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Applications of thermodynamics

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Soumith V
Soumith V

vapour power cycles, IC engines, Gas turbines Notes for GATE

Engineering
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APPLICATIONS OF THERMODYNAMICS Gate, IES Notes SouMyth
Contents Vapour power cycles...........................................................................................................................................................................................2 Specific steam consumption ..................................................................................................................................................................2 Significance of SSC.....................................................................................................................................................................................2 Work Ratio....................................................................................................................................................................................................2 Back work Ratio .........................................................................................................................................................................................2 Carnot Vapour power cycles ......................................................................................................................................................................3 Drawbacks of Carnot vapour power cycles.....................................................................................................................................3 Rankine cycle ...............................................................................................................................................................................................3 Analysis of the Cycles................................................................................................................................................................................4 Methods of improving performance of simple rankine cycle .......................................................................................................5 Increasing Boiler pressure .....................................................................................................................................................................5 Super Heating..............................................................................................................................................................................................5 Reheating.......................................................................................................................................................................................................6 Regeneration................................................................................................................................................................................................6 Gas power cycles...................................................................................................................................................................................................8 Otto Cycle.......................................................................................................................................................................................................8 Advantages of I.C engines.......................................................................................................................................................................8 Engine nomenclature....................................................................................................................................................................................8 Air Standard Cycles..........................................................................................................................................................................................10 Otto Cycle.........................................................................................................................................................................................................10 Diesel cycle......................................................................................................................................................................................................11 Comparison of Otto and diesel cycles .............................................................................................................................................12 Dual cycle.........................................................................................................................................................................................................13 Engine Performance Parameters..........................................................................................................................................................13 Gas turbines.........................................................................................................................................................................................................15 Gas turbine cycles....................................................................................................................................................................................15 Analysis of a simple gas turbine cycle............................................................................................................................................16 Efficiency of simple gas turbine cycle.............................................................................................................................................17 Methods of improving the performance of gas turbine cycle...............................................................................................18
Vapour power cycles Reasons for using water as working fluid 1. It’s cheap 2. It’s chemically stable 3. It’s not toxic Selection of a power plant Efforts to improve efficiency and thereby reducing the running cost or operating cost may be desirable, but this would lead to increase initial cist and hence efforts must be taken to optimize total cost. Specific steam consumption It is steam consumed for producing unit output of power. β†’ 𝑆𝑆𝐢 = π‘šΜ‡ 𝑠 𝑃 𝑁𝑒𝑑 = π‘šΜ‡ 𝑠 π‘šΜ‡ 𝑠 Γ— π‘Šπ‘π‘’π‘‘ = 1 π‘Šπ‘π‘’π‘‘ π‘˜π‘” 𝐾𝐽 β†’ 𝑆𝑆𝐢 = 3600 π‘Šπ‘π‘’π‘‘ π‘˜π‘” πΎπ‘Š β„Žπ‘Ÿ Significance of SSC SSC indicates the size of plant, smaller the SSC larger is the net work and hence for developing given power, mass flow rate of steam must be less. Smaller the SSC, lesser I the size of plant and hence such plants are preferable. Work Ratio It is the ratio of net work to the positive work. β†’ 𝛾 𝑀 = π‘Šπ‘π‘’π‘‘ π‘Š+𝑣𝑒 β†’ 𝛾 𝑀 = π‘Š+𝑣𝑒 βˆ’ π‘Šβˆ’π‘£π‘’ π‘Š+𝑣𝑒 = 1 βˆ’ π‘Šβˆ’π‘£π‘’ π‘Š+𝑣𝑒 β†’ 𝛾 𝑀 π‘™π‘–π‘žπ‘’π‘–π‘‘ = 1 βˆ’ π‘Šπ‘π‘’π‘šπ‘ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ β†’ 𝛾 𝑀 π‘”π‘Žπ‘  = 1 βˆ’ π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ Work done by compressor is much higher than pump, so Work ratio for liquid power cycles is greater than gas power cycles. Power plants with high work ratios are preferable. The work ratio is highest for rankine cycles among all other cycles, this is because of less work done by pump in rankine cycle. In case of gas turbine power plants, the work ratio is about 0.4 – 0.6 i.e., in gas turbine power plants compressors consumes 40 – 60% of turbine work. In Rankine cycle, work ratio is about 0.96 – 0.98, i.e., I rankine cycle, pump consumes 2-4 % of turbine work. Back work Ratio It’s the ratio of negative work to positive work. β†’ 𝛾 𝑏𝑀 = π‘Šβˆ’π‘£π‘’ π‘Š+𝑣𝑒 β†’ 𝛾 𝑀 = 1 βˆ’ 𝛾 𝑏𝑀
Carnot Vapour power cycles Drawbacks of Carnot vapour power cycles 1. Saturated vapour which is entering turbine at 1 leaves ar 2, which is in wet region. The liquid which is present at state 2 may damage turbine blades due to high velocity. 2. It is difficult to design a condenser that stops suddenly at point 3. 3. It’s difficult to design a compressor which handles both liquid and vapour. 4. As Carnot vapour cycle uses compressor, the compressor work is large and hence net work is less. β†’ πœ‚ = 𝑄 𝑆 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑄 𝑅 𝑄 𝑆 β†’ πœ‚ = 1 βˆ’ 𝑇𝑙 βˆ™ 𝑑𝑆 π‘‡β„Ž βˆ™ 𝑑𝑆 β†’ πœ‚ = 1 βˆ’ 𝑇𝑙 π‘‡β„Ž Rankine cycle β†’ πœ‚ = π‘Šπ‘π‘’π‘‘ 𝑄𝑠 = π‘Šπ‘‡ βˆ’ π‘Šπ‘ƒ 𝑄𝑠 1 βˆ’ 2 ⟹ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π‘Žπ‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ 𝑒π‘₯π‘π‘Žπ‘›π‘ π‘–π‘œπ‘› (π‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’) 2 βˆ’ 3 ⟹ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘π‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘–π‘œπ‘› (πΆπ‘œπ‘›π‘‘π‘’π‘›π‘ π‘œπ‘Ÿ) 3 βˆ’ 4 ⟹ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› (π‘ƒπ‘’π‘šπ‘) 4 βˆ’ 1 ⟹ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘π‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ β„Žπ‘’π‘Žπ‘‘ 𝑠𝑒𝑝𝑝𝑙𝑦 (π΅π‘œπ‘–π‘™π‘’π‘Ÿ)
Analysis of the Cycles Assumptions 1. Each device is treated as steady flow device. 2. Kinetic and potential energy changes are neglected. β†’ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ = β„Ž1 βˆ’ β„Ž2 β†’ 𝑄 π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘’π‘‘ = β„Ž2 βˆ’ β„Ž3 β†’ π‘Šπ‘π‘’π‘šπ‘ = β„Ž4 βˆ’ β„Ž3 We know that open system work is -Vβˆ™ dP, this equation is applicable when the flow is steady, K.E and P.E changes are neglected and when the process is reversible. Steady flow equations can be applied to reversible and irreversible processes. If the pumping process is reversible, then work obtained from Steady flow energy equation and work equal to -Vβˆ™dP can be equated. β†’ π‘Šπ‘π‘’π‘šπ‘ = β„Ž3 βˆ’ β„Ž4 = βˆ’π‘‰ βˆ™ 𝑑𝑃 β†’ 𝑄𝑠𝑒𝑝𝑝𝑙𝑖𝑒𝑑 = β„Ž1 βˆ’ β„Ž4 β†’ πœ‚ = π‘Šπ‘‡ βˆ’ π‘Šπ‘ƒ 𝑄𝑠 = (β„Ž1 βˆ’ β„Ž2) βˆ’ (β„Ž4 βˆ’ β„Ž3) β„Ž1 βˆ’ β„Ž4 Reason for higher efficiency of Carnot cycle compared to rankine cycle 1 βˆ’ 2 βˆ’ 3 βˆ’ 4 ⟹ π‘…π‘Žπ‘›π‘˜π‘–π‘›π‘’ 𝑐𝑦𝑐𝑙𝑒 1 βˆ’ 2 βˆ’ 3β€² βˆ’ 4β€² ⟹ πΆπ‘Žπ‘Ÿπ‘›π‘œπ‘‘ 𝑐𝑦𝑐𝑙𝑒 β†’ πœ‚ π‘π‘Žπ‘Ÿπ‘›π‘œπ‘‘ = 1 βˆ’ π‘‡β„Ž 𝑇𝑙 β†’ πœ‚ = 𝑓(𝑇 π‘š) β†’ 𝑄𝑠 = β„Ž1 βˆ’ β„Ž4 = 𝑇 π‘š βˆ™ 𝑑𝑆 ⟹ 𝑇 π‘š = β„Ž1 βˆ’ β„Ž4 𝑑𝑆 β†’ πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑇𝑙 βˆ™ 𝑑𝑆 𝑇 π‘š βˆ™ 𝑑𝑆 β†’ πœ‚ π‘Ÿπ‘Žπ‘›π‘˜π‘–π‘›π‘’ = 1 βˆ’ 𝑇𝑙 𝑇 π‘š In Carnot cycle, complete heat is added at single fixed temperature (Th), where as in rankine cycle, some heat is added at a temperature lower than Th and remaining heat is added at Th. therefore mean temperature of heat addition is less in rankine cycle, compared to Carnot cycle therefore the efficiency of rankine cycle is less than Carnot cycle. If the mean temperature of heat addition will be more, the efficiency will be more.
Parameter Carnot Rankine Tm more less Ξ· more less WNet less more SSC more less Size of Plant large small Ξ³w less more Methods of improving performance of simple rankine cycle Increasing Boiler pressure Effects of increasing boiler pressure 1. Increase in turbine work 2. Increase in pump work 3. Increase in net work 4. Decrease in SSC 5. Decrease in heat rejection 6. Decrease in heat supply 7. Increase in mean temperature of heat addition 8. Increase in efficiency 9. Decrease in dryness fraction at the turbine exit. Note- Due to decrease in dryness fraction at the turbine exit, the boiler pressure is limited. Super Heating
Effects of Superheating 1. Increase in Turbine work 2. No change in pump work 3. Increase in Net work 4. Decrease in SSC 5. Increase in mean temperature of heat addition 6. Increase in efficiency 7. Increase in heat supply 8. Increase in heat rejection 9. Increase in dryness fraction at turbine exit Note There is a limit to superheating because of metallurgical conditions i.e., steam turbine blades can resist up to 620⁰C. Reheating Effects of Reheating 1. Increase in dryness fraction 2. Increase in turbine work 3. Increase in net work 4. Decrease in SSC. 5. Efficiency may increase or decrease. Reheating in steam turbine may increase or decrease he efficiency if steam is expanded deeply in first turbine and then reheated, the mean temperature may decrease hence the efficiency decreases. In most of the practically reheat cycles, the steam is expanded slightly in first turbine and then reheated, this increases the mean temperature of heat addition and also the efficiency of the cycle. πœ‚ = (β„Ž1 βˆ’ β„Ž2) + (β„Ž3 βˆ’ β„Ž4) βˆ’ (β„Ž6 βˆ’ β„Ž5) (β„Ž1 βˆ’ β„Ž6) + (β„Ž3 βˆ’ β„Ž2) Regeneration
Effects of regeneration 1. Decrease in turbine work due to decrease in mass. 2. Reduction in condenser load 3. Reduction in heat supply 4. Increase in mean temperature of heat addition 5. Increase in efficiency π‘Šπ‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’ = (β„Ž1 βˆ’ β„Ž2) + (1 βˆ’ 𝑦) βˆ™ (β„Ž2 βˆ’ β„Ž3) π‘Šπ‘π‘’π‘šπ‘ = (1 βˆ’ 𝑦) βˆ™ (β„Ž5 βˆ’ β„Ž4) + (β„Ž7 βˆ’ β„Ž6) π‘Šπ‘π‘’π‘‘ = π‘Šπ‘‡ βˆ’ π‘Šπ‘ 𝑄𝑠 = β„Ž1 βˆ’ β„Ž7 𝑄 𝑅 = (1 βˆ’ 𝑦) βˆ™ (β„Ž3 βˆ’ β„Ž4) πœ‚ = π‘Šπ‘π‘’π‘‘ 𝑄 𝑆 𝑦 βˆ™ β„Ž2 + (1 βˆ’ 𝑦) βˆ™ β„Ž5 = β„Ž6 Ideal regenerative Rankine cycle leads to Carnot cycle. Need of Condenser in steam power plants If no condenser is used and if steam is allowed to expand to atmospheric pressure. The saturation temperature at 1 bar is 100⁰C, therefore of no condenser is used, steam leaves the turbine at higher temperature and this energy becomes useless. By using condenser steam can be allowed to expand to lower temperature and thereby more turbine work can be obtained, this results in higher efficiency. Therefore, condensers are used to increase turbine work and efficiency. Isentropic or adiabatic efficiency of turbine It is ratio of actual work to isentropic work β†’ π‘Šπ‘Žπ‘π‘‘π‘’π‘Žπ‘™ = β„Ž1 βˆ’ β„Ž2 β†’ π‘Šπ‘–π‘ π‘’π‘› = β„Ž1 βˆ’ β„Ž2𝑠 β†’ πœ‚ 𝑇 = β„Ž1 βˆ’ β„Ž2 β„Ž1 βˆ’ β„Ž2𝑠 Isentropic efficiency of a pump or compressor It is ratio of actual work to isentropic work β†’ π‘Šπ‘Žπ‘π‘‘π‘’π‘Žπ‘™ = β„Ž2 βˆ’ β„Ž1 β†’ π‘Šπ‘–π‘ π‘’π‘› = β„Ž2𝑠 βˆ’ β„Ž1 β†’ πœ‚ 𝑇 = β„Ž2𝑠 βˆ’ β„Ž1 β„Ž2 βˆ’ β„Ž1
Gas power cycles Otto Cycle Engine is a device which converts one form of energy into other useful form. Based on combustion, engines are classified into 1. Internal combustion engines (I.C engines) 2. External combustion engines (E.C engines) In I.C engines, burning or combustion occurs in cylinder and power is also developed in same cylinder. In E.C engines, heat is transferred from products of combustion to working fluid. Heat engine is a device which converts chemical energy of fuel into heat energy and subsequently this heat energy is converted into mechanical power. Advantages of I.C engines 1. Mechanical simplicity 2. Low initial cost due to absence of boiler, turbine, condenser etc... 3. High efficiency 4. High power to weight ratio Engine nomenclature Top dead centre It’s the dead centre when the piston is farthest from crank shaft. In case of horizontal engines TDC is known as Inner dead centre. Bottom dead centre It’s the dead centre when the piston is nearest to crank shaft. In case of horizontal engines, it is known as outer dead centre. Stroke or stroke length Distance between TDC and BDC is known as stroke length. Displacement volume or stroke volume or swept volume It’s the volume swept by a piston 𝑉𝑠 = πœ‹ 4 𝐷2 𝐿 βˆ™ π‘˜ 𝐷 β†’ πΌπ‘›π‘›π‘’π‘Ÿ π‘‘π‘–π‘Žπ‘šπ‘’π‘‘π‘’π‘Ÿ π‘œπ‘“ π‘π‘¦π‘™π‘–π‘›π‘‘π‘’π‘Ÿ 𝐿 β†’ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’ π‘™π‘’π‘›π‘”π‘‘β„Ž π‘˜ β†’ π‘›π‘œ. π‘œπ‘“ π‘π‘¦π‘™π‘–π‘›π‘‘π‘’π‘Ÿπ‘  Clearance volume It’s the volume of cylinder when piston is at top dead centre or Inner Dead centre. Clearance volume is provided to accommodate valves and to prevent damage to valves.
Compression ratio (Ξ³) It’s defined as ratio of volume before compression to volume after compression. 𝛾 = π‘‰π‘œπ‘™π‘’π‘šπ‘’ π‘π‘’π‘“π‘œπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› π‘£π‘œπ‘™π‘’π‘šπ‘’ π‘Žπ‘“π‘‘π‘’π‘Ÿ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› 𝛾 = 𝑉1 𝑉2 = 𝑉𝑐 + 𝑉𝑠 𝑉𝑠 = 1 + 𝑉𝑐 𝑉𝑠 𝛾 = 1 + 𝑉𝑐 𝑉𝑠 Mean Piston Speed In one stroke the shaft completes half revolution (180⁰C). therefore, piston covers 2L in 1 revolution. β†’ π‘€π‘’π‘Žπ‘› π‘π‘–π‘ π‘‘π‘œπ‘› 𝑠𝑝𝑒𝑒𝑑 = 2𝐿𝑁 60
Air Standard Cycles Assumptions 1. Working fluid is air and it’s treated as Ideal gas 2. Specific heat cp, cv are assumed as constants. 3. Working fluids is of fixed mass (liquid system analysis) 4. Working fluid doesn’t undergo any chemical reaction that is of constant chemical composition throughout cycle. 5. All processes are assumed to be reversible (internally reversible). Generally, cp & cv of air are taken as 25⁰C⁰this analysis is known as cold air cycle analysis. Otto Cycle 1 βˆ’ 2 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› 2 βˆ’ 3 β†’ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘£π‘œπ‘™π‘’π‘šπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Žπ‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› 3 βˆ’ 4 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ 𝑒π‘₯π‘π‘Žπ‘›π‘ π‘–π‘œπ‘› 4 βˆ’ 1 β†’ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘£π‘œπ‘™π‘’π‘šπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘–π‘œπ‘› β†’ 𝛾 = 𝑉1 𝑉2 β†’ πœ‚ = 1 βˆ’ 𝑄 π‘Ÿ 𝑄𝑠 β†’ 𝑄23 = 𝑄𝑠 = π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇3 βˆ’ 𝑇2) β†’ 𝑄41 = 𝑄 π‘Ÿ = π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) β†’ πœ‚ = 1 βˆ’ π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ (𝑇4 βˆ’ 𝑇1) (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 𝑇1 ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 ( 𝑇3 𝑇2 βˆ’ 1) ( 𝑇2 𝑇1 = π‘Ÿ π›Ύβˆ’1 = 𝑇3 𝑇4 ⟹ 𝑇4 𝑇1 = 𝑇3 𝑇2 ) πœ‚ = 1 βˆ’ 𝑇1 𝑇2 = 1 βˆ’ 1 π‘Ÿ π›Ύβˆ’1 Significance of compression ratio Compression ratio is an indicator of efficiency of engine, greater the compression ratio greater is the efficiency. This is because if compression ratio is high there will be more scope for expansion and net-work will be more. Therefore, for a given heat supply if compression ratio is more, the efficiency will be more. πœ‚ ∝ π‘Ÿ (π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘–π‘œ)
Mean effective pressure It’s hypothetical (imaginary) pressure which gives same net-work as that of actual cycle for swept volume (same size of engine). β†’ π‘Šπ‘π‘’π‘‘ = π‘ƒπ‘š βˆ™ 𝑉𝑠 β†’ 𝑃 𝑀 = π‘Šπ‘π‘’π‘‘ 𝑉𝑠 The mean effective pressure of Otto cycle can be expressed as π‘ƒπ‘š = πœ‚ π‘‘β„Ž Β· π›₯𝑃 (𝛾 βˆ’ 1) Β· (π‘Ÿ βˆ’ 1) Significance of Pm This term is used for comparing different engines of same size. That is engines with higher mean effective pressure have more net-work and for a given heat input greater the mean effective pressure, greater is efficiency. Diesel cycle 1 βˆ’ 2 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› 2 βˆ’ 3 β†’ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘π‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Žπ‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› 3 βˆ’ 4 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ 𝑒π‘₯π‘π‘Žπ‘›π‘–π‘ π‘œπ‘› 4 βˆ’ 1 β†’ πΆπ‘œπ‘ π‘›π‘‘π‘Žπ‘›π‘‘ π‘‰π‘œπ‘™π‘’π‘šπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘–π‘œπ‘› β†’ πΆπ‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘› (π‘Ÿ) = 𝑉1 𝑉2 β†’ πΆπ‘’π‘‘π‘œπ‘“π‘“ π‘Ÿπ‘Žπ‘‘π‘–π‘œ (π‘Ÿπ‘) = 𝑉3 𝑉2 = 𝑇3 𝑇2 β†’ 𝐸π‘₯π‘π‘Žπ‘›π‘–π‘ π‘–π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘–π‘œ (π‘Ÿπ‘’) = 𝑉4 𝑉3 = 𝑉1 𝑉3 β†’ π‘Ÿπ‘ βˆ™ π‘Ÿπ‘’ = π‘Ÿ β†’ πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = 𝑄 𝑁𝑒𝑑 𝑄𝑠 = 𝑄𝑠 βˆ’ 𝑄 π‘Ÿ 𝑄𝑠 = 1 βˆ’ 𝑄 π‘Ÿ 𝑄𝑠 β†’ 𝑄23 = 𝑄𝑠 = π‘š βˆ™ 𝑐 𝑝 βˆ™ (𝑇3 βˆ’ 𝑇2) β†’ 𝑄41 = 𝑄 π‘Ÿ = π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1)
β†’ πœ‚ = 1 βˆ’ π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) π‘š βˆ™ 𝑐 𝑝 βˆ™ (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) 𝑐 𝑝 βˆ™ (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 1 𝛾 βˆ™ 𝑇1 ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 ( 𝑇3 𝑇2 βˆ’ 1) ( 𝑇2 𝑇1 = π‘Ÿ π›Ύβˆ’1 ) ( 𝑉3 𝑉2 = 𝑇3 𝑇2 = π‘Ÿπ‘) ( 𝑇4 𝑇1 = π‘Ÿπ‘ 𝛾 ) β†’ πœ‚ = 1 βˆ’ 1 𝛾 βˆ™ 1 π‘Ÿ π›Ύβˆ’1 βˆ™ π‘Ÿπ‘ 𝛾 βˆ’ 1 π‘Ÿπ‘ βˆ’ 1 Efficiency of a diesel cycle depends on compression ratio and cut-off ratio. Efficiency of a diesel cycle increases with increase in compression ratio. Effect of cut-off ratio (rc) on efficiency of diesel cycle With increase in cut off ratio, both heat supply and heat rejection increases, but the increase in heat rejection is more, because heat rejection occurs at constant volume and heat supply occurs ar constant pressure and the slope of constant volume is greater than slope of constant pressure curve. Comparison of Otto and diesel cycles Case 1- Same compression ratio and heat supply πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄 𝑆 For the same compression ratio and heat supply, rejection is more in diesel cycle and hence efficiency of Otto cycle is more than efficiency of diesel cycle. π‘‚π‘‘π‘‘π‘œ β†’ 𝑄𝑠 = 𝑄 𝑉 = π‘‘π‘ˆ = π‘š Β· 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) 𝐷𝑖𝑒𝑠𝑒𝑙 β†’ 𝑄𝑠 = 𝑄 𝑃 = 𝑑𝐻 = π‘š Β· 𝑐 𝑝 Β· (𝑇3 β€² βˆ’ 𝑇2) 𝑄𝑠 𝑢𝒕𝒕𝒐 = 𝑄𝑠 π‘«π’Šπ’†π’”π’†π’ ⟹ π‘š Β· 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) = π‘š Β· 𝑐 𝑝 Β· (𝑇3 β€² βˆ’ 𝑇2) 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) = 𝑐 𝑝 Β· (𝑇3 β€² βˆ’ 𝑇2) ⟹ 𝑇3 βˆ’ 𝑇2 = 𝑇3 β€² βˆ’ 𝑇2 𝑻 πŸ‘ > 𝑻 πŸ‘ β€² Case 2 – Same compression ratio and heat rejection πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 For the same compression ratio and heat rejection, heat supply is more in Otto cycle and hence its efficiency is more. πœ‚ 𝑑𝑖𝑒𝑠𝑒𝑙 > πœ‚ π‘‚π‘‘π‘‘π‘œ Case 3 – Same maximum temperature and heat rejection For same maximum temperature and heat rejection, heat supplied is more in diesel cycle and hence it’s more efficient. In this case compression ratio of diesel cycle is more than compression ratio of Otto cycle. πœ‚ 𝑑𝑖𝑒𝑠𝑒𝑙 > πœ‚ π‘‚π‘‘π‘‘π‘œ 1-2-3-4 β†’ Otto cycle 1-2-3’-4’ β†’ Diesel cycle
Dual cycle In actual engines, heat is neither added at constant volume or at constant pressure. A dual cycle is developed and it has features of both Otto and Diesel cycle. 𝛾 = 𝑉1 𝑉2 π‘Ÿπ‘ = 𝑉4 𝑉3 π‘Ÿπ‘’ = 𝑉5 𝑉4 πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 𝑄𝑠 = 𝑄𝑠1 + 𝑄𝑠2 𝑄𝑠1 = π‘š Β· 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) ; 𝑄𝑠2 = π‘š Β· 𝑐 𝑝 Β· (𝑇5 βˆ’ 𝑇1) ; 𝑄 𝑅 = π‘š Β· 𝑐 𝑣 Β· (𝑇5 βˆ’ 𝑇1) As dual cycle has features of both Otto and diesel cycle, its efficiency lies between Otto and diesel cycle. Engine Performance Parameters Indicated power (IP) The power developed inside the cylinder is known as indicated power. In case of 2 stroke engine, 1 cycle is completed in 2 strokes of piston or 1 revolution of crank shaft. Similarly, in 4 stroke engine, 1 cycle is completed in 4 strokes or 2 revolutions of shaft. Theoretically power developed in 2 stroke engines is twice that of 4 stroke engines. Indicated power of 2 stroke engine π‘Šπ‘›π‘’π‘‘ = π‘ƒπ‘š Β· 𝑉𝑠 β§Έ 𝑐𝑦𝑐𝑙𝑒 π‘Šπ‘›π‘’π‘‘ = π‘ƒπ‘š Β· 𝑉𝑠 β§Έ π‘Ÿπ‘’π‘£π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘œπ‘€π‘’π‘Ÿ = 𝑷 π’Žπ’†π’‚π’ Β· 𝑽 𝑺 π‘Ÿπ‘’π‘£π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘› Γ— 𝑡 π‘Ÿπ‘’π‘£π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘›π‘  𝑠𝑒𝑐 (Pmβ†’indicated mean effective pressure) πΌπ‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘‘ π‘π‘œπ‘€π‘’π‘Ÿ (𝑰. 𝑷) = π‘ƒπ‘š Β· 𝑉𝑆 Β· 𝑁 60 = π‘ƒπ‘š Β· (𝐴 Β· 𝐿) Β· 𝑁 60 If there are K cylinders, then indicated power is equal to ⟹ π‘ƒπ‘š Β· ( 𝐴 Β· 𝐿) Β· 𝑁 Β· 𝑲 60 In case of 4 – stroke engine ⟹ 𝐼. 𝑃 = π‘ƒπ‘š Β· ( 𝐴 Β· 𝐿) Β· 𝑁 Β· 𝑲 60 Γ— 2 Brake power (B.P) The power available at the output shaft is known as brake power. π΅π‘Ÿπ‘Žπ‘˜π‘’ π‘ƒπ‘œπ‘€π‘’π‘Ÿ (𝑩. 𝑷) = 𝑇 Γ— πœ” 𝐡. 𝑃 = 𝑇 Γ— 2πœ‹π‘ 60 β†’ applicable for 2 or 4 stroke and any number of cylinders Frictional Power (F.P) The difference between indicated power and brake power is called frictional power. 𝐹. 𝑃 = 𝐼. 𝑃 βˆ’ 𝐡. 𝑃
Mechanical efficiency πœ‚ π‘šπ‘’π‘β„Ž = 𝐡. 𝑃 𝐼. 𝑃 Indicated thermal efficiency (Ξ·ith) πœ‚π‘–π‘‘β„Ž = 𝐼. 𝑃 π‘šΜ‡ 𝑓 Γ— 𝐢. 𝑉 π‘šΜ‡ 𝑓 β†’ π‘šπ‘Žπ‘ π‘  π‘“π‘™π‘œπ‘€ π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ 𝑓𝑒𝑒𝑙 𝐢. π‘‰βˆ’. πΆπ‘Žπ‘™π‘œπ‘Ÿπ‘–π‘“π‘–π‘ π‘£π‘Žπ‘™π‘’π‘’ Brake thermal efficiency (Ξ·bth) πœ‚ π‘π‘‘β„Ž = 𝐡. 𝑃 π‘šΜ‡ 𝑓 Γ— 𝐢. 𝑉 Indicated specific fuel consumption (isfc) 𝑖𝑠𝑓𝑐 = π‘šΜ‡ 𝑓 𝐼. 𝑃 ( π‘˜π‘” πΎπ‘Š Β· β„Žπ‘Ÿ ) Brake specific fuel consumption (bsfc) 𝑏𝑠𝑓𝑐 = π‘šΜ‡ 𝑓 𝐡. 𝑃 ( π‘˜π‘” πΎπ‘Š Β· β„Žπ‘Ÿ ) Volumetric efficiency (Ξ·vol) It represents the breathing capacity of the cylinder, it’s the ratio of actual volume of fluid entering the cylinder per cycle to the swept volume. πœ‚ π‘£π‘œπ‘™ = π΄π‘π‘‘π‘’π‘Žπ‘™ π‘‰π‘œπ‘™π‘’π‘šπ‘’ π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘›π‘” πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 = π‘šΜ‡ 𝑓 Β· 𝜐 πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 β†’ 2 βˆ’ π‘ π‘‘π‘Ÿπ‘œπ‘˜π‘’ πœ‚ π‘£π‘œπ‘™ = π΄π‘π‘‘π‘’π‘Žπ‘™ π‘‰π‘œπ‘™π‘’π‘šπ‘’ π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘›π‘” πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 2 = π‘šΜ‡ 𝑓 Β· 𝜐 πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 2 β†’ 4 βˆ’ π‘ π‘‘π‘Ÿπ‘œπ‘˜π‘’
Gas turbines Advantages of Gas turbines 1. Simple mechanism 2. Compact 3. Easy balancing 4. High speeds can be obtained Disadvantages 1. As gas turbines are rotary devices, they are subjected to higher temperatures continuously, hence costly material is required. 2. As compressor is used it consumes more work and hence the Net-work is less. 3. As speeds are high, reduction gear mechanisms must be used. Gas turbine cycles Gas turbine cycles work on Brayton or Joule cycle. Open cycle gas turbine Open cycle gas turbine is a mechanical device. Gas turbines are used in power generation, jet propulsion and aircraft engines. Closed cycle gas turbines 1-2 β†’ reversible adiabatic Compression 2-3 β†’ constant pressure heat addition 3-4 β†’ reversible adiabatic expansion 1-2 β†’ reversible adiabatic Compression 2-3 β†’ constant pressure heat addition 3-4 β†’ reversible adiabatic expansion 4-1 β†’ constant pressure heat rejection
Advantages of closed cycle gas turbines over open cycle gas turbines 1. It is compact. 2. Any working fluid can be used in closed cycle gas turbines but in open gas turbines, air is required for combustion. 3. As products of combustion do not enter turbine blades, any cheaper fuel can be used. Generally, in closed cycle gas turbines, working fluid with higher Ξ³ (higher mono-atomic gases) can be used leading to higher efficiency. Disadvantages 1. Additional heat exchanger is required. 2. Coolant is required in closed cycle gas turbine, where as in open cycle gas turbine atmosphere acts as a sink. Analysis of a simple gas turbine cycle Assumptions 1. Working fluid can be treated as ideal gas. 2. Compression and expansion are treated as steady flow device. 3. Each device is treated as steady flow device. 4. Kinetic and potential energy changes are neglected. 5. cp & cv doesn’t change with temperature. 6. Working fluid is same throughout the cycle. 1β€” 2 ⟢ Reversible Adiabatic β„Ž1 + 𝐢1 2 2 + 𝑔𝑧1 + π‘žβƒ₯ = β„Ž2 + 𝐢2 2 2 + 𝑔𝑧2 + 𝑀 β„Ž1 = β„Ž2 + 𝑀 β†’ βˆ’π‘€ = β„Ž2 βˆ’ β„Ž1 β†’ π’˜ π’„π’π’Žπ’‘π’“π’†π’”π’”π’π’“ = 𝒉 𝟐 βˆ’ 𝒉 𝟏 = 𝒄 𝒑 Β· (𝑻 𝟐 βˆ’ 𝑻 𝟏) 2β€”3 ⟢ Constant pressure heat rejection β„Ž2 + 𝐢2 2 2 + 𝑔𝑧2 + π‘žβƒ₯ = β„Ž3 + 𝐢3 2 2 + 𝑔𝑧3 + 𝑀 β„Ž2 + π‘žβƒ₯ = β„Ž3 β†’ π‘žβƒ₯ = β„Ž3 βˆ’ β„Ž2 β†’ 𝒒 π’”π’–π’‘π’‘π’π’š = 𝒉 πŸ‘ βˆ’ 𝒉 𝟐 = 𝒄 𝒑 Β· (𝑻 πŸ‘ βˆ’ 𝑻 𝟐) 3β€”4 ⟢ Reversible adiabatic expansion (turbine) β„Ž3 + 𝐢3 2 2 + 𝑔𝑧3 + π‘žβƒ₯ = β„Ž4 + 𝐢4 2 2 + 𝑔𝑧4 + 𝑀 β„Ž3 = β„Ž4 + 𝑀 β†’ 𝑀 = β„Ž3 βˆ’ β„Ž4 β†’ π’˜ π’•π’–π’“π’ƒπ’Šπ’π’† = 𝒉 πŸ‘ βˆ’ 𝒉 πŸ’ = 𝒄 𝒑 Β· (𝑻 πŸ‘ βˆ’ 𝑻 πŸ’) 4β€”1 ⟢ Constant pressure heat rejection β„Ž4 + 𝐢4 2 2 + 𝑔𝑧4 + π‘žβƒ₯ = β„Ž1 + 𝐢1 2 2 + 𝑔𝑧1 + 𝑀 β„Ž4 + π‘žβƒ₯ = β„Ž1 β†’ π‘žβƒ₯ = β„Ž1 βˆ’ β„Ž4 β†’ 𝒒 𝒓𝒆𝒋𝒆𝒄𝒕𝒆𝒅 = 𝒉 𝟏 βˆ’ 𝒉 πŸ’ = 𝒄 𝒑 Β· (𝑻 𝟏 βˆ’ 𝑻 πŸ’)
Efficiency of simple gas turbine cycle πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = 𝑄𝑠 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑐 𝑝 Β· (𝑇4 βˆ’ 𝑇1) 𝑐 𝑝 Β· (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 𝑇4 βˆ’ 𝑇1 𝑇3 βˆ’ 𝑇2 πœ‚ = 1 βˆ’ 𝑇1 Β· ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 Β· ( 𝑇3 𝑇2 βˆ’ 1) 1 β†’ 2 (reversible Adiabatic) ⟹ 𝑇2 𝑇1 = ( 𝑃2 𝑃1 ) π›Ύβˆ’1 𝛾⁄ = π‘Ÿπ‘ π›Ύβˆ’1 𝛾⁄ 3 β†’ 4 (reversible Adiabatic) ⟹ 𝑇3 𝑇4 = ( 𝑃3 𝑃4 ) π›Ύβˆ’1 𝛾⁄ = ( 𝑃2 𝑃1 ) π›Ύβˆ’1 𝛾⁄ = π‘Ÿπ‘ π›Ύβˆ’1 𝛾⁄ (𝑃2 = 𝑃3, 𝑃4 = 𝑃1) From above equations, 𝑇2 𝑇1 = 𝑇3 𝑇4 ⟹ 𝑻 πŸ’ 𝑻 𝟏 = 𝑻 πŸ‘ 𝑻 𝟐 πœ‚ = 1 βˆ’ 𝑇1 Β· ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 Β· ( 𝑇3 𝑇2 βˆ’ 1) ⟹ 𝜼 = 𝟏 βˆ’ 𝑻 𝟏 𝑻 𝟐 = 𝟏 βˆ’ 𝟏 𝒓 𝒑 πœΈβˆ’πŸ πœΈβ„ The efficiency of simple gas turbine cycle depends on pressure ratio. With increase in pressure ratio, efficiency increases because greater the pressure ratio, greater is the compression ratio and more is the scope for free expansion. Hence, for a given input, efficiency increases with increase in pressure ratio. With increase in pressure ratio, though the efficiency increases, at higher pressure ratios net-work decreases and hence there is a need to calculate a pressure ratio where the work output is maximum. At this pressure ratio, net-work is minimum is known as Optimum pressure ratio. π‘Šπ‘›π‘’π‘‘ = π‘Šπ‘‘ βˆ’ π‘Šπ‘ = 𝑐 𝑝 Β· (𝑇3 βˆ’ 𝑇4) βˆ’ 𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇1) = 𝑐 𝑝(𝑇3 βˆ’ 𝑇4 βˆ’ 𝑇2 + 𝑇1) 𝑇2 𝑇1 = 𝑇3 𝑇4 ⟹ 𝑻 πŸ’ = 𝑻 πŸ‘ Β· 𝑻 𝟏 𝑻 𝟐 π‘Šπ‘›π‘’π‘‘ = 𝑐 𝑝 Β· [𝑇3 βˆ’ 𝑇3 Β· 𝑇1 𝑇2 βˆ’ 𝑇2 + 𝑇1] πΉπ‘œπ‘Ÿ π‘šπ‘Žπ‘₯π‘–π‘šπ‘’π‘š π‘Šπ‘›π‘’π‘‘, 𝒅𝑾 𝒏𝒆𝒕 𝒅𝑻 𝟐 = 𝟎 π‘‘π‘Šπ‘›π‘’π‘‘ 𝑑𝑇2 = 𝑐 𝑝 Β· [0 + 𝑇3 Β· 𝑇1 𝑇2 2 βˆ’ 1 + 0] = 0 ⟹ 𝑇2 2 = 𝑇3 Β· 𝑇1 ⟹ 𝑻 𝟐 = βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 𝑇4 = 𝑇3 Β· 𝑇1 𝑇2 ⟹ 𝑇4 = 𝑇3 Β· 𝑇1 βˆšπ‘‡3 Β· 𝑇1 ⟹ 𝑻 πŸ’ = βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 𝑻 πŸ’ = 𝑻 𝟐 = βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 (rp)A < (rp)B < (rp)C Ξ·A < Ξ·B < Ξ·C
π‘Ÿπ‘ = 𝑃2 𝑃1 = ( 𝑇2 𝑇1 ) 𝛾 π›Ύβˆ’1 ⟹ 𝒓 𝒑, π‘œπ‘π‘‘π‘–π‘šπ‘’π‘š = ( 𝑇2 𝑇1 ) 𝛾 π›Ύβˆ’1 = ( βˆšπ‘‡3 Β· 𝑇1 𝑇1 ) 𝛾 π›Ύβˆ’1 ⟹ 𝒓 𝒑, π’π’‘π’•π’Šπ’Žπ’–π’Ž = ( 𝑻 πŸ‘ 𝑻 𝟏 ) 𝜸 𝟐·(πœΈβˆ’πŸ) β†’ π‘Šπ‘›π‘’π‘‘ = (𝑐 𝑝 Β· (𝑇3 βˆ’ 𝑇4)) βˆ’ (𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇1)) = 𝑐 𝑝(𝑇3 βˆ’ 𝑇4 βˆ’ 𝑇2 + 𝑇1) = 𝑐 𝑝(𝑇3 βˆ’ βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 βˆ’ βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 + 𝑇1) 𝑾 𝒏𝒆𝒕 = 𝒄 𝒑 Β· (βˆšπ‘» πŸ‘ βˆ’ βˆšπ‘» 𝟏) 𝟐 Maximum Pressure ratio (rp max) For maximum pressure ratio, rp, max β†’ T2=T3 π‘Ÿπ‘ = 𝑃2 𝑃1 = ( 𝑇2 𝑇1 ) 𝛾 π›Ύβˆ’1 π‘Ÿπ‘, π’Žπ’‚π’™ = ( 𝑻 𝟐 𝑇1 ) 𝛾 π›Ύβˆ’1 = ( 𝑻 πŸ‘ 𝑇1 ) 𝛾 π›Ύβˆ’1 = ( 𝑇 π‘šπ‘Žπ‘₯ 𝑇 π‘šπ‘–π‘› ) 𝛾 π›Ύβˆ’1 π‘Ÿπ‘, π‘œπ‘π‘‘π‘–π‘šπ‘’π‘š = ( 𝑇3 𝑇1 ) 𝛾 2Β·(π›Ύβˆ’1) = ( 𝑇 π‘šπ‘Žπ‘₯ 𝑇 π‘šπ‘–π‘› ) 𝛾 2Β·(π›Ύβˆ’1) 𝒓 𝒑, π’π’‘π’•π’Šπ’Žπ’–π’Ž = (𝒓 𝒑, π’Žπ’‚π’™) 𝟏 𝟐 Methods of improving the performance of gas turbine cycle Regeneration πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = π‘Šπ‘‡ βˆ’ π‘ŠπΆ 𝑄𝑠 = (β„Ž3 βˆ’ β„Ž4) βˆ’ (β„Ž2 βˆ’ β„Ž1) β„Ž3 βˆ’ β„Ž π‘₯
Effects of Regenerator 1. No change in compressor work 2. No change in turbine work 3. No change in Net-work 4. Decrease in heat supply 5. Decrease in heat rejection 6. Increase in efficiency Effectiveness of a Regenerator ∈ = Actual gain in heat Ideal gain in heat/Maximum heat transfer = π‘š π‘Ž Β· (β„Ž π‘₯ βˆ’ β„Ž2) π‘š π‘Ž+𝑔 Β· (β„Ž4 βˆ’ β„Ž2) = π‘š π‘Ž Β· 𝑐 𝑝,π‘Žπ‘–π‘Ÿ Β· (𝑇π‘₯ βˆ’ 𝑇2) (π‘š π‘Ž + π‘š 𝑔) Β· 𝑐 𝑝,π‘”π‘Žπ‘  Β· (𝑇4 βˆ’ 𝑇2) Generally, mass of the fuel is very small in comparison to mass of air. Therefore, mf can be neglected w.r.t ma if cp of air and cp, gas are almost same. Then, π‘š 𝑓 β‰ͺ π‘š π‘Ž, 𝑐 𝑝, π‘Žπ‘–π‘Ÿ β‰ˆ 𝑐 𝑝, π‘”π‘Žπ‘  ⟹ ∈= 𝑻 𝒙 βˆ’ 𝑻 𝟐 𝑻 πŸ’ βˆ’ 𝑻 𝟐 = π‘»π’‰π’†π’“π’Žπ’‚π’ π‘Ήπ’‚π’•π’Šπ’ Note 1. Regenerator will be effective only when the temperature difference between turbine exit (T4) and compressor exit (T2) is very large. 2. In case of maximum work output cycle as T2=T4, Regenerator can’t be used. Reheating π‘Šπ‘–π‘‘β„Žπ‘œπ‘’π‘‘ π‘Ÿπ‘’β„Žπ‘’π‘Žπ‘‘π‘–π‘›π‘” β†’ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ = (β„Ž3 βˆ’ β„Ž π‘Ž) + (β„Ž π‘Ž βˆ’ β„Ž4β€²) = β„Ž3 βˆ’ β„Ž4β€² π‘Šπ‘–π‘‘β„Ž π‘Ÿπ‘’β„Žπ‘’π‘Žπ‘‘π‘–π‘›π‘” β†’ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ = (β„Ž3 βˆ’ β„Ž π‘Ž) + (β„Ž 𝑏 βˆ’ β„Ž4) Effects of Reheating 1. No change in compressor work. 2. Increase in turbine work (on h-s diagram constant pressure line diverge). 3. Increase in Net work. 4. Temperature at the exit of turbine (T4) is greater than T4’ and hence with reheating, scope for regenerator increases. Therefore, reheating cycles are generally combined with regeneration. 5. Increase in heat supply. Note With reheating both heat supplied and net-work increases. Therefore, efficiency depends in relative increase of both values. πœ‚ = π‘Šπ‘‡ βˆ’ π‘Šπ‘ 𝑄𝑠 = (β„Ž3 βˆ’ β„Ž π‘Ž) + (β„Ž 𝑏 βˆ’ β„Ž4) βˆ’ (β„Ž2 βˆ’ β„Ž1) (β„Ž3 βˆ’ β„Ž2) + (β„Ž3 βˆ’ β„Ž π‘Ž) Reheating and intercooling are constant pressure processes.
Intercooling Effects of Intercooling 1. Decrease in compressor work 2. No change in turbine work 3. Increase in net-work 4. Increase in heat supply 5. With intercooling, the temperature difference between turbine exit and compressor exit increases and hence the scope for regeneration increases with intercooling. Therefore, intercooling cycles are generally combined with regeneration. Note With intercooling both net-work and heat supply increases, therefore, (increase in) efficiency depends on relative increase of both values. πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = (β„Ž3 βˆ’ β„Ž4) βˆ’ (β„Ž π‘Ž βˆ’ β„Ž1) + (β„Ž2 βˆ’ β„Ž 𝑏) β„Ž3 βˆ’ β„Ž2 Condition for minimum work input to the compressor with perfect intercooling (T1=Tb). π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ = π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ 𝟏 + π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ 𝟐 = (β„Ž π‘Ž βˆ’ β„Ž1) + (β„Ž2 βˆ’ β„Ž 𝑏) = 𝑐 𝑝 Β· (π‘‡π‘Ž βˆ’ 𝑇1) + 𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇𝑏) π‘Šπ‘ = 𝑐 𝑝 Β· 𝑇1 Β· [ π‘‡π‘Ž 𝑇1 βˆ’ 1] + 𝑐 𝑝 Β· 𝑻 𝒃 Β· [ 𝑇2 𝑇𝑏 βˆ’ 1] = 𝑐 𝑝 Β· 𝑇1 Β· [ π‘‡π‘Ž 𝑇1 βˆ’ 1] + 𝑐 𝑝 Β· 𝑻 𝟏 Β· [ 𝑇2 𝑇𝑏 βˆ’ 1] = 𝑐 𝑝 Β· 𝑇1 Β· [ π‘‡π‘Ž 𝑇1 βˆ’ 1 + 𝑇2 𝑇𝑏 βˆ’ 1] π‘Šπ‘ = 𝑐 𝑝 Β· 𝑇1 Β· [( 𝑃𝑖 𝑃1 ) π›Ύβˆ’1 𝛾 + ( 𝑃2 𝑃𝑖 ) π›Ύβˆ’1 𝛾 βˆ’ 2] = 𝑐 𝑝 Β· 𝑇1 Β· [( 𝑃𝑖 𝑃1 ) π‘₯ + ( 𝑃2 𝑃𝑖 ) π‘₯ βˆ’ 2] ( 𝛾 βˆ’ 1 𝛾 = π‘₯) πΉπ‘œπ‘Ÿ π‘šπ‘–π‘›π‘–π‘šπ‘’π‘š π‘€π‘œπ‘Ÿπ‘˜ 𝑖𝑛𝑝𝑒𝑑 π‘œπ‘“ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ, π‘‘π‘Šπ‘– 𝑑𝑃𝑖 = 0 π‘‘π‘Šπ‘– 𝑑𝑃𝑖 = 𝑐 𝑝 Β· 𝑇1 Β· [( 1 𝑃1 π‘₯ Β· π‘₯ Β· 𝑃𝑖 π‘₯βˆ’1 ) + (𝑃2 π‘₯ Β· (βˆ’π‘₯) Β· 𝑃𝑖 βˆ’π‘₯βˆ’1 ) βˆ’ 0] = 0 β†’ π‘₯ Β· 𝑃𝑖 π‘₯βˆ’1 𝑃1 π‘₯ = π‘₯ Β· 𝑃2 π‘₯ 𝑃𝑖 π‘₯+1 ⟹ 𝑃𝑖 π‘₯βˆ’1+π‘₯+1 = 𝑃1 π‘₯ Β· 𝑃2 π‘₯ = 𝑃𝑖 2π‘₯ ⟹ π‘·π’Š = βˆšπ‘· 𝟏 Β· 𝑷 𝟐
π‘Šπ‘1 = β„Ž π‘Ž βˆ’ β„Ž1 = 𝑐 𝑝 Β· (π‘‡π‘Ž βˆ’ 𝑇1) = 𝑐 𝑝 Β· 𝑇1 Β· ( π‘‡π‘Ž 𝑇1 βˆ’ 1) = 𝑐 𝑝 Β· 𝑇1 Β· [( π‘·π’Š 𝑃1 ) π›Ύβˆ’1 𝛾 βˆ’ 1] ⟹ 𝑐 𝑝 Β· 𝑇1 Β· [( 𝑷 𝟐 𝑃1 ) π›Ύβˆ’1 πŸΒ·π›Ύ βˆ’ 1] π‘Šπ‘2 = β„Ž2 βˆ’ β„Ž 𝑏 = 𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇𝑏) = 𝑐 𝑝 Β· 𝑇𝑏 Β· ( 𝑇2 𝑇𝑏 βˆ’ 1) = 𝑐 𝑝 Β· 𝑻 𝒃 Β· [( 𝑷 𝟐 𝑃𝑖 ) π›Ύβˆ’1 𝛾 βˆ’ 1] ⟹ 𝑐 𝑝 Β· 𝑻 𝟏 Β· [( 𝑷 𝟐 𝑃1 ) π›Ύβˆ’1 πŸΒ·π›Ύ βˆ’ 1] 𝑾 π’„πŸ = 𝑾 π’„πŸ For perfect intercooling and minimum work input each stage work input is same, therefore total compressor work is equal to (No. of stages (N)) Γ— (Each stage compressor work (Wc)). Brayton cycle with many stages of intercooling and many stages of reheating can be reduced to Ericsson cycle. Many stages if intercooling and reheating can be reduced to isothermal process.

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Applications of thermodynamics

  • 2. Contents Vapour power cycles...........................................................................................................................................................................................2 Specific steam consumption ..................................................................................................................................................................2 Significance of SSC.....................................................................................................................................................................................2 Work Ratio....................................................................................................................................................................................................2 Back work Ratio .........................................................................................................................................................................................2 Carnot Vapour power cycles ......................................................................................................................................................................3 Drawbacks of Carnot vapour power cycles.....................................................................................................................................3 Rankine cycle ...............................................................................................................................................................................................3 Analysis of the Cycles................................................................................................................................................................................4 Methods of improving performance of simple rankine cycle .......................................................................................................5 Increasing Boiler pressure .....................................................................................................................................................................5 Super Heating..............................................................................................................................................................................................5 Reheating.......................................................................................................................................................................................................6 Regeneration................................................................................................................................................................................................6 Gas power cycles...................................................................................................................................................................................................8 Otto Cycle.......................................................................................................................................................................................................8 Advantages of I.C engines.......................................................................................................................................................................8 Engine nomenclature....................................................................................................................................................................................8 Air Standard Cycles..........................................................................................................................................................................................10 Otto Cycle.........................................................................................................................................................................................................10 Diesel cycle......................................................................................................................................................................................................11 Comparison of Otto and diesel cycles .............................................................................................................................................12 Dual cycle.........................................................................................................................................................................................................13 Engine Performance Parameters..........................................................................................................................................................13 Gas turbines.........................................................................................................................................................................................................15 Gas turbine cycles....................................................................................................................................................................................15 Analysis of a simple gas turbine cycle............................................................................................................................................16 Efficiency of simple gas turbine cycle.............................................................................................................................................17 Methods of improving the performance of gas turbine cycle...............................................................................................18
  • 3. Vapour power cycles Reasons for using water as working fluid 1. It’s cheap 2. It’s chemically stable 3. It’s not toxic Selection of a power plant Efforts to improve efficiency and thereby reducing the running cost or operating cost may be desirable, but this would lead to increase initial cist and hence efforts must be taken to optimize total cost. Specific steam consumption It is steam consumed for producing unit output of power. β†’ 𝑆𝑆𝐢 = π‘šΜ‡ 𝑠 𝑃 𝑁𝑒𝑑 = π‘šΜ‡ 𝑠 π‘šΜ‡ 𝑠 Γ— π‘Šπ‘π‘’π‘‘ = 1 π‘Šπ‘π‘’π‘‘ π‘˜π‘” 𝐾𝐽 β†’ 𝑆𝑆𝐢 = 3600 π‘Šπ‘π‘’π‘‘ π‘˜π‘” πΎπ‘Š β„Žπ‘Ÿ Significance of SSC SSC indicates the size of plant, smaller the SSC larger is the net work and hence for developing given power, mass flow rate of steam must be less. Smaller the SSC, lesser I the size of plant and hence such plants are preferable. Work Ratio It is the ratio of net work to the positive work. β†’ 𝛾 𝑀 = π‘Šπ‘π‘’π‘‘ π‘Š+𝑣𝑒 β†’ 𝛾 𝑀 = π‘Š+𝑣𝑒 βˆ’ π‘Šβˆ’π‘£π‘’ π‘Š+𝑣𝑒 = 1 βˆ’ π‘Šβˆ’π‘£π‘’ π‘Š+𝑣𝑒 β†’ 𝛾 𝑀 π‘™π‘–π‘žπ‘’π‘–π‘‘ = 1 βˆ’ π‘Šπ‘π‘’π‘šπ‘ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ β†’ 𝛾 𝑀 π‘”π‘Žπ‘  = 1 βˆ’ π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ Work done by compressor is much higher than pump, so Work ratio for liquid power cycles is greater than gas power cycles. Power plants with high work ratios are preferable. The work ratio is highest for rankine cycles among all other cycles, this is because of less work done by pump in rankine cycle. In case of gas turbine power plants, the work ratio is about 0.4 – 0.6 i.e., in gas turbine power plants compressors consumes 40 – 60% of turbine work. In Rankine cycle, work ratio is about 0.96 – 0.98, i.e., I rankine cycle, pump consumes 2-4 % of turbine work. Back work Ratio It’s the ratio of negative work to positive work. β†’ 𝛾 𝑏𝑀 = π‘Šβˆ’π‘£π‘’ π‘Š+𝑣𝑒 β†’ 𝛾 𝑀 = 1 βˆ’ 𝛾 𝑏𝑀
  • 4. Carnot Vapour power cycles Drawbacks of Carnot vapour power cycles 1. Saturated vapour which is entering turbine at 1 leaves ar 2, which is in wet region. The liquid which is present at state 2 may damage turbine blades due to high velocity. 2. It is difficult to design a condenser that stops suddenly at point 3. 3. It’s difficult to design a compressor which handles both liquid and vapour. 4. As Carnot vapour cycle uses compressor, the compressor work is large and hence net work is less. β†’ πœ‚ = 𝑄 𝑆 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑄 𝑅 𝑄 𝑆 β†’ πœ‚ = 1 βˆ’ 𝑇𝑙 βˆ™ 𝑑𝑆 π‘‡β„Ž βˆ™ 𝑑𝑆 β†’ πœ‚ = 1 βˆ’ 𝑇𝑙 π‘‡β„Ž Rankine cycle β†’ πœ‚ = π‘Šπ‘π‘’π‘‘ 𝑄𝑠 = π‘Šπ‘‡ βˆ’ π‘Šπ‘ƒ 𝑄𝑠 1 βˆ’ 2 ⟹ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π‘Žπ‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ 𝑒π‘₯π‘π‘Žπ‘›π‘ π‘–π‘œπ‘› (π‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’) 2 βˆ’ 3 ⟹ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘π‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘–π‘œπ‘› (πΆπ‘œπ‘›π‘‘π‘’π‘›π‘ π‘œπ‘Ÿ) 3 βˆ’ 4 ⟹ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› (π‘ƒπ‘’π‘šπ‘) 4 βˆ’ 1 ⟹ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘π‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ β„Žπ‘’π‘Žπ‘‘ 𝑠𝑒𝑝𝑝𝑙𝑦 (π΅π‘œπ‘–π‘™π‘’π‘Ÿ)
  • 5. Analysis of the Cycles Assumptions 1. Each device is treated as steady flow device. 2. Kinetic and potential energy changes are neglected. β†’ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ = β„Ž1 βˆ’ β„Ž2 β†’ 𝑄 π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘’π‘‘ = β„Ž2 βˆ’ β„Ž3 β†’ π‘Šπ‘π‘’π‘šπ‘ = β„Ž4 βˆ’ β„Ž3 We know that open system work is -Vβˆ™ dP, this equation is applicable when the flow is steady, K.E and P.E changes are neglected and when the process is reversible. Steady flow equations can be applied to reversible and irreversible processes. If the pumping process is reversible, then work obtained from Steady flow energy equation and work equal to -Vβˆ™dP can be equated. β†’ π‘Šπ‘π‘’π‘šπ‘ = β„Ž3 βˆ’ β„Ž4 = βˆ’π‘‰ βˆ™ 𝑑𝑃 β†’ 𝑄𝑠𝑒𝑝𝑝𝑙𝑖𝑒𝑑 = β„Ž1 βˆ’ β„Ž4 β†’ πœ‚ = π‘Šπ‘‡ βˆ’ π‘Šπ‘ƒ 𝑄𝑠 = (β„Ž1 βˆ’ β„Ž2) βˆ’ (β„Ž4 βˆ’ β„Ž3) β„Ž1 βˆ’ β„Ž4 Reason for higher efficiency of Carnot cycle compared to rankine cycle 1 βˆ’ 2 βˆ’ 3 βˆ’ 4 ⟹ π‘…π‘Žπ‘›π‘˜π‘–π‘›π‘’ 𝑐𝑦𝑐𝑙𝑒 1 βˆ’ 2 βˆ’ 3β€² βˆ’ 4β€² ⟹ πΆπ‘Žπ‘Ÿπ‘›π‘œπ‘‘ 𝑐𝑦𝑐𝑙𝑒 β†’ πœ‚ π‘π‘Žπ‘Ÿπ‘›π‘œπ‘‘ = 1 βˆ’ π‘‡β„Ž 𝑇𝑙 β†’ πœ‚ = 𝑓(𝑇 π‘š) β†’ 𝑄𝑠 = β„Ž1 βˆ’ β„Ž4 = 𝑇 π‘š βˆ™ 𝑑𝑆 ⟹ 𝑇 π‘š = β„Ž1 βˆ’ β„Ž4 𝑑𝑆 β†’ πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑇𝑙 βˆ™ 𝑑𝑆 𝑇 π‘š βˆ™ 𝑑𝑆 β†’ πœ‚ π‘Ÿπ‘Žπ‘›π‘˜π‘–π‘›π‘’ = 1 βˆ’ 𝑇𝑙 𝑇 π‘š In Carnot cycle, complete heat is added at single fixed temperature (Th), where as in rankine cycle, some heat is added at a temperature lower than Th and remaining heat is added at Th. therefore mean temperature of heat addition is less in rankine cycle, compared to Carnot cycle therefore the efficiency of rankine cycle is less than Carnot cycle. If the mean temperature of heat addition will be more, the efficiency will be more.
  • 6. Parameter Carnot Rankine Tm more less Ξ· more less WNet less more SSC more less Size of Plant large small Ξ³w less more Methods of improving performance of simple rankine cycle Increasing Boiler pressure Effects of increasing boiler pressure 1. Increase in turbine work 2. Increase in pump work 3. Increase in net work 4. Decrease in SSC 5. Decrease in heat rejection 6. Decrease in heat supply 7. Increase in mean temperature of heat addition 8. Increase in efficiency 9. Decrease in dryness fraction at the turbine exit. Note- Due to decrease in dryness fraction at the turbine exit, the boiler pressure is limited. Super Heating
  • 7. Effects of Superheating 1. Increase in Turbine work 2. No change in pump work 3. Increase in Net work 4. Decrease in SSC 5. Increase in mean temperature of heat addition 6. Increase in efficiency 7. Increase in heat supply 8. Increase in heat rejection 9. Increase in dryness fraction at turbine exit Note There is a limit to superheating because of metallurgical conditions i.e., steam turbine blades can resist up to 620⁰C. Reheating Effects of Reheating 1. Increase in dryness fraction 2. Increase in turbine work 3. Increase in net work 4. Decrease in SSC. 5. Efficiency may increase or decrease. Reheating in steam turbine may increase or decrease he efficiency if steam is expanded deeply in first turbine and then reheated, the mean temperature may decrease hence the efficiency decreases. In most of the practically reheat cycles, the steam is expanded slightly in first turbine and then reheated, this increases the mean temperature of heat addition and also the efficiency of the cycle. πœ‚ = (β„Ž1 βˆ’ β„Ž2) + (β„Ž3 βˆ’ β„Ž4) βˆ’ (β„Ž6 βˆ’ β„Ž5) (β„Ž1 βˆ’ β„Ž6) + (β„Ž3 βˆ’ β„Ž2) Regeneration
  • 8. Effects of regeneration 1. Decrease in turbine work due to decrease in mass. 2. Reduction in condenser load 3. Reduction in heat supply 4. Increase in mean temperature of heat addition 5. Increase in efficiency π‘Šπ‘‡π‘’π‘Ÿπ‘π‘–π‘›π‘’ = (β„Ž1 βˆ’ β„Ž2) + (1 βˆ’ 𝑦) βˆ™ (β„Ž2 βˆ’ β„Ž3) π‘Šπ‘π‘’π‘šπ‘ = (1 βˆ’ 𝑦) βˆ™ (β„Ž5 βˆ’ β„Ž4) + (β„Ž7 βˆ’ β„Ž6) π‘Šπ‘π‘’π‘‘ = π‘Šπ‘‡ βˆ’ π‘Šπ‘ 𝑄𝑠 = β„Ž1 βˆ’ β„Ž7 𝑄 𝑅 = (1 βˆ’ 𝑦) βˆ™ (β„Ž3 βˆ’ β„Ž4) πœ‚ = π‘Šπ‘π‘’π‘‘ 𝑄 𝑆 𝑦 βˆ™ β„Ž2 + (1 βˆ’ 𝑦) βˆ™ β„Ž5 = β„Ž6 Ideal regenerative Rankine cycle leads to Carnot cycle. Need of Condenser in steam power plants If no condenser is used and if steam is allowed to expand to atmospheric pressure. The saturation temperature at 1 bar is 100⁰C, therefore of no condenser is used, steam leaves the turbine at higher temperature and this energy becomes useless. By using condenser steam can be allowed to expand to lower temperature and thereby more turbine work can be obtained, this results in higher efficiency. Therefore, condensers are used to increase turbine work and efficiency. Isentropic or adiabatic efficiency of turbine It is ratio of actual work to isentropic work β†’ π‘Šπ‘Žπ‘π‘‘π‘’π‘Žπ‘™ = β„Ž1 βˆ’ β„Ž2 β†’ π‘Šπ‘–π‘ π‘’π‘› = β„Ž1 βˆ’ β„Ž2𝑠 β†’ πœ‚ 𝑇 = β„Ž1 βˆ’ β„Ž2 β„Ž1 βˆ’ β„Ž2𝑠 Isentropic efficiency of a pump or compressor It is ratio of actual work to isentropic work β†’ π‘Šπ‘Žπ‘π‘‘π‘’π‘Žπ‘™ = β„Ž2 βˆ’ β„Ž1 β†’ π‘Šπ‘–π‘ π‘’π‘› = β„Ž2𝑠 βˆ’ β„Ž1 β†’ πœ‚ 𝑇 = β„Ž2𝑠 βˆ’ β„Ž1 β„Ž2 βˆ’ β„Ž1
  • 9. Gas power cycles Otto Cycle Engine is a device which converts one form of energy into other useful form. Based on combustion, engines are classified into 1. Internal combustion engines (I.C engines) 2. External combustion engines (E.C engines) In I.C engines, burning or combustion occurs in cylinder and power is also developed in same cylinder. In E.C engines, heat is transferred from products of combustion to working fluid. Heat engine is a device which converts chemical energy of fuel into heat energy and subsequently this heat energy is converted into mechanical power. Advantages of I.C engines 1. Mechanical simplicity 2. Low initial cost due to absence of boiler, turbine, condenser etc... 3. High efficiency 4. High power to weight ratio Engine nomenclature Top dead centre It’s the dead centre when the piston is farthest from crank shaft. In case of horizontal engines TDC is known as Inner dead centre. Bottom dead centre It’s the dead centre when the piston is nearest to crank shaft. In case of horizontal engines, it is known as outer dead centre. Stroke or stroke length Distance between TDC and BDC is known as stroke length. Displacement volume or stroke volume or swept volume It’s the volume swept by a piston 𝑉𝑠 = πœ‹ 4 𝐷2 𝐿 βˆ™ π‘˜ 𝐷 β†’ πΌπ‘›π‘›π‘’π‘Ÿ π‘‘π‘–π‘Žπ‘šπ‘’π‘‘π‘’π‘Ÿ π‘œπ‘“ π‘π‘¦π‘™π‘–π‘›π‘‘π‘’π‘Ÿ 𝐿 β†’ π‘†π‘‘π‘Ÿπ‘œπ‘˜π‘’ π‘™π‘’π‘›π‘”π‘‘β„Ž π‘˜ β†’ π‘›π‘œ. π‘œπ‘“ π‘π‘¦π‘™π‘–π‘›π‘‘π‘’π‘Ÿπ‘  Clearance volume It’s the volume of cylinder when piston is at top dead centre or Inner Dead centre. Clearance volume is provided to accommodate valves and to prevent damage to valves.
  • 10. Compression ratio (Ξ³) It’s defined as ratio of volume before compression to volume after compression. 𝛾 = π‘‰π‘œπ‘™π‘’π‘šπ‘’ π‘π‘’π‘“π‘œπ‘Ÿπ‘’ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› π‘£π‘œπ‘™π‘’π‘šπ‘’ π‘Žπ‘“π‘‘π‘’π‘Ÿ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› 𝛾 = 𝑉1 𝑉2 = 𝑉𝑐 + 𝑉𝑠 𝑉𝑠 = 1 + 𝑉𝑐 𝑉𝑠 𝛾 = 1 + 𝑉𝑐 𝑉𝑠 Mean Piston Speed In one stroke the shaft completes half revolution (180⁰C). therefore, piston covers 2L in 1 revolution. β†’ π‘€π‘’π‘Žπ‘› π‘π‘–π‘ π‘‘π‘œπ‘› 𝑠𝑝𝑒𝑒𝑑 = 2𝐿𝑁 60
  • 11. Air Standard Cycles Assumptions 1. Working fluid is air and it’s treated as Ideal gas 2. Specific heat cp, cv are assumed as constants. 3. Working fluids is of fixed mass (liquid system analysis) 4. Working fluid doesn’t undergo any chemical reaction that is of constant chemical composition throughout cycle. 5. All processes are assumed to be reversible (internally reversible). Generally, cp & cv of air are taken as 25⁰C⁰this analysis is known as cold air cycle analysis. Otto Cycle 1 βˆ’ 2 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› 2 βˆ’ 3 β†’ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘£π‘œπ‘™π‘’π‘šπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Žπ‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› 3 βˆ’ 4 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ 𝑒π‘₯π‘π‘Žπ‘›π‘ π‘–π‘œπ‘› 4 βˆ’ 1 β†’ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘£π‘œπ‘™π‘’π‘šπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘–π‘œπ‘› β†’ 𝛾 = 𝑉1 𝑉2 β†’ πœ‚ = 1 βˆ’ 𝑄 π‘Ÿ 𝑄𝑠 β†’ 𝑄23 = 𝑄𝑠 = π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇3 βˆ’ 𝑇2) β†’ 𝑄41 = 𝑄 π‘Ÿ = π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) β†’ πœ‚ = 1 βˆ’ π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ (𝑇4 βˆ’ 𝑇1) (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 𝑇1 ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 ( 𝑇3 𝑇2 βˆ’ 1) ( 𝑇2 𝑇1 = π‘Ÿ π›Ύβˆ’1 = 𝑇3 𝑇4 ⟹ 𝑇4 𝑇1 = 𝑇3 𝑇2 ) πœ‚ = 1 βˆ’ 𝑇1 𝑇2 = 1 βˆ’ 1 π‘Ÿ π›Ύβˆ’1 Significance of compression ratio Compression ratio is an indicator of efficiency of engine, greater the compression ratio greater is the efficiency. This is because if compression ratio is high there will be more scope for expansion and net-work will be more. Therefore, for a given heat supply if compression ratio is more, the efficiency will be more. πœ‚ ∝ π‘Ÿ (π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘–π‘œ)
  • 12. Mean effective pressure It’s hypothetical (imaginary) pressure which gives same net-work as that of actual cycle for swept volume (same size of engine). β†’ π‘Šπ‘π‘’π‘‘ = π‘ƒπ‘š βˆ™ 𝑉𝑠 β†’ 𝑃 𝑀 = π‘Šπ‘π‘’π‘‘ 𝑉𝑠 The mean effective pressure of Otto cycle can be expressed as π‘ƒπ‘š = πœ‚ π‘‘β„Ž Β· π›₯𝑃 (𝛾 βˆ’ 1) Β· (π‘Ÿ βˆ’ 1) Significance of Pm This term is used for comparing different engines of same size. That is engines with higher mean effective pressure have more net-work and for a given heat input greater the mean effective pressure, greater is efficiency. Diesel cycle 1 βˆ’ 2 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› 2 βˆ’ 3 β†’ πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘π‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Žπ‘‘π‘‘π‘–π‘‘π‘–π‘œπ‘› 3 βˆ’ 4 β†’ π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ 𝑒π‘₯π‘π‘Žπ‘›π‘–π‘ π‘œπ‘› 4 βˆ’ 1 β†’ πΆπ‘œπ‘ π‘›π‘‘π‘Žπ‘›π‘‘ π‘‰π‘œπ‘™π‘’π‘šπ‘’ β„Žπ‘’π‘Žπ‘‘ π‘Ÿπ‘’π‘—π‘’π‘π‘‘π‘–π‘œπ‘› β†’ πΆπ‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘–π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘› (π‘Ÿ) = 𝑉1 𝑉2 β†’ πΆπ‘’π‘‘π‘œπ‘“π‘“ π‘Ÿπ‘Žπ‘‘π‘–π‘œ (π‘Ÿπ‘) = 𝑉3 𝑉2 = 𝑇3 𝑇2 β†’ 𝐸π‘₯π‘π‘Žπ‘›π‘–π‘ π‘–π‘œπ‘› π‘Ÿπ‘Žπ‘‘π‘–π‘œ (π‘Ÿπ‘’) = 𝑉4 𝑉3 = 𝑉1 𝑉3 β†’ π‘Ÿπ‘ βˆ™ π‘Ÿπ‘’ = π‘Ÿ β†’ πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = 𝑄 𝑁𝑒𝑑 𝑄𝑠 = 𝑄𝑠 βˆ’ 𝑄 π‘Ÿ 𝑄𝑠 = 1 βˆ’ 𝑄 π‘Ÿ 𝑄𝑠 β†’ 𝑄23 = 𝑄𝑠 = π‘š βˆ™ 𝑐 𝑝 βˆ™ (𝑇3 βˆ’ 𝑇2) β†’ 𝑄41 = 𝑄 π‘Ÿ = π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1)
  • 13. β†’ πœ‚ = 1 βˆ’ π‘š βˆ™ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) π‘š βˆ™ 𝑐 𝑝 βˆ™ (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 𝑐 𝑣 βˆ™ (𝑇4 βˆ’ 𝑇1) 𝑐 𝑝 βˆ™ (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 1 𝛾 βˆ™ 𝑇1 ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 ( 𝑇3 𝑇2 βˆ’ 1) ( 𝑇2 𝑇1 = π‘Ÿ π›Ύβˆ’1 ) ( 𝑉3 𝑉2 = 𝑇3 𝑇2 = π‘Ÿπ‘) ( 𝑇4 𝑇1 = π‘Ÿπ‘ 𝛾 ) β†’ πœ‚ = 1 βˆ’ 1 𝛾 βˆ™ 1 π‘Ÿ π›Ύβˆ’1 βˆ™ π‘Ÿπ‘ 𝛾 βˆ’ 1 π‘Ÿπ‘ βˆ’ 1 Efficiency of a diesel cycle depends on compression ratio and cut-off ratio. Efficiency of a diesel cycle increases with increase in compression ratio. Effect of cut-off ratio (rc) on efficiency of diesel cycle With increase in cut off ratio, both heat supply and heat rejection increases, but the increase in heat rejection is more, because heat rejection occurs at constant volume and heat supply occurs ar constant pressure and the slope of constant volume is greater than slope of constant pressure curve. Comparison of Otto and diesel cycles Case 1- Same compression ratio and heat supply πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄 𝑆 For the same compression ratio and heat supply, rejection is more in diesel cycle and hence efficiency of Otto cycle is more than efficiency of diesel cycle. π‘‚π‘‘π‘‘π‘œ β†’ 𝑄𝑠 = 𝑄 𝑉 = π‘‘π‘ˆ = π‘š Β· 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) 𝐷𝑖𝑒𝑠𝑒𝑙 β†’ 𝑄𝑠 = 𝑄 𝑃 = 𝑑𝐻 = π‘š Β· 𝑐 𝑝 Β· (𝑇3 β€² βˆ’ 𝑇2) 𝑄𝑠 𝑢𝒕𝒕𝒐 = 𝑄𝑠 π‘«π’Šπ’†π’”π’†π’ ⟹ π‘š Β· 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) = π‘š Β· 𝑐 𝑝 Β· (𝑇3 β€² βˆ’ 𝑇2) 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) = 𝑐 𝑝 Β· (𝑇3 β€² βˆ’ 𝑇2) ⟹ 𝑇3 βˆ’ 𝑇2 = 𝑇3 β€² βˆ’ 𝑇2 𝑻 πŸ‘ > 𝑻 πŸ‘ β€² Case 2 – Same compression ratio and heat rejection πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 For the same compression ratio and heat rejection, heat supply is more in Otto cycle and hence its efficiency is more. πœ‚ 𝑑𝑖𝑒𝑠𝑒𝑙 > πœ‚ π‘‚π‘‘π‘‘π‘œ Case 3 – Same maximum temperature and heat rejection For same maximum temperature and heat rejection, heat supplied is more in diesel cycle and hence it’s more efficient. In this case compression ratio of diesel cycle is more than compression ratio of Otto cycle. πœ‚ 𝑑𝑖𝑒𝑠𝑒𝑙 > πœ‚ π‘‚π‘‘π‘‘π‘œ 1-2-3-4 β†’ Otto cycle 1-2-3’-4’ β†’ Diesel cycle
  • 14. Dual cycle In actual engines, heat is neither added at constant volume or at constant pressure. A dual cycle is developed and it has features of both Otto and Diesel cycle. 𝛾 = 𝑉1 𝑉2 π‘Ÿπ‘ = 𝑉4 𝑉3 π‘Ÿπ‘’ = 𝑉5 𝑉4 πœ‚ = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 𝑄𝑠 = 𝑄𝑠1 + 𝑄𝑠2 𝑄𝑠1 = π‘š Β· 𝑐 𝑣 Β· (𝑇3 βˆ’ 𝑇2) ; 𝑄𝑠2 = π‘š Β· 𝑐 𝑝 Β· (𝑇5 βˆ’ 𝑇1) ; 𝑄 𝑅 = π‘š Β· 𝑐 𝑣 Β· (𝑇5 βˆ’ 𝑇1) As dual cycle has features of both Otto and diesel cycle, its efficiency lies between Otto and diesel cycle. Engine Performance Parameters Indicated power (IP) The power developed inside the cylinder is known as indicated power. In case of 2 stroke engine, 1 cycle is completed in 2 strokes of piston or 1 revolution of crank shaft. Similarly, in 4 stroke engine, 1 cycle is completed in 4 strokes or 2 revolutions of shaft. Theoretically power developed in 2 stroke engines is twice that of 4 stroke engines. Indicated power of 2 stroke engine π‘Šπ‘›π‘’π‘‘ = π‘ƒπ‘š Β· 𝑉𝑠 β§Έ 𝑐𝑦𝑐𝑙𝑒 π‘Šπ‘›π‘’π‘‘ = π‘ƒπ‘š Β· 𝑉𝑠 β§Έ π‘Ÿπ‘’π‘£π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘› π‘ƒπ‘œπ‘€π‘’π‘Ÿ = 𝑷 π’Žπ’†π’‚π’ Β· 𝑽 𝑺 π‘Ÿπ‘’π‘£π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘› Γ— 𝑡 π‘Ÿπ‘’π‘£π‘œπ‘™π‘’π‘‘π‘–π‘œπ‘›π‘  𝑠𝑒𝑐 (Pmβ†’indicated mean effective pressure) πΌπ‘›π‘‘π‘–π‘π‘Žπ‘‘π‘’π‘‘ π‘π‘œπ‘€π‘’π‘Ÿ (𝑰. 𝑷) = π‘ƒπ‘š Β· 𝑉𝑆 Β· 𝑁 60 = π‘ƒπ‘š Β· (𝐴 Β· 𝐿) Β· 𝑁 60 If there are K cylinders, then indicated power is equal to ⟹ π‘ƒπ‘š Β· ( 𝐴 Β· 𝐿) Β· 𝑁 Β· 𝑲 60 In case of 4 – stroke engine ⟹ 𝐼. 𝑃 = π‘ƒπ‘š Β· ( 𝐴 Β· 𝐿) Β· 𝑁 Β· 𝑲 60 Γ— 2 Brake power (B.P) The power available at the output shaft is known as brake power. π΅π‘Ÿπ‘Žπ‘˜π‘’ π‘ƒπ‘œπ‘€π‘’π‘Ÿ (𝑩. 𝑷) = 𝑇 Γ— πœ” 𝐡. 𝑃 = 𝑇 Γ— 2πœ‹π‘ 60 β†’ applicable for 2 or 4 stroke and any number of cylinders Frictional Power (F.P) The difference between indicated power and brake power is called frictional power. 𝐹. 𝑃 = 𝐼. 𝑃 βˆ’ 𝐡. 𝑃
  • 15. Mechanical efficiency πœ‚ π‘šπ‘’π‘β„Ž = 𝐡. 𝑃 𝐼. 𝑃 Indicated thermal efficiency (Ξ·ith) πœ‚π‘–π‘‘β„Ž = 𝐼. 𝑃 π‘šΜ‡ 𝑓 Γ— 𝐢. 𝑉 π‘šΜ‡ 𝑓 β†’ π‘šπ‘Žπ‘ π‘  π‘“π‘™π‘œπ‘€ π‘Ÿπ‘Žπ‘‘π‘’ π‘œπ‘“ 𝑓𝑒𝑒𝑙 𝐢. π‘‰βˆ’. πΆπ‘Žπ‘™π‘œπ‘Ÿπ‘–π‘“π‘–π‘ π‘£π‘Žπ‘™π‘’π‘’ Brake thermal efficiency (Ξ·bth) πœ‚ π‘π‘‘β„Ž = 𝐡. 𝑃 π‘šΜ‡ 𝑓 Γ— 𝐢. 𝑉 Indicated specific fuel consumption (isfc) 𝑖𝑠𝑓𝑐 = π‘šΜ‡ 𝑓 𝐼. 𝑃 ( π‘˜π‘” πΎπ‘Š Β· β„Žπ‘Ÿ ) Brake specific fuel consumption (bsfc) 𝑏𝑠𝑓𝑐 = π‘šΜ‡ 𝑓 𝐡. 𝑃 ( π‘˜π‘” πΎπ‘Š Β· β„Žπ‘Ÿ ) Volumetric efficiency (Ξ·vol) It represents the breathing capacity of the cylinder, it’s the ratio of actual volume of fluid entering the cylinder per cycle to the swept volume. πœ‚ π‘£π‘œπ‘™ = π΄π‘π‘‘π‘’π‘Žπ‘™ π‘‰π‘œπ‘™π‘’π‘šπ‘’ π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘›π‘” πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 = π‘šΜ‡ 𝑓 Β· 𝜐 πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 β†’ 2 βˆ’ π‘ π‘‘π‘Ÿπ‘œπ‘˜π‘’ πœ‚ π‘£π‘œπ‘™ = π΄π‘π‘‘π‘’π‘Žπ‘™ π‘‰π‘œπ‘™π‘’π‘šπ‘’ π‘’π‘›π‘‘π‘’π‘Ÿπ‘–π‘›π‘” πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 2 = π‘šΜ‡ 𝑓 Β· 𝜐 πœ‹ 4 Β· 𝐷2 Β· 𝐿 Β· 𝑁 Β· 𝐾 2 β†’ 4 βˆ’ π‘ π‘‘π‘Ÿπ‘œπ‘˜π‘’
  • 16. Gas turbines Advantages of Gas turbines 1. Simple mechanism 2. Compact 3. Easy balancing 4. High speeds can be obtained Disadvantages 1. As gas turbines are rotary devices, they are subjected to higher temperatures continuously, hence costly material is required. 2. As compressor is used it consumes more work and hence the Net-work is less. 3. As speeds are high, reduction gear mechanisms must be used. Gas turbine cycles Gas turbine cycles work on Brayton or Joule cycle. Open cycle gas turbine Open cycle gas turbine is a mechanical device. Gas turbines are used in power generation, jet propulsion and aircraft engines. Closed cycle gas turbines 1-2 β†’ reversible adiabatic Compression 2-3 β†’ constant pressure heat addition 3-4 β†’ reversible adiabatic expansion 1-2 β†’ reversible adiabatic Compression 2-3 β†’ constant pressure heat addition 3-4 β†’ reversible adiabatic expansion 4-1 β†’ constant pressure heat rejection
  • 17. Advantages of closed cycle gas turbines over open cycle gas turbines 1. It is compact. 2. Any working fluid can be used in closed cycle gas turbines but in open gas turbines, air is required for combustion. 3. As products of combustion do not enter turbine blades, any cheaper fuel can be used. Generally, in closed cycle gas turbines, working fluid with higher Ξ³ (higher mono-atomic gases) can be used leading to higher efficiency. Disadvantages 1. Additional heat exchanger is required. 2. Coolant is required in closed cycle gas turbine, where as in open cycle gas turbine atmosphere acts as a sink. Analysis of a simple gas turbine cycle Assumptions 1. Working fluid can be treated as ideal gas. 2. Compression and expansion are treated as steady flow device. 3. Each device is treated as steady flow device. 4. Kinetic and potential energy changes are neglected. 5. cp & cv doesn’t change with temperature. 6. Working fluid is same throughout the cycle. 1β€” 2 ⟢ Reversible Adiabatic β„Ž1 + 𝐢1 2 2 + 𝑔𝑧1 + π‘žβƒ₯ = β„Ž2 + 𝐢2 2 2 + 𝑔𝑧2 + 𝑀 β„Ž1 = β„Ž2 + 𝑀 β†’ βˆ’π‘€ = β„Ž2 βˆ’ β„Ž1 β†’ π’˜ π’„π’π’Žπ’‘π’“π’†π’”π’”π’π’“ = 𝒉 𝟐 βˆ’ 𝒉 𝟏 = 𝒄 𝒑 Β· (𝑻 𝟐 βˆ’ 𝑻 𝟏) 2β€”3 ⟢ Constant pressure heat rejection β„Ž2 + 𝐢2 2 2 + 𝑔𝑧2 + π‘žβƒ₯ = β„Ž3 + 𝐢3 2 2 + 𝑔𝑧3 + 𝑀 β„Ž2 + π‘žβƒ₯ = β„Ž3 β†’ π‘žβƒ₯ = β„Ž3 βˆ’ β„Ž2 β†’ 𝒒 π’”π’–π’‘π’‘π’π’š = 𝒉 πŸ‘ βˆ’ 𝒉 𝟐 = 𝒄 𝒑 Β· (𝑻 πŸ‘ βˆ’ 𝑻 𝟐) 3β€”4 ⟢ Reversible adiabatic expansion (turbine) β„Ž3 + 𝐢3 2 2 + 𝑔𝑧3 + π‘žβƒ₯ = β„Ž4 + 𝐢4 2 2 + 𝑔𝑧4 + 𝑀 β„Ž3 = β„Ž4 + 𝑀 β†’ 𝑀 = β„Ž3 βˆ’ β„Ž4 β†’ π’˜ π’•π’–π’“π’ƒπ’Šπ’π’† = 𝒉 πŸ‘ βˆ’ 𝒉 πŸ’ = 𝒄 𝒑 Β· (𝑻 πŸ‘ βˆ’ 𝑻 πŸ’) 4β€”1 ⟢ Constant pressure heat rejection β„Ž4 + 𝐢4 2 2 + 𝑔𝑧4 + π‘žβƒ₯ = β„Ž1 + 𝐢1 2 2 + 𝑔𝑧1 + 𝑀 β„Ž4 + π‘žβƒ₯ = β„Ž1 β†’ π‘žβƒ₯ = β„Ž1 βˆ’ β„Ž4 β†’ 𝒒 𝒓𝒆𝒋𝒆𝒄𝒕𝒆𝒅 = 𝒉 𝟏 βˆ’ 𝒉 πŸ’ = 𝒄 𝒑 Β· (𝑻 𝟏 βˆ’ 𝑻 πŸ’)
  • 18. Efficiency of simple gas turbine cycle πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = 𝑄𝑠 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑄 𝑅 𝑄𝑠 = 1 βˆ’ 𝑐 𝑝 Β· (𝑇4 βˆ’ 𝑇1) 𝑐 𝑝 Β· (𝑇3 βˆ’ 𝑇2) = 1 βˆ’ 𝑇4 βˆ’ 𝑇1 𝑇3 βˆ’ 𝑇2 πœ‚ = 1 βˆ’ 𝑇1 Β· ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 Β· ( 𝑇3 𝑇2 βˆ’ 1) 1 β†’ 2 (reversible Adiabatic) ⟹ 𝑇2 𝑇1 = ( 𝑃2 𝑃1 ) π›Ύβˆ’1 𝛾⁄ = π‘Ÿπ‘ π›Ύβˆ’1 𝛾⁄ 3 β†’ 4 (reversible Adiabatic) ⟹ 𝑇3 𝑇4 = ( 𝑃3 𝑃4 ) π›Ύβˆ’1 𝛾⁄ = ( 𝑃2 𝑃1 ) π›Ύβˆ’1 𝛾⁄ = π‘Ÿπ‘ π›Ύβˆ’1 𝛾⁄ (𝑃2 = 𝑃3, 𝑃4 = 𝑃1) From above equations, 𝑇2 𝑇1 = 𝑇3 𝑇4 ⟹ 𝑻 πŸ’ 𝑻 𝟏 = 𝑻 πŸ‘ 𝑻 𝟐 πœ‚ = 1 βˆ’ 𝑇1 Β· ( 𝑇4 𝑇1 βˆ’ 1) 𝑇2 Β· ( 𝑇3 𝑇2 βˆ’ 1) ⟹ 𝜼 = 𝟏 βˆ’ 𝑻 𝟏 𝑻 𝟐 = 𝟏 βˆ’ 𝟏 𝒓 𝒑 πœΈβˆ’πŸ πœΈβ„ The efficiency of simple gas turbine cycle depends on pressure ratio. With increase in pressure ratio, efficiency increases because greater the pressure ratio, greater is the compression ratio and more is the scope for free expansion. Hence, for a given input, efficiency increases with increase in pressure ratio. With increase in pressure ratio, though the efficiency increases, at higher pressure ratios net-work decreases and hence there is a need to calculate a pressure ratio where the work output is maximum. At this pressure ratio, net-work is minimum is known as Optimum pressure ratio. π‘Šπ‘›π‘’π‘‘ = π‘Šπ‘‘ βˆ’ π‘Šπ‘ = 𝑐 𝑝 Β· (𝑇3 βˆ’ 𝑇4) βˆ’ 𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇1) = 𝑐 𝑝(𝑇3 βˆ’ 𝑇4 βˆ’ 𝑇2 + 𝑇1) 𝑇2 𝑇1 = 𝑇3 𝑇4 ⟹ 𝑻 πŸ’ = 𝑻 πŸ‘ Β· 𝑻 𝟏 𝑻 𝟐 π‘Šπ‘›π‘’π‘‘ = 𝑐 𝑝 Β· [𝑇3 βˆ’ 𝑇3 Β· 𝑇1 𝑇2 βˆ’ 𝑇2 + 𝑇1] πΉπ‘œπ‘Ÿ π‘šπ‘Žπ‘₯π‘–π‘šπ‘’π‘š π‘Šπ‘›π‘’π‘‘, 𝒅𝑾 𝒏𝒆𝒕 𝒅𝑻 𝟐 = 𝟎 π‘‘π‘Šπ‘›π‘’π‘‘ 𝑑𝑇2 = 𝑐 𝑝 Β· [0 + 𝑇3 Β· 𝑇1 𝑇2 2 βˆ’ 1 + 0] = 0 ⟹ 𝑇2 2 = 𝑇3 Β· 𝑇1 ⟹ 𝑻 𝟐 = βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 𝑇4 = 𝑇3 Β· 𝑇1 𝑇2 ⟹ 𝑇4 = 𝑇3 Β· 𝑇1 βˆšπ‘‡3 Β· 𝑇1 ⟹ 𝑻 πŸ’ = βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 𝑻 πŸ’ = 𝑻 𝟐 = βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 (rp)A < (rp)B < (rp)C Ξ·A < Ξ·B < Ξ·C
  • 19. π‘Ÿπ‘ = 𝑃2 𝑃1 = ( 𝑇2 𝑇1 ) 𝛾 π›Ύβˆ’1 ⟹ 𝒓 𝒑, π‘œπ‘π‘‘π‘–π‘šπ‘’π‘š = ( 𝑇2 𝑇1 ) 𝛾 π›Ύβˆ’1 = ( βˆšπ‘‡3 Β· 𝑇1 𝑇1 ) 𝛾 π›Ύβˆ’1 ⟹ 𝒓 𝒑, π’π’‘π’•π’Šπ’Žπ’–π’Ž = ( 𝑻 πŸ‘ 𝑻 𝟏 ) 𝜸 𝟐·(πœΈβˆ’πŸ) β†’ π‘Šπ‘›π‘’π‘‘ = (𝑐 𝑝 Β· (𝑇3 βˆ’ 𝑇4)) βˆ’ (𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇1)) = 𝑐 𝑝(𝑇3 βˆ’ 𝑇4 βˆ’ 𝑇2 + 𝑇1) = 𝑐 𝑝(𝑇3 βˆ’ βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 βˆ’ βˆšπ‘» πŸ‘ Β· 𝑻 𝟏 + 𝑇1) 𝑾 𝒏𝒆𝒕 = 𝒄 𝒑 Β· (βˆšπ‘» πŸ‘ βˆ’ βˆšπ‘» 𝟏) 𝟐 Maximum Pressure ratio (rp max) For maximum pressure ratio, rp, max β†’ T2=T3 π‘Ÿπ‘ = 𝑃2 𝑃1 = ( 𝑇2 𝑇1 ) 𝛾 π›Ύβˆ’1 π‘Ÿπ‘, π’Žπ’‚π’™ = ( 𝑻 𝟐 𝑇1 ) 𝛾 π›Ύβˆ’1 = ( 𝑻 πŸ‘ 𝑇1 ) 𝛾 π›Ύβˆ’1 = ( 𝑇 π‘šπ‘Žπ‘₯ 𝑇 π‘šπ‘–π‘› ) 𝛾 π›Ύβˆ’1 π‘Ÿπ‘, π‘œπ‘π‘‘π‘–π‘šπ‘’π‘š = ( 𝑇3 𝑇1 ) 𝛾 2Β·(π›Ύβˆ’1) = ( 𝑇 π‘šπ‘Žπ‘₯ 𝑇 π‘šπ‘–π‘› ) 𝛾 2Β·(π›Ύβˆ’1) 𝒓 𝒑, π’π’‘π’•π’Šπ’Žπ’–π’Ž = (𝒓 𝒑, π’Žπ’‚π’™) 𝟏 𝟐 Methods of improving the performance of gas turbine cycle Regeneration πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = π‘Šπ‘‡ βˆ’ π‘ŠπΆ 𝑄𝑠 = (β„Ž3 βˆ’ β„Ž4) βˆ’ (β„Ž2 βˆ’ β„Ž1) β„Ž3 βˆ’ β„Ž π‘₯
  • 20. Effects of Regenerator 1. No change in compressor work 2. No change in turbine work 3. No change in Net-work 4. Decrease in heat supply 5. Decrease in heat rejection 6. Increase in efficiency Effectiveness of a Regenerator ∈ = Actual gain in heat Ideal gain in heat/Maximum heat transfer = π‘š π‘Ž Β· (β„Ž π‘₯ βˆ’ β„Ž2) π‘š π‘Ž+𝑔 Β· (β„Ž4 βˆ’ β„Ž2) = π‘š π‘Ž Β· 𝑐 𝑝,π‘Žπ‘–π‘Ÿ Β· (𝑇π‘₯ βˆ’ 𝑇2) (π‘š π‘Ž + π‘š 𝑔) Β· 𝑐 𝑝,π‘”π‘Žπ‘  Β· (𝑇4 βˆ’ 𝑇2) Generally, mass of the fuel is very small in comparison to mass of air. Therefore, mf can be neglected w.r.t ma if cp of air and cp, gas are almost same. Then, π‘š 𝑓 β‰ͺ π‘š π‘Ž, 𝑐 𝑝, π‘Žπ‘–π‘Ÿ β‰ˆ 𝑐 𝑝, π‘”π‘Žπ‘  ⟹ ∈= 𝑻 𝒙 βˆ’ 𝑻 𝟐 𝑻 πŸ’ βˆ’ 𝑻 𝟐 = π‘»π’‰π’†π’“π’Žπ’‚π’ π‘Ήπ’‚π’•π’Šπ’ Note 1. Regenerator will be effective only when the temperature difference between turbine exit (T4) and compressor exit (T2) is very large. 2. In case of maximum work output cycle as T2=T4, Regenerator can’t be used. Reheating π‘Šπ‘–π‘‘β„Žπ‘œπ‘’π‘‘ π‘Ÿπ‘’β„Žπ‘’π‘Žπ‘‘π‘–π‘›π‘” β†’ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ = (β„Ž3 βˆ’ β„Ž π‘Ž) + (β„Ž π‘Ž βˆ’ β„Ž4β€²) = β„Ž3 βˆ’ β„Ž4β€² π‘Šπ‘–π‘‘β„Ž π‘Ÿπ‘’β„Žπ‘’π‘Žπ‘‘π‘–π‘›π‘” β†’ π‘Šπ‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’ = (β„Ž3 βˆ’ β„Ž π‘Ž) + (β„Ž 𝑏 βˆ’ β„Ž4) Effects of Reheating 1. No change in compressor work. 2. Increase in turbine work (on h-s diagram constant pressure line diverge). 3. Increase in Net work. 4. Temperature at the exit of turbine (T4) is greater than T4’ and hence with reheating, scope for regenerator increases. Therefore, reheating cycles are generally combined with regeneration. 5. Increase in heat supply. Note With reheating both heat supplied and net-work increases. Therefore, efficiency depends in relative increase of both values. πœ‚ = π‘Šπ‘‡ βˆ’ π‘Šπ‘ 𝑄𝑠 = (β„Ž3 βˆ’ β„Ž π‘Ž) + (β„Ž 𝑏 βˆ’ β„Ž4) βˆ’ (β„Ž2 βˆ’ β„Ž1) (β„Ž3 βˆ’ β„Ž2) + (β„Ž3 βˆ’ β„Ž π‘Ž) Reheating and intercooling are constant pressure processes.
  • 21. Intercooling Effects of Intercooling 1. Decrease in compressor work 2. No change in turbine work 3. Increase in net-work 4. Increase in heat supply 5. With intercooling, the temperature difference between turbine exit and compressor exit increases and hence the scope for regeneration increases with intercooling. Therefore, intercooling cycles are generally combined with regeneration. Note With intercooling both net-work and heat supply increases, therefore, (increase in) efficiency depends on relative increase of both values. πœ‚ = π‘Šπ‘›π‘’π‘‘ 𝑄𝑠 = (β„Ž3 βˆ’ β„Ž4) βˆ’ (β„Ž π‘Ž βˆ’ β„Ž1) + (β„Ž2 βˆ’ β„Ž 𝑏) β„Ž3 βˆ’ β„Ž2 Condition for minimum work input to the compressor with perfect intercooling (T1=Tb). π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ = π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ 𝟏 + π‘Šπ‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ 𝟐 = (β„Ž π‘Ž βˆ’ β„Ž1) + (β„Ž2 βˆ’ β„Ž 𝑏) = 𝑐 𝑝 Β· (π‘‡π‘Ž βˆ’ 𝑇1) + 𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇𝑏) π‘Šπ‘ = 𝑐 𝑝 Β· 𝑇1 Β· [ π‘‡π‘Ž 𝑇1 βˆ’ 1] + 𝑐 𝑝 Β· 𝑻 𝒃 Β· [ 𝑇2 𝑇𝑏 βˆ’ 1] = 𝑐 𝑝 Β· 𝑇1 Β· [ π‘‡π‘Ž 𝑇1 βˆ’ 1] + 𝑐 𝑝 Β· 𝑻 𝟏 Β· [ 𝑇2 𝑇𝑏 βˆ’ 1] = 𝑐 𝑝 Β· 𝑇1 Β· [ π‘‡π‘Ž 𝑇1 βˆ’ 1 + 𝑇2 𝑇𝑏 βˆ’ 1] π‘Šπ‘ = 𝑐 𝑝 Β· 𝑇1 Β· [( 𝑃𝑖 𝑃1 ) π›Ύβˆ’1 𝛾 + ( 𝑃2 𝑃𝑖 ) π›Ύβˆ’1 𝛾 βˆ’ 2] = 𝑐 𝑝 Β· 𝑇1 Β· [( 𝑃𝑖 𝑃1 ) π‘₯ + ( 𝑃2 𝑃𝑖 ) π‘₯ βˆ’ 2] ( 𝛾 βˆ’ 1 𝛾 = π‘₯) πΉπ‘œπ‘Ÿ π‘šπ‘–π‘›π‘–π‘šπ‘’π‘š π‘€π‘œπ‘Ÿπ‘˜ 𝑖𝑛𝑝𝑒𝑑 π‘œπ‘“ π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ, π‘‘π‘Šπ‘– 𝑑𝑃𝑖 = 0 π‘‘π‘Šπ‘– 𝑑𝑃𝑖 = 𝑐 𝑝 Β· 𝑇1 Β· [( 1 𝑃1 π‘₯ Β· π‘₯ Β· 𝑃𝑖 π‘₯βˆ’1 ) + (𝑃2 π‘₯ Β· (βˆ’π‘₯) Β· 𝑃𝑖 βˆ’π‘₯βˆ’1 ) βˆ’ 0] = 0 β†’ π‘₯ Β· 𝑃𝑖 π‘₯βˆ’1 𝑃1 π‘₯ = π‘₯ Β· 𝑃2 π‘₯ 𝑃𝑖 π‘₯+1 ⟹ 𝑃𝑖 π‘₯βˆ’1+π‘₯+1 = 𝑃1 π‘₯ Β· 𝑃2 π‘₯ = 𝑃𝑖 2π‘₯ ⟹ π‘·π’Š = βˆšπ‘· 𝟏 Β· 𝑷 𝟐
  • 22. π‘Šπ‘1 = β„Ž π‘Ž βˆ’ β„Ž1 = 𝑐 𝑝 Β· (π‘‡π‘Ž βˆ’ 𝑇1) = 𝑐 𝑝 Β· 𝑇1 Β· ( π‘‡π‘Ž 𝑇1 βˆ’ 1) = 𝑐 𝑝 Β· 𝑇1 Β· [( π‘·π’Š 𝑃1 ) π›Ύβˆ’1 𝛾 βˆ’ 1] ⟹ 𝑐 𝑝 Β· 𝑇1 Β· [( 𝑷 𝟐 𝑃1 ) π›Ύβˆ’1 πŸΒ·π›Ύ βˆ’ 1] π‘Šπ‘2 = β„Ž2 βˆ’ β„Ž 𝑏 = 𝑐 𝑝 Β· (𝑇2 βˆ’ 𝑇𝑏) = 𝑐 𝑝 Β· 𝑇𝑏 Β· ( 𝑇2 𝑇𝑏 βˆ’ 1) = 𝑐 𝑝 Β· 𝑻 𝒃 Β· [( 𝑷 𝟐 𝑃𝑖 ) π›Ύβˆ’1 𝛾 βˆ’ 1] ⟹ 𝑐 𝑝 Β· 𝑻 𝟏 Β· [( 𝑷 𝟐 𝑃1 ) π›Ύβˆ’1 πŸΒ·π›Ύ βˆ’ 1] 𝑾 π’„πŸ = 𝑾 π’„πŸ For perfect intercooling and minimum work input each stage work input is same, therefore total compressor work is equal to (No. of stages (N)) Γ— (Each stage compressor work (Wc)). Brayton cycle with many stages of intercooling and many stages of reheating can be reduced to Ericsson cycle. Many stages if intercooling and reheating can be reduced to isothermal process.