Solar collector¶
A PTSC receiver contains a parabolic reflective surface and a receiver tube, located at the surface focus line. The reflected solar energy is transfered to the transparent receiver tube, which contains an absorber tube coated with blackened nickel to ensure high absorption. The annulus gap between the absorber tube and the glass envelope is vacuumed in order to reduce heat losses to the ambient. The absorber tubes heat up the synthetic oil it contains to nearly 400 °C. The heat of the thermal oil is transfered by a heat exchanger to a water/steam cycle in order to produce electricity.
The SolarCollector model is a steady-state model (accumulation is considered in the glass), based on first principle energy balance equations and on heat transfer phenomena occurring in PTSC receiver tube. The model takes into account heat losses from the receiver to the outside by radiation, conduction, and convection.
Modelica component model¶
The equations mentioned below are implemented in the component SolarCollector, located in the Solar.Collectors sub-library. This component has 4 connectors:
AtmTemp: atmospheric temperature,
ISun: incident energy flow (Direct Normal Irradiation),
Incidence-Angle: angle between the sun and the zenith,
ITemperature: fluid temperature at the outlet.
Nomenclature¶
| Symbol | Description | Unit | Definition | Modelica name |
|---|---|---|---|---|
| \(A_{g, i}\) | External glass surface for cell \(i\) | \(\mathrm{m}^{2}\) | \(\pi \cdot\left(D_{\mathrm{g}}+2 . e\right) \cdot \frac{L}{N}\) | AGlass |
| \(A_{\mathrm{r}}\) | Reflector surface (Barakos 2006) | \(\mathrm{m}^{2}\) | \(4 f \cdot \tan \left(\frac{\varphi_{\mathrm{R}}}{2}\right) \cdot L\) | AReflector |
| \(A_{\mathrm{t}, i}\) | External pipe surface (absorber) for cell \(i\) | \(\mathrm{m}^{2}\) | \(\frac{\pi \cdot D \cdot L}{N}\) | ATube |
| \(c_{\mathrm{p}, \mathrm{g}}\) | Specific heat capacity of the glass | \(\mathrm{J} / \mathrm{kg} / \mathrm{K}\) | cp_glass | |
| \(D\) | External pipe diameter (absorber) | \(\mathrm{m}\) | DTube | |
| \(D_{\mathrm{g}}\) | Internal glass diameter | \(\mathrm{m}\) | DGlass | |
| \(e\) | Glass thickness or wall thickness | \(\mathrm{m}\) | e | |
| \(f\) | Focal length | \(\mathrm{m}\) | f | |
| \(F_{12}\) | View factor to surroundings (radiation heat loss) | \(-\) | F12 | |
| \(h_{\mathrm{c}}\) | Convective heat transfer coefficient between the ambient air and the glass envelop | \(\mathrm{W} / \mathrm{m}^{2} / \mathrm{K}\) | h | |
| \(K\) | Incidence angle modifier | \(-\) | \(\cos (\theta)\) | IAM |
| \(L\) | Absorber pipe length | \(\mathrm{m}\) | L | |
| \(m_{\mathrm{g}}\) | Glass mass for cell \(i\) | \(\mathrm{kg}\) | \(\rho_{\mathrm{g}} \cdot \frac{L}{N} \cdot \frac{\pi}{4} \cdot\) \(\left(\left(D_{\mathrm{g}}+2 e\right)^{2}-D_{\mathrm{g}}^{2}\right)\) | dM |
| \(N\) | Number of cells (segments) | \(-\) | Ns | |
| \(R\) | Mirror reflectivity | \(-\) | R | |
| \(T_{\text {atm }}\) | Atmospheric temperature | \(\mathrm{K}\) | Tatm | |
| \(T_{\mathrm{g}, i}\) | Glass temperature for cell \(i\) | \(\mathrm{K}\) | Tglass[i] | |
| \(T_{\mathrm{sky}}\) | Sky temperature | \(\mathrm{K}\) | \(0.0552 \cdot T_{\mathrm{atm}}^{1.5}\) | Tsky |
| \(T_{\mathrm{w}, i}\) | Temperature of the outer absorber surface for cell \(i\) | \(\mathrm{K}\) | Twall[i] | |
| \(W_{\text {abs }, g, i}\) | Power absorption by the glass envelop for cell \(i\) | \(\mathrm{W}\) | WAbsGlass[i] | |
| \(W_{\text {cond }, \mathrm{tg}, i}\) | Conduction power between the outer absorber surface and the inner glass surface for cell \(i\) | \(\mathrm{W}\) | WCondWall[i] | |
| \(W_{\text {conv,giair }, i}\) | Convection power loss from the outer glass envelop surface to the ambient air for cell \(i\) | \(\mathrm{W}\) | WConvWall[i] | |
| \(W_{\mathrm{rad}, \mathrm{t}, i}\) | Radiation power from the outer absorber surface and the inner glass surface for cell \(i\) | \(\mathrm{W}\) | WRadWall[i] | |
| \(W_{\text {rad, g:sky }, i}\) | Radiation power loss from the outer glass envelop surface to the sky for cell \(i\) | \(\mathrm{W}\) | WRadGlass[i] | |
| \(W_{\mathrm{t}, i}\) | Total power transferred to the pipe (absorber) for cell \(i\) | \(\mathrm{W}\) | WTube[i] | |
| \(\alpha_{\mathrm{g}}\) | Glass absorptivity | \(-\) | AlphaGlass | |
| \(\alpha_{\mathrm{t}}\) | Tube absorptivity | \(-\) | AlphaN | |
| \(\gamma\) | Interception factor | \(-\) | Gamma | |
| \(\varepsilon_{\mathrm{g}}\) | Glass emissivity | \(-\) | EpsGlass | |
| \(\varepsilon_{\mathrm{t}}\) | Tube emissivity | \(-\) | EpsTube | |
| \(\eta_{\text {opt }}\) | Optical efficiency | \(-\) | OptEff | |
| \(\theta\) | Zenith angle | \(^\circ\) | Theta | |
| \(\lambda\) | Gas thermal conductivity between the tube and the glass | \(\mathrm{W} /(\mathrm{m} \mathrm{K})\) | Lambda | |
| \(\rho_{\mathrm{g}}\) | Glass density | \(\mathrm{kg} / \mathrm{m}^{3}\) | rho_glass | |
| \(\sigma\) | Stefan-Boltzmann constant | \(\left(\mathrm{W} / \mathrm{m}^{2} \mathrm{K}^{4}\right)\) | \(5.67 \times 10^{-8}\) | sigma |
| \(\tau\) | Glass transmissivity | \(-\) | tauN | |
| \((\tau \alpha)_{n}\) | Transmissivity-absorptivity factor | \(-\) | TauAlphaN | |
| \(\varphi_{\mathrm{R}}\) | Rim angle | \(^\circ\) | RimAngle | |
| \(\phi_{\text {sun }}\) | Solar radiation (direct normal irradiance \(-\mathrm{DNI}\) ) | \(\mathrm{W} / \mathrm{m}^{2}\) | PhiSun |
Governing equations¶
Solar radiation absorbed by the receiver (reflector total power)¶
Mathematical formulation:
\[W_{\mathrm{abs}, \mathrm{t}}=\eta_{\mathrm{opt}} \cdot \phi_{\mathrm{sun}} \cdot A_{\mathrm{r}}\]
Comments:
The optical efficiency is given by \(\eta_{\mathrm{opt}}=R \cdot(\tau \alpha)_{n} \cdot \gamma \cdot K\). The transmissivity-absorptivity factor is given by \((\tau \alpha)_{n}=\tau \cdot \alpha \cdot \frac{1}{1-(1-\alpha) \cdot(1-\tau)}\), where \(\tau\) is the transmissivity and \(\alpha\) is the absorptivity.
Energy balance equation for the glass¶
Mathematical formulation:
\[m_{\mathrm{g}} \cdot c_{\mathrm{p}, \mathrm{g}} \cdot \frac{\mathrm{d} T_{\mathrm{g}, i}}{\mathrm{d} t} =W_{\mathrm{abs}, \mathrm{g}, i}+W_{\mathrm{rad}, t, i}+W_{\mathrm{cond}, \mathrm{t}: \mathrm{g}, i}-W_{\mathrm{rad}, \mathrm{g}: \mathrm{s} \mathrm{ky}, i}-W_{\mathrm{conv}, \mathrm{g}: \mathrm{air}, i}\]
Comments:
This equation calculates the glass temperature \(T_{g,i}\).
Energy balance equation for the pipe (power transferred to the absorber)¶
Mathematical formulation:
\[W_{\mathrm{t}, i}=\frac{W_{\mathrm{abs}, \mathrm{t}}}{N}-W_{\mathrm{rad}, \mathrm{g}: \mathrm{sky}, i}-W_{\mathrm{conv}, \mathrm{g}: \mathrm{air}, i}\]
Comments:
The net power received by each tube segment is equal to the total power absorbed by the receiver for that segment minus the losses by radiation to the sky and convection to the ambient for that segment.
References¶
El Hefni, Baligh and Bouskela, Daniel (2019). Modeling and Simulation of Thermal Power Plants with ThermoSysPro, sect. 16.1. Springer Nature Switzerland AG.