PHY 555 — Energy and Environment

PC6 — Solar Power Plant

Friday, November 5th 2021There are two main ways of converting solar energy into electricity:

• Photovoltaic panel (solar panels) made semiconductors absorb photons which have an energylarger than the band gap and directly generate electrons which are consequently collected

• In a thermodynamic concentrating solar power plant (also called solar-thermodynamic), thesunlight is reflected on large mirrors (collectors) and absorbed on heat-transfer fluid (absorber)which transports the heat directly to a steam generator for supplying a turbo-generator or to aheat accumulator which can produce electricity during the night or during hours less sunshine.

In this PC we will investigate both techniques.

RemindersThe spectral density of photons emitted by a surface Semit at temperature T in the direction Ω isgiven by the Planck law :

NBB(ℏω ,Ω)=Semitcosθ

4 π3ℏ3c2

×ℏ2ω2

exp(ℏω

k BT )−1

The total power emitted per unit surface is therefore

ϕE=1Semit

∬ℏω×NBB (ℏω ,Ω)dΩd(ℏω)=∫cos θdΩ⏟=π

×∫0

∞

d(ℏω)1

4π3ℏ3c2×

ℏω×ℏ2ω2

exp(ℏω

k BT )−1⏟=σ/π T4

ϕE=σT ⁴

where the Stefan's Boltzmann constant is σ=π2k B4

60 ℏ3c2

=5.67×10−8W /m2/K 4

We recall the solar constant (solar flux at normal incidence, at the top of the atmosphere - See PC2):

ϕsun≈Ωsun

πσT sun

4≈1360W /m2

1 – Photo-voltaic — the Shockley Queisser limitWe consider a semiconductor which absorbs all photons with energy ℏω≥Eg , where Eg is theenergy gap. We remind the Kirchhoff’s law of radiation: If a grey body shows an absorptivityϕabs=α(ℏ ω)ϕ in , then its thermal emission is given by

NGB(ℏω)=α(ℏω)×N BB(ℏ ω)

1. Let's consider first a solar cell in the dark, receiving radiations from the ambient thermal

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École PolytechniqueYear 2020-2021

Erik JohnsonMathieu de Naurois

Daniel Suchet

background. Estimate the dark current R0 of the cell, i.e. the number of electrons recombining perunit of time and surface.

2. Show that for a gap of ∼1.1eV , the flux of photo-generated electrons under sunlight G is muchlarger than the flux of recombined electrons R0 .

3. Considering that each absorbed photon generates an electron and that each photo-generatedelectron can provide an energy ∼Eg , show that there is an optimal gap to maximize the outputpower. What are the limitations of this model ?

4. This rough model can be improved by considering a better description of the electrons within thematerial.

(a)Considering that the recombination rate is the same for every electron, show that the steady statedensity of electrons n in the absorber under illumination is given by

n=n0R0

×(G−Iq )

where n0 is the density of electrons in the absorber in the dark and I is the extracted current.

(b)We will admit that the work provided by the cell for each electron extracted from the absorber is

w=k BT cell ln(nn0 )

where T cell is the semi-conductor temperature. Describe the I-V characteristic of a solar paneland show that there is a maximum point to operate the cell at maximum power.

(c)The electrical power expressed in the previous question depends on two parameters : the gap ofthe material and the operating voltage. Show that, if the gap is optimal, then the maximum poweroperation point is given by

V opt=Eg(1−TabsT cell )−

k BTabsq

ln (π

Ωsun )(d)We can easily calculate numerically that the optimal gap is around Eg≈1.1eV . Estimate the

maximum conversion efficiency for a simple solar cell.

(e)The best efficiency ever reached for a silicon solar cell is about 27% under sunlight. Commentthis value.

2 – Concentrating power plantsSolar collectors can be of two types, with either uni-dimensional or bi-dimensional focusing.

In plants with uni-dimensional focusing (parabolic cylinder, see Fig. 1, left), the absorbers are tubesplaced in the focus point of the parabola. Inside these tubes flows a heat-transfer fluid which transportsheat directly to a steam generator for supplying a turbo-generator or to a heat accumulator which canproduce electricity during the night or during hours less sunshine.

In two-dimensional focusing power plants (paraboloid, see Fig. 1, right), the absorber is directlycoupled to a Stirling engine placed at the focus of the reflector (hence the name Stirling dish). TheStirling engine is a closed cycle thermal machine with external heating. The working fluid is an inertgaz (usually air). Performance approach that of a Carnot cycle.

The collectors are equipped with motorized mounts that track the sun's movement (only east-westdirection, only elevation, or both) to steer them to the optimal direction.

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Les largest plants use parabolic trough collectors and are installed in Andalusia (Spain) and California(USA). The dish-Stirling collectors are mostly used to generate electricity in remote desert regions.

1. We first consider a black-body absorber of surface Sabs and temperature T abs facing the sun undernormal incidence, without any concentration. Estimate the power absorbed and emitted by theabsorber. What is the maximum absorber temperature Tmax ?

2. To convert sunlight into work, the Muser model considers an absorber heated by the solar radiation.This absorber serves as a hot source for a heat engine. Estimate the maximum efficiency of such amachine. Conclusions?

3. We now consider a light concentrator (mirror) concentrating the sunlight onto the absorber. Theconcentration ratio, C s , is defined as the ratio of the concentrator surface to the absorber surface.Why is the minimum width that the absorber must possess determined by the apparent diameter ofthe sun ( θS≈0.53º ) What other parameter can play a role?

4. Give the expression of CS for a perfect one-dimensional reflector of focal length f and width Dusing the paraxial optics (i.e., in the approximation of small angles). Interpret the variation of CS

with the relative opening a=D / f . Show that thenaive ray optics can violate thermodynamics.

5. The figure on the right represents the profile of acylindrical-parabolic collector of focal length fand width D , given by the functiony (x )=x2/ (4 f ) . (The top is placed in the origin,

the focal point is located at (0, f ) ). The parabolapoints to the sun ( y axis). Determine theminimum value of the radius of the absorber tubeas function to the relative opening a=D / f ,without using approximations of paraxial optics.Give the solar concentration ratio as function of a .

6. For what value of the relative opening a=D / f is solar concentration ratio CS of a parabolictrough collector maximum and then what is its value?

7. For a 2D concentrator (parabolic dish), what is the maximum concentration ratio?

8. What effects could lead to a solar concentration ration CS smaller than the maximum calculatedhere? Is it easy to estimate these effects priori? Specify how to do it if applicable.

9. Let's consider a Muser engine with a light concentrator of concentration factor CS . Draw the

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F

Figure 1 : Left: Parabolic Concentrator . Right: Dish-Stirling collector

shape of the overall yield.

10. For an average solar flux F S=250 W m−2 , a solar concentration ratio of CS=600 , and atemperature of the cold reservoir T 0=200°C ,verify that the overall performance is greatest whenthe temperature of the absorber is T=605 °C . What then is the value of overall yield? One can useα=95% and ϵ=10% .

11. Comment the choice of the cold reservoir temperature ( 200 °C ). Does it sound realistic? Can theyield be increased by changing this setting? Is it sounds technically feasible?

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