Turbomachinery - Cheat sheet Turbomachinery - Cheat sheet Hasan Ozcan Assistant Professor Mechanical

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Transcript of Turbomachinery - Cheat sheet Turbomachinery - Cheat sheet Hasan Ozcan Assistant Professor Mechanical

  • Turbomachinery - Cheat sheet

    Hasan Ozcan Assistant Professor

    Mechanical Engineering Department Faculty of Engineering

    Karabuk University

    1

  • 2

    Introduction to Turbomachinery 1. Coordinate System

    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

  • 3

    Introduction to Turbomachinery 1. Coordinate System

    The Velocity at the meridional direction is:

    Where x and r stand for axial and radial.

    NOTE: In purely axial flow machines Cr = 0, and in purely radial flow machines Cx=0

    Axial ViewAxial View Stream surface View

  • 4

    Introduction to Turbomachinery

    1. Coordinate System

    Total flow velocity is calculated based on below view as

    Stream surface View

    The swirl (tangential) angle is (i)

    Relative Velocities

    Relative Velocity (ii)

    Relative Flow Angle (iii)

    Combining i, ii, and iii ;

  • 5

    Introduction to Turbomachinery

    2. Fundemental Laws used in Turbomachinery

    2.1. Continuity Equation (Conservation of mass principle)

    2.2. Conservation of Energy (1st law of thermodynamics)

    Stagnation enthalpy;

    if gz = 0;

    For work producing machines

    For work consuming machines

  • 6

    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

  • 7

    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:

  • 8

    Introduction to Turbomachinery

    2. Fundemental Laws used in Turbomachinery

    2.6. Second Law of Thermodynamics

    The Clasius Inequality :

     For a reversible cyclic process:

    Entropy change of a state is , , that we can evaluate the isentropic process when the process is

    reversible and adiabatic (hence isentropic).

    Here we can re-write the above definition as and using the first law of thermodynamics:

    dQ-dW=dh=du+pdv and

  • 9

    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 po 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)

  • 10

    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: (i)

     Knowing that:

     and 𝐶𝑝

    𝐶𝑣 = 𝛾, one gets 𝛾 − 1 =

    𝑅

    𝐶𝑣  (ii)

     Replacing (ii) into (i) one obtains relation between static and stagnation temperatures: (iii)

     Applying the isentropic process enthalpy (𝐝𝐡 = 𝐝𝐩/𝛒) to the ideal gas law (𝑃𝑣 = 𝑅𝑇): 𝐝𝐩/𝛒 = 𝐑𝐝𝐓 and one

    gets: (iv)

    Integrating one obtains the relation between static and

    stagnation pressures : (v)

  • Introduction to Turbomachinery

    11

    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

  • 12

    Introduction to Turbomachinery

    2. Fundemental Laws used in Turbomachinery

    2.7. Compressible flow relations

    Relation of static-relative-stagnation

    temperatures on a T-s diagram

    Temperature (K)

    Temperature – gas properties relation

  • 13

    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: :

  • 14

    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

  • 15

    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 (h2s)

    2.8.2. Hydraulic turbines

    1. Turbine hydraulic efficiency

  • 16

    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 :

  • 17

    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 𝜼𝒑 = 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.

  • 18

    Introduction to Turbomachinery

    2. Fundemental Laws used in Turbomachinery

    2.8.5. Small Stage (Polytropic Efficiency) for an ideal gas For energy extracting devices

  • 19

    The Buckingham 𝝅 - Teorem

    Step 1. List all parameters in the studied system (dimensional parameters, non-

    dimensional variables and dimensional constants). Let ‘n’ to be the total number of

    parameters

    Step 2. List the primary dimensions of each ‘n’ parameters.

    Step 3. Guess the reduction (𝑘 = 𝑛 − 𝑗). K is equal to the expexted amount of ‘𝜋’s

    Step 4. Choose ‘j’ repeating parameters that will be used to construct each 𝜋

    Step 5. Generate the 𝜋s by grouping ‘j’ repeating parameters with the remaining

    parameters.

    Step 6. Check all 𝜋s write a functional relation between all 𝜋s such as:

    𝜋1 = 𝑓(𝜋2, 𝜋3 … . )

  • 20

    The Buckingham 𝝅 - Teorem

  • 21

    The Buckingham 𝝅 - Teorem

    V

    Lift coefficient !

  • 22

    The Buckingham 𝝅 - Teorem

    𝜇

    Reynolds #

    𝑐

    𝛼

  • 23

    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)