Thermal performance of a solar box cooker with outer reflectors: Numerical study and experimental investigation

Solar Energy 158 (2017) 347–359

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Solar Energy journal homepage: www.elsevier.com/locate/solener

Thermal performance of a solar box cooker with outer reflectors: Numerical study and experimental investigation

MARK

⁎

Zied Guidaraa, , Mahmoud Souissia,b, Alexander Morgensternc, Aref Maaleja a b c

Laboratory of Electro-Mechanic Systems (LASEM), National School of Engineers of Sfax (ENIS), University of Sfax, B.P. 1173, Road Soukra km 3.5, 3038 Sfax, Tunisia National School of Engineers of Sousse (ENISo), Sousse Technological Pole, B.P. 264, Erriadh, 4023 Sousse, Tunisia Division of Thermal Systems and Buildings, Fraunhofer Institut für Solare Energiesysteme ISE, Heidenhofstraße 2, 79110 Freiburg, Germany

A R T I C L E I N F O

A B S T R A C T

Keywords: Box-type solar cooker design Outer reflectors Thermal balances Computer simulation Experimental results Thermal performance improvement

A design of a box-type solar cooker was modeled and tested in the region of Sfax-Tunisia. This solar cooker was designed to allow reaching relatively high temperatures during days with low solar radiation owing to four outer reflectors. A mathematical model, which is based on thermal balances, was developed and implemented in a Matlab program to predict the thermal performances of the solar cooker. Several tests were effectuated in order to validate the robustness of the mathematical model and to determine the thermal performance of the solar cooker. The obtained results showed an acceptable matching between the experimental temperature values and the computed temperature values with a maximum error inferior to 4%. It was also proven that the use of four outer reflectors improved the optical efficiency of the solar cooker. Indeed, the first figure of merit passed from 0.07 to 0.14 and the maximum temperature of the absorber plate passed from 81.3 °C to 133.6 °C. The sensible loading test revealed that the calculated second figures of merit for different water loads were in the range of 0.34–0.39. Finally, two real cooking process were carried out where 72 min and 107 min were required to fully cook rice and beans respectively.

1. Introduction The ruin of the environment is a sufficient reason for humanity to reconsider the means by which it ensures its comfort. The excessive emission of greenhouse gases, that not only contribute to global warming but also to the pollution of the globe, is considered to be the direct cause of several anomalies affecting every living species on earth (Ashmore, 2017). Faced with this awful situation, human communities have decided to take initiatives and to encourage developing new technologies that are not based on polluting processes (Houghton et al., 2001). Cooking is among the most affected fields by this policy where several attempts have been made to replace traditional cooking means by other low-cost and low-emission technologies. Electric ovens may have locally low emissions, however this advantage collides when considering electricity as a secondary energy. Indeed, according to the U.S department of energy DOE, electricity production is accompanied with greater gas emissions than that accompanying burning LPG (DOE, 2010). All these facts managed to renew the interest of scientists in solar energy exploitation and allowed solar cookers to thrive and occupy the

⁎

focus of many researches (Schwarzer and da Silva, 2008; Yettou et al., 2014; Beaumont et al., 1997; Geddam et al., 2015). Yet, this technology, despite of the great issues it gives, have also its limitation such as a relatively long cooking time compared to electric or biomass stoves, especially during days with low solar radiation (Saxena et al., 2011). To overcome these weaknesses several works have been carried outduring the last few years and have resulted in a variety of technologies. In Rao and Subramanyam (2003), the influence of equipping the cooking vessel with lugs was studied and revealed that the improvement has reduced the boiling time compared with that obtained when using a conventional cooking vessel. Adding an annular central cavity, such as in Rao and Subramanyam (2005), has also shown a good impact since this change increases the heat transfer surface between the food to be cooked and the cooker. Another way to increase the heat transfer surface was proposed in Harmim et al. (2008) and which consists on using a finned cooking vessel. These technologies have resulted in decreasing the boiling time which is a good fact however constructing such complicated devices would be a burden on the cost of the cooking vessel. On the other hand, some researchers have focused their studies on

Corresponding author. E-mail address: [email protected] (Z. Guidara).

http://dx.doi.org/10.1016/j.solener.2017.09.054 Received 11 June 2017; Received in revised form 24 September 2017; Accepted 25 September 2017 0038-092X/ © 2017 Elsevier Ltd. All rights reserved.

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Nomenclature

̇ Qab ̇ Qacc ̇ Qcv Qṙ

Cp e hcv hr I L m Nu Ra S T U V

λ τ

absorbed heat rate [W] accumulated heat rate [W] convective heat rate [W] radiation heat rate [W] specific heat [J kg−1 K−1] thickness [m] convection heat transfer coefficient [W m−2 K−1] radiation heat transfer coefficient [W m−2 K−1] global solar radiation [W m−2 ] length of solar cooker [m] mass [kg] Nusselt’s number Rayleigh’s number surface [m2 ] temperature [K] overall heat transfer coefficient [W m−2 K−1] wind speed [m s−1]

thermal conductivity [W m−1 K−1] transmission coefficient

Subscripts

air amb back base face G1 G2 ins p ref rockwool s wall wood

air gap ambient back of the solar cooker solar collector base face of the solar cooker first glass second glass insulation absorber plate reflectors Rockwool cladding sky solar cookers walls wood box

Greek symbols

α

absorption coefficient

Fig. 1. Three dimensional sketch of the solar cooker boxtype: (a) solar cooker with outer reflectors, (b) solar cooker without outer reflectors, (c) cut view of the solar cooker.

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In this paper, a design of a low cost box-type solar cooker with good performance is, firstly, presented. Secondly, the developed mathematical model is detailed as well as the necessary formulations for predicting the thermal performances of the solar cooker. Finally, several experiments are presented in order to validate the mathematical model and to determine the thermal performances of the solar cooker.

Table 1 Geometrical and physical parameters: double glazing. Symbol

Description

Value

Unit

SG β eG1,G2 eG ρG

One glass surface

0.3654

Inclination Air layer thickness One glass thickness Glass density

45 15 4 2530

m2 deg mm mm

∊G αG τ Cp,G

Emissivity Absorbtivity Transmission coefficient Specific heat capacity

0.92 0.1 0.8 840

2. Description of the design

kg m−3 – – –

Fig. 1 shows different views of the design of the box-type solar cooker which is considered for this study. The design mainly consists of the following parts:

J kg−1 K−1

• A double-walled trapezoidal wooden box that has an inner volume of 7 × 10 m and a clearance of 32 mm between the wood-walls, • A Rockwool insulation which fills the clearance between the woodwalls, • Inner reflectors that are made of 316L polished stainless steel and which are encased on the inner walls of the wood-box, • A Matt black painted aluminum plate which constitutes the absorber plate and which is fixed on the bottom of the wood box, • A double glazing fixed on the inclined aperture of the wood box.

the technology of the solar cooker itself and resulted on a variety of solar cooker types namely: the box type solar cookers (Soria-Verdugo, 2015; Sethi et al., 2014), the concentrator type solar cookers (Purohit and Purohit, 2009; Negi and Purohit, 2005), the oval type solar cooker (Regattieri et al., 2016), and solar cookers with integrated collectors (Sharma et al., 2005; Esen, 2004). A biggest interest is accorded to the development of box-type solar cookers since this type of solar cookers have less heat loss by convection than that wasted with other types of solar cookers. Furthermore, boxtype solar cookers have a flexible construction which allows much possibilities of enhancement. An enhanced solar cooker box-type was presented in Harmim et al. (2010). The enhancement consists on employing a finned absorber plate to better the heat transfer between the food container and the absorber plate. This improvement had a good impact on cooking time, however, it is less appreciated when considering its impact on the temperature of the absorber plate. Some researchers suggested to boost their devices with internal reflectors (Terres et al., 2014; Kahsay et al., 2014). These studies have revealed that the use of inner reflectors with a specific inclination can improve the thermal performances of the solar cooker. On the other hand, other researchers have chosen to maximize the availability time of the device by increasing the exposure-time to solar radiation using sun-tracking systems (Al-Soud et al., 2010; Farooqui, 2013, 2015). This method requires an additional power supply for the actuators of the booster mirrors or the solar cooker which will affect the performance of the cooker. In Harmim et al. (2012a,b) a non-tracking box type solar cooker with a compound parabolic concentrator was presented. The proposed solar cooker achieved a relatively high temperature however it obtained a less appreciated boiling time. The common point in all of the cited technologies is that they require sufficient solar radiation during a relatively long period to achieve the cooking process, which makes regions with desert and semiarid climates the most suitable regions for these technologies. Tunisia, in particular, is a country in which both of the desert and semi-arid climate can be found. This country is situated in North Africa and in the southern border of the mediterranean with 1298 km coastline. Measurements show that Tunisia has an annual average solar radiation of 2000 kW h/m2/year which makes it one of the most insolated regions of the globe (Ouderni et al., 2013). Subsequently, the climatic conditions of this region make favorable the development of solarbased technologies, particularly solar cookers. A frequently asked question when modeling such devices is whether the heat transfer is done from the air inside the cooker to the interior of the cooking vessel or the inverse. Assuming that the heat transfer is done in only one direction, which is always the case, can result on errors when predicting the evolution of the temperatures of the main components. Indeed, the heat transfer can change its direction during the cooking process. The novelty of this work consists on taking in account the variations of the heat transfer coefficients. This fact resulted in accurate results of simulations. Which permitted a good sizing of a box-type solar cooker.

−2

3

The functioning of this device is based on creating a greenhouse effect inside the wood-box. Indeed, the double glazing transmits the incident solar radiation to the absorber plate which acts like a black body. The high absorptivity of the absorber plate allows it to reach relatively high temperatures. At the same time, the absorber plate emits Infrared waves. As the glazing is totally opaque to Infrared, there will be no heat wasted by radiation from the interior of the solar cooker. The Rockwool permits a sufficient insulation that prevents from unacceptable heat losses. The geometrical and physical parameters of the double glazing and the absorber plate are provided in Tables 1 and 2, respectively. In order to enlarge the solar capture area during days with low solar radiation, four removable outer reflectors are linked to the edges of the inclined aperture of the wood-box. These reflectors are made of 316L polished stainless steel which is available on the local market and which have a reflectivity superior to 60% for waves having a wave-length above 500 μm (Hernandez et al., 2009). The choice of this material for reflectors is justified by its relatively good reflectivity and its resistance against corrosion and variable climatic conditions. The cost of construction of the four reflectors did not exceed 50$ and did not affect the overall cost of the solar cooker. The generated angle between one reflector’s surface and the normal to the double glazing is adjustable owing to a double threaded tensioner. The geometrical and physical properties of the reflectors material are summarized in Table 3. This design of solar cooker presents the main advantages:

• Transportable. • Easy manufacturing. • Available construction materials in the local market. • Can be easily integrated in kitchens, which facilitates its commercialization.

Table 2 Geometrical and physical parameters: absorber plate.

349

Symbol

Description

Value

Unit

m2 mm

SG

Absorber plate surface

0.4678

ep

Absorber plate thickness

3

ρp

Glass density

2700

∊G αG Cp,G

Emissivity Absorbtivity Specific heat capacity

0.97 0.97 900

kg m−3 – –

J kg−1 K−1

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change with the variation of the ambient conditions.

Table 3 Geometrical and physical parameters: reflectors. Symbol

Description

Value

Unit

′ Sref

Outer reflector’s real surface

0.4559

m2

″ Sref

Inner reflector’s surface

0.146

m2

eref

Reflector’s thickness

0.5

mm

ρss

Stainless steel’s density

r ∊ref αref

Reflectivity Emissivity

8 × 103 0.6 0.1

kg m−3 – –

Absorbtivity

0.4

–

Cp,ref

Specific heat capacity

502

J kg−1 K−1

Sref

Outer reflector’s effective surface

0.4279

m2

• The physical and thermal properties of the components materials are supposed to be constant in the operated temperature range. • Only the double glazed panel, the cooking vessel and the aluminum plate are supposed to absorb solar radiation. • All of the transmitted solar radiation by the double glazed panel is • • •

3. Theoretical study

absorbed by the aluminum plate and the cooking vessel, owing to the inner reflectors. The solar cooker is positioned in a way to have normal incident sunrays to the double glazed panel. Heat transfer by radiation between the cooking load and the inner envelope of the cooking vessel is negligible. The lid and the pot of the cooking vessel have the same temperature Tvessel .

The modeling of the solar cooker is illustrated in Fig. 2. Where, the arrows in yellow present both of the solar radiation which is coming directly from the sun and the solar radiation which is reflected by the outer reflectors (not represented in the model) and the inner reflectors.

The modeling of the solar cooker includes heat transfer phenomena that govern the temperature evolution of the load to be cooked, the cooking vessel, the absorber plate, the air gap, and the double-glazing. This study allows determining the maximum temperatures that the mentioned components can reach for a period of the day during which the device is exposed to solar radiation.

3.2. Thermal balance equation of the first glass of the double glazing Concerning this thermal balance, the inlet heat-flow is composed of the absorbed solar radiation, the heat flows by convection and radiation between the first glass and the second glass. This inlet heat-flow is equal to the accumulated heat flow in the first glass in addition to the lost heat-flow by convection between the first glass and the ambient air and by radiation between the first glass and the sky. The thermal balance equation of the first glass of the glazing is expressed in terms of heat-

3.1. Assumptions To simplify the expressions of the energy balance equations, the following assumptions are adopted:

• The temperatures of the different components are uniform but can

Fig. 2. Modeling of the solar cooker.

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rates as follows:

̇ ,air = mair Cp,air Qacc

̇ ,G1 + Qcv ̇ ,G2,G1 + Qṙ ,G2,G1 = Qacc ̇ ,G1 + Qcv ̇ ,G1,amb + Qṙ ,G1,s Qab

(1)

The different heat rates that constitute the thermal balance Eq. (1) are given by the following expressions:

̇ ,G1 = αG1 (SG1 + 4 Sref ) I Qab

(2)

̇ ,G2,G1 = hcv,G2,G1 SG1 (TG2−TG1) Qcv

(3)

Qṙ ,G2,G1 = hr ,G2,G1 SG1 (TG2−TG1)

(4)

̇ ,G1 = mG1 Cp,G Qacc

∂TG1 ∂t

(6)

Qṙ ,G1,s = hr ,G1,s SG1 (TG1−Ts )

(7)

3.3. Thermal balance equation of the second glass of the double glazing

(11)

Qṙ ,lid,G2 = hr ,lid,G2 Slid (Tlid−TG2)

(12)

̇ ,G2 = mG2 Cp,G Qacc

∂TG2 ∂t

(19)

(20)

(21)

∂Tp (22)

∂t

̇ ,ins = Ubase,ins Sp (Tp−Tamb) Qbase

(23)

̇ ,p,pot = Up,pot Spot (Tp−Tvessel ) Qcd

(24)

3.6. Thermal balance equation of the cooking vessel The cooking vessel is made of two parts namely the cooking pot and the lid. These parts are made from aluminum and are painted in Matt black painting. The cooking vessel receives solar radiation through the double glazing and from the inner reflectors. The pot exchanges heat with the absorber plate by conduction and with the air gap inside the box by convection. The heat transfer can occur in both senses since the temperature of the vessel can either be higher or lower than the temperatures of the absorber plate and the air gap. A part of the received heat is accumulated inside the material of the vessel and the other is wasted in the favor of the load which is placed inside the cooking vessel and by radiation over the walls of the box and the second glass. Assuming that the pot and the lid have the same temperature Tvessel , the thermal balance equation can be written as follows:

(13)

3.4. Thermal balance equation of the air gap The air is in contact with the absorber plate, the vessel and the second glass, the only heat flows that are taken into account in the thermal balance equation are: the heat flow which is gained by convection with the absorber plate, the heat flow which is released by convection to the second glass and the vessel, the lost heat flow throughout the insulation, and the heat flow of accumulation in the air gap. Thus, the relative thermal balance equation to the air inside the box can be written as follows:

̇ ,p,air = Qacc ̇ ,air + Qcv ̇ ,air ,G2 + Q̇wall,ins + Qcv ̇ ,air ,pot + Qcv ̇ ,air ,lid Qcv

̇ ,air ,lid = hcv,air ,lid Slid (Tair −Tvessel ) Qcv

̇ ,p = mp Cp,p Qacc

(9)

Qṙ ,p,G2 = hr ,p,G2 (Sp−Slid ) (Tp−TG2)

(18)

̇ ,p = τ 2 αp (Sp + 4 Sref −5 Spot ) I Qab

The different heat rates that constitute the thermal balance Eq. (8) are given by the following expressions:

(10)

̇ ,air ,pot = hcv,air ,pot Spot (Tair −Tvessel ) Qcv

The different heat rates that constitute the thermal balance Eq. (20) are given by the following expressions:

(8)

̇ ,air ,G2 = hcv,air ,G2 SG2 (Tair −TG2) Qcv

(17)

̇ ,p = Qacc ̇ ,p + Qcv ̇ ,p,air + Qṙ ,p,G2 + Qbase ̇ ,ins + Qcd ̇ ,p,pot Qab

As the first glass absorbs a part of the solar radiation, the second glass absorbs only τ -times the incident solar radiation. This heat-flow is summed with the heat-flow by radiation coming from the absorber plate and the lid of the vessel. Since the air gap between the second glass and the absorber plate is hotter than the second glass, the second glass receives an additional heat flow by convection from it. An amount of the received heat-flow is accumulated in the second glass, while the remaining amount is lost by convection and radiation over the first glass. The thermal balance equation of the second glass of the glazing can be written as:

̇ ,G2 = τ αG1 (SG2 + 4 Sref ) I Qab

Q̇ wall,ins = Uwall,ins Swall (Tair −Tamb)

The absorber plate receives τ 2 times the incident solar radiation, accumulates an amount of it and releases the other amount as a convective heat flow with the air gap, a radiation heat flow over the second glass and a lost heat flow by conduction throughout the insulation which is covering the base of the box. When a cooking vessel is placed inside the box, the absorber plate heats the base of the pot of the vessel by conduction. However, the absorber plate loses a part of its surface which is hided from solar radiation by the cooking vessel. In this case, the thermal balance equation of the black aluminum plate is expressed in terms of heat rates as follows:

where Sref in Eq. (2) is the effective surface of the outer reflector when it is positioned in its optimum inclination and which can be calculated with reference to Sethi et al. (2014).

̇ ,G2 + Qcv ̇ ,air ,G2 + Qṙ ,p,G2 + Qṙ ,lid,G2 = Qacc ̇ ,G2 + Qcv ̇ ,G2,G1 + Qṙ ,G2,G1 Qab

(16)

3.5. Thermal balance equation of the black aluminum plate

(5)

̇ ,G1,amb = hcv,G1,amb SG1 (TG1−Tamb) Qcv

∂Tair ∂t

̇ ,vessel + Qcd ̇ ,p,pot + Qcv ̇ ,air ,lid + Qcv ̇ ,air ,pot = Qacc ̇ ,vessel + Qcv ̇ ,vessel,load Qab + Qṙ ,pot ,wall + Qṙ ,lid,G2

(25)

The different heat rates that constitute the thermal balance Eq. (20) are given by the following expressions:

̇ ,vessel = 5 τ 2 α vessel Svessel I Qab

̇ ,vessel = m vessel Cp,vessel Qacc

(14)

∂Tvessel ∂t

(26)

(27)

The different heat rates that constitute the thermal balance Eq. (14) are given by the following expressions:

̇ ,vessel,load = hcv,vessel,load Sload (Tvessel−Tload ) Qcv

(28)

̇ ,p,air = hcv,p,air (Sp−Slid ) (Tp−Tair ) Qcv

Qṙ ,pot ,wall = hr ,pot ,wall Spot (Tvessel−Twall )

(29)

(15) 351

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3.7. Thermal balance equation of the load to be cooked

hr ,G1,s = εG1 σ (Ts + TG1 )(Ts2 + TG21)

Assuming that the load to be cooked is homogeneous, and that it exchanges heat only by convection through the pot’s walls, its thermal balance equation can be written as:

where Ts is the temperature of the sky which is calculated by Adelard et al. (1998):

̇ ,pot ,load = Qacc ̇ ,load Qcv

1.5 Ts = 0.0552 Tamb

(30)

̇ ,load = mload Cp,load Qacc

∂Tload ∂t

(31)

1

Uwall,ins =

1 hcv,air ,ref

3.8. Heat transfer coefficients

λair eair

hcv,air ,G2 = 0.1291 Ra0.304

Ubase,ins =

where eair is the air gap mean thickness which can be calculated using the following expression:

The convective heat transfer coefficient, at the horizontal interface between the black aluminum plate and the air gap inside the box, is calculated using the following expression: ⎜

hcv,air ,pot =

1/4

⎞ ⎠

NuG1,G2 λair eG1,G2

Nuair ,pot

RaG1,G2 cosβ

) + ⎛⎝ (

1708

G1,G2 cosβ

∗

)

+

1 hcv,wood,amb

(42)

(43)

Svessel

Nuair ,pot λair Lpot

(44)

⎡ ⎤ 1/6 0.387 Raair ⎢ ⎥ ,pot = ⎢0.825 + ⎥ 9/16 8/27 0.492 ⎢ ⎡1 + ⎤ ⎥ Pr ⎣ ⎦ ⎦ ⎣

Nuair ,lid = 0.54 Ra1/4

10 4 ⩽ Ra ⩽ 107

Ra1/3

107 ⩽ Ra ⩽ 1011

Nuair ,lid = 0.15

∗ RaG1,G2 cosβ 1/3 −1⎞ 5830

(Tp + TG2 )(Tp2 + 1 1 + ε −1 εp G2

)

⎠

(45)

(46)

(36) 3.9. Discretization The previous formulations are implemented in an iterative Matlab program to predict the behavior of the different components when exposed to solar radiation. To do so, these equations are discretized using the finite differences method with first order derivatives and backward differences.

TG22 ) (37)

The convective heat transfer coefficient at the interface between the first glass of the glazing and the environment can be expressed in term of the wind velocity V, supposed to be constant during the process, with reference to Arasteh et al. (1989) as follows:

hcv,G1,amb = 5.67 + 3.86 V

erockwool λrockwool

The convective heat transfer coefficient between the lid and the air gap can be calculated using the following correlation (Churchill and Chu, 1975):

where terms (−)∗ are set to zero if they are negative. The radiation heat transfer coefficient of the aluminum plate over the second glass is expressed as:

hr ,p,G2 = σ

+

( )

(35)

NuG1,G2 = 1 + 1.44 1− Ra 1708 (sin(1.8 β ))1.6

(41)

2

where eG1,G2 is the spacing between the first glass and the second glass. The Nusselt’s number is calculated using Hollands formulation (Hollands et al., 1976) that can be written as follows:

(

1 hcv,wood,amb

(34)

The convective heat transfer coefficient at the interface between the first glass and the second glass of the glazing is written using this formulation:

hcv,G2,G1 =

+

Spot hcv,air ,pot + Slid hcv,air ,lid

⎟

L

erockwool λrockwool

where hcv,air ,pot is the convective heat transfer coefficient between the air inside the box and the lateral surface of the pot. The heat transfer on this surface can be assumed, according to Churchill and Chu (1975), to a heat transfer at the surface of a vertical plate. Thus,

(33)

hcv,p,air = 1.32 ⎛ ⎝

+

1 ewood λwood

hcv,air ,vessel =

eface + eback

Tp−Tair

ewood λwood

where all the convective heat transfer coefficients having an interface with the environment are calculated using Eq. (38). The convection heat transfer coefficient between the external envelope of the pot and the air inside the box is introduced in Harmim et al. (2012b) and which can be expressed as:

(32)

2

+

The overall heat transfer coefficient throughout the base of the box can be similarly written as:

Heat transfer coefficients have fluctuating values that depend on the different components temperatures and the thermal properties of the used materials. These fluctuations, once calculated, allow to obtain accurate results in order to predict the evolution of the temperature of the load which is the most important temperature in the model. The convective heat transfer coefficient at the interface between the inclined second glass of double glazing and the air gap can be expressed as follows:

(1−

(40)

The overall heat transfer coefficient throughout the walls of the box is expressed by the following semi-empirical expression:

where

eair =

(39)

∂TG1 SG1 ⎡ (SG1 + 4 Sref ) = αG I + hcv,G1,amb Tamb + hr ,G1,s Ts ∂t mG1 Cp,G ⎢ SG1 ⎣ + (hcv,G2,G1 + hr ,G2,G1) TG2−(hcv,G1,amb + hr ,G1,s + hcv,G2,G1

(38)

+ hr ,G2,G1) TG1⎤ ⎥ ⎦

The radiation heat transfer coefficient of the first glass G1 over the sky can be written as: 352

(47)

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∂Tvessel 1 [5 τ 2 α vessel Svessel I + hcv,vessel,load Sload Tload = ∂t m vessel Cp,vessel

SG2 ⎡ (SG2 + 4 Sref ) ∂TG2 τ αG I + (hr ,G2,G1 + hcv,G2,G1) TG1 = mG2 Cp,G ⎢ SG2 ∂t ⎣ + hcv,air ,G2 Tair + hr ,p,G2 −(

(Sp−Spot ) SG2

(Sp−Spot ) SG2

Tp + hr ,lid,G2

+ hr ,wall,pot Spot Twall + hr ,lid,G2 Slid TG2 + Up,pot Spot Tp

Slid Tvessel SG2

−(hcv,vessel,load Sload + hr ,wall,pot Spot + hr ,lid,G2 Slid

Slid + hcv,air ,G2 + hcv,G2,G1 SG2

hr ,p,G2 + hr ,lid,G2

+ Up,pot Spot ) Tvessel]

+ hr ,G2,G1) TG2⎤ ⎥ ⎦

∂Tload 1 = hcv,pot ,load Sload (Tpot −Tload ) ∂t mload Cp,load

(48)

1 ∂Tair [hcv,p,air (Sp−Spot ) Tp + hcv,air ,G2 SG2 TG2 = mair Cp,air ∂t

−(hcv,p,air (Sp−Spot ) + hcv,air ,G2 SG2 + Uwall,ins Swall + (hcv,air ,pot Spot + hcv,air ,lid Slid )) Tair ]

(49)

TGn1 = TGn1− 1 +

−(hcv,p,air + hr ,p,G2 +

Sp (Sp−Spot )

Ubase,ins +

Spot (Sp−Spot )

Δt SG1 ⎡ (SG1 + 4 Sref ) n αG I n + hcvn ,G1,amb Tamb + hrn,G1,s Tsn mG1 Cp,G ⎢ SG1 ⎣

+ (hcvn ,G2,G1 + hrn,G2,G1) TGn2− 1−(hcvn ,G1,amb + hrn,G1,s + hcvn ,G2,G1

(Sp−Spot ) ⎡ (Sp−Spot + 4 Sref ) 2 τ αp I + hcv,p,air Tair + hr ,p,G2 TG2 (Sp−Spot ) mp Cp,p ⎢ ⎣ Spot Sp Up,pot Tvessel + Ubase,ins Tamb + (Sp−Spot ) (Sp−Spot )

+ hrn,G2,G1) TGn1− 1⎤ ⎥ ⎦

(53)

Up,pot ) Tp⎤ ⎥ ⎦ (50)

32

790 780 770

30

760

28

750

26

740

24

730 720

22 13

13.2

13.4

13.6

13.8

08-03-2017

800

34

790

32

780

30

Average wind speed = 1.37 m.s -1 -2

Maxi. solar radiation = 800.4 W.m Maxi. ambient temperature = 23.88 °C

28

730 720

20 11.6

710 11.8

12

12.2

12.4

12.6

(a) 12-11-2016

(b) 01-03-2017

36

810 12-03-2017

800

34

790

32

780

30

Average wind speed = 1.04 m.s -1

28

Maxi. solar radiation = 803.9 W.m-2 Maxi. ambient temperature = 23.08 °C

770 760 750

26

740

24

730 720

22

710 11.5

12

12.5

13

13.5

700 14

Day-time (h)

(c) 08-03-2017 Fig. 3. Solar radiation and ambient temperature versus time in the region of Sfax-Tunisia.

353

750

24

Day-time (h)

20 11

760 740

22

700 14

770

26

Day-time (h)

Ambient temperature (°C)

20 12.8

710

810

36

12.8

13

700 13.2

Global solar radiation (W.m -2 )

Maxi. solar radiation = 750 W.m-2 Maxi. ambient temperature = 27.8 °C

800

Global solar radiation (W.m -2 )

34

810

Ambient temperature (°C)

Average wind speed = 2 m.s -1

12-11-2016

Global solar radiation (W.m -2 )

36

Ambient temperature (°C)

∂t

=

(52)

Thus, the temperature of each component at the nth time step can be calculated according to the temperature value at the (n−1)th time step and the geometrical and thermal properties of the different components of the system.

+ Uwall,ins Swall Tamb + (hcv,air ,pot Spot + hcv,air ,lid Slid ) Tvessel

∂Tp

(51)

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n n−1 Tvessel = Tvessel +

Δt [5 τα vessel Svessel I n + hcvn ,vessel,load Sload Tload m vessel Cp,vessel

n−1 + hrn,pot ,wall Spot Twall + hrn,lid,G2 Slid TGn2− 1 + Spot Upn,pot ) Tpn − 1

−(hcvn ,vessel,load Sload + hrn,pot ,wall Spot + hrn,lid,G2 Slid n−1 + Spot Upn,pot ) Tvessel ]

(57)

Δt n−1 n−1 hcvn ,pot ,load Sload (Tpot −Tload ) mload Cp,load

n n−1 Tload = Tload +

(58)

where the initial conditions of the problem are:

TG1 (t = 0 s ) = TG01

(59)

TG02

(60)

0 Tair (t = 0 s ) = Tair

(61)

TG2 (t = 0 s ) =

Tp (t = 0 s ) =

T p0

Tvessel (t = 0 s ) =

(62) 0 Tvessel

(63)

0 Tload (t = 0 s ) = Tload

(64)

3.10. Thermal performance indicators The thermal performance of the solar cooker was determined using Mullick method (Mullick et al., 1987). This method consists on calculating the figures of merit and the necessary time to boil an amount of water that would be placed inside the solar cooker. The first figure of merit F1 can be calculated using data from the stagnation test and by injecting them in the following equation:

F1 =

Tp,max −Tamb,stag Istag

(65)

Fig. 4. General structure of the Matlab program.

TGn2 = TGn2− 1 +

where Tp,max is the absorber plate maximum temperature, Tamb,stag and Istag are respectively the ambient temperature and the solar radiation on an horizontal surface at the stagnation time. In order to calculate the second figure of merit F2 , it is necessary to effectuate a sensible heat experiment where the water load temperature passes from Tload,i to Tload,f during a period of time Δt . Hence:

SG2 ⎡ (SG2 + 4 Sref ) τ αG I n + (hrn,G2,G1 + hcvn ,G2,G1) TGn1− 1 mG2 Cp,G ⎢ S G 2 ⎣

n−1 + hcvn ,air ,G2 Tair + hrn,p,G2

(Sp−Spot ) SG2

Tpn − 1 + hrn,lid,G2

Slid n − 1 Tvessel SG2

(Sp−Spot ) n S hr ,p,G2 + hrn,lid,G2 lid + hcvn ,air ,G2 + hcvn ,G2,G1 −⎛ S S 2 G G2 ⎝ ⎜

+ hrn,G2,G1⎞ TGn2− 1⎤ ⎥ ⎠ ⎦

F2 =

F1 mload Cp,load Sp Δt

Tload,i − Tamb,av

⎡ 1− F1 Iav ln ⎢ T −T ⎢ 1− load,f amb,av F1 Iav ⎣

⎤ ⎥ ⎥ ⎦

(66)

⎟

n n−1 Tair = Tair +

(54)

where mload is the mass of the load of water inside the solar cooker, Cp,load is the specific heat capacity of the load of water Tamb,av and Iav are respectively the average ambient temperature and the average solar radiation during the test. The required time to boil the water inside the solar cooker τboil is expressed as follows:

Δt [hcvn ,p,air (Sp−Spot ) Tpn − 1 + hcvn ,air ,G2 SG2 TGn2− 1 mair Cp,air

n n n n n−1 + Uwall ,ins Swall Tamb + (hcv,air ,pot Spot + hcv,air ,lid Slid ) Tvessel n n −(hcvn ,p,air (Sp−Spot ) + hcvn ,air ,G2 SG2 + Uwall ,ins Swall + hcv,air ,pot Spot n−1 + hcvn ,air ,lid Slid ) Tair ]

τboil = −

(55)

Sp (Sp−Spot )

n Ubase ,ins +

Spot (Sp−Spot )

F2 Sp

ln ⎡1− ⎢ ⎣

100−Tamb,av ⎤ F1 Iav ⎥ ⎦

(67)

4. Results and discussion

Δt (Sp−Spot ) ⎡ (Sp−Spot + 4 Sref ) 2 n−1 τ αp I n + hcvn ,p,air Tair (Sp−Spot ) mp Cp,p ⎢ ⎣ Spot Sp n n n−1 Upn,pot Tvessel Ubase + hrn,p,G2 TGn2− 1 + ,ins Tamb + (Sp−Spot ) (Sp−Spot )

Tpn = Tpn − 1 +

−(hcvn ,p,air + hrn,p,G2 +

F1 mload Cp,load

The reported results were obtained according to the climatic data which were recorded by the WH3081 weather station, during several days in the region of Sfax-Tunisia. At the first time, cold days were chosen in order to evaluate the performance of the cooker with maximum heat losses due to the low ambient temperature. Knowing that this technology has good performance in cold day encourages its commercialization and investing on it. Furthermore, experimenting during cold days would prove that the solar cooker is incapable of accomplishing satisfying temperatures when it is not equipped with outer reflectors. The recorded solar radiation I and ambient temperature Tamb

Upn,pot ) Tpn − 1⎤ ⎥ ⎦ (56)

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Fig. 5. Different views of the solar cooker.

done using two J-type thermocouples that are placed over the black aluminum plate and in the air gap, and two PTC sensors that are placed on the surfaces of the first and second glasses. All sensors are connected to a data acquisition device (type Agilent 34970A). The measured temperatures are transmitted to a Labview interface that permits to store data in a ’.txt-file’ and to display their evolution using a real time graphic. The two thermocouples have a temperature range of 0 to +760 °C and an accuracy of 0.75%, while the PTC sensors have a temperature range of −40 to +140 °C and an accuracy of 1%.

versus time during these days are shown in Fig. 3. The adopted approach to validate the model consists on experimenting on the solar cooker, then simulating using the same operated conditions as those of the experiments. The obtained results with the experiments are then compared to the simulation results. The accuracy of the model is judged according to the calculated error values between the experimental and the simulation results. The flowchart of the written Matlab program is presented in Fig. 4. This program uses Newton-Raphson method which is an iterative method. This method is very convenient for transient models since it allows to have a better idea on the behavior of the device. On the other hand, experiments are done using a box-type solar cooker, which is illustrated in Fig. 5, and which is designed in conformity with the geometrical parameters of the model. Acquisition is

4.1. Mathematical model validation A no load test was carried out in November 12th 2016 in order to 355

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140

T T

120

G1,exp G2,exp

Tair,exp

100

Tp,exp TG1,com

80

TG2,com T

60 40 12.6

Heat transfer coefficients (W.m-2.K-1)

Components temperatures (°C)

Z. Guidara et al.

air,com

Tp,com 12.8

13

13.2

13.4

13.6

13.8

12 h cv,G2,G1

10

h r,G2,G1

8

h r,G1,sky

6

h cv,lid,air

4

h c,pot,air

2 0 10

U base,ins

11

12

13

14

Day-time (h)

14

Day-time (h)

Fig. 8. Heat transfer coefficients during water heating process.

Fig. 6. Main components temperatures of the solar collector with four reflectors versus time during November 12th, 2016.

this stage, it is difficult to determine with accuracy the value of the wasted heat throughout the insulation. Indeed, the density of the Rockwool, inside the clearance of the wood-box, is not uniform as it is assumed. In addition, the device is not perfectly sealed which will induce some errors in the model. On the other hand, the obtained errors are within an acceptable range, which permits to confirm the validity of the developed model. The observation of the plots of the heat transfer coefficients that are depicted in Fig. 8 reveals the following results: The heat transfer through the cooking vessel’s walls is done in two directions. Indeed, at the beginning of the cooking process the air inside the box releases heat in favor of the cooking vessel because of temperature difference. As the temperature of the cooking vessel approaches the temperature of the air inside the box, the convection heat transfer coefficient decreases to reach zero. The cooking vessel temperature continues to increase and becomes superior to the temperature of the air inside the box. At this stage the heat transfer is done in the opposite direction which means that the cooking vessel releases heat in favor of the air inside the box. This is an important result especially when talking about initial conditions. Indeed, the fact that the mathematical model takes into account the direction of the heat transfer means that the model can handle

obtain the values of temperatures of the main components of the solar cooker. The recorded temperatures are the first glass temperature TG1,exp , the second glass temperature TG2,exp , the air gap temperature Tair ,exp and the absorber plate temperature Tp,exp . The climatic data is then injected in the Matlab program in order to compare the computed temperatures TG1,com,TG2,com,Tair ,com and Tp,com with the recorded experimental temperatures TG1,exp,TG2,exp,Tair ,exp and Tp,exp respectively. Both of these results are given in the same plot in Fig. 6. The estimated error between the experimental and simulation data is represented in Fig. 7. This error is none other than the difference between a reference value, which is the obtained temperature by experiment, and the value of temperature obtained by computer simulation at the same point of time. A positive error means that the obtained value by computer simulation exceeds the obtained temperature by experiments and vice versa. The highest calculated error corresponding to the absorber plate temperature is below 2%. While the highest calculated error on the air gap temperature is lower than 2.5%. The error curves have increasing shapes, which is an expected result since the heat losses increase with the increasing temperature inside the box. At

Fig. 7. Error: (a) first glass temperature, (b) second glass temperature, (c) air gap temperature, (d) aluminum plate temperature.

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Components temperatures (°C)

90

T

G1,com

80

TG2,com

70

Tair,com T

60

T

p,com G1,exp

50

T

40

Tair,exp

G2,exp

Tp,exp

30 11.5

12

12.5

13

13.5

Day-time (h) (a) Solar cooker without reflectors Components temperatures (°C)

140

T

G1,com

TG2,com

120

Tair,com

100

Tp,com

80

TG1,exp

60

T

TG2,exp T

40 11

11.5

12

12.5

13

13.5

air,exp p,exp

14

Day-time (h) (b) Solar cooker with reflectors

Water load temperature (°C)

Fig. 9. Components temperatures versus time in the region of Sfax-Tunisia.

120 100 80 60

Fig. 11. Pictorial view of rice and beans cooked during July 26th, and 27th respectively.

40

81.3 °C, when the solar cooker is used without reflectors, to 133.6 °C when it’s used with outer reflectors, which means that the use of four outer reflectors allows a gain by 64.3% of the maximum temperature of the absorber plate. As for the air gap maximum temperature, it passes from 74.1 °C to 121.7 °C, which means an improvement around 64.2% by passing from the mode of functioning without outer reflectors to the mode of functioning with outer reflectors. Furthermore, the second glass maximum temperature results for both of the two configurations show that the second glass gained 62.9% in temperature when the solar cooker is equipped with four outer reflectors. Indeed, the maximum temperature of the second glass passes from 64.8 °C to 105.6 °C when the four reflectors are took into consideration in the model. This enhancement have less impact on the maximum temperature of the first glass, which is a reasonable result since the first glass is separated from the second glass by an air layer that represents an isolating material and because of the direct exposure of the first glass to the ambient air. The maximum temperature of the first glass passes from 40.4 °C to 63.7 °C when adding the four outer reflectors into the model, which means an improvement by 57.6% compared to the solar collector used without outer reflectors. The first figure of merit F1 passes from 0.07 which is a very low value to 0.14 which is an acceptable value especially when noticing that this parameter can range from 0.12 to 0.14 (Yettou et al., 2014). These

20 0

10

10,5

11

11,5

Day-time (h) Fig. 10. Time to boil for different water loads.

different scenarios such as the effect of inserting a cold cooking vessel inside hot solar cooker or exposing both of solar cooker and the cooking vessel to solar radiation at the same time. It is to notice that, in the literature, researchers assume that the heat transfer between the air inside the box and the cooking vessel is done in only one direction. 4.2. Influence of the enhancement In order to highlight the influence of the enhancement with four reflectors, a first test was carried out in March 8th 2017 using the solar cooker not equipped with four reflectors, a second test using the solar cooker with four outer reflectors is also performed in March 12th 2017. The recorded results are plotted in Fig. 9. These results show that the use of outer reflectors represents an important enhancement for the solar cooker. Indeed, the maximum temperature of the black aluminum plate, which is an important temperature in the device, passes from 357

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values prove that the enhancement with four outer reflectors improved the optical efficiency of the solar cooker.

•

4.3. Time to boil different water loads and cooking performance Different water loads were introduced inside the solar cooker during several days in July. These tests were carried out in order to determine the influence of the water load on the thermal performance of the solar cooker when equipped with four outer reflectors. The utilized cooking vessels that are used for these experiments are chosen according to the recommendations of the US DOE (DOE, 2016). The water loads that were used are 0.5 kg, 1 kg, 1.5 kg and 2 kg. The tests were carried out during July 22nd, 23rd, 24th and 25th, 2017. These days were chosen close to each other to have approximately the same operated conditions. Indeed, the average solar radiation that was recorded during these days is in the range of 827–831 W m−2 and the average ambient temperature was about 33 °C. Fig. 10 shows that the time required to boil a mass of water increases with the increase of the load. Indeed, for a water load of 0.5 kg the water took only 47 min to boil, however, for a water load of 2 kg the water took about 1 h 28 min to boil. The computed results under the same operated conditions gave boiling times of 41 min, 1 h 13 min, 1 h 25 min and 1 h 35 min respectively. It is to notice that the utilized water is a drinking water which contains minerals, this explains the fact that the water began boil at approximately 96 °C instead of 100 °C. This fact explains why the computed boiling times were slightly different from those of the experiments. The calculated second figure of merit F2 ranged from 0.34 to 0.39 which are satisfying values and which proves that in addition to a good optical efficiency, the constructed solar cooker has a good heat exchange efficiency factor. Another way to determine the cooking performance of the solar cooker is by effectuating some real solar cooking process such as cooking rice and beans, Fig. 11. These operations were effectuated during July 26th, and 27th respectively. In order to cook the rice, 0.5 kg of water was placed inside the cooking vessel. After 42 min the water returned to boil, one cup of brown rice was poured in the boiling water and then covered with the lid. After 24 min the rice began to be tender which means that it was fully cooked. The mixture is taken out of the cooker for 5 min in order to finish absorbing the remaining liquid. This four persons meal was fully cooked in approximately 72 min which is an acceptable time. The same operation is repeated to cook beans. However, the beans became tender 45 min later than the rice, which means a total cooking time of 107 min.

•

temperatures which confirms the accuracy of the mathematical model. Two tests were carried out on the solar cooker without outer reflectors in March 8th, 2017 and on the solar cooker with outer reflectors in March 12th, 2017. The recorded results show an increase of the absorber plate temperature by 64.3% when passing from the first configuration to the second one. The maximum recorded absorber plate temperature was about 133.6 °C with the solar cooker with outer reflectors under an average global solar radiation valuing 794 W m−2 and an ambient temperature of 23.3 °C. The calculated first figure of merit was 0.14 which proves that the optical efficiency of the cooker with outer reflectors was improved. Stagnation tests with different water loads showed that the heavier is the load, the longer is the boiling time. The tests were effectuated under an average solar radiation of 828 W m−2 and an average ambient temperature of 33 °C. The utilized water loads in these tests are of 0.5 kg, 1 kg, 1.5 kg and 2 kg. their corresponding boiling times are respectively 47 min, 1 h 8 min, 1 h 18 min and 1 h 24 min. It is to notice that during the experiments the first vapor bubbles appeared when the water reached approximately 96 °C, which explains the little time-difference between the experimental values and the computed values.

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5. Conclusion A design of a box-type solar cooker, that is enhanced with four outer reflectors that concentrate solar radiation, is presented. This solar cooker allows introducing a food container using a back-door. A mathematical model that is based on thermal balances was developed to predict the different components temperatures under transient conditions. Several tests were effectuated in order to validate the model in the first step, and to highlight the influence of the enhancement with four outer reflectors. Other tests were also carried out in order to determine the time to boil different water loads. The recorded results of the tests are compared with the computed ones, where a good agreement was found. These results are summarized in the following points:

• The mathematical model validation was effectuated in November 12th, 2016 with an average global solar radiation of 729.5 W m−2 and ambient temperature of 27.7 °C. The maximum recorded absorber plate temperature was 121.4 °C. The error between the experimental and computed results is below 4% for all components

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