Introduction to Turbomachinery - Karabuk

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Introduction to Turbomachinery1. 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 Turbomachinery1. 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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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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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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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 Turbomachinery2. 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 , , 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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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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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)

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Introduction to Turbomachinery2. Fundemental Laws used in Turbomachinery

Deriving Speed of sound

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13





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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

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2. Fundamental Laws used in Turbomachinery


Relation of static-relative-stagnation

temperatures on a T-s diagram

Temperature (K)

Temperature – gas properties relation

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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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2.8.1 Steam and Gas Turbines

Temperature (K)

Enthalpy – entropy relation for turbines and compressors

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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

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

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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 (l1/D1 and l2/D2 are geometric variables for the pump (geometric variables)

Dimensional analysis can now bemade by considering above terms

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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)

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Efficiency (already non-dimensional): (ii)

Power coefficienct ( 𝑃) : (iii)

Here, (Q/(ND3)) is also regarded as a volumetric flow coefficient and (𝜌𝑁𝐷2

𝜇) is referred to as Reynolds

number. The volumetric flow coefficient is also called velocity or flow coefficient (𝜙) and can also be defined in terms of velocities:

𝜙 =𝑄

𝑁𝐷3=

𝑐𝑚

𝑈

Since the independent variables are complex, some assumptions are made for simplification. Effect ofgeometric variables by assuming similar values of these ratios are constant. In addition another

assumption is made by assuming the effect of Reynolds number (𝜌𝑁𝐷2

𝜇) is neglected for the flow. Now

the functional relationships are simpler:

(iv)

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• 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 𝑃𝑁 = 𝜌𝑔𝑄𝐻, and efficiency is the ratio of net power to the actual power 𝜂 = 𝑃𝑁/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/𝑃𝑛):

(i)

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2. Compressible Fluid Analysis

• For an ideal compressible fluid, mass flow rate is used instead of volumetric flow rate and twoadditional parameters are required specific to the incompressible fluid, namely the stagnation soundspeed (𝑎0) and the specific heat ratio (𝛾). Total power produced, efficiency, and the isentropicstagnation enthalpy change is considered as functions for non-dimensioning

• The subscript (1) represents the inlet conditions since these parameters vary through theturbomachine. 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/a01 as the Mach number, a rearrangement can be made

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• 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 bemore conveniently explained

And the power coefficient:



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-dimensionalmass 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.


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 Turbomachines3. 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.


3. Specific Speed and specific diameter

For compressible flow:

Design of Axial Flow Turbines1. 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 orbuckets (a rotor row). Fluid enters the stator with absolute velocity c1 at angle α1 and accelerates to an absolute velocity c2

at angle α2

From the velocity diagram, the rotor inlet relative velocity w2, at an angle β2, is found by subtracting, vectorially, the bladespeed U from the absolute velocity c2.

The relative flow within the rotor accelerates to velocity w3 at an angle β3 at rotor outlet; the corresponding absolute flow(c3, α3) is obtained by adding, vectorially, the blade speed U to the relative velocity w3.

Axial Flow Turbines

1. The Velocity diagram

Axial Flow Turbines2. First design Parameter: Stage Reaction





Axial Flow Turbines3. Second design Parameter: Stage Loading Coefficient (Work Coefficient)

Axial Flow Turbines3. Second design Parameter: Stage Loading Coefficient

Axial Flow TurbinesConsidering the sign convention:

Axial Flow Turbines4. Third design Parameter: Flow Coefficient

Axial Flow Turbines5. The Normalized Velocity Triangle

Axial Flow Turbines5. The Normalized Velocity Triangle

Another common way to represent a velocity triangle for axial turbines is:

ψ

φ

Axial Flow Turbines6. Special Cases






Axial Flow Turbines7. 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 :

Axial Flow Turbines7. Thermodynamics of axial flow turbines

Mollier diagramFor a turbinestage

Axial Flow Turbines8. Repeating Stage turbines

Substituting main Reaction and rothalpy definitions:

…….(i)

……..(ii)

Substituting (ii) into (1): or

And the work coefficient – reaction relation yields to:

Axial Flow Turbines9. Stage loss coefficients

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 velocitycomponents:

Axial Flow Turbines10. Preliminary axial turbine design

Number of stages:

With the continuity equation and the flow coeffMean 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:

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Introduction to Turbomachinery - Karabuk