22-Mec-B3 Energy Conversion and Power Generation · May 2014

Question 5 of 6: Graphical Representation

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 5: Graphical Representation (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.

This is a descriptive question, so the answer is given in essay form with the two required worksheets reproduced as figures. The paper's own lettering runs (a), (c), (d) in Part I — there is no part (b) printed — and that lettering is followed here.

Part I (a) — the six cycle diagrams

The three vapour cycles all share the same left-hand construction and differ only in how far the expansion line is pushed to the right. In the saturated Rankine cycle the feed pump raises saturated liquid from condenser pressure to boiler pressure along an almost vertical line hugging the saturated-liquid boundary; the boiler then heats along that boundary and evaporates horizontally across the dome to dry saturated steam; the turbine expands isentropically down to condenser temperature, ending well inside the dome; and the condenser closes the cycle with an isothermal horizontal line back to the start. Because state 3 is on the saturated-vapour line, the exhaust is very wet — typically 15 to 20 % moisture — which is the cycle's practical weakness. Adding superheat extends the boiler line beyond the dome to the right, so the expansion begins from a higher temperature and finishes drier; the enclosed area, which is the net work, grows and the mean temperature of heat addition rises. Adding reheat interrupts the expansion part-way, returns the steam to the boiler for a second constant-pressure heating at roughly the original superheat temperature, and lets it expand again, giving the characteristic double-hump outline. The Brayton cycle is drawn on the same axes but without a dome, since the working fluid stays a gas throughout: two isentropes joined by two constant-pressure lines, which on T–s axes diverge to the right, so the heat-rejection line is always longer than the heat-addition line at the same entropy interval — a direct picture of why gas-turbine exhaust is hot enough to be worth recovering.

The two reciprocating cycles are drawn on pressure–volume axes. Both begin with isentropic compression from bottom dead centre and both end with isentropic expansion and a constant-volume blowdown, but they differ in the heat-addition leg: the Otto cycle adds heat at constant volume, giving a vertical pressure spike at minimum volume, whereas the Diesel cycle adds heat at constant pressure over a finite fraction of the stroke, giving a horizontal plateau. That single difference explains the practical distinction between them: the Otto cycle's constant-volume combustion demands a homogeneous premixed charge, which limits the compression ratio to whatever the fuel's knock resistance permits, while the Diesel cycle's progressive injection tolerates compression ratios of 16 to 22 and therefore achieves higher efficiency despite being the less efficient of the two at equal compression ratio.

Part I (c) — why T–s for some cycles and p–V for others, and what is common

The choice follows what the engineer needs to read off the diagram, and that in turn follows how the machine is built. Steam and gas turbine plant is a steady-flow assembly of separate components, each of which exchanges heat or work with a stream passing through it. For such devices the quantity of interest is heat per unit mass, and on temperature–entropy axes heat is the area under the process line, $q = \int T\,ds$. A T–s diagram therefore shows at a glance how much heat went in, how much came out, and — because the Carnot argument depends on the mean temperatures of addition and rejection — roughly what efficiency to expect. It also carries the saturation dome, so the state of the steam and the wetness at turbine exhaust are visible, which is indispensable for vapour cycles. Reciprocating engines are different: they are closed-system machines in which one fixed mass of gas is compressed and expanded by a piston in a cylinder. There the quantity of interest is work, and on pressure–volume axes work is the area under the process line, $w = \int p\,dv$. The p–V diagram is also what the engine physically produces — an indicator diagram taken from a running cylinder is a p–V plot — so the abscissa maps directly onto crank position and compression ratio, and the enclosed area is the indicated work per cycle.

The parameter that is the same on both is the enclosed area: on either set of axes it is the net work of the cycle per unit mass, and it is traversed clockwise for a power cycle and anticlockwise for a refrigeration cycle. On p–V axes the area is $\oint p\,dv = w_{\text{net}}$ directly; on T–s axes the area is $\oint T\,ds = q_{\text{in}} - q_{\text{out}}$, which by the first law applied round a closed cycle is again $w_{\text{net}}$. The two diagrams are therefore alternative views of the same number, and the same cycle can legitimately be drawn on either — a Rankine cycle has a perfectly good p–V diagram, it is just a very thin one, because the liquid-side volume change is negligible against the vapour-side change.

Part I (d) — the advantages of superheating and reheating

Superheating earns its place three times over. First, it raises the mean temperature at which heat is added to the cycle, and since cycle efficiency rises with that mean temperature against a fixed sink, efficiency improves — the gain is modest, a point or two, but it is free thermodynamics. Second, and more important in practice, it moves the expansion line to the right so the turbine exhaust is drier. Wet steam erodes the low-pressure moving blades by droplet impact, and every extra per cent of moisture also costs roughly one per cent of stage efficiency through the Baumann effect, so keeping exhaust quality above about 0.88 is a mechanical necessity as much as an efficiency measure. Third, superheat increases the specific work, so a given power needs less steam, which shrinks the whole plant — boiler, piping, turbine annulus and condenser — and reduces the capital cost per kilowatt.

Reheating exists because superheat alone runs into a metallurgical ceiling. Once the throttle temperature is at the limit the boiler tube and turbine casing materials will tolerate — around 540 °C for ferritic steels, higher only with expensive austenitics — the only way to raise boiler pressure further without ending the expansion soaking wet is to split it. The steam is expanded in a high-pressure turbine to an intermediate pressure of roughly a quarter of throttle pressure, returned to a reheater in the boiler convection pass, brought back to approximately the original temperature and expanded again. The benefits are that boiler pressure can be raised well above what a single expansion would allow, that the exhaust is dry even from a supercritical throttle condition, and that the average temperature of heat addition rises again because the second heating happens at a high mean temperature. Typical gains are four to five points of cycle efficiency, and a second reheat adds perhaps one and a half more. The costs are a large reheater surface, a second set of hot steam lines with their own expansion problems, and a more complex start-up and turbine-protection scheme; a single reheat is nearly universal above about 100 MW, and double reheat is reserved for supercritical units where the fuel saving justifies it.

Part II (a) — blocking the plant in under the load curve

The rule for economic dispatch is to load plant in increasing order of marginal operating cost, from the bottom of the load curve upwards, so that the cheapest kilowatt-hours are the ones produced for the most hours. Reading the page-17 curve, the demand runs from a 40 % trough at about 06:00 through a 70 % daytime plateau to an 86 % evening peak at about 19:00, falling to 50 % at midnight. Against that shape the seven plant types dispatch as follows.

Nuclear (30 %) takes the bottom block and runs flat all day. Its fuel cost is almost nil and its capital charge is incurred whatever it does, so its marginal cost is the lowest on the system; it is also the least able to follow load, because xenon transients and thermal-stress limits on the reactor pressure vessel make deep cycling both slow and expensive. Run-of-river hydro (5 %) sits immediately above it and also runs continuously: its marginal cost is zero and water not used is water spilled. Those two blocks total 35 %, which is only slightly below the 40 % trough, so fossil plant (30 %) is the load-following block — it supplies the remaining 5 % overnight and then ramps up through the morning to carry the daytime plateau. Coal and gas-fired units are the natural regulating plant here because their fuel cost is real but their ramp rates and minimum stable loads permit two-shifting.

Solar (5 %) contributes only between about 08:00 and 17:00 on the stated sunny day, rising and falling with the sun and peaking near midday. Its marginal cost is zero, so it is taken whenever available, and its effect is to displace fossil output during exactly the hours when the daytime plateau is highest — a good match to this load shape, though it has already fallen away by the time the evening peak arrives. Pumped storage (10 %) and the gas turbines (10 %) carry the evening peak between roughly 16:00 and 21:00. Pumped storage goes first because its marginal cost is only the cost of the off-peak energy used to fill the upper reservoir divided by a round-trip efficiency of about 75 %, which on this system means cheap overnight nuclear and fossil energy; the same plant pumps during the 02:00–06:00 trough, deliberately raising the overnight demand so that the base-load blocks stay fully loaded. The gas turbines go last because their fuel is the most expensive on the system, but they are also the fastest to start and so serve as both peaking plant and spinning reserve. Wind (10 %) produces nothing, since the question specifies a day with no wind; its capacity is idle and, importantly, the system must be able to meet the peak without it, which this one can.

Part II (b) — which plant backs off when wind is available

Wind has effectively zero marginal cost once built, so whenever it blows it is taken in full and it displaces whatever is currently the most expensive plant running. In the overnight and daytime hours that is the fossil plant, which is the marginal unit for most of the day; in the evening peak hours it would displace the gas turbines first, because their fuel cost is higher still. It does not displace nuclear or hydro, which are cheaper than wind to keep running and, in the nuclear case, cannot manoeuvre quickly enough to be worth backing off. A secondary and increasingly important effect is that wind also displaces the pumping duty economics: cheap surplus wind at night is the ideal energy source for filling the pumped-storage reservoir, so with wind available the storage plant pumps on wind rather than on fossil energy, which improves the whole system's fuel bill and emissions beyond the direct displacement. The limit on all of this is that fossil units have minimum stable loads and cannot be shut down and restarted freely, so beyond a certain penetration surplus wind must be curtailed rather than absorbed.

Part II (c) — which plant increases if the peak reaches 90 % of installed capacity

At a 90 % peak the system must find another four points of output above the 86 % already carried. The cheap blocks are exhausted: nuclear (30 %), fossil (30 %) and hydro (5 %) are already fully loaded at the peak, which accounts for 65 %, and solar contributes nothing after dark. The additional output can therefore only come from the two peaking blocks. The order is set by operating cost: pumped storage is increased first, to its full 10 %, because its marginal cost is the cheapest of the remaining options provided the upper reservoir has been filled overnight; the gas turbines are then increased, towards their full 10 %, to make up the balance. Together those two blocks give 20 % against a requirement of 25 % above the 65 % of cheap plant, so the peak is met with 5 % of installed capacity still in reserve — adequate but thin, and the system operator would want firm imports or demand response available as well. Two practical caveats follow. Pumped storage is energy-limited, not just power-limited: 10 % of capacity for a five-hour peak requires the reservoir to hold that energy, and it can only be refilled if the overnight trough is deep enough. And running the gas turbines near their full output for the peak hours will lift the system marginal price sharply, because they are three to four times the fuel cost of the base-load plant — which is precisely the economic signal that justifies building the pumped storage in the first place.