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

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Turbomachinery - Cheat sheet

Hasan Ozcan Assistant Professor

Mechanical Engineering Department Faculty of Engineering

Karabuk University

1

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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

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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

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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 ;

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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

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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

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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:

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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

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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)

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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

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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: :

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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

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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 :

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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.

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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)