Research ArticlePerformance of a Pebble Bed Thermal Storage Integrated with Concentrating Parabolic Solar Collector for Cooking

Dejene Kebede Kedida , Demiss Alemu Amibe, and Yilma Tadesse Birhane

School of Mechanical and Industrial Engineering, Addis Ababa Institute of Technology, Addis Ababa University, Addis Ababa, Ethiopia

Correspondence should be addressed to Dejene Kebede Kedida; dejkebede@gmail.com

Received 4 May 2019; Revised 22 July 2019; Accepted 28 August 2019; Published 29 December 2019

Academic Editor: Jedrzej Szmytkowski

Copyright © 2019 Dejene Kebede Kedida et al. �is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Cooking using biomass, which is commonly practiced in developing countries, causes rampant deforestation and exposure to emission. Hence, utilization of solar energy for cooking is a green solution. As solar radiation is not available at every hour of the day, thermal storage is essential for availing thermal energy at required time of use. �erefore, this work investigates the e�ciency of solar cooker with parabolic concentrating collector integrated with thermal storage using 1D �nite di�erence computational model. A cook stove on packed pebble bed thermal storage having 0.3 m diameter and 0.9 m height and a storage capacity of 40.1 MJ of energy during a clear day and 12.85 MJ energy was simulated for charging and discharging (cooking), under Addis Ababa climatic condition for days, with highest and lowest solar irradiance and thermal storage e�ciency of 66.7%, cooker thermal e�ciency of 45% during discharging of heat by forced convection, and 41% during discharging of heat by conduction, were obtained for the day with the highest solar irradiance. �e overall e�ciency of the cook stove with thermal storage was 30% and 22% for discharging by forced convection and conduction, respectively. For the day with lowest beam solar irradiance, the storage, thermal and overall e�ciencies were 70.9%, 31.1% and 22.0%, respectively. Hence, it can be concluded that solar concentrating cookers with thermal storage can have an overall cooking e�ciency between 22% and 30% on a clear sky day when the Sun is overhead in tropical areas.

1. Introduction

Solar radiation is intermittent energy source from the sun. Solar energy is harnessed with solar thermal technologies such as parabolic trough, tower, and dish systems for high temperature, and �at plate collector, evacuated tube collector and box solar collectors for low temperature applications. Although solar energy is the preferred type of renewable energy for cooking next to biomass, some shortcomings are also reported in its application. �e major ones are the non-availability of solar radiation at all h of the day, vison hazard that can be caused by re�ected sun rays and its inconvenience for indoor cooking. �e utilization of thermal storage system helps avoid the limitation of solar energy to cook at all required h of the day and makes possible indoor cooking.

A lot of research on the design, application and perfor-mance of solar cookers from di�erent parametrical aspects were reported in the past [1]. Design and performance of solar

cooker such as panel cooker, parabolic cooker, funnel cooker and Sche�er cooker are some of the works. While most of these cookers are for outdoor cooking, Sche�er’s dish can be used for indoor cooking by re�ecting the sun ray and heating a thermal storage media. Apparently most of solar cookers mentioned are designed for day time use and lack introduction of thermal storage [1, 2]. Literatures classify such solar thermal storages as sensible and latent heat storages [1, 3, 4].

Zanganeh [5], Hänchen et al. [6], Zavattoni et al. [7], and Barton [8] investigated performance analysis of sensible ther-mal storage system by numerical methods. Furthermore, experimental investigations of sensible thermal storage for solar cooking were done by Okello et al. [9], Hänchen et al. [6] and Allen et al. [10].

Hänchen et al. [6] carried out the heat transfer analysis of high-temperature thermal storage using a packed bed of rocks with computational method which was validated experimentally for constant heat in�ow. �ey used transient one-dimensional

HindawiJournal of Renewable EnergyVolume 2019, Article ID 4238549, 12 pageshttps://doi.org/10.1155/2019/4238549

Journal of Renewable Energy2

two-phase energy conservation equation for combined con-vection and conduction heat transfer, and the numerical solu-tions were obtained for charging and discharging cycles with a constant heat in�ow. Okello et al. [9], and Barton [8] devel-oped computational model for pebble bed thermal storage and veri�ed them with the experimental results of Hänchen et al. [6]. Veslum et al. [11] described di�erent types of air heating absorber for parabolic solar collector for di�erent mass �ow rates and found stainless steel �ber mesh absorber and silicon carbide honeycomb monolith absorber to have better perfor-mance at a concentration factor of 300, than other alternatives. However, the cost of stainless �ber mesh absorber is lower than that of the honeycomb. In addition, Sharma and Sarma [12] conducted research on design and evaluation of open volumetric air receiver. Veremachi et al. [13] analyzed para-bolic dish concentrating collector for indirect solar cooking. Mohammed [14] also studied design and development of a parabolic dish solar water heater analytically with energy con-version model. Kenneth et al. [15] investigated convective heat transfer coe�cients and pressure drops throughout the ther-mal storage and discussed two phase energy equations through the packed bed of rocks. Okello et al. [9], carried out an exper-imental packed bed pebbles and PCM thermal storage during charging and discharging using 0.3 m diameter by 0.9 m height. A thermal storage tank was used for the packed bed pebbles. Although extensive work has been done in solar ther-mal storage systems, still there are no conclusive remarks on the overall e�ciency of solar cooker with thermal storage under actual cooking conditions. Hence, the objective of this work is to give conclusive remarks on di�erent e�ciencies of solar cooker with sensible thermal storage under climatic con-dition of Addis Ababa in Ethiopia using computational model.

2. Pebble Bed Thermal Storage Modeling

2.1. Physical Model. In this work, the heat transfer �uid that transports heat from the solar receiver to the thermal storage unit is hot air and direct normal irradiance is concentrated on the receiver by parabolic dish as shown in Figure 1. �e

thermal storage media is a packed pebble bed with assumed uniform spherical shape which is suitable for the storage due to its abundance and stability in chemical and mechanical properties.

Figure 2 shows the method of discharging heat when the cooking pot is placed on the pebble bed thermal storage and the heat is transferred to the pot with convection and conduc-tion. �ere is direct contact with the pot and the storage ther-mocline and most part of the circumferential is contact with air passing through pebbles.

2.1.1. Parabolic Solar Collector Model. �e size of parabolic solar collector is designed based on the solar radiation of Addis Ababa and the amount of energy required for the storage. Each parameter in Figure 3, is calculated and explained in Table 1.

2.1.2. Solar �ermal Storage Model. �e solar thermal energy storage is modeled when hot air heated in receiver is charged to packed pebble beds and the discharged air is recycled to receiver. �e thermal storage media is a packed pebble bed with assumed uniform spherical shape which is suitable for the storage due to its abundance stability in chemical and mechanical properties. �e size of the storage is modeled based on the amount of energy demand for the required application as follows.

Parameters and dimensions of packed bed storage stack model of Figure 4 is presented in Table 2.

2.2. Mathematical Model

2.2.1. Heat Transfer Analysis of Parabolic Solar Collector Receiver. A mathematical model for solar thermal storage integrated to parabolic solar collector as shown in Figure 5. �e beam radiation is concentrated on the receiver (absorber) which heats the air that return from the thermal storage.

(1)������� = ��������(1 − �)(���, �v − ���, �������).

Pebble bed,Ts

Cover plate

ValveCharging Absorber

Parabolic solar collector

Manual trackingChar.fan

Dischar.fan

Figure 1: Schematic diagram of pebble bed thermal storage integrated with parabolic solar collector and fan.

3Journal of Renewable Energy

�e rate of energy incident on thee receiver is equated to the rate of useful heat gained by the air plus the rate of heat loss from the receiver. �e energy balance equation at the receiver can be written as [15, 17]:

�e energy on the absorber is given as function of aperture area beam radiation and concentration e�ciency as follows

(2)�̇��� = �̇��� + �̇�.

(3)�̇��� = ��������,

where, the useful heat gain of the air through the receiver is formulated as:

And heat loss from an absorber becomes:

�e useful energy gain is elaborated by substituting Equations (4) and (5) in to Equation (2).

To characterize the thermal performance of a solar concen-trating collector, the concept of thermal e�ciency is used. �is concept refers to the ratio between useful energy carried by

(4)�̇��� = (�̇��)�.(��,��� − ��,��).

(5)�̇� = �������.(����,�v − ����).

(6)�̇� = �������� − �������(����, �v − ����) − ����������, ���������� .

Pebble bed, Ts

Cook pot

Valve Charging Absorber

Parabolic solar collector

Manual trackingChar.fan

Dischar.fan

Discharging

Figure 2: Schematic diagram of pebble bed thermal storage integrated with parabolic solar collector and cooking pot during discharging period.

dr

f

∆r

ϕ

D

R

Ѳ

H

Figure 3: Geometry and dimension of the solar collector parabolic dish.

Table 1: �e results of parameters and dimensions of parabolic so-lar collector.

Parameters Nomenclature Values UnitAperture diameter Da 2.0 [m]Focus point � 0.544 [m]diameter of receiver �r 0.05 [m]Rim angle Ø 85.17 [°]Edge radius � 1.0033 [m]Aperture area �a 3.14 [m2]Area of absorber �abs 0.00785 [m2]Concentration ratio � 400 [−]Height of dish � 0.45 [m]

Ta,in to receiver

Ta, out from receiver

m�ow

Lst

Dst

Figure 4: Schematic diagram of packed pebble bed storage stack.

Journal of Renewable Energy4

�e Reynolds number is given as follows for a �uid �ow in the pipe.

For �� numbers between 1000 and 50,000, Nusselt number is given by the following equation:

2.2.2. Packed Bed �ermal Energy Storage Mathematical Model. One dimensional two phase energy equations describe the heat exchange between air (�uid phase) and pebbles (solid phase) as shown in Figure 6 during charging and discharging, from which the transient temperature pro�les in packed pebbles during charging and discharging times are determined. While analyzing the energy equations, the temperature dependent speci�c heat capacity of air is incorporated in the model.

�e pebble bed consists of a cylindrical tank which can be divided into subdomains of packed pebbles with �xed thick-ness with the air �owing through the porous space around the pebbles.

In formulating the mathematical model of the system, it was assumed that the porosity, mass �ow rate and pebble geometry are uniform, air temperature is constant in the radial direction, and radiation heat transfer in the pebble bed (radi-ation heat transfer is neglected due to low temperature) and the conduction heat transfer in the air were neglected. From an energy balance, the change of enthalpy of air during time �� of the di�erential element is determined as negative of the sum of heat transferred from air to the pebbles by convection, energy transported by air from the control volume and heat loss to the surrounding as shown in Equation (14). Hence, the energy equation for air (�uid phase) is obtained as follows.

It shall be noted that � (porosity)is the volume fraction of air while 1 − � is the volume fraction of pebbles.

Considering energy balance of pebbles on the di�erential element of the pebble bed, the rate of change of internal energy of the pebbles is determined as the sum of the heat transferred from air to the pebbles, the heat conducted to the neighboring pebbles in the longitudinal direction. Hence, the energy equa-tion for pebbles (solid phase) is obtained as follows.

�e simpli�ed transient energy equation for the pebble bed becomes

(12)Re = v��� �������� .

(13)�� = 0.148Re0.633.

(14)�� ������� ����� + ℎv� ��(�� − ��) + �̇��� ����� + �w��(�� − ����) = 0.

(15)

��(1 − �)� ����� ����� − ℎv� ��(�� − ��) − ���(���������� ) = 0.

(16)

����� = ���1�(�� − ��)(1 − �−���(��/�)) +

����(1 − �)�����(�2����2 ),

the heat transfer �uid and the energy incident concentrator aperture:

�e outlet temperature of air from the receiver storage is given as a function of an inlet temperature of air and solar radiation as follows.

where �lr is the mean coe�cient of heat losses for the absorber, which is represented by [15, 16]:

where, the radiation heat transfer coe�cient is given by:

�e convective heat transfer coe�cient is calculated by con-sidering thermal conductivity of air (����), the receiver outer diameter (��) and Nusselt number (��).

(7)

��ℎ = �̇������ =(�̇��)�(��, ��� − ��, ��)�����

= ��� − �������(����, �v − ����)����� − ����������, ��������������̇ ���� .

(8)

��,���(�) = ��,��(�) + �op�����(�)(�̇��)� −�������(����,�v − ����)(�̇��)�

− ���������,���������̇���� ,

(9)��� = [ 1ℎ� + ℎ� ]−1,

(10)ℎ� = ���(�2��� + �2���)(���� + ����).

(11)ℎ� = ������ ×��.

Table 2: Bench mark experimental data [9] and calculated results.

Parameters Nomenclature Values UnitPorosity � 0.38 [−]Length of the storage L 0.9 [m]Diameter of the storage D 0.3 [m]Optimum diameter of pebbles � 0.02 [m]

Mass �ow rate �̇ 0.0495 [Kg/s]Inlet/outlet pipe diameter �p 0.05 [m]

�ickness of stainless steel of the storage stack

S 0.002 [m]

Pebble speci�c heat capacity �pb 880 ± 50 [J/kgk]

Pebble thermal conductivity �p 2.5 [W/mk]

Density of absorber Pabs 2940 [kg/m3]Speci�c heat capacity of absorber �p,abs 1194 [J/kgK]

In�ow temperature �in 355 [°C]Over all heat transfer coe�cient �wall 0.4 [W/m2k]

5Journal of Renewable Energy

While � and � + 1 designates the current and front spatial nodes in the longitudinal directions, � and � + 1 indicates the current and next time step.

Discretizing the partial di�erential equation of the pebble, Equation (16), approximating �rst order spatial and temporal derivatives by forward di�erence and the second order spatial derivative by central di�erence scheme, the following explicit algebraic equation is obtained for the temperature variation of the pebbles.

where, � = 1 − �−���(��/�) and �� is segmental section.Allen [15] analyzed di�erent convection heat transfer cor-

relations among which the following was found to give better convective heat transfer between the air and the pebbles.

�e volumetric convective heat transfer coe�cient becomes

�e e�ective thermal conductivity for the packed pebbles is determined considering the volume fraction of the pebble and the air as follows [6]:

(20)

��+1�, � − ���, �Δ� =

���1�(���, � − ���, �)(1 − �−���(��/�))+����(1 − �)�����

(���, �+1 − 2���, � + ���, �−1Δ�2

),��+1�, � =���, �(1 − (�/��)(Δ�/2�)�) + ���, �((�/��)(Δ�/�)�)

1 + (�/��)(�/2�)�+�����

(1 − �)�����Δ�2(���, �+1 − 2���, � + ���, �+1),

(21)ℎ� = ��� (0.345 × (2/3)0.7

� )Re0.7Pr1/3.

(22)ℎv=ℎ� × (1 − �) × 6� .

(23)���� = 1[�(1/��) + (1 − �)/���] .

where � is thermal time constant, which is given as follows:

NTU is number of transfer unit and is given as

2.3. Computational Model. Discretization of the partial di�erential equation for air Equation (14) by using forward �nite di�erence method, approximating the time and spatial �rst order derivates, the following explicit algebraic equation for air temperature variation is obtained:

(17)� = (1 − �)������ ����̇��� = ������̇��� .

(18)��� = ℎv� ����̇��� =ℎv����� .

(19)

��+1�,�+1 − ���,�Δ� =

ℎv

������(���,� − ���,�) − �̇�� ����

���,�+1 − ���,�Δ�

− �w��� �������(�����,� − ���,�).

Ta, out(t)

Ta, in(t)

Iins

x t

Receiver

Figure 5: Schematic diagram showing the heat transfer process in integrated packed bed with parabolic solar collector.

Charging

air inT

X t

a,xT

a,x+dxT p,x+dxT

p,xT

stLdx

Figure 6:  Schematic diagram showing the heat transfer process occurring within packed bed rocks of storage.

Journal of Renewable Energy6

�e cooker thermal e�ciency is evaluated as the ratio of the useful heat transferred to the cooking media during discharg-ing to the stored thermal energy for cooking.

�e overall e�ciency of the cooker is evaluated as the ratio of the useful heat in the cooking media to solar energy incident on the receiver surface during charging.

3. Verification of Computational Model

An experimental investigation of thermal storage under con-stant heat input was conducted by Okello et al. [9] with 0.048 kg/s air mass �ow rate for the pebble bed thermal storage of 0.3 m diameter and 0.9 m length as it indicated in Table 2. During the charging process constant temperature hot air with �xed mass �ow rate (heated by electric resistance) �ows from top to bottom of the thermal storage in the experimental set-up selected for veri�cation. In the computation model of this system 10 spatial nodes and 1800 time steps were used for the simulation. Strati�cation of temperature throughout the packed bed pebbles is observed. �e initial and boundary con-ditions to be used during charging of heat to packed bed peb-bles for the simulation required to verify experimental data are given in Table 3.

�e experimental results were compared with the results of simulation using the computational model developed in this work and good agreement between the two cases were observed with an average error of 1.8% as shown in Figure 7. Hence, it can be concluded that the accuracy of the model is su�cient to simulate an integrated system of concentrating parabolic dish and thermal storage during charging and cook-ing conditions.

4. Results and Discussions

4.1. Charging with Solar Energy and Discharging with Water Boiling. Figure 8 shows the maximum and minimum direct normal irradiance (DNI) for Addis Ababa selected for

(31)��� = ∫�0����� ��(1 − �)(��(�) − ��)��

∫��0 ������� .

(32)��ℎ = �w�w(�

w,� − �w,�)∫�0����� ��(1 − �)(��(�) − ��)��

.

(33)�� = �w�w(�w,� − �w,�)∫��0 ������� .

Prandtl number is given as

�e overall heat transfer coe�cient through the wall is calcu-lated from the equation below.

�e natural convection heat transfer coe�cient for the wall of the packed bed is determined from correlation of the Nusselt number as a function of the Rayleigh and Prandtl numbers as follows.

�e natural convection heat transfer coe�cient for the hori-zontal cylinder is determined from correlation of Nusselt number as function of the Rayleigh and Prandtl numbers as follows.

For hot air circulation pipe from receiver to the thermal stor-age, the following inside forced convective heat transfer coef-�cient is valid for laminar �ow

For the charging of the pebble bed the air is heated with the concentrating parabolic solar collector absorber. Hence, the air temperature coming from the pebble bed and heated in the receiver is updated as follows considering the loss from the receiver to the inlet of the pebble bed.

During discharging heat is transferred from storage to the material to be cooked. Simplifying the cooking process as water boiling, the temperature variation is given as follows from the heat transferred to the pan minus the radiation and convection losses of the pan:

�e stored thermal energy is obtained from the change in aver-age temperature of the pebble and evaluating the change in enthalpy of the pebble bed during charging. �e thermal storage e�ciency is determined as the ratio of the stored thermal energy to the solar energy incident on the receiver as follows.

(24)�� = ������ .

(25)1�w� = 1ℎ�� � +

��������� +1ℎ��� .

(26)ℎ� = ���[[[0.825 + 0.387��1/6�

(1 + (0.492/��)9/16)8/27]]]

2

.

(27)ℎ� = ���[[[0.6 + 0.387��1/4�(1 + (0.559/��)9/16)8/27

]]]

2

.

(28)�� = 1.86(RePr)1/3(�� )1/3( ���

w

)0.14 = ℎ���� .

(29)

��+1�, ��� = ���, �� + �0������(�)�̇��� −�������(����� − ��)�̇���

− �������� �, ���(����� − ��−1��� )Δ��̇���.

(30)��+1w= ��

w+�̇���(�) − (�̇�(�) + �̇�v(�))

�w��w

Δ�.

Table 3: Initial and boundary conditions.

��(�,�=0) = 355°C ��(�, � = 0) = ��(�, � = 1) Space step = 0.1 m

��(� = 0,� = �) = 23°C ��(� ,� = �) = ��(�, � = �+1) Number of space step = 10

��(� = 0,� = �) = 23°C Time step (Δ�) = 1 s Maximum time charging = 5 h

Number of time step =1800

7Journal of Renewable Energy

when the air is cycling throughout the storage and returned back to the receiver for the days with the highest and lowest beam solar irradiance.

Figures 10 and 11 show the temperature distributions in the thermocline storage and the recirculated air with respect to sunshine h during charging for the day with the highest DNI for Addis Ababa. A½er charging for 11 h, the maximum temperatures of packed pebbles and air reached 459.7°C and 468.4°C, respectively at the top surface of the storage. �e minimum temperatures of the air and the thermal storage were 383.2°C and 414.4°C, respectively.

modeling of the parabolic solar collector from Sunrise to Sunset. �e lowest DNI occurs in July and the highest is in March.

Parameters of solar collector are calculated and the results are explained as it is shown in Table 1 which is used to inves-tigate the variable charging of heat to storage.

�e charging of the pebble bed thermal storage was inves-tigated in typical days of March and July in which the highest and lowest direct normal irradiance occurs in Addis Ababa from 7 AM to 5 PM by recirculating the hot air through the receiver of the parabolic solar collector for the thermal storage in consideration and initial conditions of the pebbles given in Tables 2 and 3. Figure 9 shows the variation of air temperature at the outlet of the receiver of the parabolic solar collector

400

350

300

250

T s (°

C)

200

150

100

50

00.1 m 0.2 m 0.3 m 0.4 m 0.5 m 0.6 m

1 h ex1 h sim2 h ex2 h sim

3 h sim

3 h ex

4 h sim4 h ex

5 h sim5 h ex

0.7 m 0.8 m 0.9 mH

Figure 7: Comparison of experimental data with simulation results of temperature of thermal storage at 1 s. Time step and 0.1 m space step with 0.0048 kg/s mass �ow rate of hot air. (sim. is simulation and ex. is experimental)

1000

8:15:00 AM7:00:00 AM 9:30:00 AM 12:00:00 PM

Time (h)DNI (Max) DNI (Min)

2:30:00 PM 5:00:00 PM10:45:00 AM 1:15:00 PM 3:45:00 PM

DN

I (W

/m_2

)

900800

700

600500

400300

200

100

0

Figure 8: Direct normal irradiance for maximum and minimum in the year with respect to time (7 AM to 5 PM).

07:00

AM

08:00

AM

09:00

AM

10:00

AM

11:00

AM

12:00

PM

01:00

PM

02:00

PM

03:00

PM

04:00

PM

05:00

PM

0

100

200

300

T a (°

C)

400

500

600

Ta for lower DNI from absorberTa for higher DNI from absorber

Time (h)

Figure 9: Temperature of air cycling through the storage and receiver heating from 7 AM to 5 PM for both higher and lower direct normal irradiance.

07:00 AM08:00 AM

09:00 AM10:00 AM

11:00 AM12:00 PM

01:00 PM02:00 PM

03:00 PM04:00 PM

05:00 PM

0

100

200

300

400

500

600

X = 0.1 m X = 0.2 m X = 0.3 m X = 0.4 mX = 0.5 m

X = 0.6 m X = 0.7 m X = 0.8 m X = 0.9 m

Time (h)

T a (°

C)

Figure 10: Temperature of air cycling throughout the storage and receiver with respect to charging time from 7 AM to 5 PM for the higher DNI.

Journal of Renewable Energy8

�e discharging of heat from the storage was considered in the following two cases.

(i) When the cooking is carried out only by conduc-tion and natural convection heat transfer, the pot is placed on the top of storage as shown in Figure 2, and the heat is transferred from the storage to the pot by conduction and the colder air from the top get to be replenished by warmer air from the hot storage by natural convection.

(ii) When the cooking is carried out by forced convection circulation of air by a charging fan.

Figures 15 and 16 Compare discharging of heat during water boiling to simulate cooking with forced convection ver-sus conduction and natural convection heat transfer in the pebble bed. From the comparison, forced convection discharg-ing gives higher useful cooking energy than from discharging

Figures 12 and 13 show temperature distributions within thermocline storage and the air within storage with respect to charging h simulated for the lowest direct normal irradiance (DNI) of Addis Ababa. A½er charging for 11 h, the maximum temperatures of packed pebbles and air reached 199.75°C and 214.49°C, respectively at the top surface of the storage. �e minimum temperatures were 129.48°C and 139.16°C for packed pebbles and air, respectively.

Figure 14 shows the amount of energy stored in thermo-cline storage for the days with maximum and minimum direct normal irradiance. �e total stored energy was 40.1 MJ and 12.85 MJ for both conditions respectively.

4.2. Discharging Conditions. Heat is extracted from the storage when the ambient air is recirculated with 0.0048 kg/s mass �ow rate of air throughout the storage from bottom to top.

07:00 AM08:00 AM

09:00 AM10:00 AM

11:00 AM12:00 PM

01:00 PM02:00 PM

03:00 PM04:00 PM

05:00 PM

0

100

200

300

400

500

600

X = 0.1 mX = 0.2 mX = 0.3 mX = 0.4 mX = 0.5 m

X = 0.6 mX = 0.7 mX = 0.8 mX = 0.9 m

Time (h)

T p (°

C)

Figure 11: Temperature of thermocline storage charged with respect to charging h from 7 AM to 5 PM for the higher DNI.

07:00 AM08:00 AM

09:00 AM10:00 AM

11:00 AM12:00 PM

01:00 PM02:00 PM

03:00 PM04:00 PM

05:00 PM

0

50

100

150

200

250

X = 0.1 mX = 0.2 mX = 0.3 mX = 0.4 mX = 0.5 m

X = 0.6 mX = 0.7 mX = 0.8 mX = 0.9 m

Time (h)

T a (°

C)

Figure 12: Temperature of air charging the storage versus charging time from 7 AM to 5 PM for the lower DNI in July.

07:00 AM08:00 AM

09:00 AM10:00 AM

11:00 AM12:00 PM

01:00 PM02:00 PM

03:00 PM04:00 PM

05:00 PM

0

50

100

150

200

250

X = 0.1 mX = 0.2 mX = 0.3 mX = 0.4 mX = 0.5 m

X = 0.6 mX = 0.7 mX = 0.8 mX = 0.9 m

Time (h)

T p (°

C)

Figure 13: Temperature of thermocline storage versus charging time from 7AM to 5 PM for the lowest DNI in July.

07:00 AM08:00 AM

09:00 AM10:00 AM

11:00 AM12:00 PM

01:00 PM02:00 PM

03:00 PM04:00 PM

05:00 PM

0

5

10

15

20

25

30

35

40

45

Q_store DNI, higher Q_store DNI, lowerTime (h)

Q _

stor

ed (M

J)

Figure 14: Energy stored versus charging h for both maximum and minimum direct normal irradiance.

9Journal of Renewable Energy

pro�le. However, the thermal storage can heat a maximum of 28 liters of water, as shown in Figure 17 over a longer period of time, with considerable cooling in pebble stack. From results of the simulation, the temperature of water reached 93°C a½er 8 h discharging time and top surface of the thermal storage temperature reached 109°C. �erefore, the storage has a capac-ity of boiling 28 liters of water for the day with the highest DNI.

In the day of lowest DNI, the storage temperature degraded during boiling of 5 liters of water for 4 hours as it is shown in Figure 18. From results of the simulation, the temperature of water reached 93°C a½er 4 hours discharging time and the top surface of the thermal storage temperature reached 138°C. �erefore, the storage has a capacity of boiling only 5 liters of water during the day of lowest DNI. �e rest of energy in the storage will be available for the next day.

Figure 19 shows the temperature of storage during charging and discharges for two consecutive days. �e cooking was done

by conduction without air recirculation. For 5 liters water to boil, 53 minutes discharging time is required to reach a tem-perature 93°C for forced convection heat discharge. In case of the conduction and natural convection discharging, 5 liters water requires 56 minutes to reach 93°C and the storage tem-perature decreases gradually as it is shown in Figures 15 and 16. �e parameters and values required during the discharging condition (during water boiling) are presented in Table 4.

�e initial and boundary conditions during discharging of heat from packed bed pebbles for water boiling test simu-lation are given in Table 5.

In the previous simulation, 5 liters of water was heated in short time with insigni�cant change of pebble bed temperature

0 10 20 30 40 50 600

50

100

150

200

250

300

350

400

450

t (min)

T S (°

C)

5 L of water TS

Figure 15: Temperature of storage versus temperature of water during boiling of 5 liters of water by natural convection and conduction where the pot is put on the storage a½er charging a thermocline for 11 h for higher DNI.

0 10 20 30 40 50 53 600

50

100

150

200

250

300

350

400

450

500

T_5 L of waterTs at Top

Ts at middleTs at bottom

t (min)

T (°

C)

Figure 16:  Temperature of storage versus temperature of water during boiling of 5 liters of water with 0.0048 kg/s mass �ow rate for the maximum direct normal irradiance.

Table 4: Dimensions and values required during water boiling sim-ulation [18].

Parameters Symbols Values UnitDiameter of pan �pan 0.273 [m]Length of pan �pan 0.17 [m]Speci�c heat capacity of water �pw 4180 [J/kgK]

Stefan Boltzmann constant � 5.67 × 10−8 [W/m2K4]

Emissivity of steel pan � 0.4 [−]�ermal resistance of pan �th 0.865 W/K

Table 5: Initial and boundary conditions for discharging process.

��( � = �max , � = 0) = 20°C Space step = 0.1 m

Time step (Δ�) = 1 s

�� (�=0, �=�) = ��, 11 hours chargeNumber of

space step = 10 tmax = 11 hours

Start 01:00 02:00 03:00 04:00 05:00 06:00 07:00 08:000

50100150200250300350400450500

T_28 L of waterTs top

Ts middleTs bottom

t (h)

T (°

C)

Figure 17: Temperature of thermal storage and 28 liters of water during discharging with 0.0048 kg/s mass �ow rate of air for the higher DNI.

Journal of Renewable Energy10

5. Conclusion

�is work describes a successful modeling of sensible heat thermal storage integrated with parabolic solar collector and cook stove at the top of thermal storage. Constant heat in�ow and variable heat in�ow for heating air by receiver of parabolic solar collector with constant mass �ow approaches were used to evaluate the performance of the pebble bed thermal storage.

for 2 h from 5:00 AM to 7:00 PM and 5:00 PM to 7:00 PM for two reasonable cycles in a day. �e simulation was carried out in for March16 and March 17 of beam solar irradiance.

Table 6 explains di�erent amount of energies and e�cien-cies of cooking for the conditions of the highest and lowest DNI from sunrise to sunset (7 AM to 5 PM).

Table 7 explains the results of the temperatures and e�-ciencies when the cooking is carried out by conduction and enhanced by forced convection.

Figure 20 shows degradation of the thermal storage with-out cooking operation in 8 days with �nal average temperature of 38°C. In this context, the storage interacts with the envi-ronment through �berglass insulation having thickness of 75 mm.

Start 30 min 60 min 90 min 120 min 150 min 180 min 210 min 240 min0

50

100

150

200

250

T_5L of waterTs top

Ts middleTs bottom

Time

T (°

C)

Figure 18:  Temperature of storage versus temperature of water during boiling of 5 liters of water with 0.0048 kg/s mass �ow rate for the lower direct normal irradiance.

07: 00 AM 03: 00 AM12:00 PM

05:00 PM10:00 PM

01:00 PM06:00 PM

11:00 PM04:00 AM

050

100150200250300350400450

X = 0 mX = 0.1 mX = 0.2 mX = 0.3 mX = 0.4 m

X = 0.5 mX = 0.6 mX = 0.7 mX = 0.8 mX = 0.9 m

Time (h)

T s (°

C)

08:00 AM

Figure 19: Temperature of storage versus time for two consecutive days.

Table 6: Comparison of e�ciencies for the conditions of higher and lower DNI.

DNI, highest DNI, lowestSolar energy on aperture area [MJ] 60.1 18.1

Stored energy [MJ] 40.1 12.85Storage e�ciency (%) 66.7 70.9Volume of water boiled 5 liter for 53 min 5 liter for 4 hUseful cooking energy [MJ] 1.47 1.47�ermal e�ciency of cooker (%) 45 31.1

Overall e�ciency of cooker (%) 30 22.08

Table 7: E�ciency for the condition of highest DNI.

Conduction Forced convectionWater temperature at 53 min for 5 liters 87°C 93.0°C

Useful cooking energy [MJ] 1.344 1.463Stored energy for 11 h charging 40.1 MJ 40.1 MJ

Solar energy on aperture 60.1 60.1Storage e�ciency 66.7% 66.7%�ermal e�ciency of cooker 41% 45%Overall e�ciency of cooker 27.34% 30%

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

200

250

300

350

400

450

500

t (h)

T s (°

C)

Ts at top of storage Ts at bottom of storage

Figure 20: Temperature of storage when without operation with heat losses to the environment through insulation thickness of 75 mm.

11Journal of Renewable Energy

��: Prandtl number��: Temperature of storage�ab: density of absorber�abs: Temperature of absorberℎ�: Internal convective heat transfer coe�cientℎ�: Outer convective heat transfer coe�cient� cp: Circumferential area pan�st: Temperature of storage�w: Temperature of water�

w: Speci�c heat capacity of water at constant

pressure�w: Mass of water�th: �ermal resistance of pan�: Dynamic viscosity�op: Optical e�ciency of parabolic solar collector�abs: Absorber area�ap: Aperture area�̇

w: Heat transferred to water �

w��w(��/��)�̇pan: Heat of pan= (��� − �w)/(��ℎ)�̇�: Radiation heat loss = ��Span(T4w − T4a)�̇cv: Convective heat loss = ℎ� ���(�w − ��)�out(�): Temperature of air out from reciever as a

function of time [C]�in(�): Temperature of air inter in to reciever as a function of time�: Primeter�ins: �ickness of insulation�p,abs: Speci�c heat capacity of absorber�abs: Volume of absorber.

Data Availability

Addis Ababa solar radiations from Ethiopian Metrology Agency and MAT LAB coding.

Conflicts of Interest

�e author declares that they have no con�icts of interest.

Acknowledgments

�is research was funded by Addis Ababa University (AAU). We would like to thank the Ethiopia Metrology Agency for giving us Addis Ababa solar radiation data’s.

References

[1] R. M. Muthusivagami, R. Velraj, and R. Sethumadhavan, “Solar cookers with and without thermal storage-a review,” Renewable and Sustainable Energy Reviews, vol. 14, no. 2, pp. 691–701, 2010.

[2] S. B. Joshi and A. R. Jani, “Design, development and testing of a small scale hybrid solar cooker,” Solar Energy, vol. 122, pp. 148–155, 2015.

[3] H. Singh, R. P. Saini, and J. S. Saini, “A review on packed bed solar energy storage systems,” Renewable and Sustainable Energy Reviews, vol. 14, no. 3, pp. 1059–1069, 2010.

�e computational model was validated and the result are found to be consistent with an experimental work that was reported by Okello et al. [9].

For the actual condition, charging of thermal storage by air heating in receiver of parabolic solar collector was modeled by the computational model. Boiling of water at the top of thermal storage was simulated under forced convection and conduction heat extraction from the pebbles was performed to simulate discharging of thermal storage during cooking. From the simulation, the storage e�ciency was 66.7% for the day with highest DNI and the thermal e�ciency for cooking were for heat transfer by forced convection and natural con-vection and conduction were 45% and 41%, respectively. �e overall e�ciency of cooking were 30% and 27.3% for the above two cases, respectively. For the day with the lowest DNI, the storage e�ciency, thermal e�ciency of cooking and overall e�ciency of cooking 70.9%, 31.1% and 22.08%, respectively. Although thermal storage system for solar cookers is attractive due to no limitation on cooking time and place, heat losses from the system and unavailability of the energy below 100°C makes the overall e�ciency low. As the overall e�ciencies of the cooking stove assuming forced circulation of air through pebbles and conduction and natural convection across pebble bed represent the limits of best case and worst case e�ciency. It can be concluded that the overall e�ciency of solar concen-trating cooker with pebble bed thermal storage and air as heat transfer media is around 22% to 30% for tropical on a clear sky day when the sun is overhead at noon.

Nomenclature

�̇: Mass of air �ow��: Mass of pebbleℎv: Volumetric convective heat transfer coe�cientℎ�: Particle convective heat transfer coe�cient� cs: Crossectional area of storage��: Temperature of air��: Temperature of pebble�: Void fraction��: Density of air��: Density of pebble�pa: Speci�c heat capacity of air at constant pressure�pb: Speci�c heat capacity of pebble at constant

pressure�amb: Temperature of ambient air�: Over all heat transfer coe�cient�w: Over all heat transfer coe�cient wall�: Height of storage�eff: E�ective thermal conductivity of pebbles��: �ermal conductivity of air�ins: �ermal conductivity of insulation��: �ermal conductivity of pebble��: Biot number�: Diameter of pebbleℎ: Convective heat transfer coe�cient��: Beam Solar irradiance�: diameter of storage��: Reynold’s number

Journal of Renewable Energy12

[4] L. Nkhonjera, T. Bello-Ochende, G. John, and C. K. King’ondu, “A review of thermal energy storage designs, heat storage materials and cooking performance of solar cookers with heat storage,” Renewable and Sustainable Energy Reviews, vol. 75, pp. 157–167, 2017.

[5] G. Zanganeh, “High-temperature thermal energy storage for concentrated solar power with air as heat transfer fluid,” 2014.21802

[6] M. Hänchen, S. Brückner, and A. Steinfeld, “High-temperature thermal storage using a packed bed of rocks - heat transfer analysis and experimental validation,” Applied �ermal Engineering, vol. 31, no. 10, pp. 1798–1806, 2011.

[7] S. A. Zavattoni, M. C. Barbato, A. Pedretti, G. Zanganeh, and A. Steinfeld, “High temperature rock-bed TES system suitable for industrial-scale CSP plant - CFD analysis under charge/discharge cyclic conditions,” Energy Procedia, vol. 46, pp. 124–133, 2014.

[8] N. G. Barton, “Simulations of air-blown thermal storage in a rock bed,” Applied �ermal Engineering, vol. 55, no. 1–2, pp. 43–50, 2013.

[9] D. Okello, C. W. Foong, O. J. Nydal, and E. J. K. Banda, “An experimental investigation on the combined use of phase change material and rock particles for high temperature (~350 C) heat storage,” Energy Conversion and Management, vol. 79, pp. 1–8, 2014.

[10] K. G. Allen, T. W. Von Backström, and D. G. Kröger, “Packed rock bed thermal storage in power plants: design considerations,” Energy Procedia, vol. 49, pp. 666–675, 2013.

[11] T. S. Veslum, “Absorber for concentrating solar heat collectors,” 2011.

[12] P. Sharma and R. Sarma, “On the design and evaluation of open volumetric air receiver for process heat applications,” Energy Procedia, vol. 57, pp. 2994–3003, 2014.

[13] B. A. Veremachi, A. Zia, B. C. Cuamba, J. Lovseth, and O. J. Nydal, “Parabolic dish concentrating collector for indirect solar cooking,” Double Blind Peer Reviewed International Research Journal, vol. 17, no. 1, 2017.

[14] I. L. Mohammed, “Design and development of a parabolic dish solar water heater,” International Journal of Engineering Research and Applications, vol. 2, no. 1, pp. 822–830, 2012.

[15] K. G. Allen, “Rock bed thermal storage for concentrating solar power plants,” pp. 1–205, 2014.

[16] Y. Kadri, “Design of a solar dish concentrator according to the needed energy for a given application,” pp. 18–25, 2013.

[17] J. A. Alarcón, J. E. Hortúa, and G. A. Lopez, “Design and construction of a solar collector parabolic dish for rural zones in Colombia,” TECCIENCIA, vol. 7, no. 14, pp. 14–22, 2013.

[18] M. Mussard, A. Gueno, and O. J. Nydal, “Experimental study of solar cooking using heat storage in comparison with direct heating,” Solar Energy, vol. 98, pp. 375–383, 2013.

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