Mass Flow Rate Through Converging Nozzle - Flow Rate Equation and Critical Pressure Ratio at Chocked Condition

MASS FLOW RATE THROUGH CONVERGING NOZZLE
FLOW RATE EQUATION AT CHOCKED CONDITION OF COMPRESSIBLE FLUID
THROUGH A CONVERGENT NOZZLE & CRITICAL PRESSURE RATIO AT CHOCKED
CONDITION
Assuming the flow to take place adiabatically, then by using Bernoulli’s equation (for adiabatic
flow), we have
m=mass flow rate kg
s
A 2=area of nozzle ∈m2
p1=Inlet absolute pressure∈kPa
ρ1= Density of air
p2=outlet absolute pressure∈kPa
γ =Ratio of specific heat 1.4 for air
R=Gas constant of air
( γ
γ −1 ) p1
ρ1 g + V 1
2
2 g + z1=( γ
γ −1 ) p2
ρ2 g + V 2
2
2 g + z2
But z1 = z2 and V1 = 0
( γ
γ −1 ) p1
ρ1 g = ( γ
γ −1 ) p2
ρ2 g + V 2
2
2 g
( γ
γ −1 ) ( p1
ρ1 g − p2
ρ2 g ) =V 2
2
2 g
( γ
γ −1 ) ( p1
ρ1
− p2
ρ2 )= V 2
2
2
V 2= √( 2 γ
γ −1 ) ( p1
ρ1
− p2
ρ2 )
V 2= √( 2 γ
γ −1 ) p1
ρ1 ( 1− p2
ρ2
− p1
ρ1 ).......1
For adiabatic flow
ρ1
ρ2
=( p1
p2 ) 1
γ .......2
Put above value in 1
V 2= √( 2 γ
γ −1 ) p1
ρ1 (1−
( p2
p1 )γ−1
γ
)
The mass rate of flow of the compressible fluid,
m = 2A2V2, A2 being the area of the nozzle at the exit
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MASS FLOW RATE THROUGH CONVERGING NOZZLE
m=ρ 2 A 2 √ ( 2 γ
γ −1 ) p1
ρ1 (1− ( p2
p1 )γ−1
γ
)
m= A 2 √ ( 2 γ
γ −1 ) p1
ρ1
ρ 22
(1−
( p2
p1 )γ−1
γ
)......3
From equation 2
ρ1
ρ2
=( p1
p2 ) 1
γ
ρ2
2=ρ1
2
( p2
p1 ) 2
γ
Put above value in Eqn 3 we get
m= A 2 √ ( 2 γ
γ −1 ) p1 ρ1 ( ( p2
p1 ) 2
γ −
( p2
p1 ) γ +1
γ
)
The mass rate of flow (m) depends on the value of p2
p1
Value of p2
p1
for maximum mass flow rate:
For maximum value of m we have
d ( m )
d ( p2
p1 )=0
Put the value of m from above
As critical pressure value
( ( p2
p1 )=( 2
γ +1 ) γ
γ−1
)
Maximum value of flow rate m at critical pressure
mmax =A 2 √ ( 2 γ
γ −1 ) p1 ρ 1 (( p2
p1 )2
γ − ( p2
p1 )γ +1
γ
)
mmax =A 2 √ ( 2 γ
γ −1 ) p1 ρ 1 (( 2
γ +1 ) 2
γ−1 − ( 2
γ +1 ) γ +1
γ −1
)
Put γ =1.4for air
mmax =A 2 √ 7 p1 ρ1 ( 0.4018−0.3348 )
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MASS FLOW RATE THROUGH CONVERGING NOZZLE
mmax =0.685 A 2 √ p1 ρ1
Theoretical mass flow rate at chocked condition :-
mmax =0.685 A 2 √ p1 ρ1
INTRODUCTION
 A nozzle is a device in which the kinetic energy of a fluid is increased in an
adiabatic process. This increase involves a decrease in pressure and is
accomplished by the proper change in flow area. A diffuser is a device that has the
opposite function, namely, to increase the pressure by decelerating the fluid.
 Its principle is based on the fact that the gas flow accelerates to the critical
velocity at the nozzle throat (this being equal to the local sonic velocity). At the
critical velocity, the mass flow-rate of the gas flowing through the Venturi nozzle
is the maximum possible for the existing upstream conditions.
 In these conditions, the mass flow rate through a sonic nozzle is
determined from the stagnation pressure and temperature
measurements and the calculation of the thermodynamic
coefficients.
 Compressibility becomes important for High Speed Flows where M > 0.3
 M < 0.3 - Subsonic & incompressible
 0.3 < M < 0.8 - Subsonic & compressible
 0.8 < M < 1.2 - Transonic: shock waves appear mixed subsonic and sonic flow
regime
 1.2 < M < 3.0 - Supersonic: shock waves are present but NO subsonic flow
 M > 3.0 - Hypersonic: shock waves and other flow changes are very strong
 Significant changes in velocity and pressure result in density variations throughout a
flow field
 Large Temperature variations result in density variations
 Compressible flow is shown by:-
∂ ρ
∂t ≠ 0
Principal
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MASS FLOW RATE THROUGH CONVERGING NOZZLE
 The simplest flow system would use an inlet pressure regulator to control air
pressure and a thermocouple to measure temperature. Adjusting the pressure
regulator will change and maintain the flow through the Nozzle.
 Pressure differences within a piping system travel at the speed of sound and
generate flow. Downstream pressure disturbances cannot move upstream past
the throat of the Nozzle because the throat velocity is higher and in the opposite
direction. Since these pressure disturbances cannot move upstream past the
throat, they cannot affect the velocity or the density of the flow through the
Nozzle. This is what is referred to as a choked or sonic state of operation. This is
one of the greatest advantages of Sonic Nozzles when compared to subsonic
flow-meters (Venturis or Orifice Plates where any change in downstream
pressure will affect the differential pressure across the flow-meter, which in
turn, affects the flow).
 As a result, Sonic Nozzles are ideal for applications where steady inlet flow is
required even though there is pulsating or varying gas consumption
downstream. They are also ideal as flow limiters since with a fixed upstream
pressure both mass and volumetric flows are fixed.
 Accuracy levels of ±0.25% of reading or better can routinely be achieved since
there are no moving parts.
APPLICATIONS
• Applications where the assumptions of steady, uniform, isentropic flow are
reasonable:
• Exhaust gasses passing through the blades of a turbine.
• Diffuser near the front of a jet engine
• Nozzles on a rocket engine
• A broken natural gas line
Solution-2
Experiment -1
Aim :- To determine the effect of inlet pressure on the air flow rate (m) and
compare it with theoretical calculations.
Procedure:-
1. Close the air inlet throttle valve 3.
2. Adjust the inlet pressure to approx 700-900 kPa (gauge) using pressure
regulator and open the throttle valve. Open the back pressure valve and
then close it slightly to get a back pressure of 20-50kPa
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