Prof. and Head of Mechanical Power Engineering Department. Faculty of ...... compressor manual or calculated from the diameter at any three measuring points ...... Rajput, R. K., âMechanical Measurements and Instrumentationâ, Text Book,.
Zagazig University Faculty of Engineering Mechanical Power Engineering Department
Numerical and Experimental Study of Multi-stage Centrifugal Compressor A Thesis Submitted in Partial Fulfillment for the Requirements of The Degree of Master of Science in Mechanical Power Engineering
By:
Eng. Mohamed Said Hamed Abd El Moeti Emeara Supervisors Prof. Dr. Nabil Hassan Mostafa Professor of Turbomachinery Prof. and Head of Mechanical Power Engineering Department Faculty of Engineering, Zagazig University
Prof. Dr. Ahmed Farouk Abdel Gawad Professor of Computational Fluid Mechanics Mechanical Power Engineering Department Faculty of Engineering, Zagazig University 2011
IN THE NAME OF ALLAH
Dedicated to my daughters: Nour and Jana.
my parents, Nadia and Said, who have generously supported my underand post-graduate studies and represented an endless source of love to me till now.
my wife, Maryem, who give me unconditional love.
my sisters, Marwa and Ghada,
my brother, Ahmed.
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ACKNOWLEDGEMENTS First of all, thanks to ALLAH for supporting me to do this work. I would like to express my sincere gratitude to Prof. Dr. Nabil Hassan Mostafa for his close supervision and support. His patience and encouragement played a key role in the development of this work. Also, I would like to thank Prof. Dr. Ahmed Farouk Abdel Gawad for his continuous support and encouragement particularly in computational fluid dynamics CFD sections. I would like to send my deep thanks to Prof. Dr. Ahmed Fayez El-Sayed for his invaluable ideas and comments reading thesis topics. I would like also to thank Prof. Dr. Mohamed Raafat Shaalan for his valuable suggestions and continuous support. Finally, I would like to thank my father, late mother, my wife and sweet daughters for their prayers, help and patience.
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ABSTRACT Flow study in centrifugal compressors is the most complicated one in turbomachinery. The difficulty is mainly due to the unsymmetrical geometry of blades and volute as well as the presence of secondary flow. The situation becomes worse in the case of multi-stage centrifugal compressor. The investigation is so complicated because of the presence of return bends between stages. So, it is a large domain of study. Recently, it is common to use a computational solution to simulate the flow in the compressor. A sector-domain is used to simplify the case study. Few researchers considered the full domain of a single-stage centrifugal compressor. The present study concerns the full domain of a four-stage centrifugal compressor. Both computational and experimental investigations were utilized to study the flow behavior inside the compressor. Computational study was carried out by the commercial code “CFD-RC”. Experimental work was accomplished by the use of a data acquisition system, advanced sensors and “LabView” interface software. Comparisons between computational and experimental outputs were performed. The computational code was validated experimentally and numerically. Compressor map was drawn numerically and experimentally. Surge was unsteadily simulated. Surge predicted at mass flow rate of 0.0093 kg/s (0.00719 m3/s nominal flow rate) at 12,000 rpm. Uncertainty was studied for experimental results. Fulldomain solution is efficient. Using the parallel computational technology, “HPC” program, is recommended for future work.
iv
Contents Acknowledgement Abstract Contents Abbreviations Nomenclatures List of Figures
Chapter 1: Introduction 1.1 Motivation 1.2 Problem Overview 1.3 Case Study 1.4 Thesis Outlines
Chapter 2: Literature Review 2.1 Background 2.2 Compressor operation 2.3 Compressor Aerodynamics 2.4 Analysis of Compressor Performance 2.5 Rotating Stall 2.6 Surge 2.7 Stability of Compression System 2.8 Lumped Volume Parameter Modeling 2.9 Survey of Previous Studies
Chapter 3: Numerical Analysis 3.1 Governing Equations 3.1.1 Continuity Equation 3.1.2 Momentum Conservation Equations 3.1.3 Energy Conservation Equation 3.2 Auxiliary Equations 3.3 Turbulence Model 3.4 Numerical Discretization 3.5 Computational Code 3.6 Geometry Techniques 3.7 Grid Generation 3.8 Boundary Conditions 3.9 Solver Cases 3.11 Initial Conditions 3.12 Time dependence 3.11 Computational Run-Time
Chapter 4: Experiment Setup v
iii iv v vii viii x 1 1 1 2 4 5 5 9 10 14 15 16 16 18 21 33 33 33 34 35 36 36 38 40 42 44 45 47 47 48 48 51
4.1 Four Parameters of Compressor Map 4.1.1 Rotational Speed Measurements 4.1.2 Mass Flow Rate Measurements 4.1.3 Pressure Ratio Measurements 4.1.4 Adiabatic Efficiency Measurements 4.2 Instruments 4.3 Error Analysis
Chapter 5: Results and Discussions 5.1 Experiment Maps 5.1.1 Validation of measured Results 5.1.2 Speed Line at 12,000 rpm 5.1.3 Speed Line at 9,000 rpm 5.1.4 Speed Line at 6,000 rpm 5.1.5 Measured Complete Map 5.2 Computational Investigation 5.2.1 Validation of Sector-domain Technique 5.2.2 Validation of Full-domain Study Methods 5.2.2.1 Direct Map Method 5.2.2.2 Real Map-I Method 5.2.2.3 Real Map-II Method 5.2.2.4 Comparison of Different Computational Methods 5.2.2.5 Validation of Flow in Compressor 5.2.3 Faults in Volute Exclusion 5.2.4 Computational Surge Simulation 5.2.4.1 Compressor inlet 5.2.4.2 Impeller Passage 5.2.4.3 Compressor Exit Duct 5.2.5 Modified Case with High Pressure Ratio
Chapter 6: Conclusions and Future Work 6.1 Conclusions 6.2 Future Works
References Appendix A Appendix B
vi
51 51 51 54 55 55 64 72 72 72 73 77 81 84 89 89 89 89 90 90 90 92 92 93 97 97 98 99 107 107 108 109 114 116
Abbreviations 2D
Two dimensional
3D
Three dimensional
A/D
Analog to Digital Converter
ASME
American Society of Mechanical Engineers
BC
Boundary Condition
C-sec.
Cross Section
CCW
Counter Clock Wise
CFD
Computational Fluid Dynamics
CV
Control Volume
FFT
Fast Fourier Transformation
GEOM
Geometry
HPC
High Performance Computing
IC
Initial Conditions
k
1000
LES
Large Eddy Simulations
MO
Model Options
MPG
Micro Propulsion Group
NI
National Instruments
NPT
Negative Pressure Transducer
PDE
Partial Differential Equation
PPT
Positive Pressure Transducer
vii
PT
Problem Type
RANS
Reynolds Averaged Navier-Stokes Simulations
rpm
revolution per minutes
RW
Rotating Wall
UDM
Uniform Design Method
VC
Volume Condition
viii
Nomenclature Symbol
Description
Units
A
Area
m2
𝑪𝝁
constant for the standard 𝑘 − 𝜖 turbulence model
--
𝑪𝟏𝝐
constant for the standard 𝑘 − 𝜖 turbulence model
--
𝑪𝟐𝝐
constant for the standard 𝑘 − 𝜖 turbulence model
--
𝑪𝒊
velocity at cross-section of point (i)
m/s
𝑪𝑷
the specific heat at constant pressure
J/(kg.K)
𝑪𝑽
specific heat at constant volume
J/(kg.K)
D
diameter
m
𝒆𝑳
Linearity error for any device
---
𝒆𝑺
Sensitivity error for any device
---
𝒆𝒉
Hysteresis error for any device
---
𝒆𝑹
Repeatability (or precision) error for any device
---
k
Specific kinetic energy
𝑚2 𝑠 2
𝒎
mass flow rate
kg/s
N
rotational speed
revs. per minute
𝑷𝒅
dynamic pressure
Pa
Pi
static pressure at point (i)
Pa
Q
volume flow rate
m3/s
R
gas constant
J/(kg.K)
3D-components of the velocity vector
m/s
Instrument uncertainty of a measuring device
--
u, v, w 𝒖𝒊
ix
Greek letters 𝜶
angle of attack
degree
𝜸
Ratio between specific heats at constant pressure
--
and constant volume (gas ratio) = 𝐶𝑃 𝐶𝑉 𝑚2 𝑠 3
𝝐
dissipation rate coefficient
𝜼𝑪
Compressor adiabatic efficiency
--
𝜼𝒐𝒗
overall efficiency
--
𝜼𝒕𝒉
thermal efficiency
--
𝝁
dynamic viscosity
N.s/m2
𝝁𝒕
turbulent viscosity
N.s/m2
𝝆
density
kg/m3
𝝈𝒌
Kinetic energy constant
--
𝝈𝝐
Dissipation rate constant
--
Subscripts: 1
Inlet to compressor duct
2
Compressor inlet
3
Compressor outlet
C
Compressor
M
Motor
x
List of Figures Figure Title Page Dangerous effect of engine surge, Kirk (2006) 3 1.1 Compressor performance map by its manufacturer 3 1.2 company, Armfeild (2005) Front view of the computational domain of the present 3 1.3 compressor Illustrative diagram for energy transfer through a 6 2.1 turbomachine Early versions of turbomachines: (a) Hero’s rotating 6 2.2 sphere, 120 B.C., (b) Giovanni de Branca's turbine, 1629 A.D., Abou Rayan et al.(2006) Volumetric size range for different categories of 6 2.3 compressors, Yoshinka (1977) Cutaway view of Allied Signal TPE331-14 turboprop gas 8 2.4 engine with two centrifugal compressor stages, Stein (2000) Combined axial-centrifugal multi-stage turbocompressor, 8 2.5 Bloch (2006-a) Major elements of multi-stage centrifugal compressor: a) 8 2.6 inlet nozzle, b) inlet guide vanes, c) impeller, d) radial diffuser, e) return channel, f) collector volute, and g) discharge nozzle, Gresh (1991) Flow path in the return-bend of multi-stage centrifugal 11 2.7 compressor, Hanlon (2001) Various types of impeller blading, Boyce (2006) 11 2.8 Head flow-rate characteristics for various outlet blade 11 2.9 o o angles: forward; β2 > 90 , radial; β2 = 90 , and backward; β2 < 90o , Gorla and Khan (2003) Pressure distribution on impeller hub, vaneless diffuser, 13 2.10 and volute wall, Rangwala (2005) Streamlines at volute tongue (left), downstream of tongue 13 2.11 (right), Rangwala (2005) Flow patterns in volute, Boyce (2006) 13 2.12 Schematic of compressor performance characteristic map, 17 2.13 Abou Rayan et al. (2007) Compressor rotating stall, Mostafa (2006) 17 2.14 Schematic of mild and deep surge cycle, Mostafa (2006) 17 2.15 xi
2.16 2.17 2.18 2.19
2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27
2.28 2.29 2.30 2.31
2.32 2.33 2.34 2.35
Helmholtz-resonator-model, Greitzer (1981) Schematic of static and dynamic instability, Greitzer (1981) Elements considered in modeling compressors with vaned diffuser, Macdougal and Elder (1983) Comparison between developed model by Hassanein (1996), Hansen’s (1981) model, and steady state performance of Greitzer’s (1981) model Experimental setup of Willemes (2000) 3C-Doppler-L2F-probe, Forster et al. (1999) Impeller of the transonic centrifugal-compressor with splitter blades, Forster et al. (1999) Impeller of the transonic centrifugal-compressor with splitter blades, Forster et al. (1999) Boundary conditions of Niazi (2000) Numerical and experimental results of Mach number at 30% span at peak efficiency of Niazi (2000) Case study of Stein (2000) Computed meridional velocity vectors colored by total pressure, DLRCC operating design conditions (4.0 kg/sec), Stein (2000) Pressure distribution near hub at t=11 TR, Ginter et al. (2001) Flow sheet (top) and block scheme of signals (bottom) of the system, Tijl (2004) The computational mesh, which include the impeller, diffuser and the volute, Xu and Muller (2005) The computational results of Xu and Muller (2005). (a) pressure contours at the discharge of exit cone. (b) the velocity vectors at the volute exit Experimental set-up of Yutaka et al. (2007) Computational grid of Yutaka et al. (2007) 3D model of KJ66 for Ling et al. (2007); (left): original stage design, (right) hexahedra mesh Ling et al. (2007) comparison results between old and new compressor design; (left): power versus the mass flow rate, (right) velocity distribution in the diffuser with different mass flow rate xii
19 19 20 20
21 21 23 23 25 25 26 26
27 27 27 29
29 29 30 30
2.36 3.1 3.2 3.3
3.4
3.5
3.6
3.7
3.8 4.1
4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
(left) Surface grids of the computational domain, (right) Grids near the tip clearance, Tang et al. (2008) Flow chart of the solution procedure A schematic representation of CFD-RC code, CFD-RC (2009) First geometry technique (named as thin technique), which neglects thickness; (left) impeller disk thickness is neglected; (right) blades thickness is neglected Second geometry technique (named as thick technique), which considers thickness; (upper) blades thickness is displayed, (lower) impeller disk thickness Third geometry technique (named as outlet-duct technique), which considers a long outlet duct (5 m long) to achieve more realistic operation The generated grid for a part of six parts of the inlet of the duct to the compressor. The grid size is 9 × 19 × 1 = 171 cells with 10 × 20 × 2 = 400 nodes First boundary-condition (direct theoretical map) technique, which was used with first or second geometry techniques Real map boundary-condition techniques, which was used with fourth geometry technique Experiment setup; point (1): inlet to compressor duct, point (2): inlet to compressor, point (3): outlet of compressor Experiment flow diagram Digital positive pressure transducer; (left) picture, (right) wiring diagram Calibration curve of the digital positive pressure transducer, Bacca company Digital negative pressure transducer; (left) picture, (right) wiring diagram Sensitive fast response digital thermocouple; (left) picture, (right) wiring diagram Digital speedometer; (left) picture, (right) wiring diagram CIO-DAS1602/16 A/D card; (upper) its picture in the computer case, (lower) its rosette to simplify connection InstCall program which used to install and calibrate the A/D card xiii
31 39 41 34
34
44
45
46
47 54
57 58 58 59 59 59 60 60
4.10 4.11 4.12 4.13 4.14
4.15
5.1
Front panel of LabView program to acquire the PPT signal Block diagram of LabView program to acquire the PPT signal Front panel of LabView program to acquire the NPT signal Block diagram of LabView to acquire the NPT signal Front panel of LabView program to acquire the temperature signal; (upper) at inlet to the compressor, (lower) at the compressor outlet Block diagram of LabView program to acquire the temperature signal; (upper) at inlet to the compressor, (lower) at the compressor outlet Comparison of the measurements with manufactured data, 𝑃 static pressure ratio 3 versus mass flow rate 𝑚 at
61 61 62 62 63
64
74
𝑃2
5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11
5.12
different compressor speed Measured absolute velocities at inlet and outlet versus mass flow rate 𝑚 at 12,000 rpm Measured Mach numbers at inlet and outlet versus mass flow rate at 12,000 rpm Measured different compressor pressures versus mass flow rate at 12,000 rpm Measured absolute velocities at inlet and outlet versus mass flow rate at 9,000 rpm Measured Mach numbers at inlet and outlet versus mass flow rate at 9,000 rpm Measured of different compressor pressures versus mass flow rate at 9,000 rpm Measured absolute velocities at inlet and outlet versus mass flow rate at 6,000 rpm Measured Mach numbers at inlet and outlet versus mass flow rate at 6,000 rpm Measured different compressor pressures versus mass flow rate at 6,000 rpm Measured performance map of the compressor at different rotational speeds, static pressure ratio versus mass flow rate Measured performance map of the compressor at different rotational speeds, static pressure rise versus mass flow xiv
74 75 75 78 78 79 83 83 84 86
86
5.13
5.14 5.15 5.16 5.17 5.18 5.19 5.20
5.21 5.22 5.23 5.24
rate Measured performance map of the compressor at different rotational speeds, total pressure ratio versus mass flow rate Measured performance map of the compressor at different rotational speeds, overall efficiency versus mass flow rate Measured absolute velocities at compressor inlet and outlet versus mass flow rate, at different compressor speed Measured Mach numbers at compressor inlet and outlet versus mass flow rate, at different compressor speed Comparison of whole theoretical and measured results respect to compressor map, at 6,000 rpm Comparison of whole theoretical and measured results respect to compressor map, at 9,000 rpm Comparison of whole theoretical and measured results respect to compressor map, at 12,000 rpm Velocity vectors for the present compressor: (left) general view of the whole compressor, (right) detailed view of the return bend Velocity vectors between stages Velocity distributions for the case without volute in the first and second impeller Velocity distributions for the case without volute in the third and fourth impeller Flow paths in the case without volute which show no any reverse flow, at 12,000 rpm and 0.009 kg/s
87
87 88 88 91 91 92 93
94 94 95 95
5.25
Illustrative cut-view of the first region for surge study which is the inlet to the compressor
96
5.26
Illustrative cut-view of the second and third region for surge study which is the passage of the first impeller and section along the centerline of exit duct, respectively, of the compressor Surge simulation of the inlet flow at the entrance to centrifugal compressor at different time steps, from t= 0 sec to t= 0.11 sec, at 12,000 rpm and 0.0093 kg/s
96
5.27
xv
100
5.28
5.29
5.30 5.31 5.32
Surge simulation of the flow in a passage in the first impeller of the compressor at different time steps, from t= 0 sec to t= 0.11 sec, at 12,000 rpm and 0.0093 kg/s Surge simulation of the flow in the outlet duct of the compressor at different time steps, from t= 0 sec to t= 0.11 sec, at 12,000 rpm and 0.0093 kg/s Static pressure distribution for the first and second impeller in modified case Static pressure distribution for the third and fourth impeller in modified case Static pressure distributions for the modified case; (upper) in the volute, (lower) in the outlet duct
xvi
101 102 103 104 104 105 105
List of Tables Table 2-1 2-2
3-1 3-2 3-3 3-4 4-1 4-2 4-3 4-4 5-1
Title The advantages and disadvantages of various impellers [Boyce, 2006] Comparison between modeling techniques for predicting compressor performance during surge & rotating stall, [El-Mitwally et al., 1996] Types of Used Geometry techniques Grid generation for each type of geometry technique Types of Boundary Conditions Solver cases Comparison between Analog and Digital type Instruments The four parameters to draw the compressor map Uncertainty for the Used devices Uncertainty for the Measured Variables Comparison between different types of solving techniques
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Page 32 32
49 49 49 50 70 70 71 71 106
Chapter 1 INTRODUCTION
1.1 Motivation The study of the internal flow of compressors helps in predicting many phenomena. Some of these phenomena are steady and others are unsteady. Unsteady phenomena are more important due to the difficulty of achieving them experimentally besides their dangerous effect. The most famous unsteady phenomena are surge and rotating stall. Rotating stall reduces the compressor efficiency. Surge not only reduces efficiency but also may damage the compressor completely; Fig. 1.1 shows airplane damage due to surge phenomena, [Kirk, 2006].
1.2 Problem Overview Flow study in centrifugal compressors is the most complicated in turbomachinery analysis in both steady and unsteady performance. Also, the difficulty is due to the unsymmetrical geometry of blades and volute as well as the presence of secondary flow. The situation becomes worse in the case of fulldomain multi-stage centrifugal compressor. The investigation is so complicated because of the presence of return bends and the large domain of study. Recently, it is common to use a computational solution to simulate the flow in the compressor. Most researchers used a sector-domain to simplify the case study. Few researchers considered the full domain of a single-stage centrifugal compressor. Also, the studies of the compressor stages were separated from the
compressor volute. The present study concerns the full-domain of a four-stage centrifugal compressor combined with its volute.
1.3 Case Study A four-stage centrifugal compressor was considered as the present case study. This compressor is a product of Armfeild Company. The company provided the mechanical drawings and dimensions of the compressor. Experimental work was carried out by using of a data acquisition system, advanced sensors and “LabView” interface software. Both computational and experimental investigations were conducted to study the flow behavior inside the compressor. Computational study was carried out by the commercial code “CFD-RC”. Comparisons between computational and experimental outputs were performed. The data of the above compressor, which was taken as a case study, were provided by the manufacturer. Fig. 1.2 shows a performance map of the compressor. The compressor design point is 12,000 rpm, 0.022 (kg/s), and 1.25 static pressure ratio. The provided data were considered to draw the case-study in threedimensions. Also, the data was used to draw the four stages of the compressor and its volute. The problem geometry contains the inlet and outlet ducts, the four stages of the compressor as well as the volute. Each stage includes the impeller and the return-bend. However, the fourth stage contains only the impeller. Each impeller contains six backward blades. Fig. 1.3 shows the front view of the compressor which determined the inlet and outlet ducts of the compressor and its impeller.
2
Figure 1.1 Dangerous effect of engine surge, Kirk (2006).
Figure 1.2 Compressor performance map by its manufacturer company (C1MKII), Armfield (2005).
Figure 1.3 Front view of the Mechanical Drawing of the present compressor.
3
1.4 Thesis Outlines Chapter 1 gives an introduction and an overview of the investigated problem. Chapter 2 provides a literature review that covers the basic definitions as well as the different methods of compressor modeling. Chapter 3 is devoted to the governing equations as well as the computational aspects such as domain discretization, boundary conditions, turbulence models, etc. The experimental setup is explained in chapter 4. Different components and accessories of the experiments are listed. The results and discussions of both computational and experimental investigations are presented in chapter 5. Comparisons are also included. Chapter 6 gives the final conclusions and suggestions for future work.
4
Chapter 2 Literature Review 2.1 Background The word “turbo” is a Latin word that means rotate. Turbomachinery is a science which describes any machine which rotates and uses fluid to convert energy between two forms; fluid and mechanical as seen in Fig. 2.1. At 120 B.C., Hero of Alexandria had the first turbmachine which depends on the steam, Fig. 2.2(a). On 1629, Giovanni de Branca has suggested the idea of impulse steam turbine as seen in Fig. 2.2(b) [Abou Rayan et al., 2006]. In the Second World War, the German focused on the axial compressor, but Britin used the centrifugal compressor. Then, British engineers aquired much experience on the design of small high-speed centrifugal compressors for supercharging reciprocating engines, [Cohen et al., 1996]. Centrifugal compressors are used for low-flow rate applications, because of their ability to achieve higher pressure-rise-to-weight ratios than axial compressors. Centrifugal compressors are widely used for volumetric flow rates of 1,000 to 10,000 ft3/min (1,700 to 170,000 m3 /hr). A single-stage centrifugal compressor has recorded a pressure ratio as high as 12, whereas, axial compressor stage does not exceed two, [Stein, 2000]. In Fig. 2.3, indicates the characteristic ranges of different compressor categories. [Lapina, 1982] stated four important purposes for using centrifugal compressors in industrial processes, namely: 1. To cause positive flow through a process. Compressor elevates the pressure to overcome pressure drops due to piping, vessels, heat
5
Figure 2.1 Illustrative diagram for energy transfer through a turbomachine.
(a)
(b)
Figure 2.2 Early versions of turbomachines: (a) Hero’s rotating sphere, 120 B.C., (b) Giovanni de Branca's turbine, 1629 A.D., Abou Rayan et al.(2006).
Figure 2.3 Volumetric size range for different categories of compressors, Yoshinka (1977). 6
exchangers, valves, and fittings. 2. For separating heavy hydrocarbon mixtures under pressure. The components of a hydrocarbon mixture can be separated by cooling the mixture; heavier hydrocarbons are liquefied first. 3. To obtain supercool gases for refrigeration. Once under pressure, the gas temperature can be lowered by expansion of the gas. 4. As a means to force gas product flow through pipelines. General uses of centrifugal-compression systems include rotorcraft, and refrigeration systems that contain heat pumps, [El-Sayed, 2008]. In high technology, centrifugal compressor has applications such as micro-engines and artificial hearts. The main advantage of centrifugal over axial flow compressors is that manufacturing the centrifugal compressor is less expensive than the axial, [Shum, 2000]. Using of multi-stage centrifugal compressors is well-recognized in gas turbine engines to gain high compression ratios before the fluid enters the combustor. Fig. 2.4 shows the cutaway view of an Allied Signal TPE331-14 turboprop gas engine with two centrifugal compressor stages. Multi-stage centrifugal compressor could be used to get a pressure ratio up to 20 and the same volumetric flow rate like single-stage one, as displayed in Fig. 2.3. Fig. 2.5 indicates a picture of combined axial-centrifugal multi-stage compressor. From left to right, the figure shows intake, axial multi-stage, followed by two stages centrifugal. Configuration of multi-stage centrifugal compressor is displayed by Gresh (1991) as; 1- inlet nozzle, 2- inlet guide vanes, 3- impeller, 4- radial diffuser, 5return channel, 6- collector volute, and 7- discharge nozzle. Fig. 2.6 shows details
7
Figure 2.4 Cutaway view of Allied Signal TPE331-14 turboprop gas engine with two stages centrifugal compressor, Stein (2000).
Figure 2.5 Combined axial-centrifugal multi-stage turbocompressor.
Figure 2.6 Major elements of multi-stage centrifugal compressor: a) inlet nozzle, b) inlet guide vanes, c) impeller, d) radial diffuser, e) return channel, f) collector volute, and g) discharge nozzle, Gresh (1991).
8
of this configuration. Also, Fig. 2.7 shows the flow path in the return bend of a multi-stage centrifugal compressor. For Low pressure applications, multi-stage centrifugal blower is suitable. Multi-stage blower can be specified for coarse/fine bubble diffuser systems, reactor batch supplemental air, digester gas boosters, grit channels, and sludge. In the particulate handling market, multi-stage centrifugal blowers are used to pick up, convey and capture a myriad of materials ranging from aluminum granules to corn flakes, [Hibon and IR, 2007].
2.2 Compressor Operation Air is sucked into the impeller eye and whirled round at high speed by the vanes on the impeller disc. At any point in the flow of air through the impeller, the centripetal acceleration leads to a pressure head, so that the static pressure of the air increases from the eye to the tip of the impeller. The remainder of the static pressure rise is obtained in the diffuser, where the very high velocity of the air leaving the impeller tip is reduced to somewhere in the region of the velocity with which the air enters the impeller eye. The normal practice is to design the compressor so that about half the pressure rise occurs in the impeller and the other half in the diffuser, [El-Sayed, 2008]. Isentropic stage efficiencies for modern centrifugal compressors with vaned diffusers range from 82 to 87 percent. If vaneless diffusers are operated according to manufacturing constraints, peak efficiencies may only reach up to 80 percent. Polytropic rotor efficiencies for unshrouded impellers with 25 to 50 degrees backsweep may reach up to 93 percent, [Stein, 2000].
9
2.3 Compressor Aerodynamics Centrifugal-compressor fluid dynamics is probably the most complicated problem. The process is intimately and non-linearly linked with many channels of influence. The flow inside the impeller presents one of the most complex and unsteady patterns with transient perturbation arises in some cases. This is because the flow usually occurs in rotating passages of complex geometry with stationary shroud-wall. They are often transonic with shocks in the inducer. They have significant viscous and secondary flows. Also, they can be unsteady and may include regions of separation, [Moore, 1981]. The purpose of a centrifugal compressor is to produce a distinct increase in static pressure, thus, effectively increasing the static enthalpy. The increase in static enthalpy, h, across a compressor may be written as: 1 2 1 2 hout hin ( wout win2 ) (uout uin2 ) 2 2
(2.1)
where, win and wout are the fluid velocities measured in the relative reference frame and uin = rin ωin and uout = rout ωout are the circumferential velocities at the compressor inlet and exit, respectively, [Stein, 2000]. There are three impeller vane types, as shown in Fig. 2.8. These are defined according to the exit blade angles. Impellers with exit blade angle 𝛽2 = 90o◦ have radial vanes. Impellers with 𝛽2 < 90o◦ have backward-curved or backward-swept vanes, and for 𝛽2 > 90o◦, the vanes are forward-curved or forward-swept. Table 21 shows the advantages and disadvantages of various impellers, [Boyce, 2006]. They have different characteristics of theoretical head-flow relationship to each other, as shown in Fig. 2.9. Although in Fig. 2.9 the forward-curved head is the
10
Figure 2.7 Flow path in the return-bend of multi-stage centrifugal compressor, Hanlon (2001).
𝛽2 = 90𝑜 Radial vanes
𝛽2 < 90𝑜 Backward vanes
𝛽2 > 90𝑜 Forward vanes
Figure 2.8 Various types of impeller blading, Boyce (2006).
Figure 2.9 Head flow-rate characteristics for various outlet blade angles: forward; 𝛽2 > 90𝑜 , radial; 𝛽2 = 90𝑜 , and backward; 𝛽2 < 90𝑜 , Gorla and Khan (2003).
11
largest, in actual practice the head characteristics of all the impellers are similar to the backward-curved impeller, [Gorla and Khan, 2003]. The instantaneous pressure field on the impeller’s hub surface together with the steady pressure field on the volute hub wall is shown in Fig. 2.10. Large variations in pressure contours are not observed when crossing the boundary between the two major components, but noticeable gradients are present at the diffuser outlet because of the sudden increase in width at the volute inlet. Still heavier distortions in the pressure are present at the volute tongue because of the large incidence, creating a separation like flow on the suction side of the tongue. Flow conditions in this region are strongly influenced by the vortex flow as illustrated by the streamlines on the hub wall and at a cross section downstream of the throat as in Fig. 2.11, [Rangwala, 2005]. To define the volute section at a given angle θ, the shape and area of the section must be decided. Flow patterns in two types of volute are shown in Fig. 2.12. The flow in the asymmetrical volute has a single-vortex instead of the double vortex in the symmetrical volute. Where the impeller is discharging directly into the volute, it is better to have the volute width larger than the impeller width. This enlargement results in the flow from the impeller being bounded by the vortex generated from the gap between the impeller and the casing, [Boyce, 2006]. While the pressure rise in axial compressors relies solely on the deceleration of the fluid particles through the stator and the work performed by the rotor, centrifugal compressors experience additional pressure conversion due to the centrifugal pumping effect. This additional pressure rise takes place isentropically. This increase, however, is limited by large mechanical stresses in the rotating parts due to the centrifugal forces. Since a significant portion of the pressure rise is due
12
Figure 2.10 Pressure distribution on impeller hub, vaneless diffuser, and volute wall, Rangwala (2005).
Figure 2.11 Streamlines at volute tongue (left), downstream of tongue (right), Rangwala (2005).
Figure 2.12 Flow patterns in volute, Boyce (2006). 13
to the centrifugal pumping effect, acceptable compressor efficiencies are found even in the case of an aerodynamically poor rotor leading to local reversed flow. As a result, centrifugal compressors achieve stability over a wide range of operating points, [Stein, 2000].
2.4 Analysis of Compressor Performance For the compressible flow through the compressor many different variables can be defined. These variables are mass flow rate 𝑚∙ , inlet pressure 𝑃 𝑖𝑛 , outlet pressure 𝑃 𝑜𝑢𝑡 , inlet density 𝜌𝑖𝑛 , rotational speed N, a characteristic dimension D (such as blade height), fluid temperature 𝑇, and viscosity 𝜇. A large number of dimensionless values can be formed based on these parameters, but the most widely used minimum set are:
𝑃 𝑜𝑢𝑡 𝑃 𝑖𝑛
,
𝑚 𝑇 𝐷2 𝑃
𝑖𝑛
,
𝑁𝐷 𝑇
. The first parameter is the pressure
ratio across the compressor, the second is related to the mass flow rate and the third is related to the rotational speed of the compressor, [Ding, 2005]. A schematic of the centrifugal compressor performance map is shown in Fig. 2.13. This diagram represents the variation of the total pressure ratio with the flow rate across the compressor for a fixed rotational speed. Stable operation of a compressor is limited at both ends of the abscissa. In the Fig. 2.13, the region between surge line and chock line is stable region. There are two instability regions which are in the right to chock line and in the left to surge line. Researches are accomplished in instability regions. Efficiency is increased as moving from inner circles to outer circles of efficiency contours. Higher efficiencies could not be accomplished because the surge line intersects the higher efficiency contours. Compressor designers aim to increase stability region
14
by moving the surge line to left as possible. Generally, there are three types of instability in compressor, which are rotating stall, surge, and chocking.
2.5 Rotating Stall Flow separation occurs on the bodies which are immersed in a flowing fluid such that the static pressure increases rapidly in the streamwise direction. Flow around a symmetric airfoil is a popular case appears flow separation. As its angle of attack 𝜶 increases, the airfoil lift force increases. This relation is applied just till certain 𝜶 after which stall appears. Simply, rotating stall is a large rotating separation on the blades. Rotating stall was first observed by the group developing centrifugal compressors for the Whittle turbojet in 1938. When rotating stall occurs, one or more “stall cells” travel around the compressor annulus in the direction of rotation of the compressor, with a rotational speed which is usually close to one-half of the compressor rotational speed. Rotating stall can occur in both axial and centrifugal compressors. Rotating stall induces large vibratory stresses in the blading of compressors and is therefore undesirable for structural reasons although the compressor may continue to give acceptable performance. Rotating stall occurs in compressible as well as incompressible flows. In the development of a new compressor, the position of the stall line is a matter of great concern to the designer, and considerable effort is frequently devoted to moving the stall line away from the region of maximum efficiency, [Stenning, 1980]. [Mostafa,
2006]
defines
rotating
stall
as
instability
where
the
circumferential flow pattern is disturbed. This manifested through one or more stall
15
cells of reduced, or stalled, flow that propagate around the compressor annulus at a fraction of the rotor speed as demonstrated in Fig. 2.14. There are two patterns of rotating stall looking at the span-wise distribution of stall cells, i.e., the “partial span stall” and the “full span stall”. In the former the stall cells cover only a part of the blade span, whereas in the latter the whole span is covered by the stall cells, [Takata and Nagano, 1972]. Although rotating stall is often a precursor to surge, the global compressor flow rate is largely unaffected by this type of instability. The principal difference between rotating stall and surge is that the mean flow in rotating stall is steady in time, while during surge the mean flow is unsteady.
2.6 Surge Surge is an axisymmetrical oscillation of the flow through the compressor, and is characterized by a limit cycle in the compressor characteristic with a frequency range of 3-15 Hz. An example of such characteristic is shown as Sshape curve in Fig. 2.15. The characteristic shows the pressure rise over the compressor as a function of the mass flow rate. The surge severity can be classified in two levels: mild and deep surge [Mostafa, 2006].
2.7 Stability of Compression System [Galvas, 1973], with support of NASA, made a Fortran program to predict performance of multi-stage centrifugal compressors that are arranged in series. But this work was steady state and the program delt with compressor as a box. The program did not concern the detailed geometry of the compressor. Just inlet and outlet conditions of compressor are input to the program.
16
Figure 2.13 Schematic of compressor performance characteristic map, Abou Rayan et al. (2007).
Figure 2.14 Compressor rotating stall, Mostafa (2006).
Figure 2.15 Schematic of mild and deep surge cycle, Mostafa (2006). 17
2.8 Lumped Volume Parameter Modeling Two stability criteria for compression systems may be derived based upon the damped mass-spring system analogy. This idea of mapping a real axial or centrifugal compression system to a simple one-dimensional non-linear (Helmholtz-Resonator) model was pioneered by [Greitzer, 1980 and 1981]. The compressor is viewed as an actuator disk, the fluid inertia is exclusively contained in the pipes of length 𝐿𝐶 with flow-through area 𝐴𝐶 , and spring-like system properties are confined to the plenum with volume VP. A schematic of the Helmholtz-Resonator-Model is shown in Fig. 2.16 and Fig. 2.17. [Stenning, 1980] simplified Greitzer’s model by converting it from nonlinear to linear model. [Hassanein, 1996] proved that Stenning model is incorrect. [Hansen et al., 1981] assure Greitzer’s approach to small single-stage centrifugal compressor. They compared the experimental data with Greitzer’s model and put a new empirical value for the delay period ND (0.5 instead of 2.0). [Macdougal and Elder, 1983] and [Elder and Gill, 1985] introduced a model to simulate a large single stage centrifugal compressor. Fig. 2.18 shows the components of the compressor. The model was based on one-dimensional equation but they inserted a new equation to the system. The new equation is the steady state energy equation. [El-Mitwalli et al., 1996] introduced, for the first time, the thermodynamic effects of the real process in its model. He modified Greitzer’s model by inserting the energy equation into the model. Also, he made a fruitful comparison with Greitzer’s model and Hansen data as seen in Fig. 2.19. Then, he found that his model is better than Greitzer’s. Besides previous work, [Hassanein, 1996] applied
18
Figure 2.16 Helmholtz-resonator-model, Greitzer (1981).
Figure 2.17 Schematic of static and dynamic instability, Greitzer (1981).
19
Figure 2.18 Elements considered in modeling compressors with vaned diffuser, Macdougal and Elder (1983).
Figure 2.19 Comparison between developed model by Hassanein (1996), Hansen’s (1981) model, and steady state performance of Greitzer’s (1981) model.
20
Stenning model and proved its failure. He also made a wonderful comparison table between modeling techniques. They classified modeling techniques to; 1linear lumped parameters, 2- non-linear lumped parameters, and 3- distributed parameters. Table 2.2 shows a goaled comparison. [Willemes, 2000] extended previous works by making an experimental study to describe the behavior of compression system during fully-developed surge. He validates his work in a comparison with findings of Greitzer Lumped parameter model. Then, surge control also accomplished, experimentally, using the experimental rig shown in Fig. 2.20.
2.9 Survey of Previous Studies in Computational Methods [Forster et al., 1999] experimentally studied centrifugal-compressor flow with the aid of laser technology, such as doppler probe seen in Fig. 2.21. Transonic centrifugal-compressor with splitter blades, as in Fig. 2.22, was the case study. Flow simulation between blades of the compressor was taken, as in Fig. 2.23. Blade-to-blade solution is the old name for the computational work in compressors. Computational solution in compressor depends on transforming the fluid partial differential equations to algebraic equations. These will be understood briefly in chapter 3. [Niazi, 2000] and [Stein, 2000] were the first computational researches in compressor. They work with others in Georgia institute of technology under Prof. Sankar.
Both Niazi and Stein solved in a sector of axial and centrifugal
compressor, respectively. They coded their numerical scheme and boundary procedures in a computational flow solver, called GT-TURBO3D. Also, they worked on surge prediction and control method. 21
Figure 2.20 Experimental setup of Willemes (2000).
Figure 2.21 3C-Doppler-L2F-probe, Forster et al. (1999).
22
Figure 2.22 Impeller of the transonic centrifugal-compressor with splitter blades, Forster et al. (1999).
Figure 2.23 Impeller of the transonic centrifugal-compressor with splitter blades, Forster et al. (1999).
23
[Niazi, 2000] took a sector-domain single-stage in axial compressor as seen in Fig. 2.24. He validates his code for Rotor 67 and Rotor 37 of Nasa transonic axial fan with their experimental data. Comparing between numerical and experimental results is accomplished as Fig. 2.25. Determining the peak efficiency for Rotor 67 is accomplished. Rotating stall and modified surge are simulated. Also, surge control for open loop and closed loop are studied. [Stein, 2000] took also a sector-domain (or passage) between two blades in the impeller which displayed in Fig. 2.26 with red color. He made a good research on two models of single-stage centrifugal compressor which are; 1- NASA lowspeed centrifugal compressor, and 2- DLR high-speed centrifugal compressor. He generated a suitable grid for each compressor and carried out code validation with corresponding NASA experimental data. Also, he simulated surge in both cases and examined some parameters such as velocity and total pressure, Fig. 2.27. Finally, he studied surge control by air-injection. [Ginter et al., 2001] used a sector domain, also, for an axial compressor. But the new contribution in his work is that he considered all the 30-stage axial compressor. He divided the domain of 30-stage compressor to 60 subdomains such as 30 rotors and 30 stators. CFD code, named FENFLOSS, was used to simulate the 60 subdomains in parallel. The simulation results were compared with experiments. Fig. 2.28 shows a sample of his results. [Tijl, 2004] made pure experimental work to control surge avoidance. Transient simulation of flow parameters was accomplished. Fig. 2.29 shows the control system that was introduced to control and avoid surge. [Xu and Muller, 2005] presented a detailed flow simulation in the volute of a single-stage centrifugal compressor, Fig. 2.30. For the first time, they considered
24
Figure 2.24 Boundary conditions of Niazi (2000).
Figure 2.25 Numerical and experimental results of Mach number at 30% span at peak efficiency of Niazi (2000). 25
Figure 2.26 Case study of Stein (2000).
Near Pressure Side
Midpassage
Near Suction Side
Figure 2.27 Computed meridional velocity vectors colored by total pressure, DLRCC operating design conditions (4.0 kg/sec), Stein (2000).
26
Figure 2.28 Pressure distribution near hub at t=11 TR, Ginter et al. (2001).
Figure 2.29 Flow sheet (top) and block scheme of signals (bottom) of the system, Tijl (2004).
Figure 2.30 The computational mesh, which include the impeller, diffuser and the volute, Xu and Muller (2005). 27
a full-domain technique. Also, Fig. 2.31 shows different volute tongue geometries which were studied in details. The German von Karman Institute report [VKI, 2005], in 2005 about turbomachinery, described flow pattern in many turbomachines like centrifugal compressor. [Yutaka et al., 2007] carried out a numerical and experimental study of the centrifugal-compressor noise affected by the flow in the tapered diffuser. Experimentally, they made a wonderful and advanced experimental set-up as shown in Fig. 2.32. Numerically, 3D steady model was used to simulate flow. The numerical model utilized a full-domain mesh (single-stage) and the occurrence of sliding mesh was introduced as seen in Fig.2.33. [Ling et al., 2007] simulated a small centrifugal compressor undertaken at the Micro Propulsion Group and KJ66 gas turbine design. The compressor and its mesh are seen in Fig. 2.34. They studied improving the efficiency of the compressor by increasing the impeller diameter from 66 mm to 71 mm. They found that the performance of the new design is better than the old one within a certain operating range. Fig. 2.35 shows selected results. [Xinwei et al., 2008] optimized centrifugal compressor blade design using; 1- uniform design method, 2- computational fluid dynamics (CFD), 3- regression analysis method, 4- genetic algorithm. They used uniform design method to optimize parameters input to CFD program. Then, they made another optimization to output data but this optimization is by genetic algorithm. [Tang et al., 2008] studied numerically a 3D impeller and vaneless diffuser of a small centrifugal compressor. The influence of impeller tip clearance on the
28
(a) (b) Figure 2.31 The computational results of Xu and Muller (2005). (a) pressure contours at the discharge of exit cone. (b) the velocity vectors at the volute exit.
Figure 2.32 Experimental set-up of Yutaka et al. (2007).
Figure 2.33 Computational grid of Yutaka et al. (2007). 29
Figure 2.34 3D model of KJ66 for Ling et al. (2007); (left): original stage design, (right) hexahedra mesh.
Figure 2.35 Ling et al. (2007) comparison results between old and new compressor design; (left): power versus the mass flow rate, (right) velocity distribution in the diffuser with different mass flow rate. 30
flow field of the impeller was investigated. Then, a new partially shrouded impeller was designed. They used a sector-domain and 110,000 grid points, as showed in Fig. 2.36. Numerical results show that the secondary flow region becomes smaller at the exit of the impeller. Better performance is achieved in comparison with the unshrouded impeller.
Figure 2.36 (left) Surface grids of the computational domain, (right) Grids near the tip clearance, Tang et al. (2008).
31
Table 2-1 The advantages and disadvantages of various impellers [Boyce, 2006]. Types of Impellers
Radial vanes
Advantages 1. Reasonable compromise between lowenergy transfer and high absolute outlet velocity. 2. No complex bending stress. 3. Easy manufacturing.
Backward-curved vanes
1. Low-outlet kinetic energy = lowdiffuser inlet Mach no.
Forward-curved vanes
1. High-energy transfer.
Disadvantages 1. Surge margin is relatively narrow. 1. Low-energy transfer. 2. Complex bending stress. 3. Hard manufacturing. 1. High-outlet kinetic energy = high-diffuser inlet Mach number. 2. Surge margin is less than radial vanes. 3. Complex bending stress. 4. Hard manufacturing.
Table 2-2 Comparison between modeling techniques for predicting compressor performance during surge & rotating stall, [El-Mitwally et al., 1996]. Methods Parameters Equations Time Needed Surge Prediction Accuracy Model by Compressor Type No. of Stages Treatment Rotating Stall Non-linear Equations
Distributed Parameters continuity, momentum, and energy very high yes very accurate Davis Axial three multi-stage yes yes
small yes
Linear Lumped Parameters Analytical Lumped Equations very small yes
accurate
not acceptable
Non-linear Lumped Parameters Analytical Lumped Equations
Greitzer Axial three single yes yes
32
Hansen Elder Centrifugal one single yes no yes yes
Stenning Axial one single yes no
Chapter 3 Numerical Analysis This chapter describes the computational tools and techniques used in studying and simulating three-dimensional full-domain unsteady compressor flow. As many scientists know that the CFD is an abbreviation of computational fluid dynamics, others call it as colorful fluid dynamics. John D. Anderson, Jr., one of the pioneers in fluid dynamics, defines CFD methods as follows: Computational fluid dynamics is the art of replacing the governing partial differential equations of fluid flow with numbers, and advancing these numbers in space and/or time to obtain a final numerical description of the complete flow-field of interest, Anderson (1990).
3.1 Governing Equations The basic equations are conservation of mass, conservation of momentum, and conservation of energy. In addition to these basic equations, there are some other auxiliary equations. The set of five coupled partial differential equations PDEs for the conservation of mass, momentum and energy in fluid flows is known as the Navier-Stoke’s equations. These equations can be presented in both differential and integral forms. Some terms of the full Navier-Stoke’s equations may be simplified or ignored if certain assumptions are made, El-Sayed (2008).
3.1.1 Continuity Equation The mass conservation equation for unsteady flow is given by: 33
( Vi ) 0 t xi
(3.1)
where; Vi
:
the absolute velocity in the i th direction
xi
:
the coordinate in the i th direction
:
is the air density.
i
:
a tensor indicating 1, 2, 3.
The relative velocity Vr ,i in the rotating frame can be obtained by Vr ,i Vi e jki j x k
(3.2)
where; j
the angular velocity for the rotating frame in the j : direction
xk
: the coordinates in the rotating frame in the k direction.
j , k , i : tensors indicating 1, 2, 3. e jki
: the permutation symbol given by:
1 e jki 1 0
If j, k , i are in a repeating order as 1, 2, 3. If j, k , i are in different repeating order. If any two of j, k , i are equal.
3.1.2 Momentum Conservation Equations The conservation of momentum equation in the i th direction for unsteady flow can be written as:
34
ij ( Vi ) p ( ViV j ) t x j x i x j
(3.3)
Where p is the static pressure, and ij is the viscous stress tensor given by 𝜏𝑖𝑗 = 𝜇
𝜕𝑉𝑖 𝜕𝑉𝑗 2 𝜕𝑉𝑙 + − 𝜕𝑥𝑗 𝜕𝑥𝑖 3 𝜕𝑥𝑙
(3.4)
where;
: is the absolute viscosity.
i, j , l
: are tensor indices indicating 1, 2, 3.
3.1.3 Energy Conservation Equation The unsteady equation of conservation of energy is given by ( E ) Vi ( E p) t xi xi
T K h j J i ( Vi ij ) xi j
h
(3.5)
where; E
: is the total energy of the air.
K
: is the air thermal conductivity.
Ji
: is the diffusion flux of j th species in the i th direction.
The first three terms in the right-hand side of equation (3.5) represent the energy transfer due to conduction, species diffusion, and thermal energy created by viscous shear in the flow, respectively. The air total energy E h
p
Vi 2 2
E
is given by (3.6)
and h j is the specific enthalpy, given by 35
T
hj
c p , j dT
Tref
(3.8)
where Tref = 298.15 K
3.2 Auxiliary Equations The density of an ideal gas is computed through the equation of state. Also, the air viscosity is computed according to the Sutherland viscosity law. The formula is specified for two or three coefficients. Sutherland’s formula with three coefficients is used in the present work, and it is expressed as: 𝜇 = 𝜇𝑂
𝑇 𝑇𝑂
3/2
𝑇𝑂 + 𝑆 𝑇+𝑆
(3.10)
For air at moderate temperatures and pressures, 𝜇𝑂 = 1.7894 × 10−5 𝑃𝑎. 𝑠 , 𝑇𝑂 = 273.11 𝐾, 𝑆 = 110.56 𝐾 . The relationships for the internal energy, e, and the static enthalpy h are: (3.11) 𝑒 = 𝐶𝑉 𝑇 = 𝐶𝑃 𝑇
(3.12)
where, T is the static temperature, 𝐶𝑉 is the specific heat at constant volume and 𝐶𝑃 is the specific heat at constant pressure, respectively.
3.3 Turbulence Model Turbulent flows occur at high Reynolds numbers, when the inertia of fluid overwhelms the viscosity of fluid, causing the laminar flow motions to become unstable. Under these conditions, the flow is characterized by rapid fluctuations in pressure and velocity which are inherently three-dimensional and unsteady.
36
Turbulence has a strong influence on continuity and momentum equations. For the present problem which has high values of Reynolds number, turbulent flow must be considered. There are two main methods for studying turbulent flows; 1RANS (Reynolds Averaged Navier-Stokes Simulations), and 2- LES (Large Eddy Simulations). The most popular models, extracted from RANS method, are; 1standard k-𝜖, 2- RNG k-𝜖, 3- Kato-Lander k-𝜖, and 4- low-Reynolds number k-𝜖, [Abdel Gawad, 1998]. Unfortunately, it is the fact that no single model is universal for all problems. Therefore, the turbulence model selection is needed. There are many factors which must be considered when selecting a turbulence model. The most obvious factors are; 1- the physics of the flow, 2- the level of accuracy required, and 3- the available computational resources. For the present complex geometry, the standard k − ε model was the most suitable model to solve the present case study. The model involves solutions of transport equations for turbulent kinetic energy and its rate of dissipation. The one adopted in CFD-RC-ACE+ is based on Launder and Spalding (1974). In the model, the turbulent viscosity is expressed as: 𝜇𝑡 = 𝜌𝐶𝜇
𝑘2 𝜖
. The transport
equations for k and 𝜀 are: 𝜕 𝜕 𝜕 𝜌𝑘 + 𝜌𝑘𝑢𝑖 = 𝜕𝑡 𝜕𝑥𝑖 𝜕𝑥𝑗 𝜕 𝜕 𝜕 𝜌𝜖 + 𝜌𝜖𝑢𝑖 = 𝜕𝑡 𝜕𝑥𝑖 𝜕𝑥𝑗
𝜇+
𝜇𝑡 𝜕𝑘 − 𝜌𝜖 𝜍𝑘 𝜕𝑥𝑗
𝜇𝑡 𝜕𝜖 𝜖 𝜖2 𝜇+ + 𝐶1𝜖 𝐶3𝜖 − 𝐶2𝜖 𝜍𝜖 𝜕𝑥𝑗 𝑘 𝑘
(3.13) (3.14)
The model constants 𝐶1𝜖 , 𝐶2𝜖 , 𝐶𝜇 , 𝜍𝑘 , and 𝜍𝜖 have the following default values:
𝐶1𝜖 = 1.44, 𝐶2𝜖 = 1.92, 𝐶𝜇 = 0.09, 𝜍𝑘 = 1.0, and 𝜍𝜖 = 1.3.
default values have been determined from experiments. 37
These
This model needs two factors to be substituted in its two equations, which are; 1kinetic energy k, 2- dissipation rate 𝜖. These two factors are needed to be determined at inlet and outlet boundary conditions of the computational domain. A comprehensive investigation was needed to determine the values of turbulence intensity (k) and dissipation rate (𝜖) at inlet and outlet boundaries. At the first, the boundary values of k and 𝜖 were taken as recommended in the code manual, CFD-RC (2008). Unfortunately, these values caused the code solver to diverge, i.e., no solution was obtained. Thus, the boundary values of k and 𝜖 were altered to find the optimum values that drive the code to converge. It was found that the optimum values for the present case study are as follows: 1- turbulence kinetic energy, k= 0.06 m2/s2, 2- turbulence dissipation rate, 𝜖 = 39 m2/s3.
3.4 Numerical Discretization Analytical solution of the Navier-Stokes equations is limited to simple geometries. Therefore, for complex geometry and flows with high non-linearity, numerical techniques must be used to find approximate solutions. In numerical methods, solutions are found for discrete points at different time levels. Several techniques, such as finite difference methods, finite volume methods, and finite element methods, exist for numerically solving the Navier-Stokes equations. Fig. 3.1 refers to general idea of numerical discretization. The finite volume method is commonly used in fluid dynamic problems. In the finite volume method in three-dimensional space, the flow is divided into a finite number of hexagonal cells. The solution domain is divided into a number of cells known as control volumes. In the finite volume approach, the governing 38
Start
Read the geometry and grid.
Set the boundary conditions. Initialization
Solve the momentum equation and update the velocity field. Solve the pressure correction equation and update velocity, pressure and face mass flux
Solve the energy equation and update temperature. Solve the turbulence, other scalar equations.
Update the flow field properties. No
Converged? Yes Stop
Figure 3.1 Flow chart of the solution procedure.
39
equations are numerically integrated over each of these computational cells or control volumes.
3.5 Computational Code Computational fluid dynamics “Research Corporation” code CFD-RC is a technology leader in the field of advanced computational fluid dynamics simulation software backed by more than 20 years of research based knowledge throughout a wide range of industries. Its old name was CFD-RC. As all CFD codes, this code contains three programs; 1- pre-processing, named CFD-GEOM, 2- solver, named CFD-ACE+, and 3- post-processing, named CFD-VIEW. Fig. 3.2 illustrates a schematic representation that simplifies the code global function. CFD-GEOM consists of two main stages as follows; 1- The first stage, drawing the geometric of computational domain, 2- The second stage is grid generation. CFD-ACE+ function consists of determining five elementary requirements; 1- Problem type PT, 2- Model options MO, 3- Volume conditions VC, 4- Boundary conditions BCs, 5- Initial conditions ICs. Finally, CFD-VIEW is responsible for; 1- Domain geometry and grid display, 40
2- Vectors plots, 3- 2D and 3D surface plots, 4- Particle tracking, 5- Make movies of unsteady solutions.
Figure 3.2 A schematic representation of CFD-RC code, CFD-RC (2009).
41
3.6 Geometry Techniques To increase the range of study, four types of geometric techniques were used, Table 3.1. Differences between these types depend on four parameters; 1- wheather or not the elements thickness in study is considered, 2- wheather or not outlet duct and ended valve is used. Two types of thickness are studied. The first type of thickness is the impeller disk thickness. The second type of thickness is the blades thickness. In the thin blades mesh, full-domain of the compressor four stages combined with its volute is drawn with thicknesses of blades and impeller discs are neglected (Fig. 3.3). The blades were drawn as lines without thickness. Also, the impeller disks were drawn as lines without thickness. In the thick blades mesh, also full-domain is used but thickness was considered (Fig. 3.4). The blades and impellers disks were drawn with their thicknesses taken from compressor drawing. In the thick blades with outlet duct mesh, thicknesses were taken, fulldomain was considered, and an outlet duct five meters long was used and atmospheric pressure was applied at outlet (Fig. 3.5). Also, a geometric valve was mounted at end of duct. The geometric valve was accomplished by dividing the end wall of the outlet duct into four parts. If we want to open the valve, the four parts will determine as outlet B.C. If we want to partially open some of the valve, for example quarter of the valve, then only one part is defined as outlet BC and the oher parts determined as wall. Then, some parts were recognized by the code as outlet boundary condition and others as wall. 42
(b)
(a)
Figure 3.3 First geometry technique (named as thin technique), which neglects thickness; (left) impeller disk thickness is neglected; (right) blades thickness is neglected.
(a) (b)
(d) (c) Figure 3.4 Second geometry technique (named as thick technique), which considers thickness; (upper) blades thickness is displayed, (lower) impeller disk thickness.
43
(a)
(b) Figure 3.5 Third geometry technique (named as outlet-duct technique), which considers a long outlet duct (5 m long) to achieve more realistic operation.
3.7 Grid Generation Since structured grids were available, they were used. Mesh size (cells or nodes) differs from geometric technique to another. Fig. 3.6 shows the sample of making the grid. Also, mesh-sensitivity study was accomplished for each geometry
44
technique except the fourth technique due to the huge size of mesh in this case. Huge size needs large PC RAM memory, which was not available. Table 3.2 shows mesh sizes and their sensitivity study.
(b)
(a)
Figure 3.6 The generated grid for a part of six parts of the inlet of the duct to the compressor. The grid size is 9 × 19 × 1 = 171 cells with 10 × 20 × 2 = 400 nodes.
3.8 Boundary Conditions According to the selected boundary conditions, there were three types of boundary-condition techniques. To draw a compressor map, there are two variables, which are named as map variables. ; 1- outlet static or total pressure, 2- mass flow rate. The theoretical map is driven by either direct, real or advanced-real methods. The direct-map method is dependent on entering the compressor boundary conditions at inlet and outlet as its manufacture map. Then, the solver just 45
simulates the flow inside the compressor; see (Fig. 3.7) for thin blades and thick blades meshes.
Figure 3.7 First boundary-condition (direct theoretical map) technique, which was used with first or second geometry techniques. The real-map-I method was carried out by getting the code one parameter of the compressor performance curve. Then, the code computes the other parameter of the performance curve. The thin blades and thick blades meshes were used with this B.C. technique. Real-map-II method depends on making the solver to determine the outlet boundary of the compressor by the code itself. This is accomplished by inserting the compressor geometry into a large computational domain. This idea is like putting the geometry of an airfoil or a building into a large computational domain. Inlet and outlet boundary conditions of the computational domain were atmospheric; see Fig. 3.8 for the fourth geometry techniques. 46
Figure 3.8 Real map boundary-condition techniques, which was used with fourth geometry technique.
3.9 Solver Cases As a result of population of four geometry techniques besides two boundary condition techniques, we have eight solver cases. The two main cases are illustrated in Table 3.3. These are the settings that made the code run. It is worth mentioning that there were some developed solving techniques. These developed techniques are such as: sliding mesh and double layer wall.
3.10 Initial Conditions As popular in computational fluid dynamics CFD, the initial conditions will be minimal values. Then, the initial conditions of u, v and w were set as 1 m/s. Also, pressure P and temperature T were assumed as atmospheric. Atmospheric
47
pressure is 101325 Pa ≈ 1 bar. Also, atmospheric temperature was taken as 25𝑂 𝐶 = 298𝐾.
3.11 Time-dependence As mentioned before, the flow in the centrifugal compressor is so complicated due to the presence of secondary flow, asymmetric geometry, and centrifugal forces. Then, the use of unsteady or transient solution was a must. As in all computational studies, it was necessary to use a suitable time step ∆𝑡 to ensure numerical stability. By time-dependent study, a time step ∆𝑡 was tested as ∆𝑡 = 1 × 10−1 𝑠𝑒𝑐, 1 × 10−2 𝑠𝑒𝑐, 1 × 10−3 𝑠𝑒𝑐, and 1 × 10−4 𝑠𝑒𝑐. The first three values were refused since they cause numerical divergence, i.e., solution instability. A time step ∆𝑡 = 1 × 10−4 𝑠𝑒𝑐 was the optimum.
3.12 Computational Run Time A computer 2.2 GB Intel ® core TM 2 duo processor and 3 GB RAM was used with a Hard disk of 40 GB. For unsteady (transient) solution, a whole day (24 hours) was needed to solve only one time step. For steady solution, about 90 hours needed.
48
Table 3.1 Types of Used Geometry techniques. Technique
Named
Thickness
Domain
First Second Third Fourth
Thin Thick Sector Outlet-duct
neglected considered neglected neglected
full full sector full
Outlet duct no no no yes
ended valve no no no no
Table 3.2 Grid generation for each type of geometry technique. Geometry Named Technique First Second Third Fourth
No. of subdomain
No. of cells
No. of nodes
Mesh sensitivity till cells nodes
164 347 38
299,953 449930 148,067
385,920 584900 177,500
164
973,446
1,142,000
~450,000 ~500,000 ~700,000 ~850,000 ~300,000 ~250,000 not available (out of memory)
Thin Thick Sector Outletduct
Tables 3.3 Types of Boundary Conditions. BC Technique
Geometry Technique Used
Compressor Theoretical Map
First Second
first and second third
Direct-Map
Third
fourth
Realistic-Map
49
Inlet BC
Outlet BC
fixed mass flow rate fixed total pressure (atmospheric)
fixed back pressure fixed back pressure (atmospheric)
Table 3.4 Samples of solver cases. Solver Case Geometry Tech. PT Turbulence factors MO Time Dependence Inlet BC Outlet BC
Note
method
014 first Flow, Turbulence, Heat K= 0.06, D=39 Gravity (𝑔𝑦 = -9.81), Rotating (w=12,000 rpm) Unsteady, ∆𝑡 = 1x10-4 sec Fixed total pressure (atmospheric) Fixed back pressure, P= 113803.848 Pa, T=45.85OC mild surge occurs mass flow rate computed as 0.009 kg/s
024 third Flow, Turbulence, Heat K= 0.06, D=39 Gravity (𝑔𝑦 = -9.81), Rotating (w=12,000 rpm) Unsteady, ∆𝑡 = 1x10-4 sec Fixed total pressure (atmospheric) Fixed back pressure, P= 108540.6 Pa, T=33OC * interface → split→ double layer * RW → cyclic (periodical) * RW was not defined
mass flow rate computed as 0.002267 kg/s
Real Map Method
50
HAPTER 4 EXPERIMENT SETUP The main objective of the experimental work in this research is to draw the compressor map experimentally. This is carried out using advanced digital devices as, will be mentioned in the chapter.
4.1 Four Parameters to Draw the Compressor Map There are four main parameters needed to draw the compressor map. These four parameters are; 1- rotational speed N (rpm), 2- mass flow rate 𝑚 (kg/s), 3- pressure ratio
𝑃3 𝑃2
or
𝑃𝑡3 𝑃𝑡2
, and 4- isentropic efficiency 𝜂𝐶 .
4.1.1 Rotational Speed Measurements Since the compressor rotational speed NC is a large value, it can not be measured directly. Compressor rotational speed NC is obtained from the motor rotational speed NM multiplied by the gear box ratio. Thus, the motor rotational speed NM is measured and multiplied by the gear box ratio. The rotational speed of the motor shaft NM was measured through an analog tachometer. For digital measurements, speedometer was used to read the rotational speed electrically. The permissible fluctuation of speed-readings must be within ≤ ± 0.5 % according to Al-Sulaiman (2003).
4.1.2 Mass Flow Rate Measurements Mass flow rate is the multiplication of the static density 𝜌 and volume flow rate Q. That is 𝑚 = 𝜌 × 𝑄. Also, the density cannot be measured directly. So, static temperature T and static pressure P measurements are used to calculate 51
density, as 𝜌 =
𝑃 𝑅𝑇
. The volume flow rate Q also cannot be measured directly as
𝑄 = 𝐶𝐴. So, normal velocity C at any cross-section and its projected area A are used to calculate Q. Then, four parameters are needed to calculate mass flow rate 𝑚 which are; 1- static temperature T, 2- static pressure P, 3- normal velocity C, and 4- projected area A, i.e., 𝑚 = 𝑓 𝑇, 𝑃, 𝐶, 𝐴 . Thus, mass flow rate is: 𝑚 =
𝑃 𝑅𝑇
× 𝐶𝐴. On the other
hand, it is noted that the mass flow rate measurements can be taken at any cross-section since continuity equation applies, i.e.: 𝑚=
𝑃1 𝑃2 𝑃3 × 𝐶1 𝐴1 = × 𝐶2 𝐴2 = × 𝐶3 𝐴3 𝑅𝑇1 𝑅𝑇2 𝑅𝑇3
(4.1)
The projected area A at the measured section could be found from compressor manual or calculated from the diameter at any three measuring points 1, 2 or 3. 𝐴1 = 5.73 × 10−4 𝑚2 ,
𝐴2 = 𝐴3 = 2.04 × 10−3 𝑚2 . Thus, A is
calculated. Concerning normal velocity C, dynamic pressure is introduced. The definition of dynamic pressure comes from Bernoulli’s equation. This equation is based on the assumption that the flow is incompressible. Here, the case study is a centrifugal compressor. Generally, the flow is compressible and then Bernoulli’s equation could not be applied. But, when the compressor has a low pressure-ratio, Bernoulli’s equation between two points such as; points 1 and 2. Then, we can use the dynamic pressure which is the difference between total and static pressure, as: 1 1 𝑃 2 𝑃𝑑 = 𝑃𝑡 − 𝑃 = 𝜌𝐶 2 = 𝐶 2 2 𝑅𝑇 𝑅𝑇1 𝐶1 = 2 × × 𝑃𝑡1 − 𝑃1 𝑃1 52
(4.2) (4.3)
𝐶2 = 2 ×
𝑅𝑇2 × 𝑃2
𝑃𝑡2 − 𝑃2
(4.4)
Then, C1 and C2 could be calculated from equations (4.3) and (4.4). Static temperature T is measured by using a thermocouple as the temperature probe and read its output voltage. Thermocouple output-voltage could be read by one of three methods; 1- traditional voltmeter (avometer), 2-common thermocouple reader, or 3- A/D card in case of digital and more accurate measurements. Static pressure P was measured by using pressure taps. Compressor duct was used to accelerate the air entering to the compressor inlet. Then, the pressure is measured at: (point 1) which is inlet to compressor duct, and (point 2) which is inlet to the compressor, Fig. 4.1. The pressure signal is connected to any simple pressure measuring instruments. There are four main methods to measure pressure signal, which are; 1- a traditional water manometer, 2- mercury manometer in case of high-measured pressure, 3- inclined manometer in case of small-measured pressure, 4- pressure transducer in case of digital and more accurate measurements. The actual instrument used in measurements was the pressure transducer. Traditional pressure instruments were used to calibrate the pressure transducers.
53
Figure 4.1 Experiment setup; point (1): inlet to compressor duct, point (2): inlet to compressor, point (3): outlet of compressor.
4.1.3 Pressure Ratio Measurements The compressor pressure ratio, which is needed to draw the compressor performance map, may be defined as static pressure ratio, or may be defined as total pressure ratio. The static pressure ratio is the static pressure at the compressor exit P3 divided by static pressure at the compressor inlet P2. The total pressure ratio is the total pressure at compressor exit Pt3 over the total pressure at the compressor inlet Pt2. For the total pressure measurements, two methods can be used to find the total pressure, either by direct measurements, or by combination calculations. Direct measurements are carried out with a total pressure probe, the pitot tube, which is connected to either an analog pressure tool, or a digital one. Combination 54
calculations are accomplished by applying thermodynamics relations to get the total pressure as a function of static pressure and mass flow rate. At the beginning, total temperature is calculated from static temperature and its normal velocity, as: 𝐶2 𝑇𝑡 = 𝑇 + 2 × 𝐶𝑃
(4.7)
Then, by applying the isentropic relations of compressible ideal gases, we get Pt as: 𝛾 𝛾−1 𝑇𝑡
𝑃𝑡 = 𝑃 𝑇
(4.8)
where 𝛾 is the specific heat ratio for the air and equal to 1.4.
4.1.4 Adiabatic Efficiency Measurement To find the compressor adiabatic efficiency, the following equation is used: 𝑇𝑡2 𝜂𝐶 =
𝑃𝑡3 𝑃𝑡2
𝛾−1 𝛾
−1
(4.9)
𝑇𝑡3 − 𝑇𝑡2
Where, 𝜂𝐶 𝑇𝑡2 𝑇𝑡3 𝑃𝑡2 𝑃𝑡3 𝛾
is the compressor adiabatic efficiency. is the total temperature at compressor inlet. is the total temperature at the compressor outlet. is the total pressure at the compressor inlet. is the total pressure at the compressor outlet. is the specific heat ratio of the air (=1.4).
4.2 Instruments Fig. 4.2 shows the experiment flow diagram. The need of using digital instruments is to make accurate and fast response measurements. Any digital device has its own connecting technique (wiring diagram). Most of digital devices 55
need input signal to send output signal. Input signal to the digital device may be volt or current. Also, the digital devices send the output signal as voltage, current, resistance or frequency. Calibration must be carried out to each digital device. From the calibration curve, a device equation could be extracted. Then, any digital device has its calibration equation according to which we can calculate the required value as a function of the measured value. Positive Pressure Transducer was used to measure positive pressures. The term of “positive pressure” means the delivery side of compressor which refers to point 3, shown in Fig. 4.1. It is important to know how to connect this device. This device needs 5 volt as input and sends a voltage signal within 0.5 to 4.5 voltages as an output signal. Fig. 4.3 shows the device picture and its wiring diagram. The device calibration curve is seen in Fig. 4.4 and according to it, the calibration equation is: 𝑦 = 3919𝑥 + 0.265
(4.10)
where y refer to the output variable from the transducer which is the output voltage, x refer to the input variable to the transducer which is the input gauge positive pressure in bar. Negative Pressure Transducer is another digital device used to read the compressor suction pressures such as points 1 and 2, shown in Fig. 4.1. This device needs 12 volt input to send its output signal. The device calibration equation is: 𝑦 = 2.2 × 10−6 𝑥 − 0.068
(4.11)
where y refers to the output voltage from the negative transducer, x refers to the input gauge negative pressure. Fig. 4.5 shows the transducer picture and its wiring diagram.
56
Sensitive fast response thermocouple was used to acquire the temperature of the measurements, Fig. 4.6. Digital speedometer was used to measure the compressor rotational speed directly. Fig. 4.7 shows the transducer picture and its wiring diagram. Its idea depends on analyzing its signal and finds its frequency f. Then, the circular frequency will be: = 2𝜋𝑓 . Thus, the rotational speed N could be calculated as: 60 × 𝜔 (4.11) 2𝜋 Analyzing the output signal of the digital speedometer is accomplished by the 𝑁=
advanced technique of Fast Fourier Transformation. Fast Fourier Transformation is available in most of the advanced measurement packages such as LabView.
Figure 4.2 Experiment flow diagram.
57
Figure 4.3 Digital positive pressure transducer; (left) picture, (right) wiring diagram.
Figure 4.4 Calibration curve of the digital positive pressure transducer, Bacca Company.
58
Figure 4.5 Digital negative pressure transducer; (left) picture, (right) wiring diagram.
Figure 4.6 Sensitive fast response digital thermocouple; (left) picture, (right) wiring diagram.
Figure 4.7 Digital speedometer; (left) picture, (right) wiring diagram.
59
Figure 4.8 CIO-DAS1602/16 A/D card; (upper) its picture in the computer case, (lower) its rosette to simplify connection.
Figure 4.9 InstCall program that was used to install and calibrate the A/D card. 60
Figure 4.10 Front panel of LabView to acquire the PPT signal.
Figure 4.11 Block diagram of LabView to acquire the PPT signal. 61
Figure 4.12 Front panel of LabView to acquire the NPT signal.
Figure 4.13 Block diagram of LabView to acquire the NPT signal.
62
Figure 4.14 Front panel of LabView program to acquire the temperature signal; (upper) at inlet to the compressor, (lower) at the compressor outlet. 63
Figure 4.15 Block diagram of LabView program to acquire the temperature signal; (upper) at inlet to the compressor, (lower) at the compressor outlet.
4.3 Error Analysis Error analysis study is important to determine the range of readings around the true value. For a measured variable, there are three types of error source in measurements which are; 1- instrument error uc, 2- design stage uncertainty 𝑢𝑑 , and 3- propagation error uR, and. Instrument error represents the overall error for that device. It is calculated as; 𝑢𝐶 =
𝑒2 + 𝑒𝐿2 + 𝑒𝐾2 + 𝑒𝑅2
(4.12)
where 𝑒, 𝑒𝐿 , 𝑒𝐾 , and 𝑒𝑅 are the device hysteresis error, linearity error, sensitivity error, and repeatability error, respectively. Design stage uncertainty 𝑢𝑑 for the instrument can be approximated by 64
combining the instrument uncertainty 𝑢𝐶 with the interpolation error 𝑢𝑂 as: 𝑢𝑑 =
𝑢𝑂2 + 𝑢𝐶2
(4.13)
where 𝑢𝑂 and 𝑢𝐶 are the interpolation and instrument error, respectively. Also, interpolation error equals to half the instrument resolution, as: 1 (4.14) 𝑢𝑂 = ± 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 2 Table 4-3 shows the uncertainty for each device used in the present measurements. Errors in measured variables are the errors of their devices. Table 4-4 shows the errors values of the measured variables. For a certain measurement, there are two kinds of variables; 1- measured variable, and 2- dependent variable. Measured variable corresponds the instrument reading. Dependent variable comes by substitute in a function of measured variables such as 𝑅 = 𝑓 𝑥1 , 𝑥2 , 𝑥3 , . . . Figliola and Beasley (1995) determined the uncertainty of the dependent variable R as:
𝑢𝑅 =
𝜕𝑅 𝜕𝑥1
2
𝜕𝑅 + 𝜕𝑥2
2
𝜕𝑅 + 𝜕𝑥3
2
+⋯
(4.15)
For the present research, the dependent variables are; 1- the mass flow rate 𝑚, 2- the static pressure ratio 𝑃3 𝑃2 , 3- the static pressure rise 𝑃3 − 𝑃2 , 4- the overall efficiency 𝜂𝑜𝑣 , 5- the total pressure ratio 𝑃3𝑡 𝑃2𝑡 , 6- the compressor inlet velocity 𝐶2 , 7- the compressor inlet Mach number 𝑀2 , 8- the compressor outlet velocity 𝐶3 , and 9- the compressor outlet Mach number 𝑀3 . For the mass flow rate, the dependence relation is:
65
𝑚 = 𝐴2 2
𝑃2 × 𝑃𝑎𝑡𝑚 − 𝑃2 𝑅 × 𝑇2
(4.16)
The differential relations are: 𝜕𝑚 𝐴2 2 𝑃𝑎𝑡𝑚 − 2𝑃2 = × 𝜕𝑃2 2 𝑅 𝑇2
𝑅𝑇2 𝑃2 𝑃𝑎𝑡𝑚 − 𝑃2
(4.17)
𝜕𝑚 −𝐴2 2 𝑃𝑎𝑡𝑚 − 2𝑃2 = × 𝑃2 𝑃𝑎𝑡𝑚 − 𝑃2 𝜕𝑇2 2 𝑅 𝑇23
(4.18)
Then, the uncertainty of the mass flow rate is: 𝑢𝑚 =
2 𝑢𝑃2 𝐴2 𝑃𝑎𝑡𝑚 − 2𝑃2 2 𝑅 𝑇2
− 2 𝑢 𝑇2 𝐴2 𝑃𝑎𝑡𝑚 − 2𝑃2 𝑅𝑇2 + 𝑃2 𝑃𝑎𝑡𝑚 − 𝑃2 2 𝑅 𝑇23
𝑃2 𝑃𝑎𝑡𝑚 − 𝑃2
(4.19)
By the same way, the relations for the total temperature at compressor inlet are: 3/2
1 𝑅 𝑇2 𝑃𝑎𝑡𝑚 − 𝑃2 𝑇2𝑡 = 𝑇2 + 2𝐶𝑃 𝑃2 𝜕𝑇2𝑡 −3 𝑅 𝑇2 3/2 𝑃𝑎𝑡𝑚 = 𝜕𝑃2 4 𝐶𝑃 𝑃23 𝜕𝑇2𝑡 3 𝑅 3/2 𝑃𝑎𝑡𝑚 − 𝑃2 =1+ 𝜕𝑇2 4 𝐶𝑃 𝑃2 𝑢 𝑇2𝑡 =
(4.20)
𝑃𝑎𝑡𝑚 −1 𝑃2
(4.21)
3/2
𝑇2
𝜕𝑇2𝑡 𝜕𝑇2𝑡 × 𝑢𝑃2 + × 𝑢 𝑇2 𝜕𝑃2 𝜕𝑇2
(4.22) (4.23)
The relations for the total temperature at compressor outlet are: 1 𝑅 𝑇3 𝑃3𝑡 − 𝑃3 𝑇3𝑡 = 𝑇3 + 2𝐶𝑃 𝑃3 𝜕𝑇3𝑡 −3 𝑅 𝑇3 3/2 𝑃3𝑡 = 𝜕𝑃3 4 𝐶𝑃 𝑃33 𝜕𝑇3𝑡 3 𝑅 𝑇3 = 𝜕𝑃3𝑡 4 𝐶𝑃 𝑃3
3/2
(4.24)
𝑃3𝑡 −1 𝑃3
3/2
× 𝑃3𝑡 − 𝑃3
66
5/2
(4.25)
𝜕𝑇3𝑡 3 𝑅 3/2 𝑃3𝑡 − 𝑃3 = 1+ 𝜕𝑇3 4 𝐶𝑃 𝑃3 𝑢 𝑇3𝑡 =
−3 𝑢𝑃3 𝑅 𝑇3 4 𝐶𝑃
3/2
𝑃3𝑡 𝑃33
𝑃3𝑡 −1 𝑃3
1/2
+
3 𝑅3/2 𝑃3𝑡 − 𝑃3 + 1+ 4 𝐶𝑃 𝑃3
3/2
3𝑢𝑃3𝑡 𝑅 𝑇3 4 𝐶𝑃 𝑃3
𝑇3
(4.26)
3/2
3/2
𝑃3𝑡 − 𝑃3
5/2
(4.27)
𝑇3 × 𝑢 𝑇3
For the compressor inlet velocity 𝐶2 , the relations are: 𝐶2 =
𝑃𝑎𝑡𝑚 − 𝑃2 𝑅 × 𝑇2 𝑃2
𝜕𝐶2 −1 𝑃𝑎𝑡𝑚 = 𝜕𝑃2 2 𝑃22 𝜕𝐶2 1 = 𝜕𝑇2 2 𝑢𝐶2 =
(4.28)
𝑅𝑇2 𝑃𝑎𝑡𝑚 𝑃2 − 1
𝑃𝑎𝑡𝑚 − 𝑃2 𝑃2 𝑇2
𝜕𝐶2 𝜕𝐶2 × 𝑢𝑃2 + × 𝑢𝑇2 𝜕𝑃2 𝜕𝑇2
(4.29)
(4.30) (4.31)
For the compressor outlet velocity 𝐶3 , the relations are: 𝐶3 =
𝑃3𝑡 − 𝑃3 𝑅 × 𝑇3 𝑃3
𝜕𝐶3 −1 𝑃3𝑡 = 𝜕𝑃3 2 𝑃32 𝜕𝐶3 = 𝜕𝑃3𝑡
(4.32)
𝑅𝑇3 𝑃3𝑡 𝑃3 − 1
(4.33)
𝑅𝑇3 2𝑃3
𝑃3𝑡 − 𝑃3 𝑅 × 𝑇3 𝑃3
67
(4.34)
𝜕𝐶3 1 = 𝜕𝑇3 2 𝑢𝐶3 =
−𝑢𝑃3 𝑃3𝑡 2 𝑃32
𝑅𝑇3 𝑃3𝑡 𝑃3 − 1
𝑃3𝑡 − 𝑃3 𝑃3 𝑇3
(4.35)
𝑢𝑃3𝑡 𝑅𝑇3
+
𝑃3𝑡 − 𝑃3 𝑅 × 𝑇3 𝑃3
2𝑃3
+
𝑢𝑇3 2
𝑃3𝑡 − 𝑃3 𝑃3 𝑇3
(4.36)
The relations for the compressor inlet Mach number 𝑀2 , are: 𝑀2 =
𝐶2 𝛾𝑅𝑇2
𝑃𝑎𝑡𝑚 − 𝑃2 𝛾𝑃2
=
(4.37)
𝜕𝑀2 −1 𝑃𝑎𝑡𝑚 = 𝜕𝑃2 2 𝑃22
𝑃𝑎𝑡𝑚 − 𝑃2 𝛾𝑃2
(4.38)
−𝑢𝑃2 𝑃𝑎𝑡𝑚 2 𝑃22
𝑃𝑎𝑡𝑚 − 𝑃2 𝛾𝑃2
(4.39)
𝑢𝑀2 =
The relations for the compressor outlet Mach number 𝑀3 , are: 𝑀3 =
𝐶3 𝛾𝑅𝑇3
=
𝜕𝑀3 −1 𝑃3𝑡 = 𝜕𝑃3 2 𝑃32 𝜕𝑀3 = 𝜕𝑃3𝑡
𝑢𝑀3 =
−𝑢𝑃3 𝑃3𝑡 2 𝑃32
𝑃3𝑡 − 𝑃3 𝛾𝑃3
(4.40)
𝑃3𝑡 − 𝑃3 𝛾𝑃3
(4.41)
1 2
(4.42)
𝑃3𝑡 − 𝑃3 𝛾𝑃3
𝑃3𝑡 − 𝑃3 + 𝛾𝑃3
𝑢𝑃3𝑡 2
𝑃3𝑡 − 𝑃3 𝛾𝑃3
(4.43)
The relations for the pressure rise across the compressor 𝑃3 − 𝑃2 , are: 𝑢 𝑃3 −𝑃2 =
𝜕 𝑃3 − 𝑃2 𝜕 𝑃3 − 𝑃2 × 𝑢𝑃2 + × 𝑢𝑃3 𝜕𝑃2 𝜕𝑃3 𝑢 𝑃3 −𝑃2 = 𝑢𝑃3 − 𝑢𝑃2
The relations for the overall efficiency of the compressor 𝜂𝑜𝑣 , are: 68
(4.44) (4.45)
𝑇𝑡2 𝜂𝐶 =
𝜕𝑇2
1 𝑅 𝑇3 𝑃3𝑡 − 𝑃3 𝑇3 + 2𝐶 𝑃3 𝑃
3/2
3 R ∗T2∗(Patm −P 2) 1/2 R (Patm −P 2) × × 4Cp P2 P2
1+
=
𝜕𝜂𝐶 = 𝜕𝑇3
𝜕𝜂𝐶 = 𝜕𝑃2
1 R T2 (Patm − P2 ) × 2Cp P2
1 T3 − T2 + × 2Cp 3 R T2 (Patm − P2 ) 4 P2
1/2
3/2
3/2
𝑃3𝑡 𝑃𝑎𝑡𝑚
𝛾−1 𝛾
×
γ −1 γ
P 3t Patm
3 2
×
γ−1 γ
P3t Patm
R ∗ T3 ∗ (P3t − P3) P3
1 3 R ∗T2∗(Patm −P 2) 2 R ∗(Patm −P 2) × 4Cp P2 P2
γ−1 γ
R ∗ T2 ∗ (Patm − P2 ) 1 − 2Cp ∗ P2
3/2
+
R ∗ T2 ∗ (Patm − P2) 3Cp 1 − 4 (T2 + 2Cp × P2
𝑢𝜂 𝐶 =
3/2
P3t × Patm
γ−1 γ
×
R ∗ T3 ∗ (P3t − P3) 1 T3 − T2 + 2 ∗ Cp P3
2
1/2
× 1+
3/2
3T3 R × T3 (P3t − P3 ) × 4Cp P3
R ∗ T2 ∗ (Patm − P2) − P2
R ∗ T2 ∗ (Patm − P2) P2
1/2
R × T3 × (P3t − P3 ) 1 T3 − T2 + 2Cp × P
3/2
R ∗ T3 ∗ (P3t − P3) P3
1/2
×
1 − 2C
p
3/2
R ∗ T2 ∗ (Patm − P2) 1 − 2 ∗ Cp P2
(4.49)
2
P3t ∗ Patm
γ−1 γ
R × T2 × (Patm − P2 ) P2
R ∗ T3 ∗ (P3t − P3) −R ∗ T3 P3 − P3
(4.48)
3/2
2
3/2
R ∗ T2 ∗ (Patm − P2 ) RT × − P2− P2 2
R ∗ T2 ∗ (Patm − P2) 3 1 4 × T2 + 2Cp ∗ P2
3/2
(4.47)
3/2
× −1−
3
𝜕𝜂𝐶 = 𝜕𝑃3
−1
1 𝑅 𝑇2 𝑃𝑎𝑡𝑚 − 𝑃2 − 𝑇2 + 2𝐶 𝑃2 𝑃
R T2 ∗ (Patm − P2 ) RT P × – P2− ∗ P 3t P22 2 atm
R ∗ T3 ∗ (P3t − P3 ) 1 T3 − T2 + 2C ∗ P3 P
(4.46)
R ∗T3∗(P 3t−P 3) 3/2 R ∗T2∗(Patm −P 2) 3/2 − P3 P2
1 × 2Cp
T3−T2+
− T2 +
−1
𝑇𝑡3 − 𝑇𝑡2
1 𝑅 𝑇2 𝑃𝑎𝑡𝑚 − 𝑃2 𝑇2 + 2𝐶 𝑃2 𝑃
𝜂𝐶 =
𝜕𝜂 𝐶
𝛾−1 𝛾
𝑃𝑡3 𝑃𝑡2
(4.57) 3/2 2
2
3/2 2
𝜕𝜂𝐶 𝜕𝜂𝐶 𝜕𝜂𝐶 𝜕𝜂𝐶 𝜕𝜂𝐶 × 𝑢 𝑇2 + × 𝑢 𝑇3 + × 𝑢𝑃2 + × 𝑢𝑃3 + × 𝑢𝑃3𝑡 𝜕𝑇2 𝜕𝑇3 𝜕𝑃2 𝜕𝑃3 𝜕𝑃3𝑡
(4.58)
(4.59)
The differentials were found by the aid of computer program using Matlab; some differentials were too long. This is accomplished by a facility on Matlab named symbolic. Appendix B has the commands and results of Matlab program.
69
5 6 7
Table4-2 The four parameters to draw the compressor map No.
1
2
3
The four parameters compressor rotational speed NC mass flow rate 𝒎 static pressure ratio total pressure ratio
Direct measured parameters
Digital tool
Elementary tool
motor rotational speed NM
digital speedometer
shaft guide
P, T, C at appropriate c-section
P
positive or negative pressure transducer
T
digital fast response thermocouple
P2, P3
positive or negative pressure transducer
Pt3 digital fast response thermocouple
T2, T3
4
isentropic efficiency
positive or negative pressure transducer
P2, P3 Pt3
70
to A/D card
4
to A/D card
2 3
to A/D card
No. 1
Table 4-1 Comparison between Analog and Digital type Instruments Aspects Analog Instruments Digital Instruments Information form As the position of pointer As a number. against a calibrated scale or dial. Possibility of human error Exist. Do not exist. Best possible accuracy ±0.25%. ±0.005% or better. [Rajput, 2009] Resolution One in several hundreds. One part in several hundred thousand. Time required to observe Need time to approximate the Very fast. the reading reading. Auxiliary Power Required No power required. Any digital sensor needs power to operate it. Examples: Manometer (vertical or Pressure transducer. inclined). Speedometer (digital Tachometer. tachometer). Common voltmeter. Digital voltmeter. Common thermometers or thermocouple with digital thermocouple with common voltmeter. voltmeter.
pressure taps temperature measuring locations pressure taps pitot-tube temperature measuring locations pitot-tube
Table4-3 Uncertainty for the Used devices Device Thermocouple (analog) Thermocouple (digital) Positive Pressure transducer (PPT) Negative Pressure Transducer (NPT) Manometer
𝑹𝒆𝒔.
𝒖𝑶
𝒖𝑪
𝒖𝒅
O
0.1
±0.05
±0.1
±0.11
O
C
0.001
±0.0005
±0.001
±0.0011
Pressure
Pa
1000
±500
±100
±510
Digital
Pressure
Pa
1000
±500
±150
±522
Analog
Pressure
Pa
9.81
±4.91
±1530
±1530
Type
Variable
Analog
Temperature
Digital
Temperature
Digital
Unit C
Table4-4 Uncertainty for the Measured Variables Measured Variable 𝑃2 𝑇2 𝑃3 𝑇3 𝑃3𝑡
Unit Pa O C Pa O C Pa
Analog Device ±1530 ±0.11 ±1530 ±0.11 ±1530
71
Digital Device ±522 ±0.0011 ±510 ±0.0011 ±510
CHAPTER 5 RESULTS AND DISCUSSIONS As well-known, any numerical study must be validated firstly by comparison with either experimental results or other numerical predictions. The code is validated, and then numerical results are displayed as numerical map and surge simulation. Finally, comparisons between numerical and experimental results are shown.
5.1 Experiment Maps Experimental maps are those maps obtained experimentally by digital measuring devices. Three rotational speeds of the compressor are considered, which are 12,000, 9,000 and 6,000 rpm. For each rotational speed, some curves are presented. In all the following graphs, the symbols represent the actual measured values while the red curve represents the best fit.
5.1.1 Validation of Measured Results For validation, comparisons of the present measured with manufactured data are shown in Fig. 5.1 for the rotational speeds 12,000, 9,000, and 6,000 rpm. At 12,000 rpm, the measurements error ranges between 0.005 and 0.006 and its average equals to 0.0055. The percentage error ranges from 0.4% to 0.5% of the true value with an average of 0.45% of the true value. At 9,000 rpm, the measurements error ranges between 0.005 and 0.006 and its average value is 0.0055. The percentage error ranges from 0.44% to 0.487% of 72
the true value with an average of 0.463% of the true value. At 6,000 rpm, the measurements error ranges between 0.0049 and 0.0051 with an average of 0.005. The percentage error ranges from 0.41% to 0.49% of the true value and its average equals to 0.45%. It should be noted that the points of surge were predicted experimentally by detecting a mass flow rate fluctuation. This is well defined in Sec. 2.6.
5.1.2 Speed Line at 12,000 rpm The results at the rotational speed 12,000 rpm are displayed in Figs. 5.2 - 5.4 for the relations between; 1- the static pressure ratio 𝑃3 /𝑃2 , 2- the static pressure rise Δ𝑃 = 𝑃3 − 𝑃2 (Pa), 3- the total pressure ratio, 4- the overall efficiency 𝜂𝑜𝑣 , 5velocity at inlet and outlet (m/s), 6- Mach number at inlet and outlet, 7- different compressor pressures, and the mass flow rate 𝑚 (kg/s). Fig. 5.4 shows the relation of the velocity at inlet and outlet with the mass flow rate 𝑚 (kg/s). It seems that the ratio between the velocities 𝐶3 𝐶2 averages to 1.2. Fig. 5.3 shows the relation of the Mach number at inlet and outlet with the mass flow rate 𝑚 (kg/s). It seems that the ratio between the Mach numbers 𝑀3 𝑀2 averages to 1.16. Fig. 5.4 shows the relation between the different compressor pressures and the mass flow rate 𝑚 (kg/s). The difference between the total pressure and static pressure at compressor outlet averages to 11,000 Pa. The coefficient of determination R represents the trend of convergence between the actual measurements and the fits. The mass flow rate ranges from 0.016 to 0.032 kg/s.
73
Figure 5.1 Comparison of the measurements with manufactured data, static 𝑃 pressure ratio 3 versus mass flow rate 𝑚 at different compressor 𝑃2
speed.
Figure 5.2 Measured absolute velocities at inlet and outlet versus mass flow rate 𝑚 at 12,000 rpm. 74
Figure 5.3 Measured Mach numbers at inlet and outlet versus mass flow rate at 12,000 rpm.
Figure 5.4 Measured different compressor pressures versus mass flow rate at 12,000 rpm.
75
The static pressure ratio ranges from 1 to 1.12 according to the mass flow rate. The results can be expressed by the following relation (red curve): 𝑃3 = 1.4067 + 2.7124 𝑚 − 494.26 𝑚2 𝑃2
(5.11)
with the coefficient of determination 𝑅 = 0.999735. The static pressure rise ranges from 2500 to 32,000 Pa along the mass flow rate range. The results can be expressed by the following relation (red curve): Δ𝑃 = 39394.060 + 199246.23 𝑚
(5.12)
− 45879786.44 𝑚2 with the coefficient of determination 𝑅 = 0.999739. The total pressure ratio ranges from 1 to 1.34. The results can be expressed by the following relation (red curve): 𝑃3𝑡 = 1.42665 + 2.712 𝑚 − 494.256 𝑚2 𝑃2𝑡
(5.13)
with the coefficient of determination 𝑅 = 0.999735. The overall efficiency 𝜂𝑜𝑣 ranges from 0.05 to 0.85. The results can be expressed by the following relation (red curve): 𝜂𝑜𝑣 = −0.652698 + 166.98 𝑚 − 4629.2325 𝑚2
(5.14)
with the coefficient of determination 𝑅 = 0.999047. The obvious increase in values of 𝜂𝑜𝑣 comparing to the values of analog measurements is due to the better accuracy of digital devices. The velocity at the compressor inlet 𝐶2 (m/s) ranges from 103 to 112 m/s. The results can be expressed by the following relation (red curve): 𝐶2 = 99.748 − 9.676 𝑚 + 12389.2975 𝑚2
(5.15)
with the coefficient of determination 𝑅 = 0.999768. The Mach number at the compressor inlet 𝑀2 ranges from 0.302 to 0.325. 76
The results can be expressed by the following relation (red curve): 𝑀2 = 0.2924 + 0.02381 + 31.8059 𝑚2
(5.16)
with the coefficient of determination 𝑅 = 0.999786. The velocity at the compressor outlet 𝐶3 (m/s) ranges from 119 to 130 m/s. The results can be expressed by the following relation (red curve): 𝐶3 = 122.599 − 726.423 𝑚 + 30656.46 𝑚2
(5.17)
with the coefficient of determination 𝑅 = 0.999307. The Mach number at the compressor outlet 𝑀3 ranges from 0.342 to 0.367. The results can be expressed by the following relation (red curve): 𝑀3 = 0.35226 − 1.97676 𝑚 + 79.473 𝑚2
(5.18)
with the coefficient of determination 𝑅 = 0.999192.
5.1.3 Speed Line at 9,000 rpm The results at the rotational speed 9,000 rpm are displayed in Figs. 5.5 - 5.7 for the relations between; 1- the static pressure ratio 𝑃3 /𝑃2 , 2- the static pressure rise Δ𝑃 = 𝑃3 − 𝑃2 (Pa), 3- the total pressure ratio, 4- the overall efficiency 𝜂𝑜𝑣 , 5velocity at inlet and outlet (m/s), 6- Mach number at inlet and outlet, 7- different compressor pressures, and the mass flow rate 𝑚 (kg/s). Fig. 5.5 shows the relation of the velocity at inlet and outlet with the mass flow rate 𝑚 (kg/s). It seems that the ratio between the velocities 𝐶3 𝐶2 averages to 1.33. Fig. 5.6 shows the relation of the Mach number at inlet and outlet with the mass flow rate 𝑚 (kg/s). It seems that the ratio between the Mach numbers 𝑀3 𝑀2 averages to 1.34. Fig. 5.7 shows the relation between the different compressor pressures and the mass flow rate 𝑚 (kg/s). The difference between the total pressure and static pressure at compressor outlet averages to 9,000 Pa. 77
Figure 5.5 Measured absolute velocities at inlet and outlet versus mass flow rate at 9,000 rpm.
Figure 5.6 Measured Mach numbers at inlet and outlet versus mass flow rate at 9,000 rpm. 78
Figure 5.7 Measured of different compressor pressures versus mass flow rate at 9,000 rpm. The coefficient of determination R represents the trend of convergence between the actual measurements and the fits. The mass flow rate ranges from 0.014 to 0.028 kg/s. The static pressure ratio ranges from 1 to 1.26 according to the mass flow rate. The results can be expressed by the following relation (red curve): 𝑃3 = 1.318 + 1.7635 𝑚 − 443.34 𝑚2 𝑃2
(5.19)
with the coefficient of determination 𝑅 = 0.999688. The static pressure rise ranges from 1,500 to 2,400 Pa along the mass flow rate range. The results can be expressed by the following relation (red curve): Δ𝑃 = 31352.35 + 13138.69 𝑚 − 42206026 𝑚2 with the coefficient of determination 𝑅 = 0.999694.
79
(5.20)
The total pressure ratio ranges from 1 to 1.28. The results can be expressed by the following relation (red curve): 𝑃3𝑡 = 1.33 + 1.7635 𝑚 − 443.34 𝑚2 𝑃2𝑡
(5.21)
with the coefficient of determination 𝑅 = 0.999651. The overall efficiency 𝜂𝑜𝑣 ranges from 0.05 to 0.7. The results can be expressed by the following relation (red curve): 𝜂𝑜𝑣 = −0.4736 + 141.569 𝑚 − 4329.244 𝑚2
(5.22)
with the coefficient of determination 𝑅 = 0.998702. The velocity at the compressor inlet 𝐶2 (m/s) ranges from 83.5 to 92 m/s. The results can be expressed by the following relation (red curve): 𝐶2 = 80.327 + 37.63 𝑚 + 13126.7 𝑚2
(5.23)
with the coefficient of determination 𝑅 = 0.999699. The Mach number at the compressor inlet 𝑀2 ranges from 0.245 to 0.268. The results can be expressed by the following relation (red curve): 𝑀2 = 0.2355 + 0.14529 + 34.816 𝑚2
(5.24)
with the coefficient of determination 𝑅 = 0.999685. The velocity at the compressor outlet 𝐶3 (m/s) ranges from 117 to 126 m/s. The results can be expressed by the following relation (red curve): 𝐶3 = 119.4764 − 472.3747 𝑚 + 25030.52 𝑚2
(5.25)
with the coefficient of determination 𝑅 = 0.998286. The Mach number at the compressor outlet 𝑀3 ranges from 0.337 to 0.361. The results can be expressed by the following relation (red curve): 𝑀3 = 0.3445 − 1.30057 𝑚 + 64.914 𝑚2 80
(5.26)
with the coefficient of determination 𝑅 = 0.997824.
5.1.4 Speed Line at 6,000 rpm The results at the rotational speed 6,000 rpm are displayed in Figs. 5.8 - 5.10 for the relations between; 1- the static pressure ratio 𝑃3 /𝑃2 , 2- the static pressure rise Δ𝑃 = 𝑃3 − 𝑃2 (Pa), 3- the total pressure ratio, 4- the overall efficiency 𝜂𝑜𝑣 , 5velocity at inlet and outlet (m/s), 6- Mach number at inlet and outlet, 7- different compressor pressures and the mass flow rate 𝑚 (kg/s). Fig. 5.8 shows the relation of the velocity at inlet and outlet with the mass flow rate 𝑚 (kg/s). It seems that the ratio between the velocities 𝐶3 𝐶2 averages to 2.1. Fig. 5.9 shows the relation of the Mach number at inlet and outlet with the mass flow rate 𝑚 (kg/s). It seems that the ratio between the Mach numbers 𝑀3 𝑀2 averages to 1.8. Fig. 5.10 shows the relation between the different compressor pressures and the mass flow rate 𝑚 (kg/s). The difference between the total pressure and static pressure at compressor outlet averages to 9,600 Pa. The coefficient of determination R represents the trend of convergence between the actual measurements and the fits. The mass flow rate ranges from 0.011 to 0.025 kg/s. The static pressure ratio ranges from 1.0 to 1.19 according to the mass flow rate. The results can be expressed by the following relation (red curve): 𝑃3 = 1.236 + 0.4419 𝑚 − 380.455 𝑚2 𝑃2
(5.27)
with the coefficient of determination 𝑅 = 0.999881. The static pressure rise ranges from 2,000 to 19,000 Pa along the mass flow rate range. The results can be expressed by the following relation (red curve): 81
Δ𝑃 = 23587.545 + 20411.536 𝑚
(5.28)
− 37151295.88 𝑚2 with the coefficient of determination 𝑅 = 0.999877. The total pressure ratio ranges from 1.03 to 1.21. The results can be expressed by the following relation (red curve): 𝑃3𝑡 = 1.256 + 0.4419 𝑚 − 380.455 𝑚2 𝑃2𝑡
(5.29)
with the coefficient of determination 𝑅 = 0.999881. The overall efficiency 𝜂𝑜𝑣 ranges from 0.05 to 0.5. The results can be expressed by the following relation (red curve): 𝜂𝑜𝑣 = −0.30492 + 111.568 𝑚 − 3902.156 𝑚2
(5.30)
with the coefficient of determination 𝑅 = 0.999762. The velocity at the compressor inlet 𝐶2 (m/s) ranges from 58.5 to 67.5 m/s. The results can be expressed by the following relation (red curve): 𝐶2 = 55.278 + 115.48 𝑚 + 15198.6 𝑚2
(5.31)
with the coefficient of determination 𝑅 = 0.999613. The Mach number at the compressor inlet 𝑀2 ranges from 0.171 to 0.196. The results can be expressed by the following relation (red curve): 𝑀2 = 0.161965 + 0.36412 + 41.589 𝑚2
(5.32)
with the coefficient of determination 𝑅 = 0.999591. The velocity at the compressor outlet 𝐶3 (m/s) ranges from 117 to 123 m/s. The results can be expressed by the following relation (red curve): 𝐶3 = 116.87 − 214.393 𝑚 + 18363.8 𝑚2 with the coefficient of determination 𝑅 = 0.99776.
82
(5.33)
Figure 5.8 Measured absolute velocities at inlet and outlet versus mass flow rate at 6,000 rpm.
Figure 5.9 Measured Mach numbers at inlet and outlet versus mass flow rate at 6,000 rpm. 83
The Mach number at the compressor outlet 𝑀3 ranges from 0.338 to 0.352. The results can be expressed by the following relation (red curve): 𝑀3 = 0.338 − 0.59544 𝑚 + 64.914 𝑚2
(5.34)
with the coefficient of determination 𝑅 = 0.996908.
Figure 5.10 Measured different compressor pressures versus mass flow rate at 6,000 rpm.
5.1.5 Measured Complete Map The term complete map refers to the compressor performance that was drawn at different compressor rotational speeds N, namely 12,000, 9,000, and 6,000 rpm. The performance maps of the compressor is displayed in Figs. 5.11 5.16 for the relations of; 1- static pressure ratio 𝑃3 /𝑃2 , 2- the static pressure rise Δ𝑃 = 𝑃3 − 𝑃2 (Pa), 3- the total pressure ratio 𝑃3𝑡 𝑃2𝑡 , 4- the overall efficiency 84
𝜂𝑜𝑣 , 5- the absolute velocity at compressor inlet and outlet (𝐶2 and 𝐶3 ), 6- the Mach number at compressor inlet and outlet (𝑀2 and 𝑀3 ), versus the mass flow rate 𝑚 (kg/s). Fig. 5.11 shows the compressor performance map that represents the relation between the static pressure ratio 𝑃3 /𝑃2 and the mass flow rate 𝑚 (kg/s). As it is well known, the performance map shows that the static pressure ratio increases as the compressor rotational speed N increases. Fig. 5.12 shows the compressor performance map that represents the relation between the static pressure rise Δ𝑃 = 𝑃3 − 𝑃2 (Pa) and the mass flow rate 𝑚 (kg/s). Likewise the previous figure, it is seen that the static pressure rise increases as compressor rotational speed N increases. Fig. 5.13 shows the compressor performance map that represents the relation between the total pressure ratio 𝑃3𝑡 𝑃2𝑡 and the mass flow rate 𝑚 (kg/s). By connecting the upper points for each speed line, surge line is formed. Surge line compares well to that of Abou Rayan et al (2007), reported in Fig. 2.13. Fig. 5.14 shows the compressor performance map that represents the relation between the compressor overall efficiency 𝜂𝑜𝑣 and the mass flow rate 𝑚 (kg/s). As expected, the efficiency increases as the compressor speed N increases. As the compressor speed 𝑁 increases, inlet static pressure 𝑃2 decreases and then inlet kinetic energy increases. Thus, the compressor inlet velocity 𝐶2 increases. On the other hand, the compressor outlet static pressure 𝑃3 increases as the outlet kinetic energy is reduced in the volute. However, the outlet velocity approaches the value of inlet velocity. The outlet velocities coincided with each others for different compressor speeds 𝑁 = 6,000, 9,000 and 12,000 𝑟𝑝𝑚.This is 85
shown in Fig. 5.15. Accordingly, the Mach numbers take the same increasing trend as can be seen in Fig. 5.16.
Figure 5.11 Measured performance map of the compressor at different rotational speeds, static pressure ratio versus mass flow rate.
Figure 5.12 Measured performance map of the compressor at different rotational speeds, static pressure rise versus mass flow rate. 86
Figure 5.13 Measured performance map of the compressor at different rotational speeds, total pressure ratio versus mass flow rate.
Figure 5.14 Measured performance map of the compressor at different rotational speeds, overall efficiency versus mass flow rate. 87
Figure 5.15 Measured absolute velocities at compressor inlet and outlet versus mass flow rate, at different compressor speed.
Figure 5.16 Measured Mach numbers at compressor inlet and outlet versus mass flow rate, at different compressor speed. 88
5.2 Computational Investigation 5.2.1 Validation of Sector-domain Technique A sector geometry was discussed briefly in Sec. 3.6 and Fig. 3.5. The second boundary condition technique (Sec. 3.8 and Fig. 3.9) was used with this geometry. When the sector geometry technique is used, it makes a sever divergence at the beginning of numerical iteration. This means that the sector-domain technique is invalid for a multi-stage centrifugal compressor. In other words, it is impossible to use the sector-domain technique for a full flow study in a multi-stage centrifugal compressor.
5.2.2 Validation of Full-domain Study Methods The first (thin), second (thick), and the fourth (outlet duct) geometry techniques presented in Figs. 3.3, 3.4, and 3.6, respectively represent full-domain techniques. There are two types of boundary-condition techniques used with these three geometry techniques. These are; 1- The first B.C. technique (as in Sec. 3.8, Fig. 3.8), 2- The third B.C. technique (as in Sec. 3.8, Fig. 3.10). There were three methods to validate the full-domain technique. These methods are; 1- Direct Map Method, 2- Real map method, and 3- Advanced real-map method.
5.2.2.1 Direct Map Method In this type of validation, all boundary conditions from the experimental compressor map are put. If the code solves the case, then it means that this case is confirmed. Direct map method of validation used the first and second geometry techniques with the first B.C. technique. The compressor experimental data were supplied by the manufacturer company, Armfeild (2005). 89
It is important to mention that the present cases considered the steady solution. Also, the second geometry technique took about quarter the time taken by the first geometry technique. The second technique consumed 51 hours to converge, whereas the first technique consumed 11 hours. The reason is obviously due to the effect that the second geometry technique took into account the thickness of blades and impeller discs. The case of unsteady (transient) solution was strongly diverged. Unfortunately, this means that the direct map method can not simulate surge.
5.2.2.2 Real Map-I Method This method was carried out by introducing the code of only one parameter of the compressor performance curve. Then, the code computes the other parameters of the performance curve. The first and second geometry techniques (Figs. 3.3 and 3.4) were used with the third B.C. technique (Fig. 3.10).
5.2.2.3 Real Map-II Method Advanced real-map method depends on making the solver to determine the outlet boundary of the compressor by the code itself. This was accomplished by parallel processing as HPC program of 4 PCs. Also, CFD-RC code could not interface with HPC program. Then FLUENT program was used to interface the HPC program.
5.2.2.4 Comparison of Different Computational Methods It is obviously from Figs. 5.17-5.19 that the real map-I with thin blades mesh method was least accuracy. Also, it was appeared that the real map-I with thick blades mesh method was more accurate than the real map-I with thin blades mesh 90
method. Table 5-1 shows a fruitful comparison between the different types of solving techniques. Finally, it is important to say that the real map-II method was the best method.
Figure 5.17 Comparison of whole theoretical and measured results respect to compressor map, at 6,000 rpm.
Figure 5.18 Comparison of whole theoretical and measured results respect to compressor map, at 9,000 rpm. 91
Figure 5.19 Comparison of whole theoretical and measured results respect to compressor map, at 12,000 rpm.
5.2.2.5 Validation of Flow in Compressor Figs. 5.20-5.21 represent a good validation of the present computational scheme (code). It shows the air flows in the correct path from the compressor entrance to the exit. This correct path is a validation that the code runs correctly.
5.2.3 Faults in Volute Exclusion This case was taken to examine the possibility of surge predicting with exclude volute. Figs 5.22 and 5.23 show the gradual increase in velocity in impellers. No reverse flow was found since there is not any obstruction. Fig. 5.24 92
shows the flow path with too small mass flow rate. No reverse flow appears since flow rate was not sufficient to create surge.
5.2.4 Computational Surge Simulation Surge is usually studied by decreasing the mass flow rate until flow begins to oscillate. For the present research, the surge was predicted at 12,000 rpm and mass flow rate 𝑚 = 0.009399 kg/s. Actually, surge is simulated in three regions. The first region is the inlet to the compressor. The second region is in the passage of the first impeller. The third region is the section along the centerline of the exit duct, Figs. 5.25 and 5.26. The arrows represent the velocity vectors and the arrow length scales the velocity magnitude.
Figure 5.20 Velocity vectors for the present compressor: (left) general view of the whole compressor, (right) detailed view of the return bend.
93
Figure 5.21 Velocity vectors between stages.
Figure 5.22-a Velocity distributions for the case without volute in the first impeller, at 12,000 rpm and 0.0093 kg/s.
Figure 5.22-b Velocity distributions for the case without volute in the second impeller, at 12,000 rpm and 0.0093 kg/s.
94
Figure 5.23-a Velocity distributions for the case without volute in the third impeller, at 12,000 rpm and 0.0093 kg/s.
Figure 5.23-b Velocity distributions for the case without volute in the fourth impeller, at 12,000 rpm and 0.0093 kg/s.
Figure 5.24 Flow paths in the case without volute which show no any reverse flow, at 12,000 rpm and 0.009 kg/s.
95
Figure 5.25 Illustrative cut-view of the first region for surge study which is the inlet to the compressor.
Figure 5.26 Illustrative cut-view of the second and third region for surge study which is the passage of the first impeller and section along the centerline of exit duct, respectively, of the compressor.
96
5.2.4.1 Compressor inlet Fig. 5.27 shows the oscillation of the velocity vectors. These oscillations can be detected at compressor inlet by following the flow vectors with time steps. It is obviously seen that the velocity vectors move in the direction of partial reverse flow and then move back to the correct direction. The term partial reverse flow means that there are some velocity vectors but the overall flow moves in the correct direction. This partial reverse flow makes the flow to oscillate. At 𝑡 = 0 𝑠𝑒𝑐, the numerical solution starts and the computational domain has the initial conditions. At 𝑡 = 0.01 𝑠𝑒𝑐, the flow moves to the right and to the left to enter the impeller. At 𝑡 = 0.02 𝑠𝑒𝑐, the velocity magnitude increases. At 𝑡 = 0.03 𝑠𝑒, the velocity magnitude decreases. In the same manner, the velocity magnitude continues to increase and decrease, which makes the flow oscillate. Since the net flow does not lead to reverse flow, this phenomenon represents a mild surge as mentioned before in section 2.6 and seen in Fig. 2.15.
5.2.4.2 Impeller Passage Fig. 5.28 shows the surge simulation in a passage in the first impeller of the compressor. The impeller is backward, i.e., the impeller rotates in the CCW direction. Then, the right side of the passage is named as the positive pressure side of the blade. Also, the left side of the passage is named as the negative pressure side of the blade. At t= 0.01 sec, the flow moves in a path adjacent to the positive pressure side in the passage. Full-span rotating stall appears next to the negative pressure 97
side, as mentioned in section 2.5 and seen in Fig. 2.14. The stall is a flow separation that forms one or more vortices. The rotating stall causes a blockage from one vortex or two vortices. This blockage narrows the flow in the right side (the positive pressure side). At t= 0.02 sec, the separation blockage decreases as the main vortex is divided into two vortices. The first one is big and goes up. The second one is small and goes down. The flow path increases since the stall blockage is decreased. At t= 0.03 sec, the stall blockage was decreased also and then the flow path is increased. At t= 0.04 sec, the stall blockage increases rapidly and then the flow path decreases. At t= 0.05 sec, the stall blockage decreases and then the flow path increases. At t= 0.06 sec, t= 0.07 sec, and t= 0.08 sec, flow path continues to increase as the stall blockage continues to decrease. At t= 0.09 sec, flow path rapidly decreases as the stall blockage rapidly increases. The same state continues for the reminder time steps. From Fig. 2.27, the flow oscillation can be obviously predicted. Here, there is no net reverse flow but only flow fluctuations.
5.2.4.3 Compressor Exit Duct Also, surge is simulated in the outlet duct of the compressor, Fig. 5.29. This is a section along the centerline of the exit duct. At t= 0.01 sec, the flow moves from the sides of the duct towards the middle. There is a partial reverse flow in the middle of the duct. Two small vortices are formed in the middle. Note that the partial reverse flow is small in comparison with the total flow and then the net flow has no reverse stream. Since there is no net reverse flow, it is therefore a mild surge and not deep surge. 98
At t= 0.02 sec, the two vortices decrease and thus the partial reverse flow decreases and the net flow increases. At t= 0.03 sec, the two vortices increase and the partial reverse flow increases and then the net flow decreases. By following the time steps, we can notice the flow fluctuations as the size of the partial reverse flow fluctuates.
5.2.5 Modified Case with High Pressure Ratio Since the maximum compressor pressure ratio was 1.3 and the compressor pressure ratio should be 1.5 at least, then, a modified case was defined with 1.55 pressure ratio at 20,000 rpm. In this case, surge phenomenon could be predicted. Figs. 5.30 and 5.31 show the gradual increase in the static pressure distribution in different impellers. Also, Fig. 5.32 shows the importance of combining a volute since it is responsible for partially converting the velocities in the four stages to increase static pressure. To simulate surge, 20,000 rpm and 0.0093 kg/s were taken. Severe reverse values of mass flow rate were detected at some time steps. Then, deep surge was simulated. The same shape of Figs. 5.27 to 5.29 was noted with change of frequency. The frequency at 12,000 rpm and 0.00939 kg/s was about 100 Hz. In high pressure case, the surge frequency increased to about 250 Hz.
99
t= 0 sec
t= 0.01 sec
t= 0.02 sec
t= 0.03 sec
t= 0.04 sec
t= 0.05 sec
t= 0.06 sec
t= 0.07 sec
t= 0.08 sec
t= 0.09 sec
t= 0.10 sec
t= 0.11 sec
Figure 5.27 Surge simulation of the inlet flow at the entrance to centrifugal compressor at different time steps, from t= 0 sec to t= 0.11 sec, at 12,000 rpm and 0.0093 kg/s. 100
t= 0 sec
t= 0.01 sec
t= 0.02 sec
t= 0.03 sec
t= 0.04 sec
t= 0.05 sec
Figure 5.28 Continued.
101
t= 0.06 sec
t= 0.07 sec
t= 0.08 sec
t= 0.09 sec
t= 0.10 sec
t= 0.11 sec
Figure 5.28 Surge simulation of the flow in a passage in the first impeller of the compressor at different time steps, from t= 0 sec to t= 0.11 sec, at 12,000 rpm and 0.0093 kg/s.
102
t= 0 sec
t= 0.01 sec
t= 0.02 sec
t= 0.03 sec
t= 0.04 sec
t= 0.05 sec
t= 0.06 sec
t= 0.07 sec
Figure 5.29 Continued
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t= 0.08 sec
t= 0.09 sec
t= 0.10 sec
t= 0.11 sec
Figure 5.29 Surge simulation of the flow in the outlet duct of the compressor at different time steps, from t= 0 sec to t= 0.11 sec, at 12,000 rpm and 0.0093 kg/s.
Figure 5.30-a Static pressure distributions at the modified case for first impeller.
Figure 5.30-b Static pressure distributions at the modified case for second impeller
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Figure 5.31-a Static pressure distributions at the modified case for third impeller.
Figure 5.31-b Static pressure distributions at the modified case for fourth impeller
Figure 5.32 Static pressure distributions for the modified case; (upper) in the volute, (lower) in the outlet duct.
105
Table 5.1 Comparison between different types of computational methods Geometry Technique
B.C. technique Steady solution Unsteady solution Surge simulation Time for steady solution Time for one step of unsteady solution Facility
mesh size (nodes) no. of blocks code used
Direct Map Method
Real Map Method
Thin Thick Blades Blades Mesh Mesh first yes no not possible 11 hrs 51 hrs
Thin Thick Blades Blades Mesh Mesh third yes yes yes
not done
Advanced Real Map Method Thick Blades with outlet duct Mesh third yes not done not done
20 hrs
90 hrs
20 hrs
17 hrs
24 hrs
not done
core 2 du processor 2.2 G ram 3 G 385920 584900 385920 584900 261 347 261 347 CFD-RC CFD-RC
106
(HPC/ 4 PCs) each: dell core 2 due, p 4 G, ram 4 G 1,142,000 348 FLUENT/ HPC
CHAPTER 6 COCLUSIONS AND FUTURE WORK 6.1 Conclusions Both computational and experimental investigations were utilized to study the flow behavior inside a multi-stage centrifugal compressor. Computational study was carried out using the commercial code “CFD-RC”. Experimental work was accomplished by the use of a data acquisition system, advanced sensors and “LabView” interface software. Comparisons between computational and experimental outputs were performed. The CFD-RC code was validated experimentally and numerically. It is numerically capable, of solving the flow in the multi-stage centrifugal compressor. Also, the code was validated to simulate and predict a mild surge. From the results of the present study, the following conclusions are drwan: 1- Comparisons of the present experimental and computational results with those of other investigators validate the present techniques and methodologies. 2- Computational results in all investigated cases are in good agreement with the corresponding experiment results. This implies that digital sensors are very efficient in measuring the flow in the centrifugal compressor. 3- Results indicate that it is rather impossible to use the sector-domain technique for full flow study in a multi-stage centrifugal compressor. 4- Direct-map method is the simplest method to simulate steady flow in rotating machines. On the other side, the direct-map method could not 107
simulate the unsteady (transient) flow case especially as a surge phenomenon. 5- Flow fluctuations at compressor exit are noticed when a mild surge occurs at a speed of 12,000 rpm and mass flow rate of 0.0093 kg/s. Surge was successfully predicted by unsteady computations. 6- In spite of the small pressure ratio of the present compressor, the study covers almost all features of such multi-stage centrifugal compressors.
6.2 Future Work 1- The use of the technology of parallel-processing technology is becoming essential since one complete day was needed to solve only one time step of the present work with the present facilities. 2- Use of more popular designs like those of NASA to change the geometry of the compressor and investigate the effect of the change on surge.
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REFRENCES Al-Sulaiman, F., “Test Facility Development for Small Centrifugal Compressor”, M.Sc. Thesis, King Fahd University of Petroleum and Minerals, Saudi Arabia, June 2003. Abdel Gawad, A. F., “Computational of Turbulent Flow and Heat Transfer in Rotating Non-Circular Ducts”, Ph.D. Thesis, Zagazig University, Zagazig, EGYPT, July 1998. Abou Rayan, M., Mostafa, N. H., and Djebedjian, B., “Hydraulic Machines”, Electronic course, Zagazig University, 2007. Armfeild, Manufacture Company of the Compressor, “Compressor Performance Map Results”, technical drawing, Armfeild Company, 2005. Butterworth-Heinemann, “Compressors publishing company, Houston, TX, 1997.
Selection
and
Sizing”,
Gulf
Boyce, M. P., “Gas Turbine Engineering Hand Book”, Gulf Professional Publishing, Third Edition, 2006. Cohen, H., Rogers, G. F. C., and Saravanamutto, H. I. H., “Gas Turbine Theory”, Longman Group Limited, Fourth edition, 1996. El-Sayed, A. F., “Aircraft Propulsion and Gas Turbine Engines”, text book, CRC Press, 2008. El-Mitwally, E. S., Abou-Rayan, M., Mostafa, N. H., and Hassanein, A. H., “Modeling Techniques for Predicting Compressor Performance during Surge & Rotating Stall”, Fluid Engineering Division Conference, FED-Vol.238, ASME, Vol.3, 1996. Elder, R. L., and Gill, M. E., “A Discussion of the Factors Affecting Surge in Centrifugal Compressor”, J. Engineering for Gas Turbines and Power, Vol.107, pp.409-506, 1985. CFD-RC, Manuals of Using ESI-CFD-RC Code, Program Documentation, www.esi-group.com, 2008. 109
Figliola, R., and Beasley, D., “Theory and Design for Mechanical Measurements”, text book, Second edition, John Wiley & Sons Inc., pp. 171209. Forster, W., Karpinsky, G., Krain, H., and Schodl, R., “3-ComponentDoppler-Laser-Two-Focus Velocimetry Applied to a Transonic Centrifugal Compressor”, Institute of Propulsion Technology, German Aerospace Center (DLR), Germany, 1999. Galvas, M. R., “Fortran Program for Predicting off-Design Performance of Centrifugal Compressor”, NASA Technical Note D-7487, Nov. 1973. Ginter, F., Ruprecht, A., and Gode, E., “Numerical Simulation of Rotating Stall in an Axial Compressor”, Institute of Fluid Mechanics and Hydraulic Machinery, University of Stuttgart, 2001. Greitzer, E. M., “Review – Axial Compressor Stall Phenomena,” J. Fluids Engineering, Vol. 102, pp. 134-151, June 1980. Greitzer, E. M., “The Stability of Pumping Systems,” J. Fluids Engineering, Vol.103, pp. 193-242, June 1981. Gresh, M. T., “Compressor Performance: Selection, Operation, and Testing of Axial and Centrifugal Compressors”, Stoneham Mass: Butterworth-Heinemann, 1991. As quoted by [Hanlon, P. C., “Compressor Handbook”, McGraw-Hill, ISBN 0-07-026005-2, 2001.]. Gorla, R. S. R., and Khan, A. A., “Turbomachinery Design and Theory”, Tenth Edition, Marcel Dekker, USA, 2003. Hassanien, A. H., “Modeling and Simulation of Industrial Thermo-Fluid Systems with Application to Fertilizer Industries”, Ph.D. Thesis, Zagazig Univ., 1996. Hanlon, P. C., “Compressor Handbook”, McGraw-Hill, ISBN 0-07-026005-2, 2001.
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Hansen, K. E., Jorgensen, P., and Larsen, P. S., “Experimental and Theoretical Study of Surge in a Small Centrifugal Compressor”, ASME, Vol. 103, pp. 391-395, Sep. 1981. Hibon and IR (Ingersoll-Rand) companies, “Multistage Centrifugal Blowers”, www.hibon.com and www.ingersollrand.com, 2007. Kirk, D. R., “Air Breathing Engines; Advanced Concepts”, Presentation, Mechanical and Aerospace Engineering Department, Florida Institute Technology, Dec. 5, 2006. Lapina, R. P., “Estimating Centrifugal Compressor Performance”, Process Compressor Technology, Vol. 1, First Edition, Gulf Publishing Company, Texas, USA, 1982. Launder, B.E., and Spaulding, D.B., “The Numerical Computation of Turbulent Flows”, Comp. Methods for Appl. Mech. Eng., vol.3, pp. 269-289, 1974. As quoted by ESI (2008). Ling, J., Wong, K. C., and Armfeild, S., “Numerical Investigation of a Small Gas Turbine Compressor”, 16th Australian Fluid Mechanics Conference, Crown Plaza, Gold Coast, Australia, 2-7 Dec. 2007. Macdougal, I., and Elder, R. L., “Simulation of Centrifugal Compressor Transient Performance for Process Plant Applications”, ASME J. Engineering for Power, 83-GT-25, pp. 1-5, July 1983. Moore, J., and Moore, J. G., “Calculation of Three-Dimensional, Viscous Flow and Wake Development in a Centrifugal Impeller”, ASME J. Engineering for Power, Vol. 103, pp. 367-372, April 1981. Mostafa, N. H., “Prediction of Surge and Rotating Stall in Compressor”, Eighth International Conference in Fluid Dynamics and Propulsion (ICFDP8), Sharm El-Sheikh, Egypt, December, 2006. Niazi, S., “Numerical Simulation of Rotating Stall and Surge”, Ph.D. Thesis, Georgia Institute of Technology, July 2000. Rajput, R. K., “Mechanical Measurements and Instrumentation”, Text Book, S. K.KATARIA & SONS®, New Delhi, 2009. 111
Rangwala, A. S., “Turbo-machinery Dynamics; Design and Operation”, McGraw-Hill, First Edition, USA, 2005. Shum, Y. K. P., “Impeller-Diffuser Interaction in Centrifugal Compressor”, Ph.D. Thesis, Massachusetts Institute of Technology, Feb. 2000. Stein, A., “Computational Analysis of Stall and Separation Control in Centrifugal Compressors”, Ph.D. Thesis, Georgia Institute of Technology, USA, May 2000. Stenning, A. H., “Rotating Stall and Surge”, Transaction of the ASME, Vol. 102, March 1980. Takata, H., and Nagano, S., “Nonlinear Analysis of Rotating Stall”, J. Engineering for Power, Oct. 1972. Tang, J., Turunen-Saaresti, T., and Larjola, J., “Use of Partially Shrouded Impeller in a Small Centrifugal Compressor”, J. Thermal Science, Vol. 17, No.1, pp. 21-27, Article ID: 1003-2169(2008)01-0021-07, Nov. 2008. Tijl, P., “Modeling Simulation and Evaluation of a Centrifugal Compressor with Surge Avoidance Control”, M.Sc. Thesis, Dynamic and Control Technology group, Mechanical Engineering Department, Technical University, Eindhoven, Netherlands, March 2004. Willemes, F. P., “Modeling and Bounder Feedback Stabilization of Centrifugal Compressor Surge”, Ph.D. Thesis, Technical University, Eindhoven, Netherlands, 2000. VKI, “Flow in Turbomachines”, Von Karman Institute (VKI), Report 2005, http://www.vki.ac.be/index.html, Germany, 2005. Xinwei, S., Chuangung, G., Jun, X., and Chuang, G., “Centrifugal Compressor Blade Optimization Based on Uniform Design and Genetic Algorithm”, Front. Energy Power Eng., DOI 10.1007/s11708-008-0083-5, CHINA, 2008. Xu, C., and Muller, M., “Development and Design of a Centrifugal Compressor Volute”, Int. J. Rotating Machinery, Vol. 3, pp. 190-196, 2005. 112
Yutaka, O., Takashi, G., and Eisuke, O., “Effect of Tapered Diffuser Vane on the Flow Field and Noise of a Centrifugal Compressor”, J. Thermal Science, Vol. 16, No.4, pp. 301-308, DIO: 10.1007/s11630-007-0301-1, 2007. Yoshinka, T., “Surge Responsibility and Range Characteristics of Centrifugal Compressor”, Tokyo Joint Gas Turbine, 1977. As quoted by Stein [2000].
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Appendix A Matlab Program Commands Used in Error Analysis >> syms eta T2 Cp R Patm P2 P3t P3 P2 T3 gm >> eta=(T2+(1/2*Cp)*(R*T2*(Patm-P2)/P2)^(3/2))*((P3t/Patm)^((gm1)/gm))/((T3+(1/2*Cp)*(R*T3*(P3t-P3)/P3)^(3/2))-(T2+(1/2*Cp)*(R*T2*(PatmP2)/P2)^(3/2))) eta = (T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2)) >> diff (eta,T2) ans = (1+3/4*Cp*(R*T2*(Patm-P2)/P2)^(1/2)*R*(Patm-P2)/P2)*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))-(T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))^2*(-1-3/4*Cp*(R*T2*(Patm-P2)/P2)^(1/2)*R*(Patm-P2)/P2) >> diff (eta,T3) (∂η_C)/(∂T_3 )=-(T_2+Cp/2(R T_2 〖(P_atm-P_2)/P_2)〗^(3/2))*(P_3t/P_atm )^((γ-1)/γ)/(T_3+Cp/2**(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))^2*(1+3/4*Cp*(R*T3*(P3t-P3)/P3)^(1/2)*R*(P3t-P3)/P3) >> diff (eta,P2) ans = 3/4*Cp*(R*T2*(Patm-P2)/P2)^(1/2)*(-R*T2/P2-R*T2*(PatmP2)/P2^2)*(P3t/Patm)^((gm-1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T21/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))+3/4*(T2+1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))*(P3t/Patm)^((gm-1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)T2-1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))^2*Cp*(R*T2*(Patm-P2)/P2)^(1/2)*(R*T2/P2-R*T2*(Patm-P2)/P2^2) >> diff (eta,P3) ans =
114
-3/4*(T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))^2*Cp*(R*T3*(P3t-P3)/P3)^(1/2)*(-R*T3/P3-R*T3*(P3t-P3)/P3^2) >> diff (eta,P3t) ans = (T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm-1)/gm)*(gm1)/gm/P3t/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))-3/4*(T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))^2*Cp*(R*T3*(P3t-P3)/P3)^(1/2)*R*T3/P3 >> syms UT2 UT3 UP3 UP3t >> syms Ueta >> Ueta=diff(eta, T2)*UT2+diff(eta,T3)*UT3+diff(eta,P3)+P3+diff(eta,P3t)*UP3t Ueta = ((1+3/4*Cp*(R*T2*(Patm-P2)/P2)^(1/2)*R*(Patm-P2)/P2)*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))-(T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))^2*(-1-3/4*Cp*(R*T2*(Patm-P2)/P2)^(1/2)*R*(PatmP2)/P2))*UT2-(T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))^2*(1+3/4*Cp*(R*T3*(P3t-P3)/P3)^(1/2)*R*(P3t-P3)/P3)*UT33/4*(T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T2-1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))^2*Cp*(R*T3*(P3t-P3)/P3)^(1/2)*(-R*T3/P3-R*T3*(P3tP3)/P3^2)+P3+((T2+1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))*(P3t/Patm)^((gm1)/gm)*(gm-1)/gm/P3t/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)-T21/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))-3/4*(T2+1/2*Cp*(R*T2*(PatmP2)/P2)^(3/2))*(P3t/Patm)^((gm-1)/gm)/(T3+1/2*Cp*(R*T3*(P3t-P3)/P3)^(3/2)T2-1/2*Cp*(R*T2*(Patm-P2)/P2)^(3/2))^2*Cp*(R*T3*(P3tP3)/P3)^(1/2)*R*T3/P3)*UP3t
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Appendix B Analog to Digital Card Specifications The used A/D card in the present research was manufactured by Computer Boards company. It is 16 bit, 1 MHz and it is labeled as CIO-DAS1602/16. Its base address is 300H and it was connected based on a differential method. A rosette was used to simplify the connection between the card and sensors. Fig. 4.8 shows the A/D card picture and its rosette. Interfacing the A/D card with the computers needs three stages; first, introducing the card to the motherboard, second, reading the output signal of the card, and third, analyzing the card signal. Analyzing is accomplished by several methods such as substituting in the calibration equation, filtering, or calculating the Fast Fourier Transformation. Introducing the A/D card to the computer motherboard may be accomplished by several programs. The most popular program is InsatCall, Fig. 4.9. Its name is an abbreviation of installation and calibration of the card. Reading the signal from the card is accomplished by InstaCall also. Analyzing the card signal may be accomplished by several programming package like C, C+, Visual Basic and LabView. The latter is the most famous and best to use in this application. The main feature of LabView is that it contains two screens. The first one is the front panel screen, which is used for the user manipulation. The second screen is the block diagram, which contains the idea of programming. This screen is the program designer screen. The front panel screen contains some indicator and control fields. Indicator fields show the output data while the control fields take the input data. Then, the user will fill the control fields to get results in the indicator fields. Figs. 4.10 - 4.15 show the LabView programs to acquire sensor signals. Figs. 4.10 and 4.11 are the front panel and the block diagram of the LabView program for the Positive Pressure Transducer, respectively. Fig. 4.10 is the front panel of the Positive Pressure Transducer program. “Board Number” is the first control field which refers to the A/D card as defined to the motherboard. Any A/D card has many channels such as 8, 12 and 16 channels. “Channel” is the second control field. Also, the A/D cards differ in their input range such as ±5 volt, ±10, 116
etc. “Path” is the fourth control field that refers to the path at which reading file will be saved at. “PPT (Pa)” is the indicator field in which average pressure readings appear. Fig. 4.11 shows the blocks of the program and the wires between them. Also, the calibration equation that transforms the sensor output (voltage) to pressure (Pa) appears in the figure. In the same way, Figs. 4.12 and 4.13 show the front panel and the block diagram of the Net Positive Transducer program, respectively. Also, Figs. 4.14 and 4.15 show the front panel and the block diagram of the digital thermocouple program, respectively.
117
يهخض ػشبٍ حؼخبش دساعت انغشَاٌ فٍ ضاغط انطشد انًشكضٌ األكزش حؼمُذا و طؼىبت فٍ دساعاث االث انًىائغ .و حؼىد هزِ انظؼىبت باألعاط انً ػذو اَخظاو شكم انضاغط ووصىد يضًغ انهىاء
) (voluteرو انشكم غُش
انًخًارم باألضافت انً وصىد انغشَاٌ انزاَىٌ .و َضداد انًىلف حؼمُذا فٍ حانت ضاغط انطشد انًشكضٌ يخؼذد انًشاحم .و َشصغ رنك نىصىد اَحُاءاث فٍ انًغاس بٍُ انًشاحم
) (Return-bendsوكزنك احغاع يضال
انذساعت. و نمذ شاع اعخخذاو انحم انحغابٍ نًحاكاة انغشَاٌ خالل انضاغط .حُذ اعخخذو انكزُش يٍ انباحزٍُ انحم بطشَمت انحُض انمطاػٍ نهخبغُط .أيا انمهُم يُهى فمذ اعخخذو انحُض انكهٍ نضاغط انطشد انًشكضٌ أحادٌ انًشحهت .أيا انذساعت انحانُت فمذ أخزث فٍ األػخباس انحُض انكهٍ نضاغط انطشد انًشكضٌ سباػٍ انًشاحم. ولذ اػخًذث انذساعت انحانُت ػهً طشق انبحذ انحغابُت و انؼًهُت نًؼشفت خظائض انغشَاٌ داخم انضاغط. و لذ حًج انذساعت انحغابُت باعخخذاو انبشَايش انحغابٍ انخضاسٌ ” .“CFD-RCأيا انذساعت انؼًهُت فمذ حًج باعخخذاو َظاو انمُاط انشلًٍ ) ،(Data Acquisitionانحغاعاث انًخمذيت و بشَايش ” “LabViewنهشبط بٍُ انحغاعاث و َظاو انمُاط. و لذ حى انخحمك يٍ دلت َخائش انبشَايش انحغابٍ انًغخخذو ػًهُا و حغابُا .كًا حى سعى خشَطت أداء انضاغط ػًهُا و َظشَا .و كزنك حى أَضا يحاكاة انخًىساث يغ انضيٍ .و حى انخُبأ بحذود انخًىساث نهضاغط يحم انذساعت ػُذيا وطم يؼذل حذفك انهىاء انً
0.009395كضى/راَُت ػُذ عشػت دوساٌ 12أنف نفت فٍ
انذلُمت .كًا حى ححهُم ػذو انخأكذ فٍ انمُاعاث انؼًهُت نألصهضة انخمهُذَت و انشلًُت. و لذ حى اعخُخاس أٌ انطشَمت انًؼخًذة ػهً حم انحُض انحغابٍ انكهٍ نهضاغط هى األفضم نهضاغط يحم انذساعت .كًا حى حزكُت أٌ اعخخذاو حمُُت انحم انشلًٍ انًخىاصٌ ػٍ طشَك بشَايش
” “HPCهى خطىة
أعاعُت فٍ أٌ ػًم يغخمبهٍ خاص بانذساعاث انحغابُت انًخكايهت نهضىاغط يخؼذد انًشاحم.
ب
صايؼت انضلاصَك كهُت انهُذعت لغى هُذعت انمىي انًُكاَُكُت
دراست حسابيت وعمليت لضاغط طارد مركزى متعدد المراحل بحذ يمذو يٍ انًهُذط
محمد سعيد حامد عبد المعطي عميرة يؼُذ بمغى هُذعت انمىي انًُكاَُكُت -كهُت انهُذعت -صايؼت انضلاصَك- ضًٍ يخطهباث انحظىل ػهً دسصت انًاصغخُش فً هُذعت انمىي انًُكاَُكُت
المشرفون
أ.د .وبيل حسه مصطفي أعخار االالث انخىسبىَُت سئُظ يضهظ لغى هُذعت انمىي انًُكاَُكُت -كهُت انهُذعت صايؼت انضلاصَك
أ.د .أحمد فاروق عبد الجواد أعخار يُكاَُكا انًىائغ انحغابُت لغى هُذعت انمىي انًُكاَُكُت -كهُت انهُذعت صايؼت انضلاصَك 20 11
