met 411 basic concepts(turbo)

7/25/2019 MET 411 Basic Concepts(Turbo)

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MET

DEPA R T M

ENT

MECHAN

ICA

L

ENGINEERING

TEC

HNOLOGY

MET 411 - TURBO MACHINES

BASIC

CONCEPTS


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3/14 MET 411 TURBOMACHINES BASICS Dr.Hazim Moria

is known as radial flow or centrifugal machine. If the flow is partly axial and partly radial, the

machine is known as

mixed flow machine.

BASIC LAWS AND GOVERNING EQUATIONS

The basic laws of thermodynamics and fluid mechanics are used in turbo machines. The

important laws and governing equations used in turbo machines are as follows

:

1. The Principle of Conservation of Mass

The conservation of mass is one of the most fundamental principles in nature. Mass, like energy

is a conserved property, and it cannot be created or destroyed. The conservation of mass

principle for a controlled volume undergoing a steady flow process requires that the mass flow

rate (m) across the controlled volume remains constant. Mathematically,

m =

1

A

1

C

1 = 2

A

2

C

2

Where - subscripts 1 and 2 denote the inlet and outlet conditions respectively. The conservation

of mass equation is often referred to as the continuity equation in fluid mechanics. In

compressible flow machines, the mass flow rate (kg/s) is exclusively used while in hydraulic

machines the volume flow rate (m

3

/s) is preferred.

2. The First Law of Thermodynamics

The first law of thermodynamics which is also known as the conservation of energy principle

states that energy can neither be created nor destroyed; it can only change from one form to

another. The conservation of energy equation for a general steady flow system can be expressed

verbally as

[Heat transferred] - [Shaft work] = (Mass flow rate) [(Change in enthalpy per unit mass) + (Change

in kinetic energy per unit mass) + (Change in potential energy per unit mass)]

or

This equation is known as steady flow energy equation (SFEE). A turbo machine, being operated

essentially under the same conditions for long periods of time, can be conveniently analyzed as a

steady flow device. This equation, when applied to a turbo machine, may be simplified pertaining

to the type of turbo machine, because many of the terms are zero (or) get cancelled with others.


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3. The Newton's Second Law of Motion

According to this law, the sum of all the forces acting on a controlled volume in a particular

direction is equal to the rate of change of momentum of the fluid across the controlled volume in

the same direction.

Figure 1.1 Movement of fluid particle across a controlled volume

In turbo machines, the impellers are rotating and the power output is expressed as the product of

torque and angular velocity and so angular momentum is the prime parameter. Consider a fluid

particle moving across a controlled volume as shown in Fig 1.1. The Fluid particle travels from

point A to point B while simultaneously moving from a radius

r

1

to radius

r

2

. If C

x1

and C

x2

are

components of absolute velocities in the tangential direction, then the sum of all the torques

acting on the system is equal to the rate of change of angular momentum. Mathematically,

If the machine revolves with angular velocity , then the power (W) is

Since

This equation is known as the general form of Euler's equation. Euler's turbine equation is


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Euler's pump equation is

4. The Second Law of Thermodynamics

The second law of thermodynamics leads to the definition of Entropy, and is defined as

Q

rev

= Tds

Entropy change is caused by heat transfer, mass flow, and irreversibility. The entropy change

during a process is positive for an irreversible process or zero for a reversible process. Thus,

work producing devices such as turbines, deliver more work and work consuming devices such as

pumps and compressors consume less work when they operate reversibly.

The differential form of the conservation of energy equation for a closed stationary system (a fixed

mass) can be expressed for a reversible process as

This equation is known as the First Ids equation or Gibb's equation. The second Ids equation is

obtained by eliminating du from the first Tds equation by using the definition of enthalpy

(h = u + Pv)

Tds = du - vPdv

dh=du + Pdv + vdP

Thus

Tds =dh- vdP

The second Tds equation is extensively used in the study of compressible flow machines. In

terms of stagnation properties

Tds = dh

o

v

o

dp

o

For an incompressible fluid undergoing an isentropic process (i.e.

ds

= 0) as in fans, the ideal

change in stagnation enthalpy is

Since

v

0

= l/

o

and

o

=


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EFFICIENCIES OF COMPRESSORS

Fig. 1.2 shows the reversible and irreversible adiabatic compression processes on the enthalpy-

entropy diagram. The initial condition of the fluid is represented by state-1. The stagnation point

corresponding to this state is 01. The final condition of the fluid is denoted by state-2 and the

corresponding stagnation point is 02. If the process were reversible, the final fluid static and

stagnation conditions would be 2s and 02s respectively.

Figure 1.2 Reversible and irreversible compression processes

Process 1-2 is the actual compression process and is accompanied by an increase in entropy.

Process 1-2S is the ideal compression process. The efficiencies of compressors may be

defined in terms of either stagnation or static properties of the fluid or even a combination of

both. The following are the commonly used compressor efficiencies:

1. Total-to-Total Efficiency

It is an efficiency based on stagnation properties at entry and exit.

2. Static-to-Static Efficiency

It is an efficiency based on static properties at entry and exit.

3. Polytropic Efficiency

A compressor stage can be viewed as made up of an infinite number of small stages. To account

for a compression in an infinitesimal stage, polytropic efficiency is defined for an elemental

compression process. Consider a small compressor stage as shown in Fig. 1.3 between

pressures

p

and

p + dp.

The polytropic efficiency of a compressor stage is defined as


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Figure 1.3 Compression process in infinitesimal and finite compressor stages

For an isentropic process, the relationship between pressure and temperature is given by

Differentiating equation (1.2) and substituting equation (1.1), we get

Constant value is obtained from equation (1.2). Therefore,

Integrating between the limits of the full compression from PI to P we get

Rearranging,

If the irreversible adiabatic compression process is assumed to be equivalent to a polytropic

process with polytropic index, n, the following relationship between temperature and pressure will

exist.


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Consider now, a single stage compressor raising the fluid pressure from

P

01

to The actual work

input that would be supplied is

Where

c

is the overall compressor efficiency and W

s

is the isentropic work.

The actual work input is the same for both single stage and multistage compression processes.

Then from equ's 1.7 and 1.7a,

Since the constant pressure lines diverge in the direction of increasing entropy on h-s diagram,

the isentropic enthalpy rise across each stage increases even for a constant stagnation pressure

rise P

o

across each stage. Then, the sum of the stage isentropic enthalpy rises is greater than

the isentropic enthalpy rise in a single stage compression.

For a two stage compressor

For N stages,

Equation (1.8) can be written as


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The preheat factor is less than unity. Then, equation (1.8a) becomes

or

i.e., the overall compressor efficiency

c

is less than the compressor stage efficiencies

s

.

Consider again Fig. (1.4) for a first stage compression, state 02 may be obtained after an ideal

compression from 01 to 02, followed by preheating of the fluid from state 02S to 02 at constant

pressure (T

02

> T

02s

).

This inherent thermodynamic effect that reduces the efficiency of a multistage compressor is

called the preheat e ffect.

EFFICIENCIES OF TURBINES

The enthalpy-entropy diagram for flow both reversible and irreversible through a turbine is shown

in Fig. 1.5. The static condition of the fluid at inlet is determined by state 1, with state 01, as the

corresponding stagnation state. The final static properties are determined by the state 2, with 02,

as the corresponding stagnation state.

Figure 1.5

Reversible and irreversible expansion processes

If the process were reversible, the final fluid static state would be 2s and the stagnation state

would be 02s.

Process 1 2 is the actual expansion process and process 1 2s is the isentropic or ideal

expansion process. In turbines, the efficiencies may be defined using either the static or the

stagnation properties of the fluid or even a combination of both. The commonly used turbine

efficiencies are


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1. Total-to-Total Efficiency

It is an efficiency based on stagnation properties at inlet and outlet.

2. Total-to-Static Efficiency

It is an efficiency in which the ideal work is based on stagnation property at inlet and static

property at outlet.

3. Polytropic Efficiency

A turbine stage can be considered as made up of an infinite number of small or infinitesimal

stages. Then to account for expansion in an infinitesimal turbine

Figure 1.6 Expansion process in infinitesimal and finite turbine stages

stage, a small stage or infinitesimal stage or polytropic efficiency is defined. Consider a small

stage (Fig. 1.6) between pressures P and Pdp. The efficiency of this turbine stage is defined as

For an isentropic process

Differentiating eqn. (1.10), we get


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Integrating between the limits of the overall expansion between PI and

Rearranging,

Assuming the irreversible adiabatic expansion (1-2) as equivalent to a poly tropic process with

index

n,

the temperature and pressure are related by

Equating eqns. (1.12) and (1.13),

Comparing the powers,

Alternatively, the index of expansion in the actual process is expressed as

When

p

=1, n = r. The actual expansion of process curve (1-2) coincides with the isentropic

expansion line (1 2s).

4. Finite Stage Efficiency

The stage efficiency, considering static value of temperature and pressure (Fig. 1 .6.), is defined

as


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The stage efficiency can now be expressed in terms of polytropic efficiency

Therefore,

The same equation can be used to determine the overall efficiency of a multistage turbine, except

that the stage pressure ratio is replaced by the overall pressure ratio.

The overall efficiency, for an N-stage turbine with a constant stage pressure ratio, can be

expressed as

met 411 basic concepts(turbo)

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