general features of steady one dimensional flow
TRANSCRIPT
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