DEVELOPMENT OF A HIGH PERFORMANCE LINEAR COMPRESSOR SYSTEM

*Maurizio Lamantia, Mechanical Specialist Electrolux Compressors Via Consorziale 13, 33170 Comina (PN) ITALY; TEL. +39 0434 379 854 FAX: +39 0434 379 814

e-mail: Maurizio.Lamantia@electrolux.it Andrea Contarini, Mechanical Researcher Electrolux Compressors

Via Consorziale 13, 33170 Comina (PN) ITALY; TEL. +39 0434 379 874 e-mail: Andrea.Contarini@electrolux.it

Giovanni Strapazzon, Electric Researcher Electrolux Compressors Via Consorziale 13, 33170 Comina (PN) ITALY; TEL. +39 0434 379 875

e-mail: Strappazzon.Giovanni@electrolux.it Alberto Pezzutto, Acoustics and Vibrations Researcher Electrolux Compressors

Via Consorziale 13, 33170 Comina (PN) ITALY; TEL. +39 0434 379 879 e-mail: Alberto.Pezzutto@electrolux.it

ABSTRACT

Adams software has been successfully utilized to develop a new compressor for household appliances. Linear compressor has a linear motor coupled to a resonating mechanical system: a piston and a spring. The advantages of this compressor respect to a standard reciprocating compressor are a high reduction of energy losses and a variable cooling capacity.

The predicted high performances were confirmed from a compressor prototype produced on the basis of the numerical optimized model.

NOMENCLATURE z1 stator position b1 supporting spring damping K1 supporting spring stiffness m1 stator mass z2 piston position K2 piston spring stiffness b2 piston spring damping m2 piston mass Fel electric force Fth pressure force FT total force

i current B magnetic flux density Beff effective flux density leff effective wire lenght V voltage source V1 motion induced tension Φ magnetic flux Leq equivalent inductance Ri iron losses Lc coil inductance Rc coil resistance Req equiv. resistance P chamber pressure V chamber volume Ω section m mass transfer γ polytrophic exponent Pin pressure before valve Pout pressure after valve Tin temperature before valve ρ pressure compression ratio ε dead volume ratio p oil pressure W oil velocity vector µ oil viscosity ρ oil density δ film thickness r radial dimension of piston z piston axial dimension p1 suction pressure h piston height u oil velocity along radius

INTRODUCTION The increasing demand of high efficiency compressors and the strong diffusion of electronic control devices requests the development of better compressor systems. Linear compressors have been studied by several authors, mainly because of the interesting energy consumption capabilities. The paper presents the development of a new single cylinder electrodynamic compressor from the mathematical simulation to the experimental tests. In a linear compressor the motion of the piston is directly due to the magnetic field forces, while in standard compressor the rotational motion of the motor is transformed in an alternative motion by a crankshaft mechanism. The linear compressors have some advantages respect to the standard ones: - mechanical simplicity, as the magnetic field directly exerts an alternative force to the piston, while in

standard compressors the rotational motion of the electric motor is transformed in an alternative motion by a crankshaft mechanism, allowing thus the mechanical system to be compact and robust;

- - reduction of friction losses theoretically to zero, as all forces are directed along the piston motion’s

direction; this way there are lower power losses, there is no wear and theoretically there would not be necessity of lubrication;

- - cooling capacity variability through the piston displacement’s control, even if this possibility

introduces a potential weakness: the necessity of introducing some controls to evaluate the piston’s position.

MECHANICAL SYSTEM The mechanical system has been analyzed through an Adams multibody model. The advantage of this approach is that it allows the introduction in the model of flexible elements, taking so into account those components’ modal behavior. The system can anyway be schematized as a simple two degrees of freedom system (Fig. 1). Using the index 1 for the the stator’s properties (z1 stator position, K1 and b1 suspension spring stiffness and damping) and the index 2 for the piston’s (z2 piston position, K2 and b2 piston spring stiffness and damping), the characteristic equations of this system become:

( ) ( )( ) ( )

=−−−−=−++−++

TFzzKzzbzmzKzKKzbzbbzm

21221222

211212112111 0&&&&

&&&&

The previous differential equation system can be solved numerically. The external applied force FT depends on several term:

1. Magnetic force; 2. Pressure force applied by the gas; 3. Gravity force;

z1

z2

K2

K1

b2

b1

FT

m2

m1

Fig. 1 Mechanical system

4. Friction force; The first contribution (Fel) is due to the electric system and will be analyzed in the paragraph Electric system. The second force (Fth) depends on the thermodynamic cycle and on the valves dynamics. It will be analyzed in the paragraph thermodynamic system. The third term has impact on the static equilibrium position, but has no relevant importance for the dynamic model. The fourth term is due to friction forces along sliding parts. It will be analyzed in the paragraph friction losses. Impact forces on piston and valves have been considered in the model, using Adams features.

ELECTRIC SYSTEM The magnetic forces, according to Lorentz’s equation, can be calculated using the following equation:

∫ ∧= dlBiF

So the oscillating motion is due to the alternative current. In the hypothesis of stationary conditions, the force applied by the electric motor is:

( ) ( )tilBtF effeffel ⋅⋅=

Where: Beff is the effective magnetic flux density that acts on the coil, leff is the effective length of the coil wire. The current i(t) is obtained solving the equations of the equivalent electric circuit (Fig. 2). For a transient analysis, for example during the start up, or for a correct evaluation of higher harmonics, it is necessary to evaluate correctly the current in the circuit. In first approximation you can consider the motor as a constant resistance and inductance in series. A better model considers a resistance in parallel with the inductance of the motor (due to the iron losses). Applying Kirkhoff’s equation to the electric equivalent system you obtain:

( )

=Φ

−−−

=++

021

21

zdzd

dtdi

Lzdz

dLiR

ViRiRR

cc

i

cic

&&

Where: Ri is the equivalent of iron losses, Lc is the coil inductance and Rc is the coil resistance. The term:

zdzd

dzdL

V c &

Φ

+=1

is due to the relative motion of the two parts. In stationary conditions, it is possible to simplify the previous expression, obtaining:

222

22

ci

icceq LR

RLRR

ϖϖ

++= and

222

2

ci

cieq LR

LRL

ϖ+=

This approach can be used only in stationary conditions, as it takes into account only the first harmonic.

V

V1

Rc

Ri Lc

i1

i2

Fig. 2 Electric equivalent system

You can take into account also higher harmonics solving the equations with a Fourier series of generators of sinusoidal frequencies also for higher frequencies. It is obvious from the previous equations that higher harmonics will have significantly lower inductance, and for these the circuit will be mainly seen as the coil resistance. Then the equation of the electric circuit would become:

dtdi

LiRVV eqeq ++= 1

where: V is the voltage source. V1 is due to the relative motion of the stator and magnets. If the inductance is not related to the position, the expression of the induced motional voltage is:

( ) ( )121 zzlBtV effeff && −⋅⋅= .

The power consumption of the system is: W=VI, the model allows the evaluation of this parameter and the coefficient of performance of the system.

THERMODYNAMIC SYSTEM Force applied by the gas on piston (Fth) is proportional to the piston pressure which depends on the thermodynamic cycle. The theoretical thermodynamic cycle for a compressor (Fig. 3) is composed of two polytrophic transformation (governed by the state equation: PVγ=constant), plus suction and discharge. The polytropic equation was used to determine the pressure applied on the cylinder during compression and expansion. For a linear compressor piston displacement is not constant, however the equation can be used between two contiguous states. In first approximation, suction and discharge can considered as isobaric transformations. Anyway a better approximation of these transformations can be obtained considering an adiabatic efflux. In this case from the thermodynamic theory the mass flow through a section Ω is:

−

−

⋅=Ω

+γ

γγ

γγ

12

12

in

out

in

out

inin P

PPP

RTP

m&

Where the index in correspond to the position before the valve. This equation can be used to evaluate mass flow through valves. In this case the valves are approximated as spring–mass systems and the lift due to the pressure difference Pin-Pout is used in the equation of motion. The mass flow equation is important as a control parameter: during the discharge phase mass inside the cylinder can not be less than mass discharged through the valve from the instant it was opened, and a similar consideration can be done for the suction phase. This equation allows also to find maximum of

Ωm& varying pressure ratio to avoid high turbulence and flow drop.

The dead volume minimization is more difficult, as the piston’s displacement depends on several parameters; the volumetric efficiency µ is expressed using the following equation:

−−=

−−

= 111

01

41 γρεµvvvv

P

V

1

2 3

4 0

Pd

Ps

Fig. 3 Theoretical thermodynamic cycle

where ρ is compression ratio and ε is:

01

0

VVV−

=ε .

From the previous equation you can find that:

γ

ερ

+≤

11

in order to avoid the µ = 0 condition, as in this case the compressor would not work properly. An improved model will consider the heat exchange during the phases. This has clearly an impact on power consumption. In fig. 4 there is a comparison between the theoretical pressure-volume cycle and the cycle with simulated spring-mass valve. Obviously the main difference between the two is due to the consideration of valve dynamics.

FRICTION FORCES

Friction forces have been considered in two different models, taking into account the compressor’s lubrication system or not. As previously said, in a linear compressor all forces are directed along the piston’s motion direction. In this case friction forces depend also on the mounting of the

f r i c t i o n c o e f f i c i e n t

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.5 1 1.5 2 2.5 3

v e l o c i t y

µ

Vs

µ s

µ d

Vd

Fig. 5 Friction coefficient versus velocity

Fig. 4 Pressure-volume diagram

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Volume (m 3)

Pressure (MPa)

Cycle with valve motion Ideal cycle

compressor and gravity could be important. If there is no lubrication system, friction forces due to normal forces applied between the piston and stator and to the relative motion can be evaluated using Coulomb’s theory on friction (considering also the friction coefficient changes from static to dynamic conditions). Instead, if there is a lubrication system, friction forces can be evaluated from Navier-Stokes equations with the hypotheses of constant viscosity and incompressible fluid:

WpDt

DW 2∇+−∇= µρ

In the volume between cylinder and piston, considering single directional motion of the system and constant oil film thickness (δ) and velocity (u):

02

2

=∂∂

+−ru

dzdp

µ

Where: p is the pressure of oil film u is the oil velocity, µ and ρ are respectively the oil viscosity and density. Integrating the equation along radial direction r and imposing boundary conditions along radial direction u(0)=0 and ( )12)( zzu && −=δ ,you obtain:

( )dzdprrrzz

u

−−

−=

δµδ

δ1

212 &&

From fluid constitution equation (ru

rz ∂∂

= µτ ), considering a linear pressure distribution:

hpp

dzdp 1−

=

where: p1 is the suction pressure, h is height of the piston. Then friction force on cylinder surface is:

( ) ( ) ( ) ( )

−

+−Φ=−

+−

== ∫∫ 221

12112 pp

zzhdSh

ppzzdSF

SSrzr &&

&&δµ

πδ

δµ

δτ

where Φ is piston diameter.

MEASUREMENTS ON THE COMPRESSOR First some measurements on the compressor have been performed to validate the mechanical and electrical input parameters (mass, spring stiffness, damping, resistance, inductance, etc.) from which Adams model evaluated the dynamical parameters and efficiency of the compressor. From this first set of measurements, the mechanical resonance of the system without pressure has been measured, as it is possible to see on Fig. 6. Then a second set of measurements as been performed to evaluate the behaviour of the system considering also thermodynamic cycle. In Fig. 7 you can notice the influence of voltage and evaporating temperature on the compressor efficiency for a given working condition. As consequence it is possible to optimize the system for a given current frequency changing the voltage.

Fig. 6 Influence of current frequency and voltage on Input power-displacement ratio

Fig. 7 Efficiency Vs. evaporating temperature and voltage

As it is possible to see in Fig. 8, the compressor efficiency is high especially at lower evaporating temperatures.

CONCLUSIONS As conclusion of this study the following considerations can be made:

1. Linear compressors have higher efficiency then standard reciprocating compressors; 2. The system has high sensitivity to the driving frequency, for which it has to be designed; 3. Model has a good agreement with experimental results; 4. The applied voltage has to be controlled in order to optimize the efficiency of the compressor,

with a good knowledge of input parameters; 5. Further optimization will be performed to improve the knowledge on this system.

REFERENCES 1. Adams User’s Manual Release 11 2. R.V. Cadman, A technique for design electrodynamic oscillating compressor Ph.D. thesis, Purdue

university, 1067. 3. D.A. Coates and R. Cohen, Preliminary study of free piston stability in an electrodynamic gas compressor

XIIth. Int. Congress on Refrigeration Madrid 1967 4. E. Pollak, W Soedel Mathematical model of an electrodynamic oscillating refrigeration compressor Purdue

conference 1990 5. E.E. Staff M.I. T. Magnetic Circuits and transformers John Willey and Sons, Inc. New York 1955 6. J. Timmerman, Two electrodynamic vibrators, Philips tech Rev. 33 (1973)

Fig. 8 Efficiency Vs. evaporating temperature and frequency