hes5340 fluid mechanics 2, lab 2 - compressible flow (converging-diverging duct test) (semester 2,...
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SWINBURNE UNIVERSITY OF TECHNOLOGY (SARAWAK CAMPUS)
FACULTY OF ENGINEERING AND INDUSTRIAL SCIENCE
HES5340 Fluid Mechanics 2Semester 2, 2012
COMPRESSIBLE FLOW
Convergent-Divergent Duct Test
By
Stephen, P. Y. Bong (4209168)
Lecturer: Dr. Basil, T. Wong
Due Date:
19th
November 2012 (Monday), 12 pm
Date Performed Experiment:
2nd
November 2012 (Thursday), 3:30 – 5:30 pm
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Table of Contents1.0 Introduction ........................................................................................................................................... 3
2.0 Objectives .............................................................................................................................................. 4
3.0 Theory.................................................................................................................................................... 5
3.1 The Ideal-Gas Equation of State (or the Perfect Gas Law) ........................................................... 5
3.2 Variation of Fluid Velocity with Flow Area ................................................................................. 5
3.3 Speed of Sound and Mach number ............................................................................................... 6
3.4 Relationships of Fluid Properties in Isentropic (Reversible Adiabatic) Flow .............................. 7
3.5 Properties of Fluid at Critical State (M = 1) ................................................................................. 7
3.6 Flow of Compressible Air through Converging-Diverging Nozzles ............................................ 8
4.0 Description of Apparatus ..................................................................................................................... 10
5.0 Experimental Procedures ..................................................................................................................... 12
6.0 Results and Calculations ...................................................................................................................... 13
6.1 Absolute Pressure at Each Point along the Convergent-Divergent Duct .................................... 14
6.2 Mach number, M, at Each Point along the Convergent-Divergent Duct .................................... 15
6.3 Density of Compressible Air at Each Point along the Convergent-Divergent Duct ................... 16
6.4 Temperature Distributed at Each Point along the Convergent-Divergent Duct .......................... 17
6.5 Speed of Sound at Each Point along the Convergent-Divergent Duct ........................................ 18
6.6 Velocity of the Air Flow at each point along the Convergent-Divergent Duct .......................... 19
6.7 Mass Flow Rate of Compressible Air at Each Point along the Convergent-Divergent Duct ..... 20
6.8 Sample Calculations .................................................................................................................... 21
6.8.1 Absolute Pressure ............................................................................................................. 21
6.8.2 Mach Number ................................................................................................................... 21
6.8.3 Density .............................................................................................................................. 22
6.8.4 Temperature ...................................................................................................................... 22
6.8.5 Speed of Sound ................................................................................................................. 22
6.8.6 Velocity ............................................................................................................................ 23
6.8.7 Mass Flow Rate ................................................................................................................ 237.0 Discussions .......................................................................................................................................... 24
7.1 Experimental Results .................................................................................................................. 24
7.2 Experimental Errors .................................................................................................................... 25
8.0 Conclusion ........................................................................................................................................... 25
9.0 References ........................................................................................................................................... 26
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1.0 Introduction
According to Cengel & Boles (2007, p. 849), the term “Compressible Flow” can be interpreted as
the flow of fluid in which variation in fluid properties such as density is significant due to pressure
deviations. The mechanics of compressible flow had been extensively employed in wide range of
engineering applications and technological processes such as the converging-diverging nozzlesemployed in rocket engine (see Figure 1 below), the steam and gas turbines, propulsive system of
aircraft and spacecraft, and die casting as well as injection molding in manufacturing processes
(Genick 2007). As addressed in the Codes and Standards of American Society of Mechanical
Engineers (ASME 2012), such applications may touch upon the existence of fluid flow with high
velocity in which negative side effects such as acoustical disturbances and turbulence will be
consequence. Therefore, in order to overcome these drawbacks, the development of rightful
comprehending on the fundamental theories in the physics of compressible flow is significant.
Figure 1: Converging-diverging nozzles employed in rocket engines to provide high thrust
(Cengel & Boles 2006, p. 868)
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2.0 Objectives
The examination and analyzing of the characteristics of air as it is flowed through the convergent-
divergent duct is the primary objectives of this experiment. Apart from that, the determinations of
the effect of compressibility on flow equations and the density of air are the objectives of the
convergent-divergent duct test as well.
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3.0 Theory
In order to examine the characteristics of pressure flow of air through a convergent-divergent duct
and clearly visualize on how the properties of air being affected by Mach number, the development
of rightful comprehending on the fundamental theories behind the physics of compressible flow are
significant.
3.1 The Ideal-Gas Equation of State (or the Perfect Gas Law)
Although the calculations of compressible flow can be employed in any fluid equation of state, but
most elementary treatments are limited to the perfect gas with constant specific heat capacity.
According to Cengel & Boles (2007, p. 137), the perfect gas law is a relation that relates properties
of a substance at an equilibrium state such as pressure, temperature, and volume. The ideal-gas
equation of state can be mathematically expressed as:
pV = mRT or p = ρ RT Eq. [1]
where p = Absolute pressure (Pa)
V = Volume (m3)
m = Mass (kg)
R = Gas constant (For air, R = 287 kJ/kg·K)
T = Absolute temperature (K)
ρ = Density (kg/m3)
3.2 Variation of Fluid Velocity with Flow Area
The conservation of mass (the equation of continuity) can be used to determine the flow rate of a
fluid through a conduit of variable cross-sectional area. The conservation of mass is:
AV m ρ = Eq. [2]
where ṁ = Mass flow rate (kg/s)
ρ = Density (kg/m3)
A = Cross-sectional area of the conduit (m
2
)V = Average velocity of the fluid flow (m/s)
As mentioned by Cengel & Cimbala (2004, p. 633), the change in velocity with respect to the cross-
sectional area is given by:
1M
12−
⋅=
A
V
dA
dV Eq. [3]
where V = Velocity of the fluid flow (m/s)
A = Cross-sectional area of the duct (m2)
M = Mach number
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For incompressible flow (M < 0.3),V
A1
∝
For subsonic flow (M < 1), 0<
dA
dV and
V A
1∝
For sonic flow (M = 1), 0=
dA
dV and V = c
For supersonic flow (M > 1), 0>
dA
dV and V A∝
Based on Eq. [3] and above relations, it can be concluded that, for incompressible flow, the velocity
of the fluid flow with constant density is inversely proportional to the cross-sectional area. In
contrary, for supersonic flow, as the area increased, the velocity increases.
3.3 Speed of Sound and Mach number
One of the parameter which plays a significant role in the analysis of compressible flow is the speed
of sound or sometimes it also referred as the sonic speed. The speed of sound is the speed at which
an infinitesimal pressure pulse or wave propagates through a medium, and it is given by:
kRT c = Eq. [4]
where c = Speed of sound (m/s)
k = Specific heat ratio = c p / cv
R = Gas constant (For air, R = 287 J/kg·K)
T = Absolute temperature (K)
Apart from the speed of sound, Mach number, M, is also a crucial parameter that had been
employed in the analysis and study of compressible flow. Cengel & Boles (2007, p. 854) addressed
that Mach number can defined as the ratio of the actual velocity of fluid (or object in still air) to the
speed of sound. It can be mathematically expressed as:
c
V =M Eq. [5]
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3.4 Relationships of Fluid Properties in Isentropic (Reversible Adiabatic) Flow
The variation in Mach number in Eq. [4] will results in drastic change in the properties of fluid in
compressible flow. The properties of fluid such as pressure, temperature, and density can be related
to Mach number and their corresponding stagnation state which denoted by a subscript “ t ” by the
following equations:
2M2
11
−+=
k
T
T t Eq. [6]
( )1
2M2
11
−
−+=
k k
t k
p
p Eq. [7]
( )11
2M2
11
−
−+=
k
t k
ρ
ρ Eq. [8]
3.5 Properties of Fluid at Critical State (M = 1)
The flow of compressible air through the convergent-divergent duct is initiated by employing a
vacuum pump at the downstream end. This is due to the fact that when a suction pressure is created
by the vacuum pump, the air will through the convergent-divergent duct from the surroundings. If
the back pressure is sufficiently low, the flow of compressible air can be accelerated to a maximum
Mach number of 1 which often termed as chocked flow.
According to Cengel & Boles (2007, p. 861), critical properties can be interpreted as the propertiesof fluid at which the corresponding Mach number is unity (the critical state). This occurrence of this
phenomenon often exists at the throat which has the smallest cross-sectional area among the entire
converging-diverging duct. By substituting M = 1 and k = 1.4 (for air) into Eq. [6], [7] & [8] above
gives:
*2.1 T T = Eq. [9]
*8929.1 p pt ≈ Eq. [10]
*5774.1 ρ ρ ≈t Eq. [11]
A relation for the variation of flow area through the convergent-divergent duct relative to the throat
area A* is given by:
1
1
2
1
2M
2
11
1
2
M
1
*
−
+⋅
−+
+
=
k
k
k
k A
A Eq. [12]
Substituting k = 1.4 for air into Eq. [12] yields:
*M5 A A = Eq. [13]
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3.6 Flow of Compressible Air through Converging-Diverging Nozzles
The primary intention of the design of Laval nozzle is to compress the air flow through to become
supersonic. As the compressible air is flow through the convergent-divergent duct, there will be no
flow exists if the back pressure is same as the stagnation pressure. Nevertheless, the flow pattern of
compressible air will deviate if the back pressure at the downstream end of the convergent-divergent duct is reduced. There are five possible cases of flow pattern of compressible air at the
exit of the convergent-divergent duct as depicted in Figure 4 and 5 below:
Figure 2: Characteristics of Back Pressure
(Case (a), (b) & (c))
Case (a): p1 > p2 >> p kr
The flow of compressible air is subsonic
throughout the entire flow path and resembles flow
through a venture nozzle. The lowest pressure is
found at the throat. Based on the plot of pressure
against position as shown in Figure 4 on the right,,
the pressure decreases in the converging section,
and reaches a minimum which is the critical
pressure at the throat, and increase again due to the
retardation of velocity in the diverging section that
has a larger cross-sectional area.
Case (b): p1 >> p2 > p kr
Supersonic flow is barely can be attained at thethroat of the nozzle. On the other hand,
compression takes place again in the divergent
section. At the discharge which located at the
downstream end of the nozzle, the pressure p2 is
higher than the critical pressure, and the mass flow
rate is also the maximum.
Case (c): p2 = p2 a < p kr (Ideally-Expanded)
The pressure at the discharge of the nozzle is equal
to the pressure for which the Laval nozzle is
designed. Sonic flow is attained at the throat of the
nozzle. In the divergent section, the sonic flow is
accelerated to supersonic velocities as the pressure
reduces. The Laval nozzle is said to be “suitably
matched”.
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Figure 3: Characteristics of Back Pressure
Case (d)
Case (d): p kr > p1 > p2 a (Over-Expanded)
The counter pressure p’2 is attained before the end
of the wide divergent section. The nozzle is “not
suitably matched”. In the remaining of the
divergent section, a compression shock wave willformed after which reversible adiabatic flow is no
longer possible. The flow is then diverged from the
wall of the nozzle. A subsonic flow is produced
after the compression shock. The discharge
pressures is then between that in Cases (b) and (c).
Figure 4: Characteristics of Back Pressure
Case (e)
Case (e): p kr > p2 a > p2
The counter pressure is not yet reached at nozzle
exit. This nozzle is also “not suitably matched”.
The drop in p2 to less than p2a does not influence
the flow within the nozzle because the interference
is after the exit and upstream effect it possesses in
the supersonic flow. All that occurs after the exit is
post-expansion and gas spreading.
The compression shocks occurring in the divergent
section or behind the outlet when the counter
pressure is not adapted. Considerable flow losses
due to vibrations can be consequence.
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4.0 Description of Apparatus
The primary application of the LS-18011 Compressible Flow Bench as illustrated in Figure 2 below
is to demonstrate the characteristics of compressible air flow with high velocities. This unit allows
various experiments to be conducted for the study of the motion of compressible fluid flow at
different Mach numbers by control the velocity of the air flow with respect to their correspondingspeed of sound.
The LS-18011 Compressible Flow Bench encompasses of an air compressor which is driven by a
motor and various interchangeable test sections. Apart from that, it also consists of a compressor
operation speed display. The air compressor is a centrifugal machine incorporating aluminum
impellers. Performance test can be conducted on the compressor over a wide range of shaft
velocities.
The design of this unit is to create Mach 1 velocity at the throat which is one of the sections of the
converging-diverging duce with smallest cross-sectional area, and supersonic flow at the
downstream. A series of pressure sensing points is provided on the test section includes the throat,
and the inlet as well as outlet of the nozzle and diffuser respectively, so that the characteristics of air
at every test sections can be examined.
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Figure 5: The LS-18011 Compressible Flow Bench
Figure 6: Schematics of LS18011 Compressible Flow Bench
Legend
A Main power switch
B Emergency stop button
C Speed controllerD Air blower
E Digital display meters
FDigital differential pressure
meter
G Digital torque meter
H Pressure sensors
I Test Sections
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5.0 Experimental Procedures
1. The converging-diverging duct was connected to the inlet of the blower.
2. The joints were tightened by using the set screws and nuts provided.
3. The main power was switched “ON”.
4. The reading of measurements is zero was ensured prior to the starting of experiment.5. The air blower was started by pressing the “RUN” button, and the speed was increased slowly
to 30 Hz using the speed controller.
6. The pressure measurement points were connected to the pressure sensor by the flexible tubing.
7. The pressure readings for all 10 points were recorded.
8. The blower speed was increased from 35, 40, 45, and 50 Hz, and the pressure readings were
recorded.
9. The bypass valve was fully shut.
10. The pressure readings for all 10 points on the duct were recorded.
11. The bypass valve was fully opened after the experiment, and the speed of the blower was
decreased gradually. The main power was then switched “OFF”
12. The results are tabulated.
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6.0 Results and Calculations
The pressure distributions on the inner surface of the entire convergent-divergent duct obtained
from the experiment are tabulated in Table 1 below.
Table 1: Pressure distributions on the inner surfaceof the entire convergent-divergent duct obtained from the experiment
Point
Reference
Pressure
(Bar)
Distance, x Area, A Pressure, p (Bar)
Frequency (Hz)
(mm) (m) (mm2) (m2) 30 35 40 45 50
1 -0.0001 24 0.024 1134.26 0.00113426 p1 -0.2580 -0.3177 -0.3745 -0.4200 -0.4556
2 -0.0002 54 0.054 951.27 0.00095127 p2 -0.2555 -0.3145 -0.3701 -0.4156 -0.4510
3 -0.0032 84 0.084 613.32 0.00061332 p3 -0.2650 -0.3240 -0.3815 -0.4210 -0.4652
4 -0.0002 114 0.114 349.24 0.00034924 p4 -0.2600 -0.3216 -0.3791 -0.4250 -0.4667
5 -0.0004 144 0.144 159.03 0.00015903 p5 -0.2780 -0.3440 -0.4050 -0.4490 -0.4940
6 -0.0003 174 0.174 42.68 0.00004268 p6 -0.3485 -0.4050 -0.4560 -0.5025 -0.5371
7 -0.0027 186 0.186 28.28 0.00002828 p7 -0.3867 -0.3900 -0.3855 -0.3875 -0.3872
8 -0.0006 198 0.198 95.96 0.00009596 p8 -0.0140 -0.0135 -0.0135 -0.0140 -0.0135
9 -0.0003 228 0.228 1035.85 0.00103585 p9 0.0012 0.0010 0.0012 0.0025 0.0025
10 0.0002 258 0.258 1134.26 0.00113426 p10 -0.0004 -0.0005 -0.0001 -0.0003 0.0005
Torque (N·m) 6.2500 7.2620 8.3270 9.3090 10.1580
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6.1 Absolute Pressure at Each Point along the Convergent-Divergent Duct
The absolute pressure acting on each pressure point on the inner surface of the convergent-divergent
duct can be computed by taking the difference of the gauge and reference pressures as tabulated in
Table 1 above. The absolute pressures computed are then converted to Pascal and the results are
tabulated in Table 2 below.
Table 2: Absolute Pressure Acting on Each Point along the Convergent-Divergent Duct
Point Distance, x (m) Area, A (m2) PressureFrequency (Hz)
30 35 40 45 50
1 Inlet 0.024 0.00113426 p1 75.535 69.565 63.885 59.335 55.775
2
Converging
0.054 0.00095127 p2 75.795 69.895 64.335 59.785 56.245
3 0.084 0.00061332 p3 75.145 69.245 63.495 59.545 55.125
4 0.114 0.00034924 p4 75.345 69.185 63.435 58.845 54.675
5 0.144 0.00015903 p5 73.565 66.965 60.865 56.465 51.965
6 0.174 0.00004268 p6 66.505 60.855 55.755 51.105 47.645
7 Throat 0.186 0.00002828 p7 62.925 62.595 63.045 62.845 62.875
8Diverging
0.198 0.00009596 p8 99.985 100.035 100.035 99.985 100.035
9 0.228 0.00103585 p9 101.475 101.455 101.475 101.605 101.605
10 Outlet 0.258 0.00113426 p10 101.265 101.255 101.295 101.275 101.355
The plot of absolute pressure against the position of pressure points is depicted in Graph 1 below.
Graph 1: Plot of Absolute Pressure, p (kPa) vs. Position of Pressure Point, x (m)
45
55
65
75
85
95
105
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
A b s o l u t e P r e s s u r e , p ( k P a )
Position, x (m)
Plot of Absolute Pressure, p (kPa) vs. Position, x (m)
f = 30 Hz f = 35 Hz f = 40 Hz f = 45 Hz f = 50 Hz
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6.2 Mach number, M, at Each Point along the Convergent-Divergent Duct
Based on the absolute pressure tabulated in Table 2 above, the Mach number at each point along the
convergent-divergent duct can be computed by Eq. [7]. Whereas the absolute pressures tabulated in
Table 1 are static pressure, and the total pressure is atmospheric pressure, patm = 101.325 kPa. The
Mach numbers calculated are tabulated in Table 3 below.
Table 3: Mach number at each point along the convergent-divergent duct
PointDistance,
x (m)
Area, A
(m2)
Mach
Number,
M
Frequency (Hz)
30 35 40 45 50
1 Inlet 0.024 0.00113426 M1 0.661616 0.753107 0.839236 0.908854 0.964326
2
Converging
0.054 0.00095127 M2 0.657571 0.748096 0.832399 0.901916 0.956935
3 0.084 0.00061332 M3 0.667671 0.757964 0.845166 0.905614 0.974586
4 0.114 0.00034924 M4 0.664568 0.758874 0.846079 0.916425 0.981717
5 0.144 0.00015903 M5 0.692061 0.792522 0.885320 0.953483 1.025194
6 0.174 0.00004268 M6 0.799491 0.885473 0.964641 1.039203 1.096791
7 Throat 0.186 0.00002828 M7 0.853843 0.858872 0.852015 0.855061 0.854604
8Diverging
0.198 0.00009596 M8 0.138039 0.135418 0.135418 0.138039 0.135418
9 0.228 0.00103585 M9 0.000000 0.000000 0.000000 0.000000 0.000000
10 Outlet 0.258 0.00113426 M10 0.029090 0.031422 0.020568 0.026555 0.000000
The distribution of Mach number at each point along the convergent-divergent duct is manifested in
the plot of Mach number versus position as shown in Graph 2 below.
Graph 2: Mach number, M, at each point along the convergent-divergent duct
0.00
0.10
0.20
0.30
0.40
0.500.60
0.70
0.80
0.90
1.00
1.10
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
M a c h N u m b e r , M a
Position, x (m)
Plot of Mach Number, Ma vs. Position, x (m)
f = 30 Hz f = 35 Hz f = 45 Hz f = 50 Hz f = 40 Hz
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6.3 Density of Compressible Air at Each Point along the Convergent-Divergent
Duct
The density of compressible air at each point along the convergent-divergent duct can be
determined based on the Mach number tabulated in Table 3 above. The calculations can be
performed by Eq. [8] by taking the total density equal to density of air at a temperature of 25 °C.According to Munson, et al (2009, p. 718), the density of air at standard atmospheric pressure and
temperature of 25 °C is 1.184 kg/m3. The computed densities of air at each point along the
convergent-divergent duct are tabulated in Table 4 below.
Table 4: Density of Air at Each Point along the convergent-divergent duct
PointDistance,
x (m)
Area, A
(m2)
Density,
ρ
(kg/m3)
Frequency (Hz)
30 35 40 45 50
1 Inlet 0.024 0.00113426 ρ1 0.95991202 0.905087 0.851663 0.807881 0.772954
2
Converging
0.054 0.00095127 ρ2 0.96227095 0.908152 0.855943 0.812253 0.777601
3 0.084 0.00061332 ρ3 0.95636927 0.902111 0.847946 0.809923 0.766509
4 0.114 0.00034924 ρ4 0.95818672 0.901553 0.847373 0.80311 0.762035
5 0.144 0.00015903 ρ5 0.94196242 0.880793 0.822707 0.779773 0.73486
6 0.174 0.00004268 ρ6 0.87646739 0.822611 0.772756 0.726153 0.690686
7 Throat 0.186 0.00002828 ρ7 0.84250155 0.839343 0.843649 0.841736 0.842023
8Diverging
0.198 0.00009596 ρ8 1.17279437 1.173213 1.173213 1.172794 1.173213
9 0.228 0.00103585 ρ9 1.184 1.184 1.184 1.184 1.184
10 Outlet 0.258 0.00113426 ρ10 1.18349916 1.183416 1.18375 1.183583 1.184
Graph 3: Plot of density, ρ (kg/m3) versus position, x (m)
0.6
0.7
0.8
0.9
1
1.1
1.2
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
D e n s i t y
, ρ ( k g / m 3 )
Position, x (m)
Plot of Density, ρ (kg/m3) vs. Position, x (m)
f = 30 Hz f = 35 Hz f = 40 Hz f = 45 Hz f = 50 Hz
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6.4 Temperature Distributed at Each Point along the Convergent-Divergent Duct
The distribution of static temperatures at each point along the convergent-divergent duct can be
calculated based on the Mach number obtained in Section 6.2 as well. By setting the total
temperature equal to 298 K (25 °C), the static temperatures can be determined by Eq. [6], and the
computed temperatures are listed below.
Table 5: Temperature distributed at each point along the convergent-divergent duct
PointDistance, x
(m)
Area, A
(m2)
Temperature, T
(K)
Frequency (Hz)
30 35 40 45 50
1 Inlet 0.024 0.00113426 T 1 274.0111 267.6404 261.2057 255.7494 251.268
2
Converging
0.054 0.00095127 T 2 274.280248 268.0026 261.7301 256.3021 251.8711
3 0.084 0.00061332 T 3 273.606133 267.2881 260.7491 256.0077 250.4278
4 0.114 0.00034924 T 4 273.813996 267.2219 260.6787 255.1442 249.842
5 0.144 0.00015903 T 5 271.94997 264.7434 257.6165 252.1522 246.23946 0.174 0.00004268 T 6 264.222555 257.6044 251.2422 245.0681 240.2082
7 Throat 0.186 0.00002828 T 7 260.07815 259.6877 260.2198 259.9836 260.0191
8Diverging
0.198 0.00009596 T 8 296.868646 296.9111 296.9111 296.8686 296.9111
9 0.228 0.00103585 T 9 298 298 298 298 298
10 Outlet 0.258 0.00113426 T 10 297.949572 297.9412 297.9748 297.958 298
The distribution of static temperatures at each point along the convergent-divergent duct is
illustrated in Graph 4 below.
Graph 4: Distribution of static temperatures at each point along the convergent-divergent duct
240
250
260
270
280
290
300
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
T e m p e r a t u r e ,
T ( K )
Position, x (m)
Plot of Temperature, T (K) vs. Position, x (m)
f = 30 Hz f = 35 Hz f = 40 Hz f = 45 Hz f = 50 Hz
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6.5 Speed of Sound at Each Point along the Convergent-Divergent Duct
As mentioned in Eq. [4], the variation of temperature will results in deviation in speed of sound at
each point along the convergent-divergent duct. According to Cengel & Boles (2007, p. 910), the
specific heat ratio and gas constant of air are k = 1.4 and R = 287 J/kg·K. The speed of sound is then
computed and tabulated in Table 6 below.
Table 6: Speed of sound at each point along the convergent-divergent duct
PointDistance,
x (m)
Area, A
(m2)
Speed of
Sound, c
(m/s)
Frequency (Hz)
30 35 40 45 50
1 Inlet 0.024 0.00113426 c1 331.809674 327.9298 323.9636 320.5622 317.7412
2
Converging
0.054 0.00095127 c2 331.972595 328.1515 324.2887 320.9084 318.1223
3 0.084 0.00061332 c3 331.56439 327.7138 323.6804 320.724 317.2095
4 0.114 0.00034924 c4 331.690313 327.6733 323.6367 320.1827 316.8383
5 0.144 0.00015903 c5 330.559371 326.1501 321.7302 318.2998 314.54576 0.174 0.00004268 c6 325.829131 321.7226 317.7249 313.7967 310.6697
7 Throat 0.186 0.00002828 c7 323.263671 323.0209 323.3517 323.2049 323.227
8Diverging
0.198 0.00009596 c8 345.372005 345.3967 345.3967 345.372 345.3967
9 0.228 0.00103585 c9 346.029479 346.0295 346.0295 346.0295 346.0295
10 Outlet 0.258 0.00113426 c10 346.000199 345.9953 346.0148 346.0051 346.0295
According to speeds of sound tabulated in Table 6 above, the variation of the speed of sound with
respect to the position of each point along the convergent-divergent duct is illustrated in Graph 5
below.
Graph 5: Variation in speed of sound with respect to position of each point along the convergent-divergent duct
310
320
330
340
350
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
S p e e d o f S o u n d , c ( m / s )
Position, x (m)
Plot of Speed of Sound, c (m/s) vs. Position, x (m)
f = 30 Hz f = 35 Hz f = 40 Hz f = 45 Hz f = 50 Hz
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6.6 Velocity of the Air Flow at each point along the Convergent-Divergent Duct
With the speeds of sound listed in Table 6, the velocity of the flow of compressible air at each point
along the convergent-divergent duct can be computed by using Eq. [5]. The results are listed in
Table 7 below.
Table 7: Velocity of air at each point along the convergent-divergent duct
PointDistance,
x (m)
Area, A
(m2)
Velocity,
V (m/s)
Frequency (Hz)
30 35 40 45 50
1 Inlet 0.024 0.00113426 V 1 219.530637 246.9663 271.8819 291.3442 306.4061
2
Converging
0.054 0.00095127 V 2 218.295628 245.4889 269.9376 289.4324 304.4223
3 0.084 0.00061332 V 3 221.375875 248.3952 273.5636 290.4523 309.1481
4 0.114 0.00034924 V 4 220.430676 248.6628 273.8221 293.4235 311.0457
5 0.144 0.00015903 V 5 228.767373 258.481 284.834 303.4934 322.4703
6 0.174 0.00004268 V 6
260.497382 284.8767 306.4904 326.0985 340.7399
7 Throat 0.186 0.00002828 V 7 276.016297 277.4335 275.5005 276.36 276.2312
8Diverging
0.198 0.00009596 V 8 47.6748303 46.77275 46.77275 47.67483 46.77275
9 0.228 0.00103585 V 9 0 0 0 0 0
10 Outlet 0.258 0.00113426 V 10 10.0653141 10.87197 7.116875 9.188171 0
Based on the velocity of air at each point along the convergent-divergent duct tabulated in Table 7
above, the plot of velocity of air with respect to the position of point along the convergent-divergent
duct is shown in Graph 6 above.
Graph 6: Plot of velocity, V (m/s) versus position, x (m)
0
50
100
150
200
250
300
350
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
V e l o c i t y ,
V ( m / s )
Position, x (m)
Plot of Velocity, V (m/s) vs. Position, x (m)
f = 30 Hz f = 35 Hz f = 40 Hz f = 45 Hz f = 50 Hz
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6.7 Mass Flow Rate of Compressible Air at Each Point along the Convergent-
Divergent Duct
The mass flow rate of the compressible air at each point along the convergent-divergent duct can be
computed by Eq. [2]. As mentioned in Section 3.2, the deviations in cross-sectional area will result
in distinct velocity of air flow, in which difference mass flow rate is consequence. The mass flowrate at each point along the convergent-divergent duct are computed and tabulated in Table 8 below.
Table 8: Mass flow rate of compressible air at each point along the convergent-divergent duct
PointDistance, x
(m)
Area, A
(m2)
Mass Flow
Rate, ṁ (kg/s)
Frequency (Hz)
30 35 40 45 50
1 Inlet 0.024 0.00113426 ṁ1 0.23902272 0.253537 0.26264 0.266973 0.268636
2
Converging
0.054 0.00095127 ṁ2 0.19982334 0.212077 0.219792 0.223636 0.225184
3 0.084 0.00061332 ṁ3 0.12985032 0.137433 0.14227 0.14428 0.1453354 0.114 0.00034924 ṁ4 0.07376429 0.078294 0.081034 0.082299 0.08278
5 0.144 0.00015903 ṁ5 0.03426942 0.036206 0.037266 0.037635 0.037685
6 0.174 0.00004268 ṁ6 0.00974459 0.010002 0.010108 0.010107 0.010044
7 Throat 0.186 0.00002828 ṁ7 0.00657635 0.006585 0.006573 0.006579 0.006578
8Diverging
0.198 0.00009596 ṁ8 0.00536539 0.005266 0.005266 0.005365 0.005266
9 0.228 0.00103585 ṁ9 0 0 0 0 0
10 Outlet 0.258 0.00113426 ṁ10 0.01351163 0.014593 0.009556 0.012335 0
The variation of mass flow rate with respect to the position of points along the convergent-divergent
duct that has different cross-sectional area is depicted in Graph 7 below.
Graph 7: Plot of mass flow rate versus position of points along the convergent-divergent duct
0
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.27
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26
M a s s F l o w R
a t e ,
ṁ ( k
g / s )
Position, x (m)
Plot of Mass Flow Rate, ṁ (kg/s) vs. Position, x (m)
f = 30 Hz f = 35 Hz f = 40 Hz f = 45 Hz f = 50 Hz
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6.8 Sample Calculations
The experimental data at pressure measurement point No. 5 in the converging section at a frequency
of 40 Hz has been selected for the sample calculations of absolute pressure, Mach number, density,
temperature, speed of sound, velocity, as well as the mass flow rate.
6.8.1 Absolute Pressure
According to the experimental pressure tabulated in Table 1 in Section 6.0, the reference pressure
and gauge pressure at the pressure measurement point No. 3 with a frequency of 40 Hz are
pref = -0.004 Bar and pg = -0.4050 Bar respectively. Thus, the absolute pressure listed in Table 2
(Row 6; Column 8) can be calculated as follows:
( )[ ]
kPa60.865=
+×−−−=
+=
kPa325.101Bar1kPa100Bar004.04050.0
abs atmg p p p
6.8.2 Mach Number
As listed in Table 2, the absolute pressure at point 5 and frequency of 40 Hz is 60.865 kPa. As
mentioned, the atmospheric pressure, patm = 101.325 kPa. With k = 1.4, the Mach number can be
computed as follows:
( )1
2M2
11
−
−+=
k k
t k
p
p
Substituting k = 1.4 and pt = patm = 101.325 kPa into the above equation and re-arranging terms
gives:
0.88532=
−
=
−
= 1
kPa60.865
kPa101.325
5
11
5
1M
2
7
2
7
p
pt
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6.8.3 Density
As mentioned, the density of air at standard atmospheric pressure with temperature of 25 °C is
1.184 kg/m3. The density of the compressible air can be computed by Eq. [8]. According to Section
3.4, Eq. [8] is given by:
( )11
2M
2
11
−
−+=
k
t k
ρ
ρ
Substituting k = 1.4 and making ρ the subject yields:
( )
( )2
52
11
2 M2.01M2
11 +
=
−+
=−
t
k
t
k
ρ ρ ρ
Substituting ρ = 1.184 kg/m3
and M = 0.88532 (Point 5; f = 40 Hz) gives:
( )[ ]3
mkg0.822707=
+
=
2
52
3
0.885320.21
mkg1.184 ρ
6.8.4 Temperature
The temperature used in this experiment is 298 K (25 °C). Based on the Mach number computed,the temperature at point 5 with frequency of 40 Hz can be calculated by using Eq. [6] as follows:
2M2
11
−+=
k
T
T t
Re-arranging terms and substituting k = 1.4 and M = 0.88532 into the above equation give:
K257.6165=
+
=
+
=22
)88532.0(2.01
K298
M2.01
t T T
6.8.5 Speed of Sound
The speed of sound at point 5 with frequency of 40 Hz can be determined by Eq. [4]. As mentioned,
the specific heat and gas constant for air is 1.4 and 287 J/kg·K. Substituting T = 257.6165 K into
Eq. [4] gives:
sm321.7302=⋅== K)5K)(257.616kgJ(287)4.1(kRT c
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6.8.6 Velocity
Likewise, as done in Section 6.6, the velocity of compressible air at point 5 with a frequency of 40
Hz can be calculated by Eq. [5]. Substituting M = 0.88532 and c = 321.7302 m/s leads to:
( )( ) sm284.834==⇔= sm321.739288532.0M V cV
6.8.7 Mass Flow Rate
The mass flow rate of compressible air through the convergent-divergent duct can be computed by
Eq. [2] which is given by:
AV m ρ =
Substituting ρ = 0.822707 kg/m3, A = 0.00015903 m
2, and V = 284.834 m/s into equation above
gives:
( )( )( ) skg0.037266== sm284.834m00015903.0mkg822707.0 23m
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7.0 Discussions
Due to the shortage and insufficient of theoretical information or data, the contrast between
theoretical and experimental results cannot be conducted. Hence, the discussion had been shrunk to
the discussion of experimental results obtained, and the errors exist.
7.1 Experimental Results
According to the absolute pressure as tabulated in Table 1 in Section 3.1, it can be clearly observed
that the pressure is to be the minimum at the throat (point 7) with blower speed of 30 Hz. In
contrary, for blower speeds of 35, 40, 45, and 50 Hz, the minimum pressure occur at point 6 which
is the last point on the converging section instead of throat of the convergent-divergent duct.
Theoretically, the pressure at the throat should be the smallest in which the Mach number is unity as
addressed by Crowe (2010, p. 499) which illustrated in Figure 4 below. The discrepancies in
absolute pressures for blower speeds of 35 to 50 Hz might be due to the friction acting on the flow
path.
Another significant observations which can be clearly seen in Table 3 is that all the Mach numbers
are approximately 0.85 and hence sonic state is not attained at the throat. Apart from that, there are
significant deviations in Mach number for all blower speeds at point 9 and point 10 with blower
speed of 50 Hz. It can be clearly observed that the pressure readings at those points are greater than
the atmospheric pressure in which are abnormal. Adversely, the pressure on those points should be
equal or less than atmospheric pressure. Therefore, in order to make the calculations for other
parameters possible, the Mach numbers had been assumed to zero and the Mach number varies
inversely to pressure. Since it is improbable to have a positive gauge pressure in these cases,therefore, it can be concluded that the occurrence of these results might due to the existence of
experimental errors which going to be discussed in subsequent section.
Figure 7: Laval Nozzle
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7.2 Experimental Errors
As discussed in Section 7.1, the pressure readings at blower speeds of 35 to 50 Hz did not follow
the trend as the pressure distribution with blower speed of 30 Hz. This is due to vibrations induced
by the vacuum pump when suction process commenced. In order to diminish the probability of
occurrence of this drawback, a small vacant space should be provided between the converging-diverging duct and the pump.
In addition, the occurrence of fluctuations on the pressure measurement points also will leads to
experimental errors as well. In order to minimize the fluctuations occurred; new equipment should
be used. Apart from that, periodic maintenance should be performed to ensure that the convergent-
divergent is operating under optimum conditions. When fluctuations are a natural phenomenon
which cannot be neglected, the experiment should be conducted several times so that average data
can be obtained.
Besides, experimental errors can be arising as a result of losses due to friction as well. This is due to
the fact that, friction or viscosity always exists in real-life applications. Even the interior surface of
the convergent-divergent duct is fully furnished; there is still the existence of friction. Therefore, it
can be says that the occurrence of errors also consequence by friction as well.
8.0 Conclusion
Sonic flow is unable to be formed at the throat of the convergent-divergent duct. But, the
characteristics of pressure flow is analyzed and examined. Hence, the objectives of this experiment
are met.
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HES5340 Fluid Mechanics 2, Semester 2, 2012
9.0 References
ASME 2012, Sonic Flow Nozzles and Venturis – Critical Flow, Chocked Flow Condition, ASME
Codes & Standards, viewed on 16th
November 2012, <http://cstools.asme.org/>
Cengel, YA, Boles, MA 2007, Thermodynamics An Engineering Approach, 6th
edn, McGraw-HillEducation (Asia), Singapore.
Cengel, YA, Cimbala, JM 2010, Fluid Mechanics Fundamentals and Applications, 2nd
edn,
McGraw-Hill Education (Asia), Singapore.
Crowe, CT, Elger DF, Williams, BC, Roberson, JA 2010, Engineering Fluid Mechanics, 9th
edn,
John Wiley & Sons (Asia) Pte Ltd, Asia.
Genick, BM 2007, Gas Dynamics Tables, Version 1.3, viewed on 16th
November 2012,
< http://www.potto.org/tableGasDynamics/tableGasDynamics.php>
Munson, BR, Young, DF, Okiishi, TH, Huebsch, WW 2009, Fundamentals of Fluid Mechanics, 6th
edn, John Wiley & Sons, Inc., United States of America.