a critical investigation into the heat and mass transfer analysis of crossflow wet cooling...

A CRITICAL INVESTIGATION INTO THE HEATAND MASS TRANSFER ANALYSIS OF CROSSFLOWWET-COOLING TOWERS

J. C. KloppersSasol Technology (Pty) Ltd, Secunda, South Africa

D. G. KrogerDepartment of Mechanical Engineering, University of Stellenbosch,Stellenbosch, South Africa

The heat and mass transfer process of evaporative cooling in crossflow wet-cooling tower

fills is investigated. The governing equations of the crossflow evaporative process are

derived from first principles. A detailed account is given of how to solve these equations. The

governing equations, according to the Poppe, Merkel, and e-NTU methods of analysis, are

considered. The equations of the Poppe method of analysis are extended to give a more

detailed representation of the transfer characteristic or Merkel number. The results of a

crossflow wet-cooling tower fill analysis according to the Merket, Poppe, and e-NTU

methods of analysis are presented. The differences between the results of these methods are

evaluated.

1. INTRODUCTION

The heat and mass transfer evaporative processes according to Merkel [1],Poppe [2], and the e-NTU [3] methods are considered. Merkel [1] developed thetheory for the performance evaluation of counterow cooling towers in 1925. TheMerkel theory relies on several critical assumptions to reduce the solution of heatand mass transfer problem in wet-cooling towers to a simple hand calculation.Because of these assumptions, however, the Merkel method does not accuratelyrepresent the physics of the heat and mass transfer process in the cooling tower ll.The critical simplifying assumptions of the Merkel theory are that the Lewis factor,Lef, is equal to unity, the exiting air is saturated, and the reduction of the water owrate, due to evaporation, is neglected in the energy balance. Refer to Kroger [4] forthe derivation of the governing equations for counterow cooling towers accordingto the Merkel method.

The Poppe method was developed by Poppe and Rogener [2] in the early 1970s.This method does not make the simplifying assumptions made by Merkel. The air

Received 5 December 2003; accepted 18 June 2004.

Address correspondence to J. C. Kloppers, Sasol Technology (Pty) Ltd, Private Bag X1034,

Secunda 2302, South Africa. E-mail: [email protected]

Numerical Heat Transfer, Part A, 46: 785806, 2004

Copyright # Taylor & Francis Inc.ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/104077890504113

785

can, therefore, be unsaturated, saturated, or even supersaturated, according to thePoppe method. The e-NTU method, developed by Jaber and Webb [3], is based onthe assumptions made by Merkel. The great advantage of the e-NTU method is itssimplicity in the application of crossow congurations. For crossow, however, itmust be specied whether the air and water streams are mixed or unmixed or acombination of mixed and unmixed. Thus, there exists a choice of four possible owgeometries for crossow. The question now is which geometry will yield the mostaccurate results for a particular ll material.

During a ll performance test, the water inlet temperature, Twi, water outlettemperature, Two, water mass ow rate, mw, inlet air dry-bulb temperature, Tai, inletair wet-bulb temperature, Twb, dry air mass ow rate, ma, and the atmosphericpressure, patm, are measured. From these measurements, the transfer characteristic,or Merkel number, Me, is determined. In the subsequent cooling-tower performanceevaluation, the variables mentioned above are generally known, as well as the Merkelnumber, except for the water outlet temperature. The water outlet temperature canthen be determined by the same set of equations, solving for Two and not Me, asduring the ll performance test.

Figure 1 shows an example of an induced-draft crossow wet-cooling tower. Ina crossow tower, the ll is usually installed at some angle to the vertical to makeprovision for the inward motion of the droplets, due to drag forces caused by theentering cooling air [4].

NOMENCLATURE

a surface area per unit volume, m1

A area, m2

cp specic heat at constant pressure,

J=kg K

C uid capacity rate (Cmin=Cmax)h heat transfer coecient, W=m2 K

hd mass transfer coecient, kg=m2 s

i enthalphy, J=kg, or index in x or xdirection

j index in z or Z directionL length, m

Lef Lewis factor h=cphd, dimensionlessm mass ow rate, kg=s

Me Merkel number hdafiLfi=Gw,dimensionless

n number

p pressure, N=m2 or Pa

q heat ux, W=m2

Q heat transfer rate, W

T temperature, K or CU overall heat transfer coecient,

W=m2 K

w humidity ratio (kg water vapour=kgdry air)

x Cartesian coordinate

y Cartesian coordinate

z Cartesian coordinate

D dierentialZ non-dimensional coordinatex non-dimensional coordinate

Subscripts

a air

atm atmospheric

c convection heat transfer, or cold

e e-NTU method

ll

h hot

i inlet

m mean, or mass transfer

max maximum

min minimum

o outlet

s saturation

ss supersaturated

v vapor

w water

wb wet-bulb

x coordinate

y coordinate

z coordinate

786 J. C. KLOPPERS AND D. G. KROGER

In 1956, Zivi and Brand [5] extended the analysis of Merkel to the ll ofcrossow cooling towers. In 1976, Kelly [6] used the method of Zivi and Brand [5],along with laboratory data, to produce a volume of crossow cooling-tower char-acteristic curves to be used in graphical solutions of cooling tower performance. Thepresent analysis does not make the simplifying assumptions of Merkel and is known,as in the case with counterow towers, as the Poppe method.

2. GOVERNING EQUATIONS FOR HEAT AND MASS TRANSFER IN FILLFOR UNSATURATED AIR

Figure 2 shows a control volume in the ll of a crossow wet-cooling tower, whereima and w are the enthalphy and humidity ratio of the air, respectively, and Tw andGw are the temperature and mass velocity of the water, respectively.

A mass balance for the control volume in Figure 2 yields

GwzDxDy Gw

zDz DxDy Ga DyDz wjx Ga DyDz wjxDx 0 1

Divide Eq. (1) by DxDyDx and let Dx, Dz ! 0:

qGwqz

Ga qwqx 2

The energy balance for the control volume of the ll in Figure 2 is as follows:

Figure 1. Induced-draft crossow wet-cooling tower.

ANALYSIS OF CROSSFLOW WET-COOLING TOWERS 787

cpwTw;Gwjz DxDy cpwTwGwjzDz DxDy Ga DyDz imajx Ga DyDz imajxDx 0 3

where cpw is the specic heat of the water.Divide Eq. (3) by DxDyDz and let Dx, Dz ! 0 and nd, after using the chain

rule of dierentiation,

cpwTwqGwqz

cpwGw qTwqz Gaqimaqx

0 4

Substitute Eq. (2) into Eq. (4) to nd, upon rearrangement,

qTwqz

GaGw

Twqwqx

1cpw

qimaqx

5

The mass balance for the water stream in the control volume is expressed by

GwzDxDy Gw

zDz DxDy hd afiwsw wDxDyDz 0 6

where hd afiwsw wDxDyDz is the amount of water evaporated in the controlvolume shown in Figure 2. Refer to Kroger [4] for the derivation of this term. hd, afi,and wsw in Eq. (6) are the mass transfer coecient, area per unit volume of ll, and

Figure 2. Control volume of crossow ll.


the humidity ratio of the air at the local water temperature. It will be shown that it isnot necessary to specify hd and afi explicitly.

Divide Eq. (6) by DxDyDz, and let Dx, Dz ! 0:

qGwqz

Ga hdafiGa

wsw w 7

Ga=Ga is introduced into the left-hand side of Eq. (7) to simplify the mathematicalmanipulation later on.

Substitute Eq. (7) into Eq. (2), and rearrange to nd

qwqx

hdafiGa

wsw w 8

The sensible heat transfer to the air stream in the control volume is expressedby

qcjx DyDz qcjxDx DyDz hafiTw TaDxDyDz 0 9

where hafiTw TaDxDyDz is the amount of sensible heat transferred to the airstream in the control volume in Figure 2. Divide Eq. (9) by DxDyDz, and let Dx,Dz ! 0:

qqcqx

hafiTw Ta 10

The latent heat transfer to the air stream in the control volume is expressed by

qmjx DyDz qmjxDx DyDz ivGwzDxDy ivGwjzDx DxDy 0 11

Divide Eq. (11) by DxDyDz, let Dx, Dz ! 0, and substitute Eq. (8) into the resultantequation

qqmqx

iv qGwqz ivhdafiwsw w 12

An energy balance at the air=water interface inside the control volume yields

qqqx

qqcqx

qqmqx

13

The temperature dierential in Eq. (10) can be substituted by an enthalpydierential. Refer to Appendix A for the derivation of the temperature potential as afunction of the enthalpy potential. Substitute Eq. (A.6) into Eq. (10). Substitute theresultant equation and Eq. (12) into Eq. (13) to nd, upon rearrangement,

qqqx

hdafi hcpmahd

imasw ima 1 hcpmahd

iv wsw w

14


where h=cpmahd is the Lewis factor, Lef. The Lewis factor for unsaturated air,according to Bosnjakovic [7], is given by

Lef 0:8650:667 wsw 0:622w 0:622 1

ln

wsw 0:622w 0:622

When the air is supersaturated with water vapor, which will be considered later, theLewis factor is given by

Lefss 0:8650:667 wsw 0:622wsa 0:622 1

,ln

wsw 0:622wsa 0:622

Lefss is the Lewis factor for supersaturated air.The enthalpy transfer to the air stream from Eq. (14) is

qimaqx

1Ga

qqqx

hdafiGa

imasw ima Lef 1 imasw ima ivwsw w 15

Substitute Eqs. (8) and (15) into Eq. (5) to nd, upon rearrangement,

qTwqz

1cpw

GaGw

hdafiGa

wsw wcpwTw imasw ima Lef 1

imasw ima wsw wivg 16

Thus, the system of equations to be solved for unsaturated air for the crossowll are Eqs. (7), (8), (15), and (16).

3. GOVERNING EQUATIONS FOR HEAT AND MASS TRANSFER IN FILLFOR SUPERSATURATED AIR

The air in the ll can reach the point of saturation before leaving the ll. Sincethe temperature of the saturated air at the interface is still higher than the tem-perature of the now-saturated free-stream air, a potential for heat and mass transferwill still exist. The excess water vapor transferred to the free-stream air will condenseas a mist. The air is then in the supersaturated state.

The governing equations when the air is supersaturated can be obtained as wasdone for the unsaturated case using the same arguments:

qGwqz

hdafiwsw wsa 17

qwqx

hdafiGa

wsw wsa 18

qissqx

1Ga

qqqx

hdafiGa

imasw iss Lefss 1imasw iss ivwsw wsaLefsscpwTww wsa

19


qTwqz

1cpw

GaGw

hdafiGa

nwsw wsacpwTw imasw iss LefsscpwTww wsa

Lefss 1imasw iss wsw wsaivo

20

Bourillot [8, 9] and Grange [10] considered the case where the air can becomesupersaturated with water vapor. By considering the equations for supersaturatedair, they obtained excellent agreement between calculated and measured waterevaporation rates. According to Bourillot and Grange, the Merkel method alwaysunderpredicts the water evaporation rate, but the error decreases for increasingambient air temperatures. Kloppers and Kroger [11, 12] also found that by consideringthe equations for supersaturated air, the predicted air outlet temperatures can varysignicantly from the temperatures predicted by the Merkel method. They discussthese dierences with the aid of psychrometric charts, which are extended to includethe properties of supersaturated air.

4. SOLVING THE SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS

To simplify the solution process of the governing equations, the ll dimensionscan be nondimensionalized. Poppe and Rogener [2] presented the governing equa-tions for crossow lls in nondimensional form. In this form, the ll can be analyzedwithout any reference to ll dimensions.

Figure 3 illustrates an example of a grid of a crossow ll that is divided intofour intervals in both the vertical and horizontal directions, thus, imax jmax 5.

All the governing equations are rst-order. These rst derivatives can be ap-proximated by rst-order backward nite-dierence expressions. An example of theapplication of this nite-dierence technique to rst derivatives can be seen inFigure 4.

When the ll dimensions are nondimensional, Eqs. (7), (8), (15), and (16),which are applicable when the air is unsaturated, become, respectively,

qGwqZ

Ga hdafiGa

wsw w 21

qwqx

hdafiGa

wsw w 22

qimaqx

hdafiGa

imasw ima Lef 1 imasw ima ivwsw w 23

qTwqZ

1cpw

GaGw

hdafiGa

wsw wcpwTw imasw imaLef 1imasw ima wsw wiv

24where x x=Lx and Z z=Lz, with Lx and Lz the ll lengths in the x and zdirections, respectively.


Figure 5 illustrates an excerpt of four grid points from the example compu-tational grid in Figure 3 for generalized nondimensional coordinates. It is essentialthat the ll is divided into equal intervals in both the horizontal and vertical di-rections for the nondimensional ll analysis, and thus, DZ Dx.

By applying rst-order backward dierences and lettingMex hdafi Dx=Ga hdafi DZ=Ga, Eqs. (7), (8), (15), and (16) become, respectively,

Gwi;j Gwi;j1 GaMexwsw wja 25

Figure 3. Example of a crossow ll that is divided into four intervals in each direction.

Figure 4. An example of a rst derivative approximated as a rst-order backward nite dierence with

respect to x for an arbitrary variable u [13].


wi;j wi1;j Mexwsw wjb 26

imai;j imai1;j Mex imasw ima Lef 1 imasw ima ivwsw w

b27

Twi;j Twi;j1 1cpw

GaGw

Mexfwsw wcpwTw imasw ima

Lef 1imasw ima wsw wivgja 28

The ja and jb symbols in the last terms in Eqs. (25)(28) refer to points a and b,respectively, in Figure 5. Point a refers to the average value of the last term of Eqs.(25) and (28) between points i; j and (i, j7 1), while point b refers to the averagevalue of the last term of Eqs. (26) and (27) between points (i, j) and (i7 1, j). Take,for example, the average value of the last term of Eq. (28) between points (i, j) and(i, j7 1), i.e.,

1

cpw

GaGw

MexfwswwcpwTwimasw imaLef1imasw imawswwivgja

MexGa2

1

cpwi;jGwi;j

wswi;j wi;jcpwi;jTwi;j imaswi;j imai;j

Lefi;j 1imaswi;j imai;j wswi;j wi;jivi;j

8>>>>>>>>:

9>>>>>=>>>>>;

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCA

29

Figure 5. Four generated grid points of one cell of a crossow ll.


where Ga and Mex are constant throughout the solution domain. Equation (29) canbe substituted into Eq. (28) to obtain the value of Twi;j. Equations (25)(27) aretreated in a similar manner to obtain average values for the last terms of theseequations.

The enthalpy of the air at every point in the solution domain is given byEq. (A.4). When the air is supersaturated, the enthalpy of the air is given by

iss cpaTa wsaifgwo cpvTa w wsacpwTa 30

where wsa is the humidity ratio of saturated air at temperature Ta. The thermo-physical properties given in the governing equations and Eqs. (A.4) and (30) aregiven in Appendix B.

The governing partial dierential equations are solved by an iterative techni-que. Gw and Tw are known at the water inlet side. ima and w are known at the air inletside and Ga is constant throughout the solution domain. Equations (25) and (28) areused to solve for Gw and Tw, respectively, at the air inlet side, while Eqs. (26) and (27)are used to solve for w and ima at the water inlet side. Equations (25)(28) can besolved simultaneously throughout the rest of the domain.

If the air is supersaturated at a point in the ll, the governing equations forsupersaturated air must be employed instead of the equations for unsaturated air.Refer to Kloppers [14] for a procedure to determine whether the air is unsaturated orsupersaturated at a point in the solution domain.

The mean water outlet temperature can be obtained by integrating the watertemperature values at the water outlet side of the ll, i.e.,

Twom 1nx

Z nx0

Two dx 31

where Zx is the number of ll intervals in the x or x direction.The mean outlet air enthalpy and humidity can be obtained by integrating

these values at the air outlet side of the ll, i.e.,

imaom 1nZ

Z nZ0

imao dZ 32

wom 1nZ

Z nZ0

wo dZ 33

where nZ is the number of ll intervals in the Z or z direction.Mex in Eqs. (25)(28) can be referred to as the local Merkel number according

to the air stream in the horizontal direction, where

Mex hdafi Dx=Ga hdafi DZ=Ga 34

At every point in the solution domain the local Merkel number according to thewater stream in the vertical direction is given by


MeZi;j GaGwi;j

Mex 35

The Merkel number for the ll, Me, is obtained by integrating MeZi;j acrossthe entire ll. First, determine the average of the MeZi;j quantities at the center ofeach cell of the entire ll. The mean Merkel number, MeZmi;j at the cell center iscalculated from Figure 6 as follows:

MeZmcelli;j MeZi;j MeZi1;j MeZi;j1 MeZi1;j1=4 36

The mean quantity of all the MeZmcelli;j values is denoted by MeZm, where

MeZm P

MeZmcelli;jnxnZ

37

From the denition of the Merkel number, as given by Kroger [4], and Eqs. (34)(37), the Merkel number of the ll is given by

Me hdafiLzGw

hdafinZ DZGw

nZMeZm 38

where Lz nZ DZ.The Merkel number for a crossow ll is determined from the ll performance

experimental data by the following approach. A value for Mex is guessed. This valueis constant throughout the computational domain. The water outlet temperature isdetermined according to Eq. (31) after the governing equations have converged. Mexis varied until the water outlet temperature from Eq. (31) matches the measuredwater outlet temperature. The Merkel number is then determined according toEq. (38).

Figure 6. Average value of MeZi;j at the cell center.


5. EFFECTIVENESS-NTU METHOD

Jaber and Webb [3] developed the equations necessary to apply the heatexchanger e-NTU method for sensible heat transfer directly to counterow orcrossow cooling towers. The method is particularly useful in the latter case andsimplies the method of solution when compared to the more conventional numericalprocedure discussed above. Kroger [4] gives a detailed derivation and implementa-tion of the e-NTU method applied to evaporative air water systems.

It can be shown according to Jaber and Webb [3] that

dimasw imaimasw ima hd

dimasw=dTwmwcpw

1ma

dA 39

Equation (39) corresponds to the heat exchanger e-NTU equation, where

dTh TcTh Tc U

1

mhcph 1mccpc

dA 40

Two possible cases of Eq. (39) can be considered, where ma is greater or lessthan mwcpw=dimasw=dTw. The maximum of ma and mwcpw=dimasw=dTw is denotedby Cmax and the minimum by Cmin. The gradient of the saturated air enthalpytemperature curve is

dimaswdTw

imasw imaswoTwi Two 41

The uid capacity rate ratio is dened as

C CminCmax

42

The eectiveness is given by

e QQmax

mwcpwTwi TwoCminimaswi l imax 43

where l is a correction factor, according to Berman [15], to improve the approx-imation of the imasw-versus-Tw curve as a straight line. The correction factor, l, isgiven by

l imaswo imaswi 2imaswm4

44

where imaswm denotes the enthalpy of saturated air at the mean water temperature.The number of transfer units, NTU, for counterow cooling towers is given by

NTU 11 C ln

1 eC1 e 45


Four dierent crossow congurations are possible with the e-NTU method. Boththe air and water streams can be unmixed or mixed, or one can be mixed and theother unmixed or vice versa. NTU can be obtained by iterative means for the fourow congurations from Eqs. (46)(49).

For crossow towers with both streams unmixed, nd

e 1 exp NTU0:22 expC NTU0:78 1 =C 46For crossow with both streams mixed, nd

e 11 expNTU

C

1 expC NTU 1

NTU

147

For crossow with Cmax mixed and Cmin unmixed, nd

e 1 expfC1 expNTUg=C 48

For crossow with Cmax unmixed and Cmin mixed, nd

e 1 expf1 expC NTU=Cg 49

If ma is greater than mwcpw=dimasw=dTw, the Merkel number according to thee-NTU method is given by

Mee cpwdimasw=dTw

NTU 50

If ma is less than mwcpw=dimasw=dTw, the Merkel number according to the e-NTUmethod is given by

Mee mamw

NTU 51

6. ILLUSTRATIVE EXAMPLE

During a crossow ll performance test, the following variables are measured:

Atmospheric pressure patm 101712:27 PaAir inlet temperature Tai 9:7C 282:85KAir inlet temperature (wet-bulb) Twb 8:23C 281:38KDry air mass ow rate ma 4:134 kg=sWater inlet temperature Twi 39:67C 312:82KWater outlet temperature Two 27:77C 300:92KInlet water mass ow rate mw 3:999 kg=s

The governing partial dierential equations for the crossow conguration pre-sented above are solved by a point-by-point Gauss-Seidel [16, 17] iterative procedureacross a two-dimensional domain using the principle of nite dierences.


6.1. Poppe Method

The governing equations of the Poppe method for crossow lls are solved byan iterative technique, as discussed above. The governing equations must be satisedon each vertex in the computational domain before convergence can be obtained.Figure 7 shows the solution domain of a counterow ll for nondimensional lldimensions. The solution domain for this example problem is divided into 50intervals in both directions. It can be seen from Figure 7 in which parts of the ll theair is unsaturated and supersaturated for the particular experimental inlet and outletmeasurements. The dividing line between the unsaturated and supersaturated regionswill be smooth if the solution domain is divided into many more intervals. It can beseen from this example problem that the air becomes saturated soon after enteringthe ll, especially in the top parts of the ll. The governing equations for unsaturatedand supersaturated air are, thus, solved in the respective regions shown in Figure 7.

Figures 8a8f show the distribution of the water temperature, water massvelocity, Lewis factor, air enthalpy, air temperature, and the humidity ratio of the air,respectively, across the nondimensional solution domain of the crossow ll. Refer toFigure 7 for the coordinate system convention used in Figure 8. The water and airinlet sides of the various plots in Figure 8 are the same as those illustrated in Figure 7.

The distribution of the water temperature across a vertical section of the ll isillustrated in Figure 8a. The mean water outlet temperature is determined by Eq. (31)and is equal to the measured 300.92 K. It can be seen that water cooling is moreeective near the air inlet side. This is because the water near this location is in

Figure 7. State of air in ll for nondimensional ll dimensions.


Figure 8. Distribution of water temperature, water mass velocity, Lewis factor, air enthalpy, air tem-

perature, and humidity across a crossow ll, determined according to the Poppe method.


contact with the cool inlet air all the time it falls through the ll. The mass velocity ofthe water as it passes through the ll can be seen in Figure 8b. Approximately thesame trends can be observed, as with the water temperature in Figure 8a. The watermass velocity is reduced as it passes through the ll because of evaporation. Theevaporation loss is larger near the air inlet side because the inlet air is relatively drycompared to the air deeper into the ll. Thus, a greater potential for evaporation lossexists where the air is the driest. Figure 8c shows how the value of the Lewis factor,according to the equation of Bosnjakovic [7], is distributed across the ll. Figures 8d8f show the enthalpy, temperature, and humidity ratio of the air as it passes throughthe ll. It can be seen that the plotted contours of these three variables follow ap-proximately the same trends. The air enthalpy increases more rapidly in the top ofthe ll because the air is in contact with the hot inlet water stream all the time as itmoves through the ll. The heat rejection rate, Q, air outlet temperature, Tao, waterevaporation rate, mw(evap), and the Merkel number, Me, according to the Poppemethod, are shown in Table 1.

6.2. Merkel Method

It is assumed in the Merkel method that Lef 1 and that the water that eva-porates in neglected in the energy balance. Figures 9a and 9b show the distribution ofthe water temperature and air enthalpy according to the Merkel method. The results

Table 1. Fill performance characteristics of a crossow ll according to the Poppe, Merkel, and e-NTU

methods

Poppe Merkel e-NTU1 e-NTU2 e-NTU3 e-NTU4

Q, MW 0.2065 0.1988 0.1988 0.1988 0.1988 0.1988

Tao, K 297.84 297.43 297.43 297.43 297.43 297.43

mwevap;kg=s 0.0628 0.0540 0.0540 0.0540 0.0540 0.0540Me 0.7976 0.7395 0.7405 0.7751 0.7589 0.7486

Figure 9. Water temperature and air enthalpy distribution in a crossow ll according to the Merkel

method.


of the Merkel method can be compared to the results of the more rigorous Poppemethod presented in Figure 8. The mean water outlet temperature of methods isequal to 300.92 K, as this is the measured value. The mean outlet air enthalpy andtemperature of the Merkel method are less than that predicted by the Poppe method.

According to the Merkel method, Q, Tao, mw(evap), and Me are also shown inTable 1. The temperature of the outlet air can only be determined according to theMerkel method through the assumption that the outlet air is saturated with watervapor. Refer to Kloppers and Kroger [11, 12] for a detailed discussion of the impli-cations of this assumption.

6.3. e-NTU Method

The crossow Merkel number according to the four variants of the e-NTUmethod is not solved by two-dimensional nite dierences. The results of the dif-ferent e-NTU methods are shown in Table 1. In Table 1, e-NTU1 and e-NTU2, refer,respectively, to the crossow cases, where both the water and air streams are un-mixed and both the air and water streams are mixed. e-NTU3 refers to the crossowcase where Cmax, which is generally the water stream, is mixed, and Cmin, which isgenerally the air stream, is unmixed. Cmax is unmixed and Cmin is mixed for the e-NTU4 case. The comparison of the four dierent e-NTU methods and the com-parison to the Merkel and Poppe methods are presented in the next section.

7. COMPARATIVE RESULTS OF THE e-NTU, MERKEL, ANDPOPPE METHODS

The illustrative example shows an example of a ll performance test when thewater outlet temperature is known. The Merkel number, or transfer characteristic, isthen determined for each method of analysis. If the Merkel number is known foreach method of analysis, together with all the other variables, except for the wateroutlet temperature, the water outlet temperature can be obtained for each method byan iterative procedure. The water outlet temperature determined by each method ofanalysis should, therefore, be identical.

It is, therefore, very important that the same method of analysis be employed inthe ll performance test (when the Merkel number is determined) and in the coolingtower design calculations (when the water outlet temperature is determined). This isbecause the Merkel number diers for each of the dierent methods of analysis.

It can be seen from Table 1 that Q, Tao, and mw(evap), determined by all fourvariants of the e-NTU method, are identical. These variables are also identical to thevalues obtained by the Merkel method. This is because it is assumed, for bothmethods, that the outlet air is saturated with water vapor. The heat rejection rate iscalculated by exactly the same manner for all the variants of the e-NTU method andthe Merkel method, i.e., Q mwcpwm (Twi7Two). The water evaporation rate,according to the Merkel and e-NTU methods, is given by mw(evap)ma (wo7wi),where the outlet humidity ratio, wo, is determined by assuming that the outlet air issaturated with water vapor.

Cooling tower performance, predicted by the all the variants of the e-NTUmethod and the Merkel method, will, therefore, be practically identical if the same


method is employed in the ll performance test and in the subsequent cooling-towerperformance analysis. It is recommended that the ll performance evaluation becarried out at approximately the same conditions at which the cooling tower willoperate.

Tables 25 show more examples of ll performance test results. The casespresented in these tables were not obtained during actual ll tests, but they are givenfor the illustration of the relative dierences between the dierent methods of ana-lysis at extreme ambient conditions. The results in Tables 25 are obtained whenTwi 45C and Two 35C. The variables patm, ma, and mw are the same as in theillustrative example. The results in Tables 2 and 3 are obtained when Tai 28C.Twb 27.8C for the results presented in Table 2. The air is, therefore, almostsaturated with water vapor. Twb 10C for the results in Table 3. The air is,therefore, almost void of water vapor. Tai 7C in Tables 4 and 5. Twb 6.8C(almost saturated air) in Table 4, and Twb 1C (almost void of vapor) in Table 5.

It can be seen from Tables 2, 4, and 5 that mw(evap) and Tao, according to thePoppe method and the methods based on the assumptions of Merkel, are relativelyclose. However, for the results in Table 3, the discrepancies between the Poppemethod and the methods based on the Merkel assumptions are greater. The reasonfor this is that the outlet air is unsaturated according to the Poppe method. Refer toKloppers and Kroger [11, 12] for a detailed discussion of this point.

8. CONCLUSION

A system of equations is derived from rst principles to solve the heat and masstransfer process of evaporative cooling in crossow wet-cooling tower lls. A dif-ferent system of equations is applicable when the air is supersaturated than when theair is unsaturated. The unique solution techniques and procedures to solve a systemof equations are presented. The ll dimensions are not required in the analysis, as the


methods Tai 28C;Twb 27:8C


Q, MW 0.1752 0.1670 0.1670 0.1670 0.1670 0.1670

Tao, K 308.45 308.16 308.16 308.16 308.16 308.16

mwevap;kg=s 0.0582 0.0532 0.0532 0.0532 0.0532 0.0532Me 0.8973 0.8270 0.8083 0.9492 0.9029 0.8530


methods Tai 28C;Twb 10C


Q, MW 0.1745 0.1670 0.1670 0.1670 0.1670 0.1670

Tao, K 304.83 296.43 296.43 296.43 296.43 296.43

mwevap;kg=s 0.0621 0.0733 0.0733 0.0733 0.0733 0.0733Me 0.3897 0.3670 0.3711 0.3690 0.3678 0.3662


equations are presented in nondimensional form. The equations of the relativelysimple e-NTU method are also presented.

Results of a typical ll performance test are obtained by the Poppe, Merkel,and e-NTU methods of analysis by employing the equations presented in thisstudy. The comparison between the results of these methods is critically evaluated.The Poppe method predicts higher heat rejection rates and water evaporation ratesthan the Merkel and all variants of the e-NTU methods. The heat rejection rates,water evaporation rates, and air outlet temperatures predicted by the Merkel andthe four variants of the e-NTU method are identical. This is because all thesemethods are based on the same simplifying assumptions. It is, therefore, veryimportant that the same method of analysis be employed in the ll performancetest and in the subsequent cooling-tower performance analysis. The transfercharacteristic determined by a particular method, therefore, cannot be employed ina cooling-tower performance analysis if it was determined by employing anothermethod.

The Poppe method is relatively complex, but more accurate, than the Merkeland e-NTU methods. If the amount of water that evaporates or the air outlettemperature is an important consideration in the design of cooling towers, then thePoppe method is the preferred method of analysis. The temperature of the outletair must be determined as accurately as possible for natural-draft cooling towers,as the potential for draft is a function of the air temperature at the air outlet sideof the ll. The Poppe method is also more accurate at extreme ambient conditionswhen the air is relatively hot and dry. The Merkel and e-NTU methods are not asaccurate under these conditions. Under normal ambient conditions, the Merkeland especially the e-NTU methods of analysis can be employed if only the wateroutlet temperature is an important consideration in the design of cooling towers.This is due to the relative simplicity of these methods of analyses compared to thePoppe method.


methods Tai 7C;Twb 6:8C


Q, MW 0.1734 0.1670 0.1670 0.1670 0.1670 0.1670

Tao, K 294.99 294.63 294.63 294.63 294.63 294.63

mwevap; kg=s 0.0540 0.0417 0.0417 0.0417 0.0417 0.0417Me 0.3663 0.3457 0.3501 0.3470 0.3461 0.3449


methods Tai 7C;Twb 1C


Q, MW 0.1739 0.1670 0.1670 0.1670 0.1670 0.1670

Tao, K 291.85 291.36 291.36 291.36 291.36 291.36

mwevap; kg=s 0.0539 0.0474 0.0474 0.0474 0.0474 0.0474Me 0.3367 0.3159 0.3205 0.3165 0.3159 0.3150


REFERENCES

1. F. Merkel, Verdunstungskuhlung, VDI-Zeitschrift, vol. 70, pp. 123128, January 1925.

2. M. Poppe and H. Rogener, Berechnung von Ruckkuhlwerken, VDI-Warmeatlas, pp. Mi1Mi 15, 1991.

3. H. Jaber and R. L. Webb, Design of Cooling Towers by the Eectiveness-NTU Method,

J. Heat Transfer, vol. 111, pp. 837843, November 1989.4. D. G. Kroger, Air-Cooled Heat Exchangers and Cooling Towers, Penn Well Corp., Tulsa,

OK, 2004.

5. S. M. Zivi and B. B. Brand, An analysis of the Crossows Cooling Tower, Refrig. Eng.,vol. 64, pp. 3134, 9092, 1956.

6. N. W. Kelly Kellys Handbook of Crossow Cooling Tower Performance, Neil W. Kellyand Associates, Kansas City, MO, 1976.

7. F. Bosnjacovic, Technische Thermodinamik, Theodor Steinkopf, Dresden, 1965.8. C. Bourillot, TEFERI, Numerical Model for Calculating the Performance of an Eva-

porative Cooling Tower, EPRI Rep. CS-3212-SR, August 1983.

9. C. Bourillot, On the Hypothesis of Calculating the Water Flowrate Evaporated in a WetCooling Tower, EPRI Rep. CS-3144-SR, August 1983.

10. J. L. Grange, Calculating the Evaporated Water Flow in a Wet Cooling Tower, Proc. 9th

IAHR Cooling Tower and Spraying Pond Symp., von Karman Institute, Brussels, Belgium,September 1994.

11. J. C. Kloppers and D. G. Kroger, Cooling Tower Performance: A Critical Evaluation of

the Merkel Assumptions, South African Institution of Mechanical Engineers, R&DJournal, vol. 20, no. 1, pp. 2429, 2004.

12. J. C. Kloppers and D. G. Kroger, The Supersaturated Psychrometric Chart with an Il-lustrative Cooling Tower Example, HEFAT2004, 3rd Int. Conf. on Heat Transfer, Fluid

Mechanics and Thermodynamics, Cape Town, South Africa, 2124 June 2004.13. J. D. Anderson, Jr., Computational Fluid Dynamics, The Basics with Applications,

McGraw-Hill, New York, 1995.

14. J. C.Kloppers, ACritical Evaluation andRenement of the Performance Prediction ofWet-Cooling Towers, Ph.D. thesis, University of Stellenbosch, Stellenbosch, South Africa, 2003.

15. L. D. Berman, Evaporative Cooling of Ciruculating Water, 2nd ed., pp.9499, H. Sawis-

towski, ed., translated from Russian by R, Hardbottle, Pergamon Press, New York, 1961.16. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York, 1980.17. J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering, 2d ed.,

Prentice-Hall International, London, 1992.

APPENDIX A: EXPRESSING THE TEMPERATURE DIFFERENTIAL AS ANENTHALPY DIFFERENTIAL

The temperature dierential, Tw Ta, employed in Eq. (10) for unsaturated air canbe rewritten in terms of an enthalpy dierential imasw ima. The enthalpy of thewater vapor, iv, at the bulk water temperature, Tw, is given by

iv ifgwo cpvTw A:1

The enthalpy of saturated air evaluated at the local bulk water temperature is given by

imasw cpaTw wswifgwo cpvTw A:2

Substitute Eq. (A.1) into Eq. (A.2) and nd, upon rearrangement,


imasw cpaTw wiv wsw wiv A:3

The enthalpy of the airwater vapor mixture per unite mass of dry air, whichaccording to Kroger [4] is expressed by

ima cpaTa wifgwo cpvTa A:4

where the specic heats are evaluated at Ta 273:15=2 and the latent heat, ifgwo, isevaluated at 273.15 K according to Eq. (B.7).

The specic heat of the air water vapor mixture for unsaturated air is denedby

cpma cpa wcpv A:5

Substract Eq. (A.4) from Eq. (A.3). The resultant equation can be simplied if thesmall dierences in specic heats, which are evaluated at dierent temperatures, areignored.

Tw Ta imasw ima wsw wivcpma

A:6

where cpma is given by Eq. (A.5).

APPENDIX B: THERMOPHYSICAL PROPERTIES

The thermophysical properties summarized here are presented in Kroger [4].Refer to Kroger [4] for the ranges of applicability of the following equations of thethermophysical properties. All the temperatures are expressed in Kelvin.

The specic heat of dry air is given by

cpa 1:045356 103 3:161783 101T

7:083814 104T2 2:705209 107T3; J=kgK B:1

The vapor pressure of saturated water vapor is given by

pv 10z;N=m2 B:2

where

z 10:795861 273:16=T 5:02808 log10273:16=T

1:50474 104 1 108:29692T=273:161

4:2873 104 104:769551273:16=T 1

2:786118312

The specic heat of saturated water vapor is given by


cpv 1:3605 103 2:31334 T 2:46784 1010T 5 5:91332 1013T 6; J=kgKB:3

The specic heat of mixtures of air and water vapor is given by

cpma cpa wcpv; J=K kg dry air B:4

The humidity ratio is given by

w 2501:6 2:3263Twb 273:152501:6 1:8577T 273:15 4:184Twb 273:15

0:62509pvwbpatm 1:005pvwb

1:00416T Twb2501:6 1:8577T 273:15 4:184Twb 273:15

B:5

where pvwb is the vapor pressure from Eq. (B.2) evaluated at the wet-bulb tem-perature.

The specic heat of water is given by

cpw 8:15599 103 2:80627 10 T

5:11283 102T2 2:17582 1013T6; J=kgK B:6

The latent heat of water is given by

ifgw 3:4831814 106 5:8627703 103 T 12:139568T2

1:40290431 102T3; J=K B:7

ifgwo is obtained from Eq. (B.7), where T 273:15.


a critical investigation into the heat and mass transfer analysis of crossflow wet cooling...

Documents

merkel method

poppe method of analysis

water mass

heat transfer

wetcooling towers

merkel number

cooling towersj

thepoppe method