Introduction to Turbomachinery

1. Coordinate System Introduction to Turbomachinery Since there are stationary and rotating blades in turbomachines, they tend to form a cylindrical form, represented in three directions; 1. Axial 2. Radial 3. Tangential (Circumferential - rθ) Axial View Side View 1

1. Coordinate System Introduction to Turbomachinery The Velocity at the meridional direction is: Where x and r stand for axial and radial. NOTE: In purely axial flow machines C r = 0, and in purely radial flow machines C x =0 Axial View Axial View Stream surface View 2

Introduction to Turbomachinery 1. Coordinate System Total flow velocity is calculated based on below view as The swirl (tangential) angle is (i) Relative Velocities Relative Velocity (ii) Relative Flow Angle (iii) Stream surface View Combining i, ii, and iii ; 3

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.1. Continuity Equation (Conservation of mass principle) 2.2. Conservation of Energy (1 st law of thermodynamics) Stagnation enthalpy; if gz = 0; For work producing machines For work consuming machines 4

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.3. Conservation of Momentum (Newtons Second Law of Motion) For a steady flow process; Here, pa is the pressure contribution, where it is cancelled when there is rotational symmetry. Using this basic rule one can determine the angular momentum as The Euler work equation is: where The Euler Pump equation The Euler Turbine equation 5

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery Writing the Euler Eqution in the energy equation for an adiabatic turbine or pump system (Q=0) NOTE: Frictional losses are not included in the Euler Equation. 2.4. Rothalpy An important property for fluid flow in rotating systems is called rothalpy (I) and Writing the velocity (c), in terms of relative velocities :, simplifying; Defining a new RELATIVE stagnation enthalpy; Redefining the Rothalpy: 6

Introduction to Turbomachinery 2. Fundamental Laws used in Turbomachinery 2.6. Second Law of Thermodynamics The Clasius Inequality : For a reversible cyclic process: Entropy change of a state is, reversible and adiabatic (hence isentropic)., that we can evaluate the isentropic process when the process is Here we can re-write the above definition as dq-dw=dh=du+pdv and and using the first law of thermodynamics: 7

Introduction to Turbomachinery 8

Introduction to Turbomachinery 2. Fundamental Laws used in Turbomachinery 2.5. Bernoulli s Equation Writing an energy balance for a flow, where there is no heat transfer or power production/consumption, one obtains : Applying for a differential control volume: (where enthalpy is When the process is isentropic ), one obtains Euler s motion equation: Integrating this equation into stream direction, Bernoulli equation is obtained: When the flow is incompressible, density does not change, thus the equation becomes: where and p o is called as stagnation pressure. For hydraulic turbomachines head is defined as H= thus the equation takes the form. NOTE: If the pressure and density change is negligibly small, than the stagnation pressures at inlet and outlet conditions are equal to each other (This is applied to compressible isentropic processes) 9

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.7. Compressible flow relations For a perfect gas, the Mach # can be written as,. Here a is the speed of sound, R, T and γ are universal gas constant, temperature in (K), and specific heat ratio, respectively. Above 0.3 Mach #, the flow is taken as compressible, therefore fluid density is no more constant. With the stagnation enthalpy definition, for a compressible fluid: Knowing that: (i) and C p C v = γ, one gets γ 1 = R C v (ii) Replacing (ii) into (i) one obtains relation between static and stagnation temperatures: Applying the isentropic process enthalpy (dh = dp/ρ) to the ideal gas law (Pv = RT): dp/ρ = RdT and one gets: (iv) (iii) Integrating one obtains the relation between static and stagnation pressures : 10 (v)

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery Deriving Speed of sound 11

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery Deriving Speed of sound 12

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery Deriving Speed of sound 13

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.7. Compressible flow relations Above combinations yield to many definitions used in turbomachinery for compressible flow. Some are listed below: 1. Stagnation temperature pressure relation between two arbitrary points: 2. Capacity (non-dimensional flow rate) : 3. Relative stagnation properties and Mach #: HOMEWORK: Derive the non dimensional flow rate (Capacity) equation using equations (iii), (v) from the previous slide and the continuity equation 14

Introduction to Turbomachinery 2. Fundamental Laws used in Turbomachinery 2.7. Compressible flow relations Temperature (K) Relation of static-relative-stagnation Temperature gas properties relation temperatures on a T-s diagram 15

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.8. Efficiency definitions used in Turbomachinery 1. Overall efficiency 2. Isentropic hydraulic efficiency: 3. Mechanical efficiency: 2.8.1 Steam and Gas Turbines 1. The adiabatic total-to-total efficiency is : When inlet-exit velocity changes are small: : 16

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.8.1 Steam and Gas Turbines Temperature (K) Enthalpy entropy relation for turbines and compressors 17

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2. Total-to-static efficiency: Note: This efficiency definition is used when the kinetic energy is not utilized and entirely wasted. Here, exit condition corresponds to ideal- static exit conditions are utilized (h 2s ) 2.8.2. Hydraulic turbines 1. Turbine hydraulic efficiency 18

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.8.3. Pumps and compressors 1. Isentropic (hydraulic for pumps) efficiency 2. Overall efficiency 3. Total-to-total efficiency 4. For incompressible flow : 19

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.8.4. Small Stage (Polytropic Efficiency) for an ideal gas For energy absorbing devices integrating For and ideal compression process η p = 1, so Therefore, the compressor efficiency is: NOTE: Polytropic efficiency is defined to show the differential pressure effect on the overall efficiency, resulting in an efficiency value higher than the isentropic efficiency. 20

Introduction to Turbomachinery 2. Fundemental Laws used in Turbomachinery 2.8.5. Small Stage (Polytropic Efficiency) for an ideal gas For energy extracting devices 21

Dimensional Analysis of Turbomachines Treating a turbomachine as a pump N: Rotational speed (can be adjusted by the current) Q: Volume flow rate (can be adjusted by external vanes) For fixed values of N and Q Torque (τ), head (H) are dependent on above parameters (Control variables) Fluid density (ρ) and dynamic viscosity are (μ) specific to the utilized fluid (Fluid properties) Impeller diameter (D) and length ratios (l 1 /D 1 and l 2 /D 2 are geometric variables for the pump (geometric variables) Dimensional analysis can now be made by considering above terms 22

Dimensional Analysis of Turbomachines 1. Incompressible Fluid Analysis The net energy transfer (gh), pump efficiency (η), and pump power (P) requirement are functions of aforementioned variables and fluid properties: Using three primary dimensions (mass, length, time) or three independent variables we can form 5 dimensional groups by selecting (ρ, N, D) as repeating parameters. Using these groups it is possible to avoid appearance of fluid terms such as μ and Q The work coefficient (energy transfer coefficient - ψ) : (i) 23

Dimensional Analysis of Turbomachines 1. Incompressible Fluid Analysis Efficiency (already non-dimensional): Power coefficienct ( P) : (ii) (iii) Here, (Q/(ND 3 )) is also regarded as a volumetric flow coefficient and ( ρnd2 ) is referred to as Reynolds μ number. The volumetric flow coefficient is also called velocity or flow coefficient (φ) and can also be defined in terms of velocities: φ = Q = c m ND 3 U Since the independent variables are complex, some assumptions are made for simplification. Effect of geometric variables by assuming similar values of these ratios are constant. In addition another assumption is made by assuming the effect of Reynolds number ( ρnd2 ) is neglected for the flow. Now the functional relationships are simpler: μ (iv) 24

Dimensional Analysis of Turbomachines 1. Incompressible Fluid Analysis Using these dimensionless groups, it is now possible to write a relationship between the power, flow, head coefficients and the efficiency. Since the new hydraulic power for a pump is P N = ρgqh, and efficiency is the ratio of net power to the actual power η = P N /P. Than one can use (i), (ii), and (iii) to form a relation between these parameters: Which yields to : For an hydraulic turbine (since η = P/P n ): (i) 25

2. Compressible Fluid Analysis Dimensional Analysis of Turbomachines For an ideal compressible fluid, mass flow rate is used instead of volumetric flow rate and two additional parameters are required specific to the incompressible fluid, namely the stagnation sound speed (a 0 ) and the specific heat ratio (γ). Total power produced, efficiency, and the isentropic stagnation enthalpy change is considered as functions for non-dimensioning The subscript (1) represents the inlet conditions since these parameters vary through the turbomachine. This 8 dimensional groups may be reduced to 5 by considering the stagnation density, rotational speed, and the turbomachine diameter as repeating parameters: Taking ND/a 01 as the Mach number, a rearrangement can be made 26

2. Compressible Fluid Analysis Dimensional Analysis of Turbomachines For an ideal compressible fluid the stagnation enthalpy can be written as and knowing that: we can write And knowing the new definition is With the ideal gas law, the mass flow can be more conveniently explained And the power coefficient: 27

Dimensional Analysis of Turbomachines 2. Compressible Fluid Analysis Collecting all definitions together, and assuming the specific heat ratio (γ) and the Re # effect are dropped for simplification The first term in the function is referred to as flow capacity and is the most commonly used form of non-dimensional mass flow. For fixed sized machinery D, R and the specific heat ratio are dropped. Here, independent variables are no Longer dimensionless. A compressor efficiency can be written in terms of common performance parameters as: There is still more parameters that are needed to fix the problem in variation of density and flow Mach number which are variables. Therefore two new parameters, namely flow coefficient (φ) and stage loading (work coefficient - ψ) are defined.

Dimensional Analysis of Turbomachines 3. Specific Speed and specific diameter There may be a direct relation between three dimensionless parameters in a hydraulic turbine when Re # effects and cavitation is absent. For specific speed, D are cancelled and thus, For power specific speed : Ratio of above definitions provide: Note: D is the only common parameter for all dimensionless parameters. By eliminating the speed from flow and work coefficients one may obtain the specific diameter:

Dimensional Analysis of Turbomachines 3. Specific Speed and specific diameter Selection of pumps based on dimensionless parameters Selection of turbines based on dimensionless parameters Note: Here, N is replaced by Ω and in rad/sec, instead of rev/min.

Dimensional Analysis of Turbomachines 3. Specific Speed and specific diameter For compressible flow:

Design of Axial Flow Turbines 1. The Velocity diagram (Courtesy of Dr. Damian Vogt) Axial turbine stage comprises a row of fixed guide vanes or nozzles (often called a stator row) and a row of moving blades or buckets (a rotor row). Fluid enters the stator with absolute velocity c 1 at angle α 1 and accelerates to an absolute velocity c 2 at angle α 2 From the velocity diagram, the rotor inlet relative velocity w 2, at an angle β 2, is found by subtracting, vectorially, the blade speed U from the absolute velocity c 2. The relative flow within the rotor accelerates to velocity w 3 at an angle β 3 at rotor outlet; the corresponding absolute flow (c 3, α 3 ) is obtained by adding, vectorially, the blade speed U to the relative velocity w 3.

1. The Velocity diagram Axial Flow Turbines

2. First design Parameter: Stage Reaction Axial Flow Turbines

Axial Flow Turbines 3. Second design Parameter: Stage Loading Coefficient (Work Coefficient)

Axial Flow Turbines 3. Second design Parameter: Stage Loading Coefficient

Considering the sign convention: Axial Flow Turbines

4. Third design Parameter: Flow Coefficient Axial Flow Turbines

5. The Normalized Velocity Triangle Axial Flow Turbines

Axial Flow Turbines 5. The Normalized Velocity Triangle Another common way to represent a velocity triangle for axial turbines is: ψ φ

6. Special Cases Axial Flow Turbines

Axial Flow Turbines 7. Thermodynamics of axial flow turbines From the Euler s work eq. Since there is no work done through the nozzle row: Writing above Eqs together : From the velocity triangle, than we can re-arrange as In terms of relative stagnation enthalpy :

7. Thermodynamics of axial flow turbines Axial Flow Turbines Mollier diagram For a turbine stage

8. Repeating Stage turbines Axial Flow Turbines Substituting main Reaction and rothalpy definitions:.(i)..(ii) Substituting (ii) into (1): or And the work coefficient reaction relation yields to:

9. Stage loss coefficients Axial Flow Turbines Losses can be defined in terms of exit kinetic energy from each blade row: Adapting this into total-to-total and total-to-static efficiencies of the stage with velocity components:

10. Preliminary axial turbine design Number of stages: Axial Flow Turbines With the continuity equation and the flow coeff Mean radius can be defined: Where, here, t and h stand for tip and hub. The blade height requirement for a flow is related with flow coefficient and the mean radius as: