22-Mec-B3 Energy Conversion and Power Generation · May 2014
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. Closed book, three hours. Two sections: Section A (Calculative) Questions 1–4 and Section B (Descriptive) Questions 5–6. Candidates answer three questions from Section A and one from Section B; four questions of 15 marks each constitute a complete 60-mark paper. Reference data for particular questions are bound in as pages 10–17, reference formulae and constants as pages 18–21, and the steam tables from Granet & Bluestein are supplied. All six printed questions are solved below.
Reference texts for 22-Mec-B3 Energy Conversion and Power Generation
Units and conventions used throughout. The paper's own constant sheet (page 19) is used without substitution: $g = 9.81\ \text{m/s}^2$, $c_p$ air $= 1.005\ \text{kJ/kg}\cdot \text{K}$, $c_v$ air $= 0.718\ \text{kJ/kg}\cdot\text{K}$, $c_p$ water $= 4.190\ \text{kJ/kg} \cdot\text{K}$, $\rho_{\text{water}} = 1000\ \text{kg/m}^3$, $p_{\text{atm}} = 100\ \text{kPa}$. Monetary amounts are in Canadian dollars, as the paper is an Engineers Canada national examination.
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.
This is a descriptive question and is answered in essay form, with the quantitative comparison that Part I (c) explicitly asks to be "proven" carried out numerically at the end of that part.
Both heating values are the heat released when a unit quantity of fuel is burned completely with the products returned to the initial temperature, usually 25 °C. They differ solely in the phase in which the water formed by combustion is counted. The higher heating value assumes that all the water vapour in the products has been condensed to liquid, so the heat released includes the latent heat of condensation, approximately 2 442 kJ per kilogram of water at 25 °C. The lower heating value assumes the water leaves as vapour and therefore excludes that latent heat. The two are related by $$HHV = LHV + m_w h_{fg}$$ where $m_w$ is the mass of water formed per unit mass of fuel, including both the hydrogen burned to water and any moisture the fuel already contained. The gap is small for a fuel low in hydrogen and large for one rich in it: for anthracite, which is nearly pure carbon and produces almost no water, the two values differ by only a per cent or two, whereas for methane the ratio is about 55.5 MJ/kg higher against 50.0 MJ/kg lower, a difference of 11 %. Which one to use is decided by what happens in the plant. A conventional boiler discharges flue gas at 130 to 150 °C, well above the acid dew point, so the water leaves as vapour and the lower heating value is the energy the plant can actually reach; efficiencies quoted on LHV are therefore higher than the same plant's HHV efficiency, and the basis must always be stated. A condensing boiler or a gas turbine with a condensing heat-recovery unit does recover part of that latent heat, which is why condensing domestic boilers can be advertised at more than 100 % efficiency — on the LHV basis. Fuel is bought and sold on HHV in North America, so a Canadian station's contractual heat balance and its performance guarantee are frequently on different bases, and reconciling them is a routine source of error.
A proximate analysis is an empirical, procedural characterisation: the coal is subjected to a standard sequence of heating and weighing tests, and the result is reported as four fractions summing to 100 % — moisture (mass lost on drying at about 105 °C), volatile matter (further mass lost on heating to about 950 °C in the absence of air), fixed carbon (the combustible residue) and ash (the incombustible residue after burning off the fixed carbon). Heating value and sometimes sulphur are reported alongside. The analysis is quick, cheap and reproducible, and it is what a boiler designer and operator actually need day to day: volatile matter governs ignition stability and flame length and therefore burner design; fixed carbon governs the burnout time and hence furnace volume; moisture governs mill drying-air requirements and depresses both flame temperature and boiler efficiency; and ash governs slagging and fouling behaviour, ash-handling capacity and precipitator sizing. It is the analysis used for contractual coal specification and for day-to-day combustion control.
An ultimate analysis is a fundamental, elemental characterisation: the coal is reported as mass fractions of carbon, hydrogen, oxygen, nitrogen and sulphur, plus ash and moisture. It says nothing directly about combustion behaviour but it is exactly what is needed to write the combustion equations. From the ultimate analysis one computes the stoichiometric air requirement, the theoretical flue-gas mass and composition, the excess-air relationship to measured flue-gas oxygen, the dew point (from the hydrogen and sulphur), the heating value by a correlation such as Dulong's formula, and the carbon dioxide and sulphur dioxide emitted per unit of energy. It is therefore the analysis used for heat-balance calculations, for emission inventories and permitting, and for sizing flue-gas desulphurisation and carbon-accounting under Canadian federal and provincial reporting requirements. The two analyses are complementary rather than alternative: proximate for operating the boiler, ultimate for calculating what comes out of it.
Natural gas gives substantially lower carbon dioxide emissions for the same energy release, by roughly a factor of two. The reason is structural rather than a matter of combustion practice: in anthracite essentially all the chemical energy comes from oxidising carbon to carbon dioxide, whereas in methane a large share comes from oxidising hydrogen to water, and hydrogen carries no carbon penalty at all. Methane has the highest hydrogen-to-carbon ratio of any hydrocarbon, four, so it is the best any fossil fuel can do.
To quantify it properly, four parameters must be known for each fuel. The first is the elemental composition, from the ultimate analysis for coal or the gas chromatographic composition for natural gas, because that fixes how much carbon dioxide is produced per unit mass of fuel through the stoichiometry. The second is the heating value and its basis, HHV or LHV, because the comparison is per unit of energy and the two bases differ by 11 % for methane but only 1 to 2 % for anthracite — comparing a coal HHV figure with a gas LHV figure is the single most common way this comparison is falsified. The third is the conversion efficiency of the plant, if the comparison is to be per kilowatt-hour of electricity rather than per gigajoule of fuel. The fourth, for a complete answer, is the combustion completeness and the upstream fugitive emissions: unburned carbon in the ash for coal, and methane leakage in production and transmission for gas, methane being a far more potent greenhouse gas than carbon dioxide over a twenty-year horizon.
Given. Anthracite treated as pure carbon, $HHV = 32\,800$ kJ/kg; methane $HHV = 55\,500$ kJ/kg and $LHV = 50\,000$ kJ/kg. Relative atomic masses C 12.011, H 1.008, O 15.999.
Find. The mass of carbon dioxide released per gigajoule of fuel energy for each fuel, and hence the ratio between them.
The distinction is not one of engineering quality but of the boundary condition the working fluid imposes, and it comes down to whether the fluid can be brought to rest.
A hydro turbine works on a confined flow whose boundaries are fixed by the plant itself. The water is delivered through a penstock at a pressure set by the head, and the entire available energy is potential energy, stored as pressure. In a reaction turbine the runner converts that pressure to velocity and then absorbs the velocity, and the draft tube recovers what velocity head remains at the runner exit, so in the ideal limit the water can be discharged into the tailrace with essentially no kinetic energy left and the whole $\rho g Q H$ can appear at the shaft. Nothing in the physics requires the water to keep moving: it arrives through a pipe because the plant put it there, and once it has given up its energy it simply sits in the tailrace. Real machines reach 90 to 95 % because of friction, leakage and residual swirl, but the theoretical ceiling is 100 %.
A wind turbine works on an unconfined flow whose only means of arriving is its own momentum. The available energy is kinetic, and the rotor cannot extract all of it because the air must still be moving fast enough downstream to carry away the mass that continues to arrive upstream. Bringing the air to rest at the disc would mean zero flow through it and therefore zero power. This is the Betz argument: writing mass, momentum and energy conservation for the streamtube through the disc gives the power coefficient as $C_P = 4a(1-a)^2$, where $a$ is the fractional slowing at the disc; maximising it gives $a = 1/3$ and $$C_{P,\max} = \frac{16}{27} = 0.593$$ with the wake leaving at one third of the free-stream speed. The ceiling is therefore imposed by the supply mechanism, not by the machine, and no rotor of any design can beat it. Real machines reach 40 to 48 % because of wake swirl, aerofoil drag and tip losses. The same argument applies to any open-flow kinetic device, including tidal-stream and river-current turbines, which is why their power densities are so much lower than those of a dammed head.
Both routes deliver roughly a fifth of the incident radiation as electricity, but the losses arise at completely different points, and understanding where they sit explains why the two technologies behave so differently in service.
In a photovoltaic panel the dominant losses are quantum-mechanical and irreducible for a given material. The cell responds to individual photons through its band gap, about 1.1 eV for silicon, and this creates two large unavoidable losses. Photons with less energy than the band gap — the whole infrared tail, roughly a fifth of the solar spectrum's energy — are simply not absorbed and pass through. Photons with more energy than the gap do create a charge carrier, but the surplus energy is immediately lost as lattice heat, or thermalised; for the blue end of the spectrum more than half of each photon's energy is wasted this way. Together these two account for the bulk of the loss and are what limits a single-junction silicon cell to the Shockley–Queisser figure of about 33 %. On top of them come recombination of carriers before they reach the contacts, series resistance and the resulting fill-factor shortfall, reflection at the front surface, and grid shading, which bring a good commercial module to 20 to 22 %. Finally the cell loses about 0.4 % of its output per kelvin of temperature rise — hot panels perform worse, the opposite of a heat engine — and the inverter takes a further 2 to 4 %. Notably, none of these losses is thermodynamic in the Carnot sense; the panel is a quantum device, not a heat engine, which is why it works at part illumination and needs no minimum size.
In a solar-thermal plant the losses are almost entirely optical and thermodynamic. The concentrator loses radiation before it ever reaches the working fluid: mirror reflectivity of 90 to 94 % that degrades with soiling, imperfect tracking and slope error, spillage of the focal image past the receiver aperture, and the cosine loss that arises whenever the mirror is not normal to the sun. The receiver then loses heat by re-radiation, which climbs as the fourth power of its absolute temperature, and by convection to the wind. Those two together typically leave 55 to 65 % of the beam as useful heat. That heat is then fed to a Rankine cycle, and here the Carnot penalty dominates: because a parabolic-trough receiver is limited to roughly 400 °C by the stability of its synthetic-oil heat-transfer fluid, the steam cycle achieves only about 38 %, well below the 45 % a utility coal plant gets from 565 °C steam. Parasitic pumping and tracking loads and generator losses take the final few points, giving about 20 % overall — and the receiver-temperature limit is precisely why central-receiver towers with molten-salt loops at 565 °C, or the supercritical carbon-dioxide cycles now being developed, are the route to improving it. The compensating advantage is decisive for grid operation: the heat can be stored in molten salt far more cheaply than electricity can be stored in batteries, so a solar-thermal plant can be dispatched into the evening peak, which a photovoltaic plant cannot do without separate storage.
