Theory of turbo machinery / Turbomaskinernas teori. Chapter 2

Transcription

1 Theory of turbo mahinery / Turbomaskinernas teori Chater

2 Basi Thermodynamis, Fluid Mehanis: Definition of Effiieny Take your hoie of those that an best aid your ation. (Shakeseare, Coriolanus) The ontinuity of flow equation (mass onservation) First law of thermodynamis and the steady flow energy equation The momentum equation The seond law of thermodynamis

3 Equation of ontinuity The mass flow through a surfae element da: dm ρ ndt da n osθ So that dm dm dt (.) ρ n d A Or, if A and A are flow areas at stations and along a assage: (.) m n n ρ A ρ A ρ n A

4 The first law The first law of Thermodynamis: For a system that omletes a yle during whih heat is sulied and work is done: ( d Q dw ) 0 (.3) If a hange is done from state to state, energy differenes must be reresented by hanges in roerty internal energy: ( dq W ) E E d or (.4) de dq dw (.4a)

5 The first law Mass flow, m, enters at and exits at Energy is transferred from fluid W to the blades of the mahine, ositive work is at the rate x Heat transfer, Q, is ositive from surrounding to mahine

6 The first law The steady state energy equation beomes [( h h ) + ( ) + g( z )] Q W x m z (.5) Negleting otential energy and using total enthaly ( ) Q W x m h 0 h 0 (.6) For an adiabati work roduing mahine (turbine): Q 0 W x > 0 W x ( ) W t m h 0 h 0 (.7) And for adiabati work absorbing mahines (omressors): W x < 0 W ( ) (.8) W x m h 0 h 0

7 The momentum equation Newton's seond law: The sum of all fores ating on a mass, m, equals the time rate of momentum hange: d Σ Fx ( m x ) (.9) dt Here, only x-omonent of fore and veloity is onsidered. For steady state the equation redues to: x ( ) ΣF m (.9a) x x If shear fores (visosity) are negleted, the Euler s equation for one-dimensional flow an be obtained: d + d + g dz ρ 0 (.0)

8 The momentum equation Integrating Euler s equation in the stream diretion yields Bernoulli s equation: d + ρ + g ( z z ) 0 (.0a) FIG..3. Control volume in a streaming fluid.

9 Bernoulli s equation For an inomressible fluid (onstant density) using total or stagnation ressure: 0 + ρ ρ ( ) + g ( z z ) (.0b) Using the Head, defined as H z + 0 redues Bernoulli s eq. to: H H 0 ( ρg) (.0) For an omressible fluid, hanges in otential are negligible: d + ρ 0 (.0d) For small ressure hanges (or isentroi roesses) : (.0e)

10 Moment of momentum For a system of mass m, the sum of external fores ating on the system about the axis A-A is equal to the time rate of hange of angular momentum: d τ A m ( r θ ) (.) dt τ Where r is the distane of the mass enter from the axis of rotation and θ is the tangential veloity omonent. For one-dimensional steady flow, entering at radius r with tangential veloity θ and leaving at r with θ : ( r ) A m r θ θ (.a) Multiliation with the angular veloity Ω U/r, where U is the blade seed, yields : ( U ) τ Ω m U (.) A θ θ

11 Euler s um and turbine equations The work done on the fluid er unit mass (seifi work) beomes: 0 > Ω Δ θ θ τ U U m m W W A (.a) 0 > Δ θ θ U U m W W t t (.b) FIG..4. Control volume for a generalised turbomahine.

12 Exemel radialum

13 Hastighetstrianglar vid in- & utlo Relativ hastighet tangent till skoveln Absoluthast. ändras i rotationsriktningen (um) x [ 0] W h h U U U 0 0 θ θ θ θ

14 Rothaly Combining the first law of thermodynamis and Euler s um equation (Newton s seond law): ΔW W m U θ Uθ h0 h0 (.) Rearranging and using the definition of stagnation enthaly, allows the definition of the rothaly, I: h U h U θ θ I (.d) Where I h U θ does not hange from entrane to exit.

15 The seond law of Thermodynamis Clausius Inequality: For a system assing through a yle involving heat exhange, d T Q 0 (.3) where dq is an element of heat transferred to the system at an absolute temerature T. If the entire roess is reversible, dq dq R, equality holds true: dq R T 0 (.3a) From this, the entroy is defined. For a finite hange of state: S S dq T R or d S m ds dq T R (.4,.4a) m being the mass of the system

16 Entroy For steady one-dimensional flow in whih the fluid goes from state to state : dq T m ( s s ) (.5) For adiabati roesses, dq 0 and.5 beomes: s s For a system undergoing a reversible roess, dq dq R m T ds and dw dw R m dυ, the first law beomes: de dq -dw m T ds - m dυ or with u E / m T ds du - dυ (.7) Further, with h u + υ, dh du + dυ + υ d: T ds dh - υ d (.8)

17 Definitions of effiieny Consider a turbine: The overall effiieny an be defined as η 0 Mehanial energy available at ouling of outut shaft in unit time Maximum energy differene ossible for the fluid in unit time If mehanial losses in bearings et. are not the aim of the analyses, the isentroi or hydrauli effiieny is suitable: η t Mehanial energy sulied to the rotor in unit time Maximum energy differene ossible for the fluid in unit time The Mehanial effiieny now beomes η 0 / η t

18 Effiieny From the steady flow energy equation, (.9) and the seond law of thermodynamis, dq an be eliminated to obtain: [ ] z g h m W Q x d d d d d + + ( ) h m s mt Q d d d d υ [ ] z g m W x d d d d + + υ For a turbine (ositive work) this integrates to: ( ) ( ) d z z g m W x + + υ (.0)

19 Effiieny One more alying T ds dh - υ d 0 for the reversible adiabati roess: d W x [ dh + d g dz], max m + and hene the maximum work from state to state is: W [ dh + d + g z] m ( h h ) + g( z z ), max m x d [ ] 0 (.0a) where the subsrit s denotes an isentroi hange from state to state 0s In the inomressible ase, negleting frition losses: [ H ] W x, max mg H where gh ρ + + gz (.0b)

20 Effiieny FIG..5. Enthaly-entroy diagrams for turbines and omressors.

21 Effiieny Negleting otential energy terms, the atual turbine rotor seifi work beomes: ΔW x W x m h ( ) 0 h0 h h + And, similarly, the ideal turbine rotor seifi work beomes: ΔW W m h h h h ( ) x, max x,max 0 0s s s + where the subsrit s denotes an isentroi hange from state to state

22 Effiieny If the kineti energy an be made useful, we define the total-to-total effiieny as η Δ tt ( h h ) ( h h ) Wx ΔWx, max s Whih, if the differene between inlet and outlet kineti energies is small, redues to (.) η tt ( h )( h h ) h s (.a) If the exhaust kineti energy is wasted, it is useful to define the totalto-stati effiieny as η ts ( h ) ( h h ) h0 0 0 s (.) Sine, here the ideal work is obtained between oints 0 and s

23 Effiieny Effiienies of omressors are obtained from similar onsiderations: η Minimum adiabati work inut er unit time Atual adiabati work inut to rotor er unit time η ( h ) ( h ) h0 s 0 0 h0 (.8) Whih, if the differene between inlet and outlet kineti energies is small, redues to η ( h ) ( h ) h s h (.8a)

24 Small stage or olytroi effiieny If a omressor is onsidered to be omosed of a large number of small stages, where the roess goes from states - x - y -. -, we an define a small stage effiieny as η ( h h ) ( h h ) ( h h ) ( h h )... δwmin δw xs x ys x y x If all small stages have the same effiieny, then ΣδW ( h h ) + ( h h ) + ( h ) x y x... h η ΣδWmin ΣδW η and thus η [( h h ) + ( h h ) + ] ( h ) xs ys x... h ( h )( h ) h s h However, sine the onstant ressure urves diverge: ( h h ) + ( h h ) + > ( h ) xs ys x... s h and η > η

25 Small stage or olytroi effiieny If T ds dh - υ d, then for onstant ressure: (dh / ds) T or At equal values of T: (dh / ds) onstant For a erfet gas, h C T, (dh / ds) onstant for equal h

26 Small stage effiieny for a erfet gas For the isentroi roess T ds dh - υ d 0 and with h C T The olytroi effiieny beomes: dh η is dh υ d C dt (.3)

27 Small stage effiieny for a erfet gas Substituting υ RT / into (.3): η R C T d dt And with C γ R / (γ ): dt T γ γη d (.3) With onstant γ and effiieny, this integrates to T T ( γ ) γη (.33)

28 Small stage effiieny for a erfet gas For the ideal omression, η, and the temerature ratio beomes: ( ) ( ) γη γ γ γ η (.36) Whih is also obtainable from γ onstant and υ RT. If this is substituted into the isentroi effiieny of omression for a erfet gas, ( ) γ γ T T (.35) a relation between the isentroi and olytroi effiienies is obtained: ( )( ) T T T T s η (.34)

29 Small stage effiieny for a erfet gas

30 Small stage effiieny for a erfet gas For a turbine, similar analyses results in ( ) ( ) γ γ γ γ η η t (.38) and ( ) γ γ η T T (.37) Thus, for a turbine, the isentroi effiieny exeeds the olytroi (or small stage) effiieny.

31 Small stage effiieny for a erfet gas

32 Reheat fator For e.g. steam turbines [( h h ) + ( h h ) + ] ( h h ) RH xs x ys... s i.e. the ratio of the sum of small isentroi enthaly hanges to the overall isentroi enthaly hange. Thus: h h h h ΣΔh is η t η h h s ΣΔhis h h s R H (.39)

33 Reheat fator FIG..0. Mollier diagram showing exansion roess through a turbine slit u into anumber of small stages.

34 Nozzles Devie in whih the fluid is aelerated at the exense of ressure dro In the adiabati ase, this means that stagnation enthaly is onserved Tyially this ours in the omressor inlet and in the stationary blade rows in turbines (imulse mahines) At subsoni onditions, a nozzle reresents a ontration (derease in flow ross setional area)

35 Nozzle effiieny s s N h h h h 0 0 η Enthaly loss oeff. Nozzle effiieny h h s N ζ Veloity oeff. s N K Combining these: N N N K + ζ η FIG... Relationshi between reheat fator, ressure ratio and olytroi effiieny (n.3).

36 Nozzle effiieny FIG... Mollier diagrams for the flow roesses through a nozzle and a diffuser: (a) nozzle; (b) diffuser.

37 Diffusers FIG..3. Some subsoni diffuser geometries and their arameters: (a) two-dimensional; (b) onial; () annular.

38 Venturi tube flow meter

39 Diagram FIG..4. Variation of diffuser effiieny with stati ressure ratio for onstant values of total ressure reovery fator (γ.4).

40 ( ) ( ) ( )( ) [ ] ( ) 0 0 γ γ γ γ η D (.53) h h h h s s D η Diffuser effiieny Pressure rise oeff. C ρ Diffuser erformane

41 Diagram FIG..5. Flow regime hart for two-dimensional diffusers (adated from Sovran and Klom 967).

42 Diagram FIG..6. Tyial diffuser erformane urves for a two-dimensional diffuser, with L/W 8 (adated from Kline et al. 959).

43 Diagram FIG..7. Performane hart for onial diffusers with B 0.0 B 0.0 (adated from Sovran and Klom 967).