Phase diagram of water

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Carnot Cycle (Ideal Cycle) T Q 12 T 1 2 a W 41 W 23 Q 34 T b 4 3 S 1) Heat absorption at constant temperature, T a (boiler) 12 2) Isentropic expansion work output (turbine) 23 3) Heat rejection at constant temperature, T b (condenser) 34 4) Isentropic compression (pump) 41 Energy Conservation (1 st Law of Thermodynamics) Q 12 + W 23 + Q 34 + W 41 = 0 (Note: Q 12 > 0, W 23 < 0, Q 34 < 0, W 41 > 0) Cycle efficiency, η c = (Useful work out)/(Heat input at T a ) ie η c = (| W 23 | − W 41 )/ Q 12 | = (T a − T b )/ T a = 1 − T b / T a (Note: T measured in K (absolute temperature) – formal definition of absolute temperature scale) Practical difficulties in using a Carnot Cycle 1) Boiler operates only in wet-steam regime otherwise temperature would rise when all the water has turned to steam, violating condition for Carnot Cycle turbine expands wet steam water droplets hit turbine blades (damage) 2) Maximum temperature (T a ) is limited to ~650 K efficiency of cycle is severely constrained 3) Compression of water/steam mixture is thermodynamically unstable (water droplets) very large volume compressor (expensive) Rankine Cycle overcomes all these problems T 2b 2a T b 1b 1a 2c S Rankine Cycle Step 1: a) Condense all the steam to water in the condenser b) Pumping water to high pressure requires small volume machine and little energy T T b T a S Step 3: Expand dry steam through a turbine to generate shaft power In practice, water droplets still form in the low pressure end of the turbine, so the steam is reheated at various stages Step 2: Use 3-stage boiler (~ constant pressure) a) Economiser – water heated at constant pressure b) Evaporator – water/steam mixture heated at constant pressure c) Superheater – dry steam heated at constant pressure [Note that there is a small drop in pressure through the boiler tube in order to overcome frictional losses] T T a T b S HP from IP LP boiler reheaters HP: high pressure turbine IP: intermediate pressure turbine LP: low pressure turbine to condenser
Frictional losses across turbine blades vary like u 2 (F D =½C D ρAu 2 ) ie very large for large u (near speed of sound) Losses reduced significantly by using many stages in series (~50 stages) Other practical effects limiting efficiency The loss of kinetic energy at each stage is small and turbulence is reduced a) Boiler tubes have finite thickness, so outer wall temperature is higher than water/steam temperature b) Metallurgical limit to temperature/pressure difference boiler tubes can withstand (creep/crack formation) c) Many pipes/tubes in flow circuit frictional losses d) Condenser is a vacuum chamber air leaks in but can not condense, so ‘air blanket’ forms, preventing water vapour from condensing on cold surface of condenser tubes T W 12 2 1 3 Q 23 W 34 Q 41 4 Efficiency of Rankine Cycle S Condenser at 30 C at a pressure of 0.04 bar Compressor increases pressure to 170 bar Three-stage boiler at 170 bar a) economiser raises temperature to 352 C b) evaporator at 352 C c) superheater raises temperature to 600 C Adiabatic turbine T p h f h g s f s g Water/Steam 30 0.04 126 2566 0.436 8.452 Water/Steam 352 170 1690 2548 3.808 5.181 Dry Steam 600 170 3564 6.603 where h f and h g are the specific enthalpies and s f and s g are the specific entropies of the fluid and gas, respectively, in kJ/kg. Adiabatic compression or expansion W = W s + (p 1 v 1 − p 2 v 2 ) = ∆U = u 2 − u 1 W s = (u 2 − u 1 ) − (p 1 v 1 − p 2 v 2 ) = (u 2 + p 2 v 2 ) − (u 1 + p 1 v 1 ) = h 2 − h 1 Work done by the shaft W s on the fluid Adiabatic so Q = 0 Total work W First Law: ∆U = Q + W Note sign convention In adiabatic process work done equals change in enthalpy Specific enthalpy h = u +pv; dh = TdS + Vdp isobaric, constant pressure, dh = du + pdv = dQ isentropic dh = dW = Vdp i) W 12 = V(p 2 − p 1 ) = 10 −3 (170 − 0.04) 10 5 =17 kJ/kg 3 ii) 12 isentropic so h 2 = h 1 + W 12 = 126 + 17 = 143 kJ/kg iii) 23 isobaric so Q 23 = h 3 – h 2 = 3564 − 143 = 3421 kJ/kg iv) 34 isentropic so W 34 = h 3 – h 4 and s 3 = s 4 s 4 = (1−x)s f4 + xs g4 s 3 = 6.603 = (1−x)0.436 + 8.452 x x = 0.769 T W 12 2 1 Q 23 W 34 Q 41 4 S
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Phase diagram of water

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