DEPARTMENT OF MECHANICAL ENGINEERING …...TurboMachine:- Any devices that extracts energy from or imparts energy to a continuously moving stream of fluid can be called a Turbomachine

SAGAR INSTITUTE OF RESEARCH & TECHNOLOGY, BHOPAL

DEPARTMENT OF MECHANICAL ENGINEERING

TURBOMACHINERY

ME-5001

Richard Victor, Department of Mechanical Engineering, SIRT Bhopal

TURBOMACHINERYME-502

RICHARD VICTORDEPARTMENT OF MECHNAICAL ENGINEERING


Unit IEnergy transfer in turbo machines

1. Application of first and second laws of

thermodynamics to turbo machines,

2. Moment of momentum equation and Euler

turbine equation,

3. Principles of impulse and reaction machines,

4. Degree of reaction,

5. Energy equation for relative velocities, one dimensional analysis only




Steam turbines / Gas turbines / Compressors / Expanders



What is a Turbomachine ?

TurboMachine:- Any devices that extracts energy from or imparts energy to a continuously moving stream of fluid can be called a Turbomachine. Elaborating, a turbomachine is a power or head generating machine which employs the dynamic action of a rotating element, the rotor; the action of the rotor changes the energy level of the continuously flowing fluid through the machine. Turbines, compressors and fans are all members of this family of machines.


Classification of Turbo M/Cs

• These two types of machines are governed by the same basic relationships including Newton's second Law of Motion and Euler's pump and turbine equation for compressible fluids. Centrifugal pumps are also turbomachines that transfer energy from a rotor to a fluid, usually a liquid, while turbines and compressors usually work with a gas.


Classification

In general, the two kinds of turbomachinesencountered in practice are open and closed turbomachines.

1. Open machines such as propellers, windmills, and unshrouded fans act on an infinite extent of fluid.

2. Closed machines operate on a finite quantity of fluid as it passes through a housing or casing.



Turbomachines are also categorized according to the type of flow.

1. When the flow is parallel to the axis of rotation, they are called axial flow machines.

2. When flow is perpendicular to the axis of rotation, they are referred to as radial (or centrifugal) flow machines.

3. There is also a third category, called mixed flow machines, where both radial and axial flow velocity components are present.



Turbomachines may be further classified into two additional categories:

1. Those that absorb energy to increase the fluid pressure, i.e. pumps, fans, and compressors,

2. Those that produce energy such as turbines by expanding flow to lower pressures. Of particular interest are applications which contain pumps, fans, compressors and turbines. These components are essential in almost all mechanical equipment systems, such as power and refrigeration cycles.


INTRODUCTION

Q. Define Turbomachine and classify them on the basis of fluid movement through the machine. (Dec. 2011)

Q. What is a Turbomachine ? Classify them on the basis of work transfer. (Dec 2010)

Q. Define Turbomachines and explain the different types of turbomachines. (June 2010)


Turbomachines work on basic laws of thermodynamics and fluid mechanics.

1. Conservation of Mass.

2. Conservation of Energy.

3. Newton’s Second law of motion.


CONTINUITY EQUATION

For steady flow through the control volume, the mass flow rate, remains constant

m

222111 AVAVm


First law of Thermodynamics:-

Conservation of Energy

Energy can neither be created nor destroyed, it can only change from one form to another.

WEQ

UEEE PK

WUEEQ PK


Steady Flow Energy Equation(First law of Thermodynamics)

][ pekehmWQ Richard Victor, Department of Mechanical

Engineering, SIRT Bhopal

‘Steady Flow’ means that the rates of flow of mass and energy across the control surface are constant.

In most engineering devices, there is a constant rate of flow of mass and energy through the control surface, and the control volume in course of time attains a steady state.


In the rate form222111 dmvpdmvpWW x

d

dmvp

d

dmvp

d

Wd

d

Wd x 222

111

222111 vpwvpwd

Wd

d

Wd x

‘a’


Since there is no accumulation of energy, by the conservation of energy, the total rate of flow of all energy streams entering the control volume must equal the total rate of flow of all energy streams leaving the control volume.

Where & - energy carried into or out of the CV with unit mass of the fluid.

d

dWew

d

Qdew 2211

1e 2e


Substituting for from equation ‘a’.d

Wd

2221112211 vpwvpwd

Wdew

d

Qdew x

2222211111 vpwd

Wdew

d

Qdvpwew x


The specific energy is given by

Substituting the expression for in equation

Where

ueee pk uZgV

2

2

e

d

WdvpwuZ

Vw

d

QdvpwuZ

Vw x

22222

2

2211111

2

11

22

d

WdgZ

Vhw

d

QdgZ

Vhw x

2

2

2221

2

111

22

pvuh


Since

Dividing the Equation by

This Equation is Known as

STEADY FLOW ENERGY EQUATION

d

dmww 21

d

dm

dm

WdgZ

Vh

dm

QdgZ

Vh x 2

2

221

2

11

22


Second Law of Thermodynamics


Second Law of Thermodynamics

Clausius’ Statement of the Second Law gives:

It is impossible to construct a device which, operating in a cycle, will produce no effect other than the transfer of heat from a cooler to a hotter body.

Heat cannot flow of itself from a body at a lower temperature to a body at a higher temperature.


Second Law of ThermodynamicsEntropy

• Second law of thermodynamics states that for a fluid undergoing a reversible adiabatic process, the entropy change is zero.

• Entropy increases from inlet to the exit, if the fluid undergoes an adiabatic or any other process. Due to the increase in entropy, the power developed by a turbine is less than the ideal isentropic power developed. Similarly the work input to a pump is greater than the isentropic or ideal workinput.

dsT

q


Newton’s Second Law of Motion-Momentum Equation

The moment of momentum equation is based on Newton’s second law applied to a rotating fluid mass system.

Previous Knowledge

Moment of momentum about an axis is known as Angular Momentum.

The moment of a force about a point is torque.


The moment of momentum principle states thatIn a rotating system the torque exerted by the resultant force on the body with respect to an axis is equal to the time rate of change of angular momentum.

[Torque exerted on the fluid by the rotating element ]=[Angular momentum of fluid leaving out of CV]-[Angular momentum of fluid entering the CV]

Where Q= discharge, Vu = tangential component of absolute velocity, r= moment arm of Vu

])()[( inuoutu rVrVQT


• The Euler’s pump and turbine equations are most fundamental equations in the field of turbo-machinery . These equations govern the power, efficiencies and other factors that contribute in the design of Turbo-machines thus making them very important. With the help of these equations the head developed by a pump and the head utilised by a turbine can be easily determined. As the name suggests these equations were formulated by Leonhard Euler in the eighteenth century.These equations can be derived from the moment of momentum equation when applied for a pump or a turbine.


Euler Turbine Equation

• The Euler turbine equation relates the power added to or removed from the flow, to characteristics of a rotating blade row. The equation is based on the concepts of conservation of angular momentum and conservation of energy.



Force exerted by a Liquid Jet Striking on a Curved Vane when the Vane is moving in the Jet Direction


Force exerted by a Liquid Jet Striking on a Curved Vane when the Vane is moving in the Jet Direction

The mass flow rate of liquid striking the vane is

= Rate of Change of Momentum in the direction of the force= Mass flow rate хchange in velocity in the jet direction

)( uVam

xF

]cos)(())[(( uVuVuVa


Consider a Jet of liquid striking a movinf curved vane tangentially at one of its tips. The liquid jet strikes the vane with an absolute velocity of . Let the velocity of the vane is . Since the vane is moving with the a relative velocity

which is obtained by subtracting vectorically the velocity from .

1V u

1rVu 1V






Assumptions

1. Fluid enters and leaves the vane in a direction tangential to the vane tip at inlet and outlet.


Euler’s Turbine Equation

Angular Velocity of wheel (rotaional Speed) (rad/sec)

Tangential Momentum of the fluid at entry

N

60/2 N

mVw

1



Moment of momentum or angular momentum at entry N-m

Similarly Angular Momentum at the outlet

N-m

11 rmVw

22 rmVw


Euler’s Turbine equation

T= Torque on the wheel=change of angular momentum N-m

Workdone=rate of energy transferred

= torque х angular velocity

mrVrV ww

2211

smN

mrVrV ww /2211



But

Workdone W

If the machine is called Turbine.

If the machine is called pump, fan, compressor or blower

2211 ; UrUr

muVuV ww

2211

2211 uVuV ww

1122 uVuV ww


Impulse and Reaction machines

Impulse turbines change the direction of flow of a high velocity fluid or gas jet. Reaction turbines develop torque by reacting to the gas or fluid's pressure or mass. The pressure of the gas or fluid changes as it passes through the turbine rotor blades


Impulse and Reaction machines

A reaction turbine is a type of turbine that develops torque by reacting to the pressure or weight of a fluid; the operation of reaction turbines is described by Newton's third law of motion (action and reaction are equal and opposite). The pressure of the fluid changes as it passes through the rotor blades.


Degree of Reaction

• In turbomachinery, Degree of reaction or reaction ratio (R) is defined as the ratio of static pressure drop in the rotor to the static pressure drop in the stage or as the ratio of static enthalpy drop in the rotor to the static enthalpy drop in the stage.

• Degree of reaction (R) is an important factor in designing the blades of a turbine , compressors, pumps and other turbo-machinery. It also tells about the efficiency of machine and is used for proper selection of a machine for a required purpose.


Degree of Reaction

• Various definitions exist in terms of enthalpies, pressures or flow geometry of the device. In case of turbines, both impulse and reaction machines, Degree of reaction (R) is defined as the ratio of energy transfer by the change in static head to the total energy transfer in the rotor i.e.Isentropic enthalpy change in rotor/Isentropic enthalpy change in stage.

For a gas turbine or compressor it is defined as the ratio of isentropic heat drop in the moving blades (i.e. the rotor) to the sum of the isentropic heat drops in the fixed blades(i.e. the stator) and the moving blades i.e.


Degree of Reaction

• The degree of reaction, R is defined as the ratio of isentropic heat drop in the moving blades to the sum of the isentropic heat drops in the fixed and the moving blades i.e. in a stage.


Degree of Reaction

Isentropic heat drop in rotor/Isentropic heat drop in stage. In pumps, degree of reaction deals in static and dynamic head. Degree of reaction is defined as the fraction of energy transfer by change in static head to the total energy transfer in the rotor i.e.

Static pressure rise in rotor/Total pressure rise in stage.


Energy Equation for Relative Velocities

Euler’s Equation

If H is the head on the machine, then energy transfer can be written as

Therefore Euler’s Equation will become

mUVUVE WW)( 2211

gHmE

g

UVUVH ww 2211


Components of Energy Transfer It is worth mentioning in this context that either of the Eqs. is applicable regardless of changes in density or components of velocity in other directions. Moreover, the shape of the path taken by the fluid in moving from inlet to outlet is of no consequence. The expression involves only the inlet and outlet conditions. A rotor, the moving part of a fluid machine, usually consists of a number of vanes or blades mounted on a circular disc. Figure shows the velocity triangles at the inlet and outlet of a rotor. The inlet and outlet portions of a rotor vane are only shown as a representative of the whole rotor


• Vector diagrams of velocities at inlet and outlet correspond to two velocity triangles, where is the velocity of fluid relative to the rotor and are the angles made by the directions of the absolute velocities at the inlet and outlet respectively with the tangential direction, while and are the angles made by the relative velocities with the tangential direction. The angles and should match with vane or blade angles at inlet and outlet respectively for a smooth, shockless entry and exit of the fluid to avoid undersirable losses. Now we shall apply a simple geometrical relation as follows:

From the inlet velocity triangle, or,

Similarly from the outlet velocity triangle. or,

Invoking the expressions of and in Euler’s Eq. , we get H (Work head, i.e. energy per unit weight of fluid, transferred between the fluid and the rotor as) as

11

2

1

2

1111

2

1

2

1

2

1 22 wr VUUVCosVUUVV

2

2

1

2

1

2

111

rw

VUVVU

22

2

2

2

2222

2

2

2

2

2

2 22 wr VUUVCosVUUVV

2

2

2

2

2

2

222

rw

VUVVU

)()()[(2

1 2

2

2

1

2

2

2

1

2

2

2

1 rr VVUUVVg

H

11 wVU 22 wVU

1 2

1 2

1 2

rV


The Eq is an important form of the Euler's equation relating to fluid machines since it gives the three distinct components of energy transfer as shown by the pair of terms in the round brackets. These components throw light on the nature of the energy transfer. The first term of Eq. is readily seen to be the change in absolute kinetic energy or dynamic head of the fluid while flowing through the rotor. The second term of Eq. represents a change in fluid energy due to the movement of the rotating fluid from one radius of rotation to another.

)()()[(2

1 2

2

2

1

2

2

2

1

2

2

2

1 rr VVUUVVg

H


• Energy Transfer in Axial Flow MachinesFor an axial flow machine, the main direction of flow is parallel to the axis of the rotor, and hence the inlet and outlet points of the flow do not vary in their radial locations from the axis of rotation. Therefore, and the equation of energy transfer Eq. can be written, under this situation, as

21 UU


• Radially Outward and Inward Flow MachinesFor radially outward flow machines, , and hence the fluid gains in static head, while, for a radially inward flow machine, and the fluid losses its static head. Therefore, in radial flow pumps or compressors the flow is always directed radially outward, and in a radial flow turbine it is directed radially inward.

12 UU

12 UU


Impulse and Reaction Machines

Impulse and Reaction Machines The relative proportion of energy transfer obtained by the change in static head and by the change in dynamic head is one of the important factors for classifying fluid machines. The machine for which the change in static head in the rotor is zero is known as impulse machine . In these machines, the energy transfer in the rotor takes place only by the change in dynamic head of the fluid. The parameter characterizing the proportions of changes in the dynamic and static head in the rotor of a fluid machine is known as degree of reaction and is defined as the ratio of energy transfer by the change in static head to the total energy transfer in the rotor.

Therefore, the degree of reaction,


Impulse and Reaction MachinesFor an impulse machine R = 0 , because there is no change in static pressure in the rotor. It is difficult to obtain a radial flow impulse machine, since the change in centrifugal head is obvious there. Nevertheless, an impulse machine of radial flow type can be conceived by having a change in static head in one direction contributed by the centrifugal effect and an equal change in the other direction contributed by the change in relative velocity. However, this has not been established in practice. Thus for an axial flow impulse machine . For an impulse machine, the rotor can be made open, that is, the velocity V1 can represent an open jet of fluid flowing through the rotor, which needs no casing. A very simple example of an impulse machine is a paddle wheel rotated by the impingement of water from a stationary nozzle as shown in Fig

2121 ; rr VVUU



• A machine with any degree of reaction must have an enclosed rotor so that the fluid cannot expand freely in all direction. A simple example of a reaction machine can be shown by the familiar lawn sprinkler, in which water comes out (Fig. b) at a high velocity from the rotor in a tangential direction. The essential feature of the rotor is that water enters at high pressure and this pressure energy is transformed into kinetic energy by a nozzle which is a part of the rotor itself.

• In the earlier example of impulse machine (Fig. a), the nozzle is stationary and its function is only to transform pressure energy to kinetic energy and finally this kinetic energy is transferred to the rotor by pure impulse action. The change in momentum of the fluid in the nozzle gives rise to a reaction force but as the nozzle is held stationary, no energy is transferred by it. In the case of lawn sprinkler (Fig. b), the nozzle, being a part of the rotor, is free to move and, in fact, rotates due to the reaction force caused by the change in momentum of the fluid and hence the word reaction machine follows.


Efficiencies

The concept of efficiency of any machine comes from the consideration of energy transfer and is defined, in general, as the ratio of useful energy delivered to the energy supplied. Two efficiencies are usually considered for fluid machines-- the hydraulic efficiency concerning the energy transfer between the fluid and the rotor, and the overall efficiency concerning the energy transfer between the fluid and the shaft. The difference between the two represents the energy absorbed by bearings, glands, couplings, etc. or, in general, by pure mechanical effects which occur between the rotor itself and the point of actual power input or output.


Therefore, for a pump or compressor,


For Turbine


Principle of Similarity and Dimensional Analysis

The principle of similarity is a consequence of nature for any physical phenomenon. By making use of this principle, it becomes possible to predict the performance of one machine from the results of tests on a geometrically similar machine, and also to predict the performance of the same machine under conditions different from the test conditions. For fluid machine, geometrical similarity must apply to all significant parts of the system viz., the rotor, the entrance and discharge passages and so on. Machines which are geometrically similar form a homologous series. Therefore, the member of such a series, having a common shape are simply enlargements or reductions of each other. If two machines are kinematically similar, the velocity vector diagrams at inlet and outlet of the rotor of one machine must be similar to those of the other. Geometrical similarity of the inlet and outlet velocity diagrams is, therefore, a necessary condition for dynamic similarity.


The ratio of rotor and shaft energy is represented by mechanical efficiency


Let us now apply dimensional analysis to determine the dimensionless parameters, i.e., the π terms as the criteria of similarity for flows through fluid machines. For a machine of a given shape, and handling

compressible fluid, the relevant variables are given in Table

Variable physical parameters Dimensional formula

D = any physical dimension of the machine as a measure of the machine's size, usually the rotor diameter L

Q = volume flow rate through the machine L3 T -1

N = rotational speed (rev/min.) T -1

H = difference in head (energy per unit weight) across the machine. This may be either gained or given by the

fluid depending upon whether the machine is a pump or a turbine respectively.L

ρ=density of fluid ML-3

µ= viscosity of fluid ML-1 T -1

E = coefficient of elasticity of fluid ML-1 T-2

g = acceleration due to gravity LT -2

P = power transferred between fluid and rotor (the difference between P and H is taken care of by the hydraulic

efficiencyML2 T-3


In almost all fluid machines flow with a free surface does not occur, and the effect of gravitational force is negligible. Therefore, it is more logical to consider the energy per unit mass gH as the variable rather than H alone so that acceleration due to gravity does not appear as a separate variable. Therefore, the number of separate variables becomes eight: D, Q, N, gH, ρ, µ, E and P . Since the number of fundamental dimensions required to express these variable are three, the number of independent π terms (dimensionless terms), becomes five. Using Buckingham's π theorem with D, N and ρ as the repeating variables, the expression for the terms are obtained as,


We shall now discuss the physical significance and usual terminologies of the different π terms. All lengths of the machine are proportional to D , and all areas to D2. Therefore, the average flow velocity at any section in the machine is proportional to . Again, the peripheral velocity of the rotor is proportional to the product ND . The first π term can be expressed as

2D

Q


Documents

DEPARTMENT OF MECHANICAL ENGINEERING …...TurboMachine:- Any devices that extracts energy from or imparts energy to a continuously moving stream of fluid can be called a Turbomachine