Testing and modelling of an automotive wobble plate compressor

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 4 2 3 – 4 3 1

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Testing and modelling of an automotivewobble plate compressor

Cristian Cuevasa,*, Eric Winandyb, Jean Lebrunb,1

aDepartamento de Ingenierıa Mecanica, Facultad de Ingenierıa, Universidad de Concepcion, Casilla 160-C, Concepcion, ChilebThermodynamics Laboratory, University of Liege, Campus du Sart Tilman – Batiment B49, Parking P33, B-4000 Liege, Belgium

a r t i c l e i n f o

Article history:

Received 18 December 2006

Received in revised form

10 July 2007

Accepted 23 July 2007

Published online 2 August 2007

Keywords:

Air conditioning

Automobile

Compressor

Modelling

Experiment

Efficiency

Volumetric efficiency

Comparison

* Corresponding author. Tel.: þ56 41 2203550E-mail addresses: [email protected] (C. C

1 Tel.: þ32 4 3664801; fax: þ32 4 3664812.0140-7007/$ – see front matter ª 2007 Elsevidoi:10.1016/j.ijrefrig.2007.07.008

a b s t r a c t

Wobble plate compressors are well used in air conditioning for high-class automobiles.

They allow continuous control by automatic adjustment of the piston stroke, to keep the

low pressure above a certain limit. Here an externally controlled wobble plate compressor

is analyzed experimentally through its isentropic and volumetric effectivenesses and con-

trol characteristics. Compressor effectivenesses depend mainly on the compressor speed

and pressure ratio: there is obtained, for example, isentropic and volumetric effective-

nesses of 0.65 and 0.8 for a pressure ratio of 4 at 1000 rpm and 0.4 and 0.35 for the same

pressure ratio at 4000 rpm. This degradation is attributed to the increasing of the supply

pressure drop. The ‘‘lubricant’’ (oilþ dissolved refrigerant) mass flow rate is obtained by

minimization of the residuals of the thermal balances on the compressor, condenser and

evaporator. Here an important oil-flow circulation is obtained: between 9.5% and 12.5%

of the refrigerant flow rate. A special displacement sensor is used to measure instanta-

neous piston stroke and to relate it to overall compressor performance. This measurement

is then compared with the results obtained with a semi-empirical model, which is able to

predict, in part load, the compressor displacement. The model predicts the displacement

ratio with deviations that vary between �14.5% and þ8.1%.

ª 2007 Elsevier Ltd and IIR. All rights reserved.

Modelisation et essais d’un compresseur a plateau oscillantpour automobile

Mots cles : Conditionnement d’air ; Automobile ; Compresseur ; Modelisation ; Experimentation ; Efficacite ; Rendement volumetrique ;

Comparaison

1. Introduction

A wobble plate compressor seems one of the most appro-

priate for automotive air conditioning. Thanks to its

; fax: þ56 41 2251142.uevas), [email protected]

er Ltd and IIR. All rights

external (or internal) control, it is able to continuously

match the system-cooling demand. Its high performance

has already been demonstrated by Kishibuchi and Nosaka

(1999) and Nadamoto and Kubota (1999).

(J. Lebrun).

reserved.

mailto:[email protected]

mailto:[email protected]

http://www.iifiir.org

http://www.elsevier.com/locate/ijrefrig

Nomenclature

AU heat transfer coefficient (W K�1)

C coefficient

c specific heat (J kg�1 K�1)

DR displacement ratio_H enthalpy flow (W)

K fraction

L stroke (m)_M mass flow rate (kg s�1)

N rotational speed (s�1)

n number of tests or number of pistons_Q heat flow (W)_R residuals (W)

V volume (m3) or tension (V)

v specific volume (m3 kg�1)

w specific work (J kg�1)_W power (W)

Z coefficient

Subscripts

amb ambient

cal calorimeter

cd condenser

cp compressor

ev evaporator

ex exhaust

g gas

in internal

l liquid

loss loss

lub lubricant

meas measured

mix mixture

nom nominal

oil oil

r refrigerant

ref reference

s isentropic or swept

sh shaft

sim simulated

su supply

v volumetric

w water

wall fictitious wall

Greek symbols

a power losses parameter

d dimensionless diameter

3 effectiveness

3 error

F residual function

w error function

s standard deviation

z liquid refrigerant fraction


This kind of compressor has been already tested and mod-

elled by Dirlea et al. (1998), Delvaux et al. (2000) and Khamsi

et al. (2000). In these studies, the compressor was character-

ized experimentally in part and in full load, and by carrying

out the piston stroke measuring with a displacement trans-

ducer installed in the centre of the wobble plate. Dirlea et al.

(1998) present a method to determine oil circulation through-

out the refrigeration circuit; one of their main conclusions is

the high oil fraction circulating in the system when this kind

of compressor is used. They developed a polynomial model

to predict compressor power and refrigerant flow rate. This

model is not capable of predicting the refrigerant exhaust

temperature.

Delvaux et al. (2000) and Khamsi et al. (2000) present the

same results in both papers. The compressor modelling is

improved, they use a semi-empirical model based on the

processes occurring inside the compressor: heat transfer,

pressure drop and isentropic compression. They perform

some tests in full and in part load and then they use these

results to identify the parameters of their simulation

model.

Tian et al. (2004, 2006) present another approach for the

modelling. They combine force balance, mass and energy con-

servation equations. Isentropic and volumetric effective-

nesses are modelled by using polynomials laws identified

from measurements. Their simulated results are compared

with some measurements, developed at full and part load, to

validate their simulation model. Among their measurements,

they also measure the piston stroke.

In this study, we follow the testing methodology and mod-

elling proposed by Delvaux et al. (2000) and Khamsi et al.

(2000), as a continuation of their research.

In this study an enormous database of experimental tests

performed at both full and part loads, at different compressor

speeds and pressure ratios, are given. One compressor is

equipped with a displacement sensor to measure the piston

stroke, but in this case (compared to Delvaux et al., 2000 and

Khamsi et al., 2000) the displacement sensor was installed di-

rectly on the piston. Different orders of magnitude of isentro-

pic and volumetric effectivenesses and the main parameters

that affect them, the compressor characteristics at part load

and some orders of magnitude of oil circulation, are also given

here.

The piston stroke measuring was carried out here to verify

the performance of a semi-empirical compressor model in

part load, i.e. if it can predict with an acceptable accuracy

the compressor performance at different part-load conditions.

2. Description

2.1. Test bench description

The test bench is mainly composed of: a coaxial water heated

evaporator, a coaxial water cooled condenser, a filter, a Corio-

lis flow meter, and two thermostatic expansion valves con-

nected in parallel. All these components and connecting

Thermocouple

Pressure transducer

Refrigerant high pressure line

Refrigerant low pressure line

Water network~

Water heater

PID Calorimeter

Filter

Cor

iolis Fa

n co

il

Sight glass

Exp. valve

Load cell

Load cell

Calorimeter tank

Condenser tank

Pump

Pum

p

Com

pres

sor

PID

M

Con

dens

er

Eva

pora

tor

Fig. 1 – Test bench.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 4 2 3 – 4 3 1 425

pipes are thermally insulated to minimize ambient losses.

Fig. 1 shows a general sketch of the test bench.

This test bench was specially built to characterize automo-

tive compressors. Testing conditions: supply and exhaust

pressures, supply temperature and compressor speed can be

adjusted separately. The exhaust pressure is adjusted by

changing the condenser water flow rate and the supply pres-

sure by changing the water heater power. Compressor

supply temperature can be also slightly modified with the

thermostatic valve, which allows us to modify the refrigerant

over-heating.

The compressor here is driven by an electric motor through

an auxiliary shaft, as shown in Fig. 2. The compressor is in-

stalled on one extremity of the shaft of a bearing support (an

out-of-use electric motor), used to transmit the compressor

50 cm.

104

cm.

Electrical motor

Fan

coil

Compressor

Load cell

Refrigerant Heating lampHeating lamp

Fig. 2 – Compressor calorimeter.

shaft torque. On the other extremity of this bearing support,

there is installed a moment arm to transform this torque

into a force, which is measured with a load cell. Compressor

speed is measured on the auxiliary shaft. This is modified by

changing the motor and auxiliary shaft pulleys.

Due to the parasitic torque introduced by the pipes con-

nected to the compressor, this system must be calibrated

beforehand to perform each testing campaign.

2.2. Compressor description

The wobble plate compressor is a variable displacement com-

pressor used in car air-conditioning systems, which changes

its swept volume continuously (between 0% and 100%), thanks

to an externally controlled solenoid valve to match the

Sensor

Teflon pistonsupport

Piston Holes for thefixation of the rod

Fig. 3 – Fixation of the bar to the piston.

Sensor Fixation to the piston

Piston

Wobble plate

Pulley

Spring

Suction chamber

Discharge chamber

Sensorcoupling

Sensor rod

Compressorcylinder head

Compressorcrankcase

Compressorcylinder casing

Cylinder Bolt

Fig. 4 – Schema of the placement of the displacement sensor.

Table 1 – Measuring uncertainties

Variable Measurementrange

Uncertainty


system-cooling demand. This concept provides a continuously

operating compressor with the following advantages as

compared to an ON/OFF control capacity system: smooth con-

tinuous compressor operation (no cycling loading on engine),

no discharge temperature swinging due to compressor cy-

cling, evaporator temperature maintained just slightly above

freezing point and improved fuel economy.

The externally controlled solenoid valve regulates the

compressor capacity according to the cooling demand.

When the A/C demand is high, the valve maintains a bleed

from the crankcase to the suction cavity and no crankcase-

suction pressure differential may exist; therefore the com-

pressor stays at maximum displacement. When the A/C

demand is lower than the equivalent compressor displace-

ment, the valve responds to increase the crankcase pressure

by opening a passage from the compressor discharge plenum

to the crankcase. This way of controlling the crankcase-

suction pressure differential creates a net force on each piston

and in turn a resultant moment on the wobble plate/journal

about the journal pivot pin.

-20

-15

-10

-5

0

5

10

15

20

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Time [s]

Pist

on s

trok

e [m

m]

Lp

Tp

Fig. 5 – Evolution of compressor stroke.

This valve is integrated in the compressor and is supplied

with a pulse width modulated (PWM) tension, at a constant

frequency of 400 Hz and an amplitude of 13.5 V. The PWM sig-

nal can be modulated to obtain different ON times (TON) for the

same frequency, which is called here as ‘‘cyclic ratio’’. During

this study, the cyclic ratio was fixed at 100%, 75%, 50%

and 25%.

The compressor has seven pistons, with a bore of 32 mm

and a maximum swept volume of 160 cm3 (values given by

the manufacturer). Two compressors of the same model are

tested here; one of the two is equipped with a piston stroke

measuring system.

For the compressor without-stroke measuring system, the

refrigerant circuit is charged with 2.5 kg of refrigerant R134a

Pressures Pr,su,cp 1.63–5.25 bar �0.05 bar

Pr,ex,cp 7.51–26.69 bar �0.25 bar

Pr,crkcase,cp 1.90–6.45 bar �0.05 bar

Pr,su,cd 7.47–26.56 bar �0.25 bar

Pr,su,ev 2.67–10.22 bar �0.05 bar

Flows _Mmix;cor 4.4–107.2 g/s Between �0.1 g/s

and �0.4 g/s

Electrical

powers

_Wem 1118–9097 W Between �8 and �65 W_WEH 683–16,206 W Between �8 and �182 W

Compressor

speed

Ncp 986–3990 rpm Between �9 and �37 rpm

Note: Condenser and evaporator exhaust pressures, Pr,ex,cd and

Pr,ex,ev, are not measured; they are assumed to be equal to the con-

denser and evaporator supply pressures, and thus affected by the

same uncertainties.

0.00 0.02 0.04 0.06 0.08 0.10 0.120.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Mr,g,su,cp [kg/s]

Moi

l [kg

/s]

Compressor 1 : Compressor with stroke measurementCompressor 2 : Compressor without stroke measurement

Moil,cp1 = -0.000286 + 0.0955·Mr,g,su,cp

Moil,cp2 = 0.000216+ 0.127·Mr,g,su,cp

Fig. 6 – Oil-mass flow rate circulating in the loop.

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr,ex,cp / Pr,su,cp [-]

ε v [

-]

986 rpm2250 rpm3130 rpm3930 rpm

986 rpm

3130 rpm

Compressor 1 Compressor 2


Fig. 8 – Volumetric effectiveness.


and 0.28 kg of polyalkylene glycol ISO VG46, i.e. an initial oil

charge of 11%. For the compressor with-stroke measuring sys-

tem, the refrigerant and oil charge are 2.5 kg and 0.30 kg,

respectively, i.e. an initial oil charge of 12%.

2.3. Piston stroke measurement

One compressor is equipped with a variable resistive vector

transducer (VRVT) to measure the piston stroke. This trans-

ducer is welded to a little rod which is connected to one of

the seven pistons, as shown in Fig. 3. This system uses a piece

of flexible wire to allow the accommodation of the mobile part

inside the fixed part of the sensor to avoid the risk of heating

due to friction. This wire also allows transversal movements

of the piston without damaging the sensor. The rod is

attached to the piston by two screws.

The main advantage of this measuring technique is that

one has the piston stroke directly. Concerning the disadvan-

tages, this system could introduce some disturbances in com-

pressor performance if some precautions are not taken during

the placement of the sensor on the piston and during the

placement of the piston on the cylinder. One must be sure

that there is still an oil film around the entire piston, because

it could introduce a higher friction coefficient and the com-

pressor failure.

0 5000 10000 15000 20000 25000-20

-15

-10

-5

0

5

10

15

20

Hw,cd, Wsh,cp or Hw,ev [W]

CondenserCompressorEvaporator

Fig. 7 – Residuals for the condenser, compressor and

evaporator thermal balances.

Fig. 4 shows the location of the displacement sensor in the

compressor.

By using the measured piston stroke, the actual swept vol-

ume is determined as:

Vs ¼ npD2

4Lp; Lp ¼ C$DV (1)

where C¼ 0.0156 m V�1 is the sensor constant and Lp is the

piston stroke.

The displacement ratio (DR) is computed as the ratio be-

tween the actual swept volume and the maximum one:

DR ¼ Vs

Vs;max(2)

The displacement sensor signal is measured with an oscillo-

scope, which gives a signal as shown in Fig. 5. The value

obtained for the displacement is a perfect sinusoid, which

gives two pieces of information: the amplitude of the piston

displacement (Lp) and the compressor rotational speed (1/Tp).

Table 1 gives the uncertainties on pressures, flows,

compressor speed and electrical powers (these are total

uncertainties, i.e. transducer uncertaintyþ data acquisition

uncertainty). For the temperatures, one considers two sources

of uncertainty: one coming from the thermocouple tolerance

(�0.5 K) and other coming from the data acquisition system

(�0.3 K). Thus, there is obtained an overall absolute

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr,ex,cp / Pr,su,cp [-]

ε s [

-]

986 rpm2250 rpm3130 rpm3930 rpm

986 rpm

3130 rpm



Fig. 9 – Isentropic effectiveness.

5 10 15 20 25 300

1

2

3

4

5

6

7

8

Pr,ex,cp [bar]

P r,s

u,cp

[ba

r]

Pr,su,cp = 1.616 + 0.0350·Pr,ex,cp

Pr,su,cp = 2.424 + 0.0426·Pr,ex,cp

Pr,su,cp = 4.277 + 0.0313·Pr,ex,cp

75 %50 %25 %

1000 rpm 1000 rpm2000 rpm3000 rpm

3000 rpm

4000 rpm


75 %

50 %

25 %


Fig. 10 – Tests at different valve modulations.

Table 2 – Tests conditions

Full load Supply pressure 1.6–5.2 bar

Exhaust pressure 9.2–25.5 bar

Supply temperature �4.0 to 49.7 �C

Exhaust temperature 55.9–120.8 �C

Refrigerant mass flow rate 0.005–0.098 kg/s

Compressor shaft power 0.670–7.550 kW

Compressor speed 986–3944 rpm

Part load Supply pressure 1.7–5.2 bar

Exhaust pressure 7.5–26.7 bar

Supply temperature �10.8 to 34.6 �C

Exhaust temperature 44.4–119.7 �C

Refrigerant mass flow rate 0.010–0.046 kg/s

Compressor shaft power 0.290–4.011 kW

Compressor speed 986–3990 rpm


uncertainty of �0.6 K (the relative uncertainty is smaller). The

system used to determinate the compressor shaft torque has

been calibrated. Thus, the compressor shaft torque is mea-

sured with an accuracy of �1.4%. The load cells used to deter-

minate the calorimeter and condenser water flow rates were

not calibrated. Thus, their theoretical uncertainties are �25%

and �10%, respectively, for most of the tests. The Coriolis

mass flow measurement is only used for checking. It is not

used in the analysis because it is installed in a place where

there are high vibrations which could disturb its ability to

measure. According to the experimental results, both results

are up to �10%, which seems to be too high because of the

low measuring scattering obtained.

0.08

0.10

0.12

]

986 rpm (cp 1)2250 rpm (cp 1)3130 rpm (cp 1)3930 rpm (cp 1)

986 rpm (cp 2)

3130 rpm (cp 2)

3. Experimental results and analysis

The main difficulty in these kinds of tests is the high oil frac-

tion circulating in the refrigerant circuit. This oil fraction as

well as the refrigerant flow rate are here identified by using

an indirect method, which consists of combining the evapora-

tor, condenser and compressor energy balances. This method

was developed by Dirlea et al. (1998); they proposed to mini-

mize the residuals of the energy balance of each component

by using the following function:

F ¼� _Rcp

_Wsh;cp

�2

þ� _Rev

_Mw;evcwðtw;su;ev � tw;ex;evÞ

�2

þ� _Rcd

_Mw;cdcwðtw;ex;cd � tw;su;cdÞ

�2

ð3Þ

twall,cp

Fictitious isothermalwall

su1ex2su2

Win,cp

Isentropiccompression

ex1

Mmix = Mr,g + Mlub

Qr,wall,su,cpQr,wall,ex,cp

Qcp,ambQlup,wall,cp

Wcp,loss

su

Supplyvalve

Exhaustvalve ex

Fig. 11 – Conceptual schema of the compressor model.

where _Rcp, _Rev and _Rcd are the residuals of the compressor,

evaporator and condenser thermal balances.

Each point (test) used here is the result of the thermal bal-

ance, carried out with the averages of a 20-min steady-state pe-

riod, for an acquisition frequency of one acquisition every 4 s.

The compressor ambient losses coefficient is here esti-

mated theoretically to 2.5 W/K.

Fig. 6 shows the oil-flow rate determined with this method.

According to these results, the oil-mass flow rate is propor-

tional to the refrigerant flow rate. It should also be influenced

by the oil charge: for the second compressor, the oil-mass flow

rate is higher and, according to the values given in Section 2.2,

the oil charge was also higher.

According to Fig. 7, most of the component residuals are

lower than �7.5%.

The compressor is characterized through its isentropic and

volumetric effectivenesses, which are calculated with the

following equations:

3s ¼_Mr;cpws

_Wsh;cp

(4)

3v ¼_Mr;cpvsu;cp

NcpVs(5)

0.00 0.02 0.04 0.06 0.08 0.10 0.120.00

0.02

0.04

0.06

Mr,cp,meas [kg/s]

Mr,

cp [

kg/s

Fig. 12 – Simulated/measured refrigerant mass flow rate at

full load.

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

Wsh,cp,meas [kW]


986 rpm (cp 2)

3130 rpm (cp 2)

Wsh

,cp

[kW

]

Fig. 13 – Simulated/measured compressor shaft power at

full load.


The volumetric effectiveness is calculated by using the swept

volume announced by the compressor manufacturer as

reference.

Fig. 8 shows the volumetric effectiveness as a function of

the compressor speed and pressure ratio. At a given compres-

sor speed, the typical decrease in the volumetric effectiveness

as the pressure ratio increases is observed, due to the clear-

ance volume re-expansion. When the compressor speed in-

creases, the volumetric effectiveness decreases. It is

attributed to the increasing of the supply pressure drop, which

also produces a negative effect on the isentropic effectiveness,

as shown in Fig. 9.

Fig. 10 shows the solenoid valve characteristics in part

load. It is observed that the compressor always works over

the same ‘working line’ for each valve position (25%, 50%

and 75%), whatever the cooling demand may be.

Thus, we can establish that when the solenoid valve works

at part load, the suction and the discharge pressures are re-

lated with a linear law of the type:

Pr;su;cp ¼ aþ bPr;ex;cp (6)

With b almost constant and a decreasing function of the cyclic

ratio.

20 40 60 80 100 120 14020

40

60

80

100

120

140

tr,ex,cp,meas [°C]

t r,ex

,cp

[°C

]


986 rpm (cp 2)

3130 rpm (cp 2)

Fig. 14 – Simulated/measured compressor exhaust

temperature at full load.

4. Compressor modelling

The compressor performance is simulated by using the model

proposed by Khamsi et al. (2000) and Winandy et al. (2002). The

evolution of the refrigerant is decomposed into five steps (see

Fig. 11): supply pressure drop, supply heating-up, isentropic

compression, exhaust cooling-down and exhaust pressure drop.

This model uses a fictitious isothermal wall, which re-

ceives the compressor losses and exchanges heat with the re-

frigerant, the lubricant and the ambient. The steady-state

thermal balance of this fictitious isothermal wall is given by

the following equation:

_Wcp;loss þ _Qr;wall;ex;cp � _Qr;wall;su;cp � _Qcp;amb � _Q lub;wall;cp ¼ 0 (7)

where the supply and exhaust heat fluxes are calculated by

using the 3-NTU method. The heat flow transferred to the lu-

bricant is calculated as:

Qlub;wall;cp ¼ _Mlubclub

�tlub;ex;cp � tlub;su;cp

�(8)

Concerning the lubricant-mass flow rate modelling, here an

improvement to the original model is introduced (Khamsi

et al., 2000; Tian et al., 2004); this is here calculated thanks to

the oil-mass flow rate and to the refrigerant solubility. To sim-

plify the modelling, it is assumed that the lubricant flow rate is

the same throughout the refrigerant loop and that it corre-

sponds to the lubricant flow rate at the compressor supply:

_Mlub ¼ _Mlub;su;cp ¼ _Moil þ _Mr;l;su;cp (9)

_Mr;l;su;cp ¼zr;su;cp

1� zr;su;cp

_Moil (10)

Klub ¼_Mlub;su;cp

_Mr;g;su;cp

¼ Koil

ð1� KoilÞ�1� 2r;su;cp

�� 2r;su;cpKoil

(11)

The oil fraction Koil is determined from the polynomial laws

identified in the experimental analysis. The liquid refrigerant

fraction zr,su,cp is calculated using the relationship proposed by

Grebner and Crawford (1993) for a mixture R134a/PAG.

The compressor mechanical losses are here calculated by

using the following equation:

_Wloss;cp ¼ a _Win;cp þ _Wloss;cp;2

�Ncp

Ncp;ref

�2

(12)

where a and _Wloss;cp;2 are the parameters to be identified with

the experimental database.

The compressor model predicts the compressor exhaust

temperature, the refrigerant flow rate, the ambient losses

and the compressor shaft power. It uses three predefined pa-

rameters and eight identified.

The predefined parameters are the followings:

Nominal compressor speed: Ncp,ref¼ 24.2 Hz.

Reference mass flow rate: _Mref ¼ 0:1 kg s�1.

Ambient heat transfer coefficient: AUcp,amb¼ 2.5 W/K�1.

The compressors were characterized with 41 tests in full

load and 95 in part load at the conditions given in Table 2.

Table 3 – Model errors in full load

Variable Average error Standard deviation Minimal deviation Maximal deviation Confidence limits

tr,ex,cp �1.9 K 3.9 K �11.3 K 6.0 K �3.1 K

�0.7 K

Mr,cp �0.000 kg/s 0.002 kg/s �0.005 kg/s þ0.004 kg/s �0.001 kg/s

(�10.4%) (þ7.1%) þ0.000 kg/s

Wsh,cp �0.068 kW 0.275 kW �0.680 kW þ0.594 kW �0.152 kW

(�14.1%) (þ9.1%) þ0.016 kW


The model parameters are identified by using only the 31

tests carried out in full load and with the compressor

without piston stroke measuring system (denoted as cp 1

in Figs. 12–14) and by using the software EES (Klein and

Alvarado, 2001). The identified parameters are determined

by minimization of the function w, which depends on the

relative errors of the following variables: compressor shaft

power, refrigerant mass flow rate, compressor wall and ex-

haust temperatures. This function is defined as follows:

w ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Xm

j¼1

Xn

i¼1

�Vj;i;sim � Vj;i;meas

Vj;i;meas

�2vuut (13)

where Vj is the variable ‘‘j’’, m is the number of variables con-

sidered for the minimization and n is the number of tests.

The values obtained for the parameters are the followings:

a ¼ 0:408; AUsu;cp ¼ 22:2

� _M_Mref

�0:8

W K�1

AUex;cp ¼ 45:4

� _M_Mref

�0:8

W K�1; Cf ¼ 0:052

dsu ¼ 0:1464; DPex;cp ¼ 6:596

� _M_Mref

�2

bar

Vs ¼ 162:5 cm3; _Wloss;cp;2 ¼ 127 W

Comparisons between measured and simulated results are

shown in Figs. 12–14. It is observed that the compressor power

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Wcp,meas [kW]

Wcp

[kW

]


986 rpm (cp 2)

3130 rpm (cp 2)

Fig. 15 – Simulated/measured compressor shaft power at

part load.

and the exhaust temperature are under-predicted for the

compressor 2 for a compressor speed of 3130 rpm. This differ-

ence maybe due to the compressor manufacturing tolerance.

The model error is here defined with a method similar to

that recommended by the ASHRAE Guideline 2 (1986) for ex-

perimental data analysis. By analogy, the average error and

the standard deviation are defined as:

3 ¼ 1n

Xn

i¼1

�Vi;meas � Vi;sim

�¼ 1

n

Xn

i¼1

ð3iÞ; s ¼"

1n

Xn

i¼1

ð3i � 3iÞ2#0:5

(14)

where Vi,meas is the measured variable and Vi,sim is the simu-

lated one. When the sample (or test number) is higher than

20, the confidence limits are defined by the following

equation:

3� Zsffiffiffinp (15)

with Z¼ 1.96 for a probability of 95%.

The model errors for the 41 tests at full load are presented

in Table 3. The order of magnitude of the errors is considered

here as acceptable.

In part load, the refrigerant mass flow rate is imposed as in-

put and the model is able to predict the compressor displace-

ment. It must be remarked that when this model is

incorporated into the overall refrigeration system, this input

is replaced by the cooling demand, which in turn will allow

to determinate the necessary refrigerant flow for the

compressor.

The results are shown in Figs. 15–17. A very good agree-

ment is also observed in this case, except at 3930 rpm, where

20 40 60 80 100 120 14020

40

60

80

100

120

140

tr,ex,cp,meas [°C]

t r,ex

,cp

[°C

]


986 rpm (cp 2)

3130 rpm (cp 2)

Fig. 16 – Simulated/measured compressor exhaust

temperature at part load.

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

1.2

DRmeas [-]

DR

[-]

986 rpm (cp 2)3130 rpm (cp 2)

Fig. 17 – Simulated/measured compressor displacement

ratio.


the compressor shaft power and the exhaust temperature are

overpredicted by the model. It is attributed to the compressor

losses that are overestimated by the model. In our analysis,

the compressor losses depend on the internal work and com-

pressor speed. When both parts are separated, it is observed

that, at 3930 rpm, the term that depends on the internal

work becomes less important than the term that depends on

the compressor speed. This was not the case in full load,

where the parameters were identified.

The piston stroke is measured for two compressor speeds:

986 and 3130 rpm. According to the results shown in Fig. 17,

the model predicts the displacement ratio with deviations

that vary between �14.5% and þ8.1%. At 986 rpm, the dis-

placement ratio is slightly under-predicted, but this result is

considered here as acceptable due to the higher relative

uncertainty at part load.

5. Conclusions

Two identical wobble plate compressors were characterized

by testing and modelling. One of these compressors was

equipped with a piston stroke measuring system.

The compressor was first characterized through its volu-

metric and isentropic effectivenesses, which depend mainly

on the compressor speed and pressure ratio. Effectivenesses

are penalised by the re-expansion of the clearance volume

and by the supply valve pressure drop. Here many experimen-

tal results are given that can be used by other researchers to

estimate the compressor refrigerant flow rate and the com-

pressor shaft power. At part load, it can be observed that suc-

tion and discharge pressures are related, for a same solenoid

valve opening, with a linear law.

A semi-empirical model was tuned with the experimental

results obtained with one compressor at full load and then it

was used to simulate both compressors in full and part loads.

In full load, the compressor model gives a good agreement, ex-

cept for compressor 2 and for a rotational speed of 3130 rpm,

where the compressor shaft power and the compressor

exhaust temperature are overpredicted.

In part load, a very good agreement is observed, except at

3930 rpm, where the compressor shaft power and exhaust

temperature are overpredicted by the model. Most of the dis-

placement ratios are predicted with deviations that vary

between �14.5% and þ8.1%.

We give the experimental parameters of the semi-

empirical model, which can be used, for example, to simulate

this compressor in different conditions, by integrating the

overall air-conditioning system or by using other refrigerant.

r e f e r e n c e s

ASHRAE Guideline 2, 1986. Engineering Analysis of ExperimentalData. American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc., Atlanta, USA.

Dirlea, R., Gauthy, L., Grodent, M., Khamsi, Y., Lebrun, J.,Negoiu, D., 1998. Modeling of wobble plate compressorsused in automotive air-conditioning. In: Proceedings of theInternational Compressor Engineering Conference,Purdue, USA.

Delvaux, J., Negoiu, D., Winandy, E., 2000. Automotive air-conditioning open reciprocating compressors: experimentalanalysis and simplified modelling. In: Proceedings of theExperimental Methods and Measuring Techniques inRefrigeration, Liege, Belgium.

Grebner, J., Crawford, R., 1993. Measurement of pressure–temperature concentration relations for mixtures R12/mineraloil and R134a synthetic oil. ASHRAE Transactions 99 (1),387–396.

Kishibuchi, A., Nosaka, M., 1999. Development of ContinuousControlled Variable Displacement Compressor. SAE paper1999-01-0876.

Khamsi, Y., Lebrun, J., Negoiu, D., Winandy, E., 2000. Automotiveair conditioning open reciprocating compressors:experimental analysis and simplified modeling. In:Proceedings of the International Compressor EngineeringConferences, Purdue, USA.

Klein, S., Alvarado, F., 2001. EES – Engineering Equation Solver,Version 6.045. F-chart Software, Wisconsin, USA.

Nadamoto, H., Kubota, A., 1999. Power Saving With the Useof Variable Displacement Compressors. SAE paper1999-01-0875.

Tian, Ch., Dou, Ch., Yang, X., Li, X., 2004. A mathematical modelof variable displacement wobble plate compressor forautomotive air conditioning system. Applied ThermalEngineering 24 (17–18), 2467–2486.

Tian, Ch., Liao, Y., Yang, X., Li, X., 2006. A mathematical model ofvariable displacement swash plate compressor for automotiveair conditioning system. International Journal of Refrigeration29 (2), 270–280.

Winandy, E., Saavedra, C., Lebrun, J., 2002. Simplified modelling ofan open-type reciprocating compressor. International Journalof Thermal Sciences 42 (2), 183–192.

Documents

Testing and modelling of an automotive wobble plate compressor