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					 International Journal of Application or Innovation in Engineering & Management (IJAIEM)
        Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 11, November 2013                                       ISSN 2319 - 4847


       Dynamic Simulation & Design of Fuzzy Logic
       Active Controller for Controlling the Surge in
               the Centrifugal Compressors
                       Qasem Abdollah Nezhad1, Jafar Ghafouri 2 and Mohammad Fathi3
        1
         Department of Mechatronics Engineering, Science and Research Branch, Islamic Azad University, Kurdistan, Iran
                               2
                                   Department of Mechanical Engineering, Azad University, Tabriz, Iran
                         3
                             Department of Electrical Engineering, University of Kurdistan , Sanandaj, Iran


                                                              Abstract
The prevention of instability occurrence of the surge type in compressors appears to be one of the most critical concerns for their
manufacturers. Compressor behavior under instable conditions is highly dynamic and non-linear as well, so leads to severe
oscillation in the features of high-frequency current flow. Such oscillations in surge mode can extend to compressor return
current resulting in severe stresses on compressor vanes. Since the centrifugal compressors contribute as major and expensive
components in most processes of amplifying the pressure of gaseous fluids, it seems essential then to protect these valuable assets
against potential damages due to the surge phenomenon. Surge control system (anti-surge) is responsible for meeting this
requirement. The present paper proposes an approach based on fuzzy logic active control principles in order to prevent the
incident of surge in a typical centrifugal compressor. As the first step, the compressor system is simulated by applying Greitzer
equation and compressor dynamics, then a fuzzy logic active controller will be designed for this system. Finally, the obtained
results from the simulation will be compared for two cases (without controller and with fuzzy logic active controller). The results
of such analysis can provide convincing evidence for the significance of incorporating a surge controller into centrifugal
compressors.

Keywords: centrifugal compressors, surge, anti-surge, active control, fuzzy logic controller.

    1. INTRODUCTION
All manufacturers provide performance graphs for their compressors. The graph consists of one horizontal axis
representing the flow rate (capacitance), one vertical axis representing the head or pressure, and a set of curves which
indicate compressor performance in different turns.




                                   Figure 1 Centrifugal compressor performance curves
As can be seen from Figure 1, there is a minimum and maximum capacitance point for each turn and the performance of
compressor is stable and predictable between these two points. The maximum and minimum points are known as
stonewall and surge respectively.
Connecting the surge points in different turns, we can obtain surge boundary line. When a compressor operates on the
right side of this line its performance is in stable mode, while if it operates on the left side of this line it experiences
instability or surge mode[1 - 3].
Surge is defined as instability in current flow which fundamentally occurs in the case of dynamic compressors. This
phenomenon takes place when the compressor is unable to produce enough head-pressure required for overcoming
downstream resistance, i.e. output pressure generated by compressor is lower than that of its downstream. Under such

Volume 2, Issue 11, November 2013                                                                                   Page 408
 International Journal of Application or Innovation in Engineering & Management (IJAIEM)
        Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 11, November 2013                                       ISSN 2319 - 4847

circumstances a reciprocating (cyclical) motion will be created in the gas flow.
By creating turbulences in the flow through, surge instability can lead to sudden damage of compressor vanes and hence
to drastic reduction in its efficiency. There are two principal methods in dealing with the surge phenomenon in
centrifugal compressors:
     1. Velocity control method.
     2. Using anti-surge control valves.
Paying close attention to characteristic curve of the compressor, we can infer that by lowering the turn we can be further
away from the surge line. However, this method (velocity control) is not an effective way for this purpose and also speed
reduction may not be accomplished quickly enough to prevent the surge phenomenon from happening. A typical example
of a control system based on the velocity control method is schematically illustrated in Figure 2.




                                Figure 2 Surge control system based on speed control method
The most commonly-used method dealing with this issue is using anti-surge valves approach in which the control of
surge process is realized through returning an amount of compressor output current back to its input by incorporating an
Anti-Surge Control Valve (ASCV) or recycle valve. In this method when current flow is reduced and approximates the
surge line a signal is transmitted from controller to anti-surge valve and causes it to open slowly and increase the flow
slightly.
A drawback to this method is the fact that opening of this valve during the compressor normal operation results in
undesired energy loss. Thus, appropriate provisions must be taken into account to keep the valve closed as possible
provided that there is no risk of reaching the surge mode at the same time.
The valve needs to prevent the occurrence of surge not only during normal operation, but also when the compressor is
going to begin or stop its operation.
During start-up stage the anti-surge valve acts as a bypass valve and makes the outgoing flow form compressor to return
back to its inlet, because in the initial stages of start-up the compressor outlet is isolated from its downstream and the gas
must reach the downstream prior to be compressed in order to enable the compressor to meet the defined head-pressure.
When the output pressure reaches the desired level, the connection between compressor and its downstream is established
and the gas flows toward downstream and simultaneously the valve shuts and the system resumes its normal operation.
During normal stop stage the flow rate diminishes simultaneously as the turn decreases gradually to stop completely.
However, in the case of emergency stop the anti-surge valve must reduce compressor head-pressure immediately.
Typical configuration of surge control systems incorporating anti-surge valve is illustrated in Figure 3.




                               Figure 3 Surge control system equipped with anti-surge valve

Volume 2, Issue 11, November 2013                                                                               Page 409
 International Journal of Application or Innovation in Engineering & Management (IJAIEM)
        Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 11, November 2013                                       ISSN 2319 - 4847

The measured values of compressor upstream and downstream pressure, passing gas temperature and flow rate, and speed
of compressor axis form inputs of surge controller.
The role of surge controller in centrifugal compressor system is to open the anti-surge valve in the case of approximating
the surge line in order to ensure the highest efficiency without jeopardizing compressor performance. In the present study
the second method, using anti-surge valves, was adopted for the purpose of controlling the surge phenomenon[2, 4, 5,
14].
For the purpose of this study, after dynamic simulation of centrifugal compressor system the design of fuzzy logic active
controller is investigated. The obtained results from simulation were compared for two modes (without controller and
with fuzzy logic active controller) and repeated for five sample rates: 16000, 18000, 21000, 23000, and 25000 Rounds
Per Minute(RPM). The sample rates were chosen randomly.

    2. SIMULATION OF CENTRIFUGAL COMPRESSOR SYSTEM
Various models have been proposed in the literature of engineering field so far for explaining dynamic behavior of
compressors. But we need to use a model which can meet the requirements of the present study. In other words, the
selected model not only must provide an appropriate foundation for implementation of surge control system but also
should have applicability to analyzing system dynamic behavior.
In the present study Greitzer model (developed in 1976) was applied to describe dynamic behavior of centrifugal
compressors. Although this model was basically proposed for the case of axial compressors, founded acceptable results
Pinsely (1989) and Willems (1991) demonstrated applicability of the model to centrifugal compressors[6, 7].
As can be seen in Figure 4, the compressor system is modeled as integrated with a tube (duct) of Lc length through which
the compressed air is discharged into a voluminous chamber. This volume is generally known as Plenum in engineering
literature. Compressor throttle is also modeled as a duct of Lt length through which the compressed air is guided toward
compressor outlet.




                                              Figure 4 Greitzer system model
In fact, the actuator disc inside the compressor duct acts as the major model of compressor and directs continuous flow
current. The following assumptions are considered in design of Greitzer model:
      The flow current passing through the ducts is unidimensional and incompressible.
      The pressure is not distributed evenly inside the volume and the velocity of gas is negligible.
      The ratio of outgoing gas temperature to that of ingoing gas.
      The effect of rotor speed variations on system behavior is negligible[8 - 10].

    3. DIMENSIONLESS MODE EQUATIONS
The occurrence of surge phenomenon in centrifugal compressors is usually accompanied by certain features among which
the followings are the most common:
      Rapid reversal of compressor output flow of the order of milliseconds.
      Severe fluctuations in pressure.
      Rapid increase in temperature of both passing gas and compressor interior.
      Sever vibrations which may lead to compressor slip.
      Excessive noise.
Since pressure fluctuations and reversal of flow are dominant characteristics of surge incident, proposed equations by
Greitzer are in fact a set of dimensionless equations based on dimensionless pressure and dimensionless flow of the
system. The parameters which are involved in system dynamics are indicated in Table 1.


Volume 2, Issue 11, November 2013                                                                           Page 410
 International Journal of Application or Innovation in Engineering & Management (IJAIEM)
        Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 11, November 2013                                       ISSN 2319 - 4847


                                         Table 1: Parameters involved in system dynamics
                         Dimension                                     Concept                       Symbol
                    Kilograms per second                               Mass flow                      m
                        Pascal                                    Pressure difference                 P
               Kilograms per cubic meter                              Air density                     a
                      Meter per second                               Rotor velocity                   Ut
                        Cubic meter                               Compressor airspace                 Vp
                          Second                                       Time                            t
                          Meter                                 Compressor tube length                 Lc
                           Meter                                     Throttle length                   Lt
                       Square meter                         Surface area of compressor tube            Ac
                       Square meter                              Surface area of throttle              At
                             -                           Compressor dimensionless mass flow           c
                             -                              Throttle dimensionless mass flow          t
                             -                         Increase in volume dimensionless pressure       ψ
                             -                       Increase in compressor dimensionless pressure    c
Taking into consideration all parameters involved in dynamics of system and performing a dimensional analysis, we can
obtain desired dimensionless parameters required for dynamic analysis of the system. Dimensional analysis results in
dimensionless flow (Ø) and dimensionless pressure (ψ) which are expressed by relations (1) and (2).

            m
                                             (1)
         a AcU t
           P
                                             (2)
         1
            aU t2
         2
                                                        ~
For the purpose of obtaining dimensionless time( t )Greitzer founded the principle on the basis of Helmholtz frequency
( H ). Relations (3) and (4) represent the issue.

               Ac
  H  a                                       (3)
              V p Lc
   ~
   t  t H                                    (4)

Applying classic thermodynamic and energy laws we have:
Flow rate variations between two sections are proportional to pressure variations between the very two sections. Now, if
apply this law firstly between input and output of compressor and then between input and output of throttle relations (5)
and (6) are resulted governing compressor flow through and throttle flow through respectively[3, 11, 12].

   dc
        B c                             (5)
   d~
    t
   dt B
         C                              (6)
   d~ G
    t
where B represents Greitzer stability parameter and G represents Greitzer dimensionless parameter which are given by
equations (7) and (8).


Volume 2, Issue 11, November 2013                                                                             Page 411
 International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 11, November 2013                                       ISSN 2319 - 4847

         Ut
   B                                                     (7)
        2 H Lc

        Lt Ac
  G                                                    (8)
        Lc At

    4. DESIGN OF FUZZY LOGIC ACTIVE CONTROLLER
Introducing fuzzy sets and concept of membership degree in 1965, Dr. Lotfali Asgarzadeh founded fuzzy logic.
Fuzzy systems are based on knowledge or principles and their core is a database consisting of "if-then" rules. Fuzzy "if-
then" rule is an "if-then" expression the words of which are defined by membership functions. Fuzzy inference engine
integrate these rules in the form of a representation from fuzzy sets to fuzzy sets in output space based on fuzzy logic
principles. Fuzzy logic is used for design of expert systems. The expert systems in turn simulate real world laws. In this
section centrifugal compressor system is equipped with fuzzy logic active controller[6, 13].
The fuzzy controller has been designed using graphical relation of fuzzy logic toolbox. The method adopted for this
purpose is of Mamdani type. Figure 5 illustrates block diagram of Fuzzy Inference System (FIS).




                                      Figure 5 Fuzzy Inference System block diagram
The designed fuzzy logic active controller system has one input and output, i.e. the error is considered as input parameter
and the response to throttle excitation signal acts as output parameter. The membership functions of input and output are
defined as of triangular type as can be seen in Figure 6. These functions are defined by verbal variables: (MF1), (MF2),
(MF3), (MF4), (MF5), (MF6), (MF7), (MF8), (MF9), (MF10).




                                   Figure 6 Type of input and output membership functions
Totally 10 membership functions have been defined for input and output, thus, 10 inference rules are required.
Figure 7, illustrates these rules.
                                                                            Demux
                                                                    Rule

                                                                   Rul e1
                                   Demux
                                             Input MF                       Demux
                             1                                      Rule               max                         COA
                             In1
                                                                                                             Defuzzi ficati on1
                                                                   Rul e2

                                               surge                        Demux
                                                                    Rule            A ggMethod1

                                                                   Rul e3

                                                                            Demux
                                                                    Rule

                                           Output MF
                                                                   Rul e4                                             >                         1
                                                                            Demux                                                              Out1
                                                                                                         Zero Fi ring Strength?
                                                                    Rule
                                            khoroj i                                                     0
                                                                   Rul e5                T otal Firing                        -C-    Swi tch
                                                                                           Strength
                                                                            Demux                                         MidRange
                                                                    Rule

                                                                   Rul e6

                                                                            Demux
                                                                    Rule

                                                                   Rul e7

                                                                            Demux
                                                                    Rule

                                                                   Rul e8

                                                                            Demux
                                                                    Rule

                                                                   Rul e9

                                                                            Demux
                                                                    Rule

                                                                   Rule10



                                                          Figure 7 Fuzzy inference rules
Volume 2, Issue 11, November 2013                                                                                                                     Page 412
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As shown in Figure 7, the number of blocks equal to fuzzy inference rules and the lines from and to these blocks represent
relations associated with fuzzy logic rules. Input and output variables are depicted on left side and right side of the figure
respectively. Trial and error method has been used for obtaining both input extent and applied fuzzy rules. The level of
control for this system is illustrated in Figure 8.




                                             Figure 8 System control level
Also, the simulated model for centrifugal compressor system equipped with fuzzy logic active controller is illustrated in
Figure 9.




                                             Figure 9 Simulated System Model

    5. THE OBTAINED RESULTS FROM SIMULATION OF CENTRIFUGAL COMPRESSOR SYSTEM
        WITHOUT SURGE CONTROLLER SYSTEM
Since the applied model in this study involves Greitzer compressional system as desired dynamic model for simulation
and considering the fact that Greitzer has used dimensionless parameters of pressure and flow in mathematical model, the
provided diagrams in the present study are presented on the basis of dimensionless parameters of pressure and flow.
It is normal that the resultant diagrams form simulation performed in sections 5 and 6 are associated with five sample
turns: 16000, 18000, 21000, 23000, and 25000 Rounds Per Minute(RPM). Each set of diagrams includes dimensionless
flow fluctuations in terms of time, dimensionless pressure in terms of time, and dimensionless pressure in terms of
dimensionless flow. Simulation time was defined as a five-second duration.
 Figures 10, 11, and 12 show the results observed in simulation for sample turn 16000 Rounds Per Minute(RPM).




                                  Figure 10 Dimensionless flow diagram in terms of time



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                             Figure 11 Dimensionless pressure diagram in terms of time




                      Figure 12 Dimensionless pressure diagram in terms of dimensionless flow

Figures 13, 14, and 15 show the results observed in simulation for sample turn 18000 Rounds Per Minute(RPM).




                               Figure 13 Dimensionless flow diagram in terms of time




                             Figure 14 Dimensionless pressure diagram in terms of time




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                      Figure 15 Dimensionless pressure diagram in terms of dimensionless flow

Figures 16, 17, and 18 show the results observed in simulation for sample turn 21000 Rounds Per Minute(RPM).




                               Figure 16 Dimensionless flow diagram in terms of time




                             Figure 17 Dimensionless pressure diagram in terms of time




                      Figure 18 Dimensionless pressure diagram in terms of dimensionless flow

Figures 19, 20, and 21 show the results observed for sample turn 23000 Rounds Per Minute(RPM).
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                               Figure 19 Dimensionless flow diagram in terms of time




                             Figure 20 Dimensionless pressure diagram in terms of time




                      Figure 21 Dimensionless pressure diagram in terms of dimensionless flow

Figures 22, 23, and 24 show the results observed for sample turn 25000 Rounds Per Minute(RPM).




                               Figure 22 Dimensionless flow diagram in terms of time


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                              Figure 23 Dimensionless pressure diagram in terms of time




                       Figure 24 Dimensionless pressure diagram in terms of dimensionless flow
As can be observed, centrifugal compressor system experiences surge phenomenon when there is no surge control system
incorporated into it.

    6. THE OBTAINED RESULTS FROM SIMULATION OF CENTRIFUGAL COMPRESSOR SYSTEM
        EQUIPPED WITH SURGE CONTROLLER SYSTEM
As discussed in previous section, each set of diagrams includes dimensionless flow fluctuations in terms of time,
dimensionless pressure in terms of time, and dimensionless pressure in terms of dimensionless flow, and Simulation time
was defined as a five-second duration.
The observed results for simulation of centrifugal compressor system equipped with fuzzy logic active controller of surge
are indicated for the cases of five sample turns: 16000, 18000, 21000, 23000, and 25000 Rounds Per Minute(RPM).
Figures 25, 26, and 27 show the results observed for sample turn 16000 Rounds Per Minute(RPM).




                                Figure 25 Dimensionless flow diagram in terms of time




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                             Figure 26 Dimensionless pressure diagram in terms of time




                        Figure 27 Dimensionless pressure diagram in terms of dimensionless flow
Figures 28, 29, and 30 show the results observed for sample turn 18000 Rounds Per Minute(RPM).




                               Figure 28 Dimensionless flow diagram in terms of time




                             Figure 29 Dimensionless pressure diagram in terms of time


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                        Figure 30 Dimensionless pressure diagram in terms of dimensionless flow
Figures 31, 32, and 33 show the results observed for sample turn 21000 Rounds Per Minute(RPM).




                               Figure 31 Dimensionless flow diagram in terms of time




                             Figure 32 Dimensionless pressure diagram in terms of time




                      Figure 33 Dimensionless pressure diagram in terms of dimensionless flow

Figures 34, 35, and 36 show the results observed for sample turn 23000 Rounds Per Minute(RPM).

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                               Figure 34 Dimensionless flow diagram in terms of time




                             Figure 35 Dimensionless pressure diagram in terms of time




                      Figure 36 Dimensionless pressure diagram in terms of dimensionless flow

Figures 37, 38, and 39 show the results observed for sample turn 25000 Rounds Per Minute(RPM).




                               Figure 37 Dimensionless flow diagram in terms of time



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                               Figure 38 Dimensionless pressure diagram in terms of time




                        Figure 39 Dimensionless pressure diagram in terms of dimensionless flow

As can be seen, incorporating a fuzzy logic active controller of surge into the centrifugal compressor system, the
occurrence of surge phenomenon is prevented effectively.

    7. CONCLUSION
In the present study the surge phenomenon is introduced as the most common and most critical instability in centrifugal
compressors incident of which may leads to decreases in the system efficiency and certain failures in its mechanical
structure. Greitzer model was investigated in dealing with simulation of compressor dynamic behavior, furthermore
dimensionless equations governing system dynamics were derived on the basis of this model and considering the
dimensional analysis. Then the focus of study concentrated on evaluating the simulation of centrifugal compressor system
in two conditions (without controller and with fuzzy logic active controller). The obtained results for five sample turns
(16000, 18000, 21000, 23000, 25000 (RPM)) were presented in sections 5 and 6. It should be noted that the criterion
defined for assessment and comparison was stability of compressor.
Considering what was observed from this comparison we can conclude that under without-controller condition, system
endures a complete surge and doesn't reach stability. On contrary, when a fuzzy logic active controller is incorporated
into centrifugal compressor the occurrence of surge is prevented and system keeps its stability.
Finally, considering 5 proposed sample turns it can be obviously concluded that when centrifugal compressor system is
equipped with fuzzy logic active controller the surge phenomenon can be easily prevented from occurring, whereas when
centrifugal compressor system lacks such controller it will experience surge and cannot gain its stability. Therefore, these
evidences confirm the importance and capability of fuzzy logic controller in preventing the occurrence of surge in
centrifugal compressors, because severe fluctuations in pressure due to surge phenomenon may lead to continuous
reciprocating displacement of rotor and consequently imposing additional loads on bearings particularly on axial load
bearings. Depending on compressor rotation speed and bearing quality, after several times the surge phenomenon can
cause failures in axial load bearing.

References
 [1] A.R. Davarinia, "Evaluating Surge control Systems in Centrifugal Compressors," Iranian National Gas Corporation
      Publications, 2011.



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        Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
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[2] D. Gysling, D. Dugundji, E. Greitzer, A. Epstein, "Dynamic Control of Centrifugal Compressor Surge Using
     Tailored Structures, " ASME J. Turbomachinery, 113(4), 710-722,1991.
[3] J. Pinsley, G. Guenette, A. Epstein, E. Greitzer, "Active Stabilization of Centrifugal Compressor Surge, " ASME J.
     Turbomachinery, 113(4), 723 – 732, 1991.
[4] K.K. Botros, J.F. Henderson, "Developments in Centrifugal Compressor Surge Control – A Technology
     Assessment," Transactions of the ASME Journal of Turbomachinery, April 1994.
[5] M.H. White, "Surge Control for Centrifugal Compressors," Chemical Engineering, Dec. 1992.
[6] A. Ashrafizadeh, "Surge Active Control through Fuzzy Logic in Axial Flow Compressors," Annual Conference on
     Mechanical Engineering, Eighteenth Session, P 1, Sahrif University of Thechnology, may 2010.
[7] B. Ribi, G. Gyarmathy, "The Behaviour of a Centrifugal Compressor Stage During Mild Surge," VID
     Berichte,(1186), 1995.
[8] F. Willems, "Modeling and Control of Compressor Flow Instabilities," Eindhoven University of Technology,      pp.
     96-151, 1997.
[9] F. Willems, B. De Jager, "Modeling and Control of Compressor Flow Instabilities," IEEE Control Systems,       pp.
     8-18, 1999.
[10] K. Hansen, P. JØrgensen, P. Larsen, "Experimental and Theoretical Study of Surge in a Small Centrifugal
     Compressor, " ASME J. Fluids Engineering, 103(3), 391-395,1981.
[11] R.E. Sonntag, C. Borgnakke, G.J. Van, Wylen, "Fundamentals Thermodynamics," Sixth Edition, April, 2002.
[12] F. Willems, B, De Jager, "One – Sided Control of Surge in a Centrifugal Compressor System," In Proc. of the
     ASME Turbo Expo, Munich, Germany. Accepted for Publication, 2000.
[13] M.S. Taheri, "An Introduction to Fuzzy Sets Theory," Second Edition, Academic Jihad Publications, 1999.
[14] C. Meuleman, F. Willems, R. De Lange, B. De Jager, "Surge in a Low – Speed Radial Compressor ," ASME paper
     no. 98 – GT- 426,1998.

AUTHOR

Qasem Abdollah Nezhad was born in 1984, He received bachelor of Science in Mechanical Engineering from the
Department of Mechanical Engineering, Islamic Azad University, Tabriz branch, Iran, in 2011, and Master of Science
Mechatronics Engineering from the Department of Mechatronics Engineering, Science and Research Islamic Azad
University, Kurdistan branch, Iran, in 2013. Currently he is a lecturer, He is the author of several papers in National
Conference, His research interests includes robotics, mechatronics, image processing, control systems, fuzzy control
application.

Jafar Ghafouri He received a BSs degree in Mechanical Engineering filed of Solids Design from Tabriz University, Iran,
in 1999, and received MSs degree in Mechanical Engineering major field of Transformation of Energy from University of
Science & Technology, Tehran, Iran, in 2001. In 2008, he got a PhD in the same major from Science and Research
Islamic Azad University,Tehran Branch, Iran. He's now a member of Academic Board of Tabriz Azad University, iran,
and assistant professor at the same place. The specific fields of research in which he's interested and has presented several
articles include: internal combustion engines, combustion, heat transfer, CFD, smart systems and heat exchangers.

Mohammad Fathi He received a BSs degree in Biomedical Engineering from University of Shahid Beheshti, Tehran,
Iran, in 2000, and received MSs degree in Electrical Engineering major field of Communications from Amirkabir
University of Technology, Tehran, Iran, in 2002. Seven years later he got a PhD in the same major and from the same
university. He's now a member of Academic Board of Kurdistan University, Iran, and at the same time assistant professor
and registrar of Electrical Engineering major. His teaching career has been dealt with signal & system, communications,
engineering mathematics, communicational networks, and random processes. The special fields on which he has studied
and presented several articles include: wireless communications, optimizing communicational networks, smart systems
and smart electrical networks.




Volume 2, Issue 11, November 2013                                                                              Page 422

				
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