Blunier, Benjamin

M\u27Boua, Jeremie

Miraoui, Abdellatif

English

Purdue E-Pubs

Purdue UniversityPurdue e-PubsInternational Compressor Engineering Conference School of Mechanical Engineering2010Geometrical and Thermodynamic Model of aThree Lobes Compressor with Simulation andExperimental ValidationJeremie M'BouaUTBMBenjamin BlunierUTBMAbdellatif MiraouiUTBMFollow this and additional works at: https://docs.lib.purdue.edu/icecThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/Herrick/Events/orderlit.htmlM'Boua, Jeremie; Blunier, Benjamin; and Miraoui, Abdellatif, "Geometrical and Thermodynamic Model of a Three Lobes Compressorwith Simulation and Experimental Validation" (2010). International Compressor Engineering Conference. Paper 1951.https://docs.lib.purdue.edu/icec/1951 1153, Page 1  International Compressor Engineering Conference at Purdue, July 12-15, 2010  Geometrical and Thermodynamic Model of a Three Lobes Compressor with Simulation and Experimental Validation  Jérémie M’BOUA*, Benjamin BLUNIER, Abdellatif MIRAOUI Transport and Systems Laboratory (SeT) – EA 3317/UTBM University of Technology of Belfort-Montbéliard, France Contact Information: phone: +33 (0)3 84 58 35 08, Fax: +33 (0)3 84 58 34 13 E-mail: jeremy.m-boua@utbm.fr, benjamin.blunier@utbm.fr, abdellatif.miraoui@utbm.fr * Corresponding Author  ABSTRACT  This paper deals with the roots compressor, which is a rotary machine used for compressing air or refrigerant. It proposes an original way of describing the geometry of the three-lobe compressor in which the symmetries are exploited in order to establish the analytical expressions of volumes and leakages as a function of the orbiting angle. This approach allows the volumes and leakages to be described analytically without any special assumption. Based on the geometrical model, a thermodynamic model is built to predict pressures, temperatures of the different compressor chambers and massflows through the leakages between the chambers and at the compressor outlet and inlet. The simulation and experimentation results show that the model is able to predict these state variables but also the mechanical power and torque on the compressor shaft.  1. INTRODUCTION  This paper present a geometrical and thermodynamic model of the compressor three-lobe. The aim of the paper is to propose a new approach to describe the geometry of the three-lobe compressor where the expressions of the control volumes and leakage areas variations are described analytically as a function of the orbiting angle . This methodology has already been applied for the Scroll compressor and gave very good results (Blunier, et al., 2009).  A thermodynamic model of the control volumes and leakage areas is coupled to the geometrical one to predict the different state variables (i.e., pressures, temperatures, massflows, power, torque). The control volumes and the leakage areas components are defined to implement the thermodynamic model in a multi-domains simulation language known as VHDL-AMS described in section 4. The section 5 describes in details the test bench and the experimental procedure for the model validation. Finally, the simulation results of the compressor model are presented together with experimental results. The comparison of several simulation and experimental maps obtained shows good agreements between simulations and experimentations. This study will permit the modeling of the air supply system for fuel cell (PEMFC), (Blunier, et al., 2010) to be done and also the control strategy to be built to satisfy the fuel cell systems requirements .  2. GEOMETRICAL MODEL OF THE THREE-LOBE COMPRESSOR   2.1 Description of the chambers The aim of the geometrical description of the three-lobe compressor is to compute for any rotation angle , the control volumes and  the leakage areas between these control volumes:  The analytical expression of the control volumes: ;  The analytical expression of the leakage area: ; Where  is the orbiting angle of the rotor. The geometrical model will be coupled together with a thermodynamic model (see section 3) to compute the pressure, the temperature inside the control volumes and massflows through the leakages. For the three-lobe compressor four variables are used to describe the compressor geometry namely (see Figure 1):    The radius of the lobe r;  The diameter of suction and discharge sections ,  The orbiting angle ,  1153, Page 2  International Compressor Engineering Conference at Purdue, July 12-15, 2010  The compressor height .                                         Several models of volumetric compressors are based on the geometrical description (Blunier et al., 2006)  (Halm, 1997) which determines, according to the orbiting angle, the volumes and the leakages of the chambers.  The geometry, coupled with a thermodynamic model, allows the pressure, the temperature in each control volume of the compressor and the mass flow through the leakages to be computed. A chamber is defined as a portion of the compressor included between two rotors or between the rotor and the compressor casing.  It is important to know that only the chambers are described in this section. The control volumes, based on the chambers description and which will be used to compute the state variables, will be presented later.     As it can be seen on Figure 2, each chamber has three steps:  The air suction (i.e., an increasing chamber volume at the compressor inlet);  The air transfer (i.e., constant chamber volume),  The air discharge (i.e., an decreasing chamber volume at the compressor outlet).  The topology of the compressor allows to define six chambers instantaneously, which will be called respectively , , , , , . For example the evolution process of the chamber four is presented on Figure 2. All other chambers can be described according the same procedure. The analytical expression of the volume is computed using the symmetry of the lobe. One rotor is built from six concave parts and six convex parts as shown in Figure 1. For each orbiting angle, the volume of air between the casing and one convex part, and the volume of air between the casing and one concave. Based on this procedure, the chamber volumes can be computed analytically as a function of the orbiting angle. The evolution of the chambers volumes as a function of the orbiting angle is given in Figure 3. The complete mathematical model can be found in (M'boua, et al., 2010 (submitted))  2.2 Description of the control volumes and the compression process During a cycle, the orbiting angle  (describing the position of two rotors) varies from  to ; it can be considered that exactly 6 control volumes (different from chambers volumes) exists (See Figure 4):  One volume describing the air suction ( ),  One volume describing the air discharge ( ),  Two volumes describing the air transfer and discharge (  and ),  Two volumes describing the air suction and transfer (  and ). The number of control volumes is considered to be fixed and equal to exactly six during all the process, even if the pressures of the neighboring chambers are identical (in this case the two volumes could be considered as one single control volume). The compression process is considered as periodic, with a period of . One period can be described by the following steps:  1. In the first half period:  a. At the beginning of the process, that is , the leakage S1-3 between the control volumes  and  is at its maximal value, it is assumed that the pressure in these two volumes are identical, at this time the signal SL (L stands for left part of the compressor) (see Figure 5) is activated. b. When the signal SL is activated, the control volume  which is in a discharge process initializes its state variables ( ; ; ) with those of the volume , which is instantaneously connected to discharge pipe, and it initializes with those of the volume . The volume   initializes its state variables with the variables ( ; ; ) of volume  connected with suction pipe. Figure 1 Schematic view of three-lobe compressor.Concave part Convex part  1153, Page 3  International Compressor Engineering Conference at Purdue, July 12-15, 2010 c. The control volume  and  are now the discharge and suction control volumes, respectively. The control volume  starts the transfer process followed by the discharge process until the end of the first half period. The control volumes  and  which are respectively in the transfer and transfer-discharge process do not initialize during this period.       Figure 2: Evolution steps of the various chambers Figure 3: Chambers volumes as a function of the orbiting angle Figure 4: Presentation of control volumes  1153, Page 4  International Compressor Engineering Conference at Purdue, July 12-15, 2010 2. In the second half period: a. At this time , the signal “SR” is actived (R stands for left part of the compressor), the leakage S1-6 is at its maximal value. The control volumes  and  have the same kind reinitialization that  and  in the first half period. The volume  and  are still in the suction and discharge process, respectively. This representation will allow thereafter the dynamic behavior of the compressor to be simulated with a simulation language. As it has been previously explained, the topology of the circuit is fixed, that is the number of components and their interconnections do not change during the simulation. The compressor is represented by means of two elementary components (see Figure 5):  The control volumes represented by “rectangles” where the number of fluidic terminals can vary depending on their number of interconnection;  The leakages are assumed to be isentropic. They are represented by two opposed arcs, with two fluidic and two thermal terminals. The number of fluidic terminals can vary depending of their number of interconnection.  The leakages are assumed to be isentropic. Each control volume has two other signal inputs “left” and “right” which permit the variables  and  of each control volume to be initialized during the simulation.  2.3 Model of the leakages In a real compressor the pressures in different control volumes differ from the ideal case because the controls volumes are not completely hermetic. The leakages, due to internal clearances, increase the energy consumption of the compressor. Indeed, the gas which flows in the volume where the pressure is the highest through a volume where pressure is lower will have to be compressed again. According to the model, three kinds of leakage exist (See Figure 6): (1) Tangential leakages: the gas flows through the clearance which exists between the lobe and the casing because the contact point is not perfect; (2) Leakages between two control volumes: the leakage between the control volume “ ” and the control volume “ “ will be written ; (3) Radial leakages: the gas flows through the clearance  between the joint and along the surface of the compressor rotor. 3. THERMODYNAMICAL MODEL  3.1 Model of control volume component The model of a control volume is based on energy and mass conservation. If the kinetic energy is neglected, the internal energy in the control volume can be written, (Blunier et al., 2009):  Where  is the mass of the gas,  its specific heat at constant volume and , the temperature of the gas. The internal energy is changed due to (Gravensen, et al., 2001):  The work done by changes in the volumes:    The energy transported into and out of the chamber through its leakage areas. If it is assumed that the flow through the leakages is adiabatic, the energy inside the flow is  where  represents the enthalpy,  the mass flow where  is the specific heat of the gas at constant pressure.  Dissipation and energy losses due to the external environment, . The equation linking the pressure and the temperature of the control volume can be obtained from the following equation:  3.2 Model of the leakage component The nozzle flow equation (isentropic assumption) is used to calculate the leakages between the control volumes. It has to be noted that the suction and discharge processes are considered as leakages in the model. The mass flow through a leakage area is (Yanagisawa, et al., 1990): leakageair if   critleakage otherwise  1153, Page 5  International Compressor Engineering Conference at Purdue, July 12-15, 2010 Where  is the flow coefficient, leakage  the leakage area,  the pressure ratio. crit , and  is the upstream pressure.   Figure 5: Schematic representation of the control volumes and fluidic terminals 0 tfamb-dp T-J Tdpo--(-~· Tamb 0 t/dp-amb j T3o- Vdp, Tdp· Pdp f---oT• t!dp-3 Discharge pipe t/dp-6 P, T. 21 l ; 1 0 t/dp-1 r.,~) ( ... T,,~) (·-·· T,• o-\ (s.., T. J T3o- T1o-T, 0 t/3- dp T, v~-dP. ~r f~r 0 f'• ~r r <! r T3 T1 ~~ 1 err 0~ ~ j (l T1 T• fiL T, P, I I SL T3 p3 Sn T• P, I I SRTs Ps V3, T3, ?3 th-1 ~ tft-3 V1,Tt ,Pt tf1- 6 ~ tf6- 1 V6. T6, P, 0 ~T6~ T1~ Volume 3 -oT1 ~Tso-- Volume 1 Volume 6 PJ T, 1- 6 Hn p, To 2J l ;, 0 tfg_, <i i ). 0 tfl-2 <! t 6 /6-5 0 ~5 r,~) T, o-(·" T,~) T,o-(·· T,J Ts (~· T3 0 t/4-3 T1 th-1 tf5-0 f• ~~ 1 J I lJ ~~ I 2!1 o~r r (l T, T5 ~~ I ;1 L 1, 1'2 SL T4 p4 .>n 1 , it j j Sn T, 1', V4 ,T4, P4 tf,_, tj,_, \12, T2, P2 th- 5 ~ t/5- 2 V5. T5, fl> 0 0 0 Volume 4 -or, ~ T,o- Volume 2 -oT5 ~ T,o- Volume 5 p T. i'JT. p, T. 2J i ).. 0 t/4-sp 2J l J. 0 t/2- ><p ci l f5 - 6p 0 ~ T,p T,) ~··· T,l ~'" ThJ (IT•-• Tspc>-- Tspo-- T,p T, o tfap,'2 j T,o-V8 p, T~p· P~P --oTs t/;;p,4 Suction pipe tfsp; P .. T. 21 l ~ Tamb 0 t/sp,amb T,.•-) Tambo--(IT•-0 tfamb-sp  1153, Page 6  International Compressor Engineering Conference at Purdue, July 12-15, 2010    3.3 Model of the heat exchange with the environment  The evolution of the thermodynamic states of the gas through the compressor can be divided into three steps: Isobaric heating up in the suction pipe, isentropic compression, and isobaric cooling down in the discharge pipe. The shaft power  can be obtained using the power conservation equation:   Where  is the isentropic power,  is the mechanical losses of the compressor and , leakage represents the losses due to the air leakage and heat transfer between the air and the compressor walls. The mechanical losses  can be defined from two terms, losses proportional to the isentropic compression power and hydrodynamic mechanical losses as shown in the following formula (Winandy, et al., 2002):  Where  and  are positive parameters to be identified;  is a reference (arbitrary) rotational speed.  In general, the different heat transfers in a compressor include the one of the suction gas , the heating-up due to mechanical losses , the heat given by the high temperature discharge gas , and the heat flow to the environment . It is assumed that the definition of a fictitious wall uniform temperature  is sufficient to represent all the heat transfer modes mentioned above. The fictitious wall temperature , has no influence on the mass flow rate so that its value can be assumed equal to the mean of compressor surface temperature during all test 47 °C. The equations for the fictitious supply heat exchanger are:   Where,  The parameters  and  may be considered equals for similar gas from the point of view of thermo-physical properties (Winandy, et al., 2002).The ambient losses are given by: Figure 6 Different kinds of leakage existing in the model  1153, Page 7  International Compressor Engineering Conference at Purdue, July 12-15, 2010  The experimental test results are used to identify the parameters related to the heat power losses and transfers. (Duprez, et al., 2007).  4. SIMULATION AND EXPERIMENTATION  The compressor model has been implemented in simulation software which supports the VHDL-AMS simulation language. The IEEE 1076.1 language, informally known as VHDL-AMS, is a superset of IEEE Std 1076-1993 (VHDL) that provides capabilities for describing and simulating continuous time and mixed-signal systems with conservative and non conservative semantics for the continuous time portion of the system. The language supports many abstraction levels in electrical and nonelectrical energy domains.  The test bench has been used to validate the scroll compressor model presented in (Blunier, et al., 2009). It has been adapted to validate the three-lobes compressor (see Figure 7). The test bench has been automated with a Dspace system using Matlab/Simulink and a Python script. The tests have been done for several compressor speeds and valve positions, recording for each operating point in steady state, the average pipe pressure and temperature, the mass flow and the compressor mechanical (torque en speed) and electrical powers.      5. RESULTS   Figure 9 and Figure 10 show respectively the comparison between the experimental and the simulated mass flows as a function of the rotational speed for all operating points (pressures) and isentropic efficiency with speed. The simulation predicts relatively well the mass flow up to a speed of 13,000 rpm. It can see that it is possible to predict the behavior of the compressor by using the model. The model demonstrated thus good agreements with experimental profiles.             Figure 7 Photo of test bench 1153, Page 8  International Compressor Engineering Conference at Purdue, July 12-15, 2010   6. CONCLUSION  The model makes possible the investigation of the influence of leakages and compressor geometrical parameters on the compressor performance and can be used to improve the compressor's design. The comparison between the simulation and the experimentation maps shows good agreements. The analysis of the simulation (or experimentation) results allows the design of an adapted motor to be done. The motor will be coupled to the compressor together with its associated power converter electronic systems.  REFERENCES Blunier Benjamin et Miraoui Abdelatif A new analytical dynamical model of a scroll compressor with experimentale validation [Revue] // International journal of refrigeration. - 2009. - pp. 874-891. Blunier Benjamin et Miraoui Abdellatif Proton Exchange Membrane Fuel Cell Air Management in Automotive Applications [Revue] // Journal of Fuel Cell Science and Technology . - 2010. - pp. Volume 7, (11 pages) . Blunier Benjamin et Miraoui Abdellatif Proton Exchange Membrane Fuel Cell Air Management in Automotive Applications [Revue] // Journal of Fuel Cell Science and Technology . - August 2010. - pp. Volume 7, (11 pages) . Duprez M.E., Dumont E. et Frère M. Modeling of reciprocating and scroll compressors. [Revue] // International Jornal of Refrigeration . - 2007. - pp. 873 -886. Gravensen J. et Henriksen C. The geometry of the scroll compressor [Revue] // Society for Industrial and Applied Mathematics . - 2001. - pp. 113-126. Halm N. P. Mathematical modeling of scroll compressor [Conférence] // Purdue University. - Purdue : [s.n.], 1997. M'boua Jeremie, Blunier Benjamin et Miraoui Abdellatif A new analytical model of a three-lobes compressor Geometrical model, thermal model, simulation and experimental validation. [Revue] // International journal of refrigeration. - 2010 (submitted). Winandy E., O C S. et Lebrun J. Simplified modeling of an open-type reciprocating compressor [Revue] // International . Journal of Thermal Sciences. - 2002. - pp. 183 -192. Yanagisawa T., Cheng M., Fukuta, M. et Shimizu T. Optimum operating pressure ratio for scroll compressor [Conférence] // Proceeding of the international compressor engineering conference. - at Purdue : [s.n.], 1990. - Vol. 425-433.   Figure 8 Experiment and Simulated iso-power maps Figure 10 Comparison between experimental and simulated   variation of isentropic efficiency with speed. Figure 9 Comparison between experimental and simulated mass flow 

Geometrical and Thermodynamic Model of a Three Lobes Compressor with Simulation and Experimental Validation

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Geometrical and Thermodynamic Model of a Three Lobes Compressor with Simulation and Experimental Validation

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