Solar Energy: 22-Mec-B3 Energy Conversion and Power Generation, December 2013, Q6

Question 6: Solar Energy (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — Principles, output estimate, and suitability for large-scale generation (5 marks)

Intensity. The sun delivers about 1367 W/m² at the top of the atmosphere, the solar constant. Atmospheric absorption and scattering reduce this to roughly 1000 W/m² on a surface normal to the beam at sea level on a clear day at midday — the value adopted as standard test conditions for rating modules. The reduction depends on the air mass traversed, so intensity falls as the sun approaches the horizon, and it is further cut by cloud, humidity, aerosol and snow cover. Radiation arrives in two forms whose distinction is central to plant selection: the direct or beam component, which arrives along the line to the sun and can be concentrated by mirrors or lenses, and the diffuse component scattered by the sky, which cannot. In the cloudy coastal climate of southwestern British Columbia diffuse radiation can be half the annual total, which is precisely why concentrating solar-thermal plant is unsuited to that region while flat-plate photovoltaic modules, which use both components, still work.

Incidence. Only the component of the beam normal to the collector is captured, so the flux on a fixed surface follows Lambert's cosine law, \(G_{collected} = G_{beam}\cos\theta\) where \(\theta\) is the angle between the beam and the surface normal. Because \(\theta\) varies through the day and through the year with the sun's altitude and azimuth, a fixed collector is optimally oriented only twice a day at best. The standard fixed compromise in Canada is to face true south and tilt the collector at roughly the site latitude, which balances the summer and winter sun paths; steeper than latitude favours winter and helps shed snow, shallower favours summer. Single-axis tracking recovers perhaps 15–25% more annual energy and two-axis tracking a little more again, at the cost of mechanism, maintenance and land, and concentrating plant has no choice but to track since a concentrator cannot focus off-axis beam at all.

Duration. Even a perfectly tracked collector receives energy only while the sun is up, and only usefully while it is well above the horizon. The convenient measure is peak sun hours, the number of hours of 1000 W/m² equivalent per day: roughly 5 h/day in a southern Canadian summer against about 1.2 h/day in midwinter, a seasonal swing of more than four to one that is worst exactly when Canadian electricity demand peaks. Annual totals run near 1100–1300 kWh per installed kW_p in southern British Columbia and Ontario and 1300–1500 in southern Alberta and Saskatchewan, the sunniest parts of the country. This makes the capacity factor of solar generation about 12–17% — against 85–90% for a CANDU unit — and the intermittency is not merely diurnal but stochastic, since passing cloud can cut output by 80% in under a minute.

Efficiency and an output estimate. Efficiency is the ratio of electrical output to incident solar energy on the aperture. For a worked estimate take an array of 6000 m² of module aperture — about 1.5 acres of module, needing some 3–4 acres of site once row spacing is allowed for — at a module efficiency of 20%: \[ P_{peak} = G_{STC}\,A\,\eta_{module} = 1.0\ \text{kW/m}^2 \times 6000\ \text{m}^2 \times 0.20 = 1200\ \text{kW}_p \] So the array is nominally 1.2 MW. Its annual yield at 1100 kWh/kW_p is \(1200 \times 1100/1000 = 1320\) MWh, a capacity factor of \(1100/8760 = 12.6\%\). On a daily basis output is zero at night, rises and falls roughly as a half-sine through the day peaking near solar noon, and on the best summer day delivers about \(1200 \times 5 = 6\) MWh against roughly 1.4 MWh on a clear midwinter day and far less under overcast.

Advantages for large-scale generation. The fuel is free, inexhaustible and immune to price shocks; operation produces no CO₂, no criteria pollutants, no ash and essentially no water consumption for photovoltaic plant; there are no moving parts in a PV array, so operation and maintenance costs are very low and plant life exceeds 25 years; capacity is modular and can be added in small increments with short lead times, and can be sited at the point of consumption to avoid transmission losses; output in temperate summer correlates well with air-conditioning demand; and capital costs have fallen by roughly 90% since 2010, making utility-scale solar among the cheapest new generation in sunny regions.

Disadvantages. The energy is dilute, so land area per megawatt is one to two orders of magnitude greater than for thermal plant. It is intermittent and non-dispatchable, and it produces nothing at the winter evening peak that governs Canadian system planning, so it cannot displace firm capacity without storage or a firm backup, and the cost of that storage must be counted against the plant. Photovoltaic inverters are power-electronic rather than rotating, contributing no inertia or short-circuit strength to the grid, which constrains how much can be accommodated without synchronous condensers or grid-forming controls. Output degrades slowly with age and drops about 0.4% per °C of cell temperature rise, so hot clear days yield less than the rating suggests. Snow, soiling and shading losses are real in Canadian conditions, and at high latitudes the annual resource is simply modest. Manufacturing is energy-intensive with an energy payback of one to three years, and end-of-life module recycling is still immature. The reasonable conclusion is that solar is a valuable energy resource in the Canadian mix but not, unaided, a capacity resource.

Part (b) — Solar-thermal steam-cycle installation (5 marks)

Figure 6.1 — Concentrating solar-thermal power plant with molten-salt storage and a conventional steam Rankine cycle.

How the plant works, component by component. The concentrator field — parabolic troughs tracking on one axis, or a field of flat heliostats aimed at a central tower — reflects the direct beam onto a small receiver area, multiplying the flux by a concentration ratio of 70–100 for troughs or 500–1000 for a tower. Concentration is essential rather than cosmetic: to raise steam at turbine conditions the absorber must run at 400 °C or more, and re-radiation from a hot surface grows as \(T^4\), so a useful collection efficiency at that temperature is only achievable if the absorbing area is far smaller than the collecting area. The receiver is a selective-coated steel tube inside an evacuated glass envelope (trough) or a panel of tubes on the tower, absorbing strongly in the solar band while emitting weakly in the infrared, and the vacuum annulus suppresses convection.

A heat transfer fluid — synthetic oil to about 390 °C, or molten nitrate salt to about 565 °C in tower plant — carries the heat from the field. A separate fluid loop is used rather than direct steam generation so the field can operate at modest pressure and so the fluid can also charge thermal storage: two tanks of molten salt, hot and cold, which decouple collection from generation and let the plant generate into the evening peak. Storage is the single feature that distinguishes solar-thermal from photovoltaic economically, since it converts an intermittent output into a partly dispatchable one. The steam generator train (preheater, evaporator, superheater) transfers that heat to the water side, and from there the plant is a conventional Rankine unit: steam turbine, generator, condenser (water-cooled where water is available, air-cooled in the deserts where these plants are usually built), and feed pump returning condensate to the steam generator. A gas-fired backup boiler is normally provided for freeze protection of the salt or oil and for morning start-up.

Overall efficiency estimate. The efficiencies multiply in a cascade: \[ \eta_{overall} = \eta_{field}\,\eta_{receiver}\,\eta_{cycle}\,\eta_{parasitic} \approx 0.55 \times 0.90 \times 0.38 \times 0.92 \approx 0.17 \] so roughly 17% solar-to-electric annual average, with instantaneous peak values near 22–25%. Commercial trough plant achieves 14–16% annual and tower plant with higher-temperature salt 18–20%.

Where the losses occur. The largest single loss is the condenser, which by the second law must reject some 60% of the heat that reaches the working fluid — unavoidable for any heat engine and the reason the cycle term is only 0.38. The field accounts for the next largest group: cosine loss because the mirrors cannot all face the sun, shading and blocking between rows, mirror reflectivity of 92–94% and soiling that erodes it further, spillage of the focused image past the receiver, and tracking error. The receiver loses heat by re-radiation (dominant at tower temperatures) and by convection and conduction through supports. Piping and storage tanks lose heat to ambient, particularly overnight. Finally, parasitic power for the heat-transfer-fluid pumps, tracking drives and freeze protection consumes 5–10% of gross generation, and start-up and shutdown transients waste thermal energy every morning and evening.

Part (c) — Photovoltaic installation (5 marks)

Figure 6.2 — Grid-connected photovoltaic installation, showing how series connection raises voltage and parallel connection raises current.

How radiation becomes electricity. A photovoltaic cell is a large-area semiconductor p–n junction, almost always crystalline silicon. A photon whose energy exceeds the semiconductor band gap (1.12 eV for silicon, corresponding to a wavelength of about 1100 nm) can be absorbed by promoting an electron from the valence band to the conduction band, creating an electron–hole pair. The junction's built-in electric field, established by the space-charge region between the p-doped and n-doped material, sweeps the electron toward the n-side and the hole toward the p-side before they can recombine. This charge separation is what produces a voltage — about 0.5–0.6 V per cell at open circuit, set by the band gap and essentially independent of cell area — while the current is proportional to the illuminated area and to the irradiance. Connecting an external circuit lets the separated carriers do work; the operating point is set on the cell's current–voltage characteristic, and the inverter's maximum power point tracker continuously adjusts the load to sit at the knee of that curve, which moves with irradiance and temperature.

How a transmission-level voltage is obtained. Half a volt per cell is useless directly, so voltage is built up by series connection at three levels. Within a module, 60 or 72 cells in series give roughly 30–45 V; bypass diodes are fitted across sub-strings so that one shaded cell cannot block the whole series current or overheat as a hot spot. Modules are then wired in series into a string of typically 15–25 modules, giving 600–1500 V DC, the upper limit being set by insulation ratings and electrical code rather than by the cells. Strings are then connected in parallel through fused combiner boxes, which multiplies current at constant voltage, and here blocking diodes and string fuses prevent a faulted string from being back-fed by its neighbours. The DC bus feeds the inverter, which synthesises three-phase AC at 400–800 V by pulse-width modulation, and a conventional step-up transformer raises that to distribution voltage (25 kV) or, through a further substation transformer, to transmission voltage (138 kV or above). Series stacking therefore sets voltage and parallel stacking sets current, and the transformer does the final step — the semiconductor devices themselves never see transmission voltage.

Overall efficiency estimate. Again a multiplicative chain: \[ \eta_{overall} = 0.20 \times 0.88 \times 0.98 \times 0.97 \times 0.99 \approx 0.166 \] that is a module efficiency of 20%, a combined temperature and soiling factor of 0.88, DC wiring 0.98, inverter 0.97 and transformer 0.99, giving roughly 16.5% solar-to-grid. The industry expresses the non-module part of this as a performance ratio, typically 0.78–0.85 for a well-built plant.

Where the losses occur. By far the largest losses are inside the cell and are set by physics rather than workmanship. Photons with energy below the band gap — roughly 20% of the spectrum for silicon — pass straight through unabsorbed, while photons with energy well above it lose the excess immediately as heat when the carrier relaxes to the band edge, another 30% or so. Together with unavoidable radiative recombination and the fact that the operating voltage is well below the band-gap voltage, these set the Shockley–Queisser ceiling of about 33% for a single-junction cell, against which a production module's 20% is respectable. Beyond that: reflection at the front surface (reduced by anti-reflective coating and texturing), grid-finger shading of the active area, series resistance in the fingers and busbars, shunt leakage at cell edges, and recombination at defects and surfaces. Operationally, cell temperature is the biggest controllable term — a module in full sun runs 25–30 °C above ambient and loses about 0.4% of its output per °C, so 10–12% is typical on a hot day. Soiling, snow cover and inter-row shading add a few percent more, module mismatch and degradation (about 0.5% per year) a little further, and finally the inverter and transformer take 3–4% together. Unlike the solar-thermal plant there is no condenser loss, because there is no heat engine — which is why a PV plant at 16.5% is competitive with a thermal plant whose cycle alone discards 60%.

Topic. The solar resource and the two routes from it to electricity — a heat engine driven by concentrated thermal energy, and direct quantum conversion in a semiconductor junction.

Key relations. Solar constant 1367 W/m², standard test irradiance 1000 W/m²; cosine law \(G = G_b\cos\theta\); array rating \(P_{peak} = G_{STC}A\eta_{module}\); annual energy \(= P_{peak} \times\) yield in kWh/kW_p; capacity factor = annual yield / 8760; efficiency cascades \(\eta_{overall} = \prod \eta_i\); series connection adds voltage, parallel connection adds current; temperature coefficient about −0.4%/°C.

Why this works. Every solar estimate is a chain of independent multiplicative factors, so the discipline is to identify each link once and not double-count. Resource terms (intensity, incidence, duration) set how much energy reaches the aperture; conversion terms set what fraction leaves as electricity. The two technologies differ in exactly one structural respect that explains all their other differences: the solar-thermal plant contains a heat engine and therefore pays the Carnot penalty in a condenser but can store heat cheaply and dispatch it, whereas the photovoltaic plant has no thermal stage and so no condenser loss, but storing electricity is expensive. That single fact explains why storage is standard on one and costly on the other.

Common pitfall. Quoting the module efficiency as the plant efficiency, which ignores temperature, soiling, wiring, inverter and transformer losses and overstates annual output by 20% or more. The second is confusing peak capacity with energy — a 1.2 MW_p array does not produce 1.2 MW around the clock but averages about 150 kW over a year — and the third is claiming that concentrating solar-thermal plant would suit a cloudy Canadian coastal site, when concentrators can use only the direct beam and diffuse radiation there is roughly half the total.

Source. El-Wakil, Powerplant Technology, Ch. 15 (solar energy: radiation geometry, flat-plate and concentrating collectors, solar-thermal power plant and photovoltaic conversion); Çengel & Boles, Thermodynamics: An Engineering Approach 9th ed., Ch. 10 (Rankine cycle for the bottoming plant) and Ch. 12 (radiation fundamentals).