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

Turbomachinery - Cheat sheet

Hasan OzcanAssistant Professor

Mechanical Engineering DepartmentFaculty of Engineering

Karabuk University

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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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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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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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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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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 … . )

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V

Lift coefficient !

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𝜇

Reynolds #

𝑐

𝛼

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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 heat 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 od 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 rearrangements 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.

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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:

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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.

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3. Specific Speed and specific diameter

For compressible flow:

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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.

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Axial Flow Turbines

1. The Velocity diagram

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Axial Flow Turbines2. First design Parameter: Stage Reaction

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

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Axial Flow Turbines3. Second design Parameter: Stage Loading Coefficient

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Axial Flow Turbines4. Third design Parameter: Flow Coefficient

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Axial Flow Turbines5. The Normalized Velocity Triangle

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Axial Flow Turbines5. The Normalized Velocity Triangle

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

ψ

φ

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Axial Flow Turbines6. Special Cases

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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 :

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Axial Flow Turbines7. Thermodynamics of axial flow turbines

Mollier diagramFor a turbinestage

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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:

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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:

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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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Axial Compressors and Fans

Mean Line Analysis: As in axial flow turbines, a mean radius approximation with invariant axial flow velocity is made through analysis for simplified design conditions. Three-dimensional flow affects are not taken into account.

Fluid flowdirection

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The Velocity diagram ofthe compressor stage

Across the rotor (relative stagnation enthalpy is constant)

Across the stator (stagnation enthalpy is constant)

The minimum work requirement

The total-to-total efficiency:


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To determine the efficiency, entropy increase should be modeled:

Loss coefficients

From the 2nd Law of Thermodynamics

For an ideal gas, using the equation of state


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For the overall stage:

Efficiency can now be written in terms of losses:

For low speed machines:


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Mean line calculation for a compressor rotor (compressible case)

Inlet condition

Exit condition

In terms of losses

Annulus area


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Mean line calculation for a compressor rotor (incompressible case)

From the continuity equation

Since

The static pressure at the rotor outlet is

Remaining quantities at the rotor exit can be calculated for fixed density in incompressible flow:


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Preliminary compressor stage design

Stage loading coeff (work coeff) for a normal repeating stage:

Local diffusion factor is an important parameter effective on the stage loading


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Cascade geometryAxial Compressors and Fans

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Reaction

andSince

Rearranging from the velocity triangle: (i)

Relation between three dimensionless parameters

(ii)


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Higher reaction tends to reduce the stage loading, which is desirable for compressors.

50% reaction is widely used in compressors since rotor-stator can be built with similar shapes and enthalpy increase is equally distributed.

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Inter-stage swirl: From (i) it is obvious that positive swirl angle will reduce the stage loading and rotor inlet relative Mach number. For advanced compressors this value is generally kept between 20° to 30°.

Blade Aspect Ratio: For given stage loading, flow coefficient and reaction, it is possible to calculate the blade aspect ratio (H/l), providing a suitable compressor length.

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Analysis of Cascade Forces

Forces and velocities on a compressor blade

Assuming constant axial velocity:

Blade #


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Lift and drag forces on blade Axial and tangential forces on unit span of a blade

Drag force can also be defined with the stagnation pressure difference for incompressible flow as:

Rearranging above defined ‘D’ one obtains: (iii)

(i)

(ii)



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Rearranging (iii) one obtains: (iv)

From (i) and (iv) (v)

and (vi)


The lift coefficient is:

The drag coefficient is:

Pressure loss coefficient:

Using above definitions Drag coef. can be redefined:


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And CL :


An approximation can be made as:


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Low Speed Ducted Fans

Velocity triangles of ducted fans


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Low Speed Ducted Fans

Since , then:

The tangential blade force is:

The work coefficient is:

Note: 1) Since Cd<<<CL it is convenient to drop Cd from the equation.2) When the mean flow angle is 45°, optimum work coeff. is obtained.


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Turbomachinery

Hasan OzcanAssistant Professor

Mechanical Engineering DepartmentFaculty of Engineering

Karabuk University

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Radial Flow Pumps, Compressors and Fans

A centrifugal (radial) flow compressor or pumpconsist of ‘in general’ an impeller followed by adiffusor

A pump refers to a machine that increases thepressure of a flowing fluid

A fan also pressurize a flowing gas, that thepressure increase is low, and the flow might beconsidered incompressible.

A compressor is also a pressure-increasingmachine that the density ratio of the outlet flow toinlet flow is higher than 1.05.

Both the static pressure and velocity of the flow isincreased by the impeller with increased angularmomentum. Than the kinetic energy is convertedinto pressure energy by the diffusor. Cross view of a compressor 74


Side and front view of a compressor

Velocity Triangles of inlet and outlet

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Rothalpy for a centrifugal compressor is

Re-arranging:

Since rothalpy at inlet and outlet conditions are equal:

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For a radial machine there is no angular momentum at the inlet condition, therefore:

So the total work production can be defined as stagnation enthalpy difference between inlet and outlet conditions, and head difference (gH) for pumps.

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Area ratio notations for diffusors 78


A diffuser performance can be defined in many ways:

1. For steady and adiabatic flow:

2. In terms of pressures(for compressible flow)

3. For incompressible flow:

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Inlet Velocity limitations at the eye of a compressor or pump:

Inlet relative velocity should be minimized for optimum inlet conditions. High relative velocity may cause cavitation on blades, and pressure reduction through the impeller.

Since the inlet relative velocity corresponds to

The volume flow rate of the flow can be defined as

For maximum flow, aboce eq. can be differentiated with respect to rs1 and by equating it to zero

Re-arranging and

Here thus the OPTIMUM inlet velocity coefficient is:

Here some typical values are; and 80


Optimum pump inlet design:

Cavitation begins when the flow static pressures are approximately euqual to that of the vapor pressure (vp). At cavitation inception:

Here and lies in the range of

For optimum inlet conditions the suction specific speed is:

Where and . Therefore:

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Optimum compressor inlet design (𝜶𝟏 = 𝟎° ):

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Optimum compressor inlet design (𝜶𝟏 = 𝟎° ):

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Optimum compressor inlet design (𝜶𝟏 > 𝟎° ):

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Variation of mass flow function at different input conditions

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Model used by Stadola

Model used by Buseman

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Choking in a compressor stage

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Choking in a compressor stage

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Radial Flow Gas Turbines

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Radial outflow turbine

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Radial Inflow turbines

These turbines provide greateramount of work per stage, ease ofmanufacture, andRuggedness. Efficiency is equal to thebest of those of axial turbines.

Main disadvantage is requirement ofadvanced techniques for rotor coolingand shockresistance

Commonly used in automobileturbochargers and helicopters, andexpanders fro cryogenicprocesses and liquefaction of gases.

1. Cantilever turbine


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There is zero incidence flow due to thedirection of the relative velocity whichmakes this an inward flow turbine.

Working principle is the reverse of acentrifugal compressor with differentimpeller shapes.

1. 90° IFR (Centripetal) Turbine


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Hydraulic turbines

Efficiencies of common hydraulic turbines

Power Specific Speed

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Hydraulic turbines – The Pelton Turbine

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Sizing the penstock with Darcy’s Equation

f= friction factor (recall Moody diagram)l = Length of the piped= pipe diameterV=Average fluid velocity in the pipe

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Hydraulic turbines – Reaction Turbines

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Hydraulic turbines – Francis Turbine

Inlet-exit velocity triangles

Comparison of velocity triangles

Comparison of

efficiencies

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Hydraulic turbines – Kaplan Turbine

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Source: gepower.com

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

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