gas dynamics-fanno flow

Fanno Curve

GDJP Anna University 2

• Infinite number of downstream states 2 for a given upstream state 1

• Practical approach is to assume various values for T2, and calculate all other properties as well as friction force.

• Plot results on T-s diagram

– Called a Fanno line

• This line is the locus of all physically attainable downstream states

• s increases with friction to point of maximum entropy (Ma =1).

• Two branches, one for Ma < 1, one for Ma >1

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Fanno Curve


• The Fanno curve is a curve on the Mollier (h-s) diagram for a given upstream condition for different amount of friction (different length of pipe).

• The maximum entropy condition corresponds to the sonic condition at which the flow is choked. Friction always drive the Mach number towards 1.

• Once the sonic condition is reached at the exit, any increase in pipe length is not possible without drastic revision of the inlet condition.

• Within the framework of 1-D theory, it is not possible to first slow a supersonic flow to the sonic condition and then to further slow it to subsonic speeds also by friction.




The effects of friction on the properties of Fanno flow





Continuity Equation


m = ρAV = constant but since the flow area is constant, this reduces to

ρV = constant

We assign a new symbol G to this constant (the quantity ρV ),

which is referred to as the mass velocity, and thus

ρV = G = constant mass flow density or mass velocity




Energy Equation


We start with

h01+ q = h02 + ws

For adiabatic and no work, this becomes

h01 = h02 = h0 = const

For any given flow, ho and G known. Thus this equation establishes a unique relationship between h and ρ

2

2

02

2

0

20

21 ;

2121

ρρGhhGhh

chh

−=+=

+=




Fanno Lines in h- v plane and h-s plane


Stagnation enthalpyStagnation enthalpy




Fanno line together with typical pressure lines


• We normally feel that frictional effects will show up as an internal generation of “heat” with a corresponding reduction in density of the fluid.

• To pass the same flow rate (with constant area), continuity then forces the velocity to increase.

• This increase in kinetic energy must cause a decrease in enthalpy, since the stagnation enthalpy remains constant.

• As can be seen in Figure. this agrees with flow along the upper branch of the Fanno line. It is also clear that in this case both the static and stagnation pressure are decreasing.




1–D Flow model


wדdirection flow theagainst

acts )Stress(Shear Wall wτ




1–D Flow model


X*X2X1




Variation of Flow Properties


We have discovered the general trend of property variationsthat occur in Fanno flow. Now we wish to develop some specific workingequations for the case of a perfect gas.

These are relations between properties at arbitrary sections ofa flow system written in terms of Mach numbers and the specific heatratio.

TemperatureIn Fanno flow process ‘Stagnation Temperature’ remains same

−

+=

−

+=

−

+=

=

222

21

2

21

211

211

as written be can flow Fannofor equation energy the Hence2

11

MTMTT

MTT

TT

t

t

tt

γγ

γ

( )[ ]( )[ ] 2

2

21

1

2

2/11 2/11 ;MM

TT

−+−+

=γγ




Cont..


Pressure

Velocity

Density

Stagnation Pressure

Impulse Function

)1( 2

2

2

MpA

pAMpA

AcpAF

γ

γ

ρ

+=

+=

+=In general, the Impulse function is




Cont..


Entropy Change

Put temperature and pressure ratio in the above eqn.

Variation of Mach number with Duct length

1Mfor 0 4

2

max

_

==

M

DLfNote:




Friction Factor


Friction factor or friction coefficient for a compressible fluid flowing in a duct is a function of Reynolds number. It also depends on the roughness of the pipe surface (Nikuradse)

:table the in shown are

tiesirregulari wall ofheight averageor )roughness( absolute of values Typical

roughness relative D /

number Reynolds (Re) ; D) / (Re, f f

ε

ε

µρ

ε

=

== vD

Wall material ε (mm)Drawn tubing 0.00015Commercial Steel &Wrought Iron

0.045

Galvanized Iron 0.15Cast Iron 0.25Concrete 0.3 – 3.0Rivetted Steel 1 - 10

22/1

head dynamicstressshear

cf

wallf

w

ρτ

=

=




Moody diagram for friction factor


The relationship among f, Re and ε/D is determined experimentally and plotted on a chart, which is called Moody diagram.

Rough pipes

• For small Re (<2000) , f = 64/ Re• Re > 2000 & small ε/D, such a wall surface is said to be ultimatesmoothness

• For large Re and ε/D, friction factor is independent of Re, such asurface is said to be wholly rough.




Problem solving technique





Hydraulic mean diameter


For turbulent flows in non-circular ducts a hydraulicmean diameter may be used in place of the pipediameter (eg. Heating and ventilation ducting). Thehydraulic mean diameter is defined by

DH =

wetH P

Aerimeterwetted

areacrossD 4p

sectional 4=

×=




Problem1


Show that the Mach number corresponds to the maximum entropy point on a Fanno curve is unity.

Solution:




Cont..





Cont..


Where,k-specific heat ratioV-fluid velocityc-sound velocity




Problem2


A flow is supplied by a converging nozzle (unchoked)(a) Will the addition of diverging section increase or decrease the m?

Solution:

(b) What about adding a constant area duct? Will it increase or decreasethe m?




Cont..





Problem3





Cont..


2




Cont..





Cont..


Pipe choking pressure < Con. Nozzle choking pressure(0.4436) < (0.528)




gas dynamics-fanno flow

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