Heat Recovery Systems & ClIP Vol. 13, No. 2, pp. 97-113, 1993 0890-4332/93 $6.00+ .

00
Printed in Great Britain Pergamon Press Ltd

A THREE-ZONE SIMULATION MODEL FOR

AIR-COOLED CONDENSERS
M . L. MARTINS COSTA a n d J. A . R . PARISE*
Department of Mechanical Engineering, Pontificia Universidade Cat61ica do Rio de Janeiro,
22453 Rio de Janeiro, Brazil

(Received in revised form 10 September 1992)

Al~lzact--The present paper deals with the development of a mathematical model for the performance
prediction of air-cooled condensers. The model considers the heat exchanger as formed by three distinct
zones: de-superheater, condenser and subcooler. To make the model as generally applicable as possible,
each piece of straight tube, between two return bends, is treated as a separate heat exchanger. The coil
can thus be formed by any particular composition of straight tubes, enabling the overall effectiveness
of each zone to be determined, with the use of correlations available in the literature. Zone-ending
(starting of condensation or subcooling half-way along a tube) is also considered, with appropriate
modelling. The energy balances over each zone, for both air and refrigerant streams, together with the
effectiveness equations, form a set of non-linear equations, which are solved numerically. Despite its
simplicity the method produces results comparable with those produced by more sophisticated local
analyses. In the present paper six coil configurations are studied. Predicted results are compared with
existing experimental data, with generally good agreement. A few possible applications of the method are
also presented.

NOMENCLATURE

A a r e a ( m 2)
e specific heat (kJ kg- i K - l )
C capacity rate (kW K - 1 )
D diameter (m)
effectiveness
h~ latent heat (kJ kg- ~)
I number of rows (high)
K number of rows (deep)
l tube length (m)
rh mass flow rate (kg s- t )
M number of rows in zone z
NTU number of transfer units
heat transfer rate (kW)
R capacity rate ratio
T temperature (K)
U overall heat transfer coefficient (kW m -2 K -1 )
w width (m)
x mass flow rate fraction
Y parameter defined in equation (40)
Subscripts
a air
c condenser
cd condensation
d de-superheater
f refrigerant
i inlet
j basic module
1 liquid
m minimum
o outlet
p constant pressure
s subcooler
t total
v vapour
z zone

*To whom correspondence should be addressed.

97
98 M.L. MARTINSCOSTAand J. A. R. PARISE

1. INTRODUCTION

The present paper deals with the development of a mathematical model for the performance
prediction of air-cooled condensers. Figure 1 shows a typical set-up. One important application
of air-cooled condensers refers to vapour-compression cycles, largely employed in refrigerating and
heat pumping systems. In such applications condensers have to cope with de-superheating,
condensation and eventual subcooling of the refrigerant. As the first zone involves heat exchange
between two gases, with consequent low heat transfer rates, it is of great importance tO predict
which proportion of the condenser total area will be covered by de-superheating.
The existence of three zones represents a difficulty in the modelling of air-cooled condensers, as
conventional single-zone methods (overall effectiveness or mean temperature difference) do not
assess the actual effect that each separate zone has on condenser performance. The use of a single
overall heat exchanger conductance is certainly an over-simplification which precludes any proper
condenser analysis. More complex analyses are therefore required.
A number of papers on modelling of air-cooled condensers can be found in the literature.
Rozenman and Pundik [1], for instance, investigated the effect of unequal heat loads on the
performance of air-cooled condensers. For that, a stepwise calculation procedure, for a one-pass
or multi-pass condenser, has been presented. Simulation models have also been developed for
particular applications. Such was the case of Fleming [2] and Hogan [3] who studied models for
a variable speed heat pump and a heat pump grain drier, respectively. Breber et al. [4] employed
a simple method to study the effect of the presence of non-condensable gases in air-cooled
condensers.
Parise and Cartwright [5, 6] indicated that, even within the phase-change region, the condensing
fluid might sustain considerable changes in its film coefficient, due to the many different two-
phase flow regimes that occur within a tube. To cope with that, a local method was developed
wherein the condenser was divided into small tube sections. At each section the predominating
flow regime was identified, with the help of a two-phase flow diagram, thus enabling the
local refrigerant-side film coefficient to be determined. Proper energy balance and heat transfer
equations were applied to each section, forming the model. The same local analysis approach was
employed by Huang and Pate [7] in a study where emphasis was put on the in-tube heat transfer
enhancement.
The present model considers the heat exchanger as formed by three distinct zones: de-super-
heater, condenser and subcooler. To make the model as generally applicable as possible, each piece
of straight tube, between two return bends, is treated as a separate heat exchanger. The coil can
thus be formed by any particular combination of straight tubes, enabling the overall effectiveness
of each zone to be determined. The method consists in solving the set of non-linear equations that
result from the energy balancing and zone effectiveness equations.
The three-zone approach has already been employed in condenser modelling by a number of
authors, including Freeman et al. [8], Davis and Scott [9], Blundell [10] and Hiller and Glicksman
[11]. Simplifying assumptions, however, had to be made when determining the effectiveness (or the

REFRIGERANT
INLET

, [~FRIOER/~/"
OOLET

Fig. 1. Typicalair-cooledcondenser.
 Air-cooledcondensers 99

mean temperature difference) of each zone, so that real effect of coil configuration on condenser
performance was not taken into account.
Condenser modelling can also be found in two popular vapour-compression simulation models:
the Oak Ridge National Laboratory Heat Pump Model, described by Fischer and Rice [12], and
in an updated version by Fischer et al. [13], and the National Institute for Standards and
Technology Model, by Domanski and Didion [14, 15]. The ORNL heat pump model uses a
three-zone condenser model with parallel refrigerant circuits. On the other hand, a different
approach is used in another ORNL computer model, specifically developed for air-cooled
condensers with complex refrigerant circuiting, as reported by Ellison et al. [16]. The model adopted
a tube-by-tube computational approach for calculating the thermal and fluid-flow performance of
each tube in the heat exchanger individually, using local temperatures and heat transfer coefficients.
The NIST model [14] employs a three-zone condenser model that follows the refrigerant flow
tube-by-tube. The NIST condenser model was further adapted to accommodate the study of a
non-azeotropic refrigerant mixture [15]. Both ORLN and NIST models have become standard
reference tools in the HVAC industry in the U.S.A.
It is shown that, despite its simplicity, the present method produces results comparable with those
obtained by complex local analyses. In the present analysis six different coil configurations were
studied. The paper also compared predicted results with experimental data from an existing
condenser.

2. MODEL DESCRIPTION

The model is intended to simulate forced-convection air-cooled condensers. Air is forced past
a series of straight tubes, which can be interconnected in many different ways, forming the coil.
The condensing fluid flows inside the coil.
The model considers, as known quantities, both air and refrigerant inlet states, their respective
mass flow rates, plus the complete geometric description of the condenser, including the number
of rows, high and deep, fin and tube spacing and coil diameter. Fouling, in both sides, is also
known.
Only the condenser heat transfer total area, At, is known. It is, of course the sum of each zone
heat transfer area. Thus,
At = Ad + A¢ + As. (1)
The values of Ad, A¢ and As are to be determined by the model. The total area is given by
A, = n D w l K . (2)

2.1. Coil configurations

The tube elements can be arranged in several ways, composing a vast number of different coil
configurations. Six of them, represented in 3 x 3 configuration in Fig. 2, have been studied. They
are believed to cover the majority of basic circuitry types. In practice, coils of air-cooled condensers
can become much more complex [15, 16]. The Appendix discusses how these complex circuitry coils
could be handled by the model.
(a) Multi-circuit crossflow--air flows in a straightforward crossflow, with refrigerant being split
into K circuits.
(b) Multi-pass overall parallel flow--refrigerant flow is distributed into I circuits which follow
an overall parallel flow.
(c) Single-circuit overall parallel flow--all tubes form a single crossflow circuit, with K passes,
in overall parallel flow.
(d) Multi-pass overall counterflow--similar to configuration (b), apart from the refrigerant
overall counterflow nature.
(e) Single-circuit with compound passes of alternating parallel and counterflow.
(f) Single-circuit overall counterflow--similar to configuration (c), but with refrigerant in overall
counterflow.
100 M.L. MARTINSCOSTAand J. A. R. PARISE

,~, ~J) ,, ,b, ~ ', ,o, ,~ .

0 ~ I.,'L/~ "- -I1~ l 0 ,/,
AIR ~
: .......
H I -td~ = : (i) I ~ c:i .I i)! 1.
II I ' 'a (, r.-~

Fig. 2. Coil configurations. (a) Multi-circuit crossflow; (b) multi-pass overall parallel flow; (c) single-
circuit overall parallel flow; (d) multi-pass overall counterflow; (e) single-circuit with compound passes
of alternating parallel and counterflow; and (f) single-circuit overall counterflow.

2.2. Refrigerant-side heat balance equations

Irrespective of the coil arrangement, the following equations apply for each zone.
(i) De-superheater
Qd = Cf,(Tri - T~). (3)
(ii) Condenser
Qc = rhrhlv. (4)
(iii) Subcooler
Os = Cfl(Tcd-- Tfo), (5)
where Crv and Cn are the capacity rates for vapour and liquid refrigerant, respectively.
Cn = rhf% (6a)
Cry = rhfCpv. (6b)
2.3. Air-side heat balance equations
Contrary to the refrigerant equations (3)-(5), air balance equations may have different results
for each coil arrangement. This is because, in some cases, the air stream is split into all three zones,
with mass flow rate proportional to each zone area whilst, in other cases, the total air stream is
forced past one zone before going into the next. This is illustrated by the schematic flow diagrams
for each configuration in Fig. 3. The air total capacity rate is given by
Ca = Fna Cpa. (7)

ai ao
ai

°'1 ado aco

A~

Fq aCO aSO

(b,c) (d,f)
,o,
(a,e)

Fig. 3. Schematic flow diagrams for configurations (a)-(f).

 Air-cooled condensers 101

2.3.1. Compound and crossflow configurations (a, e).

O.d = C, xd(r, do- T.i) (8)
0.~ = C, xc(T~o -- T,a) (9)
0.~ = C a x s ( T a ~ - T.,). (1o)
For any configuration, the air stream is considered as equally divided amongst the I rows
(horizontal planes). Since, for configurations (a) and (e), each zone will occupy a proportion of
these rows, the corresponding air mass flow rate will be proportional to the zone heat transfer area.
This assumption is more approximate for (e) as back rows may have both single and two-phase
sections. The mass flow rate fractions, xd, xc and xs are, therefore, given by the ratio of each zone
area to the total heat exchanger area.
hd
Xo = A--~ (1 la)

he
xc = A---~ (1 lb)

,4s
x~ = - - . (1 lc)
At
2.3.2. Overall parallel flow configurations (b, c).
Qd = C a X d ( T a d o - Tai) (12)

(~ = C,x¢(T,¢o- Taao) (13)

Q-~s= Caxs(Tao -- Taro). (14)
Regarding configuration (b), being multi-pass overall parallel flow, each zone should receive the
whole air stream. However, there may be situations where one zone does not completely occupy
a single row. In such cases, which are likely to occur with moderate superheats or subcoolings, x
is defined as follows:

xd = ~ , ifld<w (15a)
w
lc
x¢=-, if l¢< w (15b)
w
is
xs=-, if ls < w, (15c)
w
where l is the length of each of the I tubes occupied by a zone. If, at least, one whole row of I
tubes is occupied by the zone, x becomes equal to unity.
Apart from the refrigerant pressure drop due to return bends, there should be no difference of
configuration (c) to one composed by K tubes, wI long, disposed in overall parallel flow. For this
reason, x can be defined as follows:

Xd = WI (l 6a)

l0
xc wI (16b)

xs wI (16c)

Equations (16a)-(16c) apply for l < wL

2.3.3. Overall counterflow configurations (d,f). Configurations (d) and (f) are similar to (b) and
(c), respectively, but with opposite air flow. This implies different air temperature differences across
102 M.L. MARTINSCOSTA and J. A. R. PARISE

each zone. The parameter x continues to be defined by equations (15a)-(15c) for configuration (d),
and by (16a)-(16b) for configuration (f).
Qd = Caxd(Tao - Taco) (17)

Oc = Caxc(Taco -- Taso) (18)

0s = Caxs(Taso -- Tai)" (19)

2.4. H e a t transfer rate equations

These equations are based on the heat exchanger effectiveness concept. For that, minimum
capacity rates, for both subcooler and de-superheater, are defined as follows:
Cmd = min(Crv, XdCa) (20)
Cms -~- m i n ( C n , Xs Ca). (21)

Reference to Fig. 4 gives the maximum temperature difference (i.e., the difference between the inlet
temperatures of the refrigerant and air streams) that occurs at each zone of the six configurations.
2.4.1. De-superheater.
(i) Compound, crossflow and overall parallel flow configurations (a, b, c, e)
Qd = CrndEd(Tfi -- Tai). (22a)
(ii) Overall counterflow configurations (d, f)
Qd = CmdEd(Tri-- Taco). (22b)
2.4.2. Condenser.
(i) Compound and crossflow configurations (a, e)
0,~ = XcCRE~(T~d -- Tai). (23a)
(ii) Overall parallel flow configurations (b, c)
0_~ = xcCaE~(T~ - T~do). (23b)

Cf, L

z~
Co Xz
t

MULTI-CIRCUIT CROSSFLOW

of/%

'~'~:' ], ~ , b = ~ ~L-_-::_q4_-_
.... Y ..... Y
MULTI~ FLOW

c~

co)CO. I, I--"-t,' I,-~~--J 3 ~ . . . . . . .

SINeI..E CIRCUIT
Fig. 4. Air and refrigerant temperatures for configurations (a)-(f).
 Air-cooled condensers 103

(iii) Overall counterflow configurations (d, f)

O.c = xcCcEc(T,~ - T~). (23c)
2.4.3. Subcooler.
(i) Compound, crossflow and overall counterflow configurations (a, e, d, f)
O, = Cm, e,(rcd - T,i). (24a)
(ii) Overall parallel flow configurations (b, c)
Qs = Cms~s(rcd -- T~co). (24b)

2.5. Effectiveness equations

In this section, equations are derived for the determination of the effectiveness of each zone. The
condenser effectiveness, independently of coil configuration, is given by
Ec= 1 - exp(- NTUc), (25)
where the condenser number of transfer units is defined as
UoAc
NTUc = (26)
C,x¢
Clearly, the effectiveness of the de-superheater and subcooler zones will depend on the refrigerant
circuiting configuration. Simple correlations for overall parallel and counterflow could be used, as
provided by Kays and London [17] even though the analysis of the actual effect of each type of
coil arrangement would be impaired. In the present work a more elaborate method for
the determination of the zone effectiveness is employed. Basically, each piece of straight tube,
limited by two return bends, with air in crossflow, is considered as a sole heat exchanger. The
way these tubes are arranged will determine the coil configuration, which now is considered as
an assembly of heat exchangers (i.e. straight tubes). Domingos [18] developed a method for
the determination of the effectiveness of such assemblies, in terms of the effectiveness of
each individual heat exchanger. Refrigerant, flowing inside the tube, is assumed to be fully
mixed, whereas the air stream, due to the plate fin arrangement, is considered unmixed.
With these assumptions the effectiveness of the basic module is given by Kays and London
[17].

Ej=~{1-exp[-~[1-exp(-NrUj)]]}, if Crj > C,j (27a)

NTUj Cq ,
Ej = 1 - e x p { - ~ [1 - exp( ~ ) 1 } if C,j > Cfj, (27b)

where C,j and Crj are the local air and refrigerant capacity rates. The number of transfer units is
defined as

UjAj (28)
NTUj = rain(Co ' C,j)'
where
Aj = nDw. (29)
Expressions for the effectiveness of the de-superheater and subcooler, for each coil configuration,
are derived in the following sections. Figure 5 depicts the schematic diagrams for each of the six
coil assemblies. Each small block corresponds to a simple straight tube module. It is assumed that
all straight tube modules in a single zone have the same effectiveness. This simplifies the application
of Domingos' [18] correlations.
104 M. L. MARTINSCOSTAand J. A. R. PARISE

rai

rn
~ ! r.ao Ir~ I r : "°r®
~ D AND CROSSFLOW ((~,e)

T,i ' 'rod' 'rod' 'r,o

OVERALL PARALLEL FLOW (b,c)

T;, r',,
OVERALL ~ R FLOW (d,f)

Fig. 5. Schematic flow diagrams for de-superheater and subcooler, for configurations (a)-(f).

2.5.1. Multi-circuit crossflow (a). The air and refrigerant capacity rates, per module, are given
by
Caj = x~Ca, (30)
with xz defined as in equations (11 a) and (1 l c).
Cf
Crj = (31)

with the minimum to maximum capacity rate ratio defined as

min(Gj, C,j)
(32)
Rj = max(Cfj, C,j)"
The zone effectiveness, for either de-superheater or subcooler, is thus given by

111 - - \(RJq'~lx
Ez=~jj K ]J ' ifCfj<C,j (33a)

G=I-[ 1 - ( \ R j'jK )"~]x,

] ifCfj>Caj and Rj> 1 (33b)

1
G=l-(1-q)x, ifCfj>C,j and Rj<~. (33c)

2.5.2. Multi-pass overall parallel flow (b). In this configuration both de-superheater and
subcooler are regarded as being formed by M vertical planes of I tubes in crossflows. These planes,
of course, are assembled in overall parallel flow, in relation to air. Thus, according to Domingos
[18], the effectiveness of an assembly of M heat exchangers in overall parallel flow can be given
by
1 - [1 - Ej(Rj + 1)1M (34)
G= Rj+I '
 Air-cooled condensers 105

with
C,x, (35)
C,j = I

Crj = Cf (36)
I"
The parameter x, is defined by equations (15a) and (15c).
2.5.3. Single-circuit overall parallel flow (c). Configuration (c) is considered as formed by K
tubes, wl long, arranged in overall parallel flow. With zone z occupying M vertical planes, one has
[18]
1 - [1 -- Ej(Rj -4- 1)]u
(37)
E,= Rj+I
with x, defined by equations (16a) and (16¢), and
caj = C , x , (38a)
cfj = Cr. (38b)
2.5.4. Multi-pass overall counter[low (d). Similarly to configuration (b), with M planes,
now arranged in overall counterflow, the zone effectiveness, according to Domingos [18] is given
by
Ez= Yj- 1 (39)
~-Rj'
where

L (1 - Ej) J " (40)

The local capacity rates are determined as follows:

C,x,
c,j= t ' (41)
with x, defined as in equations (153) and (15c).

Co =--.Cf (42)
I
2.5.5. Single-circuit with compound passes of parallel and counterflow (e). Configuration (e) is
characterized by alternating passes of parallel and counterflow. For this reason the de-superheater
and subcooler zones are formed by M two-plane modules, each including one parallel and one
countefflow pass. The effectiveness of the parallel pass, Et, is obtained in a similar way to
configuration (b), from equations (34) to (36). Similarly, equations (39)-(42), derived for
configuration (d), apply for the effectiveness of the counterflow pass, E2. The effectiveness of the
two-plane module, E3, has been derived by Martins Costa [19] based on Domingos' [18] method.
It gives
2(El + E2) -- E,E2Rj
E3= 4 , ifCrj>C,j (43a)

£3=EI "~- ~2 - - EIE2, if Cfj < Caj and Rj < 0.5 (43b)
El £2
~l -I- E2 ----
2Rj
E3= 2Rj , ifC o<C,j and Rj>0.5. (43c)

The zone is then formed by an assembly of M two-phase modules, with refrigerant flowing in series
and the air stream being equally divided amongst them. The resulting equations are similar to
106 M . L . MARTINSCOSTA and J. A. R. PARISE

equations (33a)-(33c), with K being replaced by M, ci replaced by e3 and the capacity rates defined
as follows,
C,
Caj = ~ (44a)

(::fj = Cf. (44b)

In case the counterflow pass is not first, equations (43a)-(43c) do not apply. This case was not
considered in the present work.
2.5.6. Single-circuit overall counterfiow (f). Similarly to configuration (c), each zone is formed
by M vertical planes with tubes wI long, allowing air and refrigerant to flow counter-currently. With
such an arrangement equations (39) and (40) will apply, for capacity rates given by
C.j = Ca. (45a)
When the zone does not occupy one plane completely, equation (45a) is replaced by equation (38).
Crj = Cf. (45b)

2.6. Further remarks

It should be remembered that the above equations, derived for all six configurations, do not cope
with the situation of zone-ending (starting of condensation or subeooling) half-way along a tube,
or a module. Such cases, more than likely to occur, required further analysis to be developed.
Basically, the remaining part is regarded as a separate heat exchanger, as shown in Fig. 6. A heat
balance is then made to evaluate the overall performance of the zone, now formed by the complete
M-row heat exchanger (DSA) and part of the next row (DSB), the area of which has to be
determined. This is made with the usual heat balance equations. Expressions for the effectiveness
of the M-row heat exchanger have been derived in the previous sections. As regards the remaining
part, the determination of its effectiveness should present no problem (straight finned tube;
crossflow; equations (27a) and (27b)). Referring to the example given in Fig. 6, the air outlet

SUPERHEATED , ~ AIR
VAPOUR Toi

DSA
(-
Ds. ,)
oF
CONDENSATION

~, Tai
DSA ]
~ Tadoo

' T°da° I T°db°

I Todo

I °° ]
Fig. 6. Heat balance over a zone with a partially occupied row.
 Air-cooled condensers 107

temperature from DSA (de-superheating region excluding the tube where condensation starts),
Tadao, is calculated from energy balance and heat transfer rate equations. The vapour quality at
the DSA exit can be calculated (refrigerant-side energy balance) so that one knows the area of the
tube required to complete de-superheating. That gives the area of DSB. The air inlet temperature
for DSB is Tad~o. The air inlet temperature for the two-phase region is /'ado, which is an
area-averaged value between T~d~oand T~dbo.Shortage of space precludes the full treatment to be
presented. It can be found in ref. [19].

2.7. Film coefficients

For the air-side film coefficient a correlation from McQuiston [20] for plate-fin and tube
surfaces, has been employed. It relates the heat transfer coefficient to the air flow Reynolds number,
tube diameter, longitudinal and transverse pitches, fin spacing and thickness, and number of tube
rows.
In the de-superheater and subcooler zones, for the refrigerant side film coefficient, the
Dittus-Boelter relation was used. The average values (local values integrated over the length of the
two-phase region) for the condenser zone heat transfer coefficient are provided by integrating the
Traviss et al. [21] equation, as calculated by Fischer and Rice [12].
Care should be taken, in the de-superheating region, when the tube wall temperature is below
the refrigerant saturation temperature. This superheated vapour should condense directly on the
surface, without having to be cooled by single-phase forced convection to the saturation
temperature, before condensation occurs (see Fischer and Rice [12], p. 53).

3. METHOD OF SOLUTION

In the previous section, the energy balances over each of the three zones, for both refrigerant
and air streams, together with the heat transfer rate and effectiveness equations, have been
developed. They form a set of equations which can be reduced, for any of the six coil arrangements,
to a single non-linear equation, which is then solved numerically. This equation, with the
de-superheater area as the independent variable, is continuous, monotonic but piecewise differen-
tiable on the interval 0 < Ad < At. This required a reliable method (bisection) to be employed.
As the equation is solved only once, for parallel (b, c) and crossflow (a) arrangements, the use
of the bisection method would not require large computing times, particularly on a
mainframe computer. Overall counterflow configurations (d, f), together with arrangement (e),
which has counterflow passes, require a first estimate of air outlet temperature to be made.
The system of equations is then solved iteratively, with updated values of the air outlet
temperature.
The area of the two-phase condenser zone is obtained from equations (25) and (26), whereas the
area of the subcooler zone comes directly from equation (1).
Table 1 summarizes the equations and unknowns for each flow configuration treated.

4. COMPARISON WITH E X P E R I M E N T A L RESULTS

The model has been applied to predict the performance of an existing air-cooled condenser, part
of a heat pump assisted grain dryer [22]. Experimental and predicted values were compared for
a number of operating conditions, with air mass flow rate acting as the varying parameter.
Figure 7 shows the comparison between predicted capacities and the experimental values of the
total heat transfer rate, evaluated at both refrigerant and air streams. There was, as seen in
Fig. 7, a discrepancy between air and refrigerant measurements (temperatures and mass flow rates).
Nevertheless, it can be observed that good agreement has been obtained, with predicted results lying
within the uncertainty of the experimental measurements, which have been estimated by Pereira
[22] to be smaller than 8.7%. Comparisons made with the condenser model from HPSIM code [14],
against two other operating conditions from the same test rig [22], are also included in Fig. 7.
Table 2 lists the design and operating characteristics of the condenser.
108 M. L. MARTINSCOSTA and J. A. R. PARtSE

Table 1. Equations and unknown variables for each flow configuration treated
Configuration Equations Unknowns
(a) (I),(3),(4),(5).(8),(9),(10),(lla),(llb),(llc)
(22a).(23a),(24a).(25). Q d , Q c , Qs,
(27) (for DS and SC), X d , X c , X s,
(33) (for DS and SC) Tado, r.co, T,,o, Tro,
(d, ~c, Es,
E~d. Ejs
(b) (1), (3), (4), (5), (12), (13), (14) a..d, 4c, A.,,
(16a), (16b), (16c), Qd, Qc, Qs,
(22a), (23b), (24b), (25). Xd,Xc,Xs,
(27) (for DS and SC), Tado , Taco, T~.o, Tro ,
(34) (for DS and SC) Ed, ~c, Es ,
~ . ~jl
(c) (1),(3).(4).(5),(12).(13),(14), 4d, 4o, A.,,
(16a), (16b). (16c) Od, Qc, O,,
(22a),(23b).(24b),(25), Xd, Xc. Xl,
(27) (for DS and SC), T, do, T,,o, T,o, Tro,
(37) (for DS and SC) Ed, Ec, £s,

(d) (1),(3),(4),(5),(15a),(15b),(15c),(17),(18),(19). 4., a.c, a,,

(22b).(23c),(24a),(25), Qd, Qc, Q,,
(27) (for DS and SC), Xd, Xc, Xs,
(39) (for DS and SC) T,~,L~,Lo, Tfo,
~d, £c, ~s,
Ejd, %
(e) (1), (3). (4). (5). (8). (9). (10), 4,. a.o.~.,.
(1 la). (11b), (1 lc), Qd, Qc, Q,,
(22a). (23a), (24a), (25), Xd,Xc,Xs,
(27) (for DS and SC), r , do, T,o~,Tuo, Tro
(43) (for DS and SC). £d, ~c, Es,
(34) (for DS and SC, E~ replacing Q). Ejd, ~js, ¢ld, ¢2d
(39) (for DS and SC, Q replacing Q). E3d, ~ls ~E2s, ~3s
(33) (for DS and SC, M replacing
K and E3 replacing Ej)
(f) (1), (3). (4), (5). (16a), (16b). (16c). (17), (18), (19),
(22b), (23c), (24a), (25), Q,,,Qc, Q,
(27) (for DS and SC). Tao, T~.,, T,m, Tfo
(39) (for DS and SC)
~.~,

19 21 2.3 2.5
2.5", ........ "~ . . . . . . . . . T . . . . . . . . . 2.5
o Refrigerant s t r e a m [pres. m o d e l ] /
• Refrigerant .stream [14J_ /
V Air s t r e a m [pres. modelJ /
• Air s t r e a m ~

2.3
8

.!2.1 2.1

1.91.9 . . . . . . . . . . . . . . . . 2.1
......... 2.S "' ' 251"9.
Predicted Capacity (kW)

Fig. 7. Comparison with experimental results.

 Air-cooled condensers 109

Table 2. Design and operating characteristics of the condenser tested [22]

Tube internal diameter 0.0079 m
Tube outer diameter 0.0125 m
Tube length (heat exchanger width) 0.31 m
Transverse pitch 0.025 m
Longitudinal pitch 0.026 m
Circuit configuration (e), singie-circuit with compounding passes
of alternating parallel and countcrtlow
Number of rows (transv. to flow) 12
Number of rows (long. to flow) 4
Fin spacing 0.002 m
Fin thickness 0.0003 m
Fin thermal conductivity 0.028 kW m - ~°C-
Refrigerant R- 12
Range of pressure 12-14 bar
Range of degree of superheat 10-15°C
Range of degree of subcooling 20-25°C

5. A P P L I C A T I O N S

The model has been applied to predict the effect that main design parameters have on the
performance of a typical air-cooled condenser. The baseline geometry and operating conditions
were as follows:
refrigerant stream
type: R-12
pressure: 13.4 bar
inlet temperature: 69.2°C
mass flow rate: 0.0161 kg s-~ (velocity: 1967 m s-1)
air stream
inlet temperature: 36.3°C
mass flow rate: 0.384 kg s- l
geometry
tube internal diameter: 0.0077 m
tube outside diameter: 0.0094 m
tube length: 0.31 m
transverse pitch: 0.023 m
longitudinal pitch: 0.009 m
fin spacing: 0.001 m
fin thickness: 0.0005 m
fin thermal conductivity: 0.228 kW m -1 °C-'
number of rows (transv. to flow): 12
number of rows (long. to flow): 4

f ~ a.ln

'o7" /~-e
,s S

02 0,3 0.4
HEAT E X ~ WIDTH (m)

Fig. 8. Influence o f c o n d e n s e r frontal area o n c o n d e n s e r capacity.

110 M . L . MARTINSCOSTA and J. A. R. PARISE

-). ~o /."
// ]
!
f" I i i

[ o /.""
;
~-~ ~ / . / ! Lb
/
1.5 2 2.5 3
FRONTAL AIR VELOCITY(m/s)

Fig. 9. Influence of air incoming ve|ocity on condenser capacity.

fouling factors
refrigerant-side: 0 m: °C W - 1
air-side: 0 m 2°C W - i.
From the baseline case, parameters were varied, one at a time, as follows:
tube length: from 0.2 to 0.4 m
air velocity: from 1.2 to 3.0ms -]
degree of superheat: from 5 to 45°C.
Figures 8-10 show the variation of the total condenser capacity with the frontal area, air
incoming velocity and refrigerant superheat, respectively. It can be said that the overall counterflow
configurations (d, f) presented a superior capacity over the entire range of the parameters, when
compared with the overall parallel flow arrangements (b, c). The crossflow (a) and compound flow
(e) configurations remained between these two extremes.
Figure 8 shows the influence of the condenser frontal area, here represented by a varying coil
width (w). As expected, there is an increase in the condenser capacity with w, as this implies a larger
heat transfer area. The rate of increase becomes smaller after the original width (0.31 m) has been
reached. This is because the air mass flow rate has been kept constant which, together with an
increasing frontal area, results in lower air velocities, with consequent poor heat transfer rates. The
discontinuity that is found for the compound configuration (e), at about 0.325 m, is due to the
occupation, by the subcooler, of a new overall counterflow pass.

.¢,~
~2.4 d,f 7 --'..~"

.~-..f.-z'..- "" ~
,,'22 ..."'f":~''~'7""

I0 20 30 40
DEaREE OF SUPERHEAT{eel

Fig. 10. Influence of r©frigcrant superheat on condenser performance.

 Air-cooled condensers 111
t0

>-

ioj J
_.1

THREE Z ~
i_ 2 I ........------~

o 20 40
DEGREE OF SUPERHEAT ('C)
Fig. l 1. Comparison between condenser capacities predicted by the single-zone ~-NTU method and the
present analysis.

It can be observed, from Fig. 9, that air incoming velocity has a great effect on condenser total
capacity. The general trend is the same as observed in Fig. 6. It is interesting to note that
configuration (e) presents, at low air velocities, a performance of the same magnitude as of the
overall parallel flow types, only to have a considerable increase at moderate velocities, reaching
the same limit value of the overall counterflow arrangements. This behaviour can, in part, be
explained by the fact that at low velocities very little area is left for the subeooler. In the present
example it occupied, initially, only part of the last pass, of the parallel flow type. As the air velocity
increases, more passes, including the more effective counterflow ones, are "activated" by the
subeooler, thus increasing its heat release. This behaviour can be observed in Fig. 9, at air velocities
of 1.6 and 2.0 m s- ~. Counterflow passes have little effect on the performance of the condenser zone
as its effectiveness is independent of the coil arrangement (equation (25)).
One characteristic of refrigerating equipment and, in particular, heat pump systems, is that the
condenser has to cope with considerable changes in the amount of incoming refrigerant superheat.
Since the de-superheater is the zone with the lowest overall heat transfer coefficient, it is of great
interest to see its effect on the condenser performance, with particular attention on the existing type
of coil arrangement. This is shown in Fig. 10, where predicted values of the condenser total output
are plotted against refrigerant superheat. The trend of greater capacity for the overall counterflow
configurations is confirmed. However, they all present approximately the same percentage increase
in capacity for a superheat increase from 0 to 20°C.
Finally, Fig. 11 compares the condenser capacity as calculated by the single-zone analysis,
equation (25) using the value of Uc, and the present three-zone method. It becomes clear that great
estimate errors are incurred when that simple analysis is employed, particularly for large degrees
of superheat.

6. CONCLUSION
The development of a simulation model for the performance prediction of air-cooled condensers
has been presented. The model employs the three-zone concept, with accurate calculations of zone
effectiveness. Despite its relative simplicity, when compared with more complex analyses [5, 6], the
method is capable of producing reasonable results, dealing with different types of coil configur-
ations. It can be employed as a useful tool in the design of air-cooled condensers. On the other
hand, it can be improved by taking into account, in the calculation, the effect of refrigerant pressure
drop.
The model compares well (in regard to the assumptions made) with the most currently used
simulation models (ORNL and NIST). By pointing to the insensitivity of a former version of the
ONRL model, by Ellison and Creswick [23], in dealing with less regular coil circuitry patterns,
Domanski and Didion [14] proposed the tube-by-tube approach, which had been used by other
authors [3, 5]. More simulation capability was obtained at the cost of increased computational time.
HRS 13/2--B
 112 M.L. MART1NSCA3STAand J. A. R. PARISE

This a p p r o a c h was t a k e n even further by Parise a n d C a r t w r i g h t [6], whose m o d e l divided each t u b e

into small sections. T h e p u r p o s e o f the p r e s e n t m o d e l was to keep the possibility o f dealing with
c o m p l e x refrigerant circuiting, while r e t a i n i n g the simplicity o f the three-zone m e t h o d s . This was
a c c o m p l i s h e d with the use o f m o r e specialized expressions for the z o n e effectiveness. T h e
c o m p u t a t i o n a l w o r k was restricted to the s o l u t i o n o f the energy b a l a n c e a n d the heat transfer
e q u a t i o n s o n l y for each o f the three zones. In other m o d e l s [5, 14] this s o l u t i o n w o u l d have to be
carried o u t for each tube.
O n the o t h e r h a n d , it s h o u l d be n o t e d that the expression for the effectiveness o f the c o n d e n s e r
(or two-phase) z o n e is the s a m e ( e q u a t i o n (25)), regardless o f the coil a r r a n g e m e n t . Therefore, the
i m p r o v e m e n t in the effectiveness c a l c u l a t i o n is restricted to the d e - s u p e r h e a t i n g a n d s u b e o o l i n g
regions.

REFERENCES

1. T. Rozenman and J. Pundik, Effect of unequal heat loads on the performance of air-cooled condensers. AIChE
Symposium Series, Heat Transfer--Research and Design, No. 133, Vol. 70, pp. 178-184 (1974).
2. H. Fleming, A variable speed heat pump. M.Sc. thesis, Manchester Polytechnic, Manchester, U.K. (1978).
3. R. H. Hogan, The development of a low-temperature heat pump grain drier. Ph.D. thesis, University of Purdue, U.S.A.
(1980).
4. G. Breber, J. W. Palen and J. Taborek, Study on noncondensable vapor accumniation in air-cooled condenser, in 7th
Int. Heat Transfer Conf., Paper HX18, Vol. 6, pp. 263-268, Munich (1982).
5. W. G. Cartwright and J. A. R. Parise, Performance estimation of a condenser at off-design conditions using a two-phase
flow diagram, in 7th Int. Heat Transfer Conf. Paper HX19, Vol. 6, pp. 269-274, Munich (1982).
6. J. A. R. Parise and W. G. Cartwright, Local analysis of three-dimensional air-cooler condenser using a two-phase flow
diagram, in 21st National Heat Transfer Conf. Heat Exchangers for Two-phase applications, ASME, HTD-Vol. 27,
Paper CE2, pp. 123-129, Seattle, U.S.A. (1983).
7. K. Huang and M. B. Pate, A model of an air-conditioning condenser and evaporator with emphasis on in-tube
enhancement, Preprints of the 1988 Purdue Int. Inst. Refrigeration (IIR) Conf., in Progress in the Design and
Construction of Refrigeration Systems, pp. 266-276. Purdue University, West Lafayette, U.S.A. (1988).
8. T. L. Freeman, J. W. Mitchell, W. A. Beckman and J. A. Duffle, Computer modelling of heat pumps and the simulation
of solar heat pump systems. ASME Paper 75-WA/Sol-3, ASME Winter Annual Meeting, Texas (1975).
9. G. L. Davis and T. C. Scott, Component modeling requirements for refrigeration system simulation, in Proc. 1976
Purdue Compressor Technology Conf., pp. 401-408. Purdue University, U.S.A. (1976).
10. C. J. Blundell, Optimizing heat exchangers for air-to-air space-heating heat pump in the U.K. Energy Res. 1, 69-74
(1977).
11. C. C. Hiller and L. R. Glicksman, Improving heat pump performance via compressor capacity control-analysis and
test. Energy Laboratory Report, MIT-EL 76-001, Vol I, Section 2.4, pp. 89-96. Massachusetts Institute of Technology,
Department of Mechanical Engineering, Energy Laboratory (1976).
12. S. K. Fischer and C. K. Rice, The Oak Ridge heat pump models: 1. a steady-state computer design model for air-to-air
heat pumps. ORNL/CON-80/R1, Oak Ridge National Laboratory, Oak Ridge, TN, U.S.A. (1983).
13. S. K. Fischer, C. K. Rice and W. L. Jackson, The Oak Ridge heat pump design model: mark III version program
documentation. ORNLfrM-10192, Oak Ridge National Laboratory, Oak Ridge, TN, U.S.A. (1988).
14. P. Domanski and D. A. Didion, Computer modeling of the vapor compression cycle with constant flow area expansion
device. NBS Building Science Series 155, National Institute of Standards and Technology, Washington, DC, U.S.A.
(1988).
15. p. Domanski, Modeling of a heat pump charged with a non-azeotropic refrigerant mixture. NBS Technical Note 1218,
National Institute of Standards and Technology, Washington, DC, U.S.A. (1986).
16. R. D. Ellison, F. A. Creswick, S. K. Fischer and W. L. Jackson, A computer model for air-cooled refrigerant condensers
with specified refrigerant circuiting. ASHRAE Trans. 87, 1106-1124 (1981).
17. W. M. Kays and A. L. London, Compact Heat Exchangers. Mc-Graw-Hill, New York (1964).
18. J. D. Domingos, Analysis of complex assemblies of heat exchangers. Int. J. Heat Mass Transfer 12, 537-548
(1969).
19. M. L. Martins Costa, Simulation of air-cooled condensers. M.Sc. thesis, Department of Mechanical Engineering,
Pontificia Univcrsidade Cattlica do Rio de Janeiro, Brazil (in Portuguese) (1986).
20. F. C. McQuiston, Correlation of heat, mass and momentum transport coefficients for plate-fin-tube heat transfer
surfaces with staggered tubes. ASHRAE Trans. 84, 294-309 (1978).
21. D. P. Traviss, W. M. Rohsenow and A. B. Baron, Forced-convection condensation inside tubes: a heat transfer equation
for condenser design. ASHRAE Trans. 79, 157-165 (1973).
22. C. A. B. Pereira, Experimental analysis of a heat pump dehumidifier running on refrigerant-12. M.SC. thesis,
Department of Mechanical Engineering, Pontificia Universidade Cat61ica do Rio de Janeiro, Brazil (in Portuguese)
(1987).
23. R. D. Ellison and F. A. Creswick, A computer simulation of steady-state performance of air-to-air heat pumps.
ORNL/CON-16, Oak Ridge National Laboratory, Oak Ridge, TN, U.S.A. (1978).
 Air-cooled condensers 113

APPENDIX: COMPLEX CIRCUITRY CONFIGURATIONS

It is possible to adapt the model to more complex configurations. Recalling Sections 2.2-2.5, the refrigerant-side energy
balance equations would, of course, remain unchanged. To establish the air-side energy balance equations (Section 2.3),
an assessment of the distribution of the three zones across the coil, in relation to the air stream (as in Fig. 3), would be
required. This would provide the definition of air-side mass flow rate fractions (x), the air inlet temperature for each zone
as well as the heat transfer rate equations (as in Section 2.4). Finally, to determine the effectiveness of the single-phase zones
(equation (25) would still apply) one should start from the basic tube (equations (27a) and (27b)). Domingos' techniques
[18] could then be applied until an expression for the zone effectiveness is obtained. For instance, a circuit confluence could
be dealt with as two heat exchangers in parallel (upstream of the confluence) followed by a single one in series (downstream).
Overall, the result would be a set of non-linear algebraic equations, similar to those presented in Table 1.
Attention should be brought to the fact that the computational effort for the solution of condensers with complex circuitry
should not differ considerably from that required for the less complex coil configurations in Fig. 2. On the other hand,
the difficulty in performing the analytical work (mainly to determine the effectiveness equations as well as to establish the
heat balance procedure for tubes with both single and two-phase regions) would increase dramatically. The assumptions
that would necessarily be made throughout the derivation of such equations might lead to the conclusion that the sequential
tube-by-tube approach [14, 15] is more suitable for complex circuitry coils.