Modeling Of Wire On Tube Heat Exchangers Using Finite Element Method

Finite Elements in Analysis and Design 38 (2002) 417–434
www.elsevier.com/locate/ÿnel

Modeling of wire-on-tube heat exchangers using ÿnite element
method
G.A. Quadir ∗ , G.M. Krishnan, K.N. Seetharamu
School of Mechanical Engineering, USM(KCP), 31750 Tronoh, Perak, Malaysia

Abstract
Wire-on-tube heat exchangers are analysed under normal operating conditions (free convection) using
ÿnite element method. Galerkin’s weighted residual method is used to minimise the errors. The e ects of
ambient temperatures and mass ow rates of the refrigerant are determined. This is used to ÿnd out the
length of tube required for phase change for its initiation and completion. The methodology adopted is
validated against the existing data for both sensible heat and latent heat transfer. The analysis also leads to
the information about the adequacy of the number of tubes for complete condensation of the refrigerant
vapour under given operating conditions di erent from that of design conditions. This methodology
can be used as a design tool for the design of wire-on-tube heat exchangers. The derating of heat
exchangers under abnormal ambient conditions can also be predicted. ? 2002 Elsevier Science B.V. All
rights reserved.
Keywords: Wire on-tube heat exchangers; Free convection; Finite element method; Phase change

1. Introduction
Wire-on-tube heat exchangers are more often used by refrigerator manufacturers as condensers
mainly due to their simple construction, ruggedness, and low cost. This type of heat exchanger
consists of a single steel tube, bent into serpentine parallel passes carrying the uid, mainly
refrigerant and solid steel wires are attached to the tube to increase the surface area. The solid
wires are spot welded on to diametrically opposite sides of the tubes as shown in Fig. 1. The
refrigerant enters the tube in a vapour state and leaves the condenser in a liquid state undergoing
a phase change. The heat transfer takes place from the outer surfaces of the wires and tubes
∗

Corresponding author.
E-mail address: (G.A. Quadir).

0168-874X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 8 7 4 X ( 0 1 ) 0 0 0 7 9 - 8


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G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

Nomenclature
A
Cp
D
H
h
hf
hfg
L
m
N
P
S
T; t
U
X
x

area, m2

speciÿc heat capacity at constant pressure, W=kg K
diameter, m
speciÿc enthalpy of saturated vapour, kJ=kg
speciÿc enthalpy of a liquid, kJ=kg
speciÿc enthalpy of saturated liquid, kJ=kg
latent heat of vaporization, kJ=kg
length, m
mass ow rate, kg=s
number, shape function
perimeter, m
pitch, m
temperature, K
overall heat transfer coe cient, W=m2 K
dryness fraction(= (H − hf )=hfg )
distance along the length of an element, m

Subscripts
r
sat
sup
t
ti
to
w
1; 2
∞

refrigerant
saturated condition
superheated condition

tube
inside tube
outside tube
wire
inlet and outlet conditions
ambient temperature

Fig. 1. Sketch of the wire-on-tube heat exchanger.


G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

419

to the external environment by free or forced convection. The analysis of heat transfer through
wire-on-tube heat exchangers is very complex in view of the number of parameters involved
such as the wire diameter Dw , the tube diameter Dt , the wire spacing Sw , the tube spacing St ,
the overall wire length Lw , the length of the tube Lt , the multilayer stacking of the coil, ow
of refrigerant, etc.
Tanda and Tagliaÿco [1] gave a Nusselt number correlation as a function of the geometric
and operating parameters to predict free convection heat transfer from a vertical wire-on-tube
heat exchanger to ambient air based on their experimental results. Hoke et al. [2] carried out
experiments to investigate the air-side convective heat transfer for wire-on-tube heat exchangers
used in most refrigerators. They were able to give a correlation valid for all the experimental data for seven wire-on-tube heat exchangers studied under forced convection regime. The
importance of angle of attack for locating the wire-on-tube exchangers that are cooled by
forced convection is also highlighted [2]. The air-side resistance to heat ow for such type
of heat exchanger is around 95 percent of the total resistance between the refrigerant and the
air when the refrigerant is in two phase region [3]. New and e ective heat exchangers with
tubes ÿnned with wires and spirals were conducted by Martynov [4] to investigate heat exchangers for cryogenic plants. Hiroshi et al. [5] performed experiments to study the ow and
heat transfer characteristics during condensation of R-11 and R-113 in the annuli of horizontal double-tube test condensers. The condensers were made up of a 19:1 mm O.D. corrugated

inner tube with ÿns soldered on the outer surface and three outer smooth tubes of 24.8, 27.2,
and 29:9 mm I.D. An empirical equation for the local heat transfer coe cient was developed,
in which the dimensionless parameters based on the surface tension controlled ow and the
vapour shear controlled ow models were introduced for the low and high vapour velocity
regimes.
The wire-on-tube heat exchangers are also commonly used in immersion cooling of electronic equipments [6]. The miniaturisation of electronic equipments has resulted in high heat
uxes being produced. The simplest type of immersion cooling system involves an external
reservoir which supplies liquid continually to the electronic enclosure. The vapour generated
inside is simply allowed to escape to the atmosphere. Due to several impracticalities associated
with this system, a more sophisticated immersion cooling system is used. In this system, the
vapour is condensed and returned to the electronic enclosure by placing the condenser either
externally or internally to the electronic enclosure. In the external arrangement, the vapour
leaving the enclosure is cooled by a cooling uid such as air or water outside the enclosure. The condensate is returned to the enclosure for reuse. With the internal arrangement,
the condenser is placed in the electronic enclosure. The cooling uid in this case circulates
through the condenser tube, removing heat from the vapour surrounding the wire-on-tube heat
exchanger. The vapour that condenses drips on top of the liquid in the enclosure and continues
to recirculate.
Finite element modeling for the analysis of general types of heat exchangers with and without
phase change has been carried out by various researchers [7–14]. However, to the authors’
knowledge, no attempt has so far been made to analyse numerically the performance of a
wire-on-tube heat exchanger either under free convection or forced convection conditions. The
present paper is an attempt in that direction. Free convection condition on the outside of the
tubes normally exists and hence this case is considered for the analysis of wire-on-tube heat


420

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

Fig. 2. (a) One tube length of the wire-on-tube heat exchanger; (b) Bare portion of tube; (c) Tube with wire

attached.

exchanger using ÿnite element method. Galerkin’s weighted residual approach is adopted to
minimise the residual errors.
2. Analysis
The numerical analysis developed is based on the ÿnite element method. FEM utilises discretisation of the domain over which the solution is sought and obtains the characteristic of one
element in the form of an element matrix depending upon the weighting function assumed for
minimising the residual error. If need arises, the assembly of the element matrix is performed
and the solution obtained for imposed boundary conditions. Otherwise step-wise solution for
each element is carried out over the entire domain.
Fig. 2(a) shows one tube length of a heat exchanger consisting of bare portions of the
tube and the wired portion. A typical one-element discretised model of the wire-on-tube heat
exchanger is shown in Figs. 2(b) and (c) depending upon whether the element refers to the bare
portion of the tube or wired portion of the tube, respectively. Every element has two nodes, one
at entry and the other at exit point of tube side uid. Refrigerant 12 is considered as the uid
entering the tube in a superheated vapour state. The uid ÿrst follows desuperheating process
bringing the state of the uid to dry saturated condition. Then the uid starts condensing in the
tube. Depending upon the length of the tube available, the uid may be subcooled at the exit
of the wire-on-tube heat exchanger.
3. Governing equations and ÿnite element formulation
Since the refrigerant in a wire-on-tube heat exchanger may follow three distinct regions as
described above, it is important to describe the governing equations and their FEM formulations
separately for each region.


G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

421

3.1. Desuperheating regime

The di erential equation governing heat transfer from superheated vapour to ambient air (i.e.
free convection condition) over an elemental area dA in terms of enthalpy H is
dH
U H
− T∞ = 0;
(1)
+
dA
mr Cpr
where, mr is the mass ow rate of the refrigerant, Cpr is the speciÿc heat of the refrigerant, U
is the overall heat transfer coe cient for the element, and T∞ is the temperature of the ambient
air. The enthalpy for the refrigerant in the element (which is fairly small; 0:0017 m) is assumed
to vary linearly as
H = N1 H1 + N2 H2 ;

(2)

where the shape functions are given by
A
A
N1 = 1 −
and N2 =
(3)
A
A
and A is the area at any given location which varies from zero to A where A is the area of
an element. Applying this approximation to governing equation (1), two equations for the two
unknowns H1 and H2 are obtained as follows:
U
dH

H
N1
− T∞
dA = 0;
(4)
+
dA
mr Cpr
dH
H
U
− T∞
dA = 0:
(5)
+
dA
mr Cpr
Minimising the residual error by Galerkin’s method, the following element matrix results:
2C − 0:5 C + 0:5
H1
U AT∞ =(2mr )
=
;
(6)
C − 0:5 2C + 0:5
U AT∞ =(2mr )
H2
N2

where

C=

U A
:
(6mr Cpr)

The above formulation is valid till the enthalpy of the refrigerant equals to that corresponding
to dry saturated condition.
3.2. Phase change regime
As the temperature of the refrigerant remains constant and equal to the saturation temperature
in this region for given pressure, the di erential equation governing heat transfer is written as
dH
U
(7)
+ (Tsat − T∞ ) = 0:
dA
mr
Assuming linear variation in enthalpy of the refrigerant in the element as before:
H = N1 H1 + N2 H2

(8)


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G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

and following the same procedure as that in earlier region, the following element matrix is
found:
+1 −1

H1
U A(Tsat − T∞ )=mr
=
:
(9)
+1 −1
H2
U A(Tsat − T∞ )=mr
The above formulation will be valid till the quality of the wet vapour is zero or its enthalpy
equals to that of saturated liquid enthalpy at the operating condenser pressure.
3.3. Sub-cooling region
The di erential equation governing heat transfer from liquid refrigerant to ambient air in this
region in terms of temperature is written as:
dT
mr Cpr
(10)
+ UP(T − T∞ ) = 0;
dx
where P is the equivalent perimeter of the tube element.
Assuming linear variation for the temperature of the refrigerant in the tube:
T = N1 T1 + N2 T2

(11)

and adopting the same procedure as for the earlier regions, the following element matrix is
found:
2C − 0:5 C + 0:5
T1
3CT∞
=

;
(12)
C − 0:5 2C + 0:5
T2
3CT∞
where, C = UP l=6mr Cpr = U A=6mr Cpr . Here Cpr should refer to the liquid condition.
It may be noted that the above element matrix is the same as that derived for the desuperheating region. The only di erence is in terms of the variable selected. A computer programme
is developed to take into consideration all the three regimes discussed earlier. The programme
is able to locate the position where the change of region=regime takes place.
4. Validation
Since the methodology used for the analysis of wire-on-tube heat exchangers consist of
sensible heat transfer and latent heat transfer, it is essential to validate the above method against
two similar heat transfer cases for which results are available. One such case is the analysis
of shell and tube heat exchanger where sensible heat transfer only takes place. The other case
considered is an evaporator where latent heat is transferred. For both the cases, the initial task
is to develop the element sti ness matrix and to assemble the global sti ness matrix which are
explained below.
4.1. Case I
In order to ensure the accuracy of the computer programme and the methodology adopted in
the present analysis, a double tube heat exchanger for which experimental data is available as


G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

423

Fig. 3. (a) Schematic diagram of an evaporator; (b) A discretised element of evaporator.

shown in Fig. 3(a) is considered [15]. Refrigerant 12 evaporates in the central tube by the hot
water owing in the annulus. This example will serve the purpose of validation in the phase

change region.
As usual, the evaporator is divided into number of elements for the purpose of discretisation.
Here every element, as shown in Fig. 3(b), has four nodes, two of the nodes at entry and exit
points of the tube side uid and other two nodes at entry and exit points of the annulus side
uid. Assuming H to represent the enthalpy of the refrigerant and hw for enthalpy of water,
the di erential equations for the heat transfer between the two uids may be written as
dH UP hw
−
− Tsat = 0
(13)
dx
mr Cpw
and
dhw UP hw
− Tsat = 0:
+
dx
mw Cpw

(14)

With the introduction of linear ÿeld variables as
H = N1 H1 + N2 H2

and

hw = N1 hw1 + N2 hw2

and using Galerkin’s approach for minimising error, the following element matrix is established:
 




−0:5 0:5 −2E
−
−E
H
B
1





 

 −0:5 0:5

H 
 
 −B 

−
E
−
2E
2


=

;
(15)



 0
0 2G − 0:5 G + 0:5  
D 
hw1 








 


D
0 0 G − 0:5 2G + 0:5
hw2
where
E=

UP l
;
6mr Cpw

B=

UP lTsat
;
2mr

G=

UP l
6mw Cpw

and

D=

UP lTsat
:
2mw


424

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

Fig. 4. Variation of dryness fraction along length for an evaporator.

The element sti ness matrices are assembled together to form the global sti ness matrix. Boundary conditions are incorporated for the evaporating uid inlet and the heating water outlet given
in the following example [15]:
Tube parameters:
Inner tube bore, m

Inner tube o.d., m
Outer tube bore, m
Refrigerant 12:
Mass ow rate of refrigerant, kg=s
Inlet pressure of refrigerant, bar
Outlet pressure of refrigerant, bar

di = 0:011 887.
do = 0:012 700.
D = 0:019 050.
mr = 0:024 570.
p1 = 4:102 21.
p2 = 3:943 625.

Water:
Mass ow rate of water, kg=s
mw = 0:653 94.
Outlet temperature of water, K
T2 = 288:67.
The enthalpy of the evaporating refrigerant is obtained along the length of the evaporator by
the proposed ÿnite element method. A constant value of U = 4000 W=m2 K was taken, which is
the average value of U in the region considered (X ¡ 0:9) and is given in Fig. 4 of [15]. Then
the dryness fraction is calculated and plotted against the length of the evaporator as shown in
Fig. 4. The results are in excellent agreement with those described in [15]. The present method,
therefore, demonstrates the validity and accuracy of the FEM formulation in the phase change
region.


G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

425

Fig. 5. (a) Shell and tube heat exchanger with discretised elements; (b) The ÿrst discretised element with nodal
points.

4.2. Case II
For the purpose of validating the numerical procedure adopted in the subcooling region where
only sensible heat is exchanged, a shell and tube heat exchanger is analysed by FEM using
Galerkin’s approach. The discretised model of a shell and tube heat exchanger (1 shell pass-2
tube passes) is shown in Fig. 5(a). The ÿrst element shown in Fig. 5(b) is considered for the
element matrix formulation. The di erential equation governing the heat transfer for the hot
uid in the element is written as
dT
mh Cph h + U (Th − tc ) = 0
(16)
dA
and
dtc
mc Cpc
− U (Th − tc ) = 0
(17)
dA
for the cold uid.
Here tc refers to the temperature of the cold uid. Assuming linear isoparametric elements
for the ÿeld variables and area,
Th = N1 T35 + N2 T36 ;

tc = N1 T1 + N2 T2

426

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

and
A = N1 Ai + N2 Ao ;

(18)

the ÿnal set of equations for element matrix formulation using Galerkin’s method for error
minimisation are:
(2S2 − Y )T1 + (Y + S2 )T2 − 2S2 T35 − S2 T36 = 0;

(19)

(S2 − Y )T1 + (Y + 2S2 )T2 − S2 T35 − 2S2 T36 = 0;

(20)

−2S2 T1 − S2 T1 + (2S2 − S1 )T35 + (S1 + S2 )T36 = 0;

(21)

−S2 T1 − 2S2 T2 + (S2 − S1 )T35 + (S1 + 2S2 )T36 = 0;

(22)

where
mh Cph
mc Cpc

U A
and Y =
:
; S2 =
6
2
2
The global sti ness matrix is formed depending upon the number of elements considered. Boundary conditions are imposed and thereafter the temperature distribution is obtained from the
solution of the assembled global matrix.
To show the prediction accuracy, the following ow and uid properties for the analysis of
shell and tube heat exchanger is considered [8].
S1 =

Heat transfer coe cient, U = 600 W=m2 K.
Heat capacity rate of cold uid = 200; 000 W.
Heat capacity rate of hot uid = 17; 000 W.
Total area of heat transfer = 64:0 m2 .
◦
Inlet temperature of hot uid = 190 C.
◦
Inlet temperature of cold uid = 35:0 C.
Number of elements = 32.
The calculated temperature distribution at di erent node points using the present model is
shown in Table 1. The table also shows three other sets of solutions corresponding to a FEM
model using subdomain collocation method [8], the model given by Gaddis and Schlunder [16]
and the analytical solution obtained from expressions given in [17,18]. It is observed that the
temperature distribution obtained by the present model based on Galerkin’s method is very
close to that given by the analytical solution. Thus the accuracy of the present method for the
sensible heat transfer region is demonstrated and validated. Since ÿnite element formulation for
desuperheating region is the same as for subcooling region, there is no need to demonstrate the

validity for the desuperheating region separately.
5. Results and discussion
The FEM formulations given earlier for both sensible and latent heat transfer are coded
to study the performance of the wire-on-tube heat exchanger. Refrigerant 12 enters in the


G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

427

Table 1
Temperature distribution in shell and tube heat exchanger (Shell side)
Node points

Analytical result [17,18]
◦
T ( C)

Present model
◦
T ( C)

Ref. [8] model
◦
T ( C)

Ref. [16] model
◦
T ( C)

33
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66

46.355
170.342
153.276
138.45
125.597
114.431
104.736
96.319
89.013
82.669
77.162
72.379

68.228
64.623
61.494
58.776
56.417

46.364
170.31
153.277
138.438
125.599
114.416
104.738
96.31
89.014
82.663
77.162
72.376
68.228
64.621
61.493
58.775
56.415

46.357
170.308
153.258
138.419
125.57
114.388

104.704
96.278
88.979
82.629
77.127
72.343
68.195
64.589
61.463
58.746
56.387

46.246
170.98
154.424
139.942
127.334
116.306
106.704
98.306
90.994
84.599
79.028
74.159
69.916
66.209
62.976
60.152
57.689

◦

superheated state (10 bar, 70:69 C). The required properties are taken from [19]. Table 2 shows
the number of runs carried out for various ambient temperatures and number of coils. The coil
dimensions are tabulated in Table 3 which are taken from Ref. [2].
The overall heat transfer coe cient was taken to be 10 W=m2 K under the free convection
condition. For any tube, the outside surface area in the wired portion of the tube is di erent
from that in the bare portion of the tube. Therefore each tube is divided into three distinct
sections. The ÿrst one is related to the bare portion of the tube whereas the second one refers
to the portion of the tube where wires are welded. The third section is similar to the ÿrst
section. It is also to be noted that due to the gap between the wires in the second section, the
outside surface area of elements are to be taken properly. For this section, two methods were
attempted during the calculations. The ÿrst took into account the exact area for each element
and the second method considered an average area for an element in that section. Calculations
were performed with 70 elements in the ÿrst, 348 elements in the second and 70 elements in
◦
the third section for coil 1 with mass ow rate of 0:0008 kg=s and ambient temperature of 10 C.
The results are shown in Table 4. The two calculations, however, show very little di erences
(less than 2%) between the two results as regards the onset of phase change and sub-cooling
as well. Therefore the average element area method in the second section was considered for
all calculations that are reported in this paper.
Further, in order to have the results independent of the element size, three di erent number
of elements (size) 70-300-70, 70-600-70 and 140-600-140 are used to calculate the enthalpies,
dryness fractions and temperatures for the coils as mentioned above for the same conditions of
mass ow rate and ambient temperature for the purpose of comparison . The notation 70-300-70
means that 70 elements are taken for the bare portions of the tube, whereas 300 refers to the
wired portion of the tube. The results are shown in Table 5. It is noticed that with the increase


428

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

Table 2
Parametric studies undertaken by varying mass ow rates of refrigerant, ambient temperatures, tube diameters and
other dimensionsa
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

24
25
26
27
28
29
30
31
32
33
34
35
a

mr

0.0002

0.0004

0.0006

0.0008

0.0010

T∞ (K)
283.00
288.00
293.00

298.00
303.00
308.00
313.00
283.00
288.00
293.00
298.00
303.00
308.00
313.00
283.00
288.00
293.00
298.00
303.00
308.00
313.00
283.00
288.00
293.00
298.00
303.00
308.00
313.00
283.00
288.00
293.00
298.00
303.00

308.00
313.00

Coil #

1

1

1

1

1

No.
36
37
38
39
40
41
42
43
44
45
46
47
48
49

50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70

mr

0.0012

0.0008

0.0008

0.0008

0.0008

T∞ (K)
283.00
288.00
293.00
298.00
303.00
308.00
313.00
283.00
288.00
293.00
298.00
303.00
308.00
313.00
283.00
288.00
293.00
298.00
303.00
308.00
313.00
283.00
288.00
293.00
298.00

303.00
308.00
313.00
283.00
288.00
293.00
298.00
303.00
308.00
313.00

Coil #

1

1∗

1∗∗

4

5

T∞ —Ambient temperature in Kelvin, mr —mass ow rate of refrigerant in kg=s.

in number of elements, the maximum di erence between the results of the coarse and ÿne
elements is less than 1.9 percent in relation to the prediction of the beginning of phase change
as well as the sub-cooling. As the computation with a ÿne element is not cost e ective, the
discretisation 70-300-70 is used for the present calculation. Having decided about the element
size and also the average area method to be considered for the wired portion of the tube, it is

thought of tabulating the calculated results for a typical set of runs numbering 22–28 of Table
2. The results in Table 6 show the values of enthalpies, dryness fractions and temperatures
depending upon whether the ow condition is in the desuperheating region, phase change region


G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

429

Table 3
Wire-on-tube heat exchanger dimensionsa
Variable

Coil #1
Ref. [10]

Coil #1∗
(Modiÿed coil #1)

Coil #1∗∗
(Modiÿed coil #1)

Coil #4
Ref. [10]

Coil #5
Ref. [10]

Dw (mm)
Sw (mm)

Nw
Dto (mm)
Dti (mm)
St (mm)
Lt (mm)
Nt

1.22
5.34
176
4.80
3.18
25.4
660
16

1.22
5.34
176
4.92
3.57
25.4
660
16

1.22
5.34
176
6.21
5.16

25.4
660
16

1.62
4.57
168
4.92
3.57
31.8
578
18

1.21
4.56
166
6.21
5.16
31.8
575
12

a

Values in bold indicate the di erence in dimensions of the coil with respect to coil #1.

Table 4
Exact and average area methoda
Mass ow rate of refrigerant, mr = 0:0008 kg=s
Ambient temperature, T∞ = 283 K

Method

Location of starting of phase
change in tube #
1.40
1.45

Exact area
Average area
a

Location of starting of
sub-cooling in tube #
12.63
12.61

1.4 indicates 40% location in tube #2 in the direction of ow of the refrigerant.

Table 5
Variation in element size
Mass ow rate of refrigerant, mr = 0:0008 kg=s
Ambient temperature, T∞ = 283 K
No.

Element members
size

1

70-300-70

(1:672=1:55=1:672) mm
70-600-70
(1:672=0:776=1:672) mm
140-600-140
(0:836=0:776=0:836) mm

2
3

Location of starting of
phase change in tube #

Location of starting of
sub-cooling in tube #

1.45

12.62

1.44

12.64

1.45

12.62

or sub-cooling region. These values are shown at the end of the tubes for a refrigerant mass
◦
◦

ow rate of 0:0008 kg=s. The variation of ambient temperature from 10 C to 40 C is considered
in order to cater to di erent climatic conditions. The location of the start of phase change or
◦
sub-cooling is also shown in this table for di erent ambient temperatures. For example at 10 C
ambient temperature, the desuperheating of refrigerant vapour continues till 1.45 tube length
(tube #2) from the start. The phase change then follows and is completed at 12.62 tube length
(tube #13). Then the sub-cooling starts and the subcooled temperature of the refrigerant at the
◦
◦
end of the heat exchanger is 293:08 K (i.e. 20:08 C) which is 10 C higher than the ambient


430

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

Table 6
Calculated values of enthalpies (H ), dryness fraction (X ) and temperatures (T ) at the end of tubes for coil #1.
◦
Refrigerant initial conditions: P = 10 Bar, Tsup = 70:69 C; Mass ow rate of refrigerant, mr = 0:0008 kg=sa
Ambient
temperatures,
T∞ (K)

283.00

288.00

293.00

298.00

303.00

308.00

313.00

Tube number

H=X=T

H=X=T

H=X=T

H=X=T

H=X=T

H=X=T

H=X=T

1
2
3
4
5
6

7
8
9
10
11
12
13
14
15
16

209.2723
0.9495
0.8604
0.7713
0.6823
0.5932
0.5041
0.4151
0.3260
0.2369
0.1478
0.0588
312.0056
303.4877
297.397
293.0831

210.6779
0.9717

0.8962
0.8208
0.7453
0.6699
0.5945
0.5190
0.4436
0.3681
0.2927
0.2173
0.1418
0.066
314.4752
306.7452

212.0934
0.9926
0.9308
0.8690
0.8072
0.7454
0.6835
0.6217
0.5599
0.4981
0.4363
0.3745
0.3127
0.2508
0.1890

0.1272

213.5126
205.5525
0.9634
0.9135
0.8671
0.8190
0.7705
0.7227
0.6745
0.6264
0.5782
0.5301
0.4819
0.4338
0.3856
0.3375

214.9137
207.7447
0.9927
0.9582
0.9237
0.8892
0.8547
0.8201
0.7856
0.7511
0.7166

0.6821
0.6475
0.6130
0.5785
0.5440

216.2972
209.9989
206.1812
0.9957
0.9748
0.9539
0.9330
0.9121
0.8913
0.8704
0.8495
0.8286
0.8077
0.7868
0.76590
0.7450

217.672
212.3564
208.8455
206.8611
205.6815
204.8169
0.9991

0.9919
0.9846
0.9773
0.9700
0.9627
0.9555
0.9482
0.9409
0.9336

Location of starting
of phase change at
tube #

1.450

1.60

1.80

2.30

2.730

3.734

6.807

Location of starting
of sub-cooling at

tube #

12.620

14.791

a

NSC

NSC

NSC

NSC

NSC

NSC—no sub-cooling takes place.

temperature. As the ambient temperature is increased, it is observed that the beginning of the
◦
phase change and sub-cooling is delayed. At 20 C, there is no sub-cooling taking place and
the refrigerant remains in two phase condition. This state of a air is repeated for other higher
ambient temperatures. This suggests that additional tube length is required in order to complete
condensation of refrigerant vapour for such cases.
Further results for other mass ow rates and di erent ambient temperatures for coil #1 (corresponding to runs 1– 42) are shown in Fig. 6. In this ÿgure PC10, PC15, etc. represent curves
◦
◦
showing the location of phase change taking place in tubes at 10 C; 15 C, etc., respectively.

The other notations SC10, SC15, etc. refer to the starting of the sub-cooling similar to the phase
change notations. Fig. 6 does not show the variation of locations for sub-cooling corresponding
◦
◦
◦
to 35 C and 40 C because at 35 C, sub-cooling takes place only for 0:0002 kg=s of refrigerant,
◦
whereas there is no sub-cooling for 40 C for all mass ow rates of refrigerant. It is observed
as expected, that phase change or sub-cooling are delayed with either the increase in mass ow


G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

431

Fig. 6. Location of starting of phase change and sub-cooling with variation in mass ow rates and ambient temperatures for coil #1.

rates or increase in ambient temperatures. The variations in this ÿgure, however, follow almost
straight line behaviour.
Next, the calculations are performed with mass ow rate of refrigerant = 0:0008 kg=s for coils
#1∗ and #1∗∗ which have the same dimensions except for the tube diameters. The results are
shown in Table 7. With a small increase (4.80 –4:92 mm) in the outer tube diameter (coil
#1∗ ), there is very slight change (less than 1%) in the prediction of location for phase change
or sub-cooling. However, with larger increase (4:80–6:21 mm) in the outer tube diameter (coil
#1∗∗ ), there is a change (10%) in the results, indicating much earlier occurrence of phase change
and sub-cooling.
Finally, calculations are carried out with mass ow rate of refrigerant, mr = 0:0008 kg=s for
coils #4 and #5 which are di erent as compared to coil #1 and the results are tabulated in Table
8. It may be pointed out that coil #4 has more number of tubes compared to coil #1 or coil
#1∗ ; its tube diameter is the same as that of coil #1∗ and the length of the tube is shorter. For

coil #5, the number of tubes is less (12 only); its tube diameter is the same as coil #1∗∗ and
the length of the tube is shorter. It is observed that for both the coils, phase change is predicted
◦
earlier as compared to coil #1 and 1∗ at all temperatures considered (10 –40 C). However, in
coil #5, the phase change is delayed as compared to coil #4 and this delay increases as the
ambient temperature becomes higher. This may be due to the smaller diameter wires and less
number of tubes. Both these factors result in smaller outside surface area, thus delaying phase
◦
change. For coil #4, the sub-cooling takes place in tube number 14 at 20 C while the same
could not take place in coil #1 for the same temperature. This may be due to the more number
◦
of tubes in coil #4. The sub-cooling does not take place for coil #5 at 15 C whereas it had
taken place at that temperature for coil #1. This may be attributed to less number of tubes in


432

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

Table 7
Variation of location of phase change and sub-cooling with coil #1∗ and coil #1∗∗ . Refrigerant initial conditions:
◦
P = 10 Bar, Tsup = 70:69 C. Mass ow rate of refrigerant, mr = 0:0008 kg=sa
Coil #1∗
No.

Ambient
temperature,
T∞ (K)

1

283.00

2

288.00

3

293.00

4

298.00

5

303.00

6

308.00

7

313.00

a

Coil #1∗∗

Location of
starting of
phase change
in tube #

Location of
starting of
sub-cooling
in tube #

Location of
starting of
phase change
in tube #

Location of
starting of
sub-cooling
in tube #

1.44
(1.45)
1.59
(1.60)
1.79
(1.80)
2.28
(2.30)

2.71
(2.73)
3.70
(3.73)
6.75
(6.81)

12.53
(12.62)
14.68
(14.79)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)

1.33
(1.45)
1.47
(1.60)
1.66
(1.80)
2.03
(2.30)

2.51
(2.73)
3.43
(3.73)
6.26
(6.81)

11.40
(12.62)
13.40
(14.79)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)

NSC—no sub-cooling takes place; Values given in parentheses are results for coil #1.

Table 8
Variation of location of phase change and sub-cooling with coil #4 and coil #5. Refrigerant initial conditions:
◦
P = 10 Bar, Tsup = 70:69 C. Mass ow rate of refrigerant, mr = 0:0008 kg=sa
Coil #4
No.

Ambient
temperature,
T∞ (K)

1

283

2

288

3

293

4

298

5

303

6

308

7

313

a

Coil #5

Location of
starting of
phase change
in tube #

Location of
starting of
sub-cooling
in tube #

Location of
starting of
phase change
in tube #

Location of
starting of
sub-cooling
in tube #

1.18
(1.45)
1.29

(1.60)
1.43
(1.80)
1.64
(2.30)
2.19
(2.73)
2.77
(3.73)
5.25
(6.81)

9.50
(12.62)
11.24
(14.79)
13.53
(NSC)
17.31
(NSC)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)

1.29
(1.45)
1.42

(1.60)
1.60
(1.80)
1.84
(2.30)
2.48
(2.73)
3.31
(3.73)
5.83
(6.81)

10.79
(12.62)
NSC
(14.79)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)
NSC
(NSC)

NSC—no sub-cooling takes place; Values given in parentheses are the results for coil #1.

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

433

Fig. 7. Location of starting of phase change and sub-cooling with variation in ambient temperatures for di erent
coils.

coil #5. At all other higher temperatures, sub-cooling does not take place for coil #5 which
was observed in coil #1 as well. It may also be noted from the above results that at lower
temperatures, the occurrence of phase change in the tubes does not vary considerably for all
the coils considered. However, this observation is not true for sub-cooling. The above results
on the location of phase change and sub-cooling with variation in ambient temperatures for
di erent coils are plotted in Fig. 7. The digits 1, 1∗ , 1∗∗ , 4 and 5 used in the notations for this
ÿgure refer to the coil number whereas PC and SC stand for phase change and sub-cooling as
mentioned earlier.
6. Conclusions
Galerkin’s ÿnite element method is formulated to analyse the performance of a wire-on-tube
heat exchanger under free convection cooling condition. The present method is validated successfully against the available data where latent heat transfer or the sensible heat transfer takes
place.
The proposed method is used to study the e ect of parameters like mass ow rates of
the refrigerant, ambient temperature, etc. on the beginning and end of the phase change or
sub-cooling taking place inside the tubes of the heat exchanger. These predictions are carried


434

G.A. Quadir et al. / Finite Elements in Analysis and Design 38 (2002) 417–434

out successfully using a locally developed computer programme. The present method can be
used to check the adequacy of the number of tubes provided in an existing wire-on-tube heat

exchanger when the operating conditions are di erent from design conditions. This methodology
can also be used as a design tool to design new wire-on-tube heat exchangers and also predict
deratings under abnormal ambient conditions.
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