Mathematical Modelling of Billboard Type Central Solar Receiver for Domestic Application

Adoption of renewable energy technologies for India's sustainable development is a critical approach in scientific research in the energy generation field, particularly technologies that generate such types of energy. According to literature surveys, a significant amount of scientific research has been conducted by Laporte (2021) in order to expand the system's functionality that generates energy from these resources. Manzoo's work on time resolution assessment was used to estimate the central solar receiver (2021).

A theoretical study of convection in volumetrically absorbing solar thermal receivers, Mishra (2020). Evaluation of three methods to establish the slant path meteorological intensification of the center receiver, Madjid (2020). Satyavan's work on the novel hybridization of the solar central receiver system (2020). A quantitative analysis of the spiral solar receiver's thermal evaluation for such a centralized transmitter system is required, Chen (2019).

An investigation into the high temperature features of chemical vapor-deposited AlN coatings for solar central receivers, Fang (2019). An air curtain-equipped solar central cavity receiver's natural convective heat loss is being investigated numerically, Ayad (2018). Create a New Detector for the Solar Energy's Prime Tower, Danyela (2018). Based on observed data of direct direct sunlight, ocular risks are assessed in a central transmitter solar power facility, Andreas (2017). Utilizing molten salt and liquid metals, solar central reception tubes’ transient CFD and FEM computations are compared, Danyela (2017). Examining the cumulative and instantaneous skin radiation exposures in central receivers solar systems, Danyela (2016). An analysis of solar levels of exposure at work in concentrating solar power systems, Reyes (2016). A pressurized carbon dioxide cycle's optimization for a cutting-edge central receiver, Alberto (2015).

A projection approach for an analytical function to determine how solar radiation will affect central receivers, J. Coventry (2015). An analysis of central receiver potassium receiver technologies, Bruno (2014). Optimization of a primary radio transmitter for environmental air volume, Maria (2014). Study and comparison of CFD simulations for molten salt external receiver and streamlined thermal performance models, Pacio (2013). An assessment of liquid metal technologies is needed, as well as future directions for using them as effective heat transfer medium in solar central receiver systems, Huang (2013). Forecasting and improving the effectiveness of the parabolic solar dish concentrating, Erminia (2012). Examination of the solar power gathered in a beam-down centralized receiver in great detail system, Qiang (2012). Performance modeling and central cavity receiver analysis.

The following transient equation that relies on the time t and space x, expresses the energy balance of the fluid circulating via the inside tube.

$\begin{aligned} & \rho_{\text {fluid }} \cdot C_{p, f l u i d} \cdot A_{f l u i d} \frac{\partial T_{\text {fluid }}}{\partial t}(x . t)+ \\ & \rho_{\text {fluid }} \cdot C_{p, f l u i d} \cdot \frac{V_{\text {fluid }}}{N_{\text {receiver }}} \cdot \frac{\partial T_{\text {fluid }}}{\partial t}(x . t)=Q_{\text {conv }}(x . t)\end{aligned}$                       (22)

Where $\rho_{\text {fluid }}$ is density of fluid, $C_{p \text {,fluid }}$ is specific heat of fluid, $A_{\text {fluid }}$ is area of fluid, $\dot{V}_{\text {fluid }}$ is velocity of the fluid, $N_{\text {receiver }}=$ number of receivers.

Heat transfer by convection between the flowing fluids inside the tube at unit length is:

$Q_{\text {conv }}(x \cdot t)=h_{\text {tube \& fluid }} \cdot A_{\text {receiver }} \cdot\left(T_{\text {receiver }}-T_{\text {fluid }}\right)$                  (23)

Where $h_{\text {tube & fluid }}$ is heat transfer coefficient by convection between the tube and the fluid, $A_{\text {receiver }}$ is area of inside surface by length of the tube, $T_{\text {receiver }}$ is temperature of receiver tube and $T_{\text {fluid }}$ is temperature of flowing fluid. The variation of the energy transfer in the heat transfer fluid is written as:

$\frac{\partial T_{\text {fluid }}}{\partial t}(x . t)=\underbrace{\frac{h_{\text {tube & fluid }} \cdot A_{\text {receiver }}}{\rho_{\text {fluid }} \cdot C_{p \text { fluid }} \cdot A_{\text {fluid }}}}_{\alpha_{11}}\left(T_{\text {receiver }}-T_{\text {fluid }}\right)$                       (24)

$\frac{\partial T_f}{\partial t}(x . t)=\alpha_{11}\left(T_{\text {receiver }}-T_{\text {fluid }}\right)$                       (25)

The above equation is a first-order differential equation describing the variation in fluid temperature as a function of time and the receiver.

4.1 Energy balance for the receiver

$\begin{aligned} & \rho_{\text {receiver }} C_{\text {p,receiver. }} A_{\text {receiver }} \frac{\partial T_{\text {receiver }}}{\partial t}(x . t)=Q_{\text {solar }}(x . t)- \\ & Q_{\text {inlet }}(x . t)-Q_{\text {conv }}(x . t)\end{aligned}$                       (26)

Where $Q_{\text {solar }}(x . t)$ represents the amount of solar energy absorbed by the collector or receiver,

$Q_{\text {inlet }}(x . t)$ is the amount of heat transfer at inlet between absorber tube and heat transfer fluid and

$Q_{\text {conv }}(x . t)$ is heat transfer through convection between absorber tube and heat transfer fluid.

$Q_{\text {solar }}(x \cdot t)=I_b \cdot r_b \cdot \tau \cdot \alpha \cdot D_{\text {eff }} \eta_{\text {opt }} \cdot \gamma$                       (27)

Where $I_b$ is the incident solar beam irradiance which is calculated for selected location, $r_b$ is tilt factor, $\rho$ is reflectivity $(0.89), \tau$ is transmissivity $(0.9), D_{e f f}$ is the effective width of the receiver, $\eta_{\text {opt }}$ is optical efficiency ( 0.98$)$ and $\gamma$ is the dirt factor on the mirrors $(0.8)$.

4.2 Heat transfer by conduction and radiation between the tube and surrounding

$Q_{\text {inlet }}{ }^{\text {conv }}(x . t)=\frac{2 \pi K_{\text {air }}\left(T_{\text {receiver }}-T_{\text {glass }}\right)}{\ln \left(\frac{R_{\text {tube }, \text { out }}}{R_{\text {tube } \text { in }}}\right)}$                      (28)

Where $Q_{\text {inlet }}{ }^{\text {rad }}(x . t)$ the heat is transfer at inlet through radiation and $Q_{\text {inlet }}{ }^{\text {conv }}(x . t)$ is the heat transfer at inlet through convection.

$Q_{\text {inlet }}{ }^{\text {rad }}(x . t)=\sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}\left(T_{\text {receiver }}{ }^4-T_a{ }^4\right)$            (29)

Where $\sigma$ is Stefan Boltzmann constant $\left(5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}^4\right)$, $A_{\text {receiver }}$ is inside surface area of the absorber tube throughout of the length and $\varepsilon_{t u b e}$ is emissivity of tube surface.

$Q_{\text {inlet }}{ }^{\text {conv }}(x . t)=\frac{2 \pi K_{\text {air }}\left(T_{\text {receiver }}-T_{\text {glass }}\right)}{\ln \left(\frac{R_{\text {tube }, \text { out }}}{R_{\text {tube }, \text { in }}}\right)}$                   (30)

Where $K_{\text {air }}$ is thermal conductivity of air and $R_{\text {tube,in }} \& R_{\text {tube,out }}$ inner and outer radius of the tube.

$\begin{aligned} & Q_{\text {inlet }}(x . t)=\sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}\left(T_{\text {receiver }}{ }^4-T_a{ }^4\right)+ \\ & \frac{2 \pi K_{\text {air }}\left(T_{\text {receiver }}-T_{\text {glass }}\right)}{\ln \left(\frac{R_{\text {tube }, \text { out }}}{R_{\text {tube } \text { in }}}\right)}\end{aligned}$                       (31)

Modeling of the heat transfer equation for the receiver is hard as well as complicated because of large number of variables and parameters. Now it has been tried to formulate the mathematical model for the heat transfer through the receiver as mentioned below.

$\begin{aligned} \rho_{\text {receiver }} C_{p, \text { receiver }} & A_{\text {receiver }} \frac{\partial T_{\text {receiver }}}{\partial t}(x \cdot t) \\ & =I_b \cdot r_b \cdot \tau \cdot \alpha \cdot D_{\text {eff }} \eta_{\text {opt }} \cdot \gamma \\ & -\sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}\left(T_{\text {receiver }}{ }^4-T_a{ }^4\right) \\ & -\frac{2 \pi K_{\text {air }}\left(T_{\text {receiver }}-T_{\text {glass }}\right)}{\ln \left(\frac{R_{\text {tube } \text { out }}}{R_{\text {tube } \text { in }}}\right)}\end{aligned}$                          (32)

Divided by $\rho_{\text {receiver }} C_{p, \text { receiver }} . A_{\text {receiver }}$ on both side

$\begin{aligned} & \frac{\partial T_{\text {receiver }}}{\partial t}(x . t) \\ & =\frac{I_b \cdot r_b \cdot \tau \cdot \alpha \cdot D_{\text {eff }} \eta_{\text {opt }} \cdot \gamma}{\rho_{\text {receiver }} C_{p, \text { receiver }} \cdot A_{\text {receiver }}} \\ & -\frac{\sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}}{\rho_{\text {receiver }} C_{p, \text { receiver }} \cdot A_{\text {receiver }}}\left(T_{\text {absorber }}{ }^4-T_{\text {glass }}{ }^4\right) \\ & -\frac{2 \pi K_{\text {air }}}{\ln \left(\frac{R_{\text {tube,out }}}{R_{\text {tube,in }}}\right) \rho_{\text {receiver }} C_{p, \text { receiver }} \cdot A_{\text {receiver }}}\left(T_{\text {absorber }}\right. \\ & \left.-T_{\text {glass }}\right)-\frac{h_{\text {tube & fluid }} \cdot A_{\text {receiver }}}{\rho_{\text {receiver }} C_{p, \text { receiver }} \cdot A_{\text {receiver }}}\left(T_{\text {receiver }}-T_{\text {fluid }}\right) \\ & \frac{\partial T_{\text {receiver }}}{\partial t}(x . t)=\underbrace{\frac{I_b \cdot r_b \cdot \tau . \alpha \cdot D_{\text {eff }} \eta_{\text {opt }} \cdot \gamma}{\rho_{\text {receiver }} C_{p, \text { receiver }} \cdot A_{\text {receiver }}}}_\psi+\end{aligned}$

$\underbrace{\frac{h_{\text {tube } \& \text { fluid }} A_{\text {receiver }}}{\rho_{\text {receiver }} C_{p, \text { receiver }} A_{\text {receiver }}}}_{\alpha_{12}} T_{\text {fluid }}-$$(\underbrace{\frac{\sigma . \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}}{\rho_{\text {receiver }} C_{p, \text { receiver }} A_{\text {receiver }}}}_{\alpha_{21}}) T_{\text {receiver }}{ }^4-$$\underbrace{\left(\frac{2 \pi K_{\text {air }}}{\ln \left(\frac{R_{\text {tube,out }}}{R_{\text {tube,in }}}\right) \rho_{\text {receiver }} C_{p, \text { receiver }} A_{\text {receiver }}}+\frac{h_{\text {tube & fluid }} \cdot A_{\text {receiver }}}{\rho_{\text {receiver }} C_{p, \text { receiver }} \cdot A_{\text {receiver }}}\right)}_{\alpha_{22}}) T_{\text {rec }}$$\underbrace{\left(\frac{\sigma \cdot \varepsilon_{\text {tube }} A_{\text {receiver }}}{\rho_{\text {receiver }} C_{p, \text { receiver }}, A_{\text {receiver }}}\right)}_{\alpha_{31}} T_{\text {fluid }}{ }^4+$    $(\underbrace{\frac{2 \pi K_{\text {air }}}{\ln \left(\frac{R_{\text {tube,out }}}{R_{\text {tube,in }}}\right) \rho_{\text {receiver }} C_{\text {peceiver. }} A_{\text {receiver }}}}_{\alpha_{32}}) T_{\text {fluid }}$                  (33)           

After solving this long theoretical equations, the receiver’s energy balance equation can be written as:

$\begin{aligned} & \frac{\partial T_{\text {receiver }}}{\partial t}(x, t)=\psi+\alpha_{12} \cdot T_{\text {fluid }}-\alpha_{21} \cdot T_{\text {receiver }}{ }^4- \\ & \alpha_{22} \cdot T_{\text {receiver }}+\alpha_{31} \cdot T_{\text {fluid }}{ }^4+\alpha_{32} \cdot T_{\text {fluid }}\end{aligned}$                    (34)

The above equation is utilized for the prediction of heat transfer from the receiver by putting characteristic parameters such as emissivity, absorptivity and transmission factors etc.

4.3 Energy balance that transforms solar energy between tube and atmosphere

There are the transformations by conduction and radiation between the air and glass envelope.

$\rho_{\text {tube }} C_{p, t u b e} \cdot A_{\text {tube }} \frac{\partial T_{\text {tube }}}{\partial t}(x . t)=Q_{\text {in }}(x . t)-Q_{\text {out }}(x . t)$                    (35)

Where $\rho_{\text {tube }}$ isdensity of tube material, $C_{p, t u b e}$ isspecific heat of tube material, $A_{\text {tube }}$ iscross sectional area of tube, $Q_{i n}(x . t)$ is the heat transfer rate at tube's inlet, and $Q_{\text {out }}(x . t)$ is the heat transfer rate at tube's outlet.

The total thermal energy output by convection and radiation between the tube and the surrounding is the sum of heat transfer rate through radiation from the tube and through convection as written as.

$Q_{\text {out }}(x . t)=Q_{\text {out }}{ }^{\text {rad }}(x . t)+Q_{\text {out }}{ }^{\text {conv }}(x . t)$                   (36)

Where $Q_{\text {out }}{ }^{\text {rad }}(x . t)$ is the heat transfer rate through radiation from the tube and $Q_{\text {out }}{ }^{\text {conv }}(x . t)$ is the heat transfer rate through convection from the tube.

$Q_{\text {out }}(x . t)=\sigma \varepsilon_{\text {tube }} \cdot A_{\text {tube }, \text { out }}\left(T_{\text {tube }}{ }^4-T_a{ }^4\right)+$ $\sigma h_{\text {tube,amb }} . A_{\text {tube,out }}\left(T_{\text {tube }}-T_a\right)$                  (37)

$\begin{aligned} & \rho_{\text {tube }} C_{p, t u b e} \cdot A_{\text {tube }} \frac{\partial T_{\text {tube }}}{\partial t}(x \cdot t)= \\ & \sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}\left(T_{\text {receiver }}{ }^4-T_a{ }^4\right)+\frac{2 \pi K_{\text {air }}\left(T_{\text {receiver }}-T_a\right)}{\ln \left(\frac{R_{\text {tube,out }}}{R_{\text {tube,in }}}\right)}- \\ & \sigma \varepsilon_{\text {tube }} \cdot A_{\text {tube,out }}\left(T_{\text {tube }}{ }^4-T_a{ }^4\right)- \\ & \sigma h_{\text {air,amb }} \cdot A_{\text {tube,out }}\left(T_{\text {tube }}-T_a\right)\end{aligned}$                 (38)

For creating the partial differential equation for the heat transfer through the tube and surrounding by dividing $: \rho_{\text {tube }} C_{p, t u b e} \cdot A_{\text {tube }}$

$\begin{aligned} & \frac{\partial T_{\text {tube }}}{\partial t}(x . t)=\underbrace{\left(\frac{\sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}}{\rho_{\text {tube }} C_{p, \text { tube }} \cdot A_{\text {tube }}}\right)}_{\beta_{11}} T_{\text {receiver }}{ }^4+ \\ & \underbrace{\left(\frac{1}{\rho_{\text {tube }} C_{p, \text { tube }} \cdot A_{\text {tube }}}\right)\left(\frac{2 \pi K_{\text {air }}}{\ln \left(\frac{R_{\text {tube }, \text { out }}}{R_{\text {tube,int }}}\right)}\right)}_{\beta_{12}} T_{\text {receiver }}- \\ & \underbrace{\left(\frac{\sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {receiver }}}{\rho_{\text {tube }} C_{p, \text { tube }} \cdot A_{\text {tube }}}+\left(\frac{\sigma \cdot A_{\text {tube,out }}}{\rho_{\text {tube }} C_{p, \text { tube }} \cdot A_{\text {tube }}}\right)\right)}_{\beta_{21}} T_a{ }^4+ \\ & \underbrace{\left(\frac{\sigma \varepsilon_{\text {tube }} \cdot A_{\text {tube }, \text { out }}}{\rho_{\text {tube }} C_{p, \text { tube }} \cdot A_{\text {tube }}}+\frac{\sigma \cdot h_{\text {tube }, \text { out }} \cdot A_{\text {tube }, \text { out }}}{\rho_{\text {tube }} C_{p, \text { tube }} \cdot A_{\text {tube }}}\right)}_{\beta_{22}} T_a+ \\ & \underbrace{\left(\frac{\sigma \cdot \varepsilon_{\text {tube }} \cdot A_{\text {tube }, \text { out }}}{\rho_{\text {tube }} C_{p, \text { tube }} \cdot A_{\text {tube }}}\right)}_{\beta_{31}} T_a{ }^4+\underbrace{\left(\frac{\sigma \cdot h_{\text {tube }, a m b} \cdot A_{\text {tube }, o u t}}{\rho_{\text {tube }} c_{p, t u b e} \cdot A_{\text {tube }}}\right)}_{\beta_{32}} T_a \\ & \end{aligned}$                      (39)

After solving these long theoretical equations energy balance that transmits solar energy between atmosphere and tube can be written as:

$\begin{aligned} & \frac{\partial T_{\text {glass }}}{\partial t}(x . t)=\beta_{11} T_{\text {receiver }}{ }^4+\beta_{12} T_{\text {receiver }}-\beta_{21} T_{\text {tube }}{ }^4+ \\ & \beta_{22} T_{\text {tube }}+\beta_{31} T_a{ }^4+\beta_{32} T_a\end{aligned}$                (40)

It has been observed from the above equations the parameters $\alpha_{i j}$ and $\beta_{i j}$ are constant depends on the characteristics of the fluid used. From these variables the matrix equation can be formed and can be calculate the energy possessed by the system.

$\begin{aligned} \frac{\partial}{\partial t}\left[\begin{array}{c}T_{\text {fluid }} \\ T_{\text {receiver }} \\ T_a\end{array}\right]=\psi+\left[\begin{array}{ccc}-\alpha_{11} & \alpha_{11} & 0 \\ \alpha_{12} & -\alpha_{22} & \alpha_{32} \\ 0 & \beta_{12} & \beta_{22}\end{array}\right]\left[\begin{array}{c}T_{\text {fluid }} \\ T_{\text {receiver }} \\ T_a\end{array}\right] \\ +\left[\begin{array}{ccc}0 & 0 & 0 \\ 0 & -\alpha_{21} & -\alpha_{31} \\ 0 & \beta_{11} & \beta_{21}\end{array}\right]\left[\begin{array}{c}T_{\text {fluid }}{ }^4 \\ T_{\text {receiver }}{ }^4 \\ T_a{ }^4\end{array}\right] \\ +\beta_{31} T_a{ }^4+\beta_{32} T_a\end{aligned}$                    (41)

In the above matrix equation, there are three variables ( $T_{\text {fluid }}, T_{\text {receiver }}, T_a$ ) will use to calculate the energy possess under first order differential equation.

$\begin{aligned} & \frac{\partial}{\partial t}\left[\begin{array}{c}T_{\text {fluid }} \\ T_{\text {receiver }} \\ T_a\end{array}\right] \\ & =0.165 \\ & +\left[\begin{array}{ccc}-(0.0055) & 0.0055 & 0 \\ 0.00733 & -(0.0085) & 0.00117 \\ 0 & 0.00117 & 420.433 \times 10^{-12}\end{array}\right]\left[\begin{array}{c}T_{\text {fluid }} \\ T_{\text {receiver }} \\ T_a\end{array}\right] \\ & +\left[\begin{array}{ccc}0 & 0 & 0 \\ 0 & -\left(4.973 \times 10^{-12}\right) & -\left(4.973 \times 10^{-12}\right) \\ 0 & 4.973 \times 10^{-12} & 9.946 \times 10^{-12}\end{array}\right]\left[\begin{array}{c}T_{\text {fluid }}{ }^4 \\ T_{\text {receiver }}{ }^4 \\ T_a{ }^4\end{array}\right] \\ & +4.973 \times 10^{-12} T_a{ }^4+415.4610^{-12} T_a\end{aligned}$

Thermal efficiency of the central tower solar receiver

$\eta_{\text {th }}=\frac{\text { h.A sur,inner } \cdot\left(\mathrm{T}_{\text {out }}-\mathrm{T}_{\text {in }}\right)}{\mathrm{I}_{\mathrm{b}}}$                   (42)

$A_{\text {sur,inner }}=2 \pi \cdot r_{\text {in }} \cdot L$

And Electrical efficiency

$\eta_{e l}=\frac{P_{\text {receiver }}}{G . A_{\text {eff }}}$                 (43)

Where $P_{\text {receiver }}$ iselectrical power in the receiver, $A_{\text {eff }}$ is effective area and $G$ is solar irradiation

The authors extend their sincere appreciation to Dr.Vishwanath Karad MIT World Peace University, Pune, Maharashtra, India, for allowing us to use their infrastructure and simulation software for this analysis.