Post on 16-Nov-2014
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Fanno Curve
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• 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
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• 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.
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The effects of friction on the properties of Fanno flow
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Continuity Equation
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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
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Energy Equation
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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
−=+=
+=
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Fanno Lines in h- v plane and h-s plane
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Stagnation enthalpyStagnation enthalpy
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Fanno line together with typical pressure lines
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• 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.
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1–D Flow model
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wדdirection flow theagainst
acts )Stress(Shear Wall wτ
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1–D Flow model
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X*X2X1
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Variation of Flow Properties
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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
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Cont..
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Pressure
Velocity
Density
Stagnation Pressure
Impulse Function
)1( 2
2
2
MpA
pAMpA
AcpAF
γ
γ
ρ
+=
+=
+=In general, the Impulse function is
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Cont..
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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:
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Friction Factor
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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
=
=
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Moody diagram for friction factor
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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.
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Problem solving technique
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Hydraulic mean diameter
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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=
×=
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Problem1
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Show that the Mach number corresponds to the maximum entropy point on a Fanno curve is unity.
Solution:
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Where,k-specific heat ratioV-fluid velocityc-sound velocity
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Problem2
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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?
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Problem3
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2
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Pipe choking pressure < Con. Nozzle choking pressure(0.4436) < (0.528)
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