theory of compressible flows- chapter 1

8/13/2019 Theory of Compressible Flows- Chapter 1

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Theory of Compressible FlowsM.S. Mechanical ProcessEngineering4thSemester


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IntroductionGAS DYNAMICS

It is the branch of fluid mechanics concerned with causes and effects

arising from the motion of compressible fluids particularly gases.Fluid Mechanics

Statics Fluid Dynamics

Aerodynamics Hydrodynamics Gas Dynamics

In this subject we are concerned with following fundamental physical laws

1. Law of conservation of mass

2. Newtons second law of motion

3. First law of thermodynamics

4. Second law of thermodynamics

Above laws are applicable to all fluids and all flow processes

Above laws are applied to a fluid utilizing the continuum concept


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Continuum

Due to elastic collisions limits of velocity are between

zero and very high velocity

Continuum concept may not be applicable for a gas

at low pressure (High altitudes)

Distance b/w two consecutive collisions is mean free path

Mean free path is inversely related to molecular diameter

All materials, solid or fluid, are composed of molecules discretely

spread and in continuous motion.

However, in dealing with fluid-flow relations on a mathematical basis, itis necessary to replace the actual molecular structure by a hypothetical

continuous medium, called the continuum.

Continuum postulate assumes that every differential element of body of

fluid contains a large number of molecules.


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Continuum, contd.For continuum postulate to hold, mean free path () 0.01 gas is a combination of discrete particles


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CompressibilityAmount by which a substance can be compressed is given by a property

compressibility

Compressibility isfractional change in

volume per unit change in

pressure

Isothermal CompressibilityRise in temp is controlled by

some heat transfer mechanism

Isentropic Compressibility

Compressibility in

the form of density

Liquid dis small

Gas d

Is small for low speed flow

Is large for high speed flowAnother index


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where

Collisions between the fluid molecules causes the propagation of sound

waves

Propagation of sound waves is affected by molecular density of the

medium Where molecular density is low, dissipation of energy occurs without

colliding with any molecule (free particle flow)

It is an important quantity in gas dynamics

We shall study further details about speed of sound in Chapter 3

For Ideal Gas

The Acoustic SpeedThe speed at which a sound wave or small pressure disturbance is propagated

in the fluid medium

s

a

1

s

sdp

d

1

s

pa

2


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It is the ratio of local fluid speed Vto its acoustic speed

Mach Number

Properties of Atmosphere


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1-7 Simple thermodynamic system

1-8 Reversible and irreversible processes, 1-9 Work, 1-10 Heat

1-11 1stlaw of thermo

1-12 2ndlaw of thermo; 1-15(a) to (c)

1-15(d) to (g)

1-18 Properties of atmosphere (Detailed)

2-3 Mathematical description of continuum

2-5, 2-6, 2-7, 2-8 (equations)

Presentation Topics


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Simple Thermodynamic System

Equation of State

Determined by experiment zis dependent andx, y are independent properties

x, y are scalars but may be intensiveor extensive

A readily measurable characteristic showing the behavior of the system

and its interaction with the environment is called a Property

It depends on the state of a system In the case of a simple system, state is fully defined by any two

independent properties

If fis a differentiable

function ofxand y, dzis

called exact differential

fis not exact differential

or inexact its written as

z

For continuous M

and N, it must satisfy

It contains

Either a restricted region in space or finite portion of matter called System

A surface that system from other space or matter called Control Surface

Everything external to control system is Surrounding


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If the reciprocal relationship exists, then

Simple Thermodynamic System, contd.

Line integral of an exact differential expression is independent of the

paths connecting the end points of the integration

Line integral of an exact differential expression taken over a closed curve

is zero

Let


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Reversible and Irreversible ProcessesA process is reversibleif its direction can be reversed at any point by an

infinitesimal change in external conditions

Such process take infinite time to complete

Irreversible Effects

1. Friction

2. Heat transfer across finite temp difference

3. Free expansion

4. Mixing

5. Inelastic deformation

Irreversible Processes

1. Viscous Momentum

Flux2. Heat Conduction

3. Mass Diffusion

Reversible processes are infinitely slow processes with maximumefficiency

These processes are always in equilibrium

These processes provide the limiting results

All real processes are irreversible in nature


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Work and HeatWorkis the energy in transit across the boundary of the system where the

sole effect external to the system could have been raising or lowering of a

weight

Heatis the energy in transit across the boundary of the system caused by atemperature difference between system and its surrounding

Work done by

the system is

positive

Heat added to the system is positive

Heat and work both are path functions and depend upon the details of the

processes involved


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First Law of ThermodynamicsEnergy can neither be created nor be destroyed but it can be

changed from one form to another form.

1. Total Energy is conserved2. This Law is based on experience and no phenomenon contradicts it

3. Heat and work are mutually inter-convertible and fixed amount per

unit mass is needed for every degree rise of temperature

Line integral vanishes for the closed

curve so the integral defines a property

For a cyclic process of closed system of mass m

Eis the stored

energy

On per unit

mass basis

The equation is valid

for both Reversibleand Irreversible

processes

For Irreversible

processes


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Internal Energy

On unit mass basis

Work required to compress

volume dV intoa region

where pressure isp

It comprises of the

1. Translational kinetic energy of the gas molecule

2. Rotational kinetic energy of the gas molecule3. Vibrational kinetic energy of the gas molecule due to atoms

4. Electronic energy due to electronsFor Irreversible

processes

Flow Work


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EnthalpyDifferentiating both sides

For a reversibleprocess

By definition Enthalpy is

Where internal energy is

Putting into enthalpy

When process is

reversible and adiabaticSpecific HeatsSpecific heat at constant volume

Specific heat at constant pressure


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2ndLaw of Thermodynamics Heat is a relatively crude form f energy so it can not be converted

into work completely

Only a portion of heat can be converted into work

Every natural system will change spontaneously and reach

equilibrium and no further change will be possible

Hence

For a closed system

or

For a closed system that undergoes

a cyclic change, it may be shown

So

If heat is added reversibly then

But

For Reversible Adiabatic Process

For Irreversible

Adiabatic Process

For a Diabetic Process


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Thermodynamic Properties of a Perfect GasThermal Equation of State

A gas for which Ris nearly

constant is called thermally

perfect gas

Rdepends upon type of gas involved

It varies with tandpfor real gases

For a homogenous system composed of

single chemical specie of gas:

Thermodynamic stat is defined byp, v , t, u, h and s

Thermal Equation of State

Compressibility Factor For t/tcr> 2 andp/pcr< 0.05; Z~ 1

Caloric Equation of State

But So

Integrating

If cvis constant

and u0and t0are zero

Caloric Equation

of State

A gas having constant cvis called calorically

perfect gas


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Specific Heat Relationships

For thermally perfect gas

For a reversible process

of Thermally perfect gas

So

For an isobaric

process, we get

Combine the

two equations

Integrating

Also

cpis also

constant for

calorically

perfect gas

Where

Thermodynamic Properties of a Perfect Gas, contdDifferentiating thermal

equation state we get

Enthalpy Change

AsDifferentiating )(pvddudh

And Rdtpvd )(

dtRcdh v )( dtcdh p

If cvis constant and

h0and t0are zero


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Isentropic Relations

Integrating both equations

Considering calorically

perfect gas

Also

Combining all the relations

Where

Thermodynamic Properties of a Perfect Gas, contdEntropy Change

So

For a reversible process

of Thermally perfect gas vdpdhtds

1

2

1

212 lnln

v

vR

t

tcss v

1

2

1

212 lnln

p

pR

t

tcss p

1

2

1

2 lnln0p

pR

t

tcp

1

1

2

1

2

tt

pp

1

R

cp

1

2

1

2 lnln0v

vR

t

tcv

1

1

1

2

1

2

t

t

v

v

1

1

1

2

1

2

t

t

1

1

2

1

2

1

2

tt

pp

Where1

1

R

cv


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Process Equations

Following processes are most significant

1. Isothermal Change (dt= 0)2. Adiabatic Change (dQ= 0)

3. Isobaric Change (dp= 0)

4. Isochoric Change (dv= 0)

5. Isentropic Change (ds= 0)

where

All these may be written

in a general form

Processes occurring rapidly can easily be

considered adiabatic but these are seldom

reversible in nature. Reversibility may be assumed as a first

approximation

We shall assume flow through nozzle as

isentropic

n= 1 is for dt= 0

n= 0 for dp= 0n= for dv= 0

n= for ds= 0

Thermodynamic Properties of a Perfect Gas, contd

constantnpv


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EntropyEnthalpy Diagram

Curves of constantv are steeper than those of constantp For constant values of s,pincreases with twhile vdecreases with t

Reference conditions may be chosen arbitrarily

Same is h-sand u-splot for ideal gas

It is also called Mollier Diagram

Lines of constant specific

volume can be found from

Thermodynamic Properties of a Perfect Gas, contd

These come out to be

Lines of constant Pressure

can be found fromThese come out to be


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Quiz 1p v

c c R is valid for which of the following process of an ideal gas

1. Isobaric Process only2. Isochoric Process only

3. Isentropic Process only

4. All of the above

5. Only 1 and 2


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Governing Equations for Compressible FlowsFollowing are the Laws

1. Law of conservation of mass

2. Newtons second law of motion3. First law of thermodynamics

4. Second law of thermodynamics

Following are the Governing Equations

1. Continuity Equation

2. Momentum Equation3. Energy Equation

4. Entropy Equation

Continuity Equation

IntegralForm

DifferentialForm

Momentum Equation

Integral

Form

Differential

Form


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Energy Equation

Let us define:

B1= Rate of heat added to fluid in control volume from surroundingsB2= Rate of work done on fluid inside control volume

B3= Rate of change of energy of fluid as it flows through control volume

Governing Equations for Compressible Flows, contdIt is based on the physical

principle Energy can neither becreated nor be destroyed, it can

only change in form

dedwdq

321 BBB

Let us analyze B1term first

Heat may be added due to

1. Nuclear Reaction

2. Chemical Reaction

3. Radiation Heating etc

Let q

volumetric rate of heat addition per unit mass

Rate of heat addition to mass of small volume is


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Now let us tackle B2term:

Governing Equations for Compressible Flows, contd

Term B3:


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Energy Equation

IntegralForm

Differential

Form

Governing Equations for Compressible Flows, contd

Entropy Equation

t

qds

For S = ms the equation becomes

t

QdS

Time rate of change of entropy within the control volume is given byt

Q

Dt

DS

Integral Form

Differential Form

theory of compressible flows- chapter 1

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