31

An improved non-dimensional
model of wet-cooling towers
B A Qureshi and S M Zubair

Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

The manuscript was received on 1 April 2005 and was accepted after revision for publication on 21 July 2005.

DOI: 10.1243/095440805X62205

Abstract: A general non-dimensional mathematical model of cooling towers is improved by

including evaporation of water. The solution still consists of adjusting the assumed straight
air-saturation line to the real air-saturation data, but a new constant (H) is added as well.
Two solutions are proposed and the accuracy of each method is checked against data from
the literature and also compared with the original solution. The first method shows a maximum
decrease of 4.4 per cent in error, whereas in the second method, the maximum error was found
to be 3.3 and 6.8 per cent when the inlet air was unsaturated and saturated, respectively.

Keywords: cooling tower, non-dimensional model, evaporation

1 INTRODUCTION fan but rely on the buoyancy effect of the heated

air and a nozzle-like (or hyperbolic) shape to cause
In cooling towers, a warm water stream is cooled air circulation. A typical schematic representation
through evaporation of some of the water into an of a counterflow wet-cooling tower is shown in Fig. 1.
air stream. These towers are often used in large ther- A basic theory of cooling tower operation was orig-
mal systems to reject the waste heat via a water loop inally proposed by Walker et al. [1], but the practical
between the two devices. One of the advantages of use of the basic differential equations, however, was
the cooling tower over the dry heat exchanger is first presented by Merkel [2], in which the enthalpy
that, through evaporation, the circulating water difference was the driving potential. With the avail-
temperature may approach the atmospheric wet- ability of fast and reliable computers, the governing
bulb temperature rather than the dry-bulb tempera- differential equations can be solved numerically
ture and that cooling takes place through both with a greater accuracy [3]. It is important to note
heat- and mass-transfer mechanisms. There are that even after this, the Merkel model still remained
several types of cooling towers. Probably, the most in wide use, although various other detailed pro-
common is the mechanical draft tower in which the cedures and solutions for the design and rating of
water enters at the top of the tower as a spray and wet counterflow and cross-flow mechanical and
flows downwards through the tower. Ambient air is natural draught cooling towers have been presented
drawn into the tower with the help of fans and by various researchers [4– 9]. However, Halasz [10]
flows in a counter or cross-current manner to presented a general non-dimensional model that
the water stream. If the fans are at the bottom of described all types of evaporative heat exchangers
the tower and blow the air upwards past the water for counterflow, cross-flow, and parallel flow direc-
flow, the tower is termed as a forced draft tower, tions of water and air. The advantage of reducing
whereas if the fans are at the top, it is an induced the number of variables and the possibility of graphi-
draft tower. Large-size atmospheric towers, e.g. at cal representation of the results are evident. This
conventional or nuclear power plants, do not use a general non-dimensional model was then applied
specifically to cooling towers [11] where the effi-
ciency was expressed as a function of only two vari-


Corresponding author: Department of Mechanical Engineering, ables and plotted in one figure for each type of
King Fahd University of Petroleum and Minerals, PO Box 1474, cooling tower. It should be noted that various
Dhahran 31261, Saudi Arabia. assumptions were employed, in which neglecting

JPME69 # IMechE 2006 Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering
32 B A Qureshi and S M Zubair

4. Longitudinal heat conduction is neglected along

the wall and inside the fluids in their direction of
flow.
5. Water –air interface temperature is assumed to be
equal to the bulk water temperature.
6. All the coefficients in the non-dimensional differ-
ential equations are constants; that is, thermal
properties of air, water, and process fluids, heat-
and mass-transfer coefficients, and their combi-
nations in non-dimensional equations.
7. Saturated humidity ratio is a linear function of
temperature.
8. The second term in parentheses of equation (29),
representing the dimensionless mass-transfer
potential, is neglected because it is small com-
pared with unity.
9. Air is unsaturated in the whole process unless
otherwise indicated.
Presently, the assumption of constant water flow rate
Fig. 1 Schematic representation of a counterflow wet made by Halasz [10, 11] is neglected and the general
cooling tower non-dimensional equations are derived again for an
improved non-dimensional model.

water evaporation and adjusting the assumed

straight air-saturation line to the real air-saturation 3 MATHEMATICAL MODEL
data are prominent. The accuracy of the model was
compared with the data available in the literature An infinitesimal control volume of a wet-cooling
and was found to be good for normal operating con- tower associated with the differential area dA is pre-
ditions, failing only when the cooling range was very sented in Fig. 2. All assumptions, which were used
large or when the air flow rate was very small. Fur- by Halasz [10, 11] to derive the modelling equations,
thermore, the air was assumed to be unsaturated have been summarized in section 2. It should be
or, as a limiting case, saturated without fog. noted that all these were employed here as well
The objective of this article is to improve the non- except for the assumption of constant water flow rate,
dimensional model developed by Halasz [11] for which is the focus of the current work. In addition,
wet-cooling towers. In this regard, the assumption it is also considered that no cross-flow occurs, so as
of constant water mass flow rate is neglected and to deal with ordinary differential equations.
the derivation is repeated to obtain a more accurate
analytical solution by introducing another constant
in the mathematical model; then, two possible
3.1 Energy and mass balance: subsystem I – air
methods are proposed to obtain its value.
The airside water – vapour mass balance (Fig. 2) gives

_ a dW ¼ hD (Ws;int  W )dA
m (1)
2 LIST OF ASSUMPTIONS
Expressing the air energy balance in terms of heat-
The following basic assumptions are relevant to the
and mass-transfer coefficients, hc and hD , respect-
mathematical model and to the general closed-
ively, for the case hfg;int  hg;int
form solution that is found subsequently.
1. The process is steady state. _ a dh ¼ hc (tint  t)dA þ hD dA(Ws;int  W )hg;int
m (2)
2. The cooling tower is insulated.
3. Mass flow rates and inlet thermal states are
constant in a plane perpendicular to the
3.2 Energy and mass balance: subsystem
direction of flow of respective fluids. If no cross-
II – water
flow occurs in the device, then it is similarly
assumed constant for all other thermal states, The mass flow of recirculating water evaporating into
not only the inlet ones. air, in terms of the mass-transfer coefficient, hD , can

Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering JPME69 # IMechE 2006
 Non-dimensional model of wet-cooling towers 33

Fig. 2 Control volume of a counterflow wet cooling tower

be written as obtained

_ w ¼ hD (Ws;int  W )dA

dm (3) _ a dW ¼ hD (Ws;w  W )dA
m (8)
Expressing the water energy balance in terms of _ a dh ¼ hc (tw  t)dA þ hD dA(Ws;w  W )hg;w
m
heat- and mass-transfer coefficients, hc and hD , (9)
respectively
_ w ¼ hD (Ws;w  W )dA
dm (10)
_ w hf;w ¼ (m
m _ w þ dm
_ w )(hf;w þ dhf;w ) þ hc (tint  t)dA _ w cp;w dtw ¼ hc (tw  t)dA  hD
m
þ hD dA(Ws;int  W )(hg;int  hf;w ) (4)  dA(Ws;w  W )(hfg;w  hf;w ) (11)

Neglecting the higher order term on the RHS and sub- It should be noted that Halasz [10, 11] neglected
stituting equation (3), after simplification results in equation (10) and considered the water flow rate as
constant. In the current work, this equation is main-
_ w dhf;w ¼ hc (tint  t)dA
m tained and it is the basis of improving the non-
 hD dA(Ws;int  W )(hg;int  2hf;w ) (5) dimensional mathematical model applied to cooling
towers.

3.3 Governing differential equations

3.4 Non-dimensional variables
The change in water enthalpy can be written in terms
of specific heat as The non-dimensional air and water temperatures,
using the difference of the inlet dry- and wet-bulb
dhf;w ¼ cp;w dtw (6) air temperatures, can be expressed as [10]

For a negligible temperature difference between the t  twb;i tw  twb;i

bulk water and water– air interface [5, 12], it is Qa ¼ ; Qw ¼ (12)
tdb;i  twb;i tdb;i  twb;i
assumed that
Similarly, the non-dimensional humidity ratios are
tint ¼ tw and Ws;int ¼ Ws;w (7)

Substituting equations (6) and (7) into equations (1) W  Ws;wb Ws;w  Ws;wb
ja ¼ ; jw ¼ (13)
to (3) and (5), the following system of equations is Ws;wb  Wi Ws;wb  Wi

JPME69 # IMechE 2006 Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering
34 B A Qureshi and S M Zubair

The denominators in equations (12) and (13) are where

related by [13]
bhfg;wb
B¼ (21)
hD hfg;wb cp;a
tdb;i  twb;i ¼ (Ws;wb  Wi )
hc
hD cp;a hfg;wb It should be noted that the values of hfg;wb and cp;a can
¼ (Ws;wb  Wi ) (14) be calculated by using the following relationships
hc hc cp;a

hfg;wb ¼ h0g þ (cp;v  cp;w )twb;i (22)

The transformation of equation (8) into a non-
dimensional form, using equation (13) and introdu- cp;a ¼ cp;da þ Wcp;v ffi cp;da þ Ws;wb cp;v (23)
cing Lewis relation defined by Halasz [10] as
Le ¼ hD cp;a =hc , gives An important point here is that b and B are variables,
but, to obtain an analytical solution, these were con-
dja sidered as constant (representative) values as recog-
¼ Le(js;w  ja ) (15) nized by Halasz [10] and in the current work. The
dX
procedure for finding the value of these parameters
where the dimensionless (common) variable X (0 4 is explained in Appendix 3. Finally, substituting
X 4 XO ) is defined as equation (20) into equation (15), gives

dja
hc A ¼ Le(ja  B Le Qw ) (24)
X¼ (16) dX
_ a cp;a
m
To transform equation (9) into a non-dimensional
Another important relationship due to an extra vari- form, the moist air enthalpy is considered as [13]
able present in equation (15) can be introduced by
combining equations (12) to (14) and using the h ¼ cp;da t þ W (h0g þ cp;v t) (25)
non-dimensional relationships of Qw and js;w .
The resulting equation is easily obtained and is The derivative of equation (25) gives
given by [10]
dh dt dW
hD hfg;wb Ws;w  Ws;wb ¼ (cp;da þ Wcp;v ) þ (h0g þ cp;v t) (26)
js;w ¼ Qw (17) dA dA dA
hc tw  twb;i
Noting that
As the objective is to obtain a purely non-
dimensional model, the undesirable terms in hg;w ¼ h0g þ cp;v tw (27)
equation (17) may be removed. For this purpose,
Halasz [10] replaced the real air-saturation line by a and substituting equations (23) and (27), as well as
straight air saturation line using the relationships (dh=dA) from equation (9) and (dW =dA) from
equation (8) into equation (26), results in
Ws;w ¼ a þ btw and Ws;wb ¼ a þ btwb;i (18)
hc hD
(tw  t) þ (Ws;w  W )(h0g þ cp;v tw )
_
ma m_a
Simplifying, the following parameter is obtained
dt
¼ (cp;da þ Wcp;v ) þ (h0g þ cp;v t)
Ws;w  Ws;wb dA
b¼ (19) hD
tw  twb;i  (Ws;w  W ) (28)
m_a

It is important to note that this linearization process Simplifying

did not introduce a large error because only a small
section of the saturation line is relevant, i.e. between

dt hc hD cp;a cp;v
tw;i and tw;o . Thus, the substitution of equation (19) ¼ (t  t) 1 þ (Ws;w  W )
dA m _ a cp;a w hc cp;a
into equation (17) gives
(29)
hD cp;a bhfg;wb Halasz [10] assumed the value of terms in paren-
js;w ¼ Qw ¼ B Le Qw (20)
hc cp;a thesis to be 1, resulting in an error of only a few per

Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering JPME69 # IMechE 2006
 Non-dimensional model of wet-cooling towers 35

cent for the air temperature change, which simplified 4 CALCULATION OF PARAMETER H
the aforementioned equation and the subsequent
transformation. Now, applying equations (12) and In Table 1, the ‘improved non-dimensional model’
(16), the aforementioned equation can be written as represents the results of the current work. At present,
two possible solutions are proposed to calculate the
dQa parameter H. In both methods, it is found that the
¼ (Qw  Qa ) (30) cooling range must be 510 8C for the methods to
dX
be applicable. Furthermore, it was noted that for
smaller cooling ranges, improvement in the predic-
Finally, the non-dimensional form of equation (11)
tion is not often required and that the original
can be obtained by applying equations (12) and (13)
solution [11] suffices, and for such cases, the value
and then simplifying
of H is taken as unity.

dQw
_ w cp;w
m (tdb;i  twb;i )
dA
¼ hc ½(Qw  Qa )(tdb;i  twb;i )  hD ½(js;w  ja ) 4.1 Method 1
 (Ws;wb  Wi )(hfg;w  hf;w ) (31) Let the quantities in the numerator of equation (34)
be evaluated at the inlet wet-bulb temperature.
Substituting equations (14) and (20) in equation (31) Therefore
and dividing both sides by (tdb;i  twb;i ), after some
simplification and use of a direction indicator for (hfg;wb  hf;wb )
water (iw ¼ +1), the following is obtained H¼ (35)
hfg;wb

dQw iw
¼ ½Qa þ H ja  (1 þ Le BH)Qw  (32) It is evident from Table 1 that this method allowed
dX C
for a maximum decrease of 4.4 per cent in error
(see no. 5.2) when compared with non-dimensional
where
solution of Halasz [11].

_ w cp;w
m
C¼ (33)
m_ a cp;a
4.2 Method 2
and In the second method, an empirical relation was
developed to calculate a representative water temp-
(hfg;w  hf;w ) erature to be used for evaluating the quantities
H¼ (34) in the numerator of equation (34). Using some
hfg;wb
sample problems, linear regression was used to
evaluate the most important variables that predicted
The direction indicator has a positive sign for the the required representative temperature. In this
parallel flow and a negative sign for the counterflow. regard, it was found that the cooling range and
It is important to note that no such indicator is mass flow ratio were the major factors involved.
required for the air, as it always flows in an upward The final empirical equation based on several
direction. numerical experiments is given by
The water-to-air heat capacity ratio can be calcu-
lated by using the water flow rate at the inlet. As sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
with the terms b and B, let H be assumed to be 3(tw;i  tw;o )2
known and constant. The procedure for finding the tw;repr ¼ (36)
m0:7 ratio
value of H will be discussed in a later section. Thus,
the original system of equations has now been trans-
formed into a non-dimensional form without It can be seen clearly from Table 1 that there is a
neglecting water evaporation. The final equations marked decrease in error for all cases where the
are (24), (30), and (32), where equations (24) and cooling range is .10 8C, especially in those cases
(30) describe the heat- and mass-transfer process where the cooling range is very large. Furthermore,
between the air and water and equation (32) is the the maximum error was calculated to be 3.3 and
water energy balance for an adiabatic evaporation 6.8 per cent when the inlet air was unsaturated and
process. saturated, respectively.

JPME69 # IMechE 2006 Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering
 36

Table 1 Results for the counterflow cooling tower from references [2, 11] and the improved model with respect to the accurate (Poppe) model [3]

Improved non-dimensional model

Input data Accurate model [3] Merkel [2] Halasz [11] Using H evaluated at twb,i Using H evaluated at tw,repr

tw,i tw,o tdb,in twb,in tdb,o NTU NTU tdb,o tdb,o tdb,o
Number (8C) (8C) (8C) (8C) m
_ a =m
_ w,i (8C) Wo (Poppe) (Merkel) DM (%) NTU DH (%) (8C) Wo NTU DI1 (%) (8C) Wo NTU DI2 (%) (8C) Wo

0.1 30 26 8 4 0.25 27.01 23.39 2.119 1.900 210.4 2.114 20.2 26.59 22.63 2.114 20.2 26.59 22.63 2.114 20.2 26.59 22.63
0.2 30 26 8 4 0.3 24.36 20.09 1.396 1.283 28.1 1.361 22.5 23.45 19.45 1.361 22.5 23.45 19.45 1.361 22.5 23.45 19.45
0.3 30 26 8 8 0.3 26.28 22.61 1.777 1.615 29.1 1.749 21.6 25.31 22.02 1.749 21.6 25.31 22.02 1.749 21.6 25.31 22.02
1.1 34 30 16 12 0.2 33.62 34.22 4.707 3.422 227.3 – – – – – – – – – – – –
1.2 34 30 16 12 0.25 30.63 28.89 1.861 1.666 210.5 1.829 21.7 30.27 28.18 1.829 21.7 30.27 28.18 1.829 21.7 30.27 28.18
1.3 34 30 16 12 0.3 28.36 25.29 1.275 1.167 28.5 1.238 22.9 27.85 24.68 1.238 22.9 27.85 24.68 1.238 22.9 27.85 24.68
1.4 34 30 16 16 0.3 30.49 28.85 1.706 1.540 29.7 1.661 22.6 29.68 28.3 1.661 22.6 29.68 28.3 1.661 22.6 29.68 28.3
2.1 34 30 24 20 0.3 32.72 32.45 2.913 2.484 214.7 2.944 1.1 32.88 31.94 2.944 1.1 32.88 31.94 2.944 1.1 32.88 31.94
2.2 34 30 24 20 0.35 31.30 29.75 1.872 1.680 210.3 1.832 22.1 31.5 29.28 1.832 22.1 31.5 29.28 1.832 22.1 31.5 29.28
2.3 34 30 24 20 0.4 30.34 27.71 1.419 1.295 28.7 1.379 22.8 30.51 27.27 1.379 22.8 30.51 27.27 1.379 22.8 30.51 27.27
213.3 23.4 23.4 23.4

Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering

2.4 34 30 24 24 0.4 32.82 32.66 2.955 2.561 2.853 32.72 32.27 2.853 32.72 32.27 2.853 32.72 32.27
3.1 34 30 32 28 0.8 32.48 31.05 2.073 1.880 29.3 – – – – – – – – – – – –
3.2 34 30 32 28 1 32.20 29.45 1.393 1.287 27.6 – – – – – – – – – – – –
3.3 34 30 32 28 1.2 32.08 28.36 1.056 0.984 26.9 – – – – – – – – – – – –
4.1 34 24 16 12 0.5 33.51 33.97 7.154 5.446 223.9 – – – – – – – – – – – –
4.2 34 24 16 12 0.8 27.52 23.98 1.564 1.456 26.9 1.538 21.7 27.48 23.43 1.587 1.5 27.67 23.69 1.613 þ3.1 27.78 23.83
4.3 34 24 16 12 1 25.11 20.63 1.086 1.020 26.1 1.051 23.3 25.09 20.19 1.078 20.8 25.25 20.41 1.097 þ1.0 25.35 20.55
B A Qureshi and S M Zubair

4.4 34 24 16 16 1 27.54 24.23 1.497 1.397 26.7 1.434 24.2 26.85 23.82 1.486 20.7 27.07 24.08 1.493 20.2 27.10 24.11
5.1 34 24 24 20 1 30.07 27.74 2.603 2.404 27.6 2.485 24.5 30.31 27.35 2.664 2.3 30.55 27.76 2.625 þ0.8 30.50 27.68
5.2 34 24 24 20 1.5 27.66 23.02 1.284 1.211 25.7 1.207 26.0 27.84 22.73 1.265 21.5 27.98 23.02 1.265 21.5 27.98 23.01
5.3 34 24 24 20 2 26.65 20.6 0.861 0.817 25.1 0.808 26.2 26.77 20.35 0.842 22.2 26.86 20.57 0.847 21.6 26.88 20.61
6.1 40 20 16 12 1.5 28.21 25.19 1.560 1.489 24.6 1.395 210.6 27.59 24.78 1.432 28.2 27.80 25.07 1.586 þ1.7 28.59 26.18
6.2 40 20 16 12 2 25.06 20.70 1.031 0.988 24.2 0.916 211.2 24.55 20.42 0.937 29.1 24.70 20.64 1.035 þ0.4 25.37 21.62
6.3 40 20 16 12 3 21.61 16.20 0.617 0.593 23.9 0.547 211.4 21.62 16.01 0.558 29.6 21.71 16.15 0.621 þ0.6 22.25 16.97
6.4 40 20 16 16 3 24.24 20.02 0.875 0.839 24.2 0.741 215.3 22.73 19.88 0.761 213.0 22.87 20.06 0.835 24.5 23.40 20.73
7.1 40 20 22 18 3 25.85 21.12 1.162 1.127 23.0 0.972 216.4 25.94 20.92 1.008 213.2 26.06 21.18 1.129 22.8 26.44 21.99
7.2 40 20 22 18 5 24.18 17.29 0.623 0.606 22.7 0.527 215.5 24.22 17.12 0.545 212.6 24.28 17.28 0.623 0.0 24.55 17.95
7.3 40 20 22 18 8 23.32 15.10 0.368 0.358 22.6 0.313 215.0 23.34 14.95 0.323 212.3 23.37 15.05 0.380 þ3.3 23.58 15.60
8.1 54 24 16 12 1 39.55 49.36 2.127 2.037 24.2 1.731 218.6 37.62 48.74 1.785 216.1 38.01 49.45 2.132 þ0.2 40.28 53.53
8.2 54 24 16 12 1.5 33.50 35.17 1.150 1.108 23.7 0.930 219.2 30.19 34.95 0.952 217.2 30.44 35.43 1.118 22.8 32.23 38.80
8.3 54 24 16 12 2 29.71 28.13 0.792 0.764 23.6 0.640 219.2 26.57 28.01 0.654 217.5 26.75 28.37 0.771 22.6 28.27 31.30
8.4 54 24 16 16 2 31.70 31.95 0.961 0.926 23.6 0.747 222.2 27.54 32.02 0.769 220.0 27.80 32.49 0.896 26.8 29.27 35.14

JPME69 # IMechE 2006

 Non-dimensional model of wet-cooling towers 37

5 CASE STUDY: COUNTERFLOW Substituting the boundary conditions (Fig. 3) in the

WET-COOLING TOWER aforementioned equations
2 3
C
2 3 2 3
In this case, water flows in a downward direction, 6 1 1 þ BH H 7 C0;1 1
and therefore, the ordinary differential equations, 6 7
6 BC 7  4 C0;2 5 ¼ 4 1 5 (44)
with the directional indicator iw ¼ 1 and Lewis 6B 1 7
4 1 þ BH 5 C0;3 Qw;i
relation equal to unity, the non-dimensional
1 em2 X O 0
equations reduce to
By inspection, the parameter z can now be rep-
d ja resented by
¼ ja þ BQw (37)
dX (1 þ BH)
z¼ (45)
dQa C
¼ (Qw  Qa ) (38)
dX which is defined as the ratio of the corrected total
dQw 1 H (1 þ BH) (i.e. sensible þ latent) heat capacity rate of the mass
¼  Qa  ja þ Qw (39) flow rate of air along its saturation line to the heat
dX C C C
capacity rate of the water mass flow rate.
Although the boundary conditions are the same as Using the parameter z, the cooling tower efficiency
used in reference [11], these are shown here again can be expressed as
for convenience (Fig. 3). It should be noted that tw;i  tw;o Qw;o
only equation (39) is different from the set of ordin- 1¼ ¼1
tw;i  twb;i Qw;i
ary differential equations derived by Halasz [11].

1  e(1z)XO H 1
The general solution of the aforementioned system ¼z þ 1 (46)
1  ze(1z)XO Qw;i (1 þ BH)
of differential equations is given by
If we substitute H ¼ 1 in equation (46), it reduces to
C the expression for efficiency originally derived by
Qa ¼ C0;1 þ C0;2 em2 X þ HC0;3 eX (40)
1 þ BH Halasz [11].
It should be noted that the aforementioned set of
BC equations are to be used if the complete process (or
ja ¼ BC0;1 þ C0;2 em2 X  C0;3 eX (41) operating) line is required. Otherwise, if only the
1 þ BH
outlet conditions are required, then simpler equations
Qw ¼ C0;1 þ C0;2 em2 X (42) that are discussed in the following section can be used.

where 5.1 Calculating outlet air conditions

(1 þ BH) Once the non-dimensional model is adjusted to the

m2 ¼ 1 (43) actual process, the dimensionless outlet air tempera-
C
ture and humidity ratio can be directly calculated
from the equations given subsequently (refer to the
Appendix 2 for additional details)


1H Qw;i 1 1 þ B XO
Qa;o ¼ þ þH e
1 þ BH z 1 þ BH
1H C
¼ þ (Qw;i  Qw;o )
1 þ BH 1 þ BH


1 þ B XO
þH e (47)
1 þ BH



1H BQw;i 1 1 þ B XO
ja;o ¼B þ  e
1 þ BH z 1 þ BH


1H BC
¼B þ (Qw;i  Qw;o )
1 þ BH 1 þ BH


Fig. 3 Boundary conditions for counterflow wet-cooling 1 þ B XO
 e (48)
tower 1 þ BH

JPME69 # IMechE 2006 Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering
38 B A Qureshi and S M Zubair

The dimensional outlet dry-bulb temperature and

humidity ratio can be calculated from the present
improved model by using the following set of
equations


1H (tw;i  tw;o )
tdb;o ¼ twb;i þ (tdb;i  twb;i ) þ
1 þ BH z

 
1þB
þH (tdb;i  twb;i )eXO (49)
1 þ BH



1H
Wo ¼ Ws;wb þ (Ws;wb  Wi ) B
1 þ BH

 
B (tw;i  tw;o ) 1 þ B XO
þ  e (50)
z (tdb;i  twb;i ) 1 þ BH

As in Halasz’s model [11], the dimensionless temp-

erature and humidity ratio are undetermined for
the case of air saturated at the inlet because (tdb;i 
twb;i ) ¼ 0 and (Ws;wb  Wi ) ¼ 0. Still, the dimensional
values of these quantities can be calculated by
h1 i
tdb;o ¼ twb;i þ(tw;i  twb;i ) (51)
z
Wo ¼ Ws;wb þ b(tdb;o  twb;i ) (52)

It is important to understand that, in this case, the

first term in brackets of equation (46) will be reduced
to zero because of the non-dimensional water temp-
erature term so that the efficiency of the system
would be calculated by an expression similar to
that of Halasz [11]. The difference lies in the fact
that z would still be calculated by equation (45).
It is again noted that if H is taken as unity in
equations (37) to (50), the solution originally derived Fig. 4 Comparison of (a) the distribution of relevant
by Halasz [11] is obtained. It is further seen that, for process lines and (b) the air process line on
the case of air saturated at the inlet, the expressions the pychrometric chart (for run 6.3 refer to
obtained are the same as that derived earlier by Table 1)
Halasz [11].

explained in the current work, provide a more accu-

6 EFFECT ON PROCESS LINE
rate solution when compared with the original one
given by Halasz [11]. Furthermore, the dimensional
Run 6.3 of Table 1 is used to illustrate the effect of H
values of the air process line were plotted on the
on the air and water process (operating) lines. It can
psychrometric chart along with the straight air-
be seen from Fig. 4(a) that the use of the empirical
saturation line (See Fig. 4(b)). It should be noted
equation (i.e. Method 2) has a marked effect on all
that as the variable H does not affect the assumed
process lines except the saturated humidity ratio
straight air-saturation line, it is the same in all the
(Ws;w ), which, in this case, overlaps the saturated
cases presented in the figure.
humidity ratio line computed from Method 1. It
should be kept in mind that the saturated humidity
ratio was calculated from equation (19) and that X
does not necessarily correspond to the physical 7 CONCLUDING REMARKS
height of the cooling tower in this figure. All process
(operating) lines are further compared with the dis- It is important to emphasize that the improved non-
tribution obtained from an accurate dimensional dimensional model covers the tower range of 10 8C or
model of cooling towers and it is seen that both higher, where the error varied from 1 to 22 per cent
methods used to obtain H in the improved model, when the original non-dimensional model was

Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering JPME69 # IMechE 2006
 Non-dimensional model of wet-cooling towers 39

used. Two possible solutions are presented to evalu- 10 Halasz, B. General mathematical model of evaporative
ate the new variable H to reduce this error, the cooling devices. Rev. Gen. Therm., 1998, 37(4), 245 – 255.
amount of which depends on the method used. The 11 Halasz, B. Application of a general non-dimensional
results from the improved model indicate that both mathematical model to cooling towers. Int. J. Therm.
Sci., 1999, 38, 75– 88.
are effective in decreasing the error in prediction of
12 Parker, R. O. and Treybal, R. E. The heat, mass transfer
the NTU, but the empirical equation (Method 2) is
characteristics of evaporative coolers. AIChE Chem.
comparatively better at higher ranges. We believe Eng. Prog. Symp. Ser, 1961, 57(32), 138 – 149.
this is an important and useful step towards a more 13 Kuehn, T. H., Ramsey, J. W., and Threlkeld, J. L.
accurate analytical solution regarding evaporative Thermal environmental engineering, 3rd edition, 1998
heat exchangers. It is further understood that the (Prentice-Hall Inc., New Jersey).
approximation of the real saturation line by a straight
line constitutes the biggest assumption in the model
and can probably be neglected to a large extent by
breaking the real saturation line into smaller parts APPENDIX 1
where each of these can then be represented by a
straight line and then solved by marching from the Notation
inlet to outlet water temperatures.
A overall surface area of water–air interface, m2
b slope of straight air-saturation line
(see equations (18) and (19)), 1/ 8C
ACKNOWLEDGEMENTS B parameter defined in equation (21)
cp specific heat at constant pressure,
The authors acknowledge the support provided by King kJ/kga per C
Fahd University of Petroleum and Minerals through C heat capacity ratio of water to air
the research project (ME/RISK-FOULING/230). C0;j constant of integration ( j ¼ 1, 2, 3) in
equations (40) to (42)
h enthalpy of moist air, kJ/kga
hc convective heat-transfer coefficient of air,
REFERENCES kW/m2 per C
hD convective mass-transfer coefficient,
1 Walker, W. H., Lewis, W. K., McAdams, W. H., and kgw/m2 per s
Gilliland, E. R. Principles of chemical engineering, 3rd hj specific enthalpy of saturated liquid
edition, 1923 (McGraw-Hill Inc., New York). water, kJ/kgw
2 Merkel, F. Verdunstungshuhlung. Zeitschrift des hfg change-of-phase enthalpy, kJ/kgw
Vereines Deutscher Ingenieure (V.D.I.), 1925, 70,
hg specific enthalpy of saturated water
123 – 128.
vapour, kJ/kgw
3 Poppe, M. Wärme- und Stoffübertragung bei der
Verdunstungskühlung im Gegen- und Kreuzstrom. h0g specific enthalpy of saturated water
VDI – Forschungsheft, 1973, 38(560), 1 – 44. vapour evaluated at 0 8C, kJ/kgw
4 Webb, R. L. A unified theoretical treatment for thermal Le Lewis relation (Le ¼ hD cp;a =hc )
analysis of cooling towers, evaporative condensers, and mratio water-to-air mass flow rate ratio
fluid coolers. ASHRAE Trans., 1984, 90(Part 2), 398 – 415. (¼m _ w;i =m
_ a)
5 Webb, R. L. and Villacres, A. Performance simulation of m_ mass flow rate, kga/s
evaporative heat exchangers (cooling towers, fluid NTU number of transfer units
coolers and condensers). Heat Transfer Eng., 1985, P pressure, kPa
6(2), 31 –38. t dry-bulb temperature of moist air, 8C
6 Mohiuddin, A. K. M. and Kant, K. Knowledge base for
W humidity ratio of moist air, kgw/kga
the systematic design of wet cooling towers. Part I:
X non-dimensional parameter defined in
selection and tower characteristics. Int. J. Refrig.,
1996, 19(1), 43 – 51. equation (16)
7 Mohiuddin, A. K. M. and Kant, K. Knowledge base for z non-dimensional parameter defined in
the systematic design of wet cooling towers. Part II: equation (45)
fill and other design parameters. Int. J. Refrig., 1996,
19(1), 52 –60.
DM percentage error in NTU prediction by
8 El-Dessouky, H. T. A., Al-Haddad, A., and Al-Juwayhel, F.
A modified analysis of counter flow cooling towers. Merkel’s model with respect to Poppe’s
ASME J. Heat Transfer, 1997, 119(3), 617 – 626. model (per cent)
9 Khan, J. R. and Zubair, S. M. An improved design and DH percentage error in NTU prediction by
rating analyses of counter flow wet cooling towers. Halasz model with respect to Poppe’s
ASME J. Heat Transfer, 2001, 123(4), 770 – 778. model (per cent)

JPME69 # IMechE 2006 Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering
40 B A Qureshi and S M Zubair

DI1 percentage error in NTU prediction by (40) and (41)

Method 1 of the present improved
Qw;i  Qw;o C0;2
non-dimensional model with respect to Qa ¼ C0;1 þ þ þ HC0;3 eX (55)
Poppe’s model (per cent) z z
DI2 percentage error in NTU prediction by B(Qw;i  Qw;o ) C0;2
ja ¼ BC0;1 þ þB  C0;3 eX
Method 2 of the present improved z z
non-dimensional model with respect to
(56)
Poppe’s model (per cent)
1 efficiency of cooling tower Now the following quantity holds
Q non-dimensional temperature defined in
equation (12) C0;2 1H
j non-dimensional humidity ratio defined C0;1 þ ¼ (57)
z 1 þ BH
in equation (13)
Substituting equation (57) into equations (55) and (56)
Subscripts and evaluating at X ¼ XO, after some simplification,
the following outlet air conditions are obtained
a (moist) air


da dry air 1H Qw;i  Qw;o 1 þ B XO
db dry-bulb Qa;o ¼ þ þH e
1 þ BH z 1 þ BH
g,w vapour at water temperature
i inlet (58)
int on the air –water interface surface
 

1H B(Qw;i  Qw;o ) 1 þ B XO
L linear ja;o ¼B þ  e
m mean 1 þ BH z 1 þ BH
max maximum (59)
o outlet
O overall value It is noted that equations (58) and (59) are the same as
repr representative value equations (47) and (48).
R real
s,w saturated moist air at water temperature
w water
wb wet-bulb
APPENDIX 3

Approximation of linear saturation line

APPENDIX 2
Halasz [10] explained that the value of the parameter B
Derivation of equations (47) and (48) is considered known and constant and that
this parameter is a function of the inlet wet-bulb temp-
First, solving the matrix represented by equation (44), erature, a representative water temperature and air
the constants C0;1 , C0;2 , and C0;3 are calculated as saturation data. This is clear that for the relationship
to be valid in the non-dimensional domain, it must
zem2 XO (H  1) þ Qw;i (1 þ BH) be adjusted to the real air-saturation data to yield the
C0;1 ¼
(1 þ BH)(1  zem2 XO ) final results. Halasz [11] further explained that, for an
z(H  1 þ Qw;i (1 þ BH)) assumed straight air-saturation line, the following
C0;2 ¼ holds
(1 þ BH)(zem2 XO  1)
1þB ðA ð tw;i
C0;3 ¼ hD A hD dA dtw
1 þ BH ¼ ¼ (60)
_ w cp;w
m 0 _ w cp;w
m tw;o hs;w  h
(53)
It is important to note that the basic idea is to obtain
Now, evaluating equation (42) at X ¼ XO and X ¼ 0 an ‘equivalent’ straight air saturation line that produces
and then subtracting the result the same integral on the RHS of the aforementioned
equation as from a real saturation line. Although this
C0;2 em2 XO ¼ Qw;i  Qw;o þ C0;2 (54) assumption has an effect on the cooling tower process
and would cause a slightly different distribution of the
Substituting equations (54) and (45) into equations air enthalpy along the surface, it is assumed that h is the

Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering JPME69 # IMechE 2006
 Non-dimensional model of wet-cooling towers 41

Table 2 Polynomial coefficients for equation (63) for a taken into account by integrating equation (62) using
total pressure of 1 bar appropriate values of hs;w (tw ; P). Now, the straight
air-saturation line is expressed by combining
tw (8C) a0 a1 a2 a3
equations (19) and (25) giving
5 , tw 4 20 20.672 969 10.0723 0.756 563 0.014 3143
20 , tw 4 40 2294.945 48.4703 20.952 424 0.040 481
40 , tw 4 60 27020.16 520.79 212.0867 0.128 689 (hs;w )L ¼ cp;da tw þ {Ws;wb þ b(tw  twb;i )}(h0g þ cp;v tw )
(64)
function of tw in both saturation lines. Therefore
ð tw;i ð tw;i Substituting equation (23) into equation (64) and then
substituting the resulting equation into the LHS of
(hs;w )L dtw ¼ (hs;w )R dtw (61)
tw;o tw;o equation (61) and considering tw;m as the arithmetic
mean of the inlet and outlet water temperatures, the
The real enthalpy of the saturated air is a known func- final expression for the parameter b reduces to
tion of water temperature. That is
ð tw;i (=(tw;i ) =(tw;o ))=(tw;i  tw;o )  (Ws;wb h0g þ cp;a tw;m )
=¼ (hs;w )R dtw ¼ =(tw;i )  =(tw;o ) (62) b¼ 2 t t 0
cp;v ((4tw;m w;i w;o )=3  twb;i tw;m ) þ hg (tw;m  twb;i )
tw;o
(65)
where the values of the functions on the RHS can be
read from Table 1 of reference [11]. The following poly- It should be emphasized that there are three limitations
nomial equation can also be used (see Table 2 for coef- because of the linearization of the saturation line. First,
ficient values) the error would be significant for the case of a very large
cooling range; secondly, the model could not describe
=(tw ) ¼ a0 þ a1 tw þ a2 tw2 þ a3 tw3 (63) cooling tower operation with a very small airflow;
thirdly, the air was assumed to be unsaturated or, as
It should be noted that the effect of elevation can be a limiting case, saturated without fog.

JPME69 # IMechE 2006 Proc. IMechE Vol. 220 Part E: J. Process Mechanical Engineering