Lecture 12: Thermodynamic cycles - gas turbines • Gas turbines are used extensively in power generation and transport. We all have seen jet engines, but they are also currently used in some high speed boats and even tanks! the primary attractiveness of gas turbines is their power density, i.e. they produce a great deal of power in a small volume. For a given power output, they are not usually more efficient than reciprocating internal combustion engines. However, when used in so-called ‘cogeneration’ of ‘combined cycle’ configurations, they can be very efficient. The turbojet is a simple kind of gas turbine. In cutaway view, a typical turbojet looks like • • • This image reveals the essential functioning of a basic gas turbine: 1. air enters the gas turbine (in this case from the left) through the compressor, and is compressed to a higher pressure and temperature by the action of shaft work onto the compressor. 2. fuel is then mixed with the air within the combustion chamber, where it is ignited. Combustion raises the temperature of the gas mixture significantly, but does not change the pressure significantly. 3. the mixture then passes through the turbine, which expands the mixture to lower temperature and pressure whilst extracting shaft work from the flow. 4. the shaft work produced by the turbine drives the compressor, and the gas mixture leaves the gas turbine through the exhaust nozzle, producing a high speed gas jet that provides thrust. • we could also conceive of a case where the turbine is connected by shaft in one direction to the compressor, but in the other direction to an electrical generator. In this instance, the gas turbine is clearly a heat engine, producing shaft work rather than thrust. It is in this configuration that gas turbines are used to produce power. The ‘Joule/Brayton’ cycle • the Joule/Brayton cycle is the ideal gas turbine cycle, and can be represented as follows. • assumptions: 1. compression and expansion (turbine) are isentropic 2. no pressure losses 3. constant specific heat c p 4. air is the working fluid Thermal efficiency • as always, the thermal efficiency ηth of a heat engine is defined as the work output wout achieved for a given heat input qin : ηth = wout ( wt − wc ) = qin qin • where wc and wt are the compressor and turbine specific work respectively. • since the compression and expansion are both isentropic, they are by definition adiabatic. Assuming low velocities, it follows from the SFEE that: wc = ∆hc = C P (T2 − T1 ) • and wt = ∆ht = c p ( T3 − T4 ' ) • no work is done on the fluid during heat addition. Therefore, from the SFEE: q = ∆hCC = c p ( T3 − T2 ' ) • and: ⎡ c p ( T3 − T4 ' ) − c p ( T2' − T1 ) ⎤ ⎦ ηth = ⎣ c p ( T3 − T2' ) • define the stagnation pressure ratio as rP = p2 / p1 . It follows that: T2' T = rP γ = 3 T1 T4 ' γ −1 • and: 1−γ 1−γ ⎡ ⎛ ⎞ ⎛ γ γ ⎢T3 ⎜ 1 − rP ⎟ − T2 ' ⎜ 1 − rP ⎜ ⎟ ⎜ ⎢ ⎠ ⎝ ηth = ⎣ ⎝ (T3 − T2 ' ) ⎞⎤ ⎟⎥ ⎟ ⎠⎥ ⎦ • collecting like terms, this becomes: 1−γ ηth = 1 − rP γ • so, the thermal efficiency of the ideal gas turbine depends only on the pressure ratio rP and the ratio of specific heats γ . • the variation in thermal efficiency with pressure ratio is as follows. 60 Ideal cycle efficiency % 50 40 30 20 10 0 0 2 4 6 8 10 Compression ratio r p Specific power output • the specific power output ( w ) is the power output per unit mass flow rate through the gas turbine ( Ws / kg ) • from the SFEE: w = wt − wc ⎛ T ⎞ ⎛T ⎞ = c pT3 ⎜ 1 − 4' ⎟ − c pT1 ⎜ 2 ' − 1⎟ T3 ⎠ ⎝ T1 ⎠ ⎝ 1−γ ⎛ = c pT3 ⎜ 1 − rP γ ⎜ ⎝ ⎞ ⎛ γ −1 ⎞ − c pT1 ⎜ rP γ − 1⎟ ⎟ ⎟ ⎜ ⎟ ⎠ ⎝ ⎠ γ −1 • where, as before: T2' T = rP γ = 3 T1 T4 ' • let: γ −1 x = rP γ • the expression for the specific work then becomes: ⎛ 1⎞ w = c pT3 ⎜ 1 − ⎟ − c pT1 ( x − 1) x⎠ ⎝ • and the maximum specific work occurs when: • i.e. dw c pT3 = 2 − c pT1 dx x x2 = T3 T1 • since: γ −1 x = rP γ = T2 ' T3 = T1 T4 ' • it follows that: T3 ⎛ T2' ⎞ ⎛ T3 ⎞ = ⎜ ⎟⎜ ⎟ T1 ⎝ T1 ⎠ ⎝ T4' ⎠ x x • giving: • or: T2 ' = T4 ' ⎛ T ⎞ 2(γ −1) rP = ⎜ 3 ⎟ ⎝ T1 ⎠ γ • in terms of the T − s diagram, the condition that T2 ' = T4 ' for maximum specific work determines the ‘shape’ of the cycle for a given pressure ratio. • Remember that, for an ideal cycle, the thermal efficiency ηth depends only on rP and γ . Thus, ‘long skinny’ ideal cycles and ‘short fat’ ideal cycles on the T − s diagram will have the same thermal efficiency if their pressure ratio is the same. • However, if we want to produce work compactly (i.e. in a small space and cheaply), we wish to maximise w ( Ws / kg ) by letting T2 ' = T4 ' . This fixes the shape of the cycle in T − s space.
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