Hikichi, Takumi

Kinjo, Kenji

Morinishi, Koji

Nakano, Akira

English

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Purdue UniversityPurdue e-PubsInternational Compressor Engineering Conference School of Mechanical Engineering2010Study of CFD Considering Valve Behavior inReciprocating CompressorKenji KinjoAppliances Development CenterAkira NakanoAppliances Development CenterTakumi HikichiAppliances Development CenterKoji MorinishiDept. of Mechanical & System EngineeringFollow 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.htmlKinjo, Kenji; Nakano, Akira; Hikichi, Takumi; and Morinishi, Koji, "Study of CFD Considering Valve Behavior in ReciprocatingCompressor" (2010). International Compressor Engineering Conference. Paper 1975.https://docs.lib.purdue.edu/icec/1975 1256, Page 1 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Study of CFD Considering Valve Behavior in Reciprocating Compressor  Kenji KINJO1*, Akira NAKANO1, Takumi HIKICHI1, Koji MORINISHI21Appliances Development Center, Corporate Engineering Division, Home Appliances Company, Panasonic Corporation 2-3-1-2 Noji-higashi, Kusatsu-shi, Shiga 525-8555, Japan Phone: +81-50-3688-7417, Fax: +81-77-563-1967, E-mail: kinjo.kenji@jp.panasonic.com2Dept. of Mechanical & System Engineering, Kyoto Institute of Technology Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan *Corresponding Author ABSTRACT It is necessary to understand the behavior of the gas flows in the compression chamber generated by the driven piston, and the behavior of the reed valve that are opened and shut by the fluid force, when we reduce the loss of suction and discharge of the reciprocating compressor.  To understand the above phenomenon, the moving object such as piston and reed valves is analyzed by applying virtual flux method and Cartesian grid that is suitable for expressing complex shape as main grid in this study. In this paper, first, we introduce our numerical analysis method.  Next, we introduce the verification by the comparison result between the above-mentioned analysis and the experiment. 1. INTRODUCTION In recent years, efforts aimed at energy saving have been accelerating throughout the world from the perspective of conservation of the global environment.  Including the Asian countries, where economic growth is marked, and the European countries, which are working positively on energy saving, energy saving regulations are becoming strict year by year.  This trend has accelerated the energy saving competition among refrigerator manufacturers in domestic market and it will be necessary to develop higher efficiency reciprocating compressors in the future. In such situation, the behavior of refrigerant in the compressor is analyzed by CFD (Computational Fluid Dynamics) to make concrete idea on the design, to minimize the test expenses and to accelerate the development when the reciprocating compressor is developed for the household refrigerator. In the reciprocating compressor, the suction reed valve and the discharge reed valve are opened and shut by the fluid force of refrigerant compressed by piston in the compression chamber.  Therefore, it is important for the improvement of performance of the reciprocating compressor to calculate the behavior of the refrigerant in the compressor with behavior of the reed valve accurately. Recently, it begins to be used FSI (Fluid Structure Interaction) analysis that considers the behavior of suction reed valve and discharge reed valve.  However, a general method of FSI analysis• •has some demerit that the mesh around the object is transformed and deteriorated. Then, in this paper, it introduces following analysis method to solve the above problem.  The three-dimensional compressible Navier-Stokes equations are used as the governing equations.  The third-order weighted ENO (Essentially Non-Oscillatory) method is used for spatial discretization.  The second-order Runge-Kutta method is used for time stepping.  In addition, by applying virtual flux method to the moving objects, the flow driven by the piston of reciprocating compressor are obtained successfully on a single Cartesian grid.  1256, Page 2 International Compressor Engineering Conference at Purdue, July 12-15, 2010 2. CONSTRUCTION OF RECIPROCATING COMPRESSOR  The cross-section of a conventional reciprocating compressor is shown in Figure 1, and the• •basic construction of the suction and discharge paths is shown in Figure 2. The piston and cylinder• •mechanism is located in the upper side of the hermetic casing, while the motor is placed in the• •lower side. When the piston of this compressor moves toward bottom dead center (suction stroke),• •the pressure in the compression chamber reduces. Then the suction reed valve opens and refrigerant in the hermetic• •casing is guided into the compression chamber through the suction muffler, valve plate and• •suction reed valve. When the piston moves toward top dead center (compression stroke), the• •refrigerant in the compression chamber is compressed. Then the discharge reed valve opens and the refrigerant in the compression chamber flows out to the refrigeration system through the valve plate, discharge reed valve and cylinder head.   3. NUMERICAL PROCEDURE  Obtaining numerical solution of the compressible Navier-Stokes equations for flows about complex geometries on a Cartesian grid, the virtual flux method is introduced in conventional spatial discretizing methods.  3.1 Governing equations Fluid flows in the reciprocating compressors may be described by the compressible Navier-Stokes equations, which can be written in the following form.             SFqt     (1) The conservative vector and the flux vectors are given with: euq      (2) Figure 1: Schematic diagram of                 a reciprocating compressor Figure 2: Close-up of suction  and discharge paths Hermetic casing Motor Cylinder blockCrankshaft Connecting rod Cylinder head Suction mufflerSuction reed valve Valve plate PistonCompression chamber Discharge reed valve 1256, Page 3 International Compressor Engineering Conference at Purdue, July 12-15, 2010              pepuIuuF 2      (3)                huS0     (4) The total energy e is defined for a perfect gas as:              2211upe      (5) The components of the viscous stress tensor and the heat flux vector, for example, xx , xy , and hx, may be written as:            zwyvxuxuxx 322     (6)            xvyuxy      (7)                 xThx      (8) In this study, Smagorinsky's eddy viscosity model is used for practical turbulent flow simulations. 3.2 Spatial discretizing method The convective terms of the compressible Navier-Stokes equations can be evaluated simply at a regular point on the Cartesian grid, for example, as:                 xxiixiixix 2/12/12/12/1~,~~,~ qqFqqFF   (9) where i is the cell index as shown in Figure 3.  The numerical flux at the cell surface is obtained as:               2/12/12/12/12/12/1~~~~21~,~ iixixixiix qqAqFqFqqF   (10) where xA  is defined as:            1xxxx RRA      (11) Figure 3: Stencil about a regular point  1256, Page 4 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Here x  and Rx are, respectively, the eigenvalue matrix and the right eigenvector matrix of the flux Jacobian matrix Ax, which is defined as:                  qFA ~xx      (12) The third order method may be obtained, for example, if 2/1~iq  are evaluated with the following reconstruction.         2/112/102/1~~21~~iiii qqqq     (13) Here 2/1~iq  are obtained with:            iii qqq~~~12/1      (14) The weights 0  and 1  for the third order linear method are:             31,3210      (15) The weights 0  and 1  are defined for a nonlinear method as:              10111000 2,22     (16) where 0  and 1  are obtained as:          2/112/10 ~1,~1ii qq    (17) Here  is a small number which prevents null division in smooth flow regions. The viscous terms of the Navier-Stokes equations are evaluated simply with the second order central difference approximation.  For example,          xxiixiixix qqSqqSS~,~~,~ 11     (18) 3.3 Virtual flux method If an immersed solid boundary is located between the points i and i+1 as shown in Figure 4, the numerical fluxes of Equation (9) must be modified so that no-slip and no-penetration velocity boundary conditions are satisfied on the solid boundary.             xxiixiixix2/12/1*2/1*2/1* ~,~~,~ qqFqqFF    (19) where 2/1*~ iq , 2/1*~ iq , and 2/1*~ iq  are reconstructed with considering the immersed solid boundary conditions.  For example,        2/112/1*02/1* ~~21~~iiii qqqq     (20)  1256, Page 5 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Figure 4: Stencil about a interfacial point Figure 5: Stencil for one-sided extrapolation where 2/1*~ iq  calculated with:          iii qqq~~~1*2/1*      (21) Here 1*~ iq  are obtained with one-sided first or second order extrapolating operators L as shown in Figure 5:         IBiii L qqqq~,~,~~ 11*      (22) or                 IBiii xL qqqq~,~,~~ 11*     (23) where IBq~  and IBxq~  are Dirichlet and Neumann boundary conditions at the immersed boundary, respectively.  The numerical viscous fluxes of Equation (18) must be also modified so that no-slip and no-penetration velocity boundary conditions are satisfied on the solid boundary as:           xxiixiixix qqSqqSS~,~~,~ 11*    (24) 3.4 Time stepping method After discretizing the spatial derivatives, the Navier-Stokes equations can be written in the form:                  qQqt     (25) The second order Runge-Kutta method is used for solving the equation as:  1256, Page 6 International Compressor Engineering Conference at Purdue, July 12-15, 2010 nnn t qQqq212/1      (26) 2/11 nnn t qQqq      (27) where the superscript n denotes the time index and t  the time step size.  4. APPLICATION TO RECIPROCATING COMPRESSOR  In a reciprocating compressor, the suction and discharge reed valves are driven by the pressure force acting on them, while the piston is driven at a constant rate.  Here the fluid-structure coupling simulation is carried out to reproduce the reed valve response and gas flows in the reciprocating compressor. 4.1 Reciprocating compressor model The computational model of reciprocating compressor is drawn in Figure 6 schematically.  The assumed motion of suction reed is described with the following equation of motion.             MkdtdI s22     (28) On the other hand, the assumed motion of discharge reed valve is described with the following equation of motion, because it thinks that it is opening and shutting a discharge reed valve to the discharge port as level approximately at the actual reciprocating compressor.             Fkdtdm d22     (29) where spring stiffness of reed valve ks and kd gave a measured value. stroke Hsuctionreed valvedischargereed valvepistonplenumchamberpdp• •DFigure 6: Shematic model of a reciprocating compressor 4.2 Comparison between the simulation result and the experimental result The computation is carried out for the piston diameter D of 20.6mm and stroke H of 18mm.  The piston is operated at the constant revolution rate of 3000 rpm.  The pressure ratio of discharge pressure pd to suction pressure p  is 10. Typical velocity vectors, colored by the pressure obtained for suction and discharge strokes, are plotted in Figure 7 and 8, respectively.  Fresh gases flow into the compression chamber through the suction port (Figure 7), and the gases are compressed and driven out through discharge port (Figure 8).  The work of reciprocating compressor is clearly reproduced in the simulation.  1256, Page 7 International Compressor Engineering Conference at Purdue, July 12-15, 2010              (a)         (b) Figure 7: Velocity vectors at an instant in suction stroke            (a)         (b) Figure 8: Velocity vectors at an instant in discharge stroke 024681012140 30 60 90 120 150 180 210 240 270 300 330 360Crank Angle [deg]Pressure ratio [-]ExperimentSimulationFigure 9: Comparison between simulation and experiment of pressure in compression chamber Next, to verify the validity of the fluid-structure coupling simulation result, it evaluated in the comparison with the simulation result and the experimental result about the pressure variation in the compression chamber as shown in Figure 9.  1256, Page 8 International Compressor Engineering Conference at Purdue, July 12-15, 2010 Figure 9 shows that the pressure variation in compression chamber with suction and compression stroke agrees about the simulation result and the experimental result, being good comparatively.  However, as for the simulation result, it shows that overpressure with discharge stroke is bigger than the experimental result, too. Probably, it thinks that the modeling of the discharge reed valve has a problem and that the above problem occurs with it and it is necessary to improve in the future. 5. CONCLUSIONS The virtual flux method has been developed for simulating fluid-structure interaction problems.  The numerical flux across the fluid-structure interface is successfully replaced with the virtual flux, so that proper interface conditions are satisfied there.  In this study, the virtual flux method is applied to numerical flow simulations about reciprocating compressors.  And it gives the fluttering reed valves of reciprocating compressor driven by the pressure force acting on them.  After the first transition cycle, the compressor comes to work at almost constant flow rate.  However, when comparing the simulation result and the experimental result, the accuracy for this simulation still has a problem.  In the future, it solves the problem and then it utilizes for the development.  NOMENCLATURE e Total energy per unit volume    F Convective flux vector F Force acting on reed valve    h Heat flux vector I Moment of inertia    k Spring stiffness M Moment of force acting on reed valve  m Mass p Pressure      q Conservative vector q~  Primitive variable    S Viscous flux vector t  Time step size     T Temperature u Velocity vector     u Velocity component v Velocity component    w Velocity component x  Mesh spacing      Ratio of specific heats  Open angle of reed valve     Thermal conductivity  Open distance of reed valve    Viscosity  Density       Viscous stress tensor REFERENCES  Jiang, G. S., Shu, C. W., 1996, Efficient Implementation of Weighted ENO Schemes, Journal of Computational Physics, vol. 126: p. 202-228. Kim, J., Kim, D., Choi, H., 2001, An Immersed-Boundary Finite-Volume Method for Simulations of Flow in Complex Geometries, Journal of Computational Physics, vol. 171: p. 132-150. Kim, J., Wang, S., Park, S., Ryu, K., La, J., 2006, Valve Dynamic Analysis of a Hermetic Reciprocating Compressor, International Compressor Engineering Conference at Purdue, C107.Le, D. V., Khoo, B.C., Peraire, J., 2006, An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, Journal of Computational Physics, vol. 220: p. 109-138. Nakano, A., Kinjo, K., 2008, CFD Applications for Development of Reciprocating Compressor, International Compressor Engineering Conference at Purdue, 1326. Peskin, C. S., 1977, Numerical analysis of blood flow in the heart, Journal of Computational Physics, vol. 25:   p. 220-243. Tanno, I., Morinishi, K., Matsuno, K., Nishida, N., Validation of Virtual Flux Method for Forced Convection Flow, 2006, JSME International Journal Series B, vol. 49, no. 4: p. 1141-1148. Udaykumar, H. S., Mittal, R., Rampunggoon, P., Khanna, A., 2001, A Sharp Interface Cartesian Grid Method for Simulating Flows with Complex Moving Boundaries, Journal of Computational Physics, vol. 174: p. 345-380. 

Study of CFD Considering Valve Behavior in Reciprocating Compressor

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Study of CFD Considering Valve Behavior in Reciprocating Compressor

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