Energy Conversion and Management 42 (2001) 783±789
www.elsevier.com/locate/enconman
On the optimum sizing of cooling towers
M.S. S
oylemez *
Department of Mechanical Engineering, University of Gaziantep, P.O. Box 300, 27310 Gaziantep, Turkey
Received 31 May 2000; accepted 27 October 2000
Abstract
The optimum heat and mass transfer area at which minimum cost exists throughout the technical life of
forced draft counter-current cooling towers is studied in the present work. Original formulae are developed
and presented for the best thermoeconomical performance as a design point. Ó 2001 Elsevier Science Ltd.
All rights reserved.
Keywords: Thermoeconomics; Cooling tower; Heat and mass transfer area; Optimization
1. Introduction
Thermoeconomic optimization of cooling towers is studied in the present work. Optimum
sizing of the cooling towers is studied, and an iterational optimization technique is developed. An
optimum heat and mass transfer area for best operating condition is determined as a result. It is
shown that the life cycle costs of the cooling tower is a minimum at this critical point. Although
there exist previous studies in the current literature about optimization of cooling towers [1±9], all
of these studies are not suciently well detailed. At the beginning, the optimum water to air mass
¯ow rate ratio is determined. The water ¯ow rate is assumed to be constant as a design requirement. The optimum mass ¯ow rate of air is then calculated, using these two speci®ed values.
Certain design values are treated as constants, yielding a lower number of unknowns in the optimization study. Pressure drop per unit height of the cooling tower is assumed initially at the
beginning of the iterations, since it depends upon two basic variables, the water loading and air
loading per unit tower cross-sectional area for each di€erent type of eliminator. The optimum heat
and mass transfer area is then determined iteratively via use of the ®nal iterated value of the
pressure drop. The ®nal iterated value of the optimum heat and mass transfer area is obtained
*
Tel.: +90-342-360-1200; fax: +90-342-360-1100.
E-mail address: sait@alpha.bim.gantep.edu.tr (M.S. SoÈylemez).
0196-8904/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 9 6 - 8 9 0 4 ( 0 0 ) 0 0 1 4 8 - 5
784
M.S. S
oylemez / Energy Conversion and Management 42 (2001) 783±789
Nomenclature
a
A
Af
Af;opt
Ai
Ai;opt
Aopt
Ca;sat
CE
CX
CV
C1
C2
d
D
G
G0
H
IC
i
ii;1
ii;2
K
L
L0
ma
MS
mw
N
n1±5
NTU
OC
P1
P2
R
RV
S
Thw
TC
V
eliminator characteristics (mÿ1 )
cross section area of cooling tower (m2 )
fan exit area (m2 )
optimum fan exit area (m2 )
heat and mass transfer area (m2 )
optimum heat and mass transfer area (m2 )
optimum cross section area of cooling tower (m2 )
speci®c heat of saturated air (J/kg K)
present price of electricity ($/kW h)
area independent ®rst cost of cooling tower ($)
®rst cost of cooling tower per unit volume ($/m3 )
constant used in Eq. (21) due to type of eliminator
constant used in Eq. (20) due to type of eliminator
market discount (interest) rate (decimal)
other costs of system ($)
air loading level (kg/m2 s)
reference air loading level (kg/m2 s)
allowable cooling tower height (m)
initial cost of cooling tower ($)
energy price rate (decimal)
enthalpy of saturated air at basin temperature (J/kg)
enthalpy of saturated air at tower temperature (J/kg)
coecient of friction due to type of eliminator
water loading level (kg/m2 s)
reference water loading level (kg/m2 s)
air rate of ¯ow (kg/s)
ratio of annual maintenance and operating expenditures to original ®rst cost
water ¯ow rate (kg/s)
technical life of cooling tower (years)
constants due to type of eliminators
number of transfer units
total present value of operational cost throughout life cycle ($)
ratio of total life cycle net cost of system to ®rst yearÕs cost
economic factor
ratio of water to air mass ¯ow rates (kg/kg)
ratio of resale value to original ®rst cost
annual total operation time (h)
dimensionless correction factor
present value of total costs of tower throughout life cycle ($)
volume of cooling tower (m3 )
M.S. S
oylemez / Energy Conversion and Management 42 (2001) 783±789
Vopt
DTw
optimum value of volume of cooling tower (m3 )
temperature di€erential of water across cooling tower (K)
Greeks
e
eopt
gf
gm
q
e€ectiveness of cooling tower
optimum value of e€ectiveness of cooling tower
eciency of the cooling fan
eciency of fan motor
density of ambient air entering fan (kg/m3 )
785
after a few iterations. The P1 ±P2 method [10] is used in the optimization study. The list of design
values includes the present price of electrical energy, energy price rate, market discount (interest)
rate, technical life of the cooling tower, characteristics of the eliminator, allowable height of the
cooling tower, number of annual operating hours, density of ambient air, eciencies of the fan
and motor, and volume dependent ®rst cost of the cooling tower.
2. Formulation of the problem
Cooling towers of forced draft type are a mixed type of heat exchangers in fact. Performances
of cooling towers can be predicted by a signi®cant parameter, i.e. the value of e€ectiveness. The
e€ectiveness of the cooling tower depends upon the water to air mass ¯ow rate ratio and the
number of transfer units mainly. In the ®rst part of this study, the optimum e€ectiveness of
the cooling tower is determined and, so, the optimum mass ¯ow rate of air. The e€ectiveness of
the counter-current forced draft cooling tower is
eˆ
1 ÿ eNTU…Rÿ1†
1 ÿ R eNTU…Rÿ1†
1†
Since NTU and R are independent variables, the optimization must be done by using a multivariable optimization method [11] as
R
o…e†
o…e†
‡ NTU
ˆ0
o…NTU†
o… R†
2†
It is shown algebraically that Eq. (2) can be satis®ed if and only if R ˆ 1. For this case, the value
of R for the optimum e€ectiveness condition is [12]:
Rˆ
ma Ca;sat
ˆ 1:0
mw Cw
3†
The optimum air ¯ow rate will then be:
ma;opt ˆ
mw Cw mw Cw DTw
ˆ
Ca;sat
ii;2 ÿ ii;1
4†
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M.S. S
oylemez / Energy Conversion and Management 42 (2001) 783±789
The right-hand side of Eq. (4) is all constant design values. On the other hand, the heat and mass
transfer area of a cooling tower may be formulated as:
Ai
Ai
V
Ai
ˆ
ˆ
H ˆ aH
5†
A
A
V
V
and so:
Aˆ
Ai
aH
6†
The power requirement of the cooling tower fan can be determined by multiplying the pressure
drop with the air volume ¯ow rate [13] as:
h
i
2
m3a a2 H 2 …6:5 ‡ K † ‡ 2… A=Af †
7†
Pˆ
2q2 A2i gf gm
The present value of the cost of operation of the cooling tower throughout its technical life,
depending on the heat and mass transfer area, can be formulated by means of the P1 ±P2 method
practically as follows:
h
i
CE P1 m3a a2 H 2 S 6:5 ‡ K ‡ 2… A=Af †2
‡D
8†
OC ˆ
2q2 A2i gf gm
The initial cost of the cooling tower is:
IC ˆ CV AH ‡ CX
The present value of the total cost of the cooling tower throughout its technical life is:
h
i
CE P1 m3a a2 H 2 S 6:5 ‡ K ‡ 2… A=Af †2
‡D
TC ˆ IC ‡ OC ˆ CV AH ‡ CX ‡
2q2 A2i gf gm
9†
10†
The optimum value of the heat and mass transfer area can be determined by taking the derivative of the total cost function with respect to the heat and mass transfer area and then setting
this equation to zero as in the following equation.
o…TC†
ˆ0
o…Ai †
11†
The optimum heat and mass transfer area then can be calculated with the aid of Eq. (12).
v
h
i
u
2
3
3
2
u
3 CE P1 m a H S 6:5 ‡ K ‡ 2… A=Af †
a
t
Ai;opt ˆ
12†
q2 gf gm CV
This is exactly a minimum cost point since the second derivative of the total cost function is
always greater than zero. The P1 and P2 values are de®ned by the following:
M.S. S
oylemez / Energy Conversion and Management 42 (2001) 783±789
(
N )
1
1‡i
P1 ˆ
;
1ÿ
d ÿi
1‡d
Pi ˆ
N
1‡i
i 6ˆ d
if i ˆ d
787
13†
14†
ÿN
P2 ˆ 1 ‡ P1 MS ÿ RV …1 ‡ d†
15†
where MS is the ratio of the annual maintenance and operation cost to the original ®rst cost, RV is
the ratio of the resale value to the ®rst cost, i is the fuel price rate, d is the market discount
(interest) rate and N is the technical life. P2 can be assumed as unity for this application. K can be
determined iteratively, since it depends on the water and the air loadings in the cooling tower. K
may be assumed to be 3.5 initially at the beginning of the iteration. Then, the actual value of K can
be determined by using the optimized value of area for calculating the air and water loadings with
the help of the following equations in sequence:
Ai;opt
16†
Vopt ˆ
a
Vopt
17†
Aopt ˆ
H
A
18†
Af;opt ˆ
A=Af †
ma
Gˆ
19†
A
mw
20†
Lˆ
A
n4 n5
L
G
21†
K ˆ C2
L0
G0
The iteration ends when the actual (®nally iterated) value of K is determined. The value of NTU
is obtained by means of Eq. (22) as:
n1 n2
L
G
NTU ˆ C1 H
Thw †n3
22†
L0
G0
and the optimum value of the e€ectiveness can be established by the following equation:
eopt ˆ
NTU
NTU ‡ 1
23†
3. Results and discussion
The results obtained from the present optimization technique are compared with the actual
sizes of two sample cooling towers. In the ®rst example, a bench top cooling tower, illustrated in
Ref. [14], is considered with the following speci®cations: a ˆ 110 mÿ1 , ma ˆ 0:06 kg/s, CE ˆ 0:1 $/
788
M.S. S
oylemez / Energy Conversion and Management 42 (2001) 783±789
Fig. 1. E€ect of heat and mass transfer area on the costs of cooling tower.
kW h, N ˆ 30 years, S ˆ 16 000 h, q ˆ 1:2 kg/m3 , gf ˆ gm ˆ 0:8, CV ˆ 750 $/m3 and …A=Af † ˆ
2:43. The optimum heat and mass transfer area is calculated as 2.09 m2 , whereas the available heat
and mass transfer area is about 2.42 m2 . This indicates that approximately 15% extra heat and
mass transfer area is used for this speci®c illustrative example. Another optimization result for the
typical cooling tower speci®ed in Ref. [1] is determined as 1917 m2 , but its heat and mass transfer
area, determined by using the method given by Ref. [15], is 972.37 m2 , which indicates that approximately 50% less heat and mass transfer area was selected. The results of cost calculations for
the various heat and mass transfer areas for this sample are depicted in Fig. 1.
Fig. 1 explains the variation of the total life cycle cost of a sample cooling tower. As the heat
and mass transfer area increases, the total cost decreases up to a local minimum point which
corresponds to the optimum heat and mass transfer area. Cooling towers should be designed close
to this minimum cost point beyond which additional heat and mass transfer area is not cost effective.
4. Summary
Economy is a vital factor and extremely signi®cant in cooling tower operation. The economics
of counter-current forced draft cooling towers is discussed, and practical formulae are presented.
The validity of the optimization formulation is tested and con®rmed with two sample problems.
The present optimization technique seems to be helpful to cooling tower designers and manufacturers.
References
[1] Kintner-Meyer M, Emery AF. Cost-optimal analysis of cooling towers, ASHRAE Trans, vol 100, 1994, paper no.
3792, p. 92±101.
[2] Leary Jr VM. Optimizing cooling tower selections, Heating/Piping/Air Conditioning 1987;59:67±8.
M.S. S
oylemez / Energy Conversion and Management 42 (2001) 783±789
789
[3] Burger R. Retro®tting cuts cooling-tower costs. Chem Engng 1986;93:117±9.
[4] Shelton SV, Joyce CT. Cooling tower optimization for centrifugal chillers. ASHRAE J 1991;93:28±36.
[5] Borukhov VT, Fisenko SP. Optimization of chimney-type cooling tower operation under conditions of external
aerodynamic action. Inzhenerno-Fizicheskii Zhurnal 1992;60(6):678±83.
[6] Burger R. Modernize your cooling tower. Chem Engng Prog 1990;86:37±40.
[7] Burger R. Select the right cooling tower ®ll. Hydrocarbon Process 1994;73:141±3.
[8] Waller B. Various ¯ow rates in condenser water cycle. ASHRAE J 1988;30:30±4.
[9] Kirsner W. 3 GPM/ton condenser water ¯ow rate: does it waste energy? ASHRAE J 1996;38:63±9.
[10] S
oylemez MS. On the optimum heat exchanger sizing for heat recovery. Energy Convers Mgmt 2000;41(13):1419±
27.
[11] Stoecker WF. Design of thermal systems. 3rd ed. McGraw Hill, New York: 1989.
[12] Kreider JF, Rabl A. Heating and cooling of buildings. 1st ed. McGraw Hill, Singapore; 1994.
[13] Mills AE. Basic heat and mass transfer. 2nd ed. Prentice Hall; 1999.
[14] S
oylemez MS. Theoretical and experimental analyses of cooling towers. ASHRAE Trans 1999;105(1):330±7.
[15] Bernier MA. Cooling tower performance: theory and experiments. ASHRAE Trans, vol 100, 1994, paper no. 3794,
p. 114±21.