Parabolic Dish Solar Cooker: An Alternative Design Approach Toward Achieving High-Grade Thermal Energy Storage Solution

Abstract

Parabolic dish concentrator-based solar cooker is a highly promising alternative green technology capable of providing clean energy solution for wide varieties of domestic and commercial culinary requirements. Since a significantly high temperature range of 250–350 °C can be easily obtained adjacent to the focal region of the parabolic concentrator, the technology is suitable for a wide variety of cooking involving grilling, frying, simmering, and boiling. The utilization of solar energy for domestic or commercial cooking also reduces dependence on fossil fuels, resulting in lower carbon footprints. However, a major limitation of the existing forms of such solar cookers is associated with the duration and availability of solar irradiation, which renders the system active only during the on-sun period. The inaccessibility of solar radiation for cooking before sunrise and after sunset can only be addressed by incorporating effective thermal energy storage (TES) systems in conjunction with parabolic dish reflectors. The development of a compact storage cooking unit involving latent heat thermal energy storage (LHTES) might play a crucial role to overcome this severe limitation. Phase change material (PCM) with suitable melting point and large latent heat of fusion can be considered as an ideal thermal energy storage medium for household or commercial cooking applications. The usage of PCM as the thermal energy storage medium also enables the storage system to deliver heat at a narrow band of temperature during the cooking. Thermal energy is stored during the melting of PCM, and the process is defined as charging of LHTES system. The stored heat can be extracted during the cooking involving heat flow from the molten PCM to the cooking unit rendering gradual solidification of the storage medium. Although phase change materials (PCM) seem to be an ideal thermal storage medium, one of the major concerns associated with them is their low thermal conductivity (0.25–0.5 W/mK). Potential phase change materials with melting temperature close to 300 °C are mostly inorganic salts in pure or multicompositional forms, having low thermal conductivity. As a result, charging duration involving complete melting process of the PCM might end up getting extended beyond a feasible time span of solar irradiation. Low thermal conductivity of storage medium also causes undesirable steep thermal gradient within the storage unit. Therefore, thermal conductivity enhancement of the PCM is of utmost importance. Usage of PCM-CEG (CEG: compressed expanded graphite) composite to enhance the thermal conductivity of the latent heat storage medium is explored and found to be a promising solution. The present chapter is aimed at providing a detailed feasibility study of solar cookers containing LHTES integrated with parabolic concentrator. Numerical studies are performed to analyze the heat transfer phenomena in PCM-CEG composite embedded inside the thermal storage container with and without circumferentially oriented radial fins. Enthalpy updating scheme addressing the inclusion of CEG matrix embedded within PCM has been adapted for the numerical prediction of melt fraction within the LHTES.

1 Introduction

Thermal energy demands associated with cooking is majorly catered by fossil fuels. The ever-increasing global demand for energy over the last century led to extensive usage of fossil fuels, causing the depletion of natural reserves at an alarming rate. It is not only the severe depletion of these natural reserves, but also the adverse effects on the environment due to the burning of fossil fuels, that led the scientific community to explore alternative renewable resources of energy with lesser environmental impact. Solar energy has a tremendous potential to substitute fossil fuels for many energy extensive applications, particularly in those areas having an abundance of solar irradiation. According to NREL (https://www.nrel.gov/gis/data_solar.html), the average Direct Normal Irradiance (DNI) at Jodhpur India is significantly high (~5 ${\text{kWh}}/{\text{m}}^{2} /{\text{day}}$). The abundance of solar irradiation in and around Jodhpur (aptly named as Sun-city) along with the entire northwestern belt of India (consisting of Rajasthan, Gujrat, Haryana Punjab and part of Madhya Pradesh) makes this region ideal for harvesting solar energy as a renewable alternative. Among various solar photovoltaic and solar thermal applications, the usage of solar thermal energy for household and commercial cooking has a huge potential to replace conventional fossil fuels to a large extent. Barring the monsoon season and few odd cloudy days over the year solar cooking can reliably cater to energy requirement for cooking with zero carbon footprint. However, to make this application acceptable and more appealing at large, we need to improve the technology in terms of flexibility in cooking schedule. Almost all the commercially available solar cookers do not offer this flexibility of cooking schedule. And the handful of those offer this flexibility are based on sensible heat storage with limited storage capacity at comparatively low temperature (~100 to 150 °C) (Cuce 2018). The energy storage density being small for sensible heat storage, large volume of the storage unit poses a big challenge. Large temperature gradient within the storage medium also hinders the optimal utilization of stored heat. Since the energy storage density of latent heat storage is significantly larger than sensible heat storage, the volume of the storage unit can be reduced to a large extent. Latent heat storage mechanism also ensures heat supply during cooking to be maintained within a narrow temperature range. All these special features associated with latent heat thermal energy storage (LHTES) makes this technology an ideal thermal storage solution for providing the flexibility of cooking schedule with concentrated solar cooker.

The latent thermal energy storage unit consists of a metallic storage container filled with PCM. Different ways of cooking require heat supply at different ranges of temperatures, namely boiling (100–150 °C), simmering (120–200 °C), frying (180–250 °C), grilling (200–300 °C), and so on. Therefore, an ideal PCM for the LHTES should have melting temperature above 250 °C to cater to all these cooking requirements. Conclusive details on various types of PCMs along with their thermophysical properties are summarized by Sharma et al. (2009), Zalba et al. (2003), Iverson et al. (2012), and Kenisarin (2010). From these data on thermophysical properties, it is evident that suitable PCMs having melting temperatures in the range of 250–300 °C are mostly pure inorganic salts or salt compositions with thermal conductivity raging between 0.25 and 0.5 ${\text{W}}/{\text{mK}}$. Therefore, thermal conductivity enhancement of the PCM is one of the key challenges in designing the LHTS for solar cooking. The charging process of the LHTES involves complete melting of the PCM inside the storage container during the on-sun period, thus storing the thermal energy in the form of latent heat slightly above the melting temperature of the PCM. On the other hand, discharging process involves heat release from the storage device during off-sun hours cooking activities, rendering the PCM to resolidify in the process. For the current analysis, the sizing of the LHTES unit is done on the basis of thermal energy requirement for cooking a typical evening meal for a six-member family, with the menu consisting of roti, dal, rice, and vegetables. The major design challenge is storing sufficient energy above 300 °C within on-sun duration (between 10:00 am and 4:00 pm) to meet the abovementioned cooking requirement.

A brief overview on existing literature reporting the usage of latent heat storage unit integrated with solar cookers is now presented. Domanski et al. (1995) presented a box-type solar cooker with concentric cylindrical vessels having the annulus portion filled with magnesium nitrate hexahydrate as the PCM. The experiments revealed that the effectiveness of this box-type solar cooker strongly depends on solar intensity, cooker medium mass, and the thermophysical properties of the PCM. The heat transfer from the PCM to the cooking pot is also observed to be slow during discharging process. Buddhi and Sahoo (1997) also reported the performance of a box-type solar cooker with stearic acid as the latent heat storage medium. One of the major outcomes of their experimental observation is the possibility of cooking two batches of about 200 gm rice per pot per day. The maximum temperature obtained in this setup was around 100–110 °C. Buddhi et al. (2003) presented a box-type solar cooker with acetanilide as a latent heat storage medium. They used ternary reflectors to increase the solar radiation. Vigneswaran et al. (2017) studied a box-type solar cooker with multiple reflectors where oxalic acid dihydrate is used as the latent heat storage medium. It was observed that time duration to reach melting temperature is reduced when multiple reflector boosters are used. Sharma et al. (2005) experimentally investigated an evacuated tube solar collector coupled with a latent heat storage unit. Commercial-grade erythritol was used as the PCM. Experimental results showed that the system can cook successfully twice, afternoon and evening daily. Temperatures reached more than 110 °C during evening cooking. Hussein et al. (2008) constructed a solar cooker with heat pipes and double flat plate reflectors. It was noted that the ratio of steel wool incorporated with the PCM can be increased to enhance its effective thermal conductivity. Coccia et al. (2018) constructed a double-walled container storage composed of two stainless steel cylindrical pots constructed concentrically. The annular area in between the pots was filled with a PCM consisting of a ternary mixture of nitrite and nitrate salts (53 wt% ${\text{KNO}}_{3}$, 40 wt% ${\text{NaNO}}_{2}$, 7 wt% ${\text{NaNO}}_{3}$). A significant improvement of load thermal stabilization during the off-sun period is reported when PCM is used as the storage medium. Lecuona et al. (2013) used a parabolic reflector integrated with LHTES having Paraffin and erythritol as PCM. It was reported that it is feasible to cook the three meals for a family during summer and also in winter. Paraffin was recommended to be a better option than erythritol. Kumaresan et al. (2015, 2016) studied a parabolic trough reflector focussing solar radiation onto the absorber tube. Therminol 55 was used as the heat transfer fluid (HTF) and encapsulated of D-mannitol was used as the PCM. It was observed that the tava cooking unit developed could cater heat comparable to an LPG stove running in the simmering mode. Veremachi et al. (2016) designed and developed a double reflector with axis tracking mechanism. A mixture of sodium nitrate and potassium nitrate at a ratio of 60:40 (mol%) was used as the PCM. A hotspot temperature of 277 °C could be attained with moderate temperatures gradient within the storage unit. Bhave and Thakare (2018) proposed a parabolic solar concentrator integrated with LHTES having magnesium nitrate hexahydrate as the storage medium and thermic mineral oil as HTF. It was reported that in about 50 min the PCM temperature reached 135 °C. Fifty grams of rice with 100 ml water were found to be completely cooked in about 30 min. El-Sebaii et al. (2009) studied degradation associated with thermal cycling of commercial-grade acetanilide and magnesium chloride hexahydrate for solar cooker application. Acetanilide was found to be a more promising PCM for this application. Magnesium chloride hexahydrate degrades easily during thermal cycling because of the phase segregation. Foong et al. (2010) performed numerical and experimental study on ${\text{KNO}}_{3}$-${\text{NaNO}}_{3}$ salt composition (60:40 weight ratio) as the latent heat storage medium. A double reflector-based parabolic dish concentrator is used for this purpose. The storage container is augmented with fins for improving the thermal performance of the storage unit. The temperature range of 227–327 °C was achieved.

The review of the existing literature on solar cookers with integrated LHTES provides a basis for categorizing such solar cookers containing various design aspects with specific salient features. The block diagram presented in Fig. 1 provides a consolidated list of existing solar cooker technologies addressing thermal energy storage. The salient features of each of these solar cookers are now discussed in brief.

Fig. 1

Classifications of solar cookers with LHTES

For direct solar cookers, incident solar radiation is directly used for cooking or storing thermal energy to be dispatched during the hour of need. Solar cookers enlisted in this category are box-type solar cooker, and solar cookers with single or double paraboloid concentrators. Box-type solar cookers can be farther classified as box-cookers without reflector and box-cookers with single or multiple reflectors.

Box-cookers without reflector: Fig. 2 shows a typical design of box-cooker without reflector (Buddhi and Sahoo 1997). A double-walled tray-type container with PCM filled within the space between the walls attributes to the key design feature. The thermal storage tray containing PCM is kept under a glass lid. The upper surface of the tray is covered with absorber material. Sunrays pass through the glass lid, strike the absorber tray, and transfer heat to the PCM underneath it for charging. The tray is surrounded by insulating material. The performance of this type of cooker depends upon the ability of the transparent glass to permit passage of shorter wavelength which forms a significant part of the solar spectrum but opaque to higher wavelength coming out from the box. The temperature inside the box rises because of the greenhouse effect. However, this setup is found to be inefficient because solar rays are not concentrated by any means, and the charging time is very long.

Fig. 2

Schematic of box-cooker without reflector having LHTES (Buddhi and Sahoo 1997)

Box-cookers with single/multiple reflectors: These box-cookers contain single/multiple reflectors (Fig. 3) to concentrate solar radiation on a vessel containing PCM (Vigneswaran et al. 2017). Cooking containers and interior of these box-cookers are made of absorbing material. The box is insulated from sides. The mirrors need to be adjusted with respect to each other to concentrate the solar radiation. Multiple reflectors provide concentrated solar energy to the cooking and storage container enabling the system to achieve temperatures in the range of 100–120 °C.

Fig. 3

Box-cooker with multiple reflector (Vigneswaran et al. 2017)

Concentrated solar cooker with single paraboloid reflector: These solar cookers contain a single paraboloid reflector to concentrate solar irradiation (Fig. 4) on a double-walled container cum cooking pot with PCM filled within the space between the walls Lecuona et al. (2013). The bottom surface of the storage cum cooking container is placed closed to the focal point of the paraboloid reflector. One of the major challenges for this design is associated with continuous solar tracking during the operation. Hence, integration of an axis tracking mechanism is a must for this design concept. Locating the heavy storage cum cooking container above the reflector renders this tracking even more challenging. However, we can obtain very high temperatures of about 250–300 °C adjacent to the focal point location, enabling the cooker to cater frying and grilling requirements unlike box-cookers adequate only for boiling-type cooking requirements.

Fig. 4

Solar cooker with paraboloid reflector (Lecuona et al. 2013)

Primary paraboloid reflector coupled with secondary reflector PCM solar cooker: Since the positioning of heavy cooking cum storage vessel over the paraboloid reflector poses considerable challenge with respect to the solar tracking requirement, arrangement of a double reflector-based design (Fig. 5) is proposed to circumvent this challenge Veremachi et al. (2016). The key feature of this design is the positioning of a secondary reflector close to the focal point of the primary reflector. The primary reflector is a paraboloid reflector which reflects solar radiation onto the secondary reflector. The secondary reflector concentrates the radiation on the storage container during the charging process. A hole at the center of the primary reflector allows the radiation from the secondary reflector to reach the storage container kept beneath the primary paraboloid reflector. This setup is ergonomically more viable for solar tracking operation as the heavy storage container sits at bottom most location. Another advantage of this mechanism is obtaining a fixed hotspot location on the storage container during charging process which is otherwise very difficult to obtain with single reflector design. However, for this design, simultaneous cooking and charging operation is limited to boiling type of cooking only. Cooking requirements like frying or grilling can only be addressed once the charging process of the storage unit is over.

Fig. 5

Primary and secondary reflector solar cooker Veremachi et al. (2016)

Next, we briefly discuss some of the existing designs of indirect solar cookers. Unlike direct solar cookers solar irradiation is not directly used for cooking or energy storage, rather a heat transfer fluid (HTF) is used as the thermal transport medium to transfer heat from the receiver to the cooking cum storage unit. The thermic fluid is continuously circulated between the receiver and storage cum cooking unit during the cooking and storage operation. Three types of receivers are particularly of interest, namely evacuated tube, parabolic trough, and flat plate collector.

Evacuated tube solar cooker: Fig. 6 depicts the key components of solar cooker with an evacuated tube receiver (Sharma et al. 2005). The evacuated tube consists of an inner metallic pipe encased within a transparent glass tube. The exterior surface of the inner metallic tube is coated with a solar selective coating having high absorptivity and low emissivity. Vacuum is maintained within the gap between the inner metallic tube and an outer glass tube to protect the solar selective coating from degradation due to contact with air. The arrangement also ensures minimum heat loss through convection and radiation mechanism. During on-sun hours, solar radiation is absorbed by the evacuated tube solar collector, and heat is transferred to the HTF circulating through the heated tube. The HTF circulation is achieved using a closed-loop pumping setup. The pump drives the HTF in a close loop through a steel tubing arrangement wrapping around the storage cum cooking container and the evacuated tube receiver. Once again the storage cum cooking container is a double-walled vessel with the gap between the two walls filled with PCM. The HTF circulation system passes through this PCM-filled domain. The HTF circulation loop transports heat from the evacuated receivers to the storage cum cooking container. The flow of hot HTF through the PCM causes the melting of PCM and stores heat in PCM in the form of latent heat, which can be used for cooking at later duration. The absence of solar irradiation concentrator restricts this system to attain high temperature and confines the temperature below 100 °C. Additional pump setup, flow meter, relief valves, and electrical power are required for this closed-loop setup.

Fig. 6

Schematic diagram of evacuated tube solar cooker with LHTES (Sharma et al. 2005)

Parabolic trough solar cooker: The functionalities of a solar cooker with parabolic trough as the receiver is quite similar to the evacuated tube solar cooker. The only difference between the two being receiver type. However, owing to the usage of a parabolic concentrator, the very high HTF temperature can be achieved by using multiple parabolic concentrators. The performance of this solar cooker can be further improved by using a combination of evacuated tube and parabolic reflector. The schematic of such an arrangement is shown in Fig. 7 (Kumaresan et al. 2015, 2016).

Fig. 7

Schematic diagram of parabolic trough solar cooker with heat transfer fluid

Flat plate collector PCM solar cooker: This setup consists of a flat plate solar collector through which the HTF is circulated (Fig. 8) before delivering the heat to the cooking cum storage container containing PCM as storage material similar to the evacuated tube and parabolic trough solar cooker systems. Once again the absence of concentrator limits the obtainable temperature below 100 °C.

Fig. 8

Schematic diagram of flat plate collector PCM solar cooker (Hussein et al. 2008)

Classification of PCMs used for solar cookers

Several types of PCM have been explored by research groups across the globe as latent heat thermal energy storage medium for solar cooking applications. A consolidated list of PCMs used as thermal energy storage medium for cooking application is shown in the chart presented by Figs. 9 and 10. The pros and cons of organic, inorganic, and eutectic PCMs are listed in Table 1.

Fig. 9

Classification of PCM (Memon 2014)

Fig. 10

Fusion enthalpy (${\text{kJ}}/{\text{l}}$) versus melting temperature (°C) of various types of PCMs (Kalnæs and Jelle 2015)

Table 1 Merits and demerits of organic, inorganic, and eutectic PCMs

Selection standards for PCM used in the PCM solar cooking device: There are specific features to be addressed while choosing PCM as thermal storage medium for solar cooking application. The preferable features can be summarized as follows (Memon 2014):

• A. Thermophysical features

• 1. Large energy storage density to ensure a compact storage unit

• 2. Reasonably high thermal conductivity to achieve rapid charging process and maintain thermal uniformity and rapid heat flow during cooking or discharging process

• 3. Large specific heats for solid and liquid phase also ensure higher storage density

• 4. Small volumetric changes during phase transitions allow simple design requirement for storage container

• 5. Thermal stability of PCM and low degradation rate allow a large number of thermal cycles and infrequent replacement requirement for thermal storage material.

• B. Chemical features

• 1. The chemical composition must remain stable and uniform over a large number of thermal cycles involving phase change process

• 2. PCM must be non-toxic, non-corrosive, and non-explosive.

• C. Economic features

• 1. The material used for the storage medium must be inexpensive

• 2. Readily accessible.

• D. Environmental features

• 1. The PCM must be non-contaminating

• 2. The PCM must be easily replenishable.

Thermophysical properties of some commonly used and potential PCMs for solar cooking applications are enlisted in Table 2.

Table 2 Property table for potential PCMs for solar cooking application

Although discussions regarding design aspects of box-type solar cookers with latent heat storage are frequently encountered (Domanski et al. 1995; Buddhi and Sahoo 1997; Buddhi et al. 2003; Vigneswaran et al. 2017; Coccia et al. 2018), and reports on dish concentrator-based solar cookers with latent heat storage are rarely available (Bhave and Thakare 2018; Foong et al. 2010). In the subsequent section, we provide a detailed modeling description of double reflector-based parabolic dish solar cookers integrated with latent heat storage unit.

2 Design Details of Double Reflector-Based Parabolic Solar Cooker Integrated with Latent Heat Storage Unit

In order to store sufficient thermal energy to cook meal for a family, the total weight requirement of the thermal storage medium is quite high (~10 kg). The weights of the container and insulation material are added over it. Placing this heavy storage unit above the parabolic dish concentrator poses serious structural constraint. For any practical purpose, such top heavy arrangement is not desirable. Continuous solar tracking requirement for the parabolic dish concentrator complicates the matter farther. Locating the bulky storage structure above, the concentrator also restricts a significant amount of sunlight to reach the parabolic dish reflector due to the shadow effect. All these structural and optical challenges can be comprehensively addressed by an alternative design approach of locating the storage unit below the parabolic dish concentrator with double reflector arrangement. Figure 11 shows such an arrangement. The solar irradiation captured by the primary reflector is first focussed on a much smaller secondary reflector (Fig. 11), which directs the focussed solar rays on the storage container located below the primary reflector. A small hole at the center of the primary reflector allows the rays from the secondary reflector to pass through and incident on the latent heat storage container. The large primary reflector is a parabolic dish and the secondary smaller reflector is a hyperbolic dish. The design constraints of these two reflectors are briefed in the subsequent discussions.

Fig. 11

Schematic diagram of double reflector-based high temperature thermal storage arrangement, a 2-D view, b 3-D view

Primary and secondary reflector design: Solar energy received from the solar irradiation needs to be concentrated to receive high heat flux at high temperature in the designated receive location of the storage container. The primary and secondary reflector arrangement ensures the concentration of high heat flux at a reasonably high temperature ($\sim 350\,^\circ {\text{C}}$) to be incident on the receiver location of the storage container. To design the reflectors, we require the knowledge of total thermal energy for cooking ($E_{c}$) the target meal proportion, target charging duration ($\Delta t_{\text{ch}}$) of the thermal storage unit, average local DNI (I), and the maximum temperature ($T_{\max } > T_{m}$) attainable by the PCM without degrading. Once the total requirement of energy for cooking and charging duration is identified, the incident area ($A$) of the primary reflector can be estimated from the following relation.

$$E_{c} = I \cdot A \cdot \Delta t_{\text{ch}} \cdot \alpha$$

(1)

where $\alpha$ represents the absorptivity of the vessel surface.

Average cooking energy ($E_{c}$) requirement above 100 °C for a six-member family is typically 1.2–1.5 ${\text{kWh}}$ (Buddhi et al. 2003). The average DNI observed at Jodhpur is approximately 600 ${\text{W}}/{\text{m}}^{2}$. However, to be on the safer side, an average DNI of 450 ${\text{W}}/{\text{m}}^{2}$ is considered. Typically, the aperture diameter of the parabolic dish reflector varies within the range of 1.2–2 m. On the other hand, the target charging time should ideally be less than 5 h. A safe value of absorptivity ($\alpha$) can be assumed to be 85%. Considering all these facts, and taking into account of central hole and shadow effect of the structural frame to support the secondary reflector, Eq. 1 can provide an accurate diameter requirement of the primary reflector. TracePro ray tracing software is used for designing the primary (paraboloid) and secondary (hyperboloid) reflectors (Fig. 12) along with the incident heat flux distribution on the receiver area of the thermal storage container.

Fig. 12

a Light path of Cassegrain model; graphical definition of the b paraboloid, c hyperboloid reflectors

The ‘classic’ Cassegrain reflector model is considered for concentrating solar energy (Fig. 12a). A hole is made in the center of the paraboloid reflector to allow the rays to pass through to the receiver location of the thermal storage container. The hyperboloid is kept above the primary reflector and aligned along the same vertical axis of the paraboloid. The sun rays are reflected from the primary paraboloid reflector to the secondary hyperboloid reflector, which in turn concentrates solar radiation upon the cooking vessel. The paraboloid reflector reflects all incoming light rays parallel to its axis of symmetry toward a single point, the focus. The converging light rays reflected by the parabolic concentrator toward its focus are intercepted by a secondary hyperboloid reflector. The hyperboloid reflector surface having two foci reflects all light rays directed toward one of its two foci in the direction of its other focal point (Fig. 12a). The folded optical path in this manner results in a very compact system; hence, hyperboloid geometry is considered for the secondary reflector (Horne 2012). The light rays reflected from the hyperboloid need not be parallel. This novel design helps in decreasing the shadow effect since the aperture of the hyperboloid is considerably smaller than the radial span of the storage container. This bottom heavy arrangement also facilitates structural stability and ergonomics of the storage cum cooking container.

Recalling the basic features of paraboloid geometry, the curved surface formed by rotation of a parabola about its axis is called a paraboloid of revolution. The mathematical equation for the paraboloid (Fig. 12b) in Cartesian coordinate system with the z-axis as the axis of the paraboloid is as follows.

$$x^{2} + y^{2} = 4fz$$

(2)

where the distance f is the focal length.

In cylindrical coordinates, the equation becomes:

$$z = \frac{{r^{2} }}{4f}$$

(3)

The aperture area of the paraboloid is given by:

$$A_{p} = \frac{{\pi D^{2} }}{4}$$

(4)

where D is the aperture diameter.

In terms of aperture diameter and focal length, the height of the dish h is:

$$h = \frac{{D^{2} }}{16f}$$

(5)

The surface area of the paraboloid is given by:

$$A_{s} = \frac{{8\pi f^{2} }}{3}\left\{ {\left[ {\left( {\frac{D}{4f}} \right)^{2} + 1} \right]^{3/2} - 1} \right\}$$

(6)

Concentration ratio is defined as the ratio of aperture area of the concentrator to the focal area of the receiver.

$${\text{CR}} = \frac{{A_{p} }}{{A_{r} }}$$

(7)

where $A_{p}$ denotes the aperture area of the concentrator and $A_{r}$ denotes the focal area on the receiver surface.

Collector Efficiency

The solar energy collection efficiency $\eta_{\text{col}}$ of thermal collectors is defined as the ratio of the rate of useful thermal energy leaving the collector, to the useable solar irradiance falling on the aperture area. Simply stated, collector efficiency is

$$\eta_{\text{col}} = \frac{{Q_{\text{useful}} }}{A.I}$$

Optical Efficiency of Collector

Optical efficiency is formulated as

$$\eta_{\text{opt}} = \tau \cdot \alpha$$

where $\tau$ is the transmittance of reflector glass and $\alpha$ is the absorptivity of receiver surface.

Recalling the hyperboloid geometry the equation representing a hyperboloid in Cartesian coordinate system is given by:

$$\frac{{x^{2} }}{{a^{2} }} + \frac{{y^{2} }}{{b^{2} }} - \frac{{z^{2} }}{{c^{2} }} = - 1$$

(8)

The intercepts of the hyperboloid with z-axis is given by (0, 0, ±c). There are no intersections with the x, y-axes.

Design of thermal storage container: Dimensions of the storage container are obtained on the basis of the required cooking energy to be stored above 300 °C. We chose the total cooking energy requirement to be 1.5 ${\text{kWh}}$ which is sufficient to cook a sumptuous Indian style meal for a six-member family (Buddhi et al. 2003). A cylindrical steel vessel of wall thickness 3 mm is considered. A ceramic coating of 2 mm thicknesses on the inner surface of the steel container is also considered. Ceramic coating is considered to prevent the corrosive effect of molten PCM at elevated temperatures on the steel container (Moreno et al. 2014). However, steels with anticorrosive properties are available, and thermal storage container made with such corrosion resistant steel may not require this protective ceramic layer.

Two different designs for thermal storage container are considered for the present study (Fig. 13). The first design (Fig. 13a) consists of a cylindrical steel vessel without fin arrangement, while the second design (Fig. 13b) involves a cylindrical steel vessel with circumferentially distributed copper fins. For the finned design of thermal storage vessel, the central copper rod is of diameter 10 mm, and each of the radially extended copper plates from the central rod has a thickness of 2 mm.

Fig. 13

Schematic diagram of storage container, a without fin, b with circumferentially distributed fins

Selection of PCM material: In the present study, ${\text{NaNO}}_{3}$-CEG (sodium nitrate-compressed expanded graphite) composite is used as latent heat storage medium with a melting temperature of 308 °C. The major drawback of pure ${\text{NaNO}}_{3}$ is that its low thermal conductivity (0.5 ${\text{W}}/{\text{mK}}$) causing accumulation of heat near the receiver area of the storage container. If pure ${\text{NaNO}}_{3}$ is used as the latent heat storage medium, the low thermal conductivity causes a large temperature gradient within the storage medium with a local hotspot near the receiver region. Very high temperature (${\sim}400\,^\circ {\text{C}}$) at the hotspot may lead to decomposition of nitrate to nitrite and degrade the thermal properties of ${\text{NaNO}}_{3}$. Therefore, enhancement of thermal conductivity is of absolute necessity for such high-temperature application. The thermal conductivity of ${\text{NaNO}}_{3}$ is enhanced by incorporating 10% graphite by volume in the form of compressed expanded graphite (CEG). ${\text{NaNO}}_{3}$-CEG composite consists of a highly porous CEG matrix impregnated with ${\text{NaNO}}_{3}$. The thermal conductivity of ${\text{NaNO}}_{3}$-CEG is higher by two orders of magnitude as compared to pure ${\text{NaNO}}_{3}$ (Kenisarin 2010). Incorporating graphite in PCM results in anisotropic thermal conductivity in the PCM domain. An interesting aspect of PCM-CEG composite is that natural convection does not play a dominant part in the heat transfer and is dominated by heat diffusion only (Py et al. 2001). The small pore size of CEG matrix allows the composite to be analyzed as a homogeneous material. The thermophysical properties have been estimated using volume averaging approach. Homogeneous distribution assumption of very small pore size allowed us to consider this volume averaged estimation of thermal conductivity approach with reasonable accuracy (Mallow et al. 2016). The major limitation of this approach is negating the existence of thermal non-equilibrium between the graphite and PCM due to large difference in thermal conductivities of these two materials. Inclusion of thermal non-equilibrium effect between the graphite and PCM in the numerical model is beyond the scope of the present work and needs to be explored farther providing a future scope for the present study. Based on the volume calculation of the storage container, the total weight of ${\text{NaNO}}_{3}$ content is approximately 7 kg, which is capable of storing approximately 1.2 ${\text{kWh}}$ energy above 100 °C. Thermophysical properties of container material, PCM (Venkateshwar et al. 2017), and CEG (Bodzenta et al. 2011) are provided in Table 3.

Table 3 Thermophysical properties of container material, PCM (Venkateshwar et al. 2017), and CEG (Bodzenta et al. 2011)

3 Numerical Modeling

In this section, we present the numerical model to capture storage dynamics during the charging cycle. The first step in this endeavor is to evaluate incident flux distribution on the receiver region of the thermal storage container. Once this incident flux profile is obtained, it is used as the boundary condition to solve the energy conservation equation in the storage domain.

Incident flux profile: The incident heat flux profile is obtained by using ray tracing TracePro software (Figs. 14 and 15). Average DNI data measured at IIT Jodhpur is used as the input to the software for attaining irradiation flux profile on the receiver surface. The flux profile obtained follows Gaussian distribution.

Fig. 14

Ray tracing diagram of the double reflector system

Fig. 15

Flux profile obtained in Ray Tracepro software

The incident flux profile $q(r)$ on the PCM vessel is formulated as

$$q\left( r \right) = A_{1} {\text{e}}^{{\left( {\frac{{r - b_{1} }}{{c_{1} }}} \right)^{2} }} + A_{2} {\text{e}}^{{\left( {\frac{{r - b_{2} }}{{c_{2} }}} \right)^{2} }} + A_{3} {\text{e}}^{{\left( {\frac{{r - b_{3} }}{{c_{3} }}} \right)^{2} }} + A_{4} {\text{e}}^{{\left( {\frac{{r - b_{4} }}{{c_{4} }}} \right)^{2} }}$$

(9)

where $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, $b_{1} ,b_{2}$, $b_{3}$, $b_{4}$, $c_{1} ,c_{2} ,c_{3} ,c_{4}$ are discrete functions of time which remain constant for each time intervals of 20 min. MATLAB has been used to obtain the approximate Gaussian distribution (Eq. 9) from the solar irradiance data with uncertainty of less than 5%.

Numerical modeling: The governing energy balance equation for the PCM-CEG domain is derived on the basis of volume averaging formulations proposed by Brent et al. (1988), Bennon and Incropera (1987), and Shrivastava and Chakraborty (2019). Since the PCM is trapped within the porous matrix of CEG in the composite, convection within PCM melt is neglected and the energy balance equation is diffusion dominated. In deriving the energy balance equation, the difference between specific heat capacities of solid and liquid phases of PCM is considered (Shrivastava and Chakraborty 2019). The sensible heating of graphite matrix is also considered (Shrivastava and Chakraborty 2019). The energy conservation equation in the PCM-CEG composite domain using temperature as the primary dependent variable can be formulated as follows (Shrivastava and Chakraborty 2019):

$$\begin{aligned} \frac{\partial \left( T \right)}{\partial t} & = \nabla .\left( {\frac{k}{{\rho c_{ps} }}\nabla T} \right) - \frac{\partial }{\partial t}\left( {\frac{{g_{l} h_{sl} }}{{c_{ps} }}} \right) - \frac{\partial }{\partial t}\left[ {g_{l} \left( {\frac{{c_{pl} }}{{c_{ps} }} - 1} \right)\left( {T - T_{m} } \right)} \right] \\ & \quad - \frac{\partial }{\partial t}\left[ {g_{g} \left( {\frac{{c_{pg} }}{{c_{ps} }} - 1} \right)T} \right] \\ \end{aligned}$$

(10)

The second, third, and fourth source terms appearing on the right-hand side of Eq. 10 represent contributions due to phase change, difference between liquid and solid phase specific heats of PCM and difference between graphite and solid phase PCM specific heat, respectively. Since the graphite matrix has anisotropic thermal conductivity, the effective thermal conductivity (k) is defined in the following manner (Shrivastava and Chakraborty 2019).

$$k_{i} = g_{l} k_{l} + g_{s} k_{s} + g_{g,i} k_{g,i}$$

(11)

where subscript i represents (r, x) direction in cylindrical coordinate system, subscripts l, s represents liquid and solid phases of PCM, and subscript g represents graphite.

The energy equation for ceramic, steel, and copper fin regions is formulated as follows.

$$\frac{\partial }{\partial t}\left( T \right) = \nabla .\left( {\frac{k}{{\rho c_{p} }}\nabla T} \right)$$

(12)

Heat loss from the circumferential surface and bottom surface is assumed to be zero, i.e., these surfaces are considered to be perfectly insulated. The surface area of the storage container pertaining to the receiver location is provided with the heat flux boundary condition given by Eq. 9.

Volume fraction updating scheme is implemented to estimate the liquid volume fraction using the following updating scheme (Shrivastava and Chakraborty 2019)

$$g_{lp}^{n + 1} = g_{lp}^{n} + \lambda \left[ {\left\{ {\frac{{A_{p} + D_{p} + C_{p} g_{lp}^{n} }}{{B_{p} }}} \right\}\left( {T - T_{m} } \right)} \right]$$

(13)

where subscript p represents the nodal point where volume fraction $g_{l}$ is being updated. $A_{p}$ represents coefficient of $T_{p}$ obtained by finite volume discretization of Eq. 10 (Patankar 2018) when contribution of only left-hand side and first term (diffusion term) of the right-hand side of Eq. 10 is considered (Chakraborty 2017). λ is an under-relaxation factor. Superscripts ‘n’ and ‘$n + 1$’ denote iteration steps during implicit calculation of T and $g_{l}$ field at a given time step. $B_{p}$, $C_{p}$ and $D_{p}$ are given as follows:

$$B_{p} = \frac{{h_{sl} }}{{c_{ps} }};\quad C_{p} = \frac{{c_{pl} }}{{c_{ps} }} - 1;\;\;D_{p} = g_{g} \left( {\frac{{c_{pl} }}{{c_{ps} }} - 1} \right)$$

(14)

The liquid volume fraction $g_{l}$ ranges within the limit: $0 \le g_{l} \le 1 - g_{g}$, where $g_{g}$ represents volume fraction of graphite at an elementary control volume. If the numerical value of $g_{lp}^{n + 1}$ from Eq. 13 comes outside the range $0 \le g_{l} \le 1 - g_{g}$, it is updated as the nearest limit such that:

$$\begin{array}{*{20}l} {g_{lp}^{n + 1} = 0} \hfill & {\text{for}} \hfill & {g_{lp}^{n + 1} < 0} \hfill \\ {g_{lp}^{n + 1} = 1 - g_{g} } \hfill & {\text{for}} \hfill & {g_{lp}^{n + 1} > 1 - g_{g} } \hfill \\ \end{array}$$

(15)

Since the energy equation solved by ANSYS-FLUENT solve (Fluent ANSYS 2015) either enthalpy or total energy as primary variable, Eq. 1 could only be solved by treating temperature T as the user-defined scalar (UDS). The enthalpy updating scheme is applied through user-defined function. The anisotropic thermal conductivity (Eq. 11) and the heat flux boundary condition (Eq. 9) are also implemented through UDFs. A grid size of 1 mm has been used for numerical simulations followed by 0.5 s time step.

4 Results and Discussions

The thermal storage dynamics during the charging process is studied for two different sets of storage configurations. The first configuration consists of thermal storage container without any fin arrangement inside the PCM-CEG domain (Fig. 13a). The second configuration contains circumferentially distributed copper fins within the storage medium (Fig. 13b).

Case study 1 (Thermal storage container without fin): This study could have been performed for 2-D axisymmetric geometry. However, the other case study involving circumferentially oriented fin cannot be resolved with 2-D axisymmetric simplification, and a 3-D approach is a must. The 3-D domain can however be reduced conveniently because the fins are oriented with regular angular intervals. In order to have a better comparison with the finned configuration, the case study involving unfinned thermal storage container is carried out in 3-D domain. One-eighth symmetry with an angular span of 45° (Fig. 16) is chosen as the reduced domain for case study 1 in order to reduce total computation time.

Fig. 16

Computational domain for no-fin storage container configuration, a schematic diagram, b hex mesh generated in ANSYS

The initial temperature of the complete system is assumed to be at ambient temperature 27 °C. The charging process of the PCM is carried out between 10 am to 1:20 pm, i.e., for 3 h and 20 min since the average DNI in this period is maximum. The maximum temperature attained in the system during charging after 12,000 s is 374 °C (647 K) which is tantalizingly close to the temperature 380 °C at which decomposition of ${\text{NaNO}}_{3}$ in ${\text{NaNO}}_{2}$ starts. The minimum temperature of the storage medium is found to be 303 °C (5 °C below the melting point temperature $T_{m} = 308\,^\circ {\text{C}}$ of ${\text{NaNO}}_{3}$) indicating incomplete melting of the storage medium.

The temperature profile evolution in time for the PCM-CEG domain within the storage unit is presented in Fig. 17 at various time intervals. The solar irradiation flux profile at the receiver region of the CEG-PCM thermal storage container is such that it creates a hotspot at the center of the vessel top surface. The non-uniform temperature distribution (Ranging over a temperature difference of ~30 to 70 °C at different time instants) within the PCM may be attributed to the low thermal conductivity of the PCM. The melting fraction evolution is shown in Fig. 18 at the same time intervals 9000, 10,000, 11,000, and 12,000 s. It is to be noted from Fig. 18, that melting of the PCM starts at around 10,000 s. The complete melting is achieved at 12,100 s, and the time required for complete melting is around 2100 s. One of the interesting non-intuitive features of the melting dynamics is found to be the location where melting process occurs at the end. We found that the location where the melting process defers till the end is adjacent to the central region of the storage container (Fig. 18d) and not the farthest region from the hotspot or receiver area. The reason for this non-intuitive melting feature is attributed to the fact that steel having reasonably larger thermal conductivity (${\sim}16\,{\text{W}}/{\text{mK}}$) carries the heat from the hotspot along the container wall, promoting faster meltdown of the PCM adjacent to the wall. On the other hand, comparatively low thermal conductivity of CEG in the axial direction due to anisotropy causes slower heat transfer from the hotspot in the axial region, rendering the central region of the storage container not to reach melting temperature till the very end.

Fig. 17

Temperature (Kelvin) profile of PCM-CEG domain without fin at: a 6000 s, b 9000 s, c 10,000 s, d 11,000 s, e 11,500 s, and f 12,000 s

Fig. 18

Melting fraction profile of PCM-CEG domain without fin at: a 9000 s, b 10,000 s, c 11,000 s, d 12,000 s

Case study 2 (Thermal storage container with fin arrangement): Case study with fin arrangement consists of three subsystem studies with three different fin orientations in the circumferential direction. These three different fin configurations consists of radially extended copper plane plate fins attached to a central copper rod with an angular pitch of 30°, 45°, and 60°. The schematic of 45° fin orientation is presented in Fig. 19a. Since the fins are located at regular angular intervals, the computation domain can be reduced comprehensively by considering the volume enclosed by two consecutive fins. Figure 19b shows the computation domain for 45° fin orientation. Figure 20 shows the mesh generated in ANSYS for three different fin arrangements. The charging process starts at 10 am and is carried out for 3 h and 20 min from the start time.

Fig. 19

a Schematic diagram of thermal storage container with 45° fin orientation, b reduced computational domain for 45° fin orientation

Fig. 20

Hex mesh generated in ANSYS for: a $30^\circ$, b $45^\circ$, and c $60^\circ$ fin orientations

Once again, the initial temperature of the complete system for all the cases is assumed to be at ambient temperature 27 °C Figs. 21, 22, 23, 24 and 25 represents temperature profile and melt fraction evolution in time for the PCM-CEG domain within the storage unit for fin orientation of 30°, 45°, and 60°, respectively. When the temperature profiles of finned containers (Figs. 21, 23 and 25) are compared with that of non-finned case study, we find that the difference between the highest and lowest temperature within the PCM-CEG domain is much smaller (${\sim}10\,^\circ {\text{C}}$ for 30°, ${\sim}13\,^\circ {\text{C}}$ for 45°, and ${\sim}16\,^\circ {\text{C}}$ for 60° fin orientations, respectively) indicating much better uniformity of temperature within the storage medium. The temperature maximums at 12,000 s are also found to be significantly lower (345, 339 and 335 °C for 30°, 45°, and 60° fin orientations, respectively) than the decomposition temperature (380 °C) of ${\text{NaNO}}_{3}$.

Fig. 21

Temperature (Kelvin) profile of PCM-CEG domain for $30^\circ$ fin orientation at: a 6000 s, b 9000 s, c 10,700 s, d 10,800 s, e 10,950 s, and f 12,000 s

Fig. 22

Melting fraction profile in PCM-CEG domain for $30^\circ$ fin orientation at: a 9000 s, b 10,700 s, c 10,800 s, and d 10,950 s

Fig. 23

Temperature (Kelvin) profile of PCM-CEG domain for $45^\circ$ fin orientation at: a 6000 s, b 9000 s, c 11,000 s, d 11,100 s, e 11,300 s, and f 12,000 s

Fig. 24

Melting fraction profile in PCM-CEG domain in $45^\circ$ fin orientation at: a 9000 s, b 11,000 s, c 11,100 s, and d 11,300 s

Fig. 25

Temperature (Kelvin) profile of PCM-CEG domain for $60^\circ$ fin orientation at: a 6000 s, b 9000 s, c 11,200 s, d 11,300 s, e 11,510 s, and f 12,000 s

A close look at maximum temperatures obtained for these three orientations reveals another non-intuitive finding. Since 30° fin orientation offers more thermal uniformity, we expect the hotspot temperature to be the lowest for this configuration among the three chosen orientations. However, the temperature profiles given by Figs. 21f, 23f, and 25f predict the maximum temperatures in an exactly reverse manner. The maximum temperature for 60° orientation is found to be the lowest ($T_{\max } = 335\,^\circ {\text{C}}$), followed by 45° ($T_{\max } = 339\,^\circ {\text{C}}$) and 30° ($T_{\max } = 345\,^\circ {\text{C}}$) fin orientations. This reverse trend might be attributed to the loss of PCM-CEG volume due to the volume occupancy of the fines. The 30° fin orientation has maximum volume loss due to the presence of maximum number of fins. Equivalent energy pertaining to the latent heat storage potential of this lost volume of PCM must be accounted for and manifested by an equivalent rise in the sensible temperature of the entire storage unit. The excess maximum temperature is essentially the manifestation of the latent heat of fusion of the lost volume of PCM. However, in this context, a question may arise: why is the no-fin arrangement manifesting the lowest maximum temperature? The answer to this question can be comprehensively addressed by temperature larger non-uniformity of the no-fin arrangement. The absence of fins causes a much larger temperature gradient from the hotspot to lowest temperature region. Comparison of minimum temperatures between non-finned and finned orientations shows that the minimum temperature is lowest (${\sim}303\,^\circ {\text{C}}$) for no-fin configuration. We must remember that the total amount of heat supplied to all these orientations is the same. Therefore, larger non-uniformity of temperature leads to larger maximum temperature.

Next we compare the melt fraction evolution presented by Figs. 22, 24, and 26. Unlike no-fin configuration, the last region to melt is found to be the farthest region from the hotspot. As is evident, 30° fin orientation promotes fastest meltdown of the entire PCM with maximum temperature uniformity (minimum difference between $T_{\max }$ and $T_{\min }$). We observe another interesting fact corresponding to the onset and termination of melting process. The earliest onset of melting occurs for the no-fin configuration (Fig. 18). Large temperature non-uniformity leads to accumulation of heat in the hotspot region leading to an early rise in temperature above the melting point ($T_{m}$) in this region. However, the same large temperature non-uniformity causes the most delayed completion of melting process. For the finned configurations (Figs. 22, 24 and 26), the onset of melting gets delays as compared to no-fin configuration because the temperature uniformity causes the entire PCM-CEG domain to reach the melting point ($T_{m}$) uniformly deferring the hotspot temperature to reach $T_{m}$ at an early stage. However, the complete melting duration is significantly smaller once onset of melting occurs.

Fig. 26

Melting fraction profile in PCM-CEG domain for $60^\circ$ fin orientation at: a 9000 s, b 11,200 s, c 11,300 s, and d 11,510 s

In continuation to the discussion regarding the effect of fin arrangement on maximum temperature ($T_{\max }$) of the domain,, the time evolution of $T_{\max }$ and $T_{\min }$ is shown in Fig. 27 for different fin configurations. The case study involving no fin predicts maximum non-uniformity of the temperature field with $\left( {T_{\max } - T_{\min } } \right)\sim 70\,{\text{K}}$. As the fin numbers are progressively increased ($60^\circ$, $45^\circ$, and $30^\circ$ fin orientations), $T_{\max } - T_{\min }$ values reduce subsequently indicating better uniformity of the thermal field. It is to be noted here, that the higher values of $T_{\max }$ associated with higher number of fins ($T_{\max @30^\circ } > T_{\max @45^\circ } > T_{\max @60^\circ }$) are also subjected to higher values of $T_{\min }$ evolution, i.e. $T_{\min @30^\circ } > T_{\min @45^\circ } > T_{\min @60^\circ }$, which is physically consistent.

Fig. 27

Variation of maximum and minimum temperatures with time for different fin configurations; smaller difference between maximum and minimum temperature denotes more uniformity of temperature field

Figure 28 shows the variation of overall melt fraction with respect to time. No-fin configuration promotes earliest onset of melting and most delayed completion of the same. The 30° fin configuration promotes next earliest onset of melting followed by the 45° and 60° fin configurations. The 30° fin configuration also has the steepest melt fraction versus time plot, depicting fastest completion of melting process followed by 45° and 60° fin configurations.

Fig. 28

Evolution of overall melt fraction with time

Figure 29 shows latent heat storage capacity of four different configurations under consideration. Although increasing the total number of fins reduces the charging time significantly, the price we pay is the reduction in total latent heat capacity of the storage. Therefore, a tradeoff must be put into place to optimize between the charging duration and latent heat storage capacity.

Fig. 29

Heat storage capacity (latent. heat) versus fin angle

5 Conclusion

An alternative design of parabolic dish solar cooker with double reflector arrangement integrated with latent heat storage unit located below the parabolic dish (primary reflector) is explored in terms of its thermal performance during the charging process. A diffusion dominated numerical model to address the melting of PCM in PCM-CEG composite is described. Effect of plate fins extended in radial direction and distributed circumferentially with prescribed angular pitch on the overall charging dynamics is studied. If fins are not incorporated, the maximum temperature (hotspot temperature) attained by the storage unit after completion of the charging process is found to be very close to the decomposition temperature of the PCM ${\text{NaNO}}_{3}$. Also, the temperature distribution in the storage medium is found to be highly non-uniform in the absence of fins. Addition of circumferentially distributed radial fins not only reduces the hotspot temperature to a safer margin, but it also establishes a much uniform thermal field within the storage medium. Three different fin orientations are studied for angular pitch of $30^\circ$, $45^\circ$, and $60^\circ$ with progressively lesser number of fins. Larger number of fins resulted in earlier attainment of complete melting. Although the onset of melting occurs at a much earlier time when fins are not added, the completion of melting can only be obtained after significantly large time duration due to large non-uniformity of the thermal field. On the other hand, addition of fins delays the onset of melting process, but once melting process starts completion of melting is attained at an incredibly short time span. When the maximum temperature evolution is compared for $30^\circ$, $45^\circ$, and $60^\circ$ fin orientations, surprisingly higher values of maximum temperature is obtained for progressively higher number of fins ($T_{\max @30^\circ } > T_{\max @45^\circ } > T_{\max @60^\circ }$). However, the same trend is also observed when minimum temperatures are compared ($T_{\min @30^\circ } > T_{\min @45^\circ } > T_{\min @60^\circ }$) with the difference between maximum and minimum temperatures progressively reducing for higher number of fines ($\left[ {T_{\max } - T_{\text{mean}} } \right]_{@30^\circ } < \left[ {T_{\max } - T_{\text{mean}} } \right]_{@45^\circ } < \left[ {T_{\max } - T_{\text{mean}} } \right]_{@60^\circ }$) indicating attainment of better uniformity in the temperature field with larger number of fins, which is physically consistent.