Solar energy

Movement of the earth

The earth makes two motions: rotation about its own axis once in 24 hours (causes day and night) and revolution around the sun in an elliptical orbit once in 365.25 days. The axis is tilted by 23.45° from the normal to the orbital plane and keeps a fixed direction in space.

The sun-earth distance varies from 1.471×10¹¹ m (about 3 January) to 1.521×10¹¹ m (about 4 July), a mean of 1.496×10¹¹ m (1 AU).

Cause of seasons

Because of the fixed tilt, the angle between the sun's rays and the surface at a place changes through the year. The declination $\delta$ (angle of the sun north or south of the equator) varies from +23.45° on 21 June (summer solstice) to −23.45° on 22 December, and is zero at the equinoxes (21 March, 23 September). When the northern hemisphere leans towards the sun it has longer days and more direct rays, i.e. summer. Distance from the sun is not the cause.

Declination (Cooper): $\delta = 23.45 \sin\left[\frac{360(284+n)}{365}\right]$, $n$ = day of year.

Definitions

• Solar constant $G_{sc}$: energy from the sun per unit time on a unit area perpendicular to the rays, at the mean earth-sun distance, outside the atmosphere. Value = 1367 W/m² (WRC value).
• Air mass $m$: ratio of the path length of rays through the atmosphere to the path length when the sun is overhead. $m \approx 1/\cos\theta_z$ (for $\theta_z < 70°$). Air mass zero (AM0) is outside the atmosphere, AM1.5 is the standard for solar cells.
• Beam (direct) radiation: reaches the surface without scattering.
• Diffuse radiation: scattered by air molecules, dust and clouds and arriving from all directions.
• Global (total) radiation: sum of beam and diffuse on a surface, $G = G_b + G_d$.

Solar radiation instruments convert incident radiation into a measurable signal, usually heat or voltage.

Pyranometer

Measures global radiation (beam + diffuse) on a horizontal surface; with a shade ring or disc it measures diffuse radiation only.

The Eppley type has a black and white sensing surface (thermopile) under two glass domes. Black parts absorb radiation and get hotter than the white parts, so the thermopile produces a small voltage (about 10 µV per W/m²) proportional to the radiation. The domes protect from wind and rain, and a desiccant keeps them dry. A photodiode type is cheaper but spectrally selective.

   glass domes
     ____
    /    \
   |B|W|B|W|  <- thermopile hot (black)
   |______|      and cold (white) junctions
   heat sink --> voltage out

Pyrheliometer

Measures beam (direct) radiation at normal incidence. A long collimating tube with a narrow field of view (about 5–6°) is kept pointed at the sun by a tracker. The radiation falls on a thermopile or a blackened receiver. In the Angstrom compensation type, one of two strips is shaded and heated electrically until the temperatures equal; the electrical power then equals the absorbed radiation. It is also the reference instrument for calibration.

Sunshine recorder

Measures the duration of bright sunshine (hours), not intensity. In the Campbell-Stokes recorder a solid glass sphere focuses the sun's rays onto a card strip; the burn trace length gives the sunshine hours. The data are used with Angstrom regression $\bar H/\bar H_0 = a + b\,(\bar n/\bar N)$ to estimate radiation where no pyranometer is available.

Data

$\phi = 27.7°$ N; 15 May is day number $n = 31+28+31+30+15 = 135$. Solar time = 10:00.

Declination

$$\begin{aligned} \delta &= 23.45 \sin\left[\frac{360(284+n)}{365}\right] \\ &= 23.45 \sin\left[\frac{360 \times 419}{365}\right] = 23.45 \sin(413.3°) \\ &= 23.45 \sin(53.3°) = 18.79° \end{aligned}$$

Hour angle

15° per hour from solar noon, negative in the morning:

$$\omega = 15(10 - 12) = -30°$$

Zenith and altitude

$$\begin{aligned} \cos\theta_z &= \cos\phi\cos\delta\cos\omega + \sin\phi\sin\delta \\ &= (0.8854)(0.9467)(0.8660) + (0.4647)(0.3222) \\ &= 0.7259 + 0.1497 = 0.8756 \end{aligned}$$ $$\theta_z = 28.88°, \qquad \alpha = 90° - \theta_z = 61.12°$$

Solar azimuth

$$\cos\gamma_s = \frac{\sin\alpha\sin\phi - \sin\delta}{\cos\alpha\cos\phi} = \frac{(0.8756)(0.4647) - 0.3222}{(0.4831)(0.8854)} = 0.1992$$

$\gamma_s = 78.5°$ measured from south, towards the east (morning). That is, azimuth is 78.5° east of south.

Sunrise hour angle and day length

$$\begin{aligned} \cos\omega_s &= -\tan\phi\tan\delta = -(0.5258)(0.3405) = -0.1790 \\ \omega_s &= 100.29° \\ N &= \frac{2\omega_s}{15} = \frac{2(100.29)}{15} = 13.37\ \text{h} \end{aligned}$$

Sunrise is at $12 - 6.69 \approx 5{:}19$ solar time.

Answer: $\delta = 18.79°$, $\omega = -30°$, $\theta_z = 28.88°$, altitude $= 61.12°$, azimuth $= 78.5°$ east of south, $\omega_s = 100.3°$, day length $= 13.37$ h.

A flat plate collector (FPC) absorbs both beam and diffuse radiation on a flat surface and transfers the heat to a fluid, without concentrating. It is used for temperatures up to about 80–100 °C.

Construction

• Absorber plate: copper or aluminium sheet with tubes (risers) attached to it, coated with a black or selective coating (high absorptance, low emittance).
• Transparent cover: one or two glass sheets; they pass short-wave solar radiation but block long-wave re-radiation from the plate (greenhouse effect) and reduce convection loss.
• Insulation: glass wool or polyurethane (50–75 mm) at back and sides.
• Casing (box): aluminium or galvanised steel, weatherproof.
• Headers: connect the riser tubes to inlet and outlet pipes.

   sun rays \ \ \
 ===== glass cover ======
   air gap
 ~~~ absorber plate + tubes ~~~
 ///// insulation //////////////
 |________ casing ____________|

Working

Radiation passes the glass and heats the plate. Water in the tubes takes this heat and leaves hotter. The useful heat is given by the Hottel-Whillier-Bliss equation $Q_u = A_c F_R\,[S - U_L (T_i - T_a)]$, and the efficiency is $\eta = Q_u/(A_c G_T)$.

Thermosyphon water heater

   ______________
  | storage tank |<--- hot water out
  |______________|
     ^        |
 hot |        | cold
     |  ____  v
     | |FPC | 
     |_|____|<---- cold feed

The storage tank is placed above the collector. Water heated in the collector becomes less dense and rises to the top of the tank, while the colder, denser water from the bottom of the tank flows down to the collector inlet. This natural circulation (thermosyphon) needs no pump and runs only while the sun shines. The tank must be at least 0.3–0.5 m above the collector top, and pipes must be short and sloped without air traps.

Hottel-Whillier-Bliss equation

$$Q_u = A_c\left[F_R(\tau\alpha)\,G_T - F_R U_L\,(T_i - T_a)\right]$$

(a) Useful heat

$$\begin{aligned} Q_u &= 2\,[0.76(800) - 4.5(30-20)] \\ &= 2\,[608 - 45] = 2 \times 563 = 1126\ \text{W} \end{aligned}$$

(b) Efficiency

$$\eta = \frac{Q_u}{A_c G_T} = \frac{1126}{2 \times 800} = 0.704 \;(70.4\ \%)$$

(c) Outlet temperature

$$\begin{aligned} Q_u &= \dot m\,c_p\,(T_o - T_i) \\ T_o - T_i &= \frac{1126}{0.03 \times 4186} = 8.97\ \text{K} \\ T_o &= 30 + 8.97 = 38.97\ \text{°C} \end{aligned}$$

(d) Stagnation condition

Useful heat becomes zero when $F_R(\tau\alpha)G_T = F_R U_L (T_i - T_a)$:

$$T_i - T_a = \frac{0.76 \times 800}{4.5} = 135.1\ \text{K}$$

Answer: $Q_u = 1126$ W; $\eta = 70.4\ \%$; $T_o = 39.0$ °C; useful heat is zero when $T_i - T_a = 135$ K (the collector is then at stagnation).

(a) Why and types

A concentrator uses mirrors or lenses to focus beam radiation from a large aperture onto a small absorber (receiver). The small receiver area gives low heat loss, so high temperatures (100–1500 °C) and good efficiency at high temperature are obtained, and less expensive absorber material is needed per unit of heat. They are used for process heat, industrial steam and solar thermal power plants.

Classification

1. By focus: line focus (parabolic trough, linear Fresnel) and point focus (parabolic dish, central receiver/heliostat field).
2. By optics: reflecting (mirrors) or refracting (Fresnel lens).
3. By tracking: single axis (trough) or two axis (dish, tower).

(b) Concentration ratio

$$C = \frac{A_a}{A_r} = \frac{W\,L}{\pi D L} = \frac{W}{\pi D} = \frac{3.0}{\pi (0.05)} = 19.1$$

If the aperture area is taken as the net opening, $(W - D)/\pi D = 18.8$. So the receiver is irradiated with about 19 times the intensity falling on the aperture. Such a value is typical of a trough working at around 200–300 °C; the ideal limit for a line-focus 2-D concentrator is $1/\sin\theta_a$ (about 213 for a half acceptance angle of 0.27°).

Answer: $C \approx 19$.