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# 011 second law_cycle_analysis

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### 011 second law_cycle_analysis

1. 1. LECTURE UNIT 009 3. Second Law of Thermodynamics is concerned with the availability of energy from a thermodynamic cycle and shows the impossibility of a perpetual motion machine. A. Kelvin-Plank Statement (Concept of thermal efficiency) It is impossible to operate an engine in a cycle that will have no other effect than to extract heat from a reservoir and produce an equivalent amount ILLUSTRATION: Rankine Cycle Steam Power Plant STEAM . TURBINE ms 1 Pi QA 2 QR BOILER BOILER FEED PUMP CONDENSER 3 B . . ms ms WP Thermal Efficiency, e: WNET e= QA * 100% B. Reeves Statement Heat flows readily from a region of high temperature to region of lower temperature. C. Clausius statement It is impossible for a self-acting machine, unaided by an external agency to convey heat from one body to another at a higher temperature. ILLUSTRATION: Refrigeration Cycle QR . mR . CONDENSER mR COMPRESSOR EXPANSION VALVE WC . EVAPORATOR . mR mR QA ROOM 4. Third Law of Thermodynamics states that the entropy of any pure substance in thermodynamic equilibrium tends to approach zero as the absolute temperature approaches zero. T2 S2 - S1 = dQ T T 1 CYCLE ANALYSIS is a series of processes that a system undergoes, whereby the system’s starting state is also its final state. 1. Use Pv and Ts diagram to illustrate the cycle. 2. PVT relationship a.) PV = mRT (for any ideal gas processes) P1V1 P2V2 b.) = (for P=C, V=C or T=C only) T1 T2 k-1 k-1 k T2 P2 V1 c.) = = (for PVk=C or S=C) T1 P1 V2 Note: Change k to n for Special polytropic process (PVn=C) “God gives us the ingredients for our daily bread, but He expects us to do the baking.”
2. 2. 3. Heat Added, QA QA = +Q CYCLE 4. Heat Rejected, QR QR = -Q CYCLE 5. Net Work, WNET WNET = QA - QR = WNF = P dV 6. Cycle thermal efficiency, e WNET e = * 100% (for any cycle) QA 7. Cycle mean effective pressure, Pm Network Pm = Volumetric displacement WNET e QA Pm = = VD (Vmax - Vmin)CARNOT CYCLE The most efficient cycle and basis of comparison for Rankine Cycle. ILLUSTRATION: QA Heat Source . . 2 ma ma 1 TH = C WT ADIABATIC ADIABATIC TURBINE COMPRESSOR S=C S=C OR EXPANDER TL = C . . ma ma Cold Souce 3 4 QR P-v diagram T-s diagram P T 1 PV = C 1 T=C 2 PVk = C TH = T1 = T2 2 WNET WNET PVk = C S=C or S=C QNET 4 PV = C 3 TL = T4 = T3 3 4 T=C Pm WNET v s Definitions Vmax a.) Expansion Ratio, re = Vmin Vmax b.) Compression Ratio, rk = Vmin Vmax c.) Cut-off Ratio, rc = Vmin Pmax d.) Pressure Ratio, rp = Pmin “Practice random acts of kindness and senseless acts of beauty.”
3. 3. Equations: 1. PVT Relationships Process 1-2 (T = C) P1 V2 = = re(T=C) isothermal expansion ratio P2 V1 Process 2-3 (S = C) 1 k-1 k-1 k-1 k T3 P3 V2 = = T2 P2 V3 1 1 k-1 k T2 P2 V3 = = = re(S=C) isentropic expansion ratio T3 P3 V2 Process 3-4 (T = C) V3 P4 = = rk(T=C) isothermal compression ratio V4 P3 Process 4-1 (S = C) 1 1 k-1 k T1 P1 V4 = = = rk(S=C) isentropic compression ratio T4 P4 V1 Since T2 = T1 and T4 = T3, Hence; V3 V4 = V2 V1 Isentropic expansion ratio = Isentropic compression ratio re(S=C) = rk(S=C) And; V3 V2 = V4 V1 re(T=C) = rk(T=C) Isothermal expansion ratio = Isothermal compression ratio 2. Heat Added, QA Process 1-2 (T = C) QA = +Q P1 P1 QA = mRT1 ln = mRT2 ln P2 P2 3. Heat Rejected, QR Process 3-4 (T = C) QR = -Q P3 P3 QR = mRT3 ln = mRT4 ln P4 P4 4. Net Work, WNET WNET = QA - -QR P1 WNET = mR (T1 - T4) ln P2 “Everything is created twice - first mentally, then physically.”
4. 4. 5. Carnot cycle thermal efficiency, e cc WNET e cc = Q * 100% A T1 - T4 e cc = * 100% T1 TH - TL e cc = * 100% TH 6. Mean effective pressure, Pm WNET Pm = VD e cc QA Pm = (V3 - V1)PROBLEM SET:1. Consider a three-process air standard power cycle in which process 1-2 is an isothermal compression, 2-3 is a constant- pressure heat addition, and 3-1 is an isentropic expansion. Given that P1 = 100 kPa, t1 = 20oC, and P2 = 600 kPa, determine: (a) The work and heat transfer for each process and the thermal efficiency of the cycle. [ ] (b) Show the cycle on Ts and Pv coordinates.2. Consider a three-process air-standard power cycle in which process 1-2 is an isentropic compression, 2-3 is a constant pressure heat addition, and 3-1 is a constant-volume heat rejection. Given that P1 = 100 kPa, T1 = 330 K, and P2 =800 kPa, determine: (a) The work and heat transfer for each process and the thermal efficiency of the cycle. [ ] (b) Show the cycle on Ts and Pv coordinates.3. A three-process cycle operating with nitrogen as the working substance has: constant temperature 1-2 (t1 = 40oC, P1 = 110 kPa); constant volume heating 2-3; and polytropic expansion 3-1 (n=1.35). The isothermal compression requires -67 kJ/kg of work. Determine: (a) P, T, and v around the cycle (b) The heat in and out [ ] (c) The net work [ ]4. Two and half kg of an ideal gas with R = 296.9 J/kg.K and cv = 0.7442 kJ/kg.K at a pressure of 827.4 kPa and a temperature of 677oC reject 132.2 kJ of heat at constant pressure. The gas is then expanded according to PV1.25 =C to a point where a constant volume process will bring the gas back to its original state. Determine P3, QA, and thepower in kJ. [ ]5. An air-standard Carnot cycle is executed in a closed system between the temperature limits of 350 and 1200 K. The pressures before and after the isothermal compression are 150 kPa and 300 kPa, respectively. If the net workoutput per cycle is 400 kJ, determine. (a) The maximum pressure in the cycle [ ] (b) The heat transferred to air [ ] (c) The mass of air [ ]6. Consider a three-process thermodynamic cycle which operates on 0.5 kg/s of air is composed of the following reversible processes: (1) constant volume heating process 1-2; (2) isentropic expansion process 2-3; (3) constant pressure heat rejection 3-1 Given that P1 = 345 kPa, t1 = 38oC and P2 = 4P1, determine: (a) the heat transfer for each process [ ] (b) the network [ ] (c) and the thermal efficiency [ ]7. A thermodynamic cycle is composed of the following processes: isothermal expansion 1-2; isometric 2-3; isentropic 3-1. The cycle runs on 0.125 kg of air and has an expansion ratio of 5.4. For P3 = 100 kPaa and t3 = 27oC, find (a) P,V and t for each point of the cycle. (b) Qin and Qout (c) WNET [ ] (d) ecyc [ ] (e) mean effective pressure [ ]8. The following thermodynamic cycle operating at 30 Hz is composed of the following reversible process; isothermal expansion 1-2; polytropic 2-3; isometric 3-1. The cycle uses 0.725 kg of air, for which P2 = 105 kPaa, t2 = 735oC,and t3 = 37oC, n = 0.75 for the polytropic process. Determine (a) the volume at each corner of the cycle (b) Qin and Qout (c)the power [ ] (d) the Pm.9. A Carnot engine rejects 1000 Btu/min at 50oF and produces 40 hp. Determine the temperature of heat addition and the amount of heat flow into the engine. [ ]10. A Carnot cycle uses air as the working substance. The heat supplied is 50 Btu. The temperature of the heat rejected is 70oF, and the isentropic compression ratio V4/V1 = 10. Determine (a) the cycle efficiency [ ] (b) the temperature ofheat added [ ] (c) the work [ ] “Either you run the day or the day runs you.”