The Physics of Energy, page 98
Example 24.5 2D Concentrator Efficiency Optimization
Consider a solar thermal electric plant built using a system of 2D concentrators that reflect sunlight to a system of pipes carrying molten salt with concentration .
What is the optimal efficiency possible (assuming an ambient temperature of 300 K and incident sunlight at1000 W/m)?
Solving eq. (24.23) numerically gives 818 K. At this temperature, Carnot efficiency is approximately 63% and about 13% of incident radiation is reradiated, giving a maximum possible efficiency of .
The overall system conversion efficiency is
(24.21)
This quantity vanishes at ambient temperature (since the efficiency goes to zero) and also vanishes at the radiative equilibrium temperature (where all incoming radiation is reradiated), and is positive between. For a given concentration and rate of incoming power, then, the theoretical maximum power output available is optimized when
(24.22)
or
(24.23)
This quintic equation does not have a simple analytic solution, but can be easily solved numerically. The system efficiency is graphed as a function of T in Figure 24.13 for several possible values of concentration C, assuming clear sky overhead sun insolation of W/m. For real systems, the thermal to electric conversion efficiency is substantially less than the Carnot maximum. Knowledge of the actual conversion efficiency η as a function of temperature can be used to optimize temperature and power output for more realistic systems.
Figure 24.13 The maximum theoretically possible system conversion efficiency for concentrated solar electrical power generation as a function of absorber temperature, assuming ambient temperature K, and incoming power per unit area W/m. The curves are labeled by the concentration C. At low temperatures, efficiency is suppressed by the Carnot limit. (The dashed line gives .) At high temperatures most energy is re-radiated by the absorbing material.
Discussion/Investigation Questions
24.1 Estimate the energy used for space and water heating in your home. How feasible would it be to get all of this energy from simple low-temperature solar thermal collectors?
24.2 Solar thermal collectors powering small off-grid electric generators have been proposed for the developing world. Research and discuss the advantages and disadvantages of this proposal. Topics might include efficiency, reliability, versatility, and water requirements and availability.
24.3 Discuss some of the possible advantages or disadvantages of the different approaches to solar thermal electricity production: parabolic trough, power tower, parabolic dish. Can you imagine other approaches that might be worthwhile pursuing?
Problems
24.1 The radius of Mars’ orbit around the Sun averages roughly km. Assuming that the surface temperature is roughly uniform over the planet's surface and constant over a Martian day, that the surface is a perfect black body, and that there is no atmosphere, estimate the average surface temperature when the planet is in radiative equilibrium.
24.2 The eccentricity of Mars’ elliptic orbit is roughly 0.0935. Estimate the range over which the solar constant and average surface temperature (in the uniform temperature blackbody approximation) vary during the year.
24.3 Suppose that the surface temperature of the Sun were to increase by 3% from the current value, assuming for simplicity that the radius of the Sun stays fixed. By how much would the solar constant on Earth increase? If, at present, Earth radiates away an amount of energy equal to the current incoming energy flux from the Sun by radiating as a black body at an effective temperature of 5℃, by what amount would the temperature have to increase to offset the increase in incident solar energy?
24.4 Consider an idealized flat-plate solar collector. Assume that there are no heat losses to conduction or convection. Assume that the insolation is 1000 W/m incident at an angle of 45° from the vertical. Compute the temperature of the black body in radiative equilibrium assuming that (a) the collector is horizontal, (b) the collector is tilted at 35° towards the Sun.
24.5 Analyze the double pane glass-covered collector described in §24.1.2 and illustrated in Figure 24.14. is assumed to be fixed at K by conduction and convection and W/m. Write the equations of radiative equilibrium and show that .
Figure 24.14 A double-pane flat-plate collector.
24.6 Consider a flat-plate collector covered by a single pane of glass exposed to perpendicular insolation at 1000 W/m. The collector operates at a net efficiency of 35% at a temperature . Assuming that the glass covering stays at the ambient temperature of 15℃, compute the net power radiated and non-radiative losses for the collector. Ignore incident infrared light from the atmosphere.
24.7 Consider the examples of single- and double-pane glass-covered collectors described in §24.2.1. Verify the quoted results for the efficiency as a function of , and . In the case of double glazing, compute the temperature of the intermediate glass plate. Plot the efficiency as a function of for when K and W/m.
24.8 Redo the computation of Problem 24.7 assuming incoming radiation at W/m, and fixing , K. Show that the efficiency decreases significantly with the reduction in incoming radiation.
24.9 Compare the total (integrated daily) insolation on a flat-plate collector located in Chicago, Illinois (latitude 41.98°) on February 1 when the collector is (a) tilted at an angle to the south, (b) tilted at an angle to the south.
24.10 Consider an idealized flat-plate solar collector with no conduction or convection losses, and with insolation 1000 W/m incident at an angle of 45° from the vertical. Assume that the collector is horizontal and is covered with a pane of glass transparent to incoming radiation but opaque to outgoing (IR) radiation. The glass is kept at 300 K by the external environment. Compute the temperature of the absorber in radiative equilibrium. Recompute the temperature if the absorber is coated with a paint that modifies its emissivity in the IR to .
24.11 Consider a parabolic trough where the height of the reflector is identical to the height of the center of the absorber, as depicted in the figure in Example 24.4. The concentrator width is 8 m and the absorbing tube of radius 0.5 m is centered along the focal line at a height of 2 m above the bottom of the trough. Incident sunlight hits the concentrator from a direction parallel to the line of symmetry, with an intensity of 1000 W/m. (a) What is the effective concentration of the concentrator? (b) What is the acceptance angle within which all light hits the absorber? (c) Assume that the tube carries a fluid that is heated and removes some of the incoming energy. The tube then radiates as a black body at a temperature of 150℃. Compute the net rate at which energy is collected and transferred to the fluid, for each meter of length of the trough.
24.12 Consider a solar thermal “power tower” with a central tower of height 60 m, surrounded by an array of planar mirrors on the ground extending to a radius of 100 m around the tower. At the top of the tower there is a cylindrical absorber with its axis aligned vertically. Each mirror is large enough so that the complete reflection of the Sun can be seen on each mirror from the center of the tower at the height of the absorber. (a) Give an upper limit on the number of mirrors that could be placed around the tower subject to these constraints. (b) How wide does the absorber have to be so that light rays coming from the center of the Sun hit the absorber, no matter which point on which mirror they are reflected from?
24.13 For the power tower geometry from the previous problem, assume that the absorber height is equal to its diameter, and that the diameter is that found in part (b) above. If all the light from an overhead sun that hits the circle of radius 100 m containing the reflecting mirrors were to be reflected to the absorber, what would be the effective concentration of the mirror configuration as a solar concentrator? Assume 1000 W/m insolation from an overhead sun. What would be the temperature of the absorber if it was in radiative equilibrium with the reflected incoming radiation? Do you think that this is a realistic answer? Why or why not?
24.14 [T] Determine the maximum concentration C of a solar concentrator satisfying , where is the solar constant, directly from the second law of thermodynamics and the surface temperature of the Sun.
24.15 [T] Prove that for the compound parabolic concentrator depicted in Figure 24.11 the line containing points Q and A is parallel to the axis of parabola 2. [Hint: consider the incoming ray that is reflected along .]
24.16 Consider a linear 2D compound parabolic concentrator built from parabolas tilted at 10° to the vertical, with a trough of width 3 m, and an absorber width of 0.5 m. Compute the concentration C of the concentrator. If the incoming radiation has intensity W/m and the (blackbody) absorber is kept at a temperature of 100℃ by circulation of a thermal fluid, compute the rate of energy transfer to the fluid. Compare this rate of energy transfer to that for a linear parabolic concentrator with the same absorber area and acceptance angle; assume that the parabolic concentrator has height equal to the focal length (as in Example 24.4), and that the absorber has an area covering the lower half of a cylinder centered on the focal line, and is kept at the same temperature of 100℃.
24.17 [T] Prove that a linear compound parabolic concentrator realizes the maximum possible concentration for a given acceptance angle θ. Suggestion: following the notation of Figure 24.11, work in a coordinate system where parabola 2 is described by the equation . Identify B and Q as the points on parabola 2 where the slopes are and , with . Then show that , which proves the desired result.
24.18 Consider a parabolic dish concentrator built from a dish with radius 3 m, height equal to the focal length, and an absorber at the focus with spherical shape and radius m. Compute the concentration and acceptance angle of this concentrator, and check that the Rabl bound is satisfied.
24.19 For a concentrator with concentration and incident sunlight at intensity W/m, compute the temperature T of the absorber at which solar to electric conversion efficiency is optimized, assuming Carnot efficiency. How does your answer change if the system has an additional rate of energy loss per unit area to conduction and convection of W/Km, with K?
CHAPTER 25
Photovoltaic Solar Cells
Photovoltaic (PV) solar cells are devices that directly convert solar radiation into electrical power. Over the last 15 years, the worldwide installed capacity of PV power has grown exponentially, doubling roughly every two years. By the end of 2016,installed PV capacity worldwide exceeded 300 GW [146], though delivered power is only about 15% of capacity [133].Given the enormous solar resource available and the potential for technological improvements and cost reductions in PV devices, solar photovoltaics seem likely to play a significant role in worldwide power generation in coming decades.
In this chapter, we give a basic introduction to the physics of photovoltaic solar cells. PV cells capture energy from electrons that are excited by absorbing individual photons as they propagate through specially designed materials. By structuring a device from these materials in an asymmetric fashion, an electric potential is produced that can drive a current in a preferred direction, allowing the energy of the captured photons to be delivered directly as electrical energy to an external circuit.
The first practical PV cells were developed in the 1950s during the electronic revolution spawned by the invention of transistors. Expensive and inefficient, early silicon solar cells found only niche applications such as powering satellites. Since then, steady increases in efficiency, reductions in material requirements, and improved manufacturing techniques have brought PVs to the point of competitiveness with traditional non-renewable electric power sources in some contexts. Many promising new technologies are currently under development and research on radically new ideas such as organic PVs is quite active. A complete survey of these new technologies is beyond the scope of this book. We focus instead on the basic physical principles of photovoltaic solar cells, particularly the traditional crystalline silicon solar cells that dominate today’s PV installations. At the end of the chapter we briefly describe some alternative materials and cell designs. Given the limitations of space, we do not discuss the design and fabrication of specific devices or other engineering issues associated with constructing robust and durable PV cells and systems. We also do not follow the production of electric power outside the boundaries of an individual cell. Converting the intermittent, low-voltage DC power produced by an individual cell into stable high-voltage AC power for practical applications is a subject in its own right, aspects of which are discussed briefly in §38 (Electricity generation and transmission).
Reader’s Guide
This chapter is devoted to the study of photovoltaic solar cells. Basic ideas from solid-state physics are introduced and the appearance of band gaps in the electronic structure of materials is explained based on quantum mechanics. The principles governing semiconductor photovoltaics are introduced and developed in the context of p-n junction-based silicon solar cells. Efficiency limits on solar cells are derived, and some advanced solar cell designs are described.
Prerequisites: §3 (Electromagnetism), §7 (Quantum mechanics), §8 (Entropy and temperature), §9 (Energy in matter),and the three previous chapters on solar energy. Quantum mechanics of a free particle (§7.7) and particles in potentials (§7.8) are particularly important for §25.2.
Material developed in this chapter reappears in §36 (Systems) and §38 (Electricity generation and transmission).
We begin with two sections introducing the quantum physics of crystalline materials. A more detailed outline of the rest of the chapter is given at the end of §25.1.
25.1Introductory Aspects of Solid-state Physics
The basic challenge of the PV device is to design a material that absorbs photons into specific electronic excitations that can drive an electric circuit. To understand how this can be done, it is necessary to extend the discussion of quantum physics beyond individual atoms and molecules to solid materials – a branch of science known as solid-state physics.
The quantum states available to an electron in an atom have specific, quantized energies. The Pauli exclusion principle (see§7.8.2) allows only two electrons (with opposite spins) in each quantum state. When the electrons fill the lowest allowed energy states of the atom, the atom is in its ground state. Incoming photons or other excitations can push one of the electrons into an excited state with higher energy (typically of order ~1 eV), after which the electron can emit this energy as a photon as it drops back to a lower-energy state. We discussed some of the basic mechanisms for such transitions in §23.4. The quantum structure of a molecule, where several atoms are held together by electromagnetic forces, is somewhat more complicated, because the quantum state depends not only on the electron configuration but also on the relative positions of the constituent atomic nuclei. Because the nuclei are much more massive than the electrons, their motion occurs on a slower time scale and involves much lower energies. It is usually a reasonable approximation to separate out the quantum excitations of the electrons from excitations of nuclei within the molecule. When we discussed energy in molecules in §9, we focused on the(vibrational) motions of the nuclei, assuming that the electrons remain in the ground state as the nuclei move. In contrast, the spectrum of electronic excitations of molecules can be estimated by assuming that the nuclei are in fixed locations corresponding to the ground state of the molecule, and solving the Schrödinger equation for the electrons in the electromagnetic potential produced by the nuclei in this configuration. The result is a spectrum of states with excitation energies at the same scale as atomic excitation energies (i.e. of order ~1 eV), but more complex.
Figure 25.1 (a) The diamond crystal structure, in which carbon atoms or silicon atoms are configured into a crystalline solid. Each atom has four nearest neighbors to which it is bonded. A single cubic cell of the crystal is depicted, which is repeated periodically in each direction over a macroscopic distance to form a solid material. (b) In the zincblende crystal structure, which characterizes several PV materials, two elements combine to produce the same basic pattern as in (a).
To understand the quantum physics of a solid, we can envision it as simply an enormous molecule. For the crystalline solids that interest us here, the atoms in the solid are held in place by bonds similar to those that hold atoms together in a molecule. In a solid, however, these bonds hold a large number of atoms together in a macroscopic three-dimensional structure. For crystalline solids, the arrangement of atoms has a regular periodicity, at least at a microscopic scale, which can be described by a space-filling pattern consisting of repeated copies of a basic cell. One of the simplest examples of such a crystalline solid is diamond. Carbon atoms, with valence four, can be configured in a regular pattern in space, known as the diamond crystal structure, so that each carbon atom is bonded to four nearest neighbors (see Figure 25.1(a)). As we discuss in more detail in §25.5, the diamond crystal structure can be described in terms of a combination of periodic space-filling lattices, which are generated by integer linear combinations of a set of basis vectors in three-dimensional space. The stability of the diamond crystal structure underlies the famous strength and hardness of diamond. Silicon, which lies directly below carbon in group IV1 of the periodic table (see Figure D.1), also has valence four, and crystallizes in the diamond structure.
In §8.7.3 and §9.2.1, where we introduced Einstein’s and Debye’s models of the specific heat of solids, we considered the quantum mechanical vibrations of a crystal lattice. Like the vibrations of molecules, these excitations of crystals can be considered as relative motions of entire atoms within the crystal. The electrons adjust their configuration to stay in the electronic ground state as the atoms move. These vibrations can propagate like waves through crystals, and the quanta associated with these waves are known as phonons (see §25.5). To understand the electronic properties of crystalline solids, on the other hand, we must examine the energy levels available to the electrons in crystals. Just as the electron energy levels of a molecule can be computed with the nuclei held in fixed positions, so the electronic structure of solids can be studied in an approximation where the atoms do not move. After the allowed electron energy levels have been determined with the nuclei in the fixed crystal structure, the exchange of energy between an electron and the crystal vibrations can be considered as a separate problem.
