General Features of General Features of Steady One Dimensional Steady One Dimensional

FlowFlow

M.S. Process & Mechanical M.S. Process & Mechanical Engineering 3rd SemesterEngineering 3rd Semester

Basic AssumptionsBasic AssumptionsFollowing assumption are mostly applied to simplify the governing equationsFlow is steady i.e. at each cross section of a flow passage the magnitudes of flow properties are invariant with timeFlow is uniform i.e. area of flow passage is either constant or changes gradually.Effects of body forces are negligible justified usually for gas flows.Ideal behavior of working fluid i.e. working fluid behaves thermodynamically in accordance with

1. Thermal Equation of state2. Caloric Equation of state The results obtained after applying above assumptions are

quite accurate for internal flows and are useful qualitative for understanding external flows

The One Dimensional Flow ConceptThe One Dimensional Flow Concept

1/

max 1n

yu u

R

2

max 21y

u uR

Velocity profiles in a duct of Radius R for Laminar flow is

Turbulent flow is

Most commonly employed assumption is that “Flow is one dimensional” i.e. all fluid properties are uniform over every cross section of flow passage.

The One Dimensional Flow Concept, The One Dimensional Flow Concept, contdcontd

One dimensional flow approximation is quite accurate for throat of converging and C-D nozzles and Diffusers.

It may lead to errors flow inside passages with complicated shapes and ducts with developing flow

One dimensional flow concept is approximation as far as flow model is concerned.

No approximation is introduced while deriving the flow equations mathematically.

Necessary condition for One dimensional flow

1. Rate of change of flow driving potentials should be small in flow direction

2. Radius of curvature of flow passage should be large

3. Profiles of flow properties should be self similar

One dimensional flow model considers changes only in average flow properties in the direction of flow.

The One Dimensional Flow Concept, The One Dimensional Flow Concept, contd.contd.

Conservation of Mass For Steady 1D Conservation of Mass For Steady 1D FlowFlow

Integral form of Law of conservation of Mass can be given as

0

Applying Eq. to Fig. shown:

Integrating above eq.

Continuity eq.

Integral form of momentum eq for xi direction is:

For steady frictionless flow the Equation becomes:

Dynamics of Steady 1D Flow Dynamics of Steady 1D Flow (Frictionless Flow)(Frictionless Flow)

0 0

For the flow model:

, , and i iu V V n i V i

On the face where mass leaves CV d dAA iOn the face where mass enters CV d dAA i

The only body force considered is that caused by gravitational attraction i.e. B g

Dynamics of Steady 1D Flow Dynamics of Steady 1D Flow (Frictionless Flow), contd.(Frictionless Flow), contd.

Component of body force in x-direction is:

Agdzi

/ sindA

Stream tube boundary area is:

At inlet properties arep, ρ, V, V2/2, pA

At outlet properties arep+dp, ρ+dρ, V+dV, V2/2+d(V2/2), pA+d(pA)

Dynamics of Steady 1D Flow Dynamics of Steady 1D Flow (Frictionless Flow), contd.(Frictionless Flow), contd.

After substitution we get:

Canceling like terms and neglecting higher order terms:

Substituting

0dp VdV gdz

We get

Bernoulli equation

Integrating:2

2

dp Vgz const

Holds for steady, 1D, frictionless flow along a streamline

Dynamics of Steady 1D FlowDynamics of Steady 1D Flow(Frictionless Flow), contd(Frictionless Flow), contd

Incompressible Fluid:

ρ = constant

Adiabatic Flow of Compressible Fluid

2

2

p Vgz const

0dp d

VdV gdzd

But2

s

p dpa

d

And assuming 0gdz

2 0d dV

MV

Dynamics of Steady 1D Flow Dynamics of Steady 1D Flow (Frictional Flow)(Frictional Flow)

Net external force in x-direction is given by:

fgAdz Adp F D

Putting it we get:

Ff is represented in terms of

1. Hydraulic characteristics of flow channel

2. Experimental friction coefficient f

2 22 1 112 22

tangential Force

( ) fFf

V WP dx VV wetted area

Dynamics of Steady 1D Flow Dynamics of Steady 1D Flow (Frictional Flow), contd.(Frictional Flow), contd.

So

2 2 4( )

2 2f

V V fdxF f WP dx A

D

Substituting for Ff

Incompressible flow

= const

D = 0

Dynamics of Steady 1D Flow Dynamics of Steady 1D Flow (Frictional Flow), contd.(Frictional Flow), contd.

2/p a

Compressible Flow of Perfect Gas

Divide by p

Note That

As V2 = M2a2 and da2/a2 = dt/t we get

Substituting

Thermodynamics of Steady 1D FlowThermodynamics of Steady 1D FlowFlow with Work and Heat Transfer (Infinitesimal CV)

Thermodynamics of Steady 1D Flow, Thermodynamics of Steady 1D Flow, contd.contd.

Canceling like terms and dividing by m we get

Flow with Work and Heat Transfer (Finite CV)

Evaluating integrals and dividing by m

Adiabatic Flow with No External Work (Infinitesimal CV)

For such flows

Adiabatic Flow with No External Work (Finite CV)

Isoenergetic FlowIsentropic Flow

Bernoulli equation

Applies to flow with or without friction

Thermodynamics of Steady 1D Flow, Thermodynamics of Steady 1D Flow, contdcontd

Adiabatic Flow of Perfect Gas

1. Ideal Gas

2. dh = cp dt

3. g dz = 0

constV

tcV

tcV

tc ppp 222

222

2

21

1

For Ideal gas RtaRcp 2 and )1/(

constaVaVaV

121212

2222

22

21

21

Thermodynamics of Steady 1D Flow, Thermodynamics of Steady 1D Flow, contdcontd

Taking 1

2

a

common

As Rta 2

Thermodynamics of Steady 1D Flow, Thermodynamics of Steady 1D Flow, contdcontd

Isentropic Discharge Speed for a Perfect Gas

Process is

1. Adiabatic

2. Frictionless

So we can apply

constV

tcV

tcV

tc ppp 222

222

2

21

1

with PpTtV 111 and ,0 and ppttVV 222 and ,

We getprocess Isentropic

gasperfect for )1/(/)1(

P

p

T

t

RcpBut

Venant-Wantzel Equation

Thermodynamics of Steady 1D Flow, Thermodynamics of Steady 1D Flow, contdcontd

Since

m

RR

2/1/)1(

11

2

P

p

m

TRV

Discharge speed can be controlled by

1.

2. mT /

Pp /

Thermodynamics of Steady 1D Flow, Thermodynamics of Steady 1D Flow, contd.contd.

22ndnd Law of Thermodynamics for Steady Law of Thermodynamics for Steady 1D Flow1D FlowEntropy Eq. for control volume is:

For infinitesimal control volume eq becomes:

Expression for Q must be derived for the particular heat transfer process

For adiabatic process

This equation determines the direction of the process

Some General Effects of Compressibility on Some General Effects of Compressibility on Fluid FlowFluid FlowSpeed of Propagation of Small Disturbance

Pressure disturbances are transmitted in a fluid as successive compressive and rarefaction waves because of elastic nature of fluid

Since a sound wave consists of a repeating pattern of high pressure and low pressure regions moving through a medium, it is sometimes referred to as a pressure wave

Some General Effects of Compressibility on Some General Effects of Compressibility on Fluid FlowFluid FlowStationary observerUnsteady Case

Observer moving with pressure waveSteady Case

Flow is 1 Dimensional

Some General Effects of Compressibility on Some General Effects of Compressibility on Fluid FlowFluid Flow

In momentum eq.

0/

0

0

tV

F

B

shear

).( AVVA dpdAA

)()]([)( cmdVcmAdpppA

dV dVdp m Ac cdV

A A

So the Eq. becomes

For the control volume shown

Simplifying

From continuity Eq. Ac d A c dV Solving for dV

ddV c

dp

cd

Put in momentum eq

Some General Effects of Compressibility on Some General Effects of Compressibility on Fluid FlowFluid Flow1. Changes within wave are slight Flow is reversible

2. No heat addition

Hence, the process inside wave is isentropic, so

s

pc

For calorically perfect gas 1 1

2 2

p

p

cop nst

So 1

s

p p p

Hence pc

Butp

RT

c RT

Some General Effects of Compressibility on Some General Effects of Compressibility on Fluid FlowFluid Flow

Isothermal compressibility1

s

s

v

v p

But21/ and /v dv d

Hence 2

1 1

( / )sss

p p

Here 2( / )sp c

1

s

c

Sos

VM V

c

Some General Effects of Compressibility on Some General Effects of Compressibility on Fluid FlowFluid FlowPressure Disturbances in a Compressible Fluid

Stationary Source

Source with subsonic Velocity

Some General Effects of Compressibility on Some General Effects of Compressibility on Fluid FlowFluid Flow

Source with sonic Velocity

Source with supersonic Velocity

MVt

at 1sin

M

1sin 1

Compressibility FactorCompressibility FactorDynamic Pressure Pressure increase caused by deceleration

Stagnation Pressure is given as

Dynamic Pressure for compressible flow

Using Binomial Theorem

Taking

2 / 2M common and noting that2 2/ 2 / 2pM V

Compressibility Factor