Modelling And Analysis of Linear Compressor Madhura Sevekari; Prof.Suneeta Phadkule; Dr. Virendra Bhojwani, Prof.Amit Jomde; Dr. Suhas Deshmukh 1 Lecturer, 2,3,4,5 Professor 1 Dept of mechanical engg, JSP, pune 2,3 Dept of mechanical engg JSCOE, pune 4,5 Dept of mechanical engg SAE, Kondhwa Pune mgs_madhura8@yahoo.com, suneetaphadkule@yahoo.com, bhojwanivk@gmail.com, ammit.jomde@gmail.com, suhas.deshmukh@gmail.com Abstract The present paper discusses the mathematical model for linear compressor [1]. The mathematical model deals with the dynamic equation of linear compressor consisting of inertia, motor force, damping. It also solves the electrical circuit equation to estimate the current flowing through the motor circuit. The modeling considers dynamic operation of suction and discharge valves. The refrigerant properties are estimated by interfacing the program with standard refrigerant property software to predict the cylinder instantaneous pressure and temperature. Finally the code predicts the net cooling capacity, compressor input power and COP for the given compressor geometry. Results generated by the mathematical model are presented. A linear compressor consists of an oscillating motor and a piston rigidly coupled to it. Oscillations of the linear motor are directly transferred to the piston. Piston performs compression and suction alternately similar to the reciprocating compressor. Since the crank and the connecting rod mechanism are absent in the linear compressor the friction losses are minimum and mechanical efficiency of linear compressor is maximum. Linear compressor is one of the highest available efficiency compression technologies. Keywords Mathematical model, Linear compressor 1. INTRODUCTION Linear compressor: Linear compressor is a gas compressor where the piston moves along a linear track. The linear compressor is driven by a linear motor directly coupled with a piston and springs for resonant operation. In a linear compressor, a resonant spring is used to obtain a piston stroke with small thrust of a linear motor. Since there is no conversion of rotary to linear motion, all the forces of the linear compressor act along a single axis i.e. the axis of piston motion. This operation along a single axis and direct coupling between the motor and piston generates minimal side loads that prevents contact between the piston and cylinder and hence reduces wear and tear of piston. This characteristic of very low side load & vibrations makes this machine very silent in operation (Generates less than half of the db noise as compared to reciprocating compressor). Figure 1: Cut-section showing details of linear compressor [2] Advantages of linear compressor: - Silent operation - No mechanical linkages (crank, connecting rod) - Minimum friction loss. - Easy part load operation & modulation. 2. MATHEMATICAL MODEL [1] The mathematical model considers free body diagram (Refer Figure 2) of piston to analyze various forces acting on the piston. Forces acting on the piston are due to mechanical springs, including the gas effects and the electrodynamics driving force. Mass of the piston includes a portion of the spring mass and the driving coil mass. www.ijmca.org Page 156
K e x C e l e is the effective length of coil wire cutting the flux at a given instant in (m). I is current in (Amp) passing through the circuit. m x The circuit equation is: B e l e + R e I + L e = V 3 F Figure 2: Free Body Diagram of Compressor Piston [1] 3. EQUATIONS CONSIDERED FOR DEVELOPING THE MODEL [1] L e is the inductance of coil, Henry. Re is the resistance of the coil, Ohm. C. Refrigerant mass flow rate through valves: The mass flow rate of refrigerant from the cylinder and to the cylinder through discharge valve and suction valve respectively, can be assumed as mass flow through nozzle. Equations required describing the linear compressor characteristics are: A. Piston Dynamics B. Motor Equation C. Mass flow estimation D. Cylinder gas temperature estimation A. Piston Dynamic equation: K e is the equivalent spring constant, C e is the viscous damping constant and M e ' is the resonating mass of the system. The equation of motion that governs mechanical part of the system is, therefore, given by: M e + C e + K e x = F..1 K e = K P + Kg K P = Mechanical spring stiffness K g = Gas spring stiffness And Motor Force, F = B e l e I..2 B. Motor Equation: Electrical circuit for linear motor (Refer Figure 3): Figure 3: Electrical circuit for linear motor [1] B e is the effective magnetic flux in (W b/m 2 ) that acts on coil winding. Figure 4: Mass flow through suction and discharge valve [ 1] P 1 = Pressure inside cylinder, bar 1 = Density of the refrigerant inside the cylinder, kg/m 3 T 1 = Refrigerant gas temperature inside the cylinder, K P 2 = Pressure outside cylinder, bar 2 = Density of the refrigerant outside the cylinder, kg/m 3 T 2 = Refrigerant gas temperature outside the cylinder, K The compressor cylinder at any instant will satisfy any one of the three conditions viz. both the valves closed, discharge valve open or suction valve open. The same has been modeled in the program as follows, CASE 1: Both the valves closed P cylinder < P discharge and P suction < P cylinder When both the valves are closed, then the mass flow into and out of the cylinder is zero or mass inside the cylinder remains constant i.e. m in = 0, m out = 0 CASE 2: P cylinder Discharge valve open > P discharge When the discharge valve get open then mass is thrown out of the cylinder. Then, m out = Gas density *A valve * valve lift Then the new mass inside the cylinder is, M cyl = M cyl - m out www.ijmca.org Page 157
CASE 3: Suction Valve open P cylinder < P suction When the suction valve get open then mass enters inside the cylinder. Then, m in = Gas density *A valve * valve lift Then the total mass inside the cylinder can be, M cyl = M cyl + m in D. Cylinder gas temperature estimation: Temperature variation at every instant of piston position is derived with the assumption of real gas. General form of the energy equation for an open system is, de Q W ( e pvˆ) inm in ( e pvˆ dt ) out m de/dt =Total energy transaction w.r.t time during the suction / discharge of the mass, W. Q =Heat transfer through the cylinder wall, W W = Work supplied to the compressor, W. e in/out =internal energy of the gas entering or leaving the cylinder, J/kg. pv in/out = flow work of the gas entering or leaving the cylinder, J/kg. m in =mass of the refrigerant entering the cylinder through suction valve, kg/s m out =mass of the refrigerant leaving the cylinder through discharge valve, kg/s 4.PROGRAMME FOR MATHEMATICAL MODELING OF LINEAR COMPRESSOR a. Define the input parameters- Motor parameters Be-magnetic flux, Tesla le-length of coil cutting the flux, m L- inductance of the coil, Henry RT-total resistance, Ohm Freq-supply frequency, Hz Compressor parameters Ke-spring stiffness, N/m Me-mass of piston, kg Zeta-damping coefficient Dcyl-diameter of cylinder, m Stroke of piston, m Thermal Parameters- Psuc-suction pressure, Pa Pdis-discharge pressure, Pa Tsuc-suction temperature, K Tdis-discharge temperature, K Tw- cylinder wall temperature, K Refrigerant-R410a out Hc-enthalpy, kj/kg b. Calculate suction volume and cylinder volume by using following formula- Vsuc = ( pi/4)*(dcyl) 2* stroke Vcyl = 0.1* Vsuc Also calculate Re i.e effective resistance and Le i.e effective inductance of coil. c. Define the definite cycles and calculate the time (sec) for completion of these defined cycles. No. of Cycles = Time * Frequency e.g. if we defined 10 cycles then calculate the time required for completion of these cycles. CASE 1 Check if Current Time = Final Time i.e. If YES END program. IF NO CHECK t=0 IF YES INITIALIZE Keep the piston position at BDC i.e. at bottom dead centre and measure the temperature and pressure i.e. T= Tsuc and P= Psuc. Calculate density from NIST subroutine after knowing temperature and pressure. After finding these parameters, Calculate Msuc-suction mass, Hin enthalpy and gama-ratio of specific heat at constant pressure to specific heat at constant volume. After knowing these parameters, solve the equations by using Runge Kutta method and then we will get piston displacement, velocity, acceleration, current, mass flow rate, temperature, discharge valve displacement, velocity, acceleration, suction valve displacement, velocity, acceleration. CASE2 If time is not reached upto the defined cycles (here 10 cycles) then program will not END. Set the time to zero. Calculate density, rho = mcyl / V mcyl - mass of cylinder, V- cylinder volume Calculate pressure from NIST subroutine with temperature and density as two input parameters to NIST. Here V1=V2 Where V1 previous calculated volume and the new volume V2 can be calculated by using formula, V2= (pi/4)( Dcyl) 2* X1(t) + Vcl And Mcyl = Mcyl Where Mcyl is the previous calculated mass of cylinder and new mass of cylinder can be calculated by using formula, Mcyl = Mcyl + Min + Mout Where Min and Mout is the mass entering and leaving the cylinder. www.ijmca.org Page 158
i.if mass entering and leaving the cylinder is zero and if Pcyl > Psuc, then both the valves are closed. ii.if mass is entering the cylinder and if Pcyl > Psuc and Pcyl < Pdis, then suction valve will open. iii.if mass is leaving the cylinder and if Pcyl > Pdis, then discharge valve will open. d. If the valve displacement is negative then forcibly keep the valve displacement to zero. And if the valve displacement is greater than the maximum value then set the valve displacement to maximum displacement. So after knowing that, once again calculate the time, if time is reached 10 cycles then the program will End. Keep the piston position at BDC i.e. at bottom dead centre and measure the temperature and pressure i.e. T=Tsuc and P= Psuc. Calculate density from NIST subroutine after knowing temperature and pressure. After finding these parameters, Calculate Msuc- suction mass, Hin enthalpy and gama-ratio of specific heat at constant pressure to specific heat at constant volume. After knowing these parameters, solve the equations by using Runge Kutta method and then we will get piston displacement, velocity, acceleration, current, mass flow rate, temperature, discharge valve displacement, velocity, acceleration, suction valve displacement, velocity, acceleration. i.e. repeat the procedure again. Input Voltage (To be optimized to ensure optimized driving force for achieving full stroke condition). For resonance Ke = me*omega 2 Mechanical spring stiffness was varied from 0.8 to 0.9 of the resonance condition (Remaining is assumed to be gas spring). 7. MATHEMATICAL SIMULATION RESULTS Following sections provide with the results for mathematical modelling of linear compressor: a. Piston stroke (mm) and velocity(m/s) Vs time(sec) Figure5. Piston stroke(mm) and velocity(m/s) Vs time(sec) b. P-V diagram 5. NUMERICAL ANALYSIS Using the equations discussed a Code was developed using C++ to analyze the linear compressor. Following equations were solved using Runge Kutta 4 th order method to estimating the compressor performance Piston dynamics, Motor equation, Cylinder gas pressure, temperature estimation, Valve dynamics, and Mass flow estimation. The numerical differentiation loop run for predefined number of cycles (20 default). Refrigerator performance (Cooling capacity, EER, Theoretical Efficiency, Isentropic efficiency etc.) is obtained for the converged cycle. 6. INPUT PARAMETERS FOR MATHEMATICAL MODELLING The mathematical model was analyzed for data reported by LG [2] to validate the model. Following is the data considered for analysis, Refrigerant: R410A Piston Diameter: 22.5 mm Piston stroke: 35 mm Operating Frequency = 60 Hz (For US Market). Mass = 2.0 kg Figure 6. P-V diagram c.valve displacement(mm) Vs time(sec) Figure7. Valve displacement(mm) Vs time(sec) www.ijmca.org Page 159
8. FLOW CHART FOR MATHEMATICAL MODELLING www.ijmca.org Page 160
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The overall performance of the Vapor compression cycle for the configuration selected is as follows: Cooling Capacity = 3729.573081 W Compressor Power = 1004.704103 W Coefficient of Performance = 3.712111 Carnot COP = 5.936269 Theoretical Efficiency = 62.532724 Volumetric Efficiency = 96.567011 Energy Efficiency Ratio= 12.669321 PV Power = 984.883893 9. CONCLUSION The mathematical model that developed explains various performance and operating parameters of the linear compressor viz. Piston stroke, velocity, cylinder pressure, temperature, mass flow rate etc. 10. REFERENCES 1. Eytan Pollok, Werner Soedel,F.J.Friedlaender,Raymond Cohen, Mathematical Model Of An Electrodynamics Oscillating Refrigeration Compressor, Ray W.Herric Laboratories, School of mechanical engineering, Purdue University, West Lafayette, Indiana 47907 2. Hyuk Lee, Sang-sub Jeong, Chel-woong Lee, Hyeong-kook Lee, Linear Compressor For Air-Conditioner, L. G. Electronics, Seoul, South Korea, C047, 17 th International Compressor Conference, Purdue, USA. www.ijmca.org Page 162