith, capacity is added, as usual, correcting by phase change:
if the square root of the external area is used as the characteristic
cs þ cf Lm
length, according to [59]. Indoors there is no forced convection, but cPCM ¼ þ ð7Þ
2 DT m
outdoors wind has been considered, as indicated in Table 1.
The convection coefficient values using Eq. (4) cannot be accu- Values used are offered in Table 1.
rately checked and improved using the experimental results di- Convective heat transfer inside the inner pot is represented by
rectly obtained in non-storage solar cookers testing campaigns the correlation offered in [64], already used for similar purposes
[15,60,19,61,62] and elsewhere. The reason is twofold; those re- in [65]. As boiling starts in the inner pot, composition of this corre-
sults incorporate radiation and, eventually, some not well known lation with the one of Rohsenow for nucleate boiling is performed
amount of water evaporation when temperatures near boiling in a similar way as in Eq. (4), following indications in Incropera and

Fig. 4. Scheme of the time evolution from start to stop of the modeling, indicating the initial conditions applicable and the end temperature of the food load. Squares on the
axis indicate time extent for cooking.
 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146 1143

De Witt [58]. It was found that the application of this scheme to the mended for generic polyurethane foam, as indicated in Table 1. The
model does not impede the growth of the temperature above the discrepancy is attributed to voids found in the foam and leaking of
saturation temperature of water at atmospheric pressure. This last steam and hot air through the box lids interstices, besides model
issue has to be corrected by introducing the extra heat loss power inaccuracies. The box external emissivity was increased over nom-
Qv associated with the pressure and mass transfer coupling with inal MylarÒ values to conservatively consider a dirty surface,
atmosphere, as Eq. (8) states: Table 1.
In the schedule proposed for the use of the utensil, 1 h is left
Q v ¼ K v ðT 1  T sat Þs for T 1 P T 0sat ¼ T sat  T sat;c ð8Þ after breakfast for cleaning or other duties. After this time 70 °C
This, on one hand empirically limits the temperature of the is imposed as initial condition for the utensil, as some heat is
cooking medium to a value slightly above Tsat (what is compatible remaining, what is indicated in Fig. 4. The system of six simulta-
with the slight overpressure imposed by the two pot lids) and on neous ordinary differential equations that generates the applica-
the other hand delivers the loss because of the progressive surface tion of Eq. (1) (outdoors) is advanced in time from 07:00 am as
evaporation and sudden growth when sub-cooled boiling sets and starting point. Lunch cooking is imposed starting at 12:00 am, indi-
saturation boiling occurs at a later stage. Test performed varying cated with a rectangular step in Figs. 4–7. Integration follows up to
the exponent and the multiplier around the selected empirical val- the time to for moving the utensil to the indoor ambience for stor-
ues of s = 3.0, Kv = 10 W K3, T sat;c ¼ 2  C revealed a low sensitivity age in the afternoon. At this time the number of elements is in-
within the expected accuracy of the model. creased from 6 to 11 (indoors) with ambient temperature for the
The cooking power when water as food equivalent is added into insulating box as initial condition. Dinner cooking is imposed at
the cooking medium Qco is distributed along Dt = 1 h, as shown in 20:00 and breakfast at 05:00 am next day, indicated as squares
Fig. 4. This corresponds to: on the time axis in Figs. 4–7. to was chosen maximizing the end
temperature of breakfast Tend,1. Changing this time ±1 h does not
cw M co change appreciable Tend,1. In winter the optimum time was found
Q co ¼ ðT 1  T amb;sc Þ ð9Þ
Dt to be about 2 h less than in summer, indicating that sun intensity
and external temperature is what drives the optimum to.
Fig. 5 shows the results obtained by the numerical model in
5. Results and discussion summer (S) for the layout called CASE I, with lower load of PCM,
whose volume is detailed in Table 1, paraffin as PCM and water
Model calibration has been performed through experimenta- as cooking medium. It can be observed that water boiling is pro-
tion. The optical efficiencies of solar cookers A and B were deter- duced through most part of the day, starting about 11:00 am, out-
mined experimentally using the temperature–time evolution doors and continuing indoors. This makes necessary steam
when exclusively cold water (T1 < Tamb) completely filled the outer evacuation from the insulating box and protection of the box
pot without the inner pot, following the procedure proposed in against humidity. The PCM partially melts by noon and attains a
[60,19]. It was heated in the focus of either of both cookers A and high temperature, above water boiling point. It was observed that
B at around noon. The uniformity of water temperature was as- cooking the breakfast ends at around Tend,1 = 80 °C for to from 14:00
sured by some stirring. When T1 = Tamb, Qr = Qc = Qco = Qv = 0. From to 18:00, thus demonstrating that cooking and storage are possible
both conditions it follows that so that F0 = 1. Thus, from Eq. (1), in in a non-critical way. During the night the PCM almost solidifies
this particular case of ie = 1 and from Eq. (2) it follows: because of water evaporation continues indoors for some time. Ta-
½ðcMÞw þ ðcMÞpot;ou dT 1 =dt ble 2 indicates that evaporation and boiling results in a loss of
go ¼ ð10Þ 2.8 MJ through the day, but eventually it could be used for extra
Aap Gb
cooking. If the conductivity of the paraffin (p) is increased tenfold
The average results of go obtained experimentally are given in Tend,1 diminishes to 76 °C and the loss by evaporation increases to
Table 1. These values are coherent with published data under lab- 9.3 MJ, boiling off all the water. Thus enhancing PCM conductivity
oratory conditions, such as [19,60,61,66,67]. No coherent tendency is deleterious for heat storage as the mass of water that evaporates
was found with solar time. The higher value experimentally found is higher. Decreasing the conductivity of the insulating box down
for cooker A seems to come from the longer focal distance that re- to 0.02 W m1 K1 does increase Tend,1 from 80 °C to 84 °C and
duces incidence angles to the reflector and seemingly because of a along the day Qv remains the same 2.8 MJ in this case, supporting
higher reflectivity. In order to study the utensil concept in a generic the fact that evaporation is the major cause of heat loss and not
way a conservative value of go = 0.4 was selected for the whole day heat conduction through the insulation. A similar magnitude of
for both solar cookers because of considering: dirty surfaces during the increase in Tend,1 was observed for the remaining cases.
operation, the presumably lower values in the early morning and Water evaporation and boiling precludes full melting with both
late afternoon and non-perfect sun tracking. This value is similar PCMs, what requires temperatures inside them higher than 100 °C.
to the one reported in [61] and even higher than the one reported This is especially notorious with the higher phase change heat and
in [22]. This common value allows considering both solar cookers better heat conducting erythritol PCM. At 18:00 erythritol only
identical in the mathematical model.
The insulating box effective thermal conductivity was obtained
Table 2
by means of an additional experimental calibration. The modeled Results of the model described for summer (S) July the 15th and winter (W) January
indoor characteristic cooling time tcoo was made identical to the the 15th. Between parentheses are the results without cooking medium.
experimental one adjusting the generic polyurethane foam ther-
PCM CASE SEASON Tend (°C) Boiling loss (MJ)
mal conductivity, under constant ambient temperature. Because
of the very slow cooling, temperatures inside the utensil are fairly p I S 80 (90) 2.8 (0)
er I S 74 (118) 7.3 (0)
homogeneous, as Figs. 5–7 indicate. The high thermal resistance of p II S 87 (88) 0.29 (0)
the insulating box controls this time. The result was that the value er II S 84 (108) 3.3 (0)
of the conductivity of the insulating material had to be increased p I W 73 (82) 0.19 (0)
for matching the measured value of tcoo = 30 h, after improving er I W 68 (106) 1.25 (0)
p II W 66 (66) 0 (0)
the box insulation over previous measurements [68,69]. The ther-
er II W 71 (99) 0.35 (0)
mal conductivity resulted to be higher to what is nominally recom-
1144 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146

Fig. 5. Time evolution of temperatures versus solar time for CASE I July the 15th PCM = paraffin. Thin continuous lines are for the five elements of the PCM i = 2–6, increasing
darkness towards outside. Thick line with cross symbols is cooking medium i = 1. Dash lines are the five elements of the insulating box i = 7–11, increasing darkness towards
outside. Dash-dot horizontal lines are phase change interval. Dot line is ambient temperature. Bottom steps are cooking intervals.

Fig. 6. Time evolution of temperatures versus solar time for CASE I July the 15th, PCM = erythritol. Same legend as Fig. 5.

shows almost full melting of the external element, Fig. 6. Boiling is There is no clear advantage in the representative day for winter,
also responsible of the early fast temperature drop indoors. as Table 2 shows. Fig. 7 depicts the most marginal case of all of
Using the possibilities of the modeling, a higher load of PCM them corresponding to winter, CASE II with paraffin as PCM. This
has been defined as a synthetic CASE II, indicated in Table 1. For figure indicates that dinner is cooked at a higher temperature
that the external dimensions of the utensil have to be increased than lunch, because of the stored heat. The jump in ambient
and for the 1D modeling here used, only the new PCM volume temperature around 14:00 is due to the switch to indoor envi-
has to be redefined for the synthetic experiment. Table 2 sum- ronment, cited in Section 4.
marizes the results for the four cases, for both PCMs and for Table 2 indicates that cooking the breakfast is possible for sun-
summer and winter seasons. The increase in PCM load signifies ny days all the year round. For CASE I, the results have been
an increase in Tend,1 in the day chosen as representative for sum- checked experimentally, validating the conclusions from the
mer, although it was sufficient with the nominal PCM load. modeling.
 A. Lecuona et al. / Applied Energy 111 (2013) 1136–1146 1145

Fig. 7. Time evolution of temperatures versus solar time for CASE II January the 15th, PCM = paraffin. Same legend as Fig. 5.

Trying to reduce losses, the model has been run without any  Heat storage and retention is much better when water is
cooking medium besides food. Table 2 summarizes the results indi- removed from the inner pot or a high boiling temperature cook-
cating both Tend,1 and the boiling loss between parentheses. It can ing medium is used instead, such as culinary oil.
be observed that higher Tend,1 is the result. The exception is CASE  The combination of cooking outdoors simultaneously to heat
II in winter, using paraffin, where an identical Tend,1 is reached as storage can make the person in charge of the cooking more
already there was no evaporation and boiling loss with water as inclined to briefly cut the solar irradiation over the utensil for
the cooking medium. The larger differences when reducing the stirring or inspecting the food, reducing hazards due to
boiling losses in the just mentioned way corresponds to the use unwanted eye or skin exposition.
of erythritol, coherently with the analysis above.
The results confirm a large cooling characteristic time for heat
retention with the proposed storage utensil. This prevents cooking
Acknowledgements
failures during sun shading, either spontaneous or purpose, and for
too long periods between sun alignment.
The technical support of the laboratory technician Mr. Manuel
There is the uncertainty of the contact thermal resistance be-
Santos and Mr. Carlos Cobos is greatly appreciated, as well as the
tween the external pot and the PCM, because of the contraction
experimental work of Mr. Oscar Cerezo-Cuesta.
and possible PCM cracking during cooling. This issue needs further
investigation.
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