Chapter SM 3 Cooling Towers
SM 3.1 Heat and Mass Transfer Processes in a Cooling Tower
The performance of a cooling tower is determined using the fundamental thermodynamic
relationships for the processes that the air and water undergo as they pass through the tower. The overall
energy and mass balances for a cooling tower were developed in Chapter 5, Section 5.7. For a given
tower the air and water outlet states depend on heat and mass transfer coefficients and surface area. The
development of the basic performance relations for a cooling tower parallels that for a cooling coil
(Section SM 2.2). In contrast to a cooling coil, a cooling tower does not have a surface that separates the
two streams. Both water vapor and energy are directly transferred from the water stream into the air
stream, which simplifies the analysis.
The control volume showing the notation and mass and energy flows for a counterflow cooling tower
section is given in Figure 3.1. A convenient coordinate for developing the performance relations is the
distance from the top of the tower, and the fill volume is used as the coordinate instead of the linear
distance. The control volume is the volume between the sections at V and at V + V shown in Figure
3.1. The relevant conservation relations are an overall mass balance and an overall energy balance, a
mass balance that relates the increase of moisture in the air flow to the rate of transfer from the water
surface, and an energy balance that relates the increase in the air enthalpy to the energy transfer due to the
evaporating water. These latter two relations bring in the heat and mass transfer coefficients, which then
relate to the area and capacity of the cooling tower.
Water stream
 w , Tw
m
V
e
m
E
e
V + V
Air stream
 a , Ta , w a
m
Figure 3.1 Mass and energy flows for a section of a counterflow cooling tower
The overall mass balance relates the decrease in the mass flow rate of water due to evaporation to the
moisture increase of the air stream. For the control volume, this balance is:
(3.1)
m w V  m a w VV  m w VV  m a wV  0
3.1
Rearranging equation 3.1, dividing by the volume increment V, and passing to the limit V  0, a
differential equation for the liquid water mass flow rate and the humidity ratio of the air stream as a
function of distance, or volume, through the tower, is obtained.
w
dm
dw
a
m
(3.2)
dV
dV
The water mass flow rate at any level in the tower is obtained by integrating equation 3.1. The water
enters at the top of the tower where the mass flow rate is the entering value and the air humidity ratio is
the leaving value. The water mass flow rate at any position in the tower is then given by:
 wm
 w,in  m
 a w out  w 
(3.3)
m
where w is the humidity ratio of the air at that position. The mass flow rate of the water decreases
continuously as it flows through the cooling tower.
The energy balance for the control volume relates the energy decrease of the water flow due to
evaporative cooling to the energy increase of the air flow. .
(3.4)
m w h w V  m a h a VV  m w h w VV  m a h a V  0
Rearranging equation 3.4, dividing by the volume increment V, passing to the limit V  0, and
representing the water enthalpy in terms of the product of temperature and specific heat yields a
differential equation for the air enthalpy and water temperature. The water flow rate and temperature
both change with position.
d  m w Tw 
d ha
c p,w
 ma
(3.5)
dV
dV
The first term in equation 3.5 can be expanded by differentiating by parts, and equation 3.2 used to
relate the derivative of water flow rate with volume to the air humidity ratio. The expression for the local
water flow rate, equation 3.3, is used for the water flow rate. The resulting differential equation is:
dT
d ha
dw
cp,w  mw,in  ma  w out  w   w  cp,w Tw ma
 ma
(3.6)
dV
dV
dV
The overall mass and energy balances are represented by equations 3.2 and 3.6. These equations
relate the decrease in water flow rate to the increase in air humidity ratio and the decrease in water
temperature to the increase in air enthalpy. Heat and mass transfer relationships are next introduced to
relate the rates of mass and energy transfer to the transfer coefficients and the exposed water surface area.
With these additional equations, the capacity of the tower can be related to the physical size of the cooling
tower.
The conservation of mass for a control volume that includes the air flow and extends to the air-water
interface brings in the mechanism equation that relates the rate of evaporation of water per unit volume
 "e' to the mass transfer conductance (Section 5.7).
m
m'''e  h m A'''  w w  w 
(3.7)
where hm is the mass transfer conductance, ww is the saturation value of the humidity ratio at the water
temperature Tw, A''' is the heat transfer area per unit volume of the fill, and hm is the convective heat and
mass transfer coefficient. The Lewis number equal to unity analogy is used to relate the mass transfer
coefficient to the heat transfer coefficient (Section 5.7).
3.2
hm 
hc
c p,m
(3.8)
where hc is convective heat transfer coefficient and cp,m is the air-water vapor mixture specific heat. The
 "e' is the negative of the rate of change of water flow rate with respect to
flow rate of evaporating water m
volume. The change in water flow rate can be related to the product of the air flow rate and the rate of
change of the humidity ratio. Combining equations 3.2 and 3.7 yields:
h
dw
(3.9)
ma
  me'''   c A'''  w w  w 
dV
cp
It is convenient to introduce a non-dimensional transfer coefficient (Ntu) into equation 3.9, which
parallels the development of the performance for sensible heat exchangers (Section 13.1) and cooling
coils (Section SM 2.2). The Ntu* is defined as:
h c A''' V
*
(3.10)
Ntu 
ma cp,m
Equation 3.9 can then be written compactly in terms of the number of transfer units as:
dw
Ntu *

(3.11)
ww  w
dV
V
The energy flow per unit volume associated with the evaporating water flow is given in terms of the
energy transfer coefficient and the potential for energy transfer. The potential is the difference between
the enthalpy of the saturated air at the water surface, which is at the water temperature, and the enthalpy
of the air flowing past the surface (Section 5.7). The energy transfer per unit volume is given as:
h
(3.12)
E '''  c A'''  h w,sat  h a 
cp,m
where hw, sat is the enthalpy of saturated air at the local temperature of the water. The energy balance for a
control volume that includes the air flow and extends to the air-water interface yields
d ha
h
ma
  E '''   c A'''  h w,sat  h a 
(3.13)
dV
c p,m
Using the definition of the Ntu* (equation 3.10) allows the air energy balance relation to be written as:
dh a
Ntu *

(3.14)
 h w,sat  h a 
dV
V
There are four coupled equations that describe the heat and mass transfer processes inside the cooling
tower. Equation 3.2 describes the water mass balance, equation 3.6 the energy balance, equation 3.11 the
rate of mass transfer, and equation 3.14 the rate of energy transfer. An exact solution for the outlet air
and water states can be obtained by simultaneous integration of the equations using the boundary
conditions on the air inlet enthalpy and water inlet temperature. The solution is complicated in that
boundary conditions are specified at both ends of the tower, and iteration is needed to converge on a
solution that satisfies both boundary conditions. The analogy solution presented in the next section
allows an accurate and neatly exact solution to be readily obtained.
SM 3.2 Cooling Tower Performance using an Analogy to Heat Transfer
3.3
The analogy approach developed in Sections SM 2.1 and SM 2.2 for cooling coils can be extended to
provide a general method for representing the performance of cooling towers. A simplifying assumption
is made that for purposes of determining the energy transfer to the air the water flow rate is constant
throughout the tower. Since the water loss is typically 1 to 5% of the total flow this assumption is
reasonable. However, the difference in water flow rate between the inlet and outlet is included in
determining the outlet water temperature. Assuming that the water flow rate is constant allows the energy
balance relation, equation 3.5, to be simplified to:
dT
dh
(3.15)
mw cp,w w  ma a
dV
dV
Equation 3.15 contains the temperature of the water and the enthalpy of the air. In order to develop
the analogy relations, the equation will be formulated in terms of one property, the enthalpy. An effective
specific heat similar to that employed in the cooling coil analysis is introduced to allow the water
temperature Tw to be replaced by the saturated air enthalpy hw,sat evaluated at the water temperature Tw.
The effective specific heat is defined so that the water temperature and the saturated air enthalpy are
related as
d h w ,sat
dT
cs w 
(3.16)
dV
dV
where hw,sat is the enthalpy of saturated air at the temperature of the water. As with the cooling coil, the
effective specific heat, cs is evaluated as the change in enthalpy with temperature along the saturation line.
The appropriate value for the entire cooling tower process is based on the water inlet and outlet states, and
is evaluated numerically as:
h

h
 dh w ,sat 
   w ,sat,in w ,sat,out 
c s  
(3.17)


 dTw  sat  Tw ,in Tw ,out 
The numerical value of cs is on the order of 1 to 2 Btu/lbm-F (4 to 8 kJ/kg-C). Evaluating cs may require
iteration since the water outlet state is usually not known at the start of the analysis. However, a
reasonable guess value for outlet temperature is the air inlet wet bulb temperature (or a typical approach
to wet bulb), allowing the actual value to be obtained quickly by iteration. Incorporating the effective
specific heat allows the overall energy balance, equation 3.15, to be rearranged and written in terms of
enthalpies as:
dh w,sat
m a cs d h a

(3.18)
dV
m w c p,w dV
It is convenient to define a mass transfer capacitance rate ratio m* that is analogous to the thermal
capacitance rate ratio C* used in sensible heat exchanger analysis, which is also equal to the m* used in
the analogy relations for the coiling coil.
m a cs
m* 
(3.19)
m w c p,w
The variables in the two analogy equations that describe the heat and mass transfer for the cooling
tower are enthalpies. These are the energy balance equation 3.18, which is rewritten using the mass
transfer capacitance rate ratio m* as:
3.4
dh a
1 dh w ,sat
 *
(3.20)
dV m
dV
and the air stream energy balance equation 3.14, repeated here:
dh a
Ntu *
(3.14)

 h w,sat  h a 
dV
V
Equations 3.14 and 3.20 are analogous to those for a sensible heat transfer exchanger with the
enthalpies replacing the temperatures. The analogy between the two types of exchanger means that the
solution for the cooling tower in terms of enthalpies is the same as the solution for the sensible heat
exchanger in terms of temperatures for the same values of the parameters. As a result, the effectivenessNtu relations that were developed for heat exchangers can also be directly used for cooling towers in the
same manner as they were for cooling coils.
In a heat exchanger, the heat transfer is given in terms of effectiveness and maximum heat transfer
rate. The total energy transfer for the tower can then also be represented by an effectiveness and a
maximum energy transfer rate. The maximum energy transfer would occur when the air leaving the
tower is saturated at the water inlet temperature, and is given by.


(3.21)
Q
max  m a h w ,sat ,in h a ,in


The tower heat transfer rate is given by the product of the effectiveness and the maximum energy transfer
rate:
Q  * ma h w,sat,in  h a,in
(3.22)


where * is an energy transfer effectiveness. The correspondence between the cooling tower and the
sensible heat exchanger parameters is given in Table 3.1:
Table 3.1 Analogous Parameters for Sensible Heat Exchangers and Cooling Towers
Parameter
Sensible Heat
Exchanger
Cooling Coil
Capacitance
rate ratio
C*
m*
Number of
Transfer Units
Ntu
Ntu*
Effectiveness
  f C* , Ntu
Maximum
heat flow
Heat flow



*  f m* , Ntu *
C min Th,i  Tc,i 

 a h w,sat,in  h a ,in 
m
ε C min Th,i  Tc,i 
* ma  h w,sat,in  h a,in

In computing the effectiveness, the parameter m* replaces C* in the effectiveness relations.
Although the numerical value of m* may be greater or less than unity the analogy allows it to replace the
3.5
term C* in the effectiveness-Ntu relations. The appropriate relation to use for effectiveness depends on
the tower geometry, which is either counter or cross flow.
The air outlet state is determined from the energy balance on the air as:

Q
(3.23)
h a ,out  h a ,in 
a
m
The air outlet state is determined only in terms of the enthalpy, and another property is needed to fix
the state. To find the humidity ratio of the outlet air, it is assumed that the humidification of the air
stream in the tower is equivalent to that of an air stream that passes over a wetted surface at a uniform
temperature. The energy equation, equation 3.14, can be integrated to yield the effective enthalpy of a
wet surface that is uniform in temperature and that yields the same enthalpy change of the air stream as
occurs in the actual process.
h
h
h s,eff  h a,in  a,out a,in
*
(3.24)
1 e Ntu
The air outlet humidity ratio is calculated assuming that the evaporation is from this saturated surface:
*
w out  w s,eff   w in  w s,eff  e  Ntu
(3.25)
where ws,eff is the saturation humidity ratio at the enthalpy hs, eff. The outlet air temperature is then found
from the enthalpy and humidity ratio.
The water outlet temperature is determined from an overall energy balance on the tower. It is
assumed that the makeup water is added at the outlet temperature, which gives the following relation for
the temperature of the water leaving the tower and entering the system.
m a  h a,out  h a,in 
Tw,out  Tw,in 
(3.26)
m w,in c p,w
If the makeup water is at a temperature different from that leaving the tower an energy balance on the
sump needs to be made. However, the makeup water is typically less than 5 % of the water flow and
equation 3.26 is sufficiently accurate.
As with the cooling coil, the analogy provides a relatively simple approach to calculate the
performance of a cooling tower. Example 3.1 illustrates the use of the analogy approach to determine the
outlet states and capacity for a cooling tower.
Example 3.1 Determine the performance of a counterflow cooling tower with a water flow rate of 30,000
lbm/hr that enters the tower at 100 F. The air flow rate is 6000 cfm and the ambient is at 85 F and 50 %
relative humidity. The tower has a fill volume of 900 ft3, a wetted area density of 3 ft2 of surface area
per cubic foot of volume, a convection coefficient of 6 Btu/hr-ft2-F and the geometry is counterflow.
"Problem specifications"
"Air side"
p_atm = 14.7 "psia"
T_a_in = 85 "F"
RH_in = 0.5
CFM = 6000 "cfm"
"Water side"
"Pressure"
"Temperature"
"Relative humidity"
"Volume flow rate"
3.6
T_w_in = 100 "F"
m_dot_w = 30000 "lbm/hr"
"Tower Parameters"
A = 3 "ft2/ft3"
h_c = 6 "Btu/hr-ft2-F"
Vol = 900 "ft3"
"Temperature"
"Mass flow rate"
"Area per unit volume"
"Convection coefficient"
"Volume"
"Determine the properties of air and the air flow rate"
w_a_in = HumRat(AirH2O,T=T_a_in, P=p_atm,R=RH_in) "lbm/lbm"
h_a_in= Enthalpy(AirH2O,T=T_a_in,P=p_atm,R=RH_in) "Btu/lbm"
cp_a = SpecHeat(AirH2O,T=T_a_in,P=p_atm,R=RH_in) "Btu/lbm-F"
rho_std = 0.075 "lbm/ft3"
m_dot_a = rho_std*CFM*convert(1/min,1/hr) "lbm/hr"
"Humidity ratio"
"Enthalpy"
"Specific heat"
"Density"
"Mass flow rate"
"Determine the properties of water and of saturated air at the water inlet and outlet temperatures. The outlet
temperature is not known and so the outlet properties need to be obtained by iteration."
cp_w = SpecHeat(Steam,T=T_w_in,P=p_atm) "Btu/lbm-F"
"Specific heat"
h_w_in= Enthalpy(AirH2O,T=T_w_in,P=p_atm,R=1) "Btu/lbm"
"Enthalpy"
h_w_out = Enthalpy(AirH2O,T=T_w_out,P=p_atm,R=1) "Btu/lbm"
"Enthalpy"
"Determine the effective specific heat c_s (equation 3.20)"
c_s = (h_w_in - h_w_out)/(T_w_in - T_w_out) "Btu/lbm-F"
"Specific heat"
"Determine the Ntu (equation 3.10), m_star (equation 3.19) and effectiveness of the tower. The UA is obtained
from the heat transfer coefficient per unit volume and the volume. The effectiveness is determined for a counterflow
arrangement (Table 13.1)."
UA = A*Vol*h_c
"Overall conductance"
Ntu = UA/(m_dot_a*cp_a)
"No. of transfer units"
m_star = m_dot_a*c_s/(m_dot_w*cp_w)
"Mass flow rate ratio"
eff= (1-exp(-Ntu*(1 - m_star)))/(1 - m_star*exp(-Ntu*(1 - m_star)) )
"Effectiveness"
"Determine the heat transfer and the outlet air enthalpy from an energy balance on the tower using equations 3.21
and 3.22."
Q_max = m_dot_a*(h_w_in - h_a_in)
Q = eff*Q_max "Btu'hr"
"Heat transfer rate"
Q = m_dot_a*(h_a_out - h_a_in) "Btu'hr"
"Heat transfer rate"
"Determine the outlet air temperature and humidity ratio using the effective surface temperature and enthalpy from
equations 3.24 and 3.25"
h_s_eff = h_a_in +(h_a_out - h_a_in)/(1 - exp(-Ntu)) "Btu/lbm"
"Enthalpy"
h_s_eff = Enthalpy(AirH2O,T=T_s_eff,P=p_atm,R=1) "Btu/lbm"
"Enthalpy"
w_s_eff = HumRat(AirH2O,T=T_s_eff, P=p_atm,H=h_s_eff) "lbm/lbm"
"Humidity ratio"
w_a_out = w_s_eff +(w_a_in - w_s_eff)*exp(-Ntu) "lbm/lbm"
"Humidity ratio"
h_a_out = Enthalpy(AirH2O,T=T_a_out,P=p_atm,W=w_a_out) "Btu/lbm"
"Enthalpy"
"Determine the water loss and outlet flow rate from a mass balance on the tower and the outlet water temperature
from an energy balance on the tower (equation 3.26)."
m_dot_w_loss = m_dot_a*(w_a_out - w_a_in) "lbm/hr"
"Mass flow rate"
m_dot_w_out = m_dot_w - m_dot_w_loss "lbm/hr"
"Mass flow rate"
T_w_out = T_w_in - m_dot_a*(h_a_out - h_a_in)/(m_dot_w*cp_w)"F"
"Determine the tower capacity, range, and approach."
Capacity = m_dot_w*cp_w*(T_w_in - T_w_out) "Btu'hr"
Range = T_w_in - T_w_out "F"
T_wb= WetBulb(AirH2O,T=T_a_in,P=p_atm,R=RH_in) "F"
Approach = T_w_out - T_wb "F"
3.7
"Heat transfer capacity"
"Temperature"
"Temperature"
"Temperature"
Results and Discussion
The Ntu for the tower depends on the air flow rate (27,000 lbm/hr) and the overall conductance
(16,200 Btu/hr-F) and is 2.44. The value of cs is found to be 1.397 Btu/lbm-F, yielding a value for m* of
1.258. Using the counterflow relations, the effectiveness is determined to be 0.644. The maximum heat
transfer rate is based on the difference between the enthalpy of saturated air at the water inlet temperature
(71.5 Btu/lbm) and the air inlet enthalpy (34.6 Btu/lbm) and is 642,684 Btu/hr.
The air outlet enthalpy is found from an energy balance to be 58.4 Btu/lbm. Assuming that the heat
and mass transfer are from a surface with a uniform effective enthalpy (determined as 60.7 Btu/lbm), the
outlet air temperature is 92.7 F and the outlet humidity ratio is 0.0328 lbmw/lbma. The water loss is
determined from the humidity rise of the air stream and is 537 lbm/hr. The outlet water temperature is
determined from an energy balance on the water flow to be 80.0 F.
The tower capacity is 599,785 Btu/hr. The tower capacity is slightly less than the energy transferred
in the tower. The capacity is the amount of heat that is transferred to a constant flow of water through the
equipment connected to the tower, such as a condenser. The approach of the tower outlet temperature to
the wet bulb temperature is 9.2 F and the range is 20.0 F. This set of values comprises the design
performance for the tower at these conditions.
The analogy approach allows a direct evaluation for any inlet condition, and readily accommodates
different flow arrangements. For example, cooling towers are often cross-flow rather than counterflow,
with the air and water both unmixed since the packing serves to prevent mixing across the streams. Using
the effectiveness for cross-flow heat exchangers in Table 13.1, the capacity would be reduced to 547,000
Btu/lb, which is about 9 % lower than that for a counter flow tower.
The analogy approach allows the performance of a cooling tower to be represented in a terms of
fundamental heat and mass transfer parameters. With these parameters, a cooling tower can be designed
to provide a desired amount of cooling at any set of operating conditions. In many situations, it is
desirable to select one that is available from a manufacturer. Manufacturers provide the information
necessary to select a tower to reject a given amount of heat rejection at different ambient and operating
conditions, but often do not give enough information to determine the energy parameters such as the Ntu
and flow rate ratio.
In the next section a procedure to select a cooling tower using catalog information will be developed.
In the following section, a method for combining the catalog information with the analogy approach will
be illustrated.
SM 3.3 Extension of Catalog Information
The use of manufacturer’s data to select a cooling tower for a given situation is illustrated in Chapter
14, Section 14.4. In Example 14.2, the desired operating conditions were those available from the
catalog. Manufacturer's information is limited in coverage, and may not give information at all design
conditions. For design purposes, a method is needed to extend the information to other conditions or
3.8
altitudes. Using the analogy method presented in Section 3.2, the available design information from a
catalog may be used to provide a general representation of the performance (Mitchell and Braun, 2008).
The approach to extending the information from one operating condition to another is based on using
the analogy approach to determine the number of transfer units (Ntu) for a given condition. With the
number of transfer units, the effectiveness and the capacity are then determined for the new operating
conditions.
If the new operating conditions are for different inlet states but the same air and water flow rates, then
the only parameter that needs to be changed is the effective specific heat cs. However, the overall
conductance is a function on the air and water flow rates, and if either the new air or new water flow rates
are different, then there is a change in the overall transfer conductance. A correlation recommended by
ASHRAE relates the overall conductance to the flow rates and design values using a power relation given
as (Braun et al., 1989):
 m
 w  m
 a 


h c A ''' V  h c A ''' V base 
 w , base  m
 a , base 
 m



n
(3.27)
where the exponent n may be determined from data at different operating conditions. If data are
not available, a value for n of 0.4 is a satisfactory approximation. The change in the Ntu due to
changes in flow rate is given by combining equations 3.10 and 3.27. For other flow rates, the
Ntu is related to the base value as
Ntu 
*

Ntu*base 

mw
 m w, base




n
 ma

 ma, base





n 1
(3.28)
Example 3.2 illustrates how catalog information at one operating condition can be used to estimate
the design performance at other ambient conditions. A representation of a cooling tower can then be
developed and the performance evaluated over the range of operating conditions.
Example 3.2 Using the design information for model 492 B in Chapter 14, Table 14.2 at a wet bulb
temperature of 75 F, operated with a range of 10 F and a 2 hp fan, and with a capacity of 107 gpm as a
base, estimate the design information for the three conditions listed below.
a) A wet bulb temperature of 64 F and a range of 10 F (131 gpm).
b) A wet bulb temperature of 64 F and a range of 15 F (96 gpm).
c) A wet bulb temperature of 64 F and a range of 15 F with 1 hp fan (75 gpm).
The base operating conditions are used to determine the base value of the Ntu. The water flow rate
capacity is 107 gpm, or 53,553 lbm/hr, which for a range of 10 F corresponds to capacity of 535,530
Btu/hr.
If the air flow rate were available then the value of m* could be computed directly from the relation
equation 3.22. However, the air flow rate for this tower is not known, and so the value of m* cannot be
determined. An assumption will then be made that the value of m* is unity. For well-design cooling
towers the value of m* is on the order of unity and so this is a reasonable assumption. This allows an air
flow to be determined from the definition of m*, equation 3.19. The effective specific heat cs is 1.384
3.9
Btu/lbm-F based on the water inlet and outlet conditions. The air flow rate is then determined using the
assumption that m* is unity to be
 c
m
53,553 (lb / hr ) * 1.00 (Btu / lb  F)
 a  m * w w 1 *
m
 38,700 lb / hr
cs
1.384 (Btu / lb  F)
With this value of air flow rate, the effectiveness can be determined from equation 3.22. Using the
enthalpies of the saturated air at the water inlet temperature of 63.2 Btu/lbm and that of the entering air of
38.4 Btu/lbm, the effectiveness is

Q
535,530 (Btu / hr )


 0.558
 a h w ,sat,in h a ,in 38,700 (lb / hr ) *63.2  38.4 (Btu / lb )
m


These are cross-flow towers with both flows unmixed, and the Ntu can then be determined from the
appropriate expression for effectiveness in Table 13.1. The value of Ntu corresponding to an m* of unity
and an effectiveness of 0.558 is 1.504. Although the values of effectiveness and Ntu are not “correct”
since they are based on the assumption of an m* of unity and the resulting air flow rate, the combination
of these values yields the correct total heat transfer.
The extension can now be made to situation a), in which the wet bulb temperature is 64 F. Using the
enthalpies of water at the new inlet and outlet conditions of 55.8 and 43.6 Btu/lbm, respectively, the new
value for cs of 1.220 Btu/lbm-F is computed. There are three coupled equations that need to be solved
simultaneously for the new condition: m* from the definition (equation 3.25), Ntu from the definition
(equation 3.16) including the effect of the new water flow rate (equation 3.31), and the expression for the
effectiveness of a cross-flow exchanger from Table 3.1. The solution of these three equations yields m* =
0.723, Ntu = 1.629, and  = 0.634. The heat transfer computed using equation 3.25 with the value of the
inlet air enthalpy of 29.2 Btu/lbm for this condition is 653,000 Btu/hr. The water flow rate determined
from the expression for capacity, equation 3.1, is 65,300 lbm/hr, which corresponds to 130.6 gpm. This is
essentially the same value given in Table 14.2 for these conditions of 131 gpm. Following the same
procedure for condition b) with a the 15 F range, the design flow rate at these conditions is 93.1 gpm,
which is within 3 % of the catalog value.
For the conditions represented by c), the air flow rate is different from the base conditions. The
actual values of the flow rates are not given, and so the fan law relation between power and flow rate is
used to estimate the relative change in flow rate. The power is proportional to the cubic power of flow
rate and the air flow rate at condition c) relative to the base case is then:
1/ 3
 P 
ma  ma, base 

 Pbase 
1/ 3
 1(hp 
 38,700(lb / hr) 

 2(hp) 
 30,700lb / hr
Following the calculations described earlier, the design capacity is 76.6 gpm, which is within 2 % of the
catalog value for that condition.
The extension of the available design information to all of the other inlet conditions and air and water
flow rates for the same tower model was carried out. The results are summarized in the following figure,
where the design capacity predicted by the analogy approach is plotted against the catalog values.
3.10
The inlet, outlet, and wet bulb temperatures and the air and water flow rates each affect the
performance through the heat transfer parameters (m*, Ntu, ), as indicated by the catalog information in
Table 14.2. However, the analogy method is able to satisfactorily extend the design information for a
given set of conditions to other conditions within about 2 %.
Example 3.2 demonstrates that the analogy method can be used to determine the basic parameters for
a cooling tower in operation. The assumption that the value of m* is unity apparently does not affect the
prediction of performance at other conditions. Although the values for the parameters m*, Ntu, and
effectiveness are not “correct,” they combine to give the actual heat transfer at the base conditions, which
then yields an accurate prediction for capacity at other conditions. If another condition had been taken as
base, the values of the parameters would be different, but the predicted capacity values would be
essentially the same.
In extrapolating the data to other conditions, it is important to ensure that the nozzles and sump are
appropriately sized. The catalog data presented in Table 14.2 for a given model apply to a fixed tower
geometry except for the orifice sizes in the distribution nozzles and the pipe diameters used for inlet and
outlet piping connections. The sizes of these components depend on the design flow rate. If during
operation the water flow is significantly higher or lower than the design flow (on the order of 10 to 20 %),
then the performance may be affected. For water flow rates lower than the design value the head over the
nozzles may be too low for uniform flow over the media and for higher water flow rates the basins may
overflow. For a given tower in which the flows vary significantly the performance may deviate from that
predicted by extension of the model relations presented here. Accurate performance at off-design
conditions needs to be obtained from the manufacturer.
The design information provided by a manufacturer is usually for sea level conditions and may not
accurately reflect the performance at high altitude conditions. There are several effects of altitude on
cooling tower performance. Lower air densities at higher altitude lead to lower air mass flow rates and
3.11
lower heat transfer coefficients for a given fan, which tends to reduce the heat rejection capacity from that
stated in the catalog. However, the driving potential for energy transfer increases with increasing altitude
due to reduced partial pressure of the water vapor, which tends to increase the capacity. Because a
cooling tower fan delivers a constant volume flow rate it is useful to consider a volumetric heat rejection
capacity defined as
(3.29)
Qv   a (h w,sat, in  h a,in )
where Qv is the heat transfer per unit air volume, a is the air density, and  is the tower effectiveness
based on the enthalpy potential (hwsat,in -ha,in).
The decrease in density with altitude is significant. For example, at 10,000 ft (3000 m), the density is
about 30% less than at sea level. Without considering other effects, equation 3.29 indicates that the
capacity of a cooling tower would decrease by about 30% at this altitude.
The enthalpy potential is the difference between the enthalpy of saturated air at the inlet water
temperature and the enthalpy of the atmospheric air. The enthalpy potential increases with altitude
because of the dependence of humidity ratio on pressure. Using property relations for humid air, the
enthalpy potential can be expressed as
h w,sat,in  h a,in  cp,m (Tw,in  Ta,in )  (w w,sat,in  w a,in )h fg
(3.30)


where cp,m is the specific heat of humid air and hfg is the latent heat of water at the water temperature. The
first term in equation 3.30 is associated with the sensible heat transfer from the water surface to the air.
For specified water and air temperatures this term increases by about 1.5% at 10,000 ft (3000 m) due to
the effect on the specific heat.
The second term in equation 3.30 is associated with the evaporation of moisture from the water
surface to the air. The humidity ratio potential (wsat,w,in – wa,in) increases with altitude due to the decrease
in pressure. The humidity ratio can be expressed in terms of the air and water vapor pressures as
pv
(3.31)
w  0.622
pA  p v
where pv is the partial pressure of the water vapor and pA is the atmospheric pressure. The vapor pressure
at the water surface is the saturation pressure at the water temperature and does not vary with altitude.
The vapor pressure in atmospheric air depends on the dry-bulb and wet-bulb temperatures also does not
vary with altitude. However, atmospheric pressure decreases significantly with altitude and therefore the
humidity ratio increases significantly with altitude. The increase in humidity ratio is greater for the vapor
at the water surface because of the higher vapor pressure and the humidity ratio potential (wsat,w,in – wa,in)
increases significantly. For example, at 10,000 ft (3000 m) the humidity ratio potential increases by up to
50 % for typical design specifications.
For turbulent flow such as found over cooling tower surfaces, the convection heat transfer coefficient
varies with Reynolds number, which is directly dependent on density. The relation between the
convection coefficient at altitude and sea level conditions can be expressed as
 
h c  h c,0  a

 a,0



m
(3.32)
3.12
where the subscript “0” denotes sea level conditions. The exponent m is typically about 0.8. Combining
equations 3.28 and 3.33 with the assumption of constant volumetric flow, the value of Ntu is related to the
value at sea level as
1 m
cp,m,0  a,0 
(3.33)
Ntu 


cp,m  a 
The exponent (1-m) is equal to 0.2 for turbulent flow. For the same volume air flow rate and the same
water mass flow rate, the cooling tower effectiveness then increases with altitude as the density decreases.
For example, at 10,000 ft (3000 m) although the heat transfer coefficient decreases by about 25 % the Ntu
increases only by 5 to 7 %.
For the same volume flow rate, an increase in altitude leads to a reduced mass flow rate, a somewhat
increased effectiveness and a significant increase in enthalpy potential. Figure 3.2 shows the impact of
altitude on the volumetric capacity for three different situations. The three cases were selected to bound
the range of performance effects of altitude for cooling towers. For low values of Ntu0 and high water
and wet bulb temperatures, the volumetric capacity increases slightly with altitude. For high values of
Ntu0 and low water and wet bulb temperature case, the volumetric capacity decreases by 10% at 10,000 ft.
However, for most design conditions, the volumetric capacity decreases only slightly with altitude from
that stated in the catalog for sea level conditions. Example 3.3 illustrates the effect of altitude on the
design information for the towers described in Table 14.2.
*
Ntu*0
Figure 3.2. Effect of Altitude on Volumetric Heat Rejection Capacity
Example 3.3 Determine the effect of an altitude of 10,000 ft on the design performance for Model 492
B with a 2 hp fan (Table 14.2) operating at a wet bulb temperature of 75 F with inlet and outlet
temperatures of 95 F and 85 F.
3.13
At 10,000 ft the pressure is 10.1 psia and the air density is 0.0476 lbm/ft3. The enthalpy potential
is the difference between the enthalpy of saturated air at the inlet water temperature and that of
atmospheric air. At sea level conditions the enthalpy potential is 24.8 Btu/lbm and at 10,000 ft it is
35.0 Btu/lbm. This is an increase of 40 %.
In Example 3.2, the Ntu of the Model 492 B at sea level was found to be 1.504. At altitude, the
Ntu is 1.603, which is an increase of 6.6 %. The effectiveness is increased by 3 % from the sea level
value of 0.558 to 0.574. The volumetric heat capacity for sea level conditions is 1.035 Btu/ft3 and
that at altitude is 0.955. There is an 8 % decrease in the design capacity with an altitude change of
10,000 ft (3000 m) for the same temperature specifications.
The implication for this change in volumetric heat capacity on the design point capacity is that the
design water flow rate would be reduced 8 % (from 107 gpm to 98 gpm) to provide the same range
for these conditions. This is a relatively small difference, but as shown in Figure 3.2, there would be
significantly greater differences for lower wet-bulb temperatures, lower water inlet temperatures, and
towers with higher Ntu and effectiveness.
SM 3.4 Nomenclature
A'''
cp,a
cp,m
cp,w
cs
C*
CW
E'''
h
hc
hm
hs,eff
hw,sat
HW
m
m*
Ntu
Ntu*
p
Q
area per unit volume
specific heat of air
specific heat of air-water vapor mixture
specific heat of water
effective specific heat
capacitance rate ratio
leaving water temperature
energy transfer per unit volume
specific enthalpy
convection heat transfer coefficient
convection mass transfer coefficient
effective surface enthalpy
saturated air enthalpy at Tw
entering water temperature
mass flow rate
mass transfer capacitance rate ratio
number of transfer units for heat transfer
number of transfer units for mass
transfer
pressure
heat flow rate
QV
heat flow rate per unit volume
T
V
w
ws,eff
ww
WB
temperature
volume
humidity ratio
effective surface humidity ratio
saturation value of humidity ratio at Tw
atmospheric wet bulb temperature



effectiveness for heat transfer
effectiveness for mass transfer
density
Subscripts
a
air
A
ambient
base base value
in
inlet
out
outlet
max
maximum
sat
saturated at water temperature
V
volume
w
water
wb
wet-bulb
0
sea level conditions
SM 3.5 References
ASHRAE, "2004 ASHRAE Handbook - HVAC Systems and Equipment," ASHRAE, Atlanta, GA, 2004
3.14
Braun, J. E., S. A. Klein, and J. W. Mitchell, "Effectiveness Models for Cooling Towers and Cooling
Coils," ASHRAE Transactions, 95, Part 2, 164, 1989
Marley Cooling Tower, Marley, Mission, Kansas. Website http://www.marleyct.com, 2003
Merkel, F., "Verdunstungskuhlung," VDI Forschungsarbeiten, No. 275, Berlin, 1925
Mitchell, J. W. and J. E. Braun “Using the Analogy Approach to Extrapolate Performance Data for
Cooling Towers,” ASHRAE Transactions, 2008.
Sutherland, J. W., "Analysis of Mechanical Draught Counterflow Air/Water Cooling Towers," Journal of
Heat Transfer, Vol 105, p 576 - 583, 1983
Webb, R. L., "A Unified Theoretical Treatment for Thermal Analysis of Cooling Towers, Evaporative
Condensers, and Fluid Coolers," ASHRAE Transactions, Vol 90, Part 2, 1984
SM 3.6 Problems
Problems in English Units
3.1 A cooling tower is designed to transfer 2,000,000 Btu/hr from the condenser water of a chiller
system. The water flow rate is 100,000 lbm/hr and the air flow rate is 20,000 cfm at design
conditions of 92 F and 50 % RH. The tower packing material has an area per unit volume of 3
ft2/ft3, the tower volume is 500 ft3, and the heat transfer coefficient is 40 Btu/hr-ft2-F.
a) For the design heat transfer rate, determine and plot the water inlet temperature and approach
over a range of air flow rates between 5,000 and 30,000 cfm. The conductance varies as the air
mass flow rate to the 0.8 power.
b) For the design water inlet temperature, determine and plot the heat transfer rate over a range of
air flow rates between 5,000 and 30,000 cfm.
c) Draw some conclusions from your results.
3.2
A cooling tower operates with a flow of condenser water of 50,000 lbm/hr that enters the tower at
95 F. The ambient air used to cool the water is at a dry bulb temperature of 70 F and a relative
humidity of 50 %. The air flow rate is 10,000 cfm.
a. Vary the overall conductance-volume product and plot the capacity and water outlet
temperature as a function of Ntu over the range of 0 to 5.
b. Plot the air and water outlet states for the range of Ntu of 0 to 5 on a psychrometric chart.
c. For an Ntu of 5, determine how much air flow would be required for a sensible heat
exchanger to achieve the same heat transfer.
d. Make some recommendations for design based on your results.
3.3
Design a cooling tower to transfer 1,000,000 Btu/hr from the condenser water of a chiller system at
design conditions of 95 F dry-bulb and 70 F wet-bulb. The tower packing material has an area per
unit volume of 4 ft2/ft3 and the heat transfer coefficient is 50 Btu/hr-ft2-F. The range should be
between 10 and 15 F, the approach should be between 6 and 12 F, and the inlet water temperature
should be less than 92 F. The air flow rate should be such that the value of m* is less than unity so
3.15
that the air flow is not the limiting factor. For your design, specify the water and air flow rates,
3.4
water loss, and the tower volume.
A cooling tower is to be designed for a 400 ton (cooling) air-conditioning system with a compressor
power of 0.6 kW/ton. The tower range is to be 12 F and the approach is to be 8 F at design ambient
conditions of 95 F dry bulb and 75 F wet bulb. The tower mass transfer conductance per unit
volume (hcA'''/cpm) is 300 lbm/hr-ft3. Design guidelines are that a tower air flow rate of 400 cfm
per ton of heat rejection is appropriate and that total fan power is 0.2 hp/1000 cfm.
a. Select a tower volume that will meet design conditions.
b. Determine the tower performance (range and approach) at average operating conditions of 85 F
dry bulb temperature and 65 F wet bulb temperature. The compressor power decreases 10 %
for a 10 F decrease in tower inlet temperature.
c. Determine the annual cost of operation for the chiller and the fan for 4000 hours per year of
operation. Electric costs are $ 0.07/kWh.
e. Draw some conclusions from your results.
3.5
Select a tower from one of those listed in Table 3.2 that will transfer 1,000,000 Btu/hr of heat at a
design ambient wet bulb temperature of 70 F. Provide reasons for the tower that you selected over
others.
a. Estimate the water inlet and outlet temperatures for this tower for the design heat transfer.
b. Estimate the water inlet and outlet temperatures at an ambient wet-bulb temperature of 60 F for
the design heat transfer.
c. Estimate the water inlet and outlet temperatures at an altitude of 10,000 ft for the design heat
transfer at an ambient wet-bulb temperature of 60 F.
d. Draw some conclusions from your results.
Problems in SI units
3.6
A cooling tower is designed to transfer 600 kW from the condenser water of a chiller system.
The water flow rate is 12 kg/s and the air flow rate is 10,000 L/s at design conditions of 33 C and
50 % RH. The tower packing material has an area per unit volume of 10 m2/m3, the tower volume
is 16 m3, and the heat transfer coefficient is 200 W/m2-C.
a) For the design heat transfer rate, determine and plot the water inlet temperature and approach
over a range of air flow rates between 2,000 and 15,000 L/s. The conductance varies as the air
mass flow rate to the 0.8 power.
b) For the design water inlet temperature, determine and plot the heat transfer rate over a range of
air flow rates between 2,000 and 15,000 L/s.
c) Draw some conclusions from your results.
3.16
3.7
A cooling tower operates with a flow of condenser water of 6 kg/s that enters the tower at 35 C.
The ambient air used to cool the water is at a dry bulb temperature of 20 C and a relative humidity
of 50 %. The air flow rate is 5,000 L/s.
a. Vary the overall conductance-volume product and plot the capacity and water outlet
temperature as a function of Ntu over the range of 0 to 5.
b. Plot the air and water outlet states for the range of Ntu of 0 to 5 on a psychrometric chart.
c. For an Ntu of 5, determine how much air flow would be required for a sensible heat
exchanger to achieve the same heat transfer.
d. Make some recommendations for design based on your results.
3.8
Design a cooling tower to transfer 300 kW from condenser water of a chiller system at design
conditions of 35 C dry-bulb and 20 C wet-bulb. The tower packing material has an area per unit
volume of 15 m2/m3 and the heat transfer coefficient is 250 W/m2-C. The range should be between
5 and 10 C, the approach should be between 4 and 8 C, and the inlet water temperature should be
less than 33 C. The air flow rate should be such that the value of m* is less than unity so that the
air flow is not the limiting factor. For your design, specify the water and air flow rates, water loss,
3.9
and the tower volume.
A cooling tower is to be designed for a 1500 kW (cooling) air-conditioning system with a COP of
6. The tower range is to be 7 C and the approach is to be 6 C at design ambient conditions of 35 C
dry bulb and 25 C wet bulb. The tower mass transfer conductance per unit volume (hcA'''/cpm) is
0.4 kg/s-m2. Design guidelines are that a tower air flow rate of 50 L/s per kW of heat rejection is
appropriate and that total fan power is 0.3 kW per 1000 L/s.
a. Select a tower volume that will meet design conditions.
b. Determine the tower performance (range and approach) at average operating conditions of 28 C
dry bulb temperature and 15 C wet bulb temperature. The compressor power decreases 10 %
for a 5 C decrease in tower inlet temperature.
c. Determine the annual cost of operation for the chiller and the fan for 4000 hours per year of
operation. Electric costs are $ 0.07/kWh.
e. Draw some conclusions from your results.
3.10
Select a tower from one of those listed in Table 3.2 that will transfer 300 kW of heat at a design
ambient wet bulb temperature of 21.1 C (70 F). Provide reasons for the tower that you selected
over others.
a. Estimate the water inlet and outlet temperatures for this tower for the design heat transfer.
b. Estimate the water inlet and outlet temperatures at an ambient wet-bulb temperature of 15 C for
the design heat transfer.
c. Estimate the water inlet and outlet temperatures at an altitude of 3,000 m for the design heat
transfer at an ambient wet-bulb temperature of 15 C.
3.17
d
Draw some conclusions from your results.
3.18