Turbomachinery Lecture Notes 7-9-4 Design of Multistage Turbine Damian Vogt Course MJ49 Nomenclature Subscripts Symbol Denotation Unit c Absolute velocity m/s c p Specific heat J/kgK h Enthalpy J/kg m& Mass flow rate kg/s p Pressure Pa r Radius m u Tangential velocity m/s v Specific volume m /kg w Relative velocity m/s z Number of stages - R Degree of reaction - R Gas constant J/kg/K T Temperature K W & Power J/s Y Radius ratio Y r s rh - Φ Flow coefficient - Ψ Loading coefficient - Ω Cross section m Absolute flow angle deg β Relative flow angle deg γ Isentropic coefficient - ε Turning deg ρ Density m /kg η Efficiency - ζ Loss coefficient - Rotational speed rad/s Total Inlet stator Outlet stator (inlet rotor) Outlet rotor n Normal s Shroud (tip) h Hub r Radial component x Axial component Engine inlet Engine outlet θ Tangential component
Turbomachinery Lecture Notes 7-9-4 System Discretization Schematic representation inlet outlet W & It is assumed that the turbine consists of one or several stages stator rotor Stage denotations stator inlet rotor inlet rotor outlet Reference radius Stage velocity triangles stator rotor w u u c c w u c
Turbomachinery Lecture Notes 7-9-4 Problem Statement The task is given to perform a preliminary design of a multistage turbine. It is thereby assumed that the following parameters are known Inlet pressure and temperature Outlet pressure Required power Furthermore the following limitations shall be made: Repetition and normal stages shall be valid. Zero exit swirl The expressions of the design parameters are given below for this special case wθ, R ( wθ, wθ, ) Eq. u u wθ, wθ, wθ, ψ u u Eq. c φ x u Eq. Note that the following relationship is valid ψ ( R) Eq. 4 Step : Determination of approximate flow parameters (isentropic expansion) As the geometry and design parameters of the turbine are not known a priori in this first step isentropic expansion shall be assumed. The efficiency is then calculated based on this first assumption and one or several iterations are performed thereafter. Determination of inlet specific volume RT v Eq. 5 p Determination of outlet specific volume Determination of change in enthalpy p γ v v Eq. 6 p γ γ p γ Δh s p v Eq. 7 γ p
Turbomachinery Lecture Notes 4 7-9-4 Determination of mass flow rate c As c Δ h s Δhs and due to the assumption of normal repetition stages c c it follows that Δh s Δh s leading to W m Δ h s & Eq. 8 Step : Determination of inlet and outlet annular geometry Assume: uconst, implying that rconst The radius ratio Y is dependent on flow coefficient and mean radius and yields from the conservation of mass m& Ω ρ cn Ω ρ φ u Eq. 9 rs with Y and r h rm rh rs it follows that rm rh and Y Yrm rs leading to Y ( r ) 4 Y s rh πrm Y Ω π Eq. Substituting this expression into the conservation of mass above it follows that Y Y m & 4πrm ρ φ u 4πrm ρ φ Y Y Eq. By expressing m& A 4πr ρ φ m the radius ratio is obtained from A Y Eq. A Choose Rotational speed Flow coefficient phi Determine Y in appropriate range Note: Too short blades give bad efficiency Too long blades shall be avoided due to high mechanical loads
Turbomachinery Lecture Notes 5 7-9-4 Step : Determination of number of stages The total change in enthalpy between inlet and outlet is distributed over an appropriate number of stages as follows z Δh ψ iu i i Eq. where z denotes the number of stages. Above the assumption of a normal repetition stage at constant radius has been made, thus Ψ i Ψ and u i u. In this first step the isentropic enthalpy change is regarded leading to Δh s z ψ u Eq. 4 The number of stages yields from the following expression z Δh Choose Loading coefficient s ψr Eq. 5 m Note: For the present special case of normal repetition stage at zero exit swirl the loading coefficient depends from the degree of reaction as ψ ( R). The choice of the loading coefficient has to be made such that the number of stages yields an integer number. In case the number of stages is non-integer it has to be rounded to the next higher integer
Turbomachinery Lecture Notes 6 7-9-4 Step 4: Determination of stage efficiency Determination of loss coefficients and total-to-total stage efficiency as follows ( ) h h w c T T R N tt ζ ζ η Eq. 6 Firstly it is assumed that. For a normal repetition stage with zero exit swirl the stage efficiency then yields from T T ( ) ( ) ( φ ζ ψ φ ζ ψ η R N tt ) Eq. 7 In a first approximation the loss coefficients can be determined from Soderberg s correlation as follows *.6.4 ε ζ Eq. 8 where ε denotes the turning, which for stator and rotor respectively yields from φ ψ ε tan N Eq. 9 φ φ ψ β β ε tan tan R Eq.
Turbomachinery Lecture Notes 7 7-9-4 Step 5: First iteration of flow parameters (polytropic expansion) With the knowledge of the approximate total-to-total efficiency the polytropic expansion between inlet and outlet can now be determined yielding the flow parameters at these stations more accurately. The change of enthalpy is now given by The outlet temperature yields from γ γ p γ Δh η tt p v Eq. γ p T Δh T Eq. c p, which allows us to determine the outlet specific volume by RT v Eq. p Finally the mass flow rate needs to be updated by the updated enthalpy difference as W m& Eq. 4 Δ h Step 6: Finalization of first iteration The finalization of the first iteration comprises the steps of determination of number of stages and loading factor. From the values obtained an updated polytropic efficiency can be determined, which could be used in another iteration step. This iteration process should be carried out until convergence is obtained. Criteria for obtaining convergence should be established by the user. Convergence could for example be measured by relating a number of parameters from two subsequent iteration steps. Usually one or two iterations will do if the change of state is close to the isentropic one.
Turbomachinery Lecture Notes 8 7-9-4 Example: Design of Multistage Turbine in Excel sheet The above equations have been implemented in an Excel sheet, which is made available with the present document. The purpose of the Excel sheet is to recognize the effects of design choices on the resulting engine as well as on engine costs. Parameters that are open to choose are the following: Thermodynamic parameters: inlet pressure and temperature, outlet pressure and poser Engine parameters: engine rotational speed and mean radius Design parameters: flow coefficient and loading factor (note: as the assumptions of normal repetition stages and zero exit swirl has been made the degree of reaction is directly related to the loading coefficient) To recognize the impact of certain parameter choices on costs a simplified cost analysis has been included. The cost analysis covers the factors of engine purchasing, engine service and engine fuel costs. Output is provided in numerical and graphical format. Have fun! isentropic st iteration --> polytropic Cost analysis Parameter Unit p [Pa],E6 5,E5,E6 5,E5 base costs -,E6 T [K] 8, 58,6 8, 565, cost per stage -,E5 v [m/kg],,,, fuel cost per l P [J/s],E6,E6 fuel heating value J/l 4,E7 m dot [kg/s],4,4,7,7 Δh [kj/kg],6e5,6e5 gamma [-],4 cp [J/kgK] 4,5 R [J/kgK] 87 phi [-],5,5 n [rpm],e4,e4,e4,e4 fixed costs om [rad/s] 4,59 4,59 4,59 4,59 machine costs,6e6 rm [m],8,8,8,8 A [-],,5,4,4 runtime # years Y [-],,7,4,8 rh [m],7,5,68,47 service (per stage & year),e5 rs [m],9,8,9, service (period) 4,6E6 s [m],,56,,65 u [m/s] 5, 5, fuel run time [h] per year 7 psi [-],,8 total energy per year 8,7E z [-],8,8 total fuel costs per year 4,8E6 R, fuel (period) 4,8E7 stat rot eps [rad] 75,96 6,87 74,45, subtotal (var, period) 4,6E7 zeta [-],7,4,7, η tt [-],898,94 total costs (period) 4,96E7 Legend of parameters to choose thermodynamic parameters fluid properties engine parameters design parameters operating parameters