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An Evaluation of Heat and Mass Transfer in Open
Counter Flow Cooling Towers
Amin Jodat Department of Mechanical Engineering,
University of Bojnord, Iran
Abstract: Cooling towers are regarded as the most important equipments in oil industry. Most of
equipments and processes need cooling to work and the main fluid required for this industrial
work is water. Cooling tower makes water cool. The results obtained from testing evaporative
cooling can be evaluated through confirmed theories that have been created using dimensionless
analyses. These basic methods and theories can be used to design and predict working conditions
in points other than working points of device. The solution of many heat and mass transfer
devices is determined through dimensionless analyses and theory methods. The present study is
an attempt to analyze heat and mass transfer process in open counter flow cooling water. After
reviewing the related literature and rewriting the governing equations, a method has been
proposed. Also, the proposed method has been presented at two states of designing and rating.
Compared to previously conducted study, this method is more comprehensive. Finally, using
functional information of an industrial cooling tower, the obtained results of the proposed method
have been analyzed in terms of thermal exchanges, efficiency and NTU changes evaluation.
Keywords: counter flow cooling tower, Merker method, Sutherland method, thermal design,
efficiency, thermal exchanges
1- Introduction
Using cooling towers to remove heat and avoid its excessive emission in the environment
dates back to the 1940s and after that. Although simple water cooling systems were
used during the previous years, these systems were gradually evolved. In the 1940s,
these systems were changed into cooling towers. In general, cooling processes used in
chemical industries are divided into three classes of cooling with fresh water, cooling
with recirculation, and combination cooling. In cooling with fresh water, water passes
through condenser only one time while in cooling with recirculation, the same water is
circulating in cooling system constantly and convection is slightly performed. In
combination cooling, there is a combination of various systems.
Cooling with recirculation, in its own right is divided into two systems of open (i.e. wet
cooling processes) and close (i.e. dry cooling processes). The name of these systems can
describe their nature. In other words, a system pertains to open atmosphere and
another is a close system. That is, in wet heat exchange, water and air are in direct
contact with each other. However, in dry heat exchange, water or stem are not in direct
contact with air. Cooling towers also need a certain amount of makeup to compensate
the consumed water.
Cooling towers are fallen into the following types:
1. Induced draft cooling tower
2. Forced draft cooling tower
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3. Atmosphere cooling tower
In general, the structure of cooling towers consists of wooden water distributing plates,
air trap, water distributing pipes, cooled water tank, and fan. Basically, cooling towers
works in the following way: warm water returning from thermal convertors enters into
the top of the tower, is pour on distributing plates; air moves upward from the bottom of
the tower; heat is exchanged between air and warm water; cooled water enters the tank,
and is transferred into thermal convertors through air pump. The amount of water
evaporation depends on the level of air and water contact. In induced draft cooling
towers, fans are installed and send the air saturated by stem from the inside of the
tower to the outside and the fresh air is placed around the tower instead of the exited
air.
In forced draft cooling towers, fans are installed at the bottom of the tower on the
ground and the air is derived upward from the bottom of the tower. In this type of
cooling water, fans are easier to be repaired due to their low height. Around forced draft
cooling tower is closed while around the induced draft cooling tower is open. Therefore,
in induced draft towers, the air is in contact with water from all directions and evenly.
However, in forced draft cooling towers, the air is not in contact with water everywhere.
Atmosphere cooling towers works with atmosphere pressure through natural air flow
and fan is not used in these towers to set up air. The efficiency of these towers depends
on wind speed and air moisture. If wind speed is high, more water is lost and if wind
speed is slight, water is not adequately warmed. A mild wind flow has a good effect.
Further, if air moisture is high, water evaporation is not well occurred and water is not
cooled.
In the present study, two independent methods have been introduced to solve a
cooling tower. In the following, it has tried to evaluate the obtained results with
the aim of mass and thermal analysis of these two methods in an accumulated
tower. Also, a model has been proposed in counter flow cooling tower.
2- Related Litrature
For the first time, Walker at l. (1923) proposed the primary theory of cooling tower.
However, Merkel (1925) was the first person who practically attempted to solve the
problem of towers with direct flow. Merkel‟s method was based on a combination of
differential equations of water heat and mass transfer within a tower. In his method,
Merkel assumed that each drop within the tower is surrounded with air flowed from the
bottom of the tower. He also assumed that heat transfer is performed in the form of
tangible and covert heat transfer due to the evaporation of a part of water drop (Robert,
2001). Later on, cooling towers were designed based on rewriting Merkel‟s relations and
justifying the differences between theoretical and experimental forecasts. For instance,
it can be referred to the works of Nottage (1941) and changing his method to a graphical
solving method by Liehtenstein as well as presenting another graphical method to
determine air performance in a cooling tower by Michely (19490.
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Caytan (1982) presented a method to solve counter and cross flow cooling towers by
writing STAR computer code. In this method, he solved 2D differential equations
resulted by fluids dynamic and thermodynamic analyses through a finite differential
method on a network with rectangular meshes (Caytan, 1982).
Sutherland (1983) proposed a method in which an accurate solution was used to solve
equations governing cooling tower. In his model, Sutherland considered a control volume
and regulated differential equations based on quantities‟ changes from the bottom of the
tower to the up the tower (Sutherland, 1983). In this model, he considered the effects of
input water evaporation and assumed Lewis number single in opposite tower. He solved
differential equations related to his accurate solving and approximate solving through
TOWER A program and TOWER B program using the 4th order Runge Kutta method,
respectively. Additionally, he showed that ignoring water evaporation for sample
functional conditions in a tower can lead to higher errors in higher ratios of water to air
flow rate (e.g. for the ratio of water to air flow rate equals 2 in about 14%).
Fujita and Tezuka (1986) solved thermal function of counter and cross cooling towers
with compulsory air flow using Enthalpy potential theory (Fujita and Tezuka, 1986).
Webb (1988) proposed an accurate design instruction for cooling towers. This method
was a 1D mode that considered water losses due to evaporation. Louis‟ number was
assumed 87% in this method (Webb, 1988). The model proposed by Jaber and Webb
(1989) presented an efficient design method based on NTU computation for counter and
cross flow cooling towers (Jaber, 1989).
Mohiuddin and Kant (1995) presented two articles. The first article explains details of a
method to design cross and counter flew cooling towers. In this article, a method has
been proposed for step by step design of cooling tower. Also, in this article, computer
codes existed have been reviewed. The second article explains and presents transfer
coefficients for various networks (Mohiuddin and Kant, 1995).
El-Dessouky et al. (1997) theoretically investigated a counter flow cooling tower with
modified concepts of NTU and thermal efficiency of the tower. In the model, some
assumptions such as water film resistance, counter unit Lewis factor, and curvature in
saturated air‟s enthalpy curve were considered. According to the research findings,
ignoring water film resistance and Lewis hypothesis causes an error that depends on
mass and heat transfer coefficients values (El-Dessouky et al., 1997). Through empirical
studies on PVC made packing, Goshayshi et al. (1999) obtained the effects of distance
and coarseness of packings on mass and heat transfer coefficients as well as the level of
their pressure loss (Goshayshi et al., 1999).
Halasaz (1999) proposed a dimensionless mathematical model for governing equations
and obtained the tower efficiency based on only two variables. Using this method, he
presented acceptable results for both counter flow and cross flow cooling towers
(Halasaz, 1999).
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Stefanović et al. (2000) constructed a laboratory cooling tower with plexi glass-
made parallel plates and obtained temperature distribution along the tower
(Stefanović et al., 2000).
Using PHOENICS, Stefanović et al. (2001) solved heat and mass transfer in cooling
tower through 3D way and compared the obtained results with their previous obtained
results (Stefanović et al., 2001).
Jameel-ur-Khan and Zubair (2001) proposed a model similar to Sutherland‟s model in
which water film resistance was considered. The results obtained from their research
indicated that ignoring water film resistance and Lewis hypothesis causes considerable
error in computing NTU values and efficiency in tower compared to previous methods
such as Sutherland‟s method (Jameel-ur-Khan and Zubair, 2001).
In another article, Jameel-ur-Khan and Zubair (2003) investigated functional properties
of counter flow wet cooling towers. In their article, thermal performance of cooling
towers as well as convective and evaporative heat transfer in line with tower‟s height
have been intangibly shown (Jameel-ur-Khan and Zubair, 2003).
Soylemez (2004) analyzed hydraulic-thermal performance of counter flow cooling towers
optimization through a simple algebraic formula. In this work, NTU efficiency method
with extracting psychometric properties of humid air were used to analyze thermal
performance of counter flow cooling towers (Soylemez, 2004).
Kloppers and Kroger (2005) proposed equations governing cooling towers using Rigorous
Pope method, Merkel method and NTU method. They solved Rigorous Pope by the 4th-
order Runge Kutta method and compared the results of the three methods (Kloppers
and Kroger, 2005).
3- Mathematical Model
3-1- Merkel‟s Model
After 1925 when the first mathematical model to solve cooling tower was proposed by
Merkel, many researchers tried to present a more comprehensive model to more
accurately solve cooling tower problem without restricting hypotheses. The basic point
in all new works has been their comparison with previous models and empirical results.
Merkel‟s model is based on a combination of differential equations of mass and heat
transfer of water and air within a tower. Figure 1 shows the schematic of a water drop
within a tower. In this method, Merkel assumed that each drop within the tower is
surrounded with air flowed from the bottom of the tower. He also assumed that heat
transfer is performed in the form of tangible and covert heat transfer due to the
evaporation of a part of water drop. Merkel‟s model includes the following hypotheses:
1. A resistance in mass transfer from water mass (it is referred to a falling drop or
drop collection that are in identical temperature of Tw). There is no common
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border (thin layer surrounding water drops in which air saturation exist with
Tw).
2. There is no temperature difference between water mass and air-water common
border.
3. Input water flow rate is not decreased due to its evaporation within tower.
4. Lewis number equals unit.
Writing survival equations and applying referred hypotheses, Merkel obtained
governing equations as following:
∫
(1)
∫
(2)
Since Merkel did not consider the effect of evaporation in his computations, the amount
of mw in the tower assumed a constant number; therefore, he defined NUT as
in his
equations.
Given to energy conservation law in the tower, air enthalpy in the tower relative to
water temperature within the tower will be as following:
(3)
With respect to enthalpy diagram, the temperature of air and water within the tower
and the assumption of air enthalpy linearity relative to water temperature and the
equality of water mass temperature with air-water common border, thus:
(4)
Finally, Eq. 3 and Eq. 4 lead to E. 5 to solve tower problem using Merkel‟s method.
و
(5)
Moreover, sorting Eq. 1, the magnitude of the tower volume is obtained as following:
∫
(6)
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3-2- Sutherland‟s Model
Sutherland differentially wrote equations governing water cooling process based on
quantities‟ changes from bottom to up the tower. In Sutherland‟s model, input water
evaporation effects as well as Lewis number in the tower have been considered opposite
the unit (0.9). Finally, using this model, tower solving equations led to Eq. 7 device.
This device includes two the 1st order differential equations that entails air enthalpy
changes and water temperature based on absolute humidity of the air within the tower:
(
) ( ) ( )
(
) ( )
(
( ))
(7)
The right side of the above equations can be shown according to what shown as
functions of enthalpy, air absolute humidity and water temperature. These equations
have solved by Sutherland, assuming Lewis number constant and only in design state
(Sutherland, 1983). In this paper, assuming Lewis factor variable, at both states of
designing and rating, the above equations have been solved.
3-3- The Proposed Model
Principally, the problem of cooling tower is propounded int two general forms:
a. Designing:
( )
a. Rating
( )
In designing, having functional information of a cooling tower, the capacity or volume of
the tower is obtained. In rating, having the volume of the tower and information related
to input conditions and air exiting as well as input warm water conditions, output cold
water is obtained.
To solve the problem of cooling tower through the proposed method, Eq. 7 is used.
In his method, Sutherland assumed Lewis number as a value opposite unit that is
considered constant in line with the tower; however, given to the relation:
[
(
)
(
)]
(8)
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Lewis number is a function of absolute moisture and molecular ratio of gas and liquid
mixture (Poppe & Ronger, 1991). Using the Eq. 8, the proposed method computes Lewis
factor independently from each section of the tower and solve it.
To determine quantities required to solve equations, the phrase of ,
from reference equations (17) given based on and constant numerical
coefficients are used. Computing thermodynamic constants such as , in
each section of tower has been performed by interpolating the information of
temperature limit related to water and air. To solve Eq. 5 and Eq. 7, and to compute
numerical integrals, the 4th order Runge Kutta and Romberg integration are employed,
respectively. Figure 3 presents the process of solving tower using the proposed method.
To evaluate the accuracy of the responses obtained from the proposed model, functional
information and empirical NTU presented in Sutherland‟s articles (1983) and Jameel-
ur-Khan and Zubair (2001) are used. The formula used for experimental NTU in the
reference (28) is as following:
(
)
(9)
Where c and n are empirical constants that are certain for each cooling tower. C and n
coefficients in Eq. 9 have been determined for four types of cooling towers using Simpson
and Sherwood‟s experimental information (7).
Computing the errors of Merkel, Sutherland and the proposed methods relative to
experimental values, the following formula has been used:
( )
(10)
As observed in Table 1, the accuracy of the results obtained from MATMER model are
consistent with the results of NTU parameter obtained from Merkel‟s model presented
in Sutherland‟s article; however, NTU of Sutherland‟s model has an error equal to 18%
compared to the proposed method.
Further, Table 2 shows that the error of Merkel‟s method, compared to the experimental
results obtained by Zaiber (28), is higher than the proposed method. In other words, the
proposed method more accurately predicts the tower volume. Both methods also predict
the tower volume less than the actual amount; however, only in low ranges values of the
present method, the tower volume is predicted more than its actual value.
In Table 1, NTU and the volume related to Merkel and Sutherland methods have been
completed and compared with experimental values. A) NTU values given in
Sutherland‟s article (17) with
and comparing it with MATMER and
MATPERS models.
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Table 1. The values comparison in Sutherland‟s article
( ) ( ) ( ) ⁄
| | %
26.67 26.67 22.11 1.2 1.054 1.0461 0.74
26.67 26.67 22.11 1.2 2.866 2.856 0.02
29.44 35.00 23.89 1.0 1.188 1.165 1.8
b. NTU values and the volume obtained from Zubair
and
comparing it with MATMER and MATPERS models.
Table 2. The values comparison in Zubair‟s article
( )
( )
( )
( )
(kg/s)
(kg/s)
(m3)
(m3)
(m3)
1 31.22 23.88 37.05 21.11 1.158 0.754 1.297 0.4965 1.8357 0.457 7.9 1.2312 0.471 5.13
2 28.72 24.22 29.00 21.11 1.187 1.259 1.745 0.6847 1.5748 0.655 4.23 1.8291 0.712 -3.98
3 24.50 26.22 30.50 21.11 1.187 1.259 1.745 0.6847 1.5107 0.619 9.59 1.7143 0.672 1.85
4 38.78 29.23 35.00 26.67 1.265 1.008 1.467 0.6134 1.7532 0.582 5.11 1.4301 0.598 2.51
5 38.78 29.23 35.00 26.67 1.250 1.008 1.467 0.6099 1.7575 0.585 4.08 1.4581 0.602 1.29
4- Thermal Design of Cooling Tower
4-1- Tower Demand and the Curve Characteristic the Tower
Liechtenstein (1943) introduced cooling tower using Merkel‟s theory and based on the
base equations to determine border conditions in cooling tower. The result of this
equation causes to obtain the relation of dimensionless numbers group and heat and
mass transfer variables in a counter flow cooling tower. Liechtenstein performed some
experiments and specified that his equation does not cover mass rate of air and water.
His experiments performed in California University revealed the changes in demand
curve with input water temperature.
Merkel‟s equation is used to compute cooling demand based on designing temperature
and water-air rate. In other words, the value of KaV/L indicates a criterion for liquid
cooling rigidity. This equation associates designing temperature and air-water rate and
cooling demand to MTD (mean of temperature difference) that is used in all heat
transfer issues. Cooling demand curves and KaV/L value have been depicted as a
function of air-water rate. Approach temperature lines (tw1-WBT) have been shown as
the main parameters in these diagrams. These diagrams include 821 curves showing
KaV/L values for 40 wet bubble temperature and 21 cooling areas as well as 35 different
approaches temperature.
4-2- The Tower Characteristic
To analyze cooling performance and capability of a certain cooling tower, a certain
equation is needed. The following equation is commonly used to design tower and covers
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most of demand curves. In this equation, the relation of KaV/L indicates air-water rate
in linear and logarithmic form.
KaV/L = C (L/G)-m (11)
Where the value of KaV/L is determined by Merkel‟s equation. The constant value of C
depends on the tower design or the place of characteristic curve in L/G=1. The value of
m also depends on the tower design and is called slope. This value is obtained through
experiment. The characteristic curve is determined through the following ways:
The performance is obtained by the curve‟s constructor (if any) using the
constructor‟ information. At all states, the slope of these curves indicates function
slope.
Through experiment at installation place and finding a characteristic point and
depicting characteristic curve passing through this point parallel to the main
characteristic curve or the line passing through this point with the appropriate
slope of -0.5 to -0.8.
Through experimental at the place for at least two points for various rates of air-
water, lines passing through these two points are characteristic curve. The slope
of this line should be in an acceptable area and can be a criterion for efficient
accuracy. The characteristic point is experimentally obtained by measuring wet
temperature and discharge air temperature as well as input and output water
temperature.
Figure 1. Demand Curve
Air-water rate is also obtained through the following ways:
Confidently, it can be assumed that discharge air of the tower is saturated. Therefore,
output air of the tower equals wet temperature. Knowing output wet temperature of the
tower, the increase of air flow enthalpy can be obtained from psychometric equations.
For even distribution, air-water discharge rate should have an appropriate proportion.
When air discharge in the tower is re-circulated, input air of the tower should be
considered 1-2 ℉ higher than wet temperature of the environment.
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Using thermal and mass balance equation, dry air-water rate can be determined. Now,
through obtaining air-water rate and characteristic curve, KaV/L value can be obtained
by depicting in diagram.
4-3- Cooling Tower‟s Performance Variables
Many parameters influence the design and performance of cooling towers. Some of these
parameters are as following: tower‟s working water is one of the most effective and key
factors. One of the practical problems is the changes in water working rate and its effect
on input and output water temperature. Assume a tower that works under the following
conditions: water discharge rate is increased up to 20000 gpm. Assume that there is no
change in input air mass, wet temperature and cooling load (practically, due to the
increase of water discharge and pressure loss, air mass discharge is decreased).
Data:
(L1): 16000 gpm = the primary water discharge rate
(G1): 808048 lb/min = input air discharge rate
80.0℉ = wet bubble temperature of the environment
Sea level = the site‟s height
(HWT, tw2): 104.0 ℉ = warm water temperature
(CWT, tw1): 89.0 ℉ = cold water temperature
(m): -0.800 = characteristic curve slope
(L2): 20000 gpm= the secondary water discharge rate
Solution:
R1 = HWT - CWT = tw2 - tw1 = 104 - 89 = 15°F
L1= water discharge rate × (500 / 60) = 16,000 × (500 / 60) = 133,333.3 lb/min
Cooling load/time rate D1 = L1 × R1 = 133,333.3 × 15 = 2,000,000 BTU/min
Air mass discharge rate G1 = 80,848 lb/min
Liquid/gas ratio = L1 / G1 = 133,333.3 / 80,848 = 1.6492
L2= water discharge × (500 / 60) = 20,000 × (500 / 60) =166,666.7lb/min
Cooling load/time rate D2 = D1 = 2,000,000 BTU/min
Air mass discharge rate G2 = G1 = 80,848 lb/min
Liquid/gas ratio = L2 / G2 = 166,666.7 / 80,848 = 2.0615
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R2 = D2 / L2 = 2,000,000 / 166,666.7 or = R1 × (L1 / L2) = 12°F
Figure 2. Kv/l characteristic curve
4-4- Pressure Loss in Cooling Tower
Sudden change in movement path or air speed leads to a high pressure loss. The areas
which cause pressure loss in cooling tower are as following:
1. Input air
2. Packings
3. Water distribution pipes
4. Eliminator
5. Fan input (it is sometimes called plenum loss).
Computing air pressure loss, except than air pressure loss in packings, can be easily
computed. The following relation is used to compute pressure loss in different areas. In
the following relation, the coefficient of k depends on its area. Also, air density is in 7°F
and equals 0.075 lb/ft3. This pressure loss is called static pressure.
(12)
The performance of fan depends on this static pressure loss. K coefficient for input air is
as following:
When shutter is used in air entrance, a number between 2-3 is considered for a large
shutter with high distance between its blades. For small shutters with low distance
between its blades, a number between 2.5-3.5 is considered.
When instead of shutter, square-section columns are used, k coefficient equals 1.5 and
when circular-section is used, k coefficient equals 1.3. For columns with conic section, k
coefficient is considered 1.2.
Computing pressure loss of water distribution is considered with eliminator and k
coefficient is usually considered 3-1.6. K coefficient of pressure loss in fan entrance is
also considered 0.1-0.3.
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Open from one side: such arrangement is an entrance for cases in which the tower is
located in a place that causes the increase of input wet bubble temperature and enters
into the tower only from one side without any disturbance. In such arrangement, it
should be considered that air distribution on packings is performed evenly and
accurately. Open from two sides: this arrangement is usually used in industrial towers.
Open from all sides: back-to-back and open from all sides: this arrangement is
appropriate for places with limit area for installation.
5- Discussion and Results
To investigate and compare the results obtained from the proposed method, functional
information of an industrial cooling tower in maximum load conditions is considered:
( ) (
)
(
)
The efficiency (ε ) and temperature ratio (κ) for a cooling tower is defined as following:
ε
(13)
κ
(14)
The efficiency of a cooling tower indicates that to what extent a tower can enhances
input air enthalpy relative to the ideal state.
6- Design
Figure 3 shows NTU values of the tower based on temperature ratio changes for various
ratios of air-water mass flow rate. The changes of temperature ratio for change in input
air humid temperature ( ) from the temperature of to the temperature of
has been obtained. With respect to the fact that this curve for mass ratio of
ma/mw equals 0.5 from the ratio of temperature 0.7 and higher, the value of NTU
approaches a very large number. It indicates that in this area, to achieve lower output
water temperature, a very larger tower should be considered. In other words, in low
ratios of air-water mass flow rate, it is not possible to increase the temperature of humid
air from a certain limit without considering the tower with very larger volume.
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Figure 3. The changes of NTU values of the tower based on temperature ratio changes
Take a careful look at Figure 4, it can seen that a main part of heat exchanged in the
tower derives from the covert temperature of water evaporation. On the other hand,
heat transfer due to displacing from bottom to up the tower with 1400 m3 has been from
air to water and after this point, its direction has been reversed and increased to up the
tower.
Figure 4. The changes of heat exchanged in the maximum load conditions
Figure 5 shows curves related to the tower efficiency with respect to the changes of
output wet bubble temperature. Taking a careful look at the curve, it can be found that
the increase of ma/mw leads to the decrease of input air bubble temperature. On the
other hand, the decrease of input air bubble temperature in a certain ration of ma/mw
leads to the decrease of the tower‟s efficiency with a mild slope.
Figure 5. The changes in the tower‟s efficiency level based on changes in input air bubble
temperature
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7- Conclusion
According to the aforementioned, it can be concluded that using computer methods in
thermal computations of cooling tower provide acceptable results compared to the
presented diagrams. Combining computer and empirical methods, through measuring
wet bubble, input and output water temperatures and the discharge of water passing
through the tower as well as various arrangements, the performance of towers can be
predicted in conditions out of the design.
Given to the information provided in Tables 1 and 2, it can be stated that MATPRES
model is more accurate than Merkel and Sutherland‟s models. With respect to the fact
that input warm water flow rate and its temperature is constant, the performed
analyses have focused on the changes in parameters such as wet and dry air
temperature, water and air flow rate, evaporation losses and efficiency in the tower.
Moreover, it was revealed that to achieve lower output water temperature in lower
ma/mw ratios, very larger towers are required. On the other hand, it was found that a
main part of heat exchanged in the tower is due to the covert heat of water evaporation
and the proportion of heat transfer due to displacement in cooling water is slight. The
decrease of input air humid temperature also leads to the decrease of efficiency
coefficient in the tower. This fact confirmed the previously reported results.
Based on the performed analyses, at rating state, it was revealed that in a certain
ma/mw, the decrease of approaches in the tower leads to the decrease of range values.
The independency of a tower‟s efficiency in a certain ma/mw ratio to input dry air
temperature changes was confirmed in the present paper.
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