Sc

OFC

a

ARRAA

KCPODE

1

uoactr

ctrpou

aatiat

h0

Computers and Chemical Engineering 71 (2014) 24–38

Contents lists available at ScienceDirect

Computers and Chemical Engineering

j our na l ho me pa g e: www.elsev ier .com/ locate /compchemeng

olving the heat and mass transfer equations for an evaporativeooling tower through an orthogonal collocation method

scar M. Hernández-Calderón ∗, Eusiel Rubio-Castro, Erika Y. Rios-Iribeacultad de Ciencias Químico Biológicas, Universidad Autónoma de Sinaloa, Av. de las Américas y Blvd. Universitarios, Ciudad Universitaria,P 80013 Culiacán, Sinaloa, Mexico

r t i c l e i n f o

rticle history:eceived 27 July 2013eceived in revised form 9 June 2014ccepted 16 June 2014vailable online 8 July 2014

eywords:ooling toweroppe methodrthogonal collocationormand–Princexplicit Jacobian

a b s t r a c t

In this paper, the orthogonal collocation technique is utilized to solve the Poppe method equations for heatand mass transfer in counter flowing wet-cooling towers. The six differential equations for unsaturatedand supersaturated air from the Poppe method are simplified, yielding three differential equations thatuse the Heaviside function. The humidity ratio is demonstrated to be a finite power series at a normalizedwater temperature. The air enthalpy is expressed as a function of the normalized water temperature andthe unknown coefficients of the expansion from the humidity ratio. The discrete formulation is solvedusing the Newton–Raphson method using an explicit Jacobian. The proposed methodology is applied toeight examples, and the results are compared to the results obtained when the governing equations areintegrated with the Dormand–Prince method. The results indicate that the accuracy is similar betweenboth techniques. However, the orthogonal collocation requires less CPU time.

© 2014 Elsevier Ltd. All rights reserved.

. Introduction

Cooling towers are commonly used for heat rejection toward ambient air during many industrial processes, particularly whensed as condensers for refrigeration systems, power stations and the textile industry (Yilmaz, 2010). A practical cooling tower the-ry was first developed by Merkel (1925), where the water lost due to evaporation and the water-film heat-transfer resistancere ignored, and the Lewis factor for moist air is assumed to be one. Due to these assumptions, the heat and mass transfer pro-esses in the cooling towers are represented by a single separable differential equation. The solution of this equation reveals onlyhe air outlet temperature and enthalpy. Moreover, the final estimations assume that the air leaving the cooling tower is satu-ated.

The Merkel method can be extended to include a finite liquid-side film resistance to heat transfer (Baker and Shryock, 1961; Maclaine-ross and Banks, 1981; Marseille et al., 1991). However, for this method, the local water bulk temperature is seldom more than 0.3 K abovehe air temperature at the air–water interface (Mills, 1999). Therefore, the above works indicate that it is safe to ignore the water filmesistance when analyzing the cooling towers (Mills, 1999; Singham, 1983). Therefore, several works related to the heat and mass transferhenomena in cooling towers have been produced based on the above assumption. For example, the e-NTU method (effectiveness-numberf transfer units), which was developed by Jaber and Webb (1989), the water loss is neglected, similar to Merkel, and the Lewis factor isnity.

Models that represent the heat and mass transfer processes in more detail may also include the effects of evaporative water lossnd fixed Lewis factors can be found with values other than unity (Sutherland, 1983; Kloppers and Kröger, 2005a, 2005b). These worksddressed the effect of the Lewis factor relative to the size of the cooling towers. However, the more complete model that circumvents

he above limitations is commonly known as the Poppe method because it was reported by Poppe and Rögener (1991); this methodnvolves a set of differential equations whose numerical solution reveals the behavior of the primary variables involved (i.e., air enthalpy,ir humidity, water temperature, flow rate, etc.) through the packing region of the cooling towers. In addition, the Poppe method provideshe air outlet conditions in terms of the temperature, enthalpy and humidity; the Merkel number, which is the number of transfer units,

∗ Corresponding author. Tel.: +52 667 713 7860.E-mail address: omhc@uas.edu.mx (O.M. Hernández-Calderón).

ttp://dx.doi.org/10.1016/j.compchemeng.2014.06.008098-1354/© 2014 Elsevier Ltd. All rights reserved.

O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38 25

Nomenclature

Cp specific heat at constant pressure (J/kg K)DPM Dormand–Prince methodE relative errorg auxiliary functionH Heaviside step functioni enthalpy (J/kg)ifg latent heat (J/kg)J JacobianL differential operatorLef Lewis factor (h/(Cphd))m mass flow rate (kg/s)N number of quadrature pointsNTU number of transfer unitsOCM orthogonal collocation methodP pressure, Legendre polynomial or functionQ auxiliary functionr residual functionR auxiliary functionSL saturation lineSRM Simpson’s 3/8 rule methodt CPU timeT temperature (◦C or K)w humidity ratio (kg water vapor/kg dry air)W coefficientx normalized water temperaturez normalized water temperature

Subscript0 referencea airgq Gauss quadratureh heat transferin inletj indexk indexm mass transfer or indexn degree of polynomial solutionma mixture of air and water vaporout outletp indexq indext totalv vaporw waterwb wet-bulbdb dry-bulb

Superscript* saturation at water temperature� saturation at dry-bulb temperature of air

Greek letters˛ coefficient

coefficientı Tw,out − Tw,in or Dirac delta function� inlet water temperature (◦C or K)

iLac

domain of integration

s calculated from the solution of a differential equation and accounts for the evaporation effect relative to the water flow rate. Theewis factor is estimated simultaneously from the solution of differential equations through a given equation (Bosnjakovic, 1965). Theforementioned characteristics make the Poppe method the more accurate methodology when calculating the heat and mass transfer inooling towers.

2

cmbemLmtwtiedMr

ReadmzGcasTmpuao

mwtStat

2

(Fv

h

afltt

6 O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38

The Merkel method (Merkel, 1925), e-NTU (Jaber and Webb, 1989) and the Poppe method (Poppe and Rögener, 1991) are the mostommonly used approaches when rating and designing cooling towers. These situations require a good representation of the heat andass transfer processes to obtain the correct designs. For example, Sutherland (1983) found that designs using the Merkel method could

e 5–15% undersized. Moreover, some works have addressed the optimal cooling towers design for economy based on the Merkel and-NTU approximated methods (Serna-González et al., 2010; Kintner-Meyer and Emery, 1995; Söylemez, 2001, 2004). Consequently, theseethods generate sub-optimal designs due to the errors introduced by neglecting the evaporative water loss while assuming that the

ewis factor is unity. To overcome the above errors, Rubio-Castro et al. (2011) developed a mixed integer non-linear programming (MINLP)odel to optimize the design for counter flow cooling towers. The MINLP is based on the Poppe method and a detailed geometric design;

he corresponding differential equations are solved using the fourth-order Runge–Kutta algorithm (Kloppers, 2003). The results of this lastork are compared to the results obtained when the cooling tower rating is modeled using the Merkel method, revealing the cases where

he Merkel method provides sub-optimal designs. However, the disadvantages of using the Poppe method for optimization problemsnclude the large number of variables and the non-linearities related to the numerical method used to solve the corresponding differentialquations (Rubio-Castro et al., 2011). In addition, the stability of the numerical method must be considered. Due to the aforementionedisadvantages, a strategy must be developed that gives good initials values in addition to lower and upper limits for the major variables.oreover, numerous computational and human efforts have been disclosed. Finally, the heat and mass transfer processes must be well

epresented to optimize the cooling tower designs; this representation is accomplished through the Poppe method.Kloppers (2003) describes the numerical methodology to solve the equations from the Poppe method through a fourth-order

unge–Kutta technique. This methodology, when applied for optimization, demands at least 25 intervals to discretize the differentialquations (Rubio-Castro et al., 2011), generating significant problems related to variables and non-linearities with considerable humannd computational demands. Utilizing an orthogonal collocation method might reduce aforementioned efforts (Finlayson, 1972) because itemands fewer discretization points than the Runge–Kutta method. The orthogonal collocation method is one of several weighted-residualethods where an approximate solution is substituted into the differential equation to form the residual. Afterwards, this residual is set to

ero at the collocation points. Choosing the collocation points is critical for collocation techniques to improve convergence and efficiency.enerally, the zeroes of the Jacobi polynomials are utilized as collocation points over the normalized interval. Therefore, the orthogonalollocation method generates a system of algebraic equations that must be solved to determine the unknown coefficients for the proposedpproximate solution (Villadsen and Stewart, 1967). Certainly, this discrete formulation can be solved more easily if the Jacobian is con-tructed. For the governing equations of the Poppe method, the associated Jacobian is simple to evaluate and can be expressed explicitly.herefore, the operating conditions in cooling towers involve the humidity ratio and the air enthalpy, which can be expressed as polyno-ials; the solutions to these polynomials generate the corresponding profiles relative to the cooling tower packing. The solution for the

olynomials could reduce the computational time relative to the case when the governing equations from the Poppe method are solvedsing the Runge–Kutta method. In summary, the orthogonal collocation method offers better stability while reducing the computationalnd human efforts relative to the fourth-order Runge–Kutta technique. This method is very important when optimizing the design detailsf counterflow cooling towers.

Therefore, this paper addresses the application of an orthogonal collocation technique to solve the governing equations of the Poppeethod. The humidity ratio and the air enthalpy are represented by a power series expansion relative to the normalized water temperature,hile the equations of the Poppe method including the unsaturated and supersaturated air are compacted using the Heaviside step function;

he number of transfer units (NTU) is determined by the Gauss quadrature method (Lanczos, 1956), and its accuracy was verified usingimpson’s 3/8 rule (Lanczos, 1956). Here, the application of Heaviside step function is proposed and developed in this work to simplifyhe system of governing equations. Moreover, the code was implemented in the MatLab software to solve eight examples, and the resultsre compared with those obtained using the Dormand–Prince Runge–Kutta integration (Dormand and Prince, 1980) in the context of CPUime and accuracy.

. Mathematical model

The equations for the evaporative cooling process of the Poppe method are adapted from Poppe and Rögener1991) and Kröger (2004). These equations are derived from the mass balance in the control volumes shown inigs. 1 and 2. Fig. 1 shows the control volume in the fill of a counter flow wet-cooling tower, and Fig. 2 shows the air-side contrololume of the fill shown in Fig. 1.

dw

dTw=

Cpwmwma

[w∗ − w� + (w� − w)H(w� − w)]

Lef (i∗ma − ima) + (1 − Lef ) [(w∗ − w)iv + (w − w�)(iv − iw)H(w − w�)] − (w∗ − w)iw(1)

dima

dTw= Cpw

mw

ma

[1 + iw[w∗ − w� + (w� − w)H(w� − w)]

Lef (i∗ma − ima) + (1 − Lef )[(w∗ − w)iv + (w − w�)(iv − iw)H(w − w�)] − (w∗ − w)iw

](2)

dNTUdTw

= Cpw

Lef (i∗ma − ima) + (1 − Lef ) [(w∗ − w)iv + (w − w�)(iv − iw)H(w − w�)] − (w∗ − w)iw(3)

ere H(x) is the Heaviside step function, which is given by

H(x) ={

0, x < 0

1, x ≥ 0(4)

nd Tw is the water temperature; Cpw is the specific heat at constant pressure at water temperature, mw is the water flow rate, ma is the airow rate, w is the humidity ratio through the cooling tower, w* is the humidity saturated ratio evaluated at the water temperature, w� ishe humidity saturated ratio evaluated at the dry-bulb temperature of the air, iw is the water enthalpy evaluated at water temperature, iv ishe water vapor enthalpy evaluated at the water temperature, i∗ma is the enthalpy of the saturated air evaluated at the water temperature,

O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38 27

Fig. 1. Control volume of the counter flow fill.

i(

wtLm

Tfl

3

fp

Fig. 2. Air-side control volume of the fill.

ma is the enthalpy of the air–water vapor mixture per mass of the dry-air. The definition of ima depends on whether the air is unsaturatedw − w� < 0) or saturated (w − w� ≥ 0),

ima ={

ia + wi�v w − w� < 0

ia + w�i�v + (w − w�)i�w w − w� ≥ 0(5)

here ia is the dry air enthalpy evaluated at the dry-bulb air temperature, i�v is the water vapor enthalpy evaluated at the dry-bulb airemperature, i�w is the water enthalpy evaluated at the dry-bulb air temperature, NTU is the number of transfer units, and Lef is theewis factor, which represents the relationship between the heat and mass transfer during an evaporative process. The Lewis factor is aathematical expression that can be used for saturated and unsaturated air:

Lef = 0.8652/3 ((w∗ + 0.622)/(w − (w − w�)H(w − w�) + 0.662) − 1)ln((w∗ + 0.622)/(w − (w − w�)H(w − w�) + 0.662))

(6)

he representation proposed for Eqs. (1)–(3) and (6) is a concise form of the governing equations for heat and mass transfer in the counterow fill for unsaturated and supersaturated air.

. Development of the numerical solution

The procedure for solving the Eqs. (1)–(3) through the orthogonal collocation technique is shown below. First, the discretization methodor the mass and heat transfer equations, as well as the iterative scheme for solving the corresponding nonlinear discrete system, arerovided. After, the method for obtaining the NTU profile from the Gauss quadrature technique is explained.

2

3

wm

C

T

Tt

wtt

T

w

S

w

Fp

N

8 O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38

.1. Discrete formulation of the enthalpy of the air–vapor mixture

The overall energy balance in the fill of cooling towers is as follows:

ima,in + mw,in

maiw,in − ima,out − mw,out

maiw,out = 0 (7)

here ima,in is the inlet air enthalpy, ima,out is the outlet air enthalpy, iw,in is the inlet water vapor enthalpy, iw,out is the outlet air enthalpy,w,in and mw,out are the inlet and outlet water flow rates, respectively. The outlet water flow rate is calculated as follows:

mw,out = mw,in + (win − wout)ma (8)

ombining Eqs. (7) and (8) generates the mathematical expression for determining the outlet air enthalpy:

ima,out = ima,in + mw,in

maiw,in −

(mw,in

ma+ win − wout

)iw,out (9)

he energy balance from the top of the cooling tower to an arbitrary level is calculated for the air enthalpy (ima):

ima + mw,in

maiw,in − ima,out − mw

maiw = 0 (10)

his arbitrary level represents any point of the fill height, allowing the last expression to be used at any level as a function of the wateremperature. The following equation is proposed for Eq. (10) after changing the next variable:

Tw = ıx + � (11)

here ı is the difference between the outlet and inlet water temperatures, � is the inlet water temperature and x is the normalized wateremperature with the domain [0,1], where x = 0 corresponds to the top of the cooling tower, and x = 1 represents the bottom of the coolingower. Therefore, these values are as follows:

ı = Tw,out − Tw,in (12)

� = Tw,in (13)

Similar to the energy balance (Eq. (10)), the mass balance from the top of the cooling tower to any level is as follows:

mw = mw,in − (wout − w)ma (14)

he humidity air ratio (w) can be expressed as an expanded power series (see Appendix A.2 in Supplementary material):

w =n∑

m=0

Wmxm (15)

here Wm are the polynomial unknown coefficients. Therefore, if Eq. (15) is substituted in Eq. (14), the Eq. (15) becomes the following:

mw = mw,in −(

wout −n∑

m=0

Wmxm

)ma (16)

imilar to the top of cooling tower, the humidity air ratio (w) equals the outlet air humidity ratio (wout):

w(x = 0) = wout (17)

W0 = wout (18)

Therefore, after combining Eqs. (18) and (16), the final expanded power series for air humidity ratio is obtained:

mw

ma= mw,in

ma+

n∑m=1

Wmxm (19)

here mw = mw(W1, W2, . . ., Wn, x). Therefore, substituting Eq. (19) in Eq. (10), generates the following:

ima = −mw,in

maiw,in + ima,out +

(mw,in

ma+

n∑m=1

Wmxm

)iw (20)

inally, substituting Eq. (9) into Eq. (20) produces a mathematical expression for the air–vapor mixture enthalpy, such as an expandedower series, where ima = ima(W0, W1, W2, . . ., Wn, x),

( ) (n

)

ima = ima,in − mw,in

ma+ win − W0 iw,out + mw,in

ma+∑m=1

Wmxm iw (21)

otably, this energy balance is valid for air in any state (unsaturated or supersaturated).

3

w

D

I

Ftfxgt

o

w

px

O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38 29

.2. Discrete formulation of the air humidity ratio

First, Eq. (1) is reformulated as follows:

R(W, x) = P(W, x)

Q (W, x)(22)

here W = (W0, W1, W2, . . ., Wn), and

R(W, x) = dw

dTw(23)

P(W, x) = Cpwmw

ma[w∗ − w� + (w� − w)H(w� − w)] (24)

Q (W, x) = Lef (i∗ma − ima) + (1 − Lef )[(w∗ − w)iv + (w − w�)(iv − iw)H(w − w�)] − (w∗ − w)iw (25)

ue to the previous change in variable, the proposed terms Cpw, w* and i∗ma are defined as follows:

Cpw = Cpw(ıx + �) (26)

w∗ = w∗(ıx + �) (27)

i∗ma = i∗ma(ıx + �) (28)

n addition, R(W, x) does not depend on W0 because

R(W1, W2, . . ., Wn, x) = 1ı

n−1∑m=0

(m + 1)Wm+1xm (29)

Therefore, Eq. (22) can be represented by the following expression:

P(W, x) − R(W1, W2, . . ., Wn, x)Q (W, x) = 0 (30)

The solution for Eq. (30) equals the roots of g(W, x) when defined as follows:

g(W, x) = P(W, x) − R(W1, W2, . . ., Wn, x)Q (W, x) (31)

or Eq. (31), the W coefficients are determined from Eq. (15), and these coefficients must be selected such that g(W, x) = 0. In this case,he collocation points are the following: the zeroes of Legendre polynomial (Pn−1) to the degree of n − 1 normalized in the [0,1] interval,or example (x1, x2, . . ., xn−2, xn−1) where Pn−1(xk) = 0 when 1 ≤ k ≤ n − 1. Furthermore, the next boundary coordinates are employed:0 = 0 and xn = 1. Afterwards, substituting the points x = (x0, x1, . . ., xn−1, xn) in Eq. (31) generates the next nonlinear algebraic system:(W, xk) = P(W, xk) − R(W1, W2, . . ., Wn, xk)Q (W, xk), where g(W, xk) = 0 when 0 ≤ k ≤ n. Here, the W coefficients can be obtained throughhe Newton–Raphson algorithm, which requires the following iterative process:

Wr+1 = Wr −[

Jg(W)∣∣Wr ,x

]−1g(Wr, x) (32)

r more explicitly:

⎡⎢⎢⎢⎢⎢⎢⎢⎣

Wr+10

Wr+11

...

Wr+1n−1

Wr+1n

⎤⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

Wr0

Wr1

...

Wrn−1

Wrn

⎤⎥⎥⎥⎥⎥⎥⎥⎦

−

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂g

∂W0

∣∣∣∣r

x0

∂g

∂W1

∣∣∣∣r

x0

. . .∂g

∂Wn−1

∣∣∣∣r

x0

∂g

∂Wn

∣∣∣∣r

x0

∂g

∂W0

∣∣∣∣r

x1

∂g

∂W1

∣∣∣∣r

x1

∂g

∂Wn−1

∣∣∣∣r

x1

∂g

∂Wn

∣∣∣∣r

x1

.... . .

...

∂g

∂W0

∣∣∣∣r

xn−1

∂g

∂W1

∣∣∣∣r

xn−1

∂g

∂Wn−1

∣∣∣∣r

xn−1

∂g

∂Wn

∣∣∣∣r

xn−1

∂g

∂W0

∣∣∣∣r

xn

∂g

∂W1

∣∣∣∣r

xn

· · · ∂g

∂Wn−1

∣∣∣∣r

xn

∂g

∂Wn

∣∣∣∣r

xn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎣

gr0

gr1

...

grn−1

grn

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(33)

here gqp is the function g evaluated using the W coefficients obtained in the iteration q using collocation point xp, and ∂g/∂Wj|qxp

is the

artial derivative of g relative to Wj and subsequently evaluated using the W coefficients obtained at iteration q through collocation pointp. To search the roots faster, the explicit Jacobian of the function g is obtained:

∂g(W, x)∂Wj

= ∂∂Wj

[P(W, x) − R(W1, W2, . . ., Wn, x)Q (W, x)] (34)

Subsequently, abbreviating the above expression and expanding the derivative respect to Wj generates

∂g

∂Wj

= ∂P

∂Wj

− R∂Q

∂Wj

− Q∂R

∂Wj

(35)

3

b

F

w

wa

W

0 O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38

To obtain an explicit form of Eq. (34), the following derivatives: (∂w/∂Wj), (∂ima/∂Wj) and (∂w∗/∂Wj) must be evaluated. Therefore,ecause w =

∑nm=0Wmxm,

∂w

∂Wj

= xj 0 ≤ j ≤ n (36)

For (∂ima/∂Wj), if Eq. (20) is a derivative relative to Wj, the next expression is acquired:

∂ima

∂Wj

= ∂ima,out

∂Wj

+ ∂∂Wj

(n∑

m=1

Wmxm

)iw (37)

Substituting Eq. (9) into (37), similar to wout = W0 generates

∂ima

∂W0= iw,out (38)

∂ima

∂Wj

= iwxj 1 ≤ j ≤ n

inally, for (∂w∗/∂Wj), firstly the derivative of Eq. (5) relative to Wj for supersaturated air is generated:

∂ima

∂Wj

= ∂ia∂Wj

+ i�v∂w�

∂Wj

+ w� ∂i�v∂Wj

+(

∂w

∂Wj

− ∂w�

∂Wj

)· i�w + (w − w�) · ∂i�w

∂Wj

(39)

While employing the chain rule with the derivatives of the enthalpies related to Wj in Eq. (39), the following is obtained:

∂ima

∂Wj

= ∂ia∂Tdb

∂Tdb

∂w�

∂w�

∂Wj

+ i�v∂w�

∂Wj

+ w� ∂i�v∂Tdb

∂Tdb

∂w�

∂w�

∂Wj

+(

∂w

∂Wj

− ∂w�

∂Wj

)· i�w + (w − w�)

∂i�w∂Tdb

∂Tdb

∂w�

∂w�

∂Wj

(40)

here ∂ia/∂Tdb = C�pa, ∂i�v/∂Tdb = C�

pv and ∂i�w/∂Tdb = C�pw (all heat capacities are evaluated at dry-bulb temperature of air). Therefore,

∂ima

∂Wj

={

i�v − i�w + [C�pa + w�C�

pv + (w − w�)C�pw]

∂Tdb

∂w�

}∂w�

∂Wj

+ i�w · ∂w

∂Wj

. (41)

here ∂Tdb/∂w� = (∂w�/∂Tdb)−1; similarly, ∂w�/∂Tdb is the derivative of the saturated humidity ratio relative to the dry-bulb air temper-ture. Hence,

∂w�

∂Wj

= (∂ima/∂Wj) − i�w(∂w/∂Wj)

i�v − i�w + [C�pa + w�C�

pv + (w − w�)C�pw](∂w�/∂Tdb)−1

(42)

Substituting Eq. (36) and Eq. (38) into Eq. (42) generates the next mathematical expression:

∂w�

∂W0= iw,out − i�w

i�v − i�w + [C�pa + w�C�

pv + (w − w�)C�pw](∂w�/∂Tdb)−1

(43)

∂w�

∂Wj

= iw − i�w

i�v − i�w + [C�pa + w�C�

pv + (w − w�)C�pw](∂w�/∂Tdb)−1

xj 1 ≤ j ≤ n

Moreover, (∂P/∂Wj), (∂Q/∂Wj) and (∂R/∂Wj) must be evaluated, initially generating expressions through the derivative of P relative toj,

∂P

∂Wj

= Cpw[w∗ − w� + (w� − w)H(w� − w)]∂

∂Wj

(mw

ma

)+ Cpw

(mw

ma

) ∂∂Wj

[w∗ − w� + (w� − w)H(w� − w)] (44)

Similar to ∂(mw/ma)/∂W0 = 0 and ∂(mw/ma)/∂Wj = ∂w/∂Wj = xj , when 1 ≤ j ≤ n

∂P

∂W0= Cpw

(mw

ma

)[− ∂w�

∂W0+(

∂w�

∂W0− 1

)H(w� − w)

](45)

∂P

∂Wj

= Cpw

{(w∗ − w�)xj −

(mw

ma

) ∂w�

∂Wj

+[

(w� − w)xj +(

mw

ma

)(∂w�

∂Wj

− xj

)]H(w� − w)

}1 ≤ j ≤ n

Regardless of (∂R/∂Wj), the derivative of R respect to Wj is found, as shown in Eq. (29)

∂R = 0 (46)

∂W0

∂R

∂Wj

= 1ı

jxj−1 1 ≤ j ≤ n

a

3

f

w

w

4

aDo

t

cee2am

crfA

O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38 31

Therefore, when ∂Lef/∂Wj ≈ 0, the derivative of Eq. (25) relative to Wj is as follows:

∂Q

∂Wj

= Lef

(−∂ima

∂Wj

)+ (1 − Lef )

[− ∂w

∂Wj

iv +(

∂w

∂Wj

− ∂w�

∂Wj

)(iv − iw)H(w − w�)

]+ iw

∂w

∂Wj

(47)

Therefore, Eq. (47) is as follows:

∂Q

∂W0= −Lef · iw,out + (1 − Lef )

[−iv +

(1 − ∂w�

∂W0

)(iv − iw)H(w − w�)

]+ iw (48)

∂Q

∂Wj

= −Lef · iwxj + (1 − Lef )

[−iv · xj +

(xj − ∂w�

∂Wj

)(iv − iw)H(w − w�)

]+ iwxj 1 ≤ j ≤ n

Additionally, w(x = 1) = win must be satisfied (i.e.,∑n

m=0Wm = win). Therefore, when x = 1, Eq. (32) must be redefined:

g(x = 1, W) =(

n∑m=0

Wm

)− win (49)

nd

∂g

∂Wj

∣∣∣∣x=1

= 1 0 ≤ j ≤ n (50)

.3. Determination of the number of transfer units (NTU)

By substituting Eq. (25) into (3) and changing the next variable (Tw = ıx + �),

dNTUdx

=ıCpw

(ıx + �

)Q (x, W)

(51)

Integrating Eq. (51) from x = 1 (bottom tower) to x = z where 0 < z ≤ 1 (an arbitrary level on the tower) generates the NTU profile asunction of the normalized water temperature:

NTU(z) =∫ z

1

ıCpw(ıx + �)

Q (x, W)dx (52)

here NTU(1) = 0. Applying the Gauss quadrature integration (see Appendix A.3 in Supplementary material) to Eq. (51) generates

NTU(z) = z − 12

N∑k=1

wgq,kıCpw(ıxk + �)

Q (xk, W)(53)

here the collocation points are given by

xk = z − 12

xgq,k + z + 12

(54)

The maximum value that NTU can reach is NTUmax = NTU(x = 0).Finally, the Equations that form the proposed model were codified in MatLab for application to previously reported examples.

. Results and discussion

To demonstrate the methodology presented in this paper, eight examples are solved, and the results are compared in terms of CPU timend accuracy relative to the results obtained from the Dormand–Prince Runge–Kutta integration (Dormand and Prince, 1980). Here, theormand–Prince method was also codified in this work. This new method is compared to the Dormand–Prince method instead of the 4thrder Runge–Kutta method because the first one is a multistep method, which guarantees a higher accuracy and allows a fair comparison.

The data for Cases 1–4 and Cases 5–8 are summarized in Tables 1 and 2, respectively. Cases 1–4 consist of water cooling process wherehe inlet air is warm and very dry, while in the Cases 5–8 the inlet air is cold and almost saturated.

During the analysis of each case, the water temperature variation is from 20 ◦C to 50 ◦C. To avoid the fouling, scaling and corrosionaused by the hot water, the temperature of the water sent to the cooling towers (usually coming of heat exchanger networks) should notxceed 50 ◦C (Douglas, 1988). The cooling towers are usually part of a closed cooling system where the cold water is returned to a heatxchanger network; in this network, the lower water temperature required at the inlet heat exchanger network is usually approximately0 ◦C (Ponce-Ortega et al., 2007; Rubio-Castro et al., 2013). However, the methodology proposed here is a systematic strategy, and theforementioned interval can be modified without issue. The relationships used to calculate the thermodynamic properties required forodeling evaporative cooling process are given in Appendix A.1 in Supplementary material.Similar to the methodology proposed in this paper, Fig. 3 provides the algorithm used to solve the model through an orthogonal

ollocation technique. Therefore, the humidity ratio is expressed using a polynomial with an order of n = 15 (i.e., 14-collocation points), theelative tolerance magnitude for solving the system of nonlinear algebraic equations using the Newton–Raphson method is fixed at 10−6

or the orthogonal collocation and the Dormand–Prince methods (see Fig. 3), and the NTU profile is generated using 14-quadrature points.dditionally, the NTU profile in Eq. (51) is calculated from Simpson’s 3/8 rule using 4000 equal subdivisions (�x = 2.5 × 10−4).

32 O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38

Table 1Parameters and results of the simulations for Cases 1–4.

Simulation parameters Case 1 Case 2 Case 3 Case 4

Data

Tw,in (◦C) 50Tw,out (◦C) 20Tdb,in (◦C) 35Twb,in (◦C) 15mw,in/ma 1.20 1.00 0.75 0.50

Results

win 2.5142×10−3

ima,in 4.1688×104

woutDPM 60.1309 × 10−3 51.2931 × 10−3 40.0985 × 10−3 13.0293 × 10−3

OCM 60.0746 × 10−3 51.2320 × 10−3 40.0433 × 10−3 13.0162 × 10−3

ima,outDPM 19.7284 × 104 17.1424 × 104 13.9064 × 104 5.3721 × 104

OCM 19.6663 × 104 17.0897 × 104 13.8678 × 104 5.3639 × 104

NTUDPM 9.4955 5.6146 3.8948 3.2755OCM 9.3244 5.5738 3.8821 3.2670SRM 9.3218 5.5579 3.8636 3.2635

tDPM/tOCM 20.1789 16.4549 14.2284 10.5561

EmDPM 2.0148 × 10−2 2.1644 × 10−4 8.6644 × 10−5 6.1006 × 10−6

OCM 1.1873 × 10−3 4.8755 × 10−4 1.0641 × 10−4 1.5085 × 10−11

EhDPM 7.2916 × 10−4 3.5139 × 10−5 1.4767 × 10−5 5.6361 × 10−6

OCM 4.7158 × 10−5 2.1983 × 10−5 6.6625 × 10−6 5.0866 × 10−13

Table 2Parameters and results of the simulations for the Cases 5–8.

Simulation parameters Case 5 Case 6 Case 7 Case 8

Data

Tw,in (◦C) 50Tw,out (◦C) 20Tdb,in (◦C) 17Twb,in (◦C) 16mw,in/ma 1.20 1.00 0.75 0.50

Results

win 10.9995 × 10−3

ima,in 4.4975 × 104

woutDPM 61.2357 × 10−3 52.5786 × 10−3 41.8816 × 10−3 31.3577 × 10−3

OCM 61.2172 × 10−3 52.5653 × 10−3 41.8683 × 10−3 31.3511 × 10−3

ima,outDPM 19.9956 × 104 17.4101 × 104 14.1788 × 104 10.9497 × 104

OCM 19.9335 × 104 17.3585 × 104 14.1407 × 104 10.9244 × 104

NTUDPM 12.9200 6.6971 4.4662 3.4347OCM 12.5883 6.6399 4.4500 3.4288SRM 12.6162 6.6329 4.4403 3.4198

tDPM/tOCM 14.6979 14.6483 12.8910 12.5077

EmDPM 4.6533 × 10−4 4.1158 × 10−4 1.3449 × 10−4 1.0402 × 10−6

−4 −4 −4 −4

T

pitamEnt

asipsa

OCM 9.4736 × 10 8.4238 × 10 6.2757 × 10 3.0446 × 10

EhDPM 4.1411 × 10−4 3.6499 × 10−5 3.4495 × 10−5 4.6955 × 10−5

OCM 2.5885 × 10−5 2.4348 × 10−5 1.8353 × 10−5 1.0231 × 10−5

Furthermore, the accuracy of the orthogonal collocation and Dorman–Prince methods are determined using the absolute relative error.his value is calculated in term of mass and heat transfer equations (Eqs. (1) and (2)) as follows:

Em =∣∣∣∣1 −

Cpwmwma

[w∗ − w� + (w� − w)H(w� − w)]

[(dw/dTw)][Lef (i∗ma − ima) + (1 − Lef )[(w∗ − w)iv + (w − w�)(iv − iw)H(w − w�)] − (w∗ − w)iw]

∣∣∣∣ (55)

Eh =∣∣∣∣1 − Cpw(mw/ma)

(dima/dTw)

[1 + iw[w∗ − w� + (w� − w)H(w� − w)]

Lef (i∗ma − ima) + (1 − Lef )[(w∗ − w)iv + (w − w�)(iv − iw)H(w − w�)] − (w∗ − w)iw

]∣∣∣∣ (56)

These expressions are obtained from Eqs. (1) and (2), and Appendix A.4 (see Supplementary material) provides the mathematicalrocedure used to generate the above expressions. Moreover, to evaluate the error of both methods, a grid with spacing equal to �x = 10−6

s employed. The grid independence was analyzed to determine this value. Specifically, the grid space was varied to determine its effect onhe mean relative error related to the heat and mass transfers; when �x was below 10−6, the mean relative error remained constant. Thebove procedure (analysis of grid independence) was applied only during the Dormand–Prince method because the grid space affects theean numerical accuracy of this method by dw/dTw evaluating and dima/dTw because these derivatives are evaluated using the forward

uler technique. For the orthogonal collocation method, the numerical accuracy remains independent of the grid spacing. Therefore, theumerical accuracy is evaluated from a series expansion of humidity ratios and air enthalpies; consequently, the grid points reveal onlyhe location for the error evaluation.

The proposed methodology is applied to the cases in Tables 1 and 2, and the profiles relative to the water temperature for humidity ratio,ir enthalpy and number of transfer units are given in Figs. 4 and 5. The first profile is presented in Figs. 4a (warm air) and 5a (cold air), theecond is shown in Figs. 4b (warm air) and 5b (cold air), and the last is presented in Figs. 4c (warm air) and 5c (cold air). Notably, Figs. 4 and 5

nclude the profiles obtained from the orthogonal collocation and Dormand–Prince methods on a psychometric chart. For all cases, therofiles for humidity ratio, moist air enthalpy and number of transfer units from orthogonal collocation and Dormand–Prince methods areimilar (see Figs. 4a–c and 5a–c). However, the number of transfer units for Cases 1 and 5, whose profiles like can be observed in Fig. 5and b, show an obvious discrepancy. Graphically, both methods represent the heat and mass processes transfer with the same accuracy.

O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38 33

Hg11

va

ac(t2

m1d

Fig. 3. Algorithm for evaluating the W coefficients.

owever, when evaluating the absolute relative error for each case through Eqs. (55) and (56), the Dormand–Prince method for each caseenerates larger errors than orthogonal collocation (see Table 1). In Tables 1 and 2, the biggest errors for both methods correspond to Cases

and 5; Figs. 4a and 5b show that these cases are the closest to the saturation line. Therefore, because the value for mw,in/ma is high, Cases and 5 exhibit the highest water–air mass flow rate ratio.

The discrepancy observed for the number of transfer units in Cases 1 and 5 occurs because the differences (w� − w) and (w∗ − w�) areery small near the saturation line. The next mathematical definition for the number of transfer units (this is obtained combining Eqs. (1)nd (3)) is as follows:

dNTUdTw

= mw

ma

1w∗ − w� + (w� − w)H(w� − w)

dw

dTw(57)

The aforementioned differences strongly impact the number of transfer units. Specifically, this expression is more sensitive toward theccuracy of the applied method relative to the humidity ratio and air enthalpy. Tables 1 and 2 show the number of transfer units for eachase from both numerical methods; the gap for each example ranges from 0.1718% to 2.5673%, and Cases 1 and 5 present the largest gaps1.8019% for Case 1 and 2.5673% for Case 5). In addition, the estimated number of transfer units is very important when designing coolingowers because the overall transfer and the loss coefficients are determined through the packing height and area (Kloppers and Kröger,003, 2005c); these values have a strong impact on the cooling tower performance and costs.

Fig. 6 shows the humidity air ratio path for Case 4, and Supplementary Fig. 1 for Cases 1 and 8; the values obtained from both numericalethods are indicated on a psychometric chart. Notably, the behavior is physically appropriated relative to the inlet air conditions (Mills,

999; Singham, 1983), and only Case 4 exhibits a perceptible difference in the path. However, when comparing the outlet air enthalpyetermined by orthogonal collocation and Dormand–Prince methods, a larger gap exists between the values for the outlet air enthalpy

34 O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38

Fig. 4. The results for the governing equations from the Poppe method that were obtained through the orthogonal collocation (OCM) and Dormand–Prince methods (DPM)wA

tabcw

ttCsasli

lomtimtdrf

bC

hile using the following process parameters: Tw,in = 50 ◦C, Tw,out = 20 ◦C, Tdb,in = 35 ◦C, Twb,in = 15 ◦C and mw,in/ma = 1.20, 1.00, 0.75 and 0.50 kg water/kg dry-air. (a)ir humidity ratio profile. (b) Moist air enthalpy profile. (c) NTU profile. SL is the saturation line.

han that related to the air humidity ratio (see Tables 1 and 2). Therefore, the gap between the air enthalpy paths is attributed to theccuracy when calculating the air enthalpy. Consequently, this accuracy could be improved through a numerical strategy, which enables aetter approximation for the first derivatives of the humidity ratio (dw/dTw) and the air enthalpy (dima/dTw). For example, the orthogonalollocation on the finite element using the cubic polynomial of Hermite (Davis, 1984) establishes a larger number of collocation points,hich guarantee the continuity of the aforementioned derivatives and therefore provide a better approximation.

As previously discussed, the accuracy of the numerical method is very important when optimizing designs (Rubio-Castro et al., 2011);herefore, in this work, the absolute relative error for each case relative to the mass and heat transfer equations are estimated for bothhe orthogonal collocation and Dormand–Prince methods. The results are shown in Fig. 7 and Supplementary Fig. 2 for the Case 1 (Fig. 7),ase 4 (Supplementary Fig. 2a) and Case 8 (Supplementary Fig. 2b). These graphics indicate the following: (a) the order of accuracy isimilar in both methods and (b) the maximum absolute relative error of orthogonal collocation method occurs when there is saturatedir. Furthermore, the mean relative error of Eqs. (1) and (2) are calculated, as shown in Tables 1 and 2. The Dormand–Prince method islightly superior to the orthogonal collocation method in this case. However, the orthogonal collocation technique exhibits errors equal orower in magnitude relative to 10−3 and 10−5 for the mass and heat transfer equations, respectively. Therefore, the orthogonal collocations accurate.

In addition, the tDPM/tOCM ratio (where tDPM is CPU time using the Dormand–Prince method and tOCM is CPU time using the orthogonal col-ocation method) is determined for all cases, as shown in Tables 1 and 2. These values must be highlighted because for similar accuracies, therthogonal collocation method requires much less computational time than the Dormand–Prince method (i.e., for Case 1 Dormand–Princeethod require 20 times more computational time to converge, while for Cases 2–8 the above relationship is 16 times, 14 times, 10

imes, 14 times, 14 times and 12 times, respectively). This difference occurs because the orthogonal collocation method requires fourteennterpolation points, while Dormand–Prince method requires sixty-one. Specifically, the problem size generated by the Dormand–Prince

ethod is considerably larger than that from the orthogonal collocation method, which is very important during optimization problemso reduce the human and computational efforts when searching for optimal designs. Therefore, the orthogonal collocation methodologyeveloped here might be a good alternative for optimizing the design of cooling towers in terms of performance and costs; this techniqueepresents the heat and mass transfer processes rigorously with fewer computational demands than the methods used previously (i.e.,ourth-order Runge–Kutta algorithm and Dormand–Prince method).

Another important aspect is related to the stability of the procedure. For high mw,in/ma values, the accuracy of both methods is affectedecause the cooling process path is close to the saturation line (see Cases 4a and 5a). Specifically, Twb is approximately equal to Tw.onsequently, Eq. (1) generates an indeterminate (0/0), causing the Dormand–Prince method to fail because the size of integration step

O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38 35

Fig. 5. The results for the governing equations from Poppe method that were obtained through the orthogonal collocation (OCM) and Dormand–Prince methods (DPM)while using the following process parameters: Tw,in = 50 ◦C, Tw,out = 20 ◦C, Tdb,in = 17 ◦C, Twb,in = 16 ◦C and mw,in/ma = 1.20, 1.00, 0.75 and 0.50 kg water/kg dry-air. (a)Air humidity ratio profile. (b) Moist air enthalpy profile. (c) NTU profile. SL is the saturation line.

FD

celDclmm

ig. 6. The path for the air in a wet-cooling tower as indicated on a psychrometric chart with the results obtained through the orthogonal collocation (OCM) andormand–Prince methods (DPM) for Case 4.

lose to saturation line is reduced below the smallest allowed value. However, the orthogonal collocation method finds the residualsqual to zero in the collocation points, affecting the accuracy for the above indeterminate only when the selected collocation points areocated too close to the saturation line. Therefore, for high mw,in/ma values, the orthogonal collocation method is more stable than theormand–Prince method. For example, Fig. 8 shows the humidity ratio profile where a high mw,in/ma value is used for specific air and wateronditions. The Dormand–Prince method does not converge, and this behavior stops when the humidity profile is too close to saturationine. For the same case, the orthogonal collocation converges without problems. Therefore, the largest discrepancy in the stability for each

ethod close to saturation line is observed (see Figs. 4c and 5c for Case 1). Therefore, this analysis reveals that the orthogonal collocationethod considers the practical problems with any mw,in/ma relationship.

36 O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38

Fig. 7. Absolute relative error of the orthogonal collocation (OCM) and Dormand–Prince methods (DPM) applied to the governing heat and mass transfer equations from thePoppe method for Case 1.

F1

oStfout

ftea

o

5

missG

ig. 8. Stability of the Dormand–Prince and orthogonal collocation methods for high values including mw,in/ma(Tw,in = 50 ◦C, Tw,out = 20 ◦C, Tdb,in = 17 ◦C, Twb,in =6 ◦C and mw,in/ma = 1.33 kg water/kg dry-air).

The relationship between the convergence and the number of collocation points in Supplementary Figs. 3 and 4 is shown the behaviorf the humidity ratio for all cases for different numbers of collocation points (n − 1). Specifically, the values for n are 3, 4, 6, 15 and 20;upplementary Figs. 3 and 4 show that all cases with five collocation points reach a numerical convergence because when n exceeds 6,he humidity ratio profile remains constant. Therefore, the orthogonal collocation requires few points to reach the convergence. However,ourteen collocation points are used in this work to increase the accuracy of the method. Specifically, even a numerical convergence isbtained from five collocation points, and for practical problems, these collocation points could be appropriated; these problems increasentil the numerical error becomes constant. When comparing the accuracy of the Dorman–Prince and orthogonal collocation methods,he latter provides a better or similar representation of the heat and mass transfer processes than Dormand–Prince.

For example, Fig. 9 shows the profiles for the error (heat and mass transfer) relative to the number of collocation points. When usingourteen collocation points, the error remains constant, except for Case 4 where the error continues to decrease. Therefore, for this case,he error can be improved further, but, if the number of collocation points is increased, the computational effort is large. However, therror presented using fourteen collocation points is very good. Consequently, this number of collocation points is used to show the bestccuracy generated by the orthogonal collocation method.

Finally, for cases where the orthogonal collocation method does not converge or a large number of collocation points is require,rthogonal collocation on a finite element is recommended (Carey and Finlayson, 1975).

. Conclusion

A numerical methodology for solving the heat and mass transfer equations during evaporative cooling processes was proposed. Theethodology is based on the orthogonal collocation method, and the heat and mass transfer is represented by the rigorous Poppe method;

n this method, the six differential equations for the humidity ratio, air enthalpy and numbers of transfer units in the unsaturated andupersaturated air region were simplified to yield three differential equations using the Heaviside step function. The expanded powereries were used to represent both the humidity ratio and the air enthalpy, while the number of transfer units was calculated using theauss quadrature method.

O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38 37

Dmcgc

r

A

A

0

Fig. 9. The mean relative error of the orthogonal collocation method (OCM) in (a) the mass and (b) heat transfer equations for Cases 1–8.

To demonstrate the proposed methodology, eight cases were studied, and the results were compared to the results obtained with theormand–Prince technique. This comparison was carried out in terms of the absolute relative error; the orthogonal collocation method isore stable than the Dormand–Prince method because the orthogonal collocation method does not have problems converging when the

ooling profile is close to the saturation line. In addition, the orthogonal collocation method demands less computational time because itenerates smaller problems. This behavior is advantageous when optimizing cooling tower designs because the rigorous Poppe methodan be solved using less computational and human efforts.

The numerical convergence of the orthogonal collocation method can be reached with five collocation points, but minimizing the errorequires at least fifteen points. Otherwise, the accuracy can be improved by performing orthogonal collocation on a finite element.

cknowledgment

The authors acknowledge the financial support by the Universidad Autónoma de Sinaloa (PROFAPI 2013/081).

ppendix A. Supplementary data

Supplementary material related to this article can be found, in the online version, at http://dx.doi.org/10.1016/j.compchemeng.2014.06.08.

3

R

BBCDDDF

JKK

KKKKKLMM

MMPPR

RSSSSSVY

8 O.M. Hernández-Calderón et al. / Computers and Chemical Engineering 71 (2014) 24–38

eferences

aker DR, Shryock HA. A comprehensive approach to the analysis of cooling tower performance. ASME J Heat Transfer 1961;83(3):339–49.osnjakovic F. Technische thermodinamik. Dresden: Theodor Steinkopf; 1965.arey GF, Finlayson BA. Orthogonal collocation on finite elements. Chem Eng Sci 1975;30:587–96.avis ME. Numerical methods and modeling for chemical engineers. 1st ed. NY: John Wiley and Sons Inc.; 1984.ormand J, Prince P. A family of embedded Runge–Kutta formulae. J Comp Appl Math 1980;6:19–26.ouglas JM. Conceptual design of chemical process. NY: McGraw-Hill; 1988.inlayson BA. The method of weighted residuals and variational principles: with applications in fluid mechanics, heat and mass transfer of Mathematics in Science and

Engineering. New York, NY: Academic Press; 1972.aber H, Webb RL. Design of cooling towers by the effectiveness-NTU method. ASME J Heat Transfer 1989;111(4):837–43.intner-Meyer M, Emery AF. Cost-optimal design for cooling towers. ASHRAE J 1995;37(4):46–55.loppers JC. [PhD thesis] A critical evaluation and refinement of the performance prediction of wet-cooling towers [PhD thesis]. Stellenbosch, Western Cape, South Africa:

University of Stellenbosch; 2003.loppers JC, Kröger DG. Loss coefficient correlation for wet-cooling tower fills. Appl Therm Eng 2003;23(17):2201–11.loppers JC, Kröger DG. The Lewis factor and its influence on the performance prediction of wet-cooling towers. Int J Heat Mass Transfer 2005a;44(9):879–84.loppers JC, Kröger DG. A critical investigation into heat and mass transfer analysis of counterflow wet cooling towers. Int J Heat Mass Transfer 2005b;48:765–77.loppers JC, Kröger DG. Refinement of the transfer characteristic correlation of wet-cooling tower fills. Heat Transfer Eng 2005c;26(4):35–41.röger DG. Air-cooled heat exchangers and cooling towers. 1st ed. Tulsa, OK: Penn Well Corp.; 2004.anczos C. Applied analysis. Reprint of 1956 Prentice-Hall ed. Mineola, NY: Dover Publications Inc; 1988.aclaine-cross IL, Banks PJ. A general theory of wet surface heat exchangers and its application to regenerative cooling. ASME J Heat Transfer 1981;103(3):579–85.arseille TJ, Schliesing JS, Bell DM, Johnson BM. Extending cooling tower thermal performance prediction using a liquid-side film resistance model. Heat Transfer Eng

1991;12(3):19–30.erkel F. Verdunstungskühlung, Zeitschrift Verein. Deutsch Ing 1925;70:123–8.ills AE. Basic heat and mass transfer. 2nd ed. Upper Saddle River, NJ: Prentice Hall; 1999.

once-Ortega JM, Serna-González M, Jiménez-Gutiérrez A. MINLP synthesis of optimal cooling networks. Chem Eng Sci 2007;62(21):5728–35.oppe M, Rögener H. Berechnung von Rückkühlwerken. VDI-Wärmeatlas; 1991. p. Mi1–15.ubio-Castro E, Serna-González M, Ponce-Ortega JM, Morales-Cabrera MA. Optimization of mechanical draft counter flow wet-cooling towers using a rigorous model. Appl

Therm Eng 2011;31(16):3615–28.ubio-Castro E, Serna-González E, Ponce-Ortega JM, El-Halwagi MM. Synthesis of cooling water systems with multiple cooling towers. Appl Therm Eng 2013;50(1):957–74.erna-González M, Ponce-Ortega JM, Jiménez-Gutiérrez A. MINLP optimization of mechanical draft counter flow wet-cooling towers. Chem Eng Res Des 2010;88(5–6):614–25.ingham JR. Cooling towers. In: Heat exchanger design handbook. USA: Hemisphere Publishing Corporation; 1983.

öylemez MS. On the optimum sizing of cooling towers. Energy Convers Manage 2001;42(7):783–9.öylemez MS. On the optimum performance of forced draft counter flow cooling towers. Energy Convers Manage 2004;45(15–16):2335–41.utherland JW. Analysis of mechanical-draught counterflow air/water cooling towers. ASME J Heat Transfer 1983;105(8):576–83.illadsen JV, Stewart WE. Solution of boundary value problems by orthogonal collocation. Chem Eng Sci 1967;22:1483–501.ilmaz A. Analytical calculation of wet cooling tower performance with large cooling ranges. J Therm Sci Technol 2010;30(2):45–56.