# Modeling Joule Heating Effect on Thermal Efficiency of Photovoltaic Thermal (PVT) Collectors with Operation Mode Factor (OMF)

## Abstract

**:**

## 1. Introduction

^{2}.°C), inlet water temperature ${T}_{i}$ (in °C), ambient air temperature ${T}_{a}$ (in °C), and irradiance $I$ (in W/m

^{2}). The collector system has high performance, indicated by the high $\tau \alpha $ and low ${U}_{L}$ (see Equation (1)). The energy gain part is ${F}_{R}(\tau \alpha )$, while the energy loss part is ${F}_{R}{U}_{L}({T}_{i}-{T}_{a})/I$. It is assumed that the PV cell in the PVT collector provides thermal resistance when electricity is not generated [9]. It is essential to determine the correlation between the PV cell without electricity generation and its resistance effect [10].

## 2. Materials and Methods

#### 2.1. Experiment Setup

^{3}/s ≈ 4 L/min), the water circulation system used a temperature-controlled water chamber connected to a pump and flow meter. The heat exchanger system took heat from the outlet water collector before entering the water circulation system. The experiments took place during sunny days in winter, with four inlet water temperatures ${T}_{i}$ = 12; 15; 20; and 25 °C. The data of irradiation $I$ from pyranometer, wind speed ${V}_{w}$ from anemometer, inlet water temperature ${T}_{i}$, outlet water temperature ${T}_{o}$, ambient air temperature ${T}_{a}$, voltage ${V}_{mpp}$, and current ${I}_{mpp}$ were collected by a data logger every 30 s. Data collection was focused during 12:00–13:00 to ensure stable irradiation and ambient air temperature. The experiments were run before and after that period to avoid the hysteresis effect on the PVT collector. Figure 1b shows the PVT array 2 × 2 connected in series for higher power output to provide a higher Joule heating effect. The experiment used polycrystalline silicon (p-Si) PV modules. Each PV module covered the top of the thermal collector. The numbers indicated the water flow from 1 (as inlet)—2—3—4 (as outlet). Figure 1c shows the construction of the PVT collector. Each PVT collector is 790 mm × 820 mm, or 0.60 m

^{2}. Each PV module area ${A}_{pv}$ is 780 mm × 770 mm or 0.58 m

^{2}. The total area of the PVT collector ${A}_{c}$ is 2.40 m

^{2}. A more detailed setup was given by [13].

#### 2.2. Construction of Operation Mode Factor (OMF)

^{2}, can be expressed as follows:

^{2}), with the heat transfer coefficient from the surface to an absorber of PVT collector ${K}_{SA}$ (W/m

^{2}.C) and the temperature difference between the surface and fluid of PVT, ${T}_{S}$ and ${T}_{f}$ (°C), respectively. Here, in steady state, the absorber temperature ${T}_{A}={T}_{f}$.

^{2}) with the overall heat loss coefficient of PVT ${U}_{L}$ (W/m

^{2}.C) and the temperature difference between ${T}_{s}$ (°C) and the ambient ${T}_{a}$ (°C).

#### 2.3. Validation

## 3. Result and Discussion

#### 3.1. Reconstruction Thermal Efficiency with Operation Mode Factor (OMF)

**,**${U}_{L,pvt}/{U}_{L,t}$

**.**

^{2}= 0.811. ${\eta}_{e}$ theorizes as

#### 3.2. Simulation

**,**${U}_{L}$

**,**${\eta}_{t}$, $({T}_{i}-{T}_{a})/I$, RMS, etc. There was no significant difference between Table 1 and Table 2. That is, the OMF component of the experiment and the simulation provided almost identical values.

## 4. Discussion

**,**as well as software development optimization parameter design [48] with advanced working fluids [49] and intelligent forecasting in operation [50].

## 5. Conclusions

**,**is responsible for changes in the thermal efficiency of T-mode and PVT-mode. From T-mode to PVT-mode, OMF increases 1.3 times, with ${K}_{SA}$ and ${U}_{L}$ by more than 4 times and 1.3 times, respectively. OMF may correlate to ${F}_{R}$, and their empirical relation may need to be studied further. The most decisive component in OMF was ${K}_{SA}$, which may reduce the absorber thermal resistance, while the thermal conductivity increases (see Table 1 and Table 2). There was additional heat from internal heating in the effective region, which caused the PVT-mode thermal efficiency to be higher than T-mode. The electrical efficiency ${\eta}_{e}$ decreases as the PV surface temperature ${T}_{S}$ increases, and the reduced temperature $({T}_{i}-{T}_{a})/I$ also increases. Thus, the Joule heating effect also decreases. This OMF thermal efficiency model can complement the previous models. The steady-state thermal efficiency model with OMF has confirmed PVT-mode and T-mode. The practical application of OMF is for domestic small- and medium-scale PVT systems, considering the electricity and low-temperature heat as equally important. For future research, the layers analysis should investigate ${K}_{SA}$ and ${U}_{L}$ values, as well as software development, for design optimization and operation.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PVT | Hybrid photovoltaic and thermal |

EVA | Ethylene Vinyl Acetate |

SC | Short-circuit |

RMS | Root mean square |

OMF | Operation mode factor |

p-Si | Polycrystalline silicon |

MPPT | Maximum power point tracking |

OC | Open-circuit |

vs. | Versus |

Nomenclatures | |

${T}_{S}$ | Collector surface temperature [°C] |

${T}_{A}$ | Absorber temperature [°C] |

${T}_{i}$ | Inlet water temperature [°C] |

${T}_{f}$ | Medium temperature of fluid [°C] |

${T}_{a}$ | Ambient air temperature [°C] |

${U}_{L}$ | Collector overall loss coefficient [W/m^{2}.°C] |

${\eta}_{e}$ | Electrical efficiency of PV [%] |

$\beta $ | Temperature coefficient of PV cell |

${\eta}_{t}$ | Thermal efficiency of a collector [%] |

${\eta}_{pvt}$ | Total efficiency of PVT collector [%] |

${\eta}_{t,sim}$ | Thermal efficiency from simulation [%] |

${\eta}_{t,reg}$ | Thermal efficiency from regression [%] |

$n$ | Number of data points observed |

$RM{S}_{r-e}$ | RMS value regression to experiment |

${Q}_{s}$ | Total solar energy strikes the system [W] |

${Q}_{th}$ | Thermal energy converted [W] |

${Q}_{ref}$ | Energy loss by reflection [W] |

${Q}_{conv}$ | Energy loss by convection [W] |

${Q}_{l,tot}$ | Total losses [W] |

${Q}_{pv}$ | Energy generated by the PV cell [W] |

${K}_{SA}$ | Cond. h. tr. coeff. Surf. – Abs. [W/m^{2}.°C] |

${Q}_{ref}$ | Energy loss by reflection [W] |

${Q}_{conv}$ | Energy loss by convection [W] |

${Q}_{u}$ | Useful heat collected by the fluid [W] |

$\dot{m}$ | Mass flow rate [m^{3}/s] or [litre/minute] |

${A}_{pv}$ | Effective PV area [m^{2}] |

${V}_{mpp}$ | Voltage output of PV at MPPT [V] |

${I}_{mpp}$ | Current output of PV at MPPT [A] |

${R}_{L}$ | Load resistor [Ω] |

${T}_{o}$ | Outlet water temperature [°C] |

${A}_{c}$ | Total area of PVT collector [m^{2}] |

${F}_{R}$ | Heat removal factor |

${\eta}_{r}$ | Reference efficiency of PV cell [%] |

${T}_{r}$ | Reference temperature of PV cell [%] |

$\tau $ | Transmittance |

$\alpha $ | Absorptance |

${\eta}_{o}$ | Optical efficiency [%] |

${\eta}_{t,\mathrm{exp}}$ | Thermal efficiency from the experiment |

$RM{S}_{s-e}$ | RMS value simulation to experiment |

$RM{S}_{s-r}$ | RMS value simulation to regression |

${Q}_{u,th}$ | Thermal energy transferred to the fluid [W] |

${Q}_{rad}$ | Energy loss radiation [W] |

${Q}_{pvt}$ | Solar energy converted by PVT [W] |

${Q}_{e}$ | Electrical energy converted by PV [W] |

${Q}_{ih}$ | Electro-thermal / internal heating energy [W] |

${Q}_{ih,th}$ | Useful heat from internal heating [W] |

${Q}_{ih,l}$ | Waste heat from internal heating [W] |

${U}_{L}$ | Overall loss coefficient [W/m^{2}.°C] |

$OM{F}_{pvt}$ | Operation Mode Factor for PVT-mode |

$OM{F}_{t}$ | Operation Mode Factor for T-mode |

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**Figure 1.**Previous outdoor experiment [13]: (

**a**) Experiment setup, (

**b**) PVT array 2 × 2, (

**c**) PVT construction.

**Figure 3.**Thermal efficiency vs. the reduced temperature for the wind loss effect: (

**a**) T-mode simulation vs. T-mode experiment; (

**b**) PVT-mode simulation vs. PVT-mode experiment.

**Figure 4.**The thermal efficiency vs. the reduced temperature for the without wind loss effect: (

**a**) T-mode (simulation vs. experiment); (

**b**) PVT-mode (simulation vs. experimentation).

**Figure 5.**The thermal efficiency vs. the reduced temperature: (

**a**) T-mode simulation vs. PVT-mode simulation for the wind loss effect; (

**b**) T-mode simulation vs. PVT-mode simulation for the without wind loss effect.

**Figure 6.**Electrical and thermal efficiency: (

**a**) PVT-mode (simulation) wind loss effect and without wind loss effect; (

**b**) the effective region for PVT-mode (simulation) vs. T-mode (simulation) for the without wind loss effect.

**Figure 7.**Simulation of thermal efficiency vs. reduced temperature: (

**a**) Without wind loss effect; (

**b**) wind loss effect.

Items | Wind Loss Effect | Without Wind Loss Effect | |||
---|---|---|---|---|---|

Parameter | Unit | T-Mode | PVT-Mode | T-Mode | PVT-Mode |

${K}_{SA}$ | W/m^{2}.°C | 44.38 | 176.11 | 46.68 | 242.95 |

${U}_{L}$ | W/m^{2}.°C | 28.24 | 36.57 | 23.34 | 32.93 |

${\eta}_{t}$$\to ({T}_{i}-{T}_{a})/I=0$ | ×100% | 0.495 | 0.567 | 0.540 | 0.603 |

$({T}_{i}-{T}_{a})/I$$\to {\eta}_{t}=0$ | °C.m^{2}/W | 0.029 | 0.019 | 0.035 | 0.021 |

${\eta}_{t}/\left[({T}_{i}-{T}_{a})/I\right]$ | %.m^{2}/°C.W | 17.26 | 30.28 | 15.58 | 29.00 |

$RM{S}_{r-e}$ | % | 16.3 | 11.3 | 14.0 | 10.2 |

$OMF\approx {F}_{R}$ | ^{-} | 0.61 | 0.83 | 0.67 | 0.88 |

$OM{F}_{pvt}/OM{F}_{t}$ | ^{-} | 1.36 | 1.31 | ||

${K}_{SA,pvt}/{K}_{SA,t}$ | - | 3.84 | 9.82 | ||

${U}_{L,pvt}/{U}_{L,t}$ | - | 1.27 | 1.43 |

Items | Wind Loss Effect | Without Wind Loss Effect | |||
---|---|---|---|---|---|

Parameter | Unit | T-Mode | PVT-Mode | T-Mode | PVT-Mode |

${K}_{SA}$ | W/m^{2}.°C | 44.39 | 180.42 | 46.68 | 257.34 |

${U}_{L}$ | W/m^{2}.°C | 28.25 | 36.70 | 23.34 | 32.96 |

${\eta}_{t}$$\to ({T}_{i}-{T}_{a})/I=0$ | ×100% | 0.495 | 0.569 | 0.540 | 0.603 |

$({T}_{i}-{T}_{a})/I$$\to {\eta}_{t}=0$ | °C.m^{2}/W | 0.029 | 0.019 | 0.035 | 0.021 |

${\eta}_{t}/\left[({T}_{i}-{T}_{a})/I\right]$ | %.m^{2}/°C.W | 17.26 | 30.28 | 15.58 | 29.22 |

$RM{S}_{(r-e)}$ | % | 16.3 | 10.9 | 14.0 | 10.2 |

$RM{S}_{(s-e)}$ | % | 0.00 | 0.82 | 0.02 | 0.74 |

${K}_{SA}/{U}_{L}$ | - | 1.57 | 4.92 | 2.00 | 7.81 |

$OMF\approx {F}_{R}$ | ^{-} | 0.61 | 0.83 | 0.67 | 0.89 |

$OM{F}_{pvt}/OM{F}_{t}$ | ^{-} | 1.36 | 1.33 | ||

${K}_{SA,pvt}/{K}_{SA,t}$ | - | 4.06 | 5.51 | ||

${U}_{L,pvt}/{U}_{L,t}$ | - | 1.30 | 1.41 |

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**MDPI and ACS Style**

Yandri, E.
Modeling Joule Heating Effect on Thermal Efficiency of Photovoltaic Thermal (PVT) Collectors with Operation Mode Factor (OMF). *Appl. Sci.* **2022**, *12*, 742.
https://doi.org/10.3390/app12020742

**AMA Style**

Yandri E.
Modeling Joule Heating Effect on Thermal Efficiency of Photovoltaic Thermal (PVT) Collectors with Operation Mode Factor (OMF). *Applied Sciences*. 2022; 12(2):742.
https://doi.org/10.3390/app12020742

**Chicago/Turabian Style**

Yandri, Erkata.
2022. "Modeling Joule Heating Effect on Thermal Efficiency of Photovoltaic Thermal (PVT) Collectors with Operation Mode Factor (OMF)" *Applied Sciences* 12, no. 2: 742.
https://doi.org/10.3390/app12020742