Theory of turbomachinery Chater
Introduction: Basic Princiles Take your choice of those that can best aid your action. (Shakeseare, Coriolanus)
Introduction Definition Turbomachinery describes machines that transfer energy between a rotor and a fluid, including both turbines and comressors (source: Wiki). Devices in which energy is transferred, either to, or from, a continuously flowing fluid by the dynamic action of one or more moving blade rows (Dixon) The words rotor and continuous searate turbomachines from e.g. recirocating (iston) engines
Introduction Why a course on turbomachines? Close to all electric ower is roduced by turbomachines They consume large arts of energy used in many industrial rocesses They are integral arts of gas turbines used in e.g. aircraft engines and as (shaft-) ower suly in oil and gas industry (for ums and comressors) as well as roulsion of shis
Samles Windower Steam turbines Hydroower Turbochargers of cars and trucks Vacuum cleaners Pums Dental drills
Samles
Introduction Classifications Energy may flow to the fluid (increasing velocity and/or ressure) or from the fluid roducing shaft ower Flowath: Axial or radial machines (mixed flow) Changes in density, comressible or incomressible analyses. Imulse or reaction machines: Does the ressure change in the rotor, or in a set of nozzles before the rotor?
Samles FIG... Fig. Examles of turbobomachines
Axial-flow turbines FIG. 4.. Large low ressure steam turbine (Siemens)
Axial-flow turbines FIG. 4.. Turbine module of a modern turbofan jet engine (RR)
Axial-flow Turbines: -D theory Note direction of α FIG. 4.3. Turbine stage velocity diagrams.
Coordinates c c + m x c r streamwise coordinate Fig.: The coordinate system
Coordinates () Fig.: The coordinate system
Velocity triangles U : Blade seed c : Absolute velocity w: Relative velocity α : Absolute flow angle β : Relative flow angle 3 Fig.3: Velocity triangles for an axial comressor stage
Fundamental laws The continuity of flow equation mass conservation First law of thermodynamics and the steady flow energy equation The law of conservation of energy states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but cannot be created or destroyed. The momentum equation, F m a The second law of thermodynamics Total entroy of an isolated system always increases over time, or remains constant in ideal cases where the system is in a steady state or undergoing a reversible rocess
System vs. Control Volume System: A collection of matter of fixed identity Always the same atoms or fluid articles A secific, identifiable quantity of matter Control Volume (CV): A volume in sace through which fluid may flow A geometric entity indeendent of mass
Equation of continuity The mass flow through a surface element da: dm ρ cndt da c n c cosθ So that dm dm dt ρ cnda Or, if A and A are flow areas at stations and along a assage: m ρ c A ρ c A ρ c n n n A Fig.4: Flow across an element of area
The first law The first law of Thermodynamics: For a system that comletes a cycle during which heat is sulied and work is done: ( d Q dw ) 0 If a change is done from state to state, energy differences must be reresented by changes in internal energy: ( dq W ) E E d or de dq dw
The first law Mass flow, m, enters at and exits at Energy is transferred from fluid to the blades of the machine, ositive work is at the rate W Heat transfer, Q x, is ositive from surrounding to machine
The first law The steady state energy equation becomes (Reynold s transort theorem) [( h h ) + ( c c ) + g( z )] Q W x m z Neglecting otential energy and using total (stagnation) enthaly: ( ) Q W x m h 0 h 0 For an adiabatic work roducing machine (turbine): Q 0 W x > 0 W x ( ) W t m h 0 h 0 And for adiabatic work absorbing machines (comressors): W x < 0 W c ( ) W x m h 0 h 0
The momentum equation Newton's second law: The sum of all forces acting on a mass, m, equals the time rate of momentum change: ΣF d dt ( ) x mc x Here, only x-comonent of force and velocity is considered. For steady state the equation reduces to: ΣF x ( c ) m c x x If shear forces (viscosity) are neglected, the Euler s equation for one-dimensional flow can be obtained: d + c dc + g dz ρ 0
( ) 0 d + + z z g c c ρ Integrating Euler s equation in the stream direction yields Bernoulli s equation: Control volume in a streaming fluid. The momentum equation
Bernoulli s equation For an incomressible fluid (constant density) using total or stagnation ressure: 0 + ρ c ρ ( ) + g ( z z ) 0 0 0 Using the Head, defined as reduces Bernoulli s eq. to: H H z + H 0 0 ( ρg) For an comressible fluid, changes in otential are often negligible: d + ρ c c 0 For small ressure changes (or isentroic rocesses) : 0 0 0
Units [( ) ( ) ( )] h h + c c + g z Q W x m m m m g kg s J kg ( h h ): W z J s ( c ) c kg m Nm J : W s s kg m Nm J ( z z ) m W : s s s s s s Newton is the force needed to accelerate kilogram of mass at the rate of meter er second squared Joule is the energy transferred to (or work done on) an object when a force of one newton acts on that object in the direction of its motion through a distance of one meter
Moment of momentum For a system of mass m, the sum of external forces acting on the system about the axis A-A is equal to the time rate of change of angular momentum: d τ A m rc θ dt τ ( ) Where r is the distance of the mass center from the axis of rotation and c θ is the tangential velocity comonent. For one-dimensional steady flow, entering at radius r with tangential velocity c θ and leaving at r with c θ : A m r ( c rc ) θ θ Multilication with the angular velocity Ω U/r, where U is the blade seed, yields : τ Ω A ( c U c ) m U θ θ
Euler s um and turbine equations (.6) The work done on the fluid er unit mass (secific work) becomes: 0 > Ω θ θ τ U c c U m m W W A c c 0 > θ θ c U U c m W W t t Control volume for a generalized turbomachine.
Examle: radial um
Velocity triangles at in- and outlet Rotor inlet: Relative velocity tangent to blade Rotor exit: tangential velocity induced W W m τ Ω m [ c ] θ U c c A U cθ Ucθ 0 θ
Rothaly Combining the first law of thermodynamics and Euler s um equation (from Newton s second law): W c W c m U cθ Ucθ h0 h0 Rearranging and using the definition of stagnation enthaly, allows the definition of the rothaly, I: h + c U c h + c U c I θ θ Where I h + c Uc θ does not change from entrance to exit.
The second law of Thermodynamics Clausius Inequality: For a system assing through a cycle involving heat exchange, d T Q 0 where dq is an element of heat transferred to the system at an absolute temerature T. If the entire rocess is reversible, dq dq R, equality holds true: dq R T 0 From this, the entroy is defined. For a finite change of state: S S dq T R or d S m ds dq T R m being the mass of the system
Entroy For steady one-dimensional flow in which the fluid goes from state to state : dq T m ( s s ) For adiabatic rocesses, dq 0 and this becomes: s s For a system undergoing a reversible rocess, dq dq R m T ds and dw dw R m dυ, the first law becomes: de dq - dw m T ds - m dυ T ds du - dυ or with u E / m Further, with h u + υ, dh du + dυ + υ d: T ds dh - υ d
Definitions of efficiency Consider a turbine: The overall efficiency can be defined as η 0 Mechanical energy available at couling of outut shaft in unit time Maximum energy difference ossible for the fluid in unit time If mechanical losses in bearings etc. are not the aim of the analyses, the isentroic or hydraulic efficiency is suitable: η t Mechanical energy sulied to the rotor in unit time Maximum energy difference ossible for the fluid in unit time The Mechanical efficiency now becomes η 0 / η t
Efficiency From the steady flow energy equation, and the second law of thermodynamics, dq can be eliminated to obtain: [ ] z g c h m W Q x d d d d d + + ( ) h m s mt Q d d d d υ [ ] z g c m W x d d d d + + υ For a turbine (ositive work) this integrates to: ( ) ( ) d z z g c c m W x + + υ
Efficiency Once more alying T ds dh - υ d 0 for the reversible adiabatic rocess: d m [ dh + dc g dz] W x, max + and hence the maximum work from state to state is: W [ dh + dc + g z] m ( h h ) + g( z z ), max m x d [ ] 0 where the subscrit s denotes an isentroic change from state to state 0s In the incomressible case, neglecting friction losses: [ H ] W x, max mg H where gh ρ + c + gz
The icture can't be dislayed. Efficiency Enthaly-entroy diagrams for turbines and comressors.
Efficiency Neglecting otential energy terms, the actual turbine rotor secific work becomes: W x W x m h ( c ) 0 h0 h h + c And, similarly, the ideal turbine rotor secific work becomes: W W m h h h h ( c c ) x, max x,max 0 0s s s + where the subscrit s denotes an isentroic change from state to state
Efficiency If the kinetic energy can be made useful, we define the total-to-total efficiency as η tt ( h h ) ( h h ) Wx Wx, max 0 0 0 0s Which, if the difference between inlet and outlet kinetic energies is small, reduces to η tt ( h ) ( h h ) h s If the exhaust kinetic energy is wasted, it is useful to define the totalto-static efficiency as η ts ( h ) ( h h ) h0 0 0 s Since, here the ideal work is obtained between oints 0 and s
Efficiency Efficiencies of comressors are obtained from similar considerations: η c Minimum work inut Actual work inut to rotor η c ( h ) ( h ) h0 s 0 0 h0 Which, if the difference between inlet and outlet kinetic energies is small, reduces to η c ( h ) ( h ) h s h
Small stage or olytroic efficiency If a comressor is considered to be comosed of a large number of small stages, where the rocess goes from states - x - y -. -, we can define a small stage efficiency 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 efficiency, then ΣδW ( h h ) + ( h h ) + ( h ) x y x... h η ΣδW min ΣδW η and thus c η [( h h ) + ( h h ) + ] ( h ) xs ys x... h ( h ) ( h ) h s h However, since the constant ressure curves diverge: ( h h ) + ( h h ) + > ( h ) xs ys x... s h and c η > η
Small stage or olytroic efficiency If T ds dh - υ d, then for constant ressure: (dh / ds) T or At equal values of T: (dh / ds) constant For a erfect gas, h C T, (dh / ds) constant for equal h
Exansion Assume two identical stages with: 0 η h0,s η const, h 0 const Define the total turbine efficiency as: 03 h h Δh h h h h 0 03 0 0 03,ss 0 03,ss By observation from the figure: ( h0 h03, s ) > ( h0, s h03, ss ) η ( h0 h03, ss ) 03 h < η > η 0 h h η const 0 0 0,s 03 03,ss 03,s 0 0 03 s Usage of olytroic and isentroic efficiencies?
Small stage efficiency for a erfect gas Now comression! For the isentroic rocess T ds dh - υ d 0 and with h C T, the olytroic efficiency becomes: dh η is dh υ d C dt
Small stage efficiency for a erfect gas η Substituting υ RT / into R C T d dt dh η is dh υ d C dt And with C γ R / (γ ): dt T γ γη d With constant γ and efficiency, this integrates to T T ( γ ) γη
Small stage efficiency for a erfect gas For the ideal comression, η, and the temerature ratio becomes: ( ) ( ) c γη γ γ γ η Which is also obtainable from γ constant and υ RT. If this is substituted into the isentroic efficiency of comression for a erfect gas, ( ) γ γ T T a relation between the isentroic and olytroic efficiencies is obtained: ( ) ( ) T T T T s c η
The icture can't be dislayed. Small stage efficiency for a erfect gas
For a turbine, similar analyses results in ( ) ( ) γ γ γ γ η η t and ( ) γ γ η T T Thus, for a turbine, the isentroic efficiency exceeds the olytroic (or small stage) efficiency. Small stage efficiency for a erfect gas
The icture can't be dislayed. Small stage efficiency for a erfect gas
Reheat factor For e.g. steam turbines i.e. the ratio of the sum of small isentroic enthaly changes to the overall isentroic enthaly change. Thus: [( h h ) + ( h h ) + ] ( h h ) RH xs x ys... s h h h h Σ h is η t η h h s Σ his h h s R H
Reheat factor Mollier diagram showing exansion rocess through a turbine slit u into a number of small stages.
The inherent Unsteadiness a: Pressure ta at * b: Static ressure measurements vs time