# Performance of a Pebble Bed Thermal Storage Integrated with Concentrating Parabolic Solar Collector for Cooking

Authors: Dejene Kebede Kedida; Demiss Alemu Amibe; Yilma Tadesse Birhane
Journal: Journal of Renewable Energy
 (2019)

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## Abstract

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. Therefore, this work investigates the efficiency of solar cooker with parabolic concentrating collector integrated with thermal storage using 1D finite difference 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 efficiency of 66.7%, cooker thermal efficiency 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. The overall efficiency 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 efficiencies 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 efficiency between 22% and 30% on a clear sky day when the Sun is overhead in tropical areas.


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## Body

## 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 flat 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. The major ones are the nonavailability of solar radiation at all h of the day, vison hazard that can be caused by reflected sun rays and its inconvenience for indoor cooking. The 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 performance of solar cookers from different parametrical aspects were reported in the past [1]. Design and performance of solar cooker such as panel cooker, parabolic cooker, funnel cooker and Scheffler cooker are some of the works. While most of these cookers are for outdoor cooking, Scheffler’s dish can be used for indoor cooking by reflecting 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 thermal 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 inflow. They used transient one-dimensional two-phase energy conservation equation for combined convection and conduction heat transfer, and the numerical solutions were obtained for charging and discharging cycles with a constant heat inflow. Okello et al. [9], and Barton [8] developed computational model for pebble bed thermal storage and verified them with the experimental results of Hänchen et al. [6]. Veslum et al. [11] described different types of air heating absorber for parabolic solar collector for different mass flow rates and found stainless steel fiber mesh absorber and silicon carbide honeycomb monolith absorber to have better performance at a concentration factor of 300, than other alternatives. However, the cost of stainless fiber 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 parabolic dish concentrating collector for indirect solar cooking. Mohammed [14] also studied design and development of a parabolic dish solar water heater analytically with energy conversion model. Kenneth et al. [15] investigated convective heat transfer coefficients and pressure drops throughout the thermal storage and discussed two phase energy equations through the packed bed of rocks. Okello et al. [9], carried out an experimental 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 thermal storage systems, still there are no conclusive remarks on the overall efficiency of solar cooker with thermal storage under actual cooking conditions. Hence, the objective of this work is to give conclusive remarks on different efficiencies of solar cooker with sensible thermal storage under climatic condition of Addis Ababa in Ethiopia using computational model.


## 2. Pebble Bed Thermal Storage Modeling

### 2.1. Physical Model

In this work, the heat transfer fluid 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 Figure1. The 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 1
Schematic diagram of pebble bed thermal storage integrated with parabolic solar collector and fan.Figure2 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 conduction. There is direct contact with the pot and the storage thermocline and most part of the circumferential is contact with air passing through pebbles.Figure 2
Schematic diagram of pebble bed thermal storage integrated with parabolic solar collector and cooking pot during discharging period.


#### 2.1.1. Parabolic Solar Collector Model

The 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 Figure3, is calculated and explained in Table 1.Figure 3
Geometry and dimension of the solar collector parabolic dish.Table 1
The results of parameters and dimensions of parabolic solar collector.
Parameters
Nomenclature
Values
Unit
Aperture diameter
Da
2.0
[m]
Focus point
f
0.544
[m]
diameter of receiver
d
r
0.05
[m]
Rim angle
∅
85.17
[°]
Edge radius
R
1.0033
[m]
Aperture area
A
a
3.14
[m2]
Area of absorber
A
abs
0.00785
[m2]
Concentration ratio
C
400
[−]
Height of dish
H
0.45
[m]


#### 2.1.2. Solar Thermal Storage Model

The 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. The 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. The size of the storage is modeled based on the amount of energy demand for the required application as follows.(1)Estored=ρpcpbVst1−εTst,av−Tst,initial.Parameters and dimensions of packed bed storage stack model of Figure4 is presented in Table 2.Figure 4
Schematic diagram of packed pebble bed storage stack.Table 2
Bench mark experimental data [9] and calculated results.
Parameters
Nomenclature
Values
Unit
Porosity
ε
0.38
[−]
Length of the storage
L
0.9
[m]
Diameter of the storage
D
0.3
[m]
Optimum diameter of pebbles
d
0.02
[m]
Mass flow rate
m
˙
0.0495
[Kg/s]
Inlet/outlet pipe diameter
d
p
0.05
[m]
Thickness of stainless steel of the storage stack
S
0.002
[m]
Pebble specific heat capacity
c
pb
880± 50
[J/kgk]
Pebble thermal conductivity
k
p
2.5
[W/mk]
Density of absorber
P
abs
2940
[kg/m3]
Specific heat capacity of absorber
C
p
,
abs
1194
[J/kgK]
Inflow temperature
T
in
355
[°C]
Over all heat transfer coefficient
U
wall
0.4
[W/m2k]


### 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 Figure5. The beam radiation is concentrated on the receiver (absorber) which heats the air that return from the thermal storage.Figure 5
Schematic diagram showing the heat transfer process in integrated packed bed with parabolic solar collector.The 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. The energy balance equation at the receiver can be written as [15, 17]:(2)Q˙abs=Q˙use+Q˙l.The energy on the absorber is given as function of aperture area beam radiation and concentration efficiency as follows(3)Q˙abs=ηopAapIb,where, the useful heat gain of the air through the receiver is formulated as:(4)Q˙use=m˙Cpa.Ta,out−Ta,in.And heat loss from an absorber becomes:(5)Q˙l=AabsUlr.Tabs,av−Tamb.The useful energy gain is elaborated by substituting Equations (4) and (5) in to Equation (2).(6)Q˙u=ηopAapIb−AabsUlrTabs,av−Tamb−ρabsVabsCp,absdTabsdt.To characterize the thermal performance of a solar concentrating collector, the concept of thermal efficiency is used. This concept refers to the ratio between useful energy carried by the heat transfer fluid and the energy incident concentrator aperture:(7)ηth=Q˙uAapIb=m˙CpaTa,out−Ta,inAapIb=ηop−AabsUlrTabs,av−TambAapIb−ρabsVabsCp,absdTabsAapIbm˙cpadt.The outlet temperature of air from the receiver storage is given as a function of an inlet temperature of air and solar radiation as follows.(8)Ta,outt=Ta,int+ηopAapIbtm˙Cpa−AabsUlrTabs,av−Tambm˙Cpa−ρabsVabsCp,absdTabsm˙cpadt,whereUlr is the mean coefficient of heat losses for the absorber, which is represented by [16, 17]:(9)Ulr=1hc+hr−1,where, the radiation heat transfer coefficient is given by:(10)hr=σεrTrec2+Tamb2Trec+Tamb.The convective heat transfer coefficient is calculated by considering thermal conductivity of airkair, the receiver outer diameter dr and Nusselt number Nu.(11)hc=kairdr×Nu.The Reynolds number is given as follows for a fluid flow in the pipe.(12)Re=vair×Doutγair.ForRe numbers between 1000 and 50,000, Nusselt number is given by the following equation:(13)Nu=0.148Re0.633.


#### 2.2.2. Packed Bed Thermal Energy Storage Mathematical Model

One dimensional two phase energy equations describe the heat exchange between air (fluid phase) and pebbles (solid phase) as shown in Figure6 during charging and discharging, from which the transient temperature profiles in packed pebbles during charging and discharging times are determined. While analyzing the energy equations, the temperature dependent specific heat capacity of air is incorporated in the model.Figure 6
Schematic diagram showing the heat transfer process occurring within packed bed rocks of storage.The pebble bed consists of a cylindrical tank which can be divided into subdomains of packed pebbles with fixed thickness with the air flowing through the porous space around the pebbles.In formulating the mathematical model of the system, it was assumed that the porosity, mass flow rate and pebble geometry are uniform, air temperature is constant in the radial direction, and radiation heat transfer in the pebble bed (radiation 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 timedt of the differential 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 (fluid phase) is obtained as follows.(14)εAcsρacpa∂Ta∂t+hvAcsTa−Tp+m˙cpa∂Ta∂x+UwPLTa−Tamb=0.It shall be noted thatεporosityis the volume fraction of air while 1−ε is the volume fraction of pebbles.Considering energy balance of pebbles on the differential 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 equation for pebbles (solid phase) is obtained as follows.(15)ρp1−εAcscpb∂Tp∂t−hvAcsTa−Tp−∂∂xkeffA∂Tp∂x=0.The simplified transient energy equation for the pebble bed becomes(16)∂Tp∂t=Lxi1τTa−Tp1−e−NTUxi/L+keff1−ερpcpb∂2Tp∂x2,whereτ is thermal time constant, which is given as follows:(17)τ=1−ερpcpbAcsLm˙cpa=mpcpbm˙cpa.NTU is number of transfer unit and is given as(18)NTU=hvAcsLm˙cpa=hvLGcpa.


### 2.3. Computational Model

Discretization of the partial differential equation for air Equation (14) by using forward finite difference method, approximating the time and spatial first order derivates, the following explicit algebraic equation for air temperature variation is obtained:(19)Ta,i+1n+1−Ta,inΔt=hvερacpaTp,in−Ta,in−m˙εAcsρaTa,i+1n−Ta,inΔx−UwPεAcsρacpaTamb,in−Ta,in.Whilei and i+1 designates the current and front spatial nodes in the longitudinal directions, n and n+1 indicates the current and next time step.Discretizing the partial differential equation of the pebble, Equation (16), approximating first order spatial and temporal derivatives by forward difference and the second order spatial derivative by central difference scheme, the following explicit algebraic equation is obtained for the temperature variation of the pebbles.(20)Tp,in+1−Tp,inΔt=Lxi1τTa,in−Tp,in1−e−NTUxi/L+keff1−ερpcpbTp,i+1n−2Tp,in+Tp,i−1nΔx2,Tp,in+1=Tp,in1−L/xiΔt/2τη+Ta,inL/xiΔt/τη1+L/xiΔt/2τη+keffΔt1−ερpcpbΔx2Tp,i+1n−2Tp,in+Tp,i+1n,where,η=1−e−NTUxi/L and xi is segmental section.Allen [15] analyzed different convection heat transfer correlations among which the following was found to give better convective heat transfer between the air and the pebbles.(21)hp=kad0.345×2/30.7εRe0.7Pr1/3.The volumetric convective heat transfer coefficient becomes(22)hv=hp×1−ε×6d.The effective thermal conductivity for the packed pebbles is determined considering the volume fraction of the pebble and the air as follows [6]:(23)keff=1ε1/ka+1−ε/kpb.Prandtl number is given as(24)pr=cpaμka.The overall heat transfer coefficient through the wall is calculated from the equation below.(25)1UwA=1hiAi+xinskinsA+1hoAo.The natural convection heat transfer coefficient 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.(26)ho=kaD0.825+0.387RaD1/61+0.492/pr9/168/272.The natural convection heat transfer coefficient for the horizontal cylinder is determined from correlation of Nusselt number as function of the Rayleigh and Prandtl numbers as follows.(27)ho=kaD0.6+0.387Rad1/41+0.559/pr9/168/272.For hot air circulation pipe from receiver to the thermal storage, the following inside forced convective heat transfer coefficient is valid for laminar flow(28)Nu=1.86RePr1/3DL1/3μbμw0.14=hiDka.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.(29)Ta,outn+1=Ta,inn+η0pAapIbtm˙cpa−AabsUrlTabsn−Tam˙cpa−ρabsVabsCp,absTabsn−Tabsn−1Δtm˙cpa.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:(30)Twn+1=Twn+Q˙pant−Q˙rt+Q˙cvtmwcpwΔt.The stored thermal energy is obtained from the change in average temperature of the pebble and evaluating the change in enthalpy of the pebble bed during charging. The thermal storage efficiency is determined as the ratio of the stored thermal energy to the solar energy incident on the receiver as follows.(31)ηTS=∫0LρpcpAcs1−εTpx−Todx∫0tcIbAapdt.The cooker thermal efficiency is evaluated as the ratio of the useful heat transferred to the cooking media during discharging to the stored thermal energy for cooking.(32)ηth=mwcwTw,f−Tw,i∫0LρpcpAcs1−εTpx−Todx.The overall efficiency 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.(33)ηo=mwcwTw,f−Tw,i∫0tcIbAapdt.


## 2.1. Physical Model

In this work, the heat transfer fluid 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 Figure1. The 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 1
Schematic diagram of pebble bed thermal storage integrated with parabolic solar collector and fan.Figure2 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 conduction. There is direct contact with the pot and the storage thermocline and most part of the circumferential is contact with air passing through pebbles.Figure 2
Schematic diagram of pebble bed thermal storage integrated with parabolic solar collector and cooking pot during discharging period.


### 2.1.1. Parabolic Solar Collector Model

The 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 Figure3, is calculated and explained in Table 1.Figure 3
Geometry and dimension of the solar collector parabolic dish.Table 1
The results of parameters and dimensions of parabolic solar collector.
Parameters
Nomenclature
Values
Unit
Aperture diameter
Da
2.0
[m]
Focus point
f
0.544
[m]
diameter of receiver
d
r
0.05
[m]
Rim angle
∅
85.17
[°]
Edge radius
R
1.0033
[m]
Aperture area
A
a
3.14
[m2]
Area of absorber
A
abs
0.00785
[m2]
Concentration ratio
C
400
[−]
Height of dish
H
0.45
[m]


### 2.1.2. Solar Thermal Storage Model

The 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. The 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. The size of the storage is modeled based on the amount of energy demand for the required application as follows.(1)Estored=ρpcpbVst1−εTst,av−Tst,initial.Parameters and dimensions of packed bed storage stack model of Figure4 is presented in Table 2.Figure 4
Schematic diagram of packed pebble bed storage stack.Table 2
Bench mark experimental data [9] and calculated results.
Parameters
Nomenclature
Values
Unit
Porosity
ε
0.38
[−]
Length of the storage
L
0.9
[m]
Diameter of the storage
D
0.3
[m]
Optimum diameter of pebbles
d
0.02
[m]
Mass flow rate
m
˙
0.0495
[Kg/s]
Inlet/outlet pipe diameter
d
p
0.05
[m]
Thickness of stainless steel of the storage stack
S
0.002
[m]
Pebble specific heat capacity
c
pb
880± 50
[J/kgk]
Pebble thermal conductivity
k
p
2.5
[W/mk]
Density of absorber
P
abs
2940
[kg/m3]
Specific heat capacity of absorber
C
p
,
abs
1194
[J/kgK]
Inflow temperature
T
in
355
[°C]
Over all heat transfer coefficient
U
wall
0.4
[W/m2k]


## 2.1.1. Parabolic Solar Collector Model

The 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 Figure3, is calculated and explained in Table 1.Figure 3
Geometry and dimension of the solar collector parabolic dish.Table 1
The results of parameters and dimensions of parabolic solar collector.
Parameters
Nomenclature
Values
Unit
Aperture diameter
Da
2.0
[m]
Focus point
f
0.544
[m]
diameter of receiver
d
r
0.05
[m]
Rim angle
∅
85.17
[°]
Edge radius
R
1.0033
[m]
Aperture area
A
a
3.14
[m2]
Area of absorber
A
abs
0.00785
[m2]
Concentration ratio
C
400
[−]
Height of dish
H
0.45
[m]


## 2.1.2. Solar Thermal Storage Model

The 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. The 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. The size of the storage is modeled based on the amount of energy demand for the required application as follows.(1)Estored=ρpcpbVst1−εTst,av−Tst,initial.Parameters and dimensions of packed bed storage stack model of Figure4 is presented in Table 2.Figure 4
Schematic diagram of packed pebble bed storage stack.Table 2
Bench mark experimental data [9] and calculated results.
Parameters
Nomenclature
Values
Unit
Porosity
ε
0.38
[−]
Length of the storage
L
0.9
[m]
Diameter of the storage
D
0.3
[m]
Optimum diameter of pebbles
d
0.02
[m]
Mass flow rate
m
˙
0.0495
[Kg/s]
Inlet/outlet pipe diameter
d
p
0.05
[m]
Thickness of stainless steel of the storage stack
S
0.002
[m]
Pebble specific heat capacity
c
pb
880± 50
[J/kgk]
Pebble thermal conductivity
k
p
2.5
[W/mk]
Density of absorber
P
abs
2940
[kg/m3]
Specific heat capacity of absorber
C
p
,
abs
1194
[J/kgK]
Inflow temperature
T
in
355
[°C]
Over all heat transfer coefficient
U
wall
0.4
[W/m2k]


## 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 Figure5. The beam radiation is concentrated on the receiver (absorber) which heats the air that return from the thermal storage.Figure 5
Schematic diagram showing the heat transfer process in integrated packed bed with parabolic solar collector.The 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. The energy balance equation at the receiver can be written as [15, 17]:(2)Q˙abs=Q˙use+Q˙l.The energy on the absorber is given as function of aperture area beam radiation and concentration efficiency as follows(3)Q˙abs=ηopAapIb,where, the useful heat gain of the air through the receiver is formulated as:(4)Q˙use=m˙Cpa.Ta,out−Ta,in.And heat loss from an absorber becomes:(5)Q˙l=AabsUlr.Tabs,av−Tamb.The useful energy gain is elaborated by substituting Equations (4) and (5) in to Equation (2).(6)Q˙u=ηopAapIb−AabsUlrTabs,av−Tamb−ρabsVabsCp,absdTabsdt.To characterize the thermal performance of a solar concentrating collector, the concept of thermal efficiency is used. This concept refers to the ratio between useful energy carried by the heat transfer fluid and the energy incident concentrator aperture:(7)ηth=Q˙uAapIb=m˙CpaTa,out−Ta,inAapIb=ηop−AabsUlrTabs,av−TambAapIb−ρabsVabsCp,absdTabsAapIbm˙cpadt.The outlet temperature of air from the receiver storage is given as a function of an inlet temperature of air and solar radiation as follows.(8)Ta,outt=Ta,int+ηopAapIbtm˙Cpa−AabsUlrTabs,av−Tambm˙Cpa−ρabsVabsCp,absdTabsm˙cpadt,whereUlr is the mean coefficient of heat losses for the absorber, which is represented by [16, 17]:(9)Ulr=1hc+hr−1,where, the radiation heat transfer coefficient is given by:(10)hr=σεrTrec2+Tamb2Trec+Tamb.The convective heat transfer coefficient is calculated by considering thermal conductivity of airkair, the receiver outer diameter dr and Nusselt number Nu.(11)hc=kairdr×Nu.The Reynolds number is given as follows for a fluid flow in the pipe.(12)Re=vair×Doutγair.ForRe numbers between 1000 and 50,000, Nusselt number is given by the following equation:(13)Nu=0.148Re0.633.


### 2.2.2. Packed Bed Thermal Energy Storage Mathematical Model

One dimensional two phase energy equations describe the heat exchange between air (fluid phase) and pebbles (solid phase) as shown in Figure6 during charging and discharging, from which the transient temperature profiles in packed pebbles during charging and discharging times are determined. While analyzing the energy equations, the temperature dependent specific heat capacity of air is incorporated in the model.Figure 6
Schematic diagram showing the heat transfer process occurring within packed bed rocks of storage.The pebble bed consists of a cylindrical tank which can be divided into subdomains of packed pebbles with fixed thickness with the air flowing through the porous space around the pebbles.In formulating the mathematical model of the system, it was assumed that the porosity, mass flow rate and pebble geometry are uniform, air temperature is constant in the radial direction, and radiation heat transfer in the pebble bed (radiation 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 timedt of the differential 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 (fluid phase) is obtained as follows.(14)εAcsρacpa∂Ta∂t+hvAcsTa−Tp+m˙cpa∂Ta∂x+UwPLTa−Tamb=0.It shall be noted thatεporosityis the volume fraction of air while 1−ε is the volume fraction of pebbles.Considering energy balance of pebbles on the differential 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 equation for pebbles (solid phase) is obtained as follows.(15)ρp1−εAcscpb∂Tp∂t−hvAcsTa−Tp−∂∂xkeffA∂Tp∂x=0.The simplified transient energy equation for the pebble bed becomes(16)∂Tp∂t=Lxi1τTa−Tp1−e−NTUxi/L+keff1−ερpcpb∂2Tp∂x2,whereτ is thermal time constant, which is given as follows:(17)τ=1−ερpcpbAcsLm˙cpa=mpcpbm˙cpa.NTU is number of transfer unit and is given as(18)NTU=hvAcsLm˙cpa=hvLGcpa.


## 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 Figure5. The beam radiation is concentrated on the receiver (absorber) which heats the air that return from the thermal storage.Figure 5
Schematic diagram showing the heat transfer process in integrated packed bed with parabolic solar collector.The 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. The energy balance equation at the receiver can be written as [15, 17]:(2)Q˙abs=Q˙use+Q˙l.The energy on the absorber is given as function of aperture area beam radiation and concentration efficiency as follows(3)Q˙abs=ηopAapIb,where, the useful heat gain of the air through the receiver is formulated as:(4)Q˙use=m˙Cpa.Ta,out−Ta,in.And heat loss from an absorber becomes:(5)Q˙l=AabsUlr.Tabs,av−Tamb.The useful energy gain is elaborated by substituting Equations (4) and (5) in to Equation (2).(6)Q˙u=ηopAapIb−AabsUlrTabs,av−Tamb−ρabsVabsCp,absdTabsdt.To characterize the thermal performance of a solar concentrating collector, the concept of thermal efficiency is used. This concept refers to the ratio between useful energy carried by the heat transfer fluid and the energy incident concentrator aperture:(7)ηth=Q˙uAapIb=m˙CpaTa,out−Ta,inAapIb=ηop−AabsUlrTabs,av−TambAapIb−ρabsVabsCp,absdTabsAapIbm˙cpadt.The outlet temperature of air from the receiver storage is given as a function of an inlet temperature of air and solar radiation as follows.(8)Ta,outt=Ta,int+ηopAapIbtm˙Cpa−AabsUlrTabs,av−Tambm˙Cpa−ρabsVabsCp,absdTabsm˙cpadt,whereUlr is the mean coefficient of heat losses for the absorber, which is represented by [16, 17]:(9)Ulr=1hc+hr−1,where, the radiation heat transfer coefficient is given by:(10)hr=σεrTrec2+Tamb2Trec+Tamb.The convective heat transfer coefficient is calculated by considering thermal conductivity of airkair, the receiver outer diameter dr and Nusselt number Nu.(11)hc=kairdr×Nu.The Reynolds number is given as follows for a fluid flow in the pipe.(12)Re=vair×Doutγair.ForRe numbers between 1000 and 50,000, Nusselt number is given by the following equation:(13)Nu=0.148Re0.633.


## 2.2.2. Packed Bed Thermal Energy Storage Mathematical Model

One dimensional two phase energy equations describe the heat exchange between air (fluid phase) and pebbles (solid phase) as shown in Figure6 during charging and discharging, from which the transient temperature profiles in packed pebbles during charging and discharging times are determined. While analyzing the energy equations, the temperature dependent specific heat capacity of air is incorporated in the model.Figure 6
Schematic diagram showing the heat transfer process occurring within packed bed rocks of storage.The pebble bed consists of a cylindrical tank which can be divided into subdomains of packed pebbles with fixed thickness with the air flowing through the porous space around the pebbles.In formulating the mathematical model of the system, it was assumed that the porosity, mass flow rate and pebble geometry are uniform, air temperature is constant in the radial direction, and radiation heat transfer in the pebble bed (radiation 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 timedt of the differential 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 (fluid phase) is obtained as follows.(14)εAcsρacpa∂Ta∂t+hvAcsTa−Tp+m˙cpa∂Ta∂x+UwPLTa−Tamb=0.It shall be noted thatεporosityis the volume fraction of air while 1−ε is the volume fraction of pebbles.Considering energy balance of pebbles on the differential 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 equation for pebbles (solid phase) is obtained as follows.(15)ρp1−εAcscpb∂Tp∂t−hvAcsTa−Tp−∂∂xkeffA∂Tp∂x=0.The simplified transient energy equation for the pebble bed becomes(16)∂Tp∂t=Lxi1τTa−Tp1−e−NTUxi/L+keff1−ερpcpb∂2Tp∂x2,whereτ is thermal time constant, which is given as follows:(17)τ=1−ερpcpbAcsLm˙cpa=mpcpbm˙cpa.NTU is number of transfer unit and is given as(18)NTU=hvAcsLm˙cpa=hvLGcpa.


## 2.3. Computational Model

Discretization of the partial differential equation for air Equation (14) by using forward finite difference method, approximating the time and spatial first order derivates, the following explicit algebraic equation for air temperature variation is obtained:(19)Ta,i+1n+1−Ta,inΔt=hvερacpaTp,in−Ta,in−m˙εAcsρaTa,i+1n−Ta,inΔx−UwPεAcsρacpaTamb,in−Ta,in.Whilei and i+1 designates the current and front spatial nodes in the longitudinal directions, n and n+1 indicates the current and next time step.Discretizing the partial differential equation of the pebble, Equation (16), approximating first order spatial and temporal derivatives by forward difference and the second order spatial derivative by central difference scheme, the following explicit algebraic equation is obtained for the temperature variation of the pebbles.(20)Tp,in+1−Tp,inΔt=Lxi1τTa,in−Tp,in1−e−NTUxi/L+keff1−ερpcpbTp,i+1n−2Tp,in+Tp,i−1nΔx2,Tp,in+1=Tp,in1−L/xiΔt/2τη+Ta,inL/xiΔt/τη1+L/xiΔt/2τη+keffΔt1−ερpcpbΔx2Tp,i+1n−2Tp,in+Tp,i+1n,where,η=1−e−NTUxi/L and xi is segmental section.Allen [15] analyzed different convection heat transfer correlations among which the following was found to give better convective heat transfer between the air and the pebbles.(21)hp=kad0.345×2/30.7εRe0.7Pr1/3.The volumetric convective heat transfer coefficient becomes(22)hv=hp×1−ε×6d.The effective thermal conductivity for the packed pebbles is determined considering the volume fraction of the pebble and the air as follows [6]:(23)keff=1ε1/ka+1−ε/kpb.Prandtl number is given as(24)pr=cpaμka.The overall heat transfer coefficient through the wall is calculated from the equation below.(25)1UwA=1hiAi+xinskinsA+1hoAo.The natural convection heat transfer coefficient 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.(26)ho=kaD0.825+0.387RaD1/61+0.492/pr9/168/272.The natural convection heat transfer coefficient for the horizontal cylinder is determined from correlation of Nusselt number as function of the Rayleigh and Prandtl numbers as follows.(27)ho=kaD0.6+0.387Rad1/41+0.559/pr9/168/272.For hot air circulation pipe from receiver to the thermal storage, the following inside forced convective heat transfer coefficient is valid for laminar flow(28)Nu=1.86RePr1/3DL1/3μbμw0.14=hiDka.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.(29)Ta,outn+1=Ta,inn+η0pAapIbtm˙cpa−AabsUrlTabsn−Tam˙cpa−ρabsVabsCp,absTabsn−Tabsn−1Δtm˙cpa.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:(30)Twn+1=Twn+Q˙pant−Q˙rt+Q˙cvtmwcpwΔt.The stored thermal energy is obtained from the change in average temperature of the pebble and evaluating the change in enthalpy of the pebble bed during charging. The thermal storage efficiency is determined as the ratio of the stored thermal energy to the solar energy incident on the receiver as follows.(31)ηTS=∫0LρpcpAcs1−εTpx−Todx∫0tcIbAapdt.The cooker thermal efficiency is evaluated as the ratio of the useful heat transferred to the cooking media during discharging to the stored thermal energy for cooking.(32)ηth=mwcwTw,f−Tw,i∫0LρpcpAcs1−εTpx−Todx.The overall efficiency 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.(33)ηo=mwcwTw,f−Tw,i∫0tcIbAapdt.


## 3. Verification of Computational Model

An experimental investigation of thermal storage under constant heat input was conducted by Okello et al. [9] with 0.048 kg/s air mass flow 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 fixed mass flow rate (heated by electric resistance) flows from top to bottom of the thermal storage in the experimental set-up selected for verification. In the computation model of this system 10 spatial nodes and 1800 time steps were used for the simulation. Stratification of temperature throughout the packed bed pebbles is observed. The initial and boundary conditions to be used during charging of heat to packed bed pebbles for the simulation required to verify experimental data are given in Table 3.Table 3
Initial and boundary conditions.
T
a
t
,
x
=
0
= 355°C
T
P
t
,
x
=
0
=
T
p
t
,
x
=
1
Space step = 0.1 m
T
a
t
=
0
,
x
=
L
= 23°C
T
P
t
,
x
=
L
=
T
p
t
,
x
=
L
+
1
Number of space step = 10
T
P
t
=
0
,
x
=
L
= 23°C
Time stepΔt= 1 s
Maximum time charging = 5 h
Number of time step =1800The 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 Figure7. Hence, it can be concluded that the accuracy of the model is sufficient to simulate an integrated system of concentrating parabolic dish and thermal storage during charging and cooking conditions.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 flow rate of hot air. (sim. is simulation and ex. is experimental)


## 4. Results and Discussions

### 4.1. Charging with Solar Energy and Discharging with Water Boiling

Figure8 shows the maximum and minimum direct normal irradiance (DNI) for Addis Ababa selected for modeling of the parabolic solar collector from Sunrise to Sunset. The lowest DNI occurs in July and the highest is in March.Figure 8
Direct normal irradiance for maximum and minimum in the year with respect to time (7 AM to 5 PM).Parameters of solar collector are calculated and the results are explained as it is shown in Table1 which is used to investigate the variable charging of heat to storage.The charging of the pebble bed thermal storage was investigated 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 Tables2 and 3. Figure 9 shows the variation of air temperature at the outlet of the receiver of the parabolic solar collector 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.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.Figures10 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. After 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. The minimum temperatures of the air and the thermal storage were 383.2°C and 414.4°C, respectively.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.Figure 11
Temperature of thermocline storage charged with respect to charging h from 7 AM to 5 PM for the higher DNI.Figures12 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. After 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. The minimum temperatures were 129.48°C and 139.16°C for packed pebbles and air, respectively.Figure 12
Temperature of air charging the storage versus charging time from 7 AM to 5 PM for the lower DNI in July.Figure 13
Temperature of thermocline storage versus charging time from 7AM to 5 PM for the lowest DNI in July.Figure14 shows the amount of energy stored in thermocline storage for the days with maximum and minimum direct normal irradiance. The total stored energy was 40.1 MJ and 12.85 MJ for both conditions respectively.Figure 14
Energy stored versus charging h for both maximum and minimum direct normal irradiance.


### 4.2. Discharging Conditions

Heat is extracted from the storage when the ambient air is recirculated with 0.0048 kg/s mass flow rate of air throughout the storage from bottom to top. The discharging of heat from the storage was considered in the following two cases.(i)
When the cooking is carried out only by conduction and natural convection heat transfer, the pot is placed on the top of storage as shown in Figure2, 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.Figures15 and 16 Compare discharging of heat during water boiling to simulate cooking with forced convection versus conduction and natural convection heat transfer in the pebble bed. From the comparison, forced convection discharging gives higher useful cooking energy than from discharging by conduction without air recirculation. For 5 liters water to boil, 53 minutes discharging time is required to reach a temperature 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 temperature decreases gradually as it is shown in Figures 15 and 17. The parameters and values required during the discharging condition (during water boiling) are presented in Table 4.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 after charging a thermocline for 11 h for higher DNI.Figure 16
Temperature of storage versus temperature of water during boiling of 5 liters of water with 0.0048 kg/s mass flow rate for the maximum direct normal irradiance.Table 4
Dimensions and values required during water boiling simulation [18].
Parameters
Symbols
Values
Unit
Diameter of pan
d
pan
0.273
[m]
Length of pan
L
pan
0.17
[m]
Specific heat capacity of water
C
pw
4180
[J/kgK]
Stefan Boltzmann constant
σ
5.67 × 10−8
[W/m2K4]
Emissivity of steel pan
ϵ
0.4
[−]
Thermal resistance of pan
R
th
0.865
W/KThe initial and boundary conditions during discharging of heat from packed bed pebbles for water boiling test simulation are given in Table5.Table 5
Initial and boundary conditions for discharging process.
T
a
t
=
t
max
,
x
=
0
= 20°C
Space step = 0.1 m
Time stepΔt = 1 s
T
p
t
=
0
,
x
=
L
=
T
p,11 hours charge
Number of space step = 10
tmax = 11 hoursIn the previous simulation, 5 liters of water was heated in short time with insignificant change of pebble bed temperature profile. However, the thermal storage can heat a maximum of 28 liters of water, as shown in Figure17 over a longer period of time, with considerable cooling in pebble stack. From results of the simulation, the temperature of water reached 93°C after 8 h discharging time and top surface of the thermal storage temperature reached 109°C. Therefore, the storage has a capacity of boiling 28 liters of water for the day with the highest DNI.Figure 17
Temperature of thermal storage and 28 liters of water during discharging with 0.0048 kg/s mass flow rate of air for the higher 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 Figure18. From results of the simulation, the temperature of water reached 93°C after 4 hours discharging time and the top surface of the thermal storage temperature reached 138°C. Therefore, the storage has a capacity of boiling only 5 liters of water during the day of lowest DNI. The rest of energy in the storage will be available for the next day.Figure 18
Temperature of storage versus temperature of water during boiling of 5 liters of water with 0.0048 kg/s mass flow rate for the lower direct normal irradiance.Figure19 shows the temperature of storage during charging and discharges for two consecutive days. The cooking was done 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. The simulation was carried out in for March16 and March 17 of beam solar irradiance.Figure 19
Temperature of storage versus time for two consecutive days.Table6 explains different amount of energies and efficiencies of cooking for the conditions of the highest and lowest DNI from sunrise to sunset (7 AM to 5 PM).Table 6
Comparison of efficiencies for the conditions of higher and lower DNI.
DNI, highest
DNI, lowest
Solar energy on aperture area [MJ]
60.1
18.1
Stored energy [MJ]
40.1
12.85
Storage efficiency (%)
66.7
70.9
Volume of water boiled
5 liter for 53 min
5 liter for 4 h
Useful cooking energy [MJ]
1.47
1.47
Thermal efficiency of cooker (%)
45
31.1
Overall efficiency of cooker (%)
30
22.08Table7 explains the results of the temperatures and efficiencies when the cooking is carried out by conduction and enhanced by forced convection.Table 7
Efficiency for the condition of highest DNI.
Conduction
Forced convection
Water temperature at 53 min for 5 liters
87°C
93.0°C
Useful cooking energy [MJ]
1.344
1.463
Stored energy for 11 h charging
40.1 MJ
40.1 MJ
Solar energy on aperture
60.1
60.1
Storage efficiency
66.7%
66.7%
Thermal efficiency of cooker
41%
45%
Overall efficiency of cooker
27.34%
30%Figure20 shows degradation of the thermal storage without cooking operation in 8 days with final average temperature of 38°C. In this context, the storage interacts with the environment through fiberglass insulation having thickness of 75 mm.Figure 20
Temperature of storage when without operation with heat losses to the environment through insulation thickness of 75 mm.


## 4.1. Charging with Solar Energy and Discharging with Water Boiling

Figure8 shows the maximum and minimum direct normal irradiance (DNI) for Addis Ababa selected for modeling of the parabolic solar collector from Sunrise to Sunset. The lowest DNI occurs in July and the highest is in March.Figure 8
Direct normal irradiance for maximum and minimum in the year with respect to time (7 AM to 5 PM).Parameters of solar collector are calculated and the results are explained as it is shown in Table1 which is used to investigate the variable charging of heat to storage.The charging of the pebble bed thermal storage was investigated 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 Tables2 and 3. Figure 9 shows the variation of air temperature at the outlet of the receiver of the parabolic solar collector 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.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.Figures10 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. After 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. The minimum temperatures of the air and the thermal storage were 383.2°C and 414.4°C, respectively.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.Figure 11
Temperature of thermocline storage charged with respect to charging h from 7 AM to 5 PM for the higher DNI.Figures12 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. After 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. The minimum temperatures were 129.48°C and 139.16°C for packed pebbles and air, respectively.Figure 12
Temperature of air charging the storage versus charging time from 7 AM to 5 PM for the lower DNI in July.Figure 13
Temperature of thermocline storage versus charging time from 7AM to 5 PM for the lowest DNI in July.Figure14 shows the amount of energy stored in thermocline storage for the days with maximum and minimum direct normal irradiance. The total stored energy was 40.1 MJ and 12.85 MJ for both conditions respectively.Figure 14
Energy stored versus charging h for both maximum and minimum direct normal irradiance.


## 4.2. Discharging Conditions

Heat is extracted from the storage when the ambient air is recirculated with 0.0048 kg/s mass flow rate of air throughout the storage from bottom to top. The discharging of heat from the storage was considered in the following two cases.(i)
When the cooking is carried out only by conduction and natural convection heat transfer, the pot is placed on the top of storage as shown in Figure2, 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.Figures15 and 16 Compare discharging of heat during water boiling to simulate cooking with forced convection versus conduction and natural convection heat transfer in the pebble bed. From the comparison, forced convection discharging gives higher useful cooking energy than from discharging by conduction without air recirculation. For 5 liters water to boil, 53 minutes discharging time is required to reach a temperature 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 temperature decreases gradually as it is shown in Figures 15 and 17. The parameters and values required during the discharging condition (during water boiling) are presented in Table 4.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 after charging a thermocline for 11 h for higher DNI.Figure 16
Temperature of storage versus temperature of water during boiling of 5 liters of water with 0.0048 kg/s mass flow rate for the maximum direct normal irradiance.Table 4
Dimensions and values required during water boiling simulation [18].
Parameters
Symbols
Values
Unit
Diameter of pan
d
pan
0.273
[m]
Length of pan
L
pan
0.17
[m]
Specific heat capacity of water
C
pw
4180
[J/kgK]
Stefan Boltzmann constant
σ
5.67 × 10−8
[W/m2K4]
Emissivity of steel pan
ϵ
0.4
[−]
Thermal resistance of pan
R
th
0.865
W/KThe initial and boundary conditions during discharging of heat from packed bed pebbles for water boiling test simulation are given in Table5.Table 5
Initial and boundary conditions for discharging process.
T
a
t
=
t
max
,
x
=
0
= 20°C
Space step = 0.1 m
Time stepΔt = 1 s
T
p
t
=
0
,
x
=
L
=
T
p,11 hours charge
Number of space step = 10
tmax = 11 hoursIn the previous simulation, 5 liters of water was heated in short time with insignificant change of pebble bed temperature profile. However, the thermal storage can heat a maximum of 28 liters of water, as shown in Figure17 over a longer period of time, with considerable cooling in pebble stack. From results of the simulation, the temperature of water reached 93°C after 8 h discharging time and top surface of the thermal storage temperature reached 109°C. Therefore, the storage has a capacity of boiling 28 liters of water for the day with the highest DNI.Figure 17
Temperature of thermal storage and 28 liters of water during discharging with 0.0048 kg/s mass flow rate of air for the higher 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 Figure18. From results of the simulation, the temperature of water reached 93°C after 4 hours discharging time and the top surface of the thermal storage temperature reached 138°C. Therefore, the storage has a capacity of boiling only 5 liters of water during the day of lowest DNI. The rest of energy in the storage will be available for the next day.Figure 18
Temperature of storage versus temperature of water during boiling of 5 liters of water with 0.0048 kg/s mass flow rate for the lower direct normal irradiance.Figure19 shows the temperature of storage during charging and discharges for two consecutive days. The cooking was done 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. The simulation was carried out in for March16 and March 17 of beam solar irradiance.Figure 19
Temperature of storage versus time for two consecutive days.Table6 explains different amount of energies and efficiencies of cooking for the conditions of the highest and lowest DNI from sunrise to sunset (7 AM to 5 PM).Table 6
Comparison of efficiencies for the conditions of higher and lower DNI.
DNI, highest
DNI, lowest
Solar energy on aperture area [MJ]
60.1
18.1
Stored energy [MJ]
40.1
12.85
Storage efficiency (%)
66.7
70.9
Volume of water boiled
5 liter for 53 min
5 liter for 4 h
Useful cooking energy [MJ]
1.47
1.47
Thermal efficiency of cooker (%)
45
31.1
Overall efficiency of cooker (%)
30
22.08Table7 explains the results of the temperatures and efficiencies when the cooking is carried out by conduction and enhanced by forced convection.Table 7
Efficiency for the condition of highest DNI.
Conduction
Forced convection
Water temperature at 53 min for 5 liters
87°C
93.0°C
Useful cooking energy [MJ]
1.344
1.463
Stored energy for 11 h charging
40.1 MJ
40.1 MJ
Solar energy on aperture
60.1
60.1
Storage efficiency
66.7%
66.7%
Thermal efficiency of cooker
41%
45%
Overall efficiency of cooker
27.34%
30%Figure20 shows degradation of the thermal storage without cooking operation in 8 days with final average temperature of 38°C. In this context, the storage interacts with the environment through fiberglass insulation having thickness of 75 mm.Figure 20
Temperature of storage when without operation with heat losses to the environment through insulation thickness of 75 mm.


## 5. Conclusion

This 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 inflow and variable heat inflow for heating air by receiver of parabolic solar collector with constant mass flow approaches were used to evaluate the performance of the pebble bed thermal storage. The 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 efficiency was 66.7% for the day with highest DNI and the thermal efficiency for cooking were for heat transfer by forced convection and natural convection and conduction were 45% and 41%, respectively. The overall efficiency of cooking were 30% and 27.3% for the above two cases, respectively. For the day with the lowest DNI, the storage efficiency, thermal efficiency of cooking and overall efficiency 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 efficiency low. As the overall efficiencies 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 efficiency. It can be concluded that the overall efficiency of solar concentrating 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.



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*Source: 4238549-2019-12-29.xml*