1. Introduction

2. Materials and Methods

2.1. Numerical Model of Photovoltaic-Thermal System

The energy balance equations for unsteady state of the various layers of the PVT are discussed as follows under the following assumptions:

i. One dimensional (1D) thermal model is considered.

ii. The fraction of solar irradiance that is not converted into electrical energy by the PV cell is transferred to the system.

iii. The optical properties of glass cover, the PV cells and EVA layer are constant.

iv. The PV cells and EVA layer temperatures are the same.

v. The properties of the materials are considered homogenous and constant.

vi. Perfect contact between PVT components.

Figure 1. Detailed parts of water-based and equivalent electrical circuit of PVT system [20].

Figure 1. Detailed parts of water-based and equivalent electrical circuit of PVT system [20].

2.1.1. Glass Energy Balance Equations

At the glass layer, the energy is accumulated by the glass, solar irradiance absorbed, and energy lost by convection, radiation and conduction as demonstrated below.

$$\begin{gathered} \left({\rho }_g{\delta }_g{C}_g\right)\frac{d{T}_g}{dt}={\alpha }_{g}G+{hr}_{g.sky}\left(T_{sky}-T_g\right)+{hr}_{g.gr}\left(T_{gr}-T_g\right)+{hv}_{g,a}\left(T_a-T_g\right) \\ +{hc}_{pv,g}PF\left(T_{pv}-T_g\right)+{hc}_{Ted,g}\left(1-PF\right)\left(T_{Ted}-T_g\right) \end{gathered}$$

(1)

where ${\alpha }_{g}G$ is the absorbed solar irradiance by the glass (W/m²); $hr$, $hv$ and $hc$ are heat transfer of radiation, convection, and conduction (W/m². K) respectively.

PF is the packing factor which is the ratio between the surface covered by the PV cells and the total surface of the panel been considered.

The radiative exchange of coefficient depends on $T_{sky}, T_{gr}$ and view factors $F_{g,sky} and F_{g,gr}, as:$

$${hr}_{g,sky}={\sigma \epsilon }_g{F}_{g,sky}A\left(T_{sky}-T_g\right).\left(T_{sky}^2+T_g^2\right)$$

(2)

$${hr}_{g,gr}={\sigma \epsilon }_g{F}_{g,gr}A\left(T_{gr}+T_g\right).\left(T_{gr}^2+T_g^2\right)$$

(3)

$$F_{g,sky}=\frac{1-cos\theta }{2}$$

(4)

$$F_{g,gr}=\frac{1+cos\theta }{2}$$

(5)

$${hv}_{forced}=5.7w+11.4$$

(6)

where w is the wind speed in (m/s).

2.1.2. PV Cell Energy Balance Equations

The PV cells are affected by solar radiation transmitted through the glass $\left({\tau }_{g}G\right),$ to convert the solar radiation into electrical energy $\left({\eta }_{e}G\right),$ and transfer heat by conduction to the down layers.

$$PF\left({\rho }_{PV}{\delta }_{PV}{C}_{PV}\right)\frac{d{T}_{PV}}{dt}=\left[\left({\tau }_g{\alpha }_{PV}-{\eta }_e\right)G+{hc}_{PV,g}\left(T_g-T_{PV}\right)+{hc}_{Ted,PV}\left(T_{Ted}-T_{PV}\right)\right]PF$$

(7)

2.1.3. Tedlar Energy Balance Equations

The Tedlar is in contact with the PV cells at the upper layer and the remaining part (1-PF) is considered in contact with the glass, while below is totally in contact with the absorber which is aluminium in this case. Therefore, Tedlar has been affected by the solar radiation that passes through the glass to the PV cells.

$$\begin{gathered} \left({\rho }_{Ted}{\delta }_{Ted}{C}_{Ted}\right)=\frac{d{T}_{Ted}}{dt}\left(1-PF\right){\tau }_g{\alpha }_{Ted}G+\left(1-PF\right){hc}_{Ted,g}\left(T_g-T_{Ted}\right) \\ +PF.{hc}_{Ted,PV}\left(T_{PV}-T_{Ted}\right)+{hc}_{absh,Ted}\left(T_{absh}-T_{Ted}\right) \end{gathered}$$

(8)

2.1.4. Absorber Energy Balance Equations

The absorber consists of upper and lower plate surface, the energy balance at the upper surface transmit heat by conduction with the Tedlar layer, the energy exchange between this plate and the fluid. The upper layer and lower layer are radiative in corresponding with the channel housing the working fluid and is conductive to the channels as well.

The surface where the fluid flow and the rest area of the absorber is known as percentage of the channel (PC) and is shown below the equation.

$$\begin{gathered} \left({\rho }_{absh}{\delta }_{absh}{C}_{absh}\right)\frac{d{T}_{absh}}{dt}={hc}_{absh,Ted}\left(T_{Ted}-T_{absh}\right)+\left(1-PC\right){hc}_{absl,absh}\left(T_{absl}-T_{absh}\right)+ \\ PC.{hc}_{absl,absh}\left(T_{absl}-T_{absh}\right)+{hc}_{f,absh}\left(T_f-T_{absh}\right) \end{gathered}$$

(9)

The radiative heat transfer coefficient between the two aluminium plates is calculated based on Ben Cheikh correlation [20]

$${hr}_{absl,absh}=\frac{4\sigma T^3}{\left(\frac{1}{{\epsilon }_{abs}}+\frac{1}{{\epsilon }_{abs}}-1\right)}$$

(10)

where $\sigma , T and {\epsilon }_{abs}$ are Stefan-Boltzmann constant, mean plate temperature and plate emissivity.

The energy balance at the lower absorber plate is considered as the amount of energy transferred with the fluid, the upper absorber plate, and the heat exchange with outer surface (ground and air) is shown in Equation (11).

$$\begin{gathered} \left({\rho }_{absl}{\delta }_{absl}{C}_{absl}\right)\frac{d{T}_{absl}}{dt}={hc}_{absl,f}\left(T_f-T_{absl}\right)+\left(1-PC\right){hc}_{absl,absh}\left(T_{absh}-T_{absl}\right)+ \\ PC.{hr}_{absl,absh}\left(T_{absh}-T_{absl}\right)+{hv}_{a,absl}\left(T_a-T_{absl}\right)+{hr}_{sky,absl}\left(T_{sky}-T_{absl}\right)+ \\ {hr}_{gr,absl}\left(T_{gr}-T_{absl}\right) \end{gathered}$$

(11)

The convective heat transfer coefficient between low absorber plate and outer surrounding area is calculated as a function of thermal air conductivity and Nusselt number in case of free convection [20].

$${Nu}_{free}={\left\{0.825+\frac{{0.387Ra}^{1/6}}{{\left[1+{\left(\frac{0.492}{Pr}\right)}^{9/16}\right]}^{8/27}}\right\}}^2$$

(12)

In force convection:

$${hv}_{a,absl}=5.7w_{bc}$$

(13)

The radiative exchange coefficients depend by $T_{sky}, T_{gr}$ and the view factors $F_{sky,absl}, and F_{gr,absl}$ as:

$${hr}_{sky,absl}={\sigma \epsilon }_{absl}{F}_{sky,absl}A\left(T_{sky}-T_{absl}\right).\left({T^2}_{sky}-{T^2}_{absl}\right)$$

(14)

$${hr}_{gr,absl}={\sigma \epsilon }_{absl}{F}_{gr,absl}A\left(T_{gr}+T_{absl}\right).\left({T^2}_{gr}+{T^2}_{absl}\right)$$

(15)

$$F_{sky,absl}=\frac{1-cos\theta }{2}$$

(16)

$$F_{gr,absl}=\frac{1+cos\theta }{2}$$

(17)

2.1.5. Fluid Energy Balance Equations

The heat exchange between the coolant fluid and the absorber is governed by Equation (18):

$$\left({\rho }_f{\delta }_f{C}_f\right)\frac{d{T}_f}{dt}={hc}_{f.absh}\left(T_{absh}-T_f\right)+{hc}_{absl.f}\left(T_{absl}-T_f\right)+\dot{m}{C}_f\left(T_{out}-T_{in}\right)$$

(18)

where $\dot{m}$ is the mass flow rate (kg/s), $C_f$ is the fluid heat capacity (J/kg. K), $T_{in} and T_{out}$ are inlet and outlet temperature of the fluid in (K). the mean temperature between the inlet and outlet temperature of the fluid is given as:

$$T_f=\frac{T_{out}+T_{in}}{2}$$

(19)

The water from the mains at temperature T_sup = 20^oC in this case, flows into the tank where it is mixed with already heated water at temperature T_t. The hot water demand is drawn from the tank at temperature T_t.

The energy balance considering the heat supplied by the panels, the heat supplied to household demand and the heat losses from the tank to environment.

$$\dot{\dot{Q}=\dot{m}}{C}_f\left(T_{in}-T_{out}\right)$$

(20)

According to El Fouas et al. [20], the temperature at the outlet of the solar tank T_t is calculated using the effectiveness $\left({\epsilon }_H\right)$ of the heat exchanger.

$$T_{t,out}=T_{t,in}-{\epsilon }_H\left(T_{t,in}-T_t\right)$$

(21)

The energy balance equation of the tank becomes:

$$\left(M_t{C}_f\right)\frac{d{T}_t}{dt}={\epsilon }_H\dot{m}{C}_f\left(T_{t,out}-T_t\right)-\dot{{\dot{m}}_l{C}_w\left(T_t-T_{sup}\right)-U_t{S}_t\left(T_t-T_a\right)}$$

(22)

where ${\dot{m}}_l$ is the mass flowrate of the thermal load, $C_w$ is the specific heat of water, $U_t$ is the heat loss coefficient of the tank to the environment (W/K m²), $S_t$ is the external surface of the tank.

2.3. PVT System Design and Description

The system components include PVT panel with 1.64m² area, a water tank of volume capacity of 200L, water pump, a charge controller, and a storage battery. The PV panel is integrated with a heat exchanger that circulate water as coolant. The fluid is circulated in the rear of the PV panel as soon as the surface temperature of the PV is above 35^oC to remove the excess heat at the rear of the PV panel. This drop down the temperature of the PV panel and increase the power output and electrical efficiency of the system. The electrical power produced is sent to the controller which distribute the energy to demand and charge the battery. Figure 2 shows the block diagram of PVT system understudy.

Figure 2. Block diagram of PVT system design.

Figure 2. Block diagram of PVT system design.

2.3.1. TRNSYS Simulation Model

TRNSYS is a transient simulation software program that simulate the electrical and thermal behaviour of the PVT through the introduction of the meteorological weather data with solar irradiance, ambient temperature, and wind speed. TRNSYS 18 version of software was used for the modelling of the PVT system. The mathematical models of the components connected in the PVT are formulated as equations by FORTRAN code to make up its physical behaviour. The simulation time is set from 6:00am to 6:00pm daily for all the twelve (12) months in the year. The TRNSYS model for this study is shown in Figure 3.

Figure 3. The TRNSYS model of PVT system.

Figure 3. The TRNSYS model of PVT system.

A Polycrystalline (FT250Cs) PV module at standard test condition (STC) with irradiation of 1000W/m², ambient temperature of 25^oC and Air Mass of AM1.5 is used in the design. The parameters of the PV panel used in the simulation are shown in Table 1.

This paper investigates the potentials and benefits of using PVT technology in sub-Saharan climate such as Nigeria in the West African region, which have not extensively studied. The three sites chosen for comparative study are: Maiduguri in the North-East, Makurdi in the North-central and Port Harcourt in the South-South of the country. The system simulates the performance of PVT in terms of electricity and hot water production in the selected regions in the country.

A typical off-gride households in developing countries could have an average domestic hot water demand of 5.6L per person per day which is about 28L per day for a four-person household. While in developed countries would be 30L per person per day or 120L per day for a four-person household [21]. A 200L tank to produced hot water for a family of four estimated at 50liters per person with flowrate at 100L/h was selected by experimenting different flowrate from 10L/h to 200L/h.

3. Simulation Results and Analysis

3.1. Solar Radiation and Ambient Temperature of the Three Cities Understudy

The output performance of PV panel is directly proportional with the rate of irradiation presence at a particular area. The surface temperature of PV and the ambient temperature of a place are radiation parameters. The higher the solar radiation of a place, the higher the output power of the PV system and the higher the PV surface temperature. Higher ambient temperature lowers the efficiency of the PV panel. Figure 4 shows the ambient temperature and solar radiations in the three cities in Nigeria.

Figure 4. Solar radiation and ambient temperature of the three cities in Nigeria.

Figure 4. Solar radiation and ambient temperature of the three cities in Nigeria.

3.2. Electrical Power Outputs of the System

The combine results of the three cities under study in Nigeria show results of electrical power production, Maiduguri having the highest peak power and total annual electrical power output of 184.02kWh/kWp and 1906.87kWh/kWp respectively, which signifying higher solar irradiation in the region. Makurdi producing the second highest peak power and total annual electrical power of 160.5kWh/kWp and 1541.82kWh/kWp respectively. And for Port Harcourt, the lowest peak power and total annual electrical power yield of 147.53kWh/kWp and 1355.16kWh/kWp respectively is produced which signified low solar irradiation in the region. This total annual electrical power output differences by the PVT in the country was confirmed by the study of Adun et al. [18]. Figure 5 present the monthly peak electrical power outputs of the three cities understudy.

Figure 5. Monthly peak electrical power yield in three cities in Nigeria.

Figure 5. Monthly peak electrical power yield in three cities in Nigeria.

3.3. Thermal Output of the System

The thermal outputs in Figure 6 shows Maiduguri producing the highest peak thermal and total annual thermal of 26.99kWp and 287.09kWp respectively. Makurdi having peak thermal and annual thermal output of 25.03kWp and 249.47kWp respectively. While Port Harcourt again having lowest peak thermal and total annual thermal output of 23.13kWp and 224.44kWp respectively because of low solar radiation and the ambient temperature which is depicted in Figure 4.

Figure 6. Monthly peak thermal outputs generated in the three cities in Nigeria.

Figure 6. Monthly peak thermal outputs generated in the three cities in Nigeria.

3.4. Electrical Efficiency of the System

The electrical efficiency of the three cities shows Port Harcourt having the highest monthly average electrical efficiency peak of 14.89% because of low radiation and low ambient temperature which contribute to less PV surface temperature. Makurdi with second highest monthly average electrical efficiency peak of 14.87% and the least is Maiduguri with monthly average electrical efficiency peak of 14.84% because of high solar radiation and high ambient temperature which produced high PV surface temperature in the region. Figure 7 shows the monthly average electrical efficiency of the three cities understudy.

Figure 7. Monthly average electrical efficiencies of the three cities in Nigeria.

Figure 7. Monthly average electrical efficiencies of the three cities in Nigeria.

3.4. Thermal Efficiency of the System

Figure 8, shows the monthly average thermal efficiency of the three cities, presenting Port Harcourt with the highest thermal efficiency of 67.23%. Makurdi with the second highest thermal efficiency peak of 60.26 and for Maiduguri, the lowest thermal efficiency peak of 58.65% is recorded Maiduguri respectively.

Figure 8. Monthly average thermal efficiencies of the three cities in Nigeria.

Figure 8. Monthly average thermal efficiencies of the three cities in Nigeria.

3.4. Outlet Fluid Temperature from PVT

Maiduguri recorded a monthly average fluid outlet temperature of 29.79^oC, while Makurdi record 29.04^oC and for Port Harcourt, 27.38^oC is the monthly average outlet temperature. Figure 9 shows Maiduguri with the highest outlet fluid temperature against the rest of the cities and is because of variation in radiation and ambient temperature that exist between the three cities as shown in Figure 4.

Figure 9. Monthly average outlet fluid temperatures of the three cities in Nigeria.

Figure 9. Monthly average outlet fluid temperatures of the three cities in Nigeria.

3.4. PV Cell Surface Temperature

The PV cell surface temperature of the three cities at monthly average is 38.39^oC for Maiduguri, 36.43^oC for Makurdi and 34.36^oC for Port Harcourt. Figure 10 shows the results of PV surface temperature of both cities. Maiduguri record the highest PV cell temperatures because of high solar radiation and ambient temperature in the region.

Figure 10. Monthly average PV surface temperature of the three cities in Nigeria.

Figure 10. Monthly average PV surface temperature of the three cities in Nigeria.

3.4. The Comparative Performance Evaluation of the Conventional PV and the PVT System

The conventional PV system produce total electrical power output of 1526.83W in a day while the PVT system generate a total power output of 1566.82W in the same day. Electrical efficiency of 13.99% is recorded at maximum for the conventional PV system, while the PVT system recorded electrical efficiency of 15.01% at maximum value. The conventional PV system record a cell surface temperature of 49.25^oC at peak sunny time of the day while the cooled PVT system recorded a surface temperature of 38.38^oC at peak sunny time of the day. Figure 11a–c shows the comparison results of PV and PVT in terms of Power output, efficiency, and cell surface temperature respectively.

Figure 11. a). Hourly power outputs of PV and PVT in a day. b). Hourly electrical efficiency of PV and PVT in a day. c). Hourly surface temperature of PV and PVT in a day.

Figure 11. a). Hourly power outputs of PV and PVT in a day. b). Hourly electrical efficiency of PV and PVT in a day. c). Hourly surface temperature of PV and PVT in a day.

4. Conclusions

Nomenclature

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