H. Jaber

B. L. Webb

Department of Mechanical Engineering, The Pennsylvania State University,

University Park, PA 16802

Design of Cooling Towers by the Effectiveness-NTU Method This paper develops the effectiveness-NTU design method for cooling towers. The definitions for effectiveness and NTU are totally consistent with the fundamental definitions used in heat exchanger design. Sample calculations are presented for counter and crossflow cooling towers. Using the proper definitions, a person com­petent in heat exchanger design can easily use the same basic method to design a cooling tower of counter, cross, or parallel flow configuration. The problems as­sociated with the curvature of the saturated air enthalpy line are also treated. A "one-increment" design ignores the effect of this curvature. Increased precision can be obtained by dividing the cooling range into two or more increments. The standard effectiveness-NTU method is then used for each of the increments. Calculations are presented to define the error associated with different numbers of increments. This defines the number of increments required to attain a desired degree of precision. The authors also summarize the LMED method introduced by Berman, and show that this is totally consistent with the effectiveness-NTU method. Hence, using proper and consistent terms, heat exchanger designers are shown how to use either the standard LMED or effectiveness-NTU design methods to design cooling towers.

Introduction Berman (1961) described how the "log-mean enthalpy

method" (LMED) may be applied to cooling tower design. He also developed a correction factor to account for the curvature of the saturated air enthalpy curve. In their 1940 publication, London et al. introduced definitions of e and NTU to use in plotting cooling tower test data. However, these definitions are not generally consistent with the basic defintions used today in heat exchanger design. They developed empirical curve fits of their e-NTU curves for design purposes. Moffatt (1966) is apparently the first to derive the e-NTU equation for a cooling tower (counterflow). As will be shown later, his definitions do not agree with the basic definitions of e and NTU for certain combinations of water and air flow rate. Others have used their e and/or NTU definitions for graphic representation of test data. Other than Moffatt (1966), no authors have at­tempted to employ these definitions actually to design a cooling tower.

The F-LMTD and e-NTU methods have long been used for design of heat exchangers. It is desirable to apply the basic concepts of the F-LMTD and e-NTU methods to the design of evaporative heat exchangers (cooling towers and evaporative fluid coolers or condensers). The objective of this paper is to show how the e-NTU method may be applied to cooling tower design. The present development will observe the precise con­cept definitions used in the e-NTU method for heat exchangers and be applicable to all cooling tower operating conditions.

Traditional cooling tower design methods typically use a method and nomenclature that are unlike the traditional F-LMTD and e-NTU heat exchanger design methods. Hence, heat exchanger designers cannot successfully translate their design methodology to cooling tower design. A key benefit of the present development is that heat exchanger designers will clearly understand how to apply their understanding of heat exchanger design to cooling towers.

Simultaneous heat and mass transfer processes occur in cool­ing towers. Hence, the design equations must account for both energy transfer processes. The complexity of the design equa-

Contributed by the Heat Transfer Division and presented at the ASME Winter Annual Meeting, Boston, Massachusetts, December 13-18, 1987. Manuscript received by the Heat Transfer Division April 6,1988; revision received November 15,1988. Paper No. 87-WA/CRTD-2. Keywords: Heat Exchangers, Mass Trans­fer.

tions may be simplified using the moist air enthalpy potential proposed by Merkel (1926). The enthalpy potential method is approximate, and combines the driving potential of the heat and mass transfer processes into a single enthalpy driving po­tential. The driving potential is the enthalpy difference of the moist air at the water film-air interface and the bulk air stream. Webb (1988) presents a critical discussion of precise and ap­proximate design methods. Attempts to apply the F-LMED or e-NTU methods to cooling tower design must use the enthalpy driving potential. Thus, the "log-mean enthalpy difference" (LMED) corresponds to the "log-mean temperature differ­ence" (LMTD) of heat exchanger design. One problem as­sociated with use of the F-LMED or e-NTU methods for cooling tower design is that the slope of the saturated air enthalpy curve (4) versus temperature is a curved line. So, use of the F-LMED method will involve errors associated with approx­imating this curve with a straight line. Berman (1961) rigorously applied the F-LMED method to cooling towers, and defined a correction factor (8) to correct for the curvature of the is versus T curve. The correction factor essentially provides a two-increment design (N= 2). The derivation of this correction factor is presented in the Appendix.

Traditional cooling tower design methods typically use an incremental method, which approximates the 4 versus Tcurve into N segments, where N may be in the range of 4 or more. Each segment is a straight line approximation to the 4 versus T curve. The simplest application of the F-LMED or e-NTU methods to cooling tower design would use one segment (N = 1). It will be shown that one may use N > 1 if increased accuracy is desired.

Application of the e-NTU method to cooling tower design requires physical and algebraic definition of the effectiveness (e) and the "number of transfer units" (NTU). A number of attempts have been made to define e and NTU for cooling tower design. However, virtually all of these definitions are flawed, in the sense that they are inconsistent with the cor­responding basic definitions used for heat exchanger design. Baker (1984) and the ASHVE Guide (1941) have defined e and NTU strictly for convenience, and they have not attempted to apply these definitions to the actual e-NTU design method. Baker (1984) essentially dismissed his definition as being of "no value." The e-NTU design method for heat exchangers involves use of the "capacity rate ratio" (CR = Cmin/Cmax).

Journal of Heat Transfer NOVEMBER 1989, Vol. 111 / 837 Copyright © 1989 by ASME

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< X

<

AIR FLOW

TEMPERATURE

Fig. 1 Water operating line on enthalpy-temperature diagram

Whillier (1976) defined a term "tower capacity factor," which was intended to correspond to the capacity rate ratio used in heat exchanger design. His definition was not consistent with the definition of CR. Nor did he attempt to establish an al­gebraic relationship for e-NTU. Moffatt (1966) was apparently the first to attempt to establish an algebraic e-NTU relation­ship. As will be shown later, his method works if the water is the minimum capacity rate fluid, but fails if it is not. Except for Moffatt's work, the rash of NTU definitions reported in the literature have contributed little but a myriad of conflicting definitions.

The Driving Potential As presented by Webb (1988) the driving potential is

dq= [aaLe2'3(r,- T) + hgiKm( W,- W)]dA (1)

SPECIFIC HUMIDITY

V/

WATER FILM

Fig. 2 Gravity-drained water film with temperature, velocity, and hu­midity ratio profiles

The first term accounts for the single-phase heat transfer from the water-air interface to the air, and the second term is the water evaporated at the interface. Webb (1988) shows that equation (1) may be approximated as the enthalpy driving potential, given by

dq = Km(ii~i)dA (2)

The present e-NTU formulation uses (/,• — /) as the driving potential. This potential corresponds to (TH - Tc) used in heat exchanger design.

Figure 1 shows a plot of air enthalpy versus water temper­ature for a counterflow cooling tower. The curved line is the enthalpy of saturated air (is) and the straight line is the "water operating line." The driving pontential (*',- i) is illustrated by the dashed lines. Typically, one assumes that the water-air interface temperature is equal to the local mixed water tem­perature, which is an approximation. Actually, the interface temperature is less than the local mixed water temperature. The water film-air process at the interface is illustrated in Fig. 2. The dashed line in Fig. 1 of slope -a.JKm defines the interface temperature. Since the water film heat transfer coef-

A A a

CP

cR

c d E

f

h i

h

A/

= heat transfer area, m2

= approach = Tw2 -T„b, °C = heat and mass transfer area

per unit volume, m2/m3

= fluid specific heat, kJ/kg-°C = capacity rate ratio = mmin/

"'max = fluid capacity rate = mcp,

kJ/s-°C = differential element = error defined as (1 - N T U /

NTU) x 100, percent = slope of saturated air en­

thalpy versus temperature curve = di/dT, kJ/kg-°C

= enthalpy of water, kJ/kg = enthalpy of moist air at bulk

condition, kJ/kg = enthalpy of moist air at inter­

face condition: /,-, at water in­let, ii2 at water outlet, ilav at average temperature of incre­ment, kJ/kg

= enthalpy difference between

Km —

Le = m =

m+ = N =

NTU =

q = R =

T = U =

V =

W =

z =

air at interface and local bulk air, A/, = ia - iu M2 = in

- i2, AI,„ defined by equa­tion (3) (or (5)) mass transfer coefficient, kg/ m2-s Lewis number of moist air fluid mass flow rate, kg/s water side capacity rate, kg/s number of increments number of transfer units, as defined by equation (16) heat exchange, kW cooling range = Twl - Tw2, °C temperature, °C overall heat transfer coeffi­cient, kW/m2K cooling tower packing vol­ume, m3

humidity ratio = kg of steam/kg of dry air packing depth (in air flow di­rection), m

a

13 5

e

=

= =

=

heat transfer coefficient, kW/m2-s water film thickness, m enthalpy correction factor, kJ/kg thermal effectiveness = qact/ Qmax

Subscripts 1 2 a

act av C f g

H i

m s w

wb

= = = = = = = = = = = = = =

air or water inlet conditions air or water outlet conditions air actual heat transfer average cold fluid liquid water saturated water vapor hot fluid at air-water interface moist air saturated air water wet bulb

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m w ' V

1 1

1 • v d m

w h f-dh f

mQ, 1 +di W + dW

t <tz

*

m0,i ,w

Fig. 3 Control volume illustrating the heat and mass transfer processes at the air-water interface

ficient is unknown, one typically assumes the path of the OLJ Km = oo line, which is the vertical dashed line. The assumption aw/Km = e» will be employed in the present analysis. However, this assumption is not necessary for the e-NTU equations to be developed.

F-LMED Formulation

Assuming a linear variation of is versus T, one may define the log-mean enthalpy difference (LMED) for the cooling tower process illustrated in Fig. 1 as

A/m = A/ 2 -A/ !

ln(AI2/AIl) (3)

where AT) = ia - ij and AI2 = in - i2. The F-LMTD method of heat exchanger design uses the UA

value of the heat exchanger. The corresponding value for cool­ing tower design is K„A. As seen in Fig. 1, the is versus T curve is not a straight line. Hence, equation (3) introduces an approximation. Berman (1961) developed an analytical based correction factor to correct for the nonlinearity of the is versus T curve, which is derived in the Appendix. The correction factor 8 is given by

« = ('/i + » n - 2 / | J / 4 (4)

Note that the enthalpy correction factor is independent of Twb, A (approach), and mw/ma. It is a function of the cooling range only. Figure 4 presents the enthalpy correction factor versus the exit water temperature (Tw2) for different values of the cooling range.

Introducing equation (4) in equation (3) gives the corrected LMED

A/ra A/2-A/,

l n [ ( A / 2 - 5 ) / ( A / 1 - 6 ) ] (5)

UA and LMTD are used in heat exchanger design; the cor­responding definitions for cooling tower design are K„A and LMED. The flow configuration correction factor (F) used in heat exchanger design applies equally well to cooling tower design.

Effectiveness-NTU Method

Figure 3 shows a control volume on a differential element of the cooling tower. Equation (1) is the transport equation for the energy transfer from water to air. An energy balance on the water film and the air, over the length dz, gives

dq = mwcpwdTw = madi (6)

The e-NTU derivation is performed for a counterflow cool­ing tower, using the terminology defined in Fig. 1. The deri­vation essentially parallels that for a counterflow heat exchanger. It is necessary to express dq in equation (1) in terms of the air enthalpy. The slope of the is versus T curve is defined as

in-

2 0 -

1 0 i

O l

SYMBOL

f 1 — • ! — i -

R , C

/ 5 / 10 / 15 / 20 25 , /

/

y'

,**' y

.-" -.--

. — — —

i ' • . i i • . • . 1

10 20 30 40

EXIT WATER TEMPERATURE, C

Fig. 4 Enthalpy correction factor & versus the exit water temperature TM for different cooling ranges R

Substitution of dTw from equation (7) into equation (6) and solving for dit gives

dq=(mwcpw/f')dii (8)

Using di = dq/ma from equation (6) and equation (8) one may write di, - di = rf(i, - /) as

d(i, - /) = dq[ {/' /m^cpn) - \/ma] (9)

When one solves equation (9) for dq and substitutes the result into equation (2), the result is

7 7 T 7 ? = KmUf'/myfip,,) - Uma]dA (10)

The corresponding equation that occurs in the e-NTU de­velopment for a heat exchanger with CH = Cmin is

d(TH-Tc) {TH-TC)

= Ull/mfjCpH-l/mcCpddA 01)

Notice that equation (10) contains the term ma, as compared to the "capacity rate" mcCpCm equation (11). By analogy with equation (11), we will define ma as the air capacity rate for a cooling tower, and the water capacity rate as

ml=mwcpJf (12)

Consistent with heat exchanger design terminology, we will define

CR = Wmin/^raax (13)

There are two possible cases: m£ < ma and « + > ma.

Case 1: m^ < ma. After substituting m% = wminand ma = /ftmax in equation (10) one obtains

dUi-i) KmdA

Vi-i) ITlrr ( 1 - C j ) (14)

Equation (14) corresponds to the heat exchanger e-NTU equa­tion

d(TH-Tc) = UdA (TH—TC) Cmin a-cR) (15)

In heat exchanger design, the term UA/Cmm is defined as the "number of transfer units," NTU. The analogous defi­nition for the NTU of a cooling tower is

f'=di/dTw (7) N T U =

K„A (16)

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Previous works on cooling towers, in which NTU was de­fined, have not observed the precise definition of equation (16). Some authors (London et al., 1940; Moffatt, 1966; Zivi and Brand, 1956) have defined KmA/ma as the "NTU of the air," while others (Kelly, 1976; Keyes, 1972; Majumdar and Singhal, 1983; Baker and Shryock, 1961) have defined KJ\./ mw as "the NTU of the water." The authors assert that equa­tion (16) is the only correct and consistent definition.

Next, it is necessary to define the heat exchange "effective­ness" e. This will be defined identically to that used for heat exchanger design

= 9act/<7, act' ymax

where

<7max — mmmUi\ — ' l )

(17)

(18)

Referring to Fig. 1 and integrating equation (14) between the entering and leaving air states, ;'i and i2, respectively, gives

' / 2 - * i

'ii - h

It can be shown that

ig-h

= e x p [ - N T U ( l - C * ) ]

in-ii eCfl-1

(19)

(20)

Equating equations (19) and (20) gives the final e-NTU equa­tion for the counterflow cooling tower

l - e x p [ - N T U ( l - Q ) ]

' l - C / j e x p I - N T U a - C * ) ] (21)

Equation (21) is identical to the e-NTU expresssion for a counterflow heat exchanger.

For case 1 where mmin — mj,, if the exit water temperature is assumed to be equal to the air entering wet bulb temperature, then one can write qmaxas m* (Twi - Twb). Under such con­ditions one can show that the effectiveness can be expressed as

T — T R

R+A (22)

Case 2: ma<m%. For this case, ma = mmin and m% = mmm. Substitution of these expressions into equation (10) gives equation (14). Continuing as for Case 1 leads to equation (21).

For Case 2, one cannot write the effectiveness in terms of temperatures, as was done in equation (22) for Case 1. For Case 2, one may express the effectiveness by

mmmUn-S-'i) (23)

Discussion of the e-NTU Design Method

Using the definitions for effectiveness and NTU described above, the resulting e-NTU equations for a counterflow cooling tower have been shown to be identical to those for heat ex­changer design. It may also be shown that the e-NTU equations for crossflow heat exchangers are also applicable to crossflow cooling towers. Use of the unmixed/unmixed e-NTU equation is recommended. Similarly, a parallel flow cooling tower would be designed using the e-NTU equation for a parallel flow heat exchanger.

The definition for effectiveness satisfies the thermodynamic definition, e = qaa/qm!a, and the NTU must be defined as K,„A/mmin. The myriad of definitions in the cooling tower literature for effectiveness and NTU are generally inconsistent with those used here.

The heat exchanger designer should have no difficulty in understanding cooling tower analysis, since precisely the same

basic definitions for effectiveness and NTU are used, and the same algebraic equation for the e-NTU relationship applies. A one-increment design (N= 1) may be performed very quickly.

The e-NTU (or the F-LMED) method is subject to approx­imations involved in linearizing the is versus Tcurve as a straight line. However, the desired accuracy can be obtained by break­ing the design down into N increments. Traditional cooling tower design methods typically use an incremental method. One may use the correction factor (5) given by equation (4) for the e-NTU method, which essentially gives a two-increment design. To do this, one redefines /,, and ia as (in — 5) and (ii2

— S), respectively. Hence, the definition of e is rewritten as

mwcpw \ Tv\ — T„2)

/ W m i n ( ' i l - 5 - ' l ) (24)

A typical problem that often arises in cooling tower design is the determination of the NTU when Twb, R, A, and ma/mw

are given. The traditional method of solution is to use the curves given in publications by Kelly (1976) and The Cooling Tower Institute (1967). These curves are based on use of the Merkel method, and were generated for a wide range of prac­tical operating conditions. The Cooling Tower Institute curves (1976) were generated using the Tchebychev integration method with three increments (N = 3). One may very simply use the e-NTU graphs (or equations) for the particular flow configu­ration desired (counter, cross, or parallel flow), with three increments, to design for any of the operating conditions. A simple procedure for a one-increment design using the enthalpy correction factor is outlined below:

1 Calculate the slope of the saturation line, / ' = Ai/R. 2 Calculate m£ = mwcpw/f and compare to ma to deter­

mine CR = mmin/mmax. 3 Find Ai = (mw/ma)cpwR. 4 Calculate the effectiveness e = {maAi) / [mmin (in - 5

5 Read (or calculate) the e-NTU from chart (or equation).

One should note that, for two or more increments, it is possible for the minimum capacity rate fluid to change over the length of the water temperature range. This is because of the change of slope of the is versus Tcurve (see equation (12)). If this happens, one merely redefines CR and mmin for the increment.

Illustration of the 6-NTU Design Method This section presents numerical results for counterflow cool­

ing towers using the e-NTU method. The calculations presented here are performed for a range of practical operating condi­tions, as a function of number of increments (iV) between 1 and 10. By varying the number of increments, one may estimate the number of increments required to attain a particular desired degree of accuracy. All calculations were performed assuming aJKm = oo and neglecting the effect of evaporation on the air enthalpy leaving the increment. However, the first as­sumption has no bearing on use or applicability of the e-NTU method.

Calculations were performed for the following operating conditions listed in Table 1. The number of increments was varied between 1 and 10. The value calculated is the NTU required to perform the cooling duty corresponding to the operating conditions listed in Table 1. The calculated results are presented in Figs. 5-8 in the form of Error (E) = (1 -NTU/NTU) x 100 percent versus the number of increments (TV), where NTUr is the "precise" NTU. NTUr is calculated using the traditional design method of integrating equation (25) using Simpson's rule with 10 increments.

K„A Ai (25)

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- 2 -

- 4 -

,-C - 6 -

•"• - 8 -

g g -10 -

- 1 2 -

-14 -

-16-

! 1

_ ^ ^ - ^ = = = =

1 1

SYM80L

T 1 1 1

TVIB, C

5 15 2 5 , 3 5

Fig. 5 Error = 12°C,A =

2 3 4 5 6 7 8 9

NUMBER OF INCREMENTS (N)

£ versus the number of increments N for mjm, 8°C

1.0, B

- 2 -

- 4 -

- 6 -

- 8 -

- 1 0 -

- 1 2 -

- 1 4 -

- 1 6 -

SYMBOL A,C

5 10 15

1 2 3 4 5 6 7 8 9

NUMBER OF INCREMENTS (N)

Fig. 7 Error E versus the number of increments W for mjm, = 10°C, T„b = 20 °C

1.0, Ft

Fig. 6 Error = 20 °C, A --

3 4 5 6 7

NUMBER OF INCREMENTS (N)

E versus the number of increments N for mjm, = 8°C

i.o, r„,

- 2 -

- 4 -

- 6 -

- 8 -

- 1 0 -

- 1 2 -

- 1 4 -

- 1 6 -

/{! •'/

1 <

i •" r

SYMBOL m w / m a

0 . 5 1.0 1.5

1 2 3 4 5 6 7 8

NUMBER OF INCREMENTS (N)

Fig. 8 Error E versus the number of increments A/ for = 10°C, A = 8°C

T„b = 20°C, H

Figures 5-8 show that the e-NTU method underpredicts the NTU. Table 2 gives the minimum required number of incre­ments to achieve a design, within 2 percent of the 10-increment Simpson integration reference. Table 3 shows the effect of the correction factor on the accuracy of a one-increment design. For the range of parameters considered in Table 1, a one-increment design with an enthalpy correction factor will pro­duce less than 3 percent error. The tower characteristic was also calculated using the Tchebychev integration technique. This method gave approximately 1 percent error with respect to the Simpson reference for the range of operating conditions listed in Table 1. A similar result can be achieved using the e-NTU method with three increments.

Examples of Sizing and Rating Calculations

A sizing problem determines the NTU (size of the tower) for given air and water conditions. A rating calculation de­termines the leaving water temperature for given air and water inlet conditions and the tower characteristic.

Counterflow Sizing Calculation. The following example presents a one-increment sizing calculation for a counterflow cooling tower. The same procedure would be used per incre­ment for a multi-increment design. Both the e-NTU and LMED methods will be illustrated.

Consider the following operating conditions: Water enters at a temperature of 35°C and is cooled to 30°C. The corre­sponding saturation enthalpies are /sl = 129.54 kJ/kg and is2

= 99.96 kJ/kg. The air entering is 25°C (wet bulb) with an enthalpy of /, = 76.6 kJ/kg. The mass flow rate ratio is mw/ ma = 1.0. The average water temperature is T„„„ = (30 + 35)/ 2 = 32.5°C and the corresponding saturation enthalpy is isav

= 113.92 kJ/kg. The enthalpy correction factor is calculated using equation (4) giving a value of 0.414 kJ/kg.

165

155 -

145-

135

125-CT>

\ 1 15 i - 3

105 -

>-^ 95^

< fE 85-z W 75-

65-

55-

45-

35-

LINE E - ^ //

V / / / / //

/ S / \ /

/ T„2 —

*

AREA A, AREA

* ~ T w a v

1

/

/

l\ //

A,

1

f 6

-~T„|

10 35 40 15 20 25 30

TEMPERATURE, C

Fig. 9 Graphic representation of the enthalpy correction factor S

The air enthalpy change is found from the energy balance, equation (6): A/' = (mw/ma)cpwR - 20.93 kJ/kg; the air exit enthalpy is i2 = 97.53 kJ/kg.

7 The LMED Method.

A/, = isl-ix = 99.96-76.6 = 23.36 kJ/kg

M2= isl-i2= 129.54-96.54 = 33.09 kJ/kg

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Table 1 Operating conditions for calculations

< wb< R, °C m„/m„ Figure 5-35 20 20 20

5-15

12 10, 20 10 20

1.0 1.0 1.0 0.5-1.5

5 6 7 8

1 2

Table 2 Number of increments necessary to achieve a design within 2 percent of the 10-increment Simpson integration, with­out using the enthalpy correction factor 5

A, °C R, °C mjma, °C N Figure 5-35 20 20 20

5-15

12 10-20 10 10

1.0 1.0 1.0 0.5-1.5

3 3 2 2

5 6 7 8

Table 3 Effect of the enthalpy correction factor (5) on a one-increment design using the e-NTU method TWb> C

5-35 20 20 20

A, °C

8 8 5-15 8

R, °C

12 10-20 10 10

mw/ma

1.0 1.0 1.0 0.5-1.5

Error, percent

<3 <3 < 2 < 3

Figure

5 6 7 8

Using equation (5), AIm = 27.56 kJ/kg and hence KmA/m„ = 0.76.

2 The e-NTU Method. Applying equation (7) we get: / ' = (129.54 - 99.96)/5 = 5.916 kJ/kg-°C

If myjCpvl/ma < / ' , then m£ < ma (Case 1). The terms e, CR, and NTU are calculated using equations (17), (18), (13), and (16), respectively

e = [ma(i2-ii)]/[mmin(isl-5-ii)] = 0.555

CR = mmin/mmm = 0.708

NTU =KmA/mmin= 1.051 or A'm^/wM, = 0.74

The small difference in numerical values obtained by both methods is due to round-off errors.

Counterflow Rating Calculation. Assume that the given is KmA/rrin = 1.87 and mw/ma = 1.0. The air and water inlet conditions are Twl = 35°C and T„b = 20°C, respectively. The corresponding air and water enthalpies are 129.54 kJ/kg and 57.544 kJ/kg (the correct exit water temperature is 25°C; how­ever, for the sake of illustration it will be assumed that this fact is not known). An exit water temperature is assumed, say T„2 = 29°C, and hence a mean water temperature is calculated as Twau = (35 + 29)/2 = 32°C. The exit and mean water en­thalpies are then is2 = 94.851 kJ/kg and ism = 110.95 kJ/kg, respectively. The following steps are carried out exactly as in a sizing problem:

5 = Osi + is2 ~ 2isa„)/4 = 0.623 kJ/kg

/ ' = (i,i - /* ) / (T w i - Tw2) = 5.782 kJ/kg-°C

&i = mwcp„AT„/ma= 25.121 kJ/kg

If mwcpvl/ma < / ' , then m% < ma (Case 1). Hence

Cr = Mmin/mmm = w + //«„ = mwcpw/f ma = 0.724

e = (m a A/) / [ f f j+a , -5- / 1 ) ] = 0.486

At the calculated values of CR and e, one obtains NTU = KmA/m^ = 0.84 and K^/rriy, = 0.608, as compared to the given value of 1.87. One may continue the iterative calculation using a bi-section method until the calculated tower charac­teristic agrees with the given value.

The counterflow rating calculations may be performed with­out iterations using the following procedure:

Specify the leaving water temperature. Set several A Tw increments and calculate the K„,A/ma

required for each increment. Sum the KmA/m,, values for each increment.

3 When the calculations for the last increment yields Y,K„A/ ma greater than the given value, decrease the ATW for the last increment and continue until T,KmA/ma equals the given value.

Crossflow Sizing Calculation. The operating conditions are considered to be the same as those in the counterflow sizing example. All the parameters are calculated the same way and have the same numerical values as before, except for the value of the NTU, which depends on the crossflow e-NTU relation used. It is recommended to use the unmixed/unmixed relation. The NTU is found from tables given in the Kays and London book (1984) to be 1.169 and the tower characteristic as K,„A/ ma = 0.827.

Comparison With Moffatt's Analysis The one-increment e-NTU design method developed by Mof-

fatt (1966) provides results in agreement with the present Case 1 (wmin = mi,)- Moffatt defined effectiveness as R/(R+A), which agrees with equation (22). His NTU was defined &sK,„A/ ma. Moffatt's analysis will not give the correct answer if (mmin = ma).

Conclusions 1 The analysis presented herein shows how the e-NTU the­

ory of heat exchanger design may be applied to cooling towers. The effectiveness and NTU are defined by equations (17) and (16), respectively. The effectiveness and NTU definitions are in precise agreement with those used for heat exchanger design, and are applicable to all cooling tower operating conditions.

2 One-increment sizing calculations may be quickly per­formed for any flow configuration. The calculations are im­proved by using multi-increments and/or the enthalpy correction factor.

3 The influence of the four independent variables, R, A, Twb, and m„/ma on the accuracy of the e-NTU method is evaluated as a function of the number of increments used. The ranges of the parameters considered is given in Table 1. The calculations show that:

(a) The NTU is underpredicted when the e-NTU method is used. For J? < 20°C and Twb > 15°C, the underpre-diction is 4-8 percent for a one-increment design, and less than 3 percent for a two-increment design. The un-derprediction is highest for low wet bulb temperatures and high cooling ranges.

(b) Use of the enthalpy correction factor reduces the error associated with a lower number of increments. More­over, a one-increment design with this correction factor is equivalent to a two-increment design without the cor­rection.

4 Using the methods outlined herein, a person competent in the e-NTU (or the F-LMTD) method of heat exchanger design can use the same procedure to design cooling towers of any flow configuration.

5 Using the e-NTU curve for the appropriate flow config­uration, one may quickly calculate the required NTU for spec­ified operating conditions. This negates the need for the extensive sets of curves given by Kelly (1976) and the Cooling Tower Institute (1967).

842/Vol. 111, NOVEMBER 1989 Transactions of the ASME

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References ASHVE, 1941, ASHVE Heating, Ventilation, Airconditioning Guide, 19th

ed., Chap. 26, pp. 522-523. Baker, D. R., and Shryock, H. A., 1961, "A Comprehensive Approach to

the Analysis of Cooling Tower Performance," ASME JOURNAL OF HEAT TRANS­FER, Vol. 83, pp. 339-350.

Baker, D., 1984, Cooling Tower Performance, Chemical Publishing Co., New York, Chap. 6, p. 101.

Berman, L. D., 1961, in: Evaporative Cooling of Circulating Water, 2nd ed., Henryck Sawistowski, ed., Pergamon Press, New York, Chap. 2, pp. 94-99; translated from Russian by R. Hardbottle.

Cooling Tower Institute, 1967, Cooling Tower Institute Performance Curves, The Cooling Tower Institute, Houston, TX.

Kays, W. M., and London, A. L., 1984, in: Compact Heat Exchangers, 3rd ed., McGraw-Hill, New York, Chap. 2, p. 51.

Kelly, N. W., 1976, Kelly's Handbook of Crossflow Cooling Tower Per­formance, Neil W. Kelly & Associates, Kansas City, MO.

Keyes, R. E., 1972, "Methods of Calculation for Natural Draft Cooling Towers," presented at the 13th National Heat Transfer Conference, Denver, CO, Aug. 6-9.

London, A. L., Mason, W. F., and Boelter, L. M. K., 1940, "Performance Characteristics of a Mechanically Induced Draft, Counterflow, Packed Cooling Tower," Trans. ASME, Vol. 62, pp. 41-50.

Majumdar, A. K., Singhal, A. K., and Spalding, D. B., 1983, "Numerical Modeling of Wet Cooling Towers—Part 1: Mathematical and Physical Models," ASME JOURNAL OF HEAT TRANSFER, Vol. 105, pp. 728-735.

Merkel, F., 1926, "Verdunstungskuhlung," VDI Zeitschrift deutscher Ingen-ieure, Vol. 70, pp. 123-128.

Moffatt, R. J., 1966,' 'The Periodic Flow Cooling Tower: A Design Analysis,'' Technical Report No. 62, Dept. Mechanical Engineering, Stanford University, CA.

Webb, R. L., 1988, " A Critical Review of Cooling Tower Design Methods," in: Heal Transfer Equipment Design, R. K. Shah, E. C. Subbarao, and R. A. Mashelkar, eds., Hemisphere Pub. Corp., Washington, DC, pp. 547-558.

Whillier, A., 1976, "A Fresh Look at the Performance of Cooling Towers," ASHRAE Trans., Vol. 82, pp. 269-282.

Zivi, S. M., and Brand, B. B., 1956, "Analysis of the Cross-Flow Cooling Tower," Refrigeration Engineering, Vol. 64, No. 8, pp. 31-34.

A P P E N D I X The derivation of the enthalpy correction factor is given

below with reference to Fig. 9. The saturation line is divided into two straight line segments resulting in two trapezoids whose areas are Ax and A2, respectively; hence

A i = (i,-i + ijav) (Twau - Tw2)/2 (26)

A1=(ia + ii„v){Twl-Twav)/2 (27)

Twav is the average water temperature, Twav = (Twl + Tw2)/ 2. Line E is drawn such that the area under it is equal to Ax

+ A2. Denoting this area by A, we have

A = Wa-S) + (/a -&)](Twl - Tw2)/2 (28)

where 5 is a correction factor. Setting A = At + A2 and using the definition of Twav, the final desired result is

6 = (/,, + i B - 2 i t o ) / 4 (29)

Journal of Heat Transfer NOVEMBER 1989, Vol. 111 / 843

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