International Journal on Architectural Science, Volume 1, Number 3, p.120-122, 2000
THE DEVELOPMENT OF A MATHEMATICAL MODEL WITH AN
ANALYTICAL SOLUTION OF THE COUNTERFLOW CLOSED CIRCUIT
COOLING TOWERS
Y.A. Li and M.Z. Yu
Shandong Institute of Architecture and Engineering, Ji’nan, 250014, P.R. China
F.W. Shang
Shandong Xinlu Construction Development Co. Ltd, Ji’nan, 250061, P.R. China
P. Xie
Shanghai Jiaotong University, Shanghai, 200030, P.R. China
ABSTRACT
This paper describes the counterflow closed circuit cooling towers developed by the authors. The new cooling
towers have many desirable features, including pure water, low noise, safety, energy effective and so on.
The mathematical models of the counterflow closed circuit cooling towers are established in terms of mass and
heat balance. An analytical solution of the counterflow closed circuit cooling towers is carried out. Performance
curve of the cooling towers is drawn and it is mainly calculated that the outlet temperature of cooling water
varies with the spray water flow rate. Results of the theoretical calculation are found to be close to the
experimental data.
1.
INTRODUCTION
absorb the heat from the cooling water when it
evaporates.
In air conditioning work, water is frequently used
to absorb the heat rejected at the refrigeration
condenser. Since economy of operation normally
requires that this water be used again and again, it
must be continually recooled, and various devices,
such as cooling ponds, natural draught cooling
towers, mechanical draught cooling towers and
hyperbolic towers, are available to enable this to be
carried out by evaporative method. Because the
cooling water in these equipments is exposed to the
air, it is polluted and so the cool capacity of
refrigerators reduces greatly. It is important for us
to research into and develop a new cooling tower
that not only cools water, but also keeps cooling
water from pollution.
2.
STRUCTURE AND PRINCIPLE
The counterflow closed circuit cooling tower
consists of closed circuit heat exchange coil, spray
water distributor, fan, pump and so forth, as shown
in Fig. 1.
Cooling water flows in the closed circuit heat
exchange coil, meanwhile, spray water flows on the
surface of the coil. The temperature of cooling
water could be reduced because spray water would
tf1
tf2
Fig. 1: Counterflow closed circuit cooling tower
3.
MATHEMATICAL MODELS
The heat loss of cooling water may be expressed as
[1]:
w f ⋅ c ⋅ dt f = −k ⋅ (t f − t p ) ⋅ dx
(1)
The heat loss of spray water is given by:
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International Journal on Architectural Science
w p ⋅ c ⋅ dt p = −σ ⋅ (ib − i )dx + k ⋅ (t f − t p ) ⋅ dx
(2)
β 2 + (b1 + b4 ) β + (b1b4 − b2b3 ) = 0
The heat gain of air can be written as follows:
M ⋅ di = −σ ⋅ (ib − i )dx
Here, β1, β2 are the roots of the following equation:
(3)
where
(13)
The boundary conditions of the top of the cooling
tower are:
tf = ti , tp = tpi , i = io , ib = ibi
ib = f (t p )
(4)
boundary condition,
t pi = t po
(5)
so,
yi = (tpi - tfi) = c1 + c2
z i = (i o − ibi ) = −
c1 ( β 1 + b1 ) c 2 ( β 2 + b1 )
−
b2
b2
(14)
(15)
let y = ti - tp , z = i - ib
The boundary conditions of the bottom of the
cooling tower are:
ib can be approximately considered as one time
function of spray water temperature tp, then:
tf = tfo , tp = tpo = tpi , i = ii , ib = ibo = ibi
dt f
dt b
=m
dx
dx
so,
(6)
z o = (ii − ibi ) = −
We obtain that:
 σ 
dy  k
k 
⋅ z = 0
+
+
⋅ y + −


 wpc 
dx  w f c w p c 



k 
dz 
σ
σ
+ −m
⋅ y + m
−



wpc 
dx 
 wpc M

⋅ z = 0


(7)
(8)
k
k
σ
σ
, σ2 =
, σ3 =
, σ4 =
wf c
wpc
wpc
M
c1 ( β 1 + b1 ) β1F c 2 ( β 2 + b1 ) β 2 F
e
e
−
b2
b2
(17)
c1 =
( t pi − t fo ) − ( t pi − t fi )e β 2 F
e β1F − e β1F
(t pi − t fo ) − (t pi − t fi )e β1F
e β 2 F − e β1F
(18)
(19)
Equations (15) and (17) yield:
b1 = (σ 1 + σ 3 ) , b2 = −σ 2 , b3 = −mσ 3 ,
c1 = −
(i − i ) − (i − ibi )e β 2 F
b2
⋅ o biβ F
β1 + b1
e 1 − e β2F
(20)
c2 = −
(i − i ) − (i − i )e β 2 F
b2
⋅ o bi β F i β bi
β 2 + b1
e 2 − e 1F
(21)
b4 = (mσ 2 − σ 4 )
Then, equations (7) and (8) turn into:
dy
+ b1 y + b2 z = 0
dx
(16)
Equations (14) and (16) yield:
c2 =
It is convenient to introduce:
σ1 =
y0 = (t pi − t fo ) = c1e β1F + c2 e β 21F
(9)
From equations (18) to (21), we can obtain that:
 b2 (ii − ibi ) + ( β1 + b1 )(t pi − t fo ) 
ln 

β 2  b2 (io − ibi ) + ( β1 + b1 )(t pi − t fi ) 
1
dz
+ b3 y + b4 z = 0
dx
(10)
 b2 (ii − ibi ) + ( β 2 + b1 )(t pi − t fo ) 
ln 
=

β1  b2 (io − ibi ) + ( β 2 + b1 )(t pi − t fi ) 
From the above equations, we can obtain:
y = c1e β1 x + c 2 e β 2 x
z=−
121
c1 ( β1 + b1 ) β1 x c2 ( β 2 + b1 ) β 2 x
⋅e −
⋅e
b2
b2
(22)
1
(11)
(12)
Equation (22) is a formula of the cooling water
outlet temperature tfo. To a given cooling tower, the
cooling water outlet temperature is a function of
inlet air wet bulb temperature, spray water rate,
International Journal on Architectural Science
increases with spray water rate increases. But
when spray water rate increases too much, the
rate change has only a little effect on the
cooling capacity. The best ratio of spray
water flow rate and cooling capacity is about
0.08 m3kw-1h-1.
spray water temperature, air flow rate, cooling
water inlet temperature, and cooling water rate.
4.
EXPERIMENTAL STUDY
Some experiments of thermodynamic performance
of DBL-20 (2) closed circuit cooling tower was
carried out in August 1995 [2] (the experiment
measure will be explained in another paper). Test
results have shown that the higher the spray water
flow rate, the better the cooling effect. The
resistance of air will greatly increase if spray water
flow rate is too fast. The best results of spray water
flow is 0.07 ~ 0.09 m3kw-1h-1.
Fig. 2 shows the influence of spray water rate on
the outlet cooling water temperature. As shown in
the figure, the cooling capacity of the cooling tower
increases with larger spray water rate. The reason is
that the more the spray water, the more water
evaporation and the more heat the water absorbs.
But the increase of spray water rate has a little
effect on cooling water when it exceeds a certain
rate.
o
Cooling water outlet temperature(C) ( C)
The new cooling tower developed by the authors is
conferred patent right, No. 93230206.8, by China
Patent Bureau since it is different from traditional
cooling towers in structure [2].
Experimental data
Calculated results
50
NOMENCLATURE
M
wf
wp
σ
k
F
tf
tp
i
ib
m
c
air flow rate, kgs-1
cooling water rate, kgs-1
spray water rate, kgs-1
mass transfer coefficient, from spray water to
air flow, kgm-2s -1
heat transfer coefficient, from cooling coil to
spray, reference surface is out surface,
Jm-2s-1oC-1
area of cooling coil, m2
cooling water temperature, oC
spray water temperature, oC
enthalpy of air, Jkg-1
enthalpy of wet saturated air, Jkg-1
slop of saturation curve in the hythergraph of
wet air, Jkg-1oC-1
specific heat capacity of water, Jkg-1oC-1
Subscripts
i
o
input
output
REFERENCES
45
1.
Y.A. Li et al., “Study on closed circuit cooling
tower in air conditioning systems”, Refrigeration
Journal, No. 1 (1997).
2.
Y.A. Li et al., “Dynamic simulation and
experimental study of thermodynamic performance
of closed circuit cooling tower in air conditioning
systems”, Refrigeration Journal, No. 4 (1998).
40
35
30
20
40
60
80
100
120
140
160
180
200
220
Spray water rate(% )
Fig. 2: Effect of spray water rate on cooling
water outlet temperature
5.
CONCLUSIONS
•
Main factors that affect thermodynamic
performance of the counterflow closed circuit
cooling tower are spray water rate, air flow
rate, and air wet bulb temperature if its
structure is designed.
•
Experiment and theory study show that the
cooling capacity of the cooling tower
122