a first-principles simulation model for the start-up and

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 ) 1 3 4 1 – 1 3 5 7

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A first-principles simulation model for the start-up andcycling transients of household refrigerators

Christian J.L. Hermes*, Claudio Melo

POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Federal University of Santa Catarina,

88040-970 Florianopolis, SC, Brazil

a r t i c l e i n f o

Article history:

Received 25 October 2007

Received in revised form

24 March 2008

Accepted 22 April 2008

Published online 1 May 2008

Keywords:

Domestic refrigerator

Modelling

Simulation

Steady state

Comparison

Energy consumption

* Corresponding author. Tel.: þ55 48 3234E-mail addresses: [email protected] (C

0140-7007/$ – see front matter ª 2008 Elsevidoi:10.1016/j.ijrefrig.2008.04.003

a b s t r a c t

A first-principles model for simulating the transient behavior of household refrigerators is

presented in this study. The model was employed to simulate a typical frost-free 440-l

top-mount refrigerator, in which the compressor is on–off controlled by the freezer tem-

perature, while a thermo-mechanical damper is used to set the fresh-food compartment

temperature. Innovative modeling approaches were introduced for each of the refrigerator

components: heat exchangers (condenser and evaporator), non-adiabatic capillary tube,

reciprocating compressor, and refrigerated compartments. Numerical predictions were

compared to experimental data showing a reasonable level of agreement for the whole range

of operating conditions, including the start-up and cycling regimes. The system energy con-

sumption was found to be within �10% agreement with the experimental data, while the

air temperatures of the compartments were predicted with a maximum deviation of �1 �C.

ª 2008 Elsevier Ltd and IIR. All rights reserved.

Modele de simulation fonde sur les principes fondamentauxutilise pour etudier les phenomenes transitoires lors dudemarrage et du cyclage des refrigerateurs domestiques

Mots cles : Refrigerateur domestique ; Modelisation ; Simulation ; Regime transitoire ; Comparaison ; Consommation d’energie

1. Introduction

A household refrigerator is basically composed of a thermally

insulated cabinet and a vapor–compression refrigeration loop,

as illustrated in Fig. 1. The energy consumption of a typical re-

frigerator is around 1 kWh/day, which is equivalent to the en-

ergy consumption of a 40 W light-bulb continuously running.

5691; fax: þ55 48 3234 516.J.L. Hermes), melo@polo

er Ltd and IIR. All rights

Although the energy consumption of a unitary refrigerator is

reasonably low, commercial and household refrigeration ap-

pliances are responsible for 11% of the total energy produced

annually in Brazil (PROCEL, 1998), which amounts to

2.86 TWh/year. Such a high energy consumption may be

easily accounted for considering that there is a large amount

of household refrigerators currently in use, and their

6..ufsc.br (C. Melo).

reserved.

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

http://www.sciencedirect.com

Nomenclature

A area, m2

Amin minimum flow passage, m2

c specific heat, J kg�1 K�1

C thermal capacity, W K�1

cp specific heat at constant pressure, J kg�1 K�1

D diameter, m

G mass flux, kg s�1 m�2

h specific enthalpy, J kg�1

k thermal conductivity, W m�1 K�1

L length, m

M mass, kg

N number of control volumes of the cabinet wall

n number of control volumes of the coil

NTU number of transfer units

P pitch, m

p pressure, Pa

q heat flux, W m�2

Q heat transfer rate, W

T temperature, K

t time, s

u specific internal energy, J kg�1

UA overall conductance, W K�1

v specific volume, m3 kg�1

V volume, m3

w mass flow rate, kg s�1

W power, W

x normal coordinate of the cabinet walls, m

z axial coordinate of the coil, m

Greek symbols

a thermal diffusivity, m2/s

g void fraction, dimensionless

3 temperature effectiveness, dimensionless

h efficiency, dimensionless

q crank angle, rad

l heat transfer coefficient, W m�2 K�1

m viscosity, Pa s

r specific mass, kg m�3

s solubility, dimensionless

s shear stress, Pa

f partial derivative of the density with respect to the

specific internal energy

j partial derivative of the density with respect to the

pressure

u angular speed, rad s�1

Subscripts

a air-side

c compressor, condenser

ct capillary tube

d discharge

e entrance, external, evaporator

en entrance

es outer liner

ex exit

f fin

hx heat exchanger

i inlet, internal

in inflow

k k-th control volume

l saturated liquid

is inner liner

o oil, outlet

r refrigerant-side, radiation

rc refrigerated compartment

s suction

sl suction line

t tube

tp two-phase

v saturated vapor

w tube wall, cabinet walls

Superscript_y time-derivative (¼dy/dt)


thermodynamic efficiency is intrinsically low, barely reaching

15% of Carnot’s COP. The major part of the energy is wasted by

the system components (compressor, condenser, evaporator

and capillary tube) due to irreversible losses. Studies carried

out to understand such thermodynamic losses shall lead to

the development of higher efficiency products.

The performance of a household refrigerator is usually

assessed using one of the following approaches: (i) simplified

calculations based on component characteristic curves; (ii)

numerical analysis via commercial CFD packages; and (iii)

standardized experiments. Although the first two techniques

play important roles in component design, they do not provide

enough information about component matching and system

behavior, which are only obtained by testing the refrigerator

in a controlled environment chamber. However, such tests

are time consuming and expensive. A faster and less costly al-

ternative is the use of first-principles models to simulate the

thermal- and fluid-dynamic behavior of refrigeration systems.

Steady-state and transient approaches can both be used. The

former is mainly applied for component matching, whilst the

second is essential to define the controlling strategies and to

optimize the system performance.

Former transient models for refrigeration systems date

back to the early 80s and were mostly focused on heat pump

and air conditioning equipment (Dhar, 1978; Chi and Didion,

1982; Yasuda et al., 1983; MacArthur, 1984; Murphy and Gold-

schmidt, 1985; Sami et al., 1987; MacArthur and Grald, 1989;

Wang, 1991; He et al., 1994; Vargas and Parise, 1995; Rossi

and Braun, 1999; Browne and Bansal, 2002; Kim et al., 2004;

Lei and Zaheeruddin, 2005) (see Table 1). The development

of dynamic models for household refrigerators was stimu-

lated by the CFC-12 phase-out in the late 80s (Melo et al.,

1988; Jansen et al., 1988, 1992; Lunardi, 1991; Chen and Lin,

1991; Yuan et al., 1991; Vidmar and Gaspersic, 1991). These

models were developed based on the experience acquired

for large systems (Dhar, 1978; Chi and Didion, 1982; Yasuda

et al., 1983; MacArthur, 1984; Murphy and Goldschmidt,

1985; Sami et al., 1987; MacArthur and Grald, 1989), although

refrigerator modeling strategies may differ substantially

from those adopted for air conditioning/heat pump (AC/HP)

evaporatorcondenser

compressor

freezer

fresh-food

discharge line

suction line

CT-SL HX

refrigerationsystem

dryer

refrigeratedcabinet

accumulator

12

3

4

5

Fig. 1 – Schematic of a top-mount refrigerator.

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 ) 1 3 4 1 – 1 3 5 7 1343

equipment. Firstly, household refrigerators require at least 10

times less refrigerant charge than AC/HP equipment, reason

why the refrigerators’ models are more sensitive to the refrig-

erant charge predictions than those developed for heat pumps

and air conditioners. For the same reason, the amount of re-

frigerant dissolved in the compressor oil must be accounted

for in refrigerator models in order to predict the equalizing

pressure during the cycling transients accurately. Moreover,

the simulation of household refrigerators requires an addi-

tional model for the refrigerated compartments, which adds

another time-scale to the set of differential equations – a refrig-

erator requires hours to reach its periodic steady-state operat-

ing condition, whereas AC/HP equipment achieves it in just

a few minutes. From the numerical stand-point, the solution

of such a stiff set of equations produces numerical instabilities

and convergence issues, which require tailored numerical

schemes. Finally, it is not usual for heat pumps and air condi-

tioners to employ non-adiabatic capillary tubes, whose numer-

ical simulation is time consuming and may lead to undesired

convergence problems. Table 2 summarizes the models avail-

able in the open literature for the transient simulation of

household refrigerators (Melo et al., 1988; Jansen et al., 1988,

1992; Lunardi, 1991; Chen and Lin, 1991; Yuan et al., 1991;

Vidmar and Gaspersic, 1991; Yuan and O’Neil, 1994; Jakobsen,

1995; Yu et al., 1995; Chen et al., 1995; Xu, 1996; Ploug-Sørensen

et al., 1997; Radermacher et al., 2005). Very few approaches are

able to simulate the refrigerator cycling behavior, and none of

them were validated against experimental data on energy con-

sumption. This is therefore the main focus of the present study.

2. Mathematical model

The overall system modeling requires the development of

sub-models for each of the cycle components. Basically, the

fluid and heat flows in these components are modeled based

on mass, momentum and energy conservation laws. These

equations, however, are quite complex to be solved in their

complete form. To keep the complexity at a reasonable level,

several assumptions are usually invoked according to the

characteristics of each component, as described below. More

detailed information can be found in Hermes (2006).

2.1. Heat exchangers: condenser and evaporator

A heat exchanger sub-model provides the overall heat trans-

fer rate, the refrigerant pressure, the enthalpy of the refriger-

ant at the coil outlet, and the temperature of the air exiting the

finned-region. Several approaches for heat exchanger model-

ing are available in the open literature. Most of them are based

on the application of the conservation principles to each of the

three heat exchanger sub-domains, namely refrigerant flow,

finned-walls, and air flow. According to the modeling strategy

adopted, the heat exchanger models may be classified as

global, nodal, moving boundaries, and distributed. Global

models consider the whole heat exchanger as an even lump

(Melo et al., 1988; Jakobsen, 1995). The nodal approach treats

each flow region as a single node in which the properties are

regarded as uniform (Lunardi, 1991), whilst the moving

boundary formulation assumes a linear property variation

along the superheating and subcooling regions (He et al.,

1994; Jansen et al., 1988). The distributed approach, on the

one hand, divides the domain into non-overlapping one-

dimensional control volumes (Yu et al., 1995; Chen et al.,

1995; Xu, 1996; Ploug-Sørensen et al., 1997), providing accurate

predictions of the evaporator superheating and condenser

subcooling degrees. On the other hand, such a method has

presented the following numerical issues: (i) there is not an

evolving equation for pressure computation, so it must be

solved iteratively, requiring a large computational effort; and

(ii) some flow properties are not continuous from one flow re-

gion to another (for instance, the refrigerant-side heat transfer

Table 1 – Summary of transient simulation models for refrigeration systems

Author (year) Origin Refrigeration

equipment

Cooling

capacity, TR

Refrigerant Heat

exchangers

Expansion

device

Compression

process

Compressor

shell

Refrigerated

room

Void fraction

model

Time

integration

Transient

regime

Validation

Dhar (1978) USA Unitary air

conditioner

Not available HCFC-22 Lumped Empirical Isentropic With oil Do not have Homogeneous Explicit, Euler Start-up No

Chi and Didion

(1982)

USA Air-source heat

pump

4 HCFC-22 Lumped Linear quality Polytropic Do not have Do not have Not available Explicit, Euler Start-up Yes

Yasuda et al.

(1983)

Holland Breadboard 0.3–1.4 CFC-12 Moving

boundaries

Orifice

formulation

Polytropic Do not have Do not have Hughmark Not available Start-up Yes

MacArthur

(1984)

USA Air-to-water

heat pump

3 HCFC-22 Distributed,

uniform

pressure

Orifice

formulation

Isentropic Without oil Do not have 2-Fluid model Implicit,

Crank–

Nicolson

Cycling Steady-state

Murphy and

Goldschmidt

(1985)

USA Unitary air

conditioner

3 Not available Quasi-steady Adiabatic

capillary tube

Empirical Do not have Do not have Not available Semi-

analytical

Start-up shut-

down

Cycling

Sami et al.

(1987)

Canada Water-source

heat pump

3 HCFC-22 Distributed Empirical Polytropic Do not have Do not have Homogeneous Implicit, Euler Start-up Yes

MacArthur and

Grald (1989)

USA Air-to-water

heat pump

2.5–6 Not available Distributed,

uniform

pressure

Orifice

formulation

Isentropic Without oil Do not have Zivi Implicit, Euler Cycling Yes

Wang (1991) Holland Frigorific

chamber

2.5 CFC-12 Distributed Empirical Isentropic Quasi-steady Distributed, 3-D 2-Fluid model Explicit Cycling Yes

He et al. (1994) USA Unitary air

conditioner

1 HCFC-22 Moving

boundaries

Orifice

formulation

Polytropic Do not have Do not have Fixed, 0.98 Linearized

reduced

model

Transients

next to

equilibrium

No

Vargas and

Parise (1995)

Brazil Air-source heat

pump

0.25 CFC-12 Lumped Orifice

formulation

Polytropic Do not have Lumped Not available Explicit,

Runge–Kutta-

Fehlberg

Cycling No

Rossi and Braun

(1999)

USA Unitary air

conditioner


uniform

pressure

Orifice

formulation

Isentropic Without oil Do not have Zivi explicit,

Runge-Kutta

Start-up Yes

Browne and

Bansal (2002)

New Zealand Water-source

chiller

85 HFC-134a Quasi-steady Orifice

formulation

Isentropic Do not have Do not have Fixed mass

distribution

Not available Start-up Yes

Kim et al. (2004) South Korea Water-source

chiller


uniform

pressure

Orifice

formulation

Polytropic Do not have Do not have Baroczy Implicit, Euler Start-up Yes

Lei and

Zaheeruddin

(2005)

China/Canada Water-source

chiller

Not available HCFC-22 Moving

boundaries

Orifice

formulation

Polytropic Do not have Do not have Not available Not available Start-up No

in

te

rn

at

io

na

ljo

ur

na

lo

fr

ef

rig

er

at

io

n3

1(2

00

8)

13

41

–1

35

71

34

4

Table 2 – Summary of transient simulation models for household refrigerators

Author (year) Origin Refrigerator

type

Refrigerant Evaporator Condenser Cabinet Heat

exchangers

Capillary tube Compression

process

Compressor

shell

Void fraction

model

Time

integration

Model

validation

Melo et al.

(1988)

Brazil 2-Door

refrigerator

CFC-12 Forced

convection

Forced

convection

Lumped Lumped Adiabatic Polytropic With oil Not available Explicit, Euler Start-up

Jansen et al.

(1988)

Holland Upright freezer CFC-12 Natural

convection

Natural

convection

Lumped Moving

boundaries

Adiabatic,

empirical

Empirical With oil Premoli Implicit, Euler Start-up

Lunardi (1991) Brazil 2-Door

refrigerator

CFC-12 Forced

convection

Forced

convection

Lumped Lumped Adiabatic Polytropic With oil Not available Explicit,

Runge–Kutta

Start-up

Chen and Lin

(1991)

China 2-Door

refrigerator

CFC-12 Natural

convection

Natural

convection

Not available Distributed Non-

adiabatic

Energy

balance

Without oil Not available Implicit, Euler Start-up

Yuan et al.

(1991)

China 2-Door

refrigerator

CFC-12 Forced

convection

Natural

convection

Lumped Lumped Adiabatic Isentropic Without oil Not available Explicit, Euler Start-up

Vidmar and

Gaspersic

(1991)

Yugoslavia 2-Door

refrigerator

CFC-12 HFC-

134a

Not available Not available Not available Distributed,

uniform

pressure

Adiabatic Energy

balance

Not available Not available Implicit, Euler First 10 min

Jansen et al.

(1992)

Holland Upright freezer CFC-12 Natural

convection

Natural

convection

Lumped Moving

boundaries

Adiabatic Empirical With oil Premoli

modified

Implicit, Euler Cycling

Yuan and

O’Neil (1994)

USA Upright freezer Not available Not available Not available Not available Distributed,

uniform

pressure

Adiabatic Polytropic Without oil Not available Implicit, Euler Start-up

Jakobsen (1995) Denmark All-refrigerator HC-600a Natural

convection

Forced

convection

Lumped Lumped Adiabatic,

correction

multiplier

Isentropic Without oil Fixed, 0.8 DALI Cycling

Yu et al. (1995) China 2-Door

refrigerator

Not available Not available Not available Lumped Distributed Non-

adiabatic

Not available Not available Not available Implicit, Euler Cycling

Chen et al.

(1995)

China 2-Door

refrigerator

HFC-134a HFC-

152a

Natural

convection

Natural

convection

Not available Distributed Non-

adiabatic

Energy

balance

Without oil Not available Implicit, Euler Cycling

Xu (1996) France 2-Door

refrigerator

HFC-134a Natural

convection

Natural

convection

Lumped Distributed Non-

adiabatic

Polytropic With oil Zivi Implicit, Euler First 3 min

Ploug-Sørensen

et al. (1997)

Denmark 2-Door

refrigerator

HC-600a Natural

convection

Natural

convection

Lumped Distributed Non-

adiabatic

Isentropic Not available Not available Implicit, Euler Cycling

Radermacher

et al. (2005)

USA 2-Door

refrigerator

HC-134a Forced

convection

Forced

convection

Lumped Lumped Non-

adiabatic,

empirical

Isentropic Without oil Not available Implicit, Euler Start-up,

cycling

in

te

rn

at

io

na

ljo

ur

na

lo

fr

ef

rig

er

at

io

n3

1(2

00

8)

13

41

–1

35

71

34

5


coefficient is risen by at least two orders of magnitude from

the single- to the two-phase flow region), which aggravates

the pressure iterative calculation.

The approach adopted in the present study was developed

based on the work of Rossi and Braun (1999), which introduced

a time-explicit distributed formulation that provided an evolv-

ing equation for the pressure, avoiding all the numerical issues

mentioned above. Additionally, the refrigerant flow was

modeled based on the following simplifying assumptions:

(i) one-dimensional flow; (ii) straight, horizontal and constant

cross-sectional tubes; (ii) negligible diffusion effects; and

(iv) negligible pressure drop. The governing equations, derived

from the mass and energy conservation principles, were

Z→

e 1 2 k-1 k…

Δzk

flow →a

inlet outletb

c

refrigerant outlet

wires

tubes

Fig. 2 – Schematic of heat exchangers sub-models: (a) finite-vol

evaporator; and (c) wire-and-tube condenser.

26666666664

V1ðr1 � pf1=r1Þ 0 0 0 �V�Dh1V1f1 V2ðr2 � pf2=r2Þ 0 0 �V2pj2=

« « 1 0�Dh1V1f1 �Dh2V2f2 / Vnðrn � pfn=rnÞ �Vnpjn=rn

V1f1 V2f2 / Vnfn

P

applied to each of the control volumes illustrated in Fig. 2a,

yielding

Vk _rk þwk �wk�1 ¼ 0 (1)

Vk

�uk _rk þ rk _uk

�þwkhk �wk�1hk�1 ¼ Qk (2)

whereQ¼ l(Tr� Tw).Anupwindschemewasusedto interpolate

the flow properties, and the averaged heat transfer term was in-

tegrated using a second-order approximation (trapezoidal rule).

As the refrigerant pressure through the coil was regarded as

uniform, the set of 2n dynamic ordinary differential equations

(ODEs) may be re-organized into a set of nþ 1 linear equations,

with n equations for _uk and one equation for _p, as follows:

n-1 nn-2…acumul at or

ordrye r

accumulatoror

dryer

airflow outlet

airflow inlet

outletintlet

refrigerant inlet

ume discretization of the heat exchanger coil; (b) tube–fin

1pj1=r1

r2 � Dh2V1j1

«� Dhn

Pn�1j¼1 Vjjj

nj¼1 Vjjj

37777777775

8>>>><>>>>:

_u1

_u2

«_un

_p

9>>>>=>>>>;¼

8>>>><>>>>:

Q1 �weDh1

Q2 �weDh2

«Qn �weDhn

we �wn

9>>>>=>>>>;

(3)


The peculiar shape of the coefficient matrix, with nonzero

terms located in the lower band and in the last row (nþ 1), per-

mitted its analytical inversion by LU-decomposition (Press

et al., 1995). Such an approach not only reduced the number

of ODEs, but also provided an explicit and evolving equation

for the pressure calculation. The solution of the linear set of

equations gave _p and _uk for each control volume, k¼ 1, n,

whereas _rk was obtained from the following thermodynamic

relation:

_rk ¼ fk _uk þ jk _p (4)

where fk¼ (vr/vu)p and jk¼ (vr/vp)u. The unknowns p, uk and rk

were calculated by the time integration of _p, _uk and _rk, while

the local mass flow rates, wk, were computed afterwards by

the following mass balance:

wk ¼ wi �Xk

j¼1

�Vjfj _uj

��

0@Xk

j¼1

Vjjj

1A _p (5)

where wi is the refrigerant mass flow rate at the coil inlet.

The refrigerant properties were calculated in advance using

the REFPROP software (Lemmon et al., 1998) and stored in

the computer memory in the form of cubic splines (Press

et al., 1995).

As the model was regarded as one-dimensional, the phe-

nomena related to the refrigerant-to-wall interface were in-

corporated into the model by empirical correlations.

Gnielinski’s (1976) correlation was used to estimate the sin-

gle-phase flow heat transfer coefficients. The correlations pro-

posed by Jung et al. (2003) and Wongwises et al. (2000) were

adopted for the condensing and evaporating heat transfer

coefficients, respectively.

The two-phase density was computed based on a local void

fraction model, gk,

rtp;k ¼ gkrv þ rlð1� gkÞ (6)

The void fraction models proposed by Baroczy (1965) and

Yashar et al. (2001) were used to estimate the refrigerant masses

in the condensing and evaporating regions, respectively.

The heat exchanger sub-model also accounted for the heat

transfer between the finned-walls and the internal and exter-

nal fluid streams. The following simplifying assumptions

were considered: (i) negligible heat conduction in the tube;

(ii) tube-by-tube discretization (Domanski, 1991), i.e., one con-

trol volume per tube; and (iii) fin efficiency calculated by

Schmidt’s (1945) procedure. The wall temperature of the k-th

control volume (Fig. 2a) is then given by

_Tw;k ¼AilrðTr;k � Tw;kÞ þ

�At þ hfAf

�laðTa;k � Tw;kÞ

cw

�Mt þ hfMf

� (7)

from condenser capillary

LhLen

to compressor

Fig. 3 – Schematic of a concentric capillar

The air flow through the evaporator was modeled as quasi-

steady, neglecting the presence of moisture. The evaporator

air temperature was obtained from an energy balance, consid-

ering the tube-by-tube approach, which was integrated

according to the following second-order scheme:

Ta;k ¼

hwacpa � 1

2la

�At þ hfAf

�iTa;k�1 þ la

�At þ hfAf

�Tw;k

wacpa þ 12la

�At þ hfAf

� (8)

The evaporator is a continuous flat finned-coil heat ex-

changer, where the refrigerant circuit is arranged according

to a 10-row, 2-column staggered array. In the first column,

the refrigerant flow is top-down oriented while the air flows

in the opposite direction. In the second column, both air and

refrigerant flows are in the bottom-up direction (see Fig. 2b).

The air-side heat transfer coefficient was computed using

a correlation derived using the wind-tunnel calorimeter facil-

ity described in Barbosa et al. (2008):

laDt

ka¼ 0:125

�wa

Amin

Dt

ma

�0:654�macpa

ka

�1=3

(9)

where la is the convective heat transfer coefficient

[W m�2 K�1], Amin is the minimum flow area [m2], and Dt is

the tube diameter [m]. Eq. (9) fitted the experimental data

within �5% error bands (Hermes, 2006).

The condenser is a natural draft wire-and-tube heat ex-

changer, where the air-side temperature was assumed to be

uniform (see Fig. 2c). The combined radiation and natural con-

vection heat transfer of the wire-and-tube condenser was

computed using the correlation due to Hermes and Melo

(2007),

la

lr¼ 5:68

�Af

At þ Af

�0:60�Pt � Dt

Dt

��0:28�Pf � Df

Dt

�0:49�Tw � Ta

Tfilm

�0:08

(10)

where la is the combined heat transfer coefficient (¼lcþ lr)

[W m�2 K�1]; lr is the linearized radiation heat transfer coeffi-

cient [W m�2 K�1]; At and Af are the tube and fin surface areas

[m2], respectively; pt and pf are the tube and fin pitches [m], re-

spectively; Dt and Df are the tube and fin diameter [m], respec-

tively. Eq. (10) fitted the experimental data within �10% error

bands (Hermes and Melo, 2007).

2.2. Capillary tube/suction line heat exchanger

In household refrigerators, the capillary tube forms a counter-

flow heat exchanger with the suction line, in order to increase

refrigerating capacity and to prevent slugging of the compres-

sor. Two types of the so-called capillary tube to suction line

heat exchanger (CT/SL HX) are usually found: lateral and

to evaporatorsuction line

tube

x Lex

from evaporator

y tube–suction line heat exchanger.

0 2 64 8 10 12 14 16 18

Measured mass flow rate [kg/h]

0

2

4

6

8

10

12

14

16

18

Pre

dict

ed m

assf

low

rat

e [k

g/h]

HC-600a / adiabaticHC-600a / non-adiabaticHFC-134a / adiabaticHFC-134a / non-adiabatic

+10%

-10%

Fig. 4 – Validation of the capillary tube sub-model.


concentric. In the lateral configuration the capillary tube is

brazed to the suction line whereas it passes inside the suction

line in the concentric arrangement. Fig. 3 shows a schematic

diagram of a concentric CT/SL HX, where three distinct flow

regions can be observed: entrance region (Len), heat exchanger

region (Lhx), and exit region (Lex). Both Len and Lex are usually

assumed to be adiabatic.

The CT/SL HX sub-model was based on the work of Hermes

et al. (2007), where the following simplifying assumptions

were adopted: (i) one-dimensional, viscous, compressible, ho-

mogeneous, and quasi-steady flow; (ii) negligible diffusion ef-

fects; (iii) negligible heat conduction in the tube walls; (iv)

straight, horizontal and constant cross-sectional area tube;

(v) negligible pressure drops at the tube inlet and outlet; and

(vi) negligible metastable flow effects. The refrigerant flow

through the capillary tube is governed by the mass, energy

and momentum principles, which provided the following set

of differential equations, written here in the pressure domain:

dzdp¼ �Di

4

G2hvðvv=vhÞpþðvv=vpÞh

iþ 1

sh1þ G2vðvv=vhÞp

iþ qGðvv=vhÞp

(11)

dhdp¼ �

s�G2vðvv=vpÞh

þ qG�1

�1þ G2ðvv=vpÞh

sh1þ G2vðvv=vhÞp

iþ qGðvv=vhÞp

(12)

where Di is the capillary inner diameter [m], G is the mass flux

[kg s�1 m�2], s is the shear stress on tube walls [Pa], and q is the

heat flux [W m�2]. The friction factor for both single and two-

phase flow regions was calculated from Churchill’s (1977) cor-

relation. The two-phase flow Reynolds number was calculated

using an empirical equivalent viscosity proposed by Cicchitti

et al. (1960), mtp¼ xmvþ (1� x)ml, that offered the best agree-

ment with experimental data.

Oil-refrigerant mix

compressor power,We

ws

suction

suction valve

dead volume, zo

piston displacement, z

axisradius, R

wo

Fig. 5 – Schematic of the co

From these equations the tube length and refrigerant en-

thalpy were calculated as a function of the pressure drop in

any flow region. As there is no explicit equation for the mass

flow rate, its calculation followed an iterative procedure gov-

erned by the calculated and actual tube lengths (Hermes

et al., 2007). The boundary conditions are the pressure and en-

thalpy at the inlet of the capillary tube and the exit pressure.

The choked flow at the capillary exit was identified by an infin-

ite pressure gradient criterion, i.e., dp/dz /�N (Fauske, 1962).

Eqs. (11) and (12) were solved numerically by a second-order

Heun scheme (Press et al., 1995). Satisfactory results were

shell, Tc

ture

heat transfer rate,Qc

connecting rod, L

crank angle, θ = ω t

piston

cylinder

discharge chamberdischarge valve

wd

discharge

mpressor sub-model.


obtained using only 50 integration points, a figure dramati-

cally smaller than that reported in the literature (w104) (Meza-

vila and Melo, 1996).

The suction line sub-model was based on the following

simplifying assumptions (Hermes et al., 2007): (i) flow of only

superheated vapor along the suction line; (ii) no heat transfer

to the surroundings; and (iii) negligible pressure drop. Since

the heat flux was assumed to be uniform, the suction line

exit temperature was calculated by means of a temperature

effectiveness, i.e., To,sl¼ Ti,slþ 3ct-sl(Ti,ct� Ti,sl). The effective-

ness was obtained from the theoretical 3–NTU relationship

for double-pipe counter-flow heat exchangers with parallel

temperature profiles, 3ct-sl¼NTU/(NTUþ 1), where

NTU¼ (4lslDoLhx)/(cp,slGDi2) is the number of transfer units,

and lsl is the convective heat transfer coefficient between

the refrigerant in the tube annulus and the capillary tube ex-

ternal walls. The convective heat transfer coefficient between

Measured heat transfer rate [W]

0

20

40

60

80

100

120

140

Pre

dict

ed h

eat

tran

sfer

rat

e [W

]

+10

c

a

0 20 40 60 80 100 120 140

0 50 100 150 200 250 300 350 400 450

Measured compression power [W]

0

50

100

150

200

250

300

350

400

450

Pre

dict

ed c

ompr

essi

on p

ower

[W

]

-10

-10

+10

Fig. 6 – Validation of the compressor sub-model: (a) heat transf

(d) discharge temperature.

the refrigerant and the suction line wall was estimated by the

correlation proposed by Gnielinski (1976) using an equivalent

diameter for laminar flows in tube annulus.

The CT/SL HX sub-model predictions were compared with

more than 1000 experimental data points for adiabatic and

non-adiabatic flows of HFC-134a and HC-600a. It was found

that the experimental data are reasonably predicted by the

model, with 85% of all data points falling within an error

band of �10%, as shown in Fig. 4.

2.3. Reciprocating compressor

The compressor sub-model was divided into two domains,

namely compressor shell and compression process, as illus-

trated in Fig. 5. The compression process sub-model provides

the compression power and the discharged refrigerant mass

flow rate and temperature, whilst the shell sub-model

Measured mass flow rate [kg/h]

0

5

10

15

20

25

Pre

dict

ed m

ass

flow

rat

e [k

g/h]

d

b

0 5 10 15 20 25

70 80 90 100 110 120 130

Measured discharge temperature [°C]

70

80

90

100

110

120

130

Pre

dict

ed d

isch

arge

tem

pera

ture

[°C

]

+2°C

-2°C

-10

+10

er rate; (b) mass flow rate; (c) compression power; and


provides the refrigerant mass flow rate aspired from the evap-

orator, the amount of refrigerant that is migrating to/from the

oil, and the rate of heat rejected to the surroundings. Com-

pressor models available in the literature span a wide range

of sophistication, from standardized polynomial fits (ASHRAE

Standard S23, 1993) to first-principles models (Prata et al.,

1994).

The compressor shell sub-model adopted the following

simplifying assumptions: (i) uniform shell temperature; (ii)

no oil circulation outside the compressor; (iii) shell pressure

is equal to the evaporating pressure; and (iv) the refrigerant

is aspired directly from the suction line to the compression

chamber. The refrigerant mass flow rate at the compressor in-

let was calculated through a mass balance within the com-

pressor shell,

ws ¼ wd �wo � Vc _s (13)

The amount of refrigerant absorbed/released from the lubri-

cating oil, wo, was given by

wo ¼ �Moð1� sÞ�2 _s (14)

where s corresponds to the mass fraction of refrigerant HFC-

134a dissolved in the ISO10 oil. Since s is a function of the

evaporating pressure and the compressor shell temperature,

s¼ s( pe,Tc), its time-derivative can be calculated from

_s ¼ ðvr=vTÞp _Tc þ ðvr=vpÞT _pe (15)

where _pe was calculated by the evaporator sub-model,

whereas _Tc was obtained through an energy balance in the

compressor shell,

_Tc ¼ ½wshs �wdhd þWc �UAcðTc � TaÞ�$C�1c (16)

The compressor overall thermal conductance, UAc, was

obtained from experimental data supplied by a hot-gas calo-

rimeter test facility (Hermes and Melo, 2006). A linear relation-

ship, expressed as UAc¼ 3.061–4.61� 10�3pe, with UAc given in

W/K and pe in kPa, was found between the overall thermal

conductance and the evaporation pressure. This equation fitted

theexperimentaldata within anerrorband of�10% (seeFig. 6a).

l

Environment

Aes

Ta≈Tesk-1 k+1kk=1 k=N

z

Fig. 7 – Schematic of the finite-volume

The compression process was modeled following the work

of Hermes and Melo (2006), where the following simplifying

assumptions were adopted: (i) homogeneous properties and

an adiabatic process within the cylinder; (ii) suction and dis-

charge valves modeled as two-position elements (fully open

and fully closed); (iii) valve dynamics were neglected; and

(iv) effective flow areas were approximated by the orifice

cross-sectional areas. The discharged mass flow rate, wd,

and the compression power, Wc, were calculated based on

an energy balance in the cylinder,

McvdTdq¼ ðh� � hÞdM

dq� Tðvp=vTÞv

�dVdq� v

dMdq

�(17)

where dV/dq was obtained from the piston kinematics, and

dM/dq was calculated by8<:

dMdq¼ ws=u; h� ¼ hs ðsuctionÞ

dMdq¼ �wd=u; h� ¼ h ðdischargeÞ

dMdq¼ 0 ðcompression and expansionÞ

(18)

The compression power and the refrigerant flow rate were

computed through the following integral equations:

wd ¼ 0:942

0B@ 1

2p

Z qo

qi

ðdM=dqÞu dq

1CA� 0:469 (19)

Wc ¼ �1:315

0@ 1

2p

Z 2p

0

�p� pe

�ðdV=dqÞu dq

1Aþ 11:2 (20)

where qi and qo are the discharge valve opening and closing

points, and wd is expressed in kg/h and Wc in W. The theoret-

ical values for wd and Wc were corrected using empirical data

(Hermes and Melo, 2006).

In addition, the compressor discharge temperature Td was

calculated by

Td ¼1� 3

qo � qi

Z qo

qi

TðqÞdqþ 3Tc (21)

RefrigeratedCompartment

Tis≈Ti

Rk-1

Ais

Δz

k-th layer

Tk

Ck

Tk+1Tk-1

Rk

discretization of the cabinet walls.


where 3c¼ 1� exp(�5.24/wd) is a semi-empirical temperature

effectiveness.

Eqs. (19)–(21) were integrated using the semi-analytical

method proposed in Hermes and Melo (2006). As can be seen

in Fig. 6, the compressor semi-empirical model predicted ex-

perimental data within �10% error bands for both mass flow

rate (Fig. 6b) and power consumption (Fig. 6c), and the com-

pressor discharge temperature was predicted within �2 �C er-

ror bands (Fig. 6d). It should be emphasized that the

compressor semi-empirical model required only two experi-

mental data points to be ‘calibrated’, instead of the 10-data

points required by the standardized polynomial fits (ASHRAE

Standard S23, 1993).

0 20 40 60 80 100

0

2

4

6

8

10

12

Air

flo

w r

ate

[lilt

re/s

]

totalfreezerfresh-food

-6 -4 -2 0 2 4 6 8 10 12

Temperature [°C]

0

10

20

30

40

50

60

70

80

90

100

Opening [ ]

Ope

ning

[]

MaxMin

a

b

Fig. 8 – Damper model: (a) air flow vs. damper position and

(b) damper position vs. bulb temperature.

2.4. Refrigerated compartments

The refrigerated compartments’ sub-model provides the in-

stantaneous thermal load required to estimate the variation

of the internal air temperatures with time. The thermal load

was divided into four components: (i) heat conduction through

the insulated walls; (ii) heat transmission through the gasket

region; (iii) internal energy generation, and (iv) air infiltration.

The cabinet walls are composed of three layers: inner liner,

insulating foam and outer liner. A scale analysis showed that

the thermal resistance of the insulation is 10, 102 and 105

times higher than the thermal resistances due to the air con-

vection, the plastic, and the steel liners, respectively. A similar

analysis focused on the thermal capacities showed that all

three layers have similar scales. Based on these observations,

the following assumptions were adopted (Hermes, 2006): (i)

the heat conduction through the cabinet walls was regarded

as one-dimensional; (ii) the thermal resistances due to the in-

ner and outer liners and to the internal and external air con-

vection were neglected; and (iii) the wall thicknesses were

considered uniform for both fresh-food and freezer compart-

ments. Thus, equivalent thicknesses were determined from

a reverse heat leakage test (Hermes, 2006), taking into account

not only the heat transmission through the walls, but also the

heat gain in the gasket region. The insulation density was cor-

rected to conserve the overall mass of the wall.

The cabinet walls were modeled following a one-dimen-

sional finite-volume scheme (see Fig. 7), according to which

the temperature at each k-th wall layer was given by

_Tk ¼ awðTkþ1 þ Tk�1 � 2TkÞ=Dx2 (22)

The air-side thermal resistances were neglected, and there-

fore the internal air and the inner liner temperatures were

considered to be the same, yielding

_Trc¼hwrccpðTin � TrcÞ þ 2kwAis

�Tk¼N

w � Trc

�.DxþWgen

i

��Cis þMcpðTa=TrcÞ

�1 ð23Þ

where Trc may be either fresh-food or freezer compartment

temperature [K], McpTa/Trc represents the thermal inertia aug-

mentation due to air infiltration from the surroundings [J K�1],

Cis is the thermal inertia of the plastic liner [J K�1], and

Wgen¼Wfan¼ 7 W for the freezer compartment and Wgen¼ 0

for the fresh-food compartment.

The air temperature at the evaporator inlet was averaged

based on the temperatures and the air flow rates in the freezer

and fresh-food compartments. The air flow rates supplied to

each compartment were measured in a wind-tunnel test facil-

ity (Hermes, 2006), as a function of the damper position (see

Fig. 8a). Since the damper position is a linear function of

the temperature of the fresh-food compartment (see Fig. 8b),

the dynamic response of the damper was also accounted for

by the model.

3. Numerical scheme

The set of dynamic equations was composed of 239 ordinary

differential equations (ODEs): 65 of the evaporator sub-model,

77 of the condenser sub-model, 12 of the compressor sub-

model, and 85 of the refrigerated compartments’ sub-model.


The numerical solution was carried out considering two dif-

ferent time-scales. The fastest was related to the refrigerant

mass migration from one component to another, taking just

a few minutes to reach the steady-state condition. The slow-

est concerned the cabinet thermal inertia and consequently

the rate of heat transfer to the evaporator, taking hours to sta-

bilize. It can be noted that the former is inherent to the refrig-

eration system ODEs, whilst the latter is linked to the cabinet

ODEs. Such a behavior encouraged a solution scheme involv-

ing the segregation of the cabinet and system ODEs, solving

Fig. 9 – Information flow diagram

the air-side ODEs by an explicit Euler method (first-order)

with a time-step. The refrigerant-side ODEs were also inte-

grated explicitly, following an Adams predictor–corrector

method (DE/STEP), with both order and step-size controllers

(Shampine and Gordon, 1975).

The cycle components were coupled to each other by the

mass flow rate/enthalpy pair, which is transported from one

component to another according to the refrigerant flow

direction. The model requires only two initial conditions: am-

bient temperature (32 �C) and refrigerant charge (85 g of

of the solution algorithm.

40

Damper MAXexperimental

a


HFC-134a), since the equalizing pressure is internally calcu-

lated by the model. Fig. 9 illustrates the solution algorithm

in an information flow diagram.

0 1 2 3 4 5 6

Time [h]

-10

-20

-30

0

10

20

30

Air

tem

pera

ture

[°C

]

predicted

freezer

fresh-food

damper action

250

Damper MAXexperimentalpredicted

b

4. Results and discussion

4.1. Experimental work

The refrigerator was tested in an environmental chamber, in

which both air temperature (32� 0.5 �C) and relative humidity

(50� 5%) were controlled (Hermes, 2006). Absolute pressure

transducers (�200 Pa uncertainty) were installed at the com-

pressor suction and discharge ports. The outside air tempera-

ture was measured by five T-type thermocouples (�0.2 �C

uncertainty) placed around the refrigerator. Twelve T-type

thermocouples were used to measure the internal air temper-

ature at several locations within the freezer and the fresh-

food compartments. Air-side temperatures at the condenser

and evaporator inlet and outlet ports were also measured.

Thirteen T-type thermocouples were distributed along the re-

frigeration loop to monitor the refrigerant temperatures at the

inlet and outlet ports of each of the cycle components. The

compressor and evaporator fan power consumption were

also monitored (�0.1% uncertainty).

0 1 2 3 4 5 6

Time [h]

50

100

150

200

Com

pres

sor

pow

er[W

]

Fig. 10 – Model validation in the start-up transients: (a)

compartments’ temperatures and (b) compression power.

4.2. Start-up transients

Fig. 10a shows the time variation of the air temperatures

during the refrigerator start-up, with the damper at the max-

imum cooling position. A reasonable agreement between cal-

culated and measured values was observed for the whole

start-up period. During the first 2.5 h, before the damper was

fully closed, the model tended to overestimate the freezer

cooling rate, providing a temperature 5 �C lower than that ob-

served in the experiments. An inflexion was also observed just

after the damper was completely closed, which was followed

by a decrease in the freezer temperature due to an increase in

the air flow rate supplied to this compartment. The cause this

small discrepancy is the abrupt variation in the air flow rate

near the closing point (see Fig. 9a), which was not observed

in the experimental profile since the damper was not com-

pletely closed due to air leakage.

Fig. 10b shows the time variation of the compression power

during the start-up, with the damper at the maximum cooling

position. It can be noted that the model predicted the power

consumption satisfactorily, within �5% error bands for the

whole start-up period. A tenacious observation also reveals

that the model predicted quite well the primary power peak

which took place just after the compressor start.

Table 3 – Steady-state model validation

Damper Fresh-food compartmenttemperature (�C)

Freezertemp

Measured Calculated Difference Measured Ca

MAX 1.6 1.7 þ0.1 �28.0

MIN 9.8 10.8 þ1.0 �29.7

In all cases, the model predictions tended toward the ex-

perimental curves as the simulation approached the steady-

state regime. Table 3 compares the experimental data with

model predictions under the steady-state condition, showing

compartmenterature (�C)

Compressor power (W)

lculated Difference Measured Calculated Difference

�27.7 þ0.3 106.4 108.7 �2.2%

�28.7 þ1.0 101.2 105.7 þ4.4%

tmen

t(�

C) D

iffe

ren

ce

0.9

1.1

0.4

0.2


maximum deviations of �1 �C and �5% for the air tempera-

ture and compressor power, respectively. It can be noted

that the model is able to estimate the pull-down time (i.e.,

the time needed to reach 5 �C in the fresh-food compartment)

with deviations within �5%.

Ru

nti

me

rati

o(d

imen

sio

nle

ss)

Fre

sh-f

oo

dco

mp

art

men

tte

mp

era

ture

(�C

)Fre

eze

rco

mp

ar

tem

pera

ture

lcu

late

dM

ea

sure

dD

iffe

ren

ceC

alc

ula

ted

Mea

sure

dD

iffe

ren

ceC

alc

ula

ted

Mea

sure

d

0.4

30.4

23.2

%11.6

10.8

0.8

�14.8

�15.7

0.5

20.5

03.2

%4.1

4.5

�0.4

�15.4

�16.5

0.5

40.5

6�

4.1

%11.5

10.4

1.1

�21.5

�21.9

0.6

80.6

62.4

%3.0

2.9

0.1

�22.2

�22.4

4.3. Cycling transients

Cycling simulations were carried out with the controlling

devices positioned at their extremes and, thus, four tests

were actually carried out (thermostat–damper): MAX–MAX,

MAX–MIN, MIN–MAX, MIN–MIN. Table 4 compares the model

predictions with experimental data in terms of energy con-

sumption, runtime ratio, and air temperatures. It can be

observed that the model was able to predict the overall energy

consumption and the average air temperatures with maxi-

mum deviations of �10% and �1 �C, respectively. The effect

of the control limits on the predicted average air temperatures

is better illustrated by the control-chart of Fig. 11, where it can

be noted a good agreement between predicted and measured

cycle-averaged compartment temperatures.

Fig. 12 illustrates the suction and discharge pressure pro-

files through a typical cycle with the controls at the MAX–

MAX position. It can be noted that the numerical results

were in close agreement with the experiment data for the

whole period, including the pressure equalization, the equal-

ized period, and the period after compressor restart. During

the pressure equalization period, the model underestimated

the condensing pressure because the heat rejection rate was

overestimated by the condenser sub-model. The plateau ob-

served in the equalizing period relates to the boiling of the re-

sidual liquid in the condenser. After this point on, a good

match between the experimental and calculated pressures

was observed until the compressor restart. At the beginning

of the on-cycle, the model slightly underpredicted both evap-

orating and condensing pressures, although they tended to-

ward the experimental values as the simulation evolved.

Ta

ble

4–

Mo

del

va

lid

ati

on

for

en

erg

yco

nsu

mp

tio

n

Th

erm

ost

at–

da

mp

er

En

erg

yco

nsu

mp

tio

n(k

Wh

/mo

nth

)

Ca

lcu

late

dM

ea

sure

dD

iffe

ren

ceC

a

MIN

–MIN

40.8

44.1

�7.6

%

MIN

–MA

X49.0

51.4

�4.8

%

MA

X–M

IN48.3

53.2

�9.1

%

MA

X–M

AX

61.2

62.9

�2.7

%

4.4. Model potentials

A great potential of the model is the analysis of the refrigerant

migration during the cycling operation, which is complex to

be carried out experimentally. Just after the compressor is

switched off, the condenser holds most of the refrigerant

(w55 g), as illustrated in Fig. 13. During the pressure equaliza-

tion process, refrigerant migrates from the condenser to the

evaporator. The amount of refrigerant within the evaporator

rapidly increases to 55 g, decreasing slightly thereafter due

to its migration to the compressor shell. The amount of free

refrigerant in the compressor shell or dissolved in the oil in-

creases continuously with the evaporating pressure until the

compressor is on again. Just before compressor restarts, there

is approximately 10 g of refrigerant in the condenser, 50 g in

the evaporator, and 25 g in the compressor. After compressor

restarts, the refrigerant rapidly moves from the evaporator

and from the compressor shell to the condenser. Over time,

the refrigerant returns to the evaporator through the capillary

tube. The amount in the compressor decreases continuously

with decreasing evaporating pressure and increasing shell

2 4 6 8 10 12

Fresh-food temperature [°C]

-24

-22

-20

-18

-16

-14

Fre

ezer

tem

pera

ture

[°C

]

experimentalpredicted

Fig. 11 – Model validation in the cycling transients:

averaged temperatures for various control settings.

0 20 40 60 80 100

5

10

15

20

25

30

35

40

45

50

55

60

65

Mas

s of

ref

rige

rant

[g]

Normalized period [ ]

- off - -on-

restart

evaporatorcondensercompressor

Fig. 13 – Refrigerant migration in the cycling operation.


temperature, as both tend to reduce the amount of refrigerant

dissolved in the lubricant oil.

5. Final remarks

A first-principles methodology for modeling and simulating

the dynamic behavior of domestic refrigerators was intro-

duced. Innovative features were incorporated into all

0 20 40 60 80 100

Normalized period [ ]

0

4

8

12

16

Pre

ssur

e [b

ar]

suctiondischarge

- on -- off -

start-up

Fig. 12 – Model validation in the cycling transients: suction

and discharge pressures.

component sub-models in order to guarantee both physical

reliability and numerical robustness. Numerical predictions

were compared to experimental data showing a satisfactory

level of agreement for the whole range of operating condi-

tions, including start-up and cycling regimes. The energy con-

sumption and the air temperature maximum deviations from

the experimental data were found to be �10% and �1 �C, re-

spectively. The code predicts 12 h of testing using only

30 min of CPU time (Pentium M 2.13 GHz; 1024 Mb RAM). Fur-

thermore, the model can be easily adjusted to any kind of cab-

inet model, and it can be applied to assess the refrigerator

energy performance in several real-life engineering situa-

tions, which will be explored in a further publication.

Acknowledgments

Financial support from Whirlpool S.A. and the Brazilian fund-

ing agencies, CAPES and CNPq, is duly acknowledged. The first

author is grateful to the National Institute of Standards and Tech-

nology, particularly to Dr Piotr A. Domanski for hosting him

during his sabbatical stay at NIST, during which this work

was partially carried out.

r e f e r e n c e s

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a first-principles simulation model for the start-up and

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