# Design Limitations of Laboratory Solar Chimney for Engineering Education
## 3 Mathematical Model and Numerical Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**System** & **Height, \(H\) [m]** & \begin{tabular}{c} **Tower Radius,** \\ \(d\) **[m]** \\ \end{tabular} & \begin{tabular}{c} **Collector Radius,** \\ \(r\) **[m]** \\ \end{tabular} & \begin{tabular}{c} **Inlet Height,** \\ \(h\) **[m]** \\ \end{tabular} & \begin{tabular}{c} **H/_d_ \\ \end{tabular} & \
\begin{tabular}{c} **H/_r_ \\ \end{tabular} &
\begin{tabular}{c} _h/r_ \\ \end{tabular} \\ \hline Manzanares & 200.0 & 10.00 & 120.0 & 2.000 & 20.00 & 1.660 & 0.016 \\ Simulation & 0.500 & 0.050 & 0.375 & 0.005 & 10.00 & 1.330 & 0.013 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometry of the analyzed model in comparison to Manzanares prototype. Figure 1: Mesh model of solar chimney power plant in ANSYS: (**a**) Cross section of 3D model. (**b**) Air inlet, i.e., suction part of the collector inlet, and (**c**) around the tower air outlet area. The simulation results for the temperature of the air inside the tower, the pressure under standard conditions, and the speed of the air are shown in Figure 2. Additionally, key simulation parameters are summarized in Table 2. As seen in Figure 2a, the temperature is highest at the base of the tower and gradually decreases toward the tower outlet. Figure 2b indicates that the pressure gradient is upward along the tower. These findings are in accordance with the physical laws as the warm air rises, i.e., the particles spontaneously move in the direction of the pressure gradient. However, in Figure 2c, it can be seen that the speed of air is minimal at the base of the tower and that the particles accelerate in the highest section of the tower, when they should behave oppositely. This behavior indicates that the pressure difference is not enough to establish continuous airflow. To further investigate, the simulation results were compared with the model, Equations (1)-(3), and shown in Figure 3. In Figure 3, the dashed line represents the analytical results, while the bold line represents the results obtained numerically in the ANSYS software for (a) the mass flow rate of the air within the tower, (b) the pressure difference, and (c) the speed of the air. The mass flow rates obtained analytically and numerically have approximately the same slopes, which implies that the ANSYS simulation approximates the real event well. Figure 3b shows that the simulated pressure difference also has the same physical dependence as the model. However, as the simulation in Figure 2b illustrates, the pressure difference is much less than what is required (dashed line) to establish spontaneous flow. The increase in temperature difference would usually increase the pressure difference, and the particles would accelerate as shown in Figure 3c with the dashed line. However, the small geometry causes high-energy particles to interact with boundaries more frequently, hindering upward drafts (Figure 3c, bold line). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Max Temperature, [K]** & **Outlet Temperature, [K]** & **Max Pressure, [Pa]** & **Min Pressure, [Pa]** & **Max Speed, [m/s]** \\ \hline
346 & 322 & 0.77 & 0.07 & 1.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Extreme simulation values of the temperature, pressure, and speed of air inside the chimney. Figure 2: ANSYS solar chimney power plant simulation of: (**a**) Temperature distribution through the model. (**b**) Pressure distribution in the collector and chimney area. (**c**) Speed of air inside the chimney.