Accepted Manuscript Title: Modeling and simulation of counterflow wet-cooling towers and the accurate calculation and correlation of mass transfer coefficients for thermal performance prediction Author: Mario Llano-Restrepo, Robinson Monsalve-Reyes PII: DOI: Reference:
S0140-7007(16)30347-4 http://dx.doi.org/doi: 10.1016/j.ijrefrig.2016.10.018 JIJR 3461
To appear in:
International Journal of Refrigeration
Received date: Revised date: Accepted date:
4-6-2016 26-9-2016 23-10-2016
Please cite this article as: Mario Llano-Restrepo, Robinson Monsalve-Reyes, Modeling and simulation of counterflow wet-cooling towers and the accurate calculation and correlation of mass transfer coefficients for thermal performance prediction, International Journal of Refrigeration (2016), http://dx.doi.org/doi: 10.1016/j.ijrefrig.2016.10.018. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Modeling and simulation of counterflow wet-cooling towers and the accurate calculation and correlation of mass transfer coefficients for thermal performance prediction
Mario Llano-Restrepo*, Robinson Monsalve-Reyes School of Chemical Engineering, Universidad del Valle, Ciudad Universitaria Melendez, Building 336, Apartado 25360, Cali, Colombia
* Corresponding author. Tel: + 57-2-3312935; fax: + 57-2-3392335 E-mail address:
[email protected] (M. Llano-Restrepo)
Highlights A detailed mathematical derivation is provided for a continuous differential air-water contactor (CDAWC) model of cooling towers.
The CDAWC model is valid for a bulk gas phase of unsaturated or supersaturated air.
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The CDAWC model equations are mathematically much simpler than those of previous models.
A simulation method is proposed for the accurate calculation of volumetric mass transfer coefficients in cooling towers.
The simulation-based volumetric mass transfer coefficients can be correlated accurately in terms of cooling-tower inlet quantities.
Abstract This work provides a detailed mathematical derivation of a steady-state one-dimensional continuous differential air-water contactor (CDAWC) model that describes the material and energy balances in a counterflow wet-cooling tower. The model consists of four ordinary differential equations that describe the changes (along the packed height) of the liquid water temperature, dry-bulb temperature of moist air, liquid water mass flowrate, and moist-air humidity mass ratio. The model is formulated for the cases of unsaturated and supersaturated air, and the model equations are compared to those of previous works. It is shown that the equations of some previous models are approximately equivalent to the equations of the CDAWC model. However, the formulation of the CDAWC model is simpler and the resulting equations have a more general form. A simulation method is proposed to determine accurate values of the volumetric mass transfer coefficient by matching the experimental thermal performance of counterflow wet-cooling towers.
Keywords: Cooling towers; evaporative cooling; direct-contact cooling; water cooling; mass transfer; temperature profiles.
Nomenclature a
interfacial area per packing unit volume [m2 m3]
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aM
mass-transfer interfacial area per packing unit volume [m2 m3]
aH
heat-transfer interfacial area per packing unit volume [m2 m3]
ARD
absolute relative deviation
AARD
average absolute relative deviation
AT
cooling-tower cross section area [m2]
Ci
adjustable parameter ( i 1,
C p ,B
specific heat capacity of dry air [J (kgda)1 K1]
C p ,G
specific heat capacity of moist air per unit mass of dry air [J (kgda)1 K1]
5)
sat
specific heat capacity of saturated moist air [J (kgda)1 K1]
(g)
specific heat capacity of water vapor [J (kgw)1 K1]
C p ,w
(L)
specific heat capacity of liquid water [J (kgw)1 K1]
de
packing equivalent diameter [m]
d
packing geometric diameter [m]
C p ,G C p ,w
p
D W ,G
diffusivity of water vapor in air [m2 s1]
GB
mass flowrate of dry air per unit cross section area [kgda s1 m2]
hG
gas-phase convective heat transfer coefficient [W m2 K1]
HB
specific enthalpy of dry air [J (kgda)1]
HG
specific enthalpy of moist air per unit mass of dry air [J (kgda)1]
HG
specific enthalpy of saturated air per unit mass of dry air [J (kgda)1]
HL
specific enthalpy of the liquid phase [J (kgw)1]
HR
percentage relative humidity of moist air
sat
(g)
specific enthalpy of water vapor [J (kgw)1]
Hw
(L)
specific enthalpy of liquid water [J (kgw)1]
kY
mass transfer coefficient [kgw s1 m2 (kgw (kgda)1)1]
kY aM
volumetric mass transfer coefficient [kgw s1 m3 (kgw (kgda)1)1]
L
mass flowrate of liquid water per unit cross section area [kgw s1 m2]
Le
Lewis number
Hw
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Le
f
Lewis factor
M
B
molar mass of dry air [kgda kmolda1]
M
G
molar mass of moist air [kgma kmolma1]
MW
molar mass of water [kgw kmolw1]
Me
Merkel number
NW
interphase mass-transfer flux of water [kgw s1 m2]
P
absolute barometric pressure [Pa]
P
sat w
qC r
2
vapor pressure of liquid water [Pa] interphase convective heat-transfer flux [W m2] coefficient of determination
R
universal gas constant [Pa m3 kmol1 K1]
RD
relative deviation
TG
dry-bulb temperature of moist air [°C]
TL
temperature of liquid water [°C]
T0
reference temperature [°C]
T wb
wet-bulb temperature of moist air [°C]
vG
superficial gas velocity of the gas phase [m s1]
vG / L
relative superficial velocity [m s1]
xW
mole fraction of water vapor
YW
humidity mass ratio of moist air [kgw (kgda)1]
YW , I
humidity mass ratio at gas-phase side of interface [kgw (kgda)1]
YW , S
humidity mass ratio of saturated moist air at T G [kgw (kgda)1]
YW
humidity mass ratio of saturated moist air at T wb [kgw (kgda)1]
z
vertical distance from the bottom of packed section [m]
ZT
packed height of cooling tower [m]
sat
Greek symbols
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H vap , w
specific heat of vaporization of liquid water [J (kgw)1]
z
height of infinitesimal packed section element [m]
G
viscosity of moist air [Pa s]
L
viscosity of liquid water [Pa s]
B
thermal conductivity of dry air [W m1 K1]
G
thermal conductivity of moist air [W m1 K1]
W
thermal conductivity of water vapor [W m1 K1]
B
mass density of dry air [kg m3]
G
mass density of moist air [kg m3]
Subscripts da
dry air
ma
moist air
w
water
B
dry air
G
moist air
W
water vapor
1
at the bottom of the packed section
2
at the top of the packed section
VITAE Mario Llano-Restrepo is Professor of Chemical Engineering at Universidad del Valle in Cali, Colombia. He received his Ph.D. degree (in chemical engineering) from Rice University in 1994. His main research interests are modeling and simulation of separation processes, modeling of phase and
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chemical equilibria, and molecular simulation. In 2009, he earned a Distinguished Professor recognition from Universidad del Valle for service and excellence in teaching since 1994. Some of the courses he has taught are mass transfer, separation operations, separation process modeling and simulation, chemical reaction engineering, chemical thermodynamics, statistical thermodynamics, and molecular simulation.
Robinson Monsalve-Reyes received his undergraduate degree (B.S) in chemical engineering from Universidad del Valle, in 2016, and will be pursuing studies for a master's degree in chemical engineering. His main research interests are separation process modeling and simulation.
1. Introduction Cooling of water by means of sensible and evaporative heat transfer to atmospheric air in counterflow wet packed towers has found a widespread use in many chemical and petrochemical process industries, steam power plants, and large air-conditioning units.
In the process industries, cooling water is used either in distillation column condensers (in which the vapor stream leaving the top of the column is condensed by indirect contact with cooling water in order to yield the liquid distillate product) or in some tube-and-shell heat exchangers (in which a hot liquid stream is cooled by indirect contact with cooling water).
In the low-pressure surface condensers of steam power plants, cooling water is used to condense (by indirect contact) the steam after its expansion in the turbines. In large air-conditioning units, cooling water is used for the condensation of the hot working fluid.
The surface of the packing or fill of a wet-cooling tower facilitates the direct contact of water and air, and by means of this contact the hot liquid water stream (fed to the top of the tower and flowing down through the packing) transfers sensible and latent heat to the atmospheric air stream (fed to the bottom of the tower and flowing up through the packing), and because of the
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7 continued rejection of heat to the air, the water temperature decreases from the top to the bottom of the tower, and because of the continued evaporation of liquid water to the air, the air humidity increases from the bottom to the top of the tower.
The outlet water temperature will always be greater than the inlet air wet-bulb temperature, and the corresponding temperature difference is known as cooling-tower approach. The difference between the inlet and outlet water temperatures is known as cooling-tower range. The temperature that the cooled water reaches at the outlet of the tower is an important economic factor in the design of modern chemical process and steam power plants.
1.1 Review of previous works The first model for cooling towers was proposed by Merkel (1926). Merkel’s model has been discussed in several classical papers (London et al., 1940; Hutchison and Spivey, 1942; Lichtenstein, 1943; Simpson and Sherwood, 1946; Baker and Shryock, 1961) and textbooks (Kern, 1950; Badger and Banchero, 1955; Foust, 1960; Norman, 1962; Sherwood et al., 1975; Treybal, 1980; Coulson et al., 1990; Geankoplis, 2003). By making several assumptions and approximations, it can be shown that Merkel’s model leads to the following equation: kY aM Z T L
(L)
TL , 2
C p , w (T L ) dT L
T L ,1
[ H G (T L ) H G (T G , YW )]
sat
(1)
where k Y [kgw s1 m2 (kgw (kgda)1)1] is the convective mass transfer coefficient of water vapor and subscript Y stands for the humidity mass ratio [kgw (kgda)1] of moist air, a M (m2 m3) is the mass-transfer interfacial area per packing unit volume, Z T (m) is the packed height of the cooling tower, L [kgw s1 m2] is the mass flowrate of liquid water per unit tower cross section area, T L (°C) is the liquid water temperature, C p( L, w) (T L ) [J (kgw)1 K1] is the specific heat capacity of liquid water evaluated at T L , H Gsat (T L ) [J (kgda)1] is the specific enthalpy of saturated moist air evaluated at T L , and H G (T G , YW ) is the specific enthalpy of the moist air (in
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8 contact with the liquid water at T L ) corresponding to its dry-bulb temperature T G and humidity mass ratio Y W . The integral on the right-hand side of Eq. (1) is computed from the outlet water temperature T L ,1 (i.e., at the bottom of the tower) to the inlet water temperature T L , 2 (i.e., at the top of the tower). In the original derivation and use of Eq. (1) by Merkel (1926), a constant value, (L)
C p , w , was assumed for the specific heat capacity of liquid water in order to calculate the integral.
Since T L is the integration variable in Eq. (1), an energy balance on an envelope bound by the bottom of the tower is employed to calculate the required specific enthalpy H G (T G , YW ) by neglecting the change of the water mass flowrate due to evaporation (one of the basic assumptions of Merkel’s model), as follows:
H G (T G , YW ) H G (T G ,1 , YW ,1 )
LC
(L) p ,w
GB
(T L T L ,1 )
(2)
where G B [kgda s1 m2] is the mass flowrate of dry air per unit tower cross section area, and T G ,1 and Y W ,1 are the inlet (i.e., at the bottom of the tower) air dry-bulb temperature and humidity mass ratio, respectively. Eq. (2) shows that H G (T G , YW ) in Eq. (1) is a function only of the integration variable T L .
Eq. (1) is known as the Merkel equation and its left-hand side is known as the Merkel number ( Me ) or cooling-tower transfer characteristic. To compute the value of the integral, an accurate numerical technique such as the Chebyshev’s equal weight four-point integration formula (Abramowitz and Stegun, 1972) can be employed, as explained in detail by Mohiuddin and Kant (1996).
Classical early papers (London et al., 1940; Hutchison and Spivey, 1942; Lichtenstein, 1943; Simpson and Sherwood, 1946) are important sources for experimental data of the performance of mechanical-draft counterflow wet-cooling towers, and they also illustrate the application of Merkel’s model.
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London et al. (1940) measured the performance of an induced-draft counterflow wet-cooling tower with a packed height of 1.75 m (the packing being staggered redwood decks with a streamlined shape), and used Merkel’s model and the concept of cooling-tower effectiveness to analyze their measurements (a total of 60 experimental runs). Quantities of interest such as the water and air mass flowrates, the inlet and outlet water and air (dry-bulb and wet-bulb) temperatures, and the mass transfer coefficient were fully reported by London et al. (1940).
Hutchison and Spivey (1942) measured the performance of three forced-draft counterflow wetcooling towers. Two of the towers were packed with horizontally staggered decks made up of parallel triangular bars, the packed heights being 3.4 and 6.7 m, respectively. The other tower was packed with wooden grids either in a crossed or in a parallel arrangement, the packed height being about 3.6 m. Merkel’s model was used to analyze the measurements (a total of 34 experimental runs), by making the assumption that the air leaving the tower was saturated (another basic assumption of Merkel’s model) so that for the outlet air, only the dry-bulb temperature had to be reported. Other quantities of interest such as the inlet and outlet water temperatures, inlet air dry-bulb and wet-bulb temperatures, water and air volumetric flowrates, and the mass transfer coefficient were fully reported by Hutchison and Spivey (1942).
Lichtenstein (1943) measured the performance of a forced-draft counterflow wet-cooling tower packed with wooden slats, the maximum packed height being 10.7 m, and used Merkel’s model to analyze his measurements (a total of 59 experimental runs). Even though some quantities of interest such as the cooling-tower range and approach, the water and air mass flowrates, the inlet air wet-bulb temperature, and the Merkel number were reported by Lichtenstein (1943), other important quantities such as the inlet air dry-bulb temperature and the outlet air dry-bulb and wet-bulb temperatures were not reported.
Simpson and Sherwood (1946) measured the performance of several induced-draft counterflow wet-cooling towers with a packed height of 1.05 m, the packing being either redwood slats (towers R-1 and R-2) or Masonite sheets (towers M-1 and M-2), and used Merkel’s model to
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10 analyze their measurements (a total of 119 experimental runs). Quantities of interest such as the inlet and outlet water and air (dry-bulb and wet-bulb) temperatures, water and air mass flowrates, and the mass transfer coefficient were fully reported by Simpson and Sherwood (1946).
For a cooling tower with a given packed height, Eq. (1) can be used to calculate the volumetric mass transfer coefficient k Y a M [kgw s1 m3 (kgw (kgda)1)1], as follows:
kY aM
L ZT
(L)
TL , 2
T L ,1
C p , w (T L ) dT L [ H G (T L ) H G (T G , YW )] sat
(3)
As compared to k Y , the volumetric mass transfer coefficient k Y a M is a more convenient quantity because of the practical difficulty to measure the mass-transfer interfacial area (per packing unit volume) a M . Due to the several assumptions and approximations involved in the derivation of the Merkel equation (see Section 3 below), and regardless the accuracy of the numerical computation of the integral, the value of k Y a M calculated from Eq. (3) should be regarded only as a first estimate of the actual value. In the classical early papers (London et al., 1940; Hutchison and Spivey, 1942; Lichtenstein, 1943; Simpson and Sherwood, 1946), Eq. (3) was used for the calculation of the reported values of k Y a M .
Once the volumetric mass transfer coefficient k Y a M has been determined from Eq. (3), the coupled volumetric heat transfer coefficient h G a H is usually estimated from the revised Lewis relation (Lewis, 1922, 1933): hG a H kY aM
C p ,G
(4)
where C p .G is the specific heat capacity of moist air [J (kgda)1 K1)]. The left-hand side of Eq. (4) is called the apparent psychrometric ratio (Hensel and Treybal, 1952); the mass-transfer interfacial area a M may be assumed to be identical to the heat-transfer interfacial area a H only if the cooling tower packing is completely wetted by the liquid. Eq. (4) is only an approximation.
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11 As explained in Appendix A.9, a more general form of the revised Lewis relation is given by the Chilton-Colburn analogy (Chilton and Colburn, 1934). From the consideration of the approximate nature of Merkel’s model, some other alternative methods to analyze the performance of cooling towers have been developed in the past, such as the effectiveness-NTU methods (Jaber and Webb, 1989; Braun et al., 1989), and the methods by Osterle (1991) and Poppe (Poppe, 1973; Poppe and Rögener, 2006; Kloppers and Kröger, 2005a). However, some assumptions or approximations are also involved in those three alternative methods.
The effectiveness-NTU method (Jaber and Webb, 1989) is based on the same simplifying assumptions (Kloppers and Kröger, 2005a, 2005b) as Merkel’s model. Although the model by Osterle (1991) revised the assumption of constant water flowrate made in the derivation of Merkel’s model, it still involves several mathematical approximations such as those of making use of the approximate Lewis relation, Eq. (4), with a H a M , neglecting the temperature dependence of the specific heat capacities of liquid water, dry air, and water vapor, and the dependence of the moist-air specific heat capacity on the humidity mass ratio, and ignoring the contribution of a humidity-mass-ratio driving-force term to the total heat transfer rate (see Section 4.1 below and Appendix B in the Supplementary Data). Even the more rigorous model by Poppe (Poppe, 1973; Poppe and Rögener, 2006) still involves some mathematical approximations (Kloppers and Kröger, 2005a), such as that of neglecting the temperature dependence of the specific heat capacities of air, liquid water, and water vapor (see Section 4.3 below and Appendix D in the Supplementary Data).
For these reasons, the formulation of a fundamental model of a cooling tower with the least possible number of assumptions or mathematical approximations is highly desirable. Since the temperatures of the liquid water and air streams, the humidity of air, and the water flowrate continuously change along the height of the packing inside the cooling tower, the most suitable kind of model is a continuously distributed one, consisting of differential equations that describe these changes. If only changes in the vertical direction are considered, then a set of ordinary
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12 differential equations (ODEs), with the packed height as the integration (or independent) variable, would be appropriate. A complete simulation of the cooling tower performance should include the calculation of the outlet water temperature, outlet air dry-bulb temperature, outlet air wet-bulb temperature, and outlet water mass flowrate for given values of the packed height, inlet air mass flowrate, inlet air dry-bulb temperature, inlet air wet-bulb temperature, inlet water mass flowrate, and inlet water temperature.
In the models by Osterle (1991) and Poppe (Poppe, 1973; Poppe and Rögener, 2006; Kloppers and Kröger, 2005a) and in six previous modeling and simulation works (Sutherland, 1983; Milosavljevic and Heikkilä, 2001; Papaefthimiou et al., 2006; Elsarrag, 2006; Klimanek and Bialecki, 2009; Klimanek, 2013), a set of ODEs was used either to analyze or to simulate the performance of cooling towers.
In the models by Osterle (1991) and Poppe (Poppe, 1973; Poppe and Rögener, 2006; Kloppers and Kröger, 2005a), three ODEs were formulated to describe the changes of the moist-air specific enthalpy and humidity mass ratio, and the Merkel number with respect to the liquid water temperature, which was chosen as the integration variable. In the performance-analysis model by Sutherland (1983), two ODEs were formulated to describe the changes of the specific enthalpy of moist air and the liquid water temperature with respect to the humidity mass ratio of moist air, which was chosen as the integration variable. However, the choice of either the liquid water temperature or the air humidity mass ratio as the integration variable, instead of the cooling-tower packed height, turns out to be less convenient because it makes the mathematical formulation of the equations more difficult (see Appendices B, C, and D). Furthermore, the choice of the air humidity mass ratio as the integration variable poses an additional difficulty, which is the specification of an appropriate integration step size during the implementation of the numerical solution (Sutherland, 1983).
In contrast, in the other five of the six previous modeling and simulation works (Milosavljevic and Heikkilä, 2001; Papaefthimiou et al., 2006; Elsarrag, 2006; Klimanek and Bialecki, 2009;
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13 Klimanek, 2013), the packed height was chosen as the integration variable so that the formulation of the equations became relatively simpler.
Milosavljevic and Heikkilä (2001) experimentally tested seven cooling-tower film-type packings (fluted, smooth or corrugated plates, honeycomb fillings, and some mixed arrangements), determined the volumetric heat transfer coefficients for those packings, and formulated a set of ODEs to simulate the performance of a cooling tower filled with one of those packings (chosen for its high heat transfer coefficient).
Papaefthimiou et al. (2006) formulated a set of ODEs to simulate the performance of a cooling tower originally measured and reported by Simpson and Sherwood (1946) (which had also been previously simulated by Khan et al., 2003), and also studied the effect of atmospheric air conditions, water mass flowrate, and inlet water temperature on the thermal performance of a reference-case cooling tower.
Elsarrag (2006) measured the performance of an induced-draft counterflow wet-cooling tower filled with a ceramic tile packing, determined the corresponding mass and heat transfer coefficients by means of the Merkel equation and the Lewis relation, respectively, by use of dimensional analysis developed a correlation for the mass transfer coefficient that involves the equivalent diameter of the packing d e , and formulated a set of ODEs to simulate the performance of the cooling tower.
Klimanek and Bialecki (2009) formulated a set of ODEs to describe counterflow wet cooling towers for the cases of unsaturated and supersaturated air, simulated the performance of a reference-case cooling tower and validated their model from a comparison with some benchmark results reported by Kloppers (2003).
Klimanek (2013) gave more mathematical and notational details about the derivation of the model formulated by Klimanek and Bialecki (2009), reviewed the derivations of the Merkel and
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14 Poppe methods, and provided some numerical examples of the simulation of counterflow wet cooling towers for the cases of unsaturated and supersaturated air.
Table 1 provides a list of works in which power-law correlations for the mass transfer coefficient in cooling towers have been reported by several authors for various kinds of packings.
Despite of the contributions made by the six previous modeling and simulation works (Sutherland, 1983; Milosavljevic and Heikkilä, 2001; Papaefthimiou et al., 2006; Elsarrag, 2006; Klimanek and Bialecki, 2009 Klimanek, 2013), the detailed mathematical derivation of a model in which the set of ordinary differential equations turns out to be formulated in the more general possible way (i.e., without making any unnecessary mathematical approximations and free of any notational ambiguity), is still lacking in the cooling tower literature. Even of more concern is that in three of those works some inconsistencies (either missing or extraneous terms) are involved in the differential equations that were formulated and used to obtain the reported simulation results (see Section 4 below).
1.2 Overview
The outline of the paper is as follows. In Section 2, the detailed mathematical derivation of the continuous differential air-water contactor (CDAWC) model is provided for the two cases of unsaturated and supersaturated air as the bulk gas phase. That derivation yields a clear and consistent formulation of the differential equations that describe the material and energy balances in the cooling tower. In Section 3, to emphasize the approximate nature of Merkel’s model, we show how the CDAWC model has to be simplified in order to obtain Eq. (1). In Section 4, a critical review of previous models is provided by making a detailed comparison between the CDAWC model equations formulated in Section 2 and the corresponding equations reported in previous works, in order to show that the formulation of the CDAWC model is simpler and the resulting equations
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15 have a more general form, and also to point out and correct some inconsistencies found in some of those previous works.
In Section 5, a validation test of the CDAWC model is carried out by comparing simulation results (for the two cases of unsaturated and supersaturated air) with the results available from some previous works. Also, four simulation tasks are formulated and sequentially undertaken. The CDAWC model is numerically solved to calculate the volumetric mass transfer coefficients associated to the performance of an induced-draft counterflow wet-cooling tower experimentally studied by Simpson and Sherwood (1946), and these simulation-based mass-transfer coefficient values are not only compared to those calculated by Simpson and Sherwood (1946) from the Merkel equation, but also they are accurately correlated in terms of four independent coolingtower inlet quantities. An assessment of the predictive capability of the simulated model is also made in Section 5.
Appendix A (Supplementary Data) presents the correlations and methods that were used for the calculation of properties. Appendices BE (Supplementary Data) contain the detailed analyses of the model equations by Osterle (1991), Sutherland (1983), Khan et al. (Khan and Zubair, 2001; Khan et al. 2003), Poppe (Poppe, 1973; Poppe and Rögener, 2006; Kloppers and Kröger, 2005a), and Klimanek and Bialecki (2009). For the two latter models the analyses are made for the two cases of unsaturated and supersaturated air as bulk gas phase. Lastly, Appendix F (Supplementary Data) compiles all the tabular and graphical results for the four simulations tasks undertaken and discussed in Section 5.
2. Derivation of the CDAWC model Our continuous differential air-water contactor (CDAWC) model of counterflow wet-cooling towers is based upon the following seven assumptions: (1) The cooling tower is in steady-state operation; that is, liquid-water and moist-air properties at a given height of the tower (with a uniform cross section) are timeindependent.
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16 (2) Properties change only in the vertical direction (along the cooling tower packed height); that is, variations in the other directions are neglected. (3) Axial dispersion is negligible for both mass and heat transfer. (4) Vapor-liquid equilibrium for water is assumed at the liquid-gas interface. (5) Liquid-phase resistance to heat transfer is negligible as compared to the gas-phase resistance. Therefore, the gas-liquid interface is at the same temperature as the bulk liquid phase. (6) The cooling tower operates adiabatically; that is, heat losses through the tower walls are neglected. (7) The gas phase can be regarded as an ideal solution for the calculation of its enthalpy and heat capacity.
Assumption (5) was the matter of some controversy in the past, which merits a brief review. In an early study, from experimental measurements of heat and mass transfer between water and air in a wetted-wall tower, Barnet and Kobe (1941) found that the liquid-phase resistance to heat transfer was either zero or negligibly small. Later on, by combining experimental measurements of the gas-phase mass or heat transfer coefficients and a trial-and-error procedure first proposed by McAdams et al. (1949) and later refined by Mickley (1949), some authors like McAdams et al. (1949), Yoshida and Tanaka (1951), Thomas and Houston (1959a, 1959b), and Nori and Ishii (1982) calculated the liquid-phase heat transfer coefficient h L for air-water contact in columns packed with Raschig rings (McAdams et al., 1949; Yoshida and Tanaka, 1951; Nori and Ishii, 1982) or criss-cross slats (Thomas and Houston, 1959a, 1959b), and correlated the h L values in terms of either the inlet water mass flowrate only (Yoshida and Tanaka, 1951), both the inlet water and dry air mass flowrates (McAdams et al., 1949; Thomas and Houston, 1959a, 1959b) or the Reynolds and Prandtl numbers for the liquid phase (Nori and Ishii, 1982). From the experimental measurements reported by London et al. (1940) and Simpson and Sherwood (1946), and a trial-and-error graphical procedure of his own, Inazumi (1955) also calculated h L and correlated it in terms of either the inlet water mass flowrate only or both the inlet water and dry air mass flowrates.
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17 However, since McAdams et al. (1949) had already stated that their own results were questionable and the error associated to them was unknown, Surosky and Dodge (1950) critically analized the method followed by McAdams et al. (1949) to calculate h L and arrived at the conclusion that in view of the large uncertainty involved in those calculations, it was better to follow the earlier findings of Barnet and Kobe (1941) of a negligibly small liquid-phase heat transfer resistance. Jackson (1958) critically examined the values of h L reported by both McAdams et al. (1949) and Yoshida and Tanaka (1951), and calculated the resulting temperature differences for water between the bulk and the interface, arriving at the conclusion that they were so large that they were extremely improbable, and in consequence, the values of h L reported and correlated by McAdams et al. (1949) and Yoshida and Tanaka (1951) were quite doubtful. Cribb (1959) critically examined the values of h L reported by McAdams et al. (1949) and Inazumi (1955) and arrived at the conclusion that although evidence of a liquid-phase heat transfer resistance in cooling towers follows from those two works, for practical design purposes that resistance may be ignored. Some time later, Webb (1984) stated that assumption (5) is indeed a reasonable approximation for cooling towers. For an assessment of the validity of this assumption in the present work, see Section 5.2.3 below.
In previous works (Sutherland, 1983; Khan and Zubair, 2001; Milosavljevic and Heikkilä, 2001; Khan et al., 2003; Kloppers, 2003; Kloppers and Kröger, 2005a, 2005b; Poppe and Rögener, 2006; Papaefthimiou et al., 2006; Elsarrag, 2006; Klimanek and Bialecki, 2009; Klimanek, 2013), the differential-change approach was used to formulate the mass and energy balances; in that way, the balances were, at once, expressed in terms of differential changes. However, for the sake of clarity, we decided to use the finite-difference approach to formulate all balances, by writing them for an infinitesimal section of the continuous air/water contactor (see Fig. 1). In this way, the balances are expressed in terms of infinitesimal changes, and after taking the limit as the height of the infinitesimal section goes to zero, the balances yield the corresponding ordinary differential equations of the model.
Let us consider the infinitesimal element of packed cooling tower shown in Fig. 1. Let z be the vertical distance from the bottom of the cooling-tower packed section upward. Let subscripts B
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18 and W stand for dry air and water vapor, respectively. From assumptions (1) and (2), the element is at steady state and properties change only along the axial coordinate z.
2.1 Case of unsaturated air as bulk gas phase
The case of unsaturated air as the bulk gas phase in a counterflow wet-cooling tower is considered first for the derivation of the governing mass and energy balance equations.
2.1.1 Material balance in the gas phase Let G B [kgda s1 m2 ] be the constant mass flowrate of dry air per unit area of cooling tower cross section, Y W [kgw (kgda)1] the humidity mass ratio (i.e., the mass ratio of water vapor to dry air) of moist air, N W [kgw s1 m2] the interphase mass-transfer flux of water (from the liquid to the gas phase), a M (m2 m3) the mass-transfer interfacial area per unit volume of packing, and z
the height of the tower element.
From assumption (3), axial dispersion of mass is neglected; therefore, a steady-state mass balance on water in the gas phase (see Fig. 1) is given by G B YW
z
N W a M z G B YW
z z
(5)
Dividing both sides of Eq. (5) by z and taking the limit of the rearranged equation as z 0 yields dY W dz
NW aM GB
(6)
A humidity mass ratio difference can be used as the driving force to write the interphase masstransfer flux, as follows: N W k Y (Y W , I Y W )
(7)
Page 18 of 92
19
where k Y [kgw s1 m2 (kgw (kgda)1)1] is the convective mass transfer coefficient of water vapor, Y W is the humidity mass ratio in the bulk gas phase, and Y W , I is the humidity mass ratio in the gas-phase side of the liquid-gas interface.
From assumption (4), the gas-phase side of the interface consists of saturated moist air. In the case of unsaturated air as the bulk gas phase, YW , I YW ; therefore, from Eq. (7), N W 0 , and in consequence, from Eq. (6), it follows that dY W / dz 0 ; that is, Y W increases as z increases (the air stream gets more humid as it goes upward in the tower).
Substitution of Eq. (7) into Eq. (6) leads to the final expression: dY W
k Y a M (Y W , I Y W )
dz
(8)
GB
Eq. (8) is an ordinary differential equation that describes the change of the humidity mass ratio of the air stream along the packed height of a counterflow wet-cooling tower with unsaturated air as the bulk gas phase. An expression to calculate Y W , I is given in Appendix A.4.
2.1.2 Material balance in the liquid phase Let L [kgw s1 m2] be the liquid water mass flowrate per unit area of cooling tower cross section. A steady-state mass balance in the liquid phase (see Fig. 1) is given by L
z z
N W a M z L
z
(9)
Dividing both sides of Eq. (9) by z and taking the limit of the rearranged equation as z 0 yields dL dz
NW aM
(10)
Page 19 of 92
20
Substitution of Eq. (7) into Eq.(10) yields the final expression: dL dz
k Y a M (Y W , I Y W )
(11)
Eq. (11) is an ordinary differential equation that describes the change (along the packed height of the cooling tower) of the liquid water mass flowrate (per unit area of cooling tower cross section) for the case of unsaturated air as the bulk gas phase. Since YW , I YW in that case, from Eq. (11), it follows that dL / dz 0 ; that is, L decreases as z decreases (the mass flowrate of the water stream decreases as it goes downward in the tower).
2.1.3 Energy balance in the gas phase Let T G and H G [J (kgda)1] be the temperature and the total specific enthalpy of the gas phase (per unit mass of dry air), respectively, and let H w( g,C) [J (kgw)1] be the specific enthalpy of the water vapor (w, g) being transferred by convection (C) to the bulk gas phase. Since the water vapor is transferred from the gas-phase side of the interface, H w( g,C) should be calculated at the temperature of the liquid-gas interface. However, from assumption (5), since the heat transfer resistance of the liquid phase is considered to be negligible, the interface is at the same temperature T L as the bulk liquid phase; in consequence, from heretofore H w( g,C) will be written as H
(g) w
(T L ) .
If assumptions (3) and (6) for negligible axial dispersion of heat and for an adiabatic operation are taken into consideration, and q C [W m2] and a H [m2 m3] stand for the interphase convective heat-transfer flux (from the liquid to the gas phase) and the heat-transfer interfacial area (per unit volume of packing), respectively, then the steady-state energy balance in the gas phase (see Fig. 1) is given by
Page 20 of 92
21 ( N W a M z ) H w (T L ) q C a H z G B H G (g)
GB H G
z
z z
(12)
Dividing both sides of Eq. (12) by z and taking the limit of the rearranged equation as z 0 yields dH
G
(NW aM )
dz
GB
qC a H
H w (T L ) (g)
(13)
GB
From assumption (7), the gas phase is regarded as an ideal solution for the calculation of its enthalpy; therefore, the total specific enthalpy (per unit mass of dry air) of the bulk gas phase of unsaturated air is given by the additivity rule, as follows: H G (T G , Y W ) H B (T G ) Y W H
(g) w
(14)
(T G )
where H B [J (kgda)1] is the specific enthalpy of dry air and H w( g ) [J (kgw)1] is the specific enthalpy of the water vapor contained in the moist air. In Eq. (14), both H B and H w( g ) are calculated at the moist-air dry-bulb temperature T G .
Taking the derivative with respect to z at both sides of Eq. (14) yields dH
G
dH B (T G )
dz
dz
YW
dH
(g) w
(T G )
dz
H w (T G ) (g)
dY W
(15)
dz
Equating the right-hand sides of Eqs. (13) and (15) leads to: dH B (T G ) dz
YW
dH
(g) w
dz
(T G )
H w (T G ) (g)
dY W dz
(NW aM ) GB
H w (T L ) (g)
qC a H
(16)
GB
From the definition for the specific isobaric heat capacity of dry air, C p , B , it follows that TG
H B (T G ) H B (T 0 ) C p , B (T ) dT
(17)
T0
Page 21 of 92
22 where T 0 is an arbitrary (yet constant) reference temperature.
Taking the derivative with respect to z at both sides of Eq.(17) and applying the Leibniz rule of integral calculus, yields dH
B
(T G )
dz
C p , B (T G )
dT G
(18)
dz
From the definitions for the specific heat of vaporization of liquid water, H vap , w , and the specific isobaric heat capacity of water vapor, C p( g, w) , it follows that TG
H w (T G ) H w (T 0 ) H vap , w (T 0 ) C p , w (T ) dT (g)
(L)
(g)
(19)
T0
Taking the derivative with respect to z at both sides of Eq. (19) and applying the Leibniz rule of integral calculus, yields dH
(g) w
(T G )
dz
C p , w (T G ) (g)
dT G
(20)
dz
Substitution of Eqs. (6), (18), and (20) into the left-hand side of Eq. (16), followed by a rearrangement of the resulting expression, leads to the following equation: dT G dz
q C a H ( N W a M )[ H w (T L ) H w (T G )] (g)
(g)
G B [ C p , B (T G ) YW C p , w (T G )] (g)
(21)
From assumption (7), the gas phase is regarded as an ideal solution for the calculation of its heat capacity; therefore, the total specific heat capacity (per unit mass of dry air) of the bulk gas phase of unsaturated air is given by the additivity rule, as follows: C p , G C p , B YW C p , w (g)
(22)
Page 22 of 92
23 From the definition for the specific isobaric heat capacity of water vapor, C p( g, w) , it follows that TL
H w (T L ) H w (T G ) C p , w (T ) dT (g)
(g)
(g)
(23)
TG
From assumption (5), since the heat transfer resistance of the liquid phase is considered to be negligible, the interphase convective heat-transfer flux (from the liquid to the gas phase) may be written as follows: q C h G (T L T G )
(24)
where h G [W m2 K1] is the gas-phase convective heat transfer coefficient.
Substitution of Eqs. (7), (23), and (24) into Eq. (21) leads to the final expression: TL
dT G dz
h G a H (T L T G ) k Y a M (YW , I YW ) C p , w (T ) dT
TG
G B [ C p , B (T G ) Y W C
(g) p ,w
(g)
(25)
(T G )]
Eq. (25) is an ordinary differential equation that describes the change (along the packed height of the cooling tower) of the moist-air dry bulb temperature for the case of unsaturated air as the bulk gas phase. From Eq. (22), the factor in square brackets in the denominator at the right-hand side of Eq. (25) is the total specific heat capacity (per unit mass of dry air) of the bulk gas phase of unsaturated air evaluated at the dry-bulb temperature T G .
2.1.4 Energy balance in the liquid phase
Let T L and H L be the temperature and the specific enthalpy of the liquid phase. If assumptions (3) and (6) for negligible axial dispersion of heat and for an adiabatic operation, respectively, are taken into consideration, the steady-state energy balance in the liquid phase (see Fig. 1) is given by
Page 23 of 92
24 ( LH L )
z z
( LH L )
( N W a M z ) H w (T L ) q C a H z (g)
z
(26)
Dividing both sides of Eq. (26) by z and taking the limit of the rearranged equation as z 0 yields
L
dH L (T L ) dz
H L (T L )
dL dz
( N W a M ) H w (T L ) q C a H (g)
(27)
Rewriting H L (T L ) as H w( L ) (T L ) followed by substitution of Eq. (10) into Eq. (27) yields
L
dH
L
(T L )
dz
( N W a M ) [ H w (T L ) H w (T L )] q C a H (g)
(L)
(28)
From the definition for the specific isobaric heat capacity of liquid water, C p( L, w) , it follows that TL
H L (T L ) H L (T 0 ) C p , w (T ) dT (L)
(29)
T0
Taking the derivative with respect to z at both sides of Eq.(29) and applying the Leibniz rule of integral calculus, yields
dH
L
(T L )
dz
C p , w (T L ) (L)
dT L
(30)
dz
Substitution of Eq. (30) into the left-hand side of Eq. (28), followed by a rearrangement of the resulting expression leads to the following equation:
dT L dz
q C a H ( N W a M ) [ H w (T L ) H w (T L )] (g)
(L)
(L)
(31)
L C p , w (T L )
From the definition for the specific heat of vaporization of liquid water, H vap , w , it follows that H vap , w (T L ) H w (T L ) H w (T L ) (g)
(L)
(32)
Page 24 of 92
25
Substitution of Eqs. (7), (24), and (32) into Eq. (31) leads to the final expression: dT L
h G a H (T L T G ) k Y a M (YW , I YW ) H vap , w (T L ) (L)
dz
(33)
L C p , w (T L )
Eq. (33) is an ordinary differential equation that describes the change (along the packed height of the cooling tower) of the liquid water temperature for the case of unsaturated air as the bulk gas phase.
2.2 Case of supersaturated air as bulk gas phase
The case of supersaturated air (i.e., an equilibrium mixture of saturated air and mist) as the bulk gas phase in a counterflow wet-cooling tower is considered next for the derivation of the governing mass and energy balance equations.
2.2.1 Material balance in the gas phase
For the case of supersaturated air as the bulk gas phase, Eq. (7) for the interphase mass transfer flux takes the following form: N W k Y (Y W , I Y W , S )
(34)
where Y W , S is the humidity mass ratio of saturated air calculated at the dry-bulb temperature T G . Since Y W , I is the humidity mass ratio of saturated air calculated at the interface temperature T L , then Y W , I Y W , S and N W 0 as long as T L T G .
Substitution of Eq. (34) into Eq. (6) leads to the final expression:
Page 25 of 92
26 dY W
k Y a M (Y W , I Y W , S )
dz
(35)
GB
Eq. (35) describes the change of the humidity mass ratio of the supersaturated air stream along the packed height of the cooling tower. Y W , S can be calculated from Eq. (A.7).
2.2.2 Material balance in the liquid phase
For the case of supersaturated air as the bulk gas phase, Eq. (11) takes the following form: dL dz
k Y a M (Y W , I Y W , S )
(36)
In that case, Eq. (36) describes the change of the liquid water mass flowrate (per unit area of cooling tower cross section) along the packed height of the tower.
2.2.3 Energy balance in the gas phase
For the case of supersaturated air, the total specific enthalpy (per unit mass of dry air) of the gas phase is given by the following expression: H G ( T G , Y W ) H B (T G ) Y W , S H w (T G ) ( Y W Y W , S ) H w (T G ) (g)
(L)
(37)
Taking the derivative with respect to z at both sides of Eq. (37) yields dH dz
G
dH B (T G ) dz
YW , S
dH
(g) w
dz
(T G )
H
(g) w
(T G )
dY W , S dz
(Y W Y W , S )
dH
(L) w
(T G )
dz
dY W , S dY (L) H w (T G ) W dz dz
(38) From the definition for the specific isobaric heat capacity of liquid water, C p( L, w) , it follows that
Page 26 of 92
27 TG
H w (T G ) H w (T 0 ) C p , w (T ) dT (L)
(L)
(L)
(39)
T0
Taking the derivative with respect to z at both sides of Eq.(39) and applying the Leibniz rule of integral calculus, yields dH
(L) w
(T G )
dz
C p , w (T G ) (L)
dT G
(40)
dz
Substitution of Eqs. (18), (20), and (40) into Eq. (38) yields dH
G
dz
[ C p , B (T G ) YW , S C p , w (T G ) (YW YW , S ) C p , w (T G )] (g)
(L)
dT G dz
H w (T G ) (L)
dY W dz
[ H w (T G ) H w (T G )] (g)
(L)
dY W , S dz
(41) From the definition for the specific heat of vaporization of liquid water, H vap , w , it follows that H vap , w (T G ) H w (T G ) H w (T G ) (g)
(L)
(42)
Substituting Eq. (42) into Eq. (41) and equating the right-hand sides of Eqs. (13) and (41) yields
C p , G (T G )
dT G dz
H w (T G ) (L)
dY W dz
H vap , w (T G )
dY W , S
(NW aM )
dz
GB
H w (T L ) (g)
qC a H
(43)
GB
where C p , G C p , B YW , S C p , w (YW YW , S ) C p , w (g)
(L)
(44)
is the total specific heat capacity (per unit mass of dry air) of the gas phase of supersaturated air.
Substitution of Eq. (6) into Eq. (43) followed by a rearrangement of the resulting expression, leads to the following equation:
Page 27 of 92
28
q C a H G B [ H w (T L ) H w (T G )] (g)
dT G
(L)
dY W
dz G B C p , G (T G )
dz
G B H vap , w (T G )
dY W , S dz
(45)
Substitution of Eqs.(23) and (42) into the identity H
(g) w
(T L ) H
(L) w
(T G ) [ H
(g) w
(T L ) H
(g) w
(T G )] [ H w (T G ) H (g)
(L) w
(T G )]
(46)
leads to the following expression: TL
H w (T L ) H w (T G ) C p , w (T ) dT H vap , w (T G ) (g)
(L)
(g)
(47)
TG
Substitution of Eqs. (24), (35), and (47) into Eq. (45) yields TL
dT G
h G a H (T L T G ) k Y a M (YW , I YW , S ) [ C p , w (T ) dT H vap , w (T G )] G B H vap , w (T G ) (g)
TG
dz
dY W , S dz
G B C p , G (T G )
(48) From the chain rule of differential calculus, dY W , S dz
dY W , S dT G dT G
(49)
dz
Substitution of Eqs. (44) and (49) into Eq. (48) followed by a rearrangement of the resulting expression leads to the final equation: TL
dT G dz
h G a H (T L T G ) k Y a M (Y W , I Y W , S ) [ C p , w (T ) dT H vap , w (T G )]
(g)
TG
dY W , S (g) (L) G B C p , B (T G ) Y W , S C p , w (T G ) (Y W Y W , S ) C p , w (T G ) H vap , w (T G ) dT G
(50)
Eq. (50) describes the change (along the packed height of the cooling tower) of the moist-air dry bulb temperature for the case of supersaturated air as the bulk gas phase. From Eq. (44), the sum
Page 28 of 92
29 of the first three terms in the factor in square brackets in the denominator at the right-hand side of Eq. (50) is the total specific heat capacity (per unit mass of dry air) of the bulk gas phase of supersaturated air calculated at the dry-bulb temperature T G . The derivative dY W , S / dT G is calculated from Eq. (A.8).
2.2.4 Energy balance in the liquid phase
Eq. (31) is also valid for the case of supersaturated air as the bulk gas phase. Substitution of Eqs. (24), (32), and (34) into Eq. (31) yields the following equation dT L
h G a H (T L T G ) k Y a M (YW , I YW , S ) H vap , w (T L )
dz
(L)
(51)
L C p , w (T L )
Eq. (51) describes the change (along the packed height of the cooling tower) of the liquid water temperature for the case of supersaturated air as the bulk gas phase.
3. Derivation of Merkel’s equation from the CDAWC model It is instructive to show how the Merkel equation can be obtained by simplification of the CDAWC model presented in Section 2. The following six further assumptions or approximations are necessary to obtain Eq. (1), which is the Merkel equation, from Eq. (33): (8) The Lewis relation h G k Y C p ,G is valid for the air/water system. (9) The interfacial area for heat transfer a H is identical to the interfacial area for mass transfer a M (i.e., the cooling tower packing is completely wetted by the liquid). (10) The specific heat capacities of dry air ( C p , B ), water vapor ( C p( g, w) ), and liquid water ( C p( L, w) ) are independent of temperature. (11) The reference temperature T 0 is equal to the liquid phase temperature T L .
Page 29 of 92
30 (12) The liquid water mass flowrate L remains constant along the cooling tower packed height (i.e, water loss by evaporation is neglected). (13) The volumetric mass transfer coefficient k Y a M is constant along the tower packed height. From assumptions (8) and (9), Eq. (4) follows. Substitution of Eq. (4) into Eq. (33) followed by a rearrangement of the resulting expression leads to the following equation:
(L)
L C p ,w
dT L dz
k Y a M [ C p , G (T L T G ) (YW , I Y W ) H vap , w (T L )]
(52)
Substitution of Eqs. (17) and (19) into Eq. (14) yields TG
TG
H G (T G , YW ) H B (T 0 ) C p , B (T ) dT YW [ H w (T 0 ) H vap , w (T 0 ) C p , w (T ) dT ] (L)
T0
(g)
(53)
T0
From assumptions (4) and (5), given in Section 2 for the CDAWC model, the gas-phase side of the interface consists of saturated moist air at the liquid phase temperature; therefore, the following expression may be written from Eq. (53): TL
TL
H G (T L ) H B (T 0 ) C p , B (T ) dT YW , I [ H w (T 0 ) H vap , w (T 0 ) C p , w (T ) dT ] sat
(L)
T0
(g)
(54)
T0
Subtracting Eq. (53) from Eq. (54) and considering that H w( L ) (T 0 ) 0 at the reference temperature T 0 , it follows that TL
T0
TG
TG
TL
H G (T L ) H G (T G , YW ) C p , B (T ) dT (YW , I YW ) H vap , w (T 0 ) YW C p , w (T ) dT YW , I C p , w (T ) dT sat
(g)
(g)
T0
(55) From assumption (10), it follows that
Page 30 of 92
31 TL
C p , B (T ) dT C p , B (T L T G )
(56)
TG
T0
C p , w (T ) dT C p , w (T 0 T G ) (g)
(g)
(57)
TG
TL
C p , w (T ) dT C p , w (T L T 0 ) (g)
(g)
(58)
T0
Substitution of Eqs. (56)(58) into Eq. (55) leads to the following expression: H G (T L ) H G (T G , YW ) C p , B (T L T G ) (YW , I YW ) H vap , w (T 0 ) YW C p , w (T 0 T G ) YW , I C p , w (T L T 0 ) sat
(g)
(g)
(59)
From assumption (11), Eq. (59) takes the following form: H G (T L ) H G (T G , YW ) ( C p , B YW C p , w )( T L T G ) (YW , I YW ) H vap , w (T L ) sat
(g)
(60)
Substitution of Eq. (22) into Eq. (60) yields H G (T L ) H G (T G , YW ) C p , G (T L T G ) (YW , I YW ) H vap , w (T L ) sat
(61)
Substitution of Eq. (61) into Eq. (52) yields
(L)
L C p ,w
dT L dz
k Y a M [ H G (T L ) H G (T G , Y W )] sat
(62)
Eq. (62) can be integrated from the bottom of the tower ( z 0 , T L T L ,1 ) to the top of the tower ( z Z T , T L T L , 2 ). If assumptions (12) and (13) are taken into consideration, the integration yields the final expression: kY aM Z T L
TL , 2
T L ,1
(L)
C p , w (T L ) dT L [ H G (T L ) H G (T G , Y W )] sat
(63)
Page 31 of 92
32
which is the Merkel equation, Eq. (1).
In the derivation of the Merkel equation given in classical papers and textbooks, besides assumptions (8)-(10), (12), and (13), some other approximations are usually made. For instance, in the derivation given in the textbook by Treybal (1980) two further approximations are made: sensible heat terms like C p( g, w) (T G T 0 ) and C p( L, w) (T L T 0 ) are ignored in comparison with the latent heat term H vap , w (T 0 ) , and the moist-air specific heat capacity C p ,G is assumed to be independent of the humidity mass ratio YW . It is noteworthy that in our derivation of the Merkel equation, none of these drastic approximations had to be made. Therefore, a conclusion to be drawn is that such drastic approximations are not really necessary for the derivation of the Merkel equation. However, instead of those drastic approximations, assumption (11), that the reference temperature T 0 is equal to the liquid water temperature T L , had to be made. This assumption deserves some discussion in order to judge whether it is consistent or not. It happens that although the reference temperature T 0 is arbitrary, it should be set to a constant value. However, the liquid water temperature T L is not a constant quantity because it varies continuously along the packed height of the cooling tower. Therefore, the choice of setting T 0 equal to T L is not consistent with the definition of a constant reference temperature. Moreover, such a choice is also inconsistent with the form of the energy balance for the gas phase, because the derivation of Eqs. (18), (20), and (25) holds only if T 0 is a truly constant value, not a varying value like T L .
In consequence, the approximate nature of the Merkel equation is due not only to the commonly discussed approximations involved in assumptions (8) and (12), but also to the formal inconsistency introduced by assumption (11).
As mentioned in Section 1.1, to compute the value of the integral at the right-hand side of Eq. (63), an accurate numerical technique such as the Chebyshev’s equal weight four-point
Page 32 of 92
33 integration formula (Abramowitz and Stegun, 1972) can be employed, as explained in detail by Mohiuddin and Kant (1996).
In the Merkel method of cooling tower analysis, the following further assumption is made in order to specify the thermodynamic state of the outlet air:
(14) The air is saturated at the outlet of the cooling tower packing.
However, this assumption was not required for the derivation of Eq. (63) from Eq. (33).
4. Critical review of previous models 4.1 Osterle’s model
Although the model by Osterle (1991) revised the assumption of constant water flowrate made in the derivation of Merkel’s model, it still involves the following mathematical approximations: the approximate Lewis relation, Eq. (4), with a H a M , is taken as valid, the specific heat capacities of liquid water, dry air, and water vapor are regarded as constant, the moist-air specific heat capacity is regarded as independent of the humidity mass ratio, and the contribution of a humidity-mass-ratio driving-force term to the total heat transfer rate [see Eqs. (12) and (13) in Osterle's paper] is neglected. Osterle’s model consists of three ODEs that describe the changes of the moist-air specific enthalpy H G and humidity mass ratio Y W , and the Merkel number Me k Y a M Z T / L with
Page 33 of 92
34 respect to the liquid water temperature T L , which is chosen as the integration variable. In Appendix B (Supplementary Data), we show in detail how these equations can be rewritten with the cooling tower packed height z as the integration variable, leading to the following expressions for the energy balances [see Eqs. (B.22) and (B.33)]:
dT G dz
dT L
h G a (T L T G )
(64)
G B ( C p , B YW C p , w ) (g)
h G a (T L T G ) k Y a (YW , I YW ) { H vap , w (T L ) [ C p , w T L C p , w T 0 ]} (g)
(L)
(L)
dz
(65)
L C p ,w
A comparison of Eq.(64) with Eq.(25) indicates that the second term in the numerator at the right-hand side of Eq. (25) is missing in Eq. (64). A comparison of Eq.(65) with Eq.(33) indicates that besides involving the usual assumption that a H a M a (i.e., the packing is completely wetted by the liquid), Eq. (65) has an extraneous additional term [ C p( g, w) T L C p( L, w) T 0 ] being subtracted from the heat of vaporization term. Therefore, the two energy balances of Osterle’s model, given in their final form by Eqs. (64) and (65), are incorrect, as a consequence of the approximations involved in the derivation made by Osterle (1991), the most serious one being
the
neglectance
of
the
contribution
of
the
term
[ H w (T L ) H vap , w (T 0 ) C p , w T 0 ] (YW , I YW ) to the total heat transfer rate [see Eqs.(12) and (g)
(g)
(13) in Osterle's paper]. 4.2 Sutherland's and Khan et al.’s models
In the performance-analysis model by Sutherland (1983), instead of the cooling tower packed height z, the moist-air humidity mass ratio Y W was chosen as the integration variable of the two ODEs constituting that model (one equation for the change of the specific enthalpy of moist air and the other equation for the change of the liquid water temperature). In the performanceanalysis model by Khan and Zubair (2001) and Khan et al. (2003), one ODE is obtained to
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35 describe the change of the moist-air specific enthalpy with respect to the moist-air humidity mass ratio.
For the derivation and implementation of their models, Sutherland (1983), Khan and Zubair (2001), and Khan et al. (2003) made several approximations such as those of regarding the specific heat capacities of liquid water, dry air, and water vapor as constant (also setting the reference temperature T 0 to the constant value of 0°C), and considering the Lewis factor as a constant. In Appendix C1 (Supplementary Data), we show in detail how the ODEs of Sutherland’s model can be rewritten with the cooling tower packed height z as the integration variable, leading to the following expressions for the energy balances [see Eqs. (C1.23) and (C1.30)]:
dT G
(g)
dz
dT L dz
h G a (T L T G ) k Y a (Y W , I Y W ) C p , w (T L T G )
G B ( C p , B YW C p , w ) (g)
h G a (T L T G ) k Y a (YW , I YW ) H vap , w (T L ) (L)
(66)
(67)
L C p ,w
A comparison of Eq.(66) with Eq.(25) for the CDAWC model indicates that due to the approximation made in Sutherland’s model (Sutherland, 1983) that the specific heat capacities of water vapor, C p( g, w) , and dry air, C p , B , are independent of temperature (see footnote 1 on p. 578 of Sutherland’s paper for the constant values given to C p , B and C p( g, w) ), the product C p( g, w) (T L T G ) appears in the numerator at the right-hand side of Eq.(66) instead of the integral of C p( g, w) (T ) from T G to T L in Eq.(25), and also no value of temperature is specified for the evaluation of C p , B and (g)
C p , w in the denominator at the right-hand side of Eq.(66), in contrast to the T G value that is
specified in Eq.(25). Due to these approximations and the assumption made in Sutherland’s model that a H a M a (i.e., the packing is completely wetted by the liquid), it turns out that Eq.(66) is only an approximate version of Eq.(25) for the CDAWC model.
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36
A comparison of Eq.(67) with Eq.(33) for the CDAWC model indicates that due to the approximation made in Sutherland’s model (Sutherland, 1983) that the specific heat capacity of liquid water, C p( L, w) , is independent of temperature (see footnote 1 on p. 578 of Sutherland’s paper for the constant value given to C p( L, w) ), no value of temperature is specified for the evaluation of (L)
C p , w in the denominator at the right-hand side of Eq.(67), in contrast to the T L value that is
specified in Eq.(33). Due to this approximation and the assumption made in Sutherland’s model that a H a M a (i.e., the packing is completely wetted by the liquid), it turns out that Eq.(67) is only an approximate version of Eq.(33) for the CDAWC model. In Appendix C2 (Supplementary Data), we show in detail how the ODE of Khan et al.’s model can be rewritten with the cooling tower packed height z as the integration variable, leading to the following expression for the energy balance of the gas phase [see Eq. (C2.13]: dT G dz
h G a (T L T G ) k Y a (Y W , I Y W ) C p , w (T L T 0 ) (g)
G B ( C p , B YW C p , w ) (g)
(68)
Due to the approximation made in Khan et al.’s model that the specific heat capacities of dry air, C p ,B
, and water vapor, C p( g, w) , are independent of temperature, no value of temperature is
specified for the evaluation of C p , B and C p( g, w) in the denominator at the right-hand side of Eq.(68), in contrast to the T G value that is specified in Eq.(25) of the CDAWC model. A comparison of Eq.(68) with Eq.(66) for Sutherland’s model indicates that instead of the difference T L T G in the last term of the numerator at the right-hand side of Eq.(66), the difference T L T 0 appears in the numerator at the right-hand side of Eq.(68); therefore, a comparison of Eq.(68) with Eq.(25) for the CDAWC model indicates that Eq.(68) is incorrect. This is a direct consequence of the additional approximation made in Khan et al.’s model (relative to Sutherland’s model) that the moist-air specific heat capacity is independent of the humidity mass ratio.
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37 4.3 Poppe’s model
In the Poppe model (Poppe, 1973; Poppe and Rögener, 2006, Kloppers, 2003; Kloppers and Kröger, 2005a), instead of the cooling tower packed height z, the liquid water temperature T L was chosen as the integration variable of the three ODEs constituting that model (two equations for the changes of the moist-air specific enthalpy H G and humidity mass ratio Y W , and the other equation for the change of the Merkel number Me ). A detailed derivation of Poppe’s model equations was given by Kloppers (2003) and Kloppers and Kröger (2005a) for the two cases of unsaturated and supersaturated air as the bulk gas phase. In Appendix D (Supplementary Data), we show in detail how the ODEs of Poppe’s model can be rewritten with the cooling tower packed height z as the integration variable, and in spite of the intrinsic complexity of Poppe’s model equations, how it is possible, after some lengthy algebraic simplification, to write those equations in the much simpler form obtained for the equations of the CDAWC model.
For the case of a bulk gas phase of unsaturated air, the lengthy algebraic simplification process shown in Appendix D.1 (Supplementary Data) leads to the following expressions for the energy balances [see Eqs. (D1.31) and (D1.46)]: dT G
(g)
dz
dT L dz
h G a (T L T G ) k Y a (Y W , I Y W ) C p , w (T L T G )
G B ( C p , B YW C p , w ) (g)
h G a (T L T G ) k Y a (YW , I YW ) H vap , w (T L ) (L)
(69)
(70)
L C p ,w
Eqs.(69) and (70) are identical to Eqs.(66) and (67) for Sutherland’s model (see Section 4.2).
A comparison of Eq.(69) with Eq.(25) for the CDAWC model indicates that due to the approximation made in Poppe’s model (Kloppers, 2003; Kloppers and Kröger, 2005a) that the
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38 specific heat capacities of water vapor, C p( g, w) , and dry air, C p , B , are independent of temperature [see the text below Eq.(11) in Kloppers and Kröger’s paper], the product C p( g, w) (T L T G ) appears in the numerator at the right-hand side of Eq.(69) instead of the integral of C p( g, w) (T ) from T G to (g)
T L in Eq.(25), and also no value of temperature is specified for the evaluation of C p , B and C p , w
in the denominator at the right-hand side of Eq.(69), in contrast to the T G value that is specified in Eq.(25). Due to these approximations and the assumption made in Poppe’s model that a H a M a (i.e., the packing is completely wetted by the liquid), it turns out that Eq.(69) is
only an approximate version of Eq.(25) for the CDAWC model.
A comparison of Eq.(70) with Eq.(33) for the CDAWC model indicates that due to the approximation made in Poppe’s model (Kloppers, 2003; Kloppers and Kröger, 2005a) that the specific heat capacity of liquid water, C p( L, w) , is independent of temperature [see the text below Eq.(11) in Kloppers and Kröger’s paper], no value of temperature is specified for the evaluation of C p( L, w) in the denominator at the right-hand side of Eq.(70), in contrast to the T L value that is specified in Eq.(33). Due to this approximation and the assumption made in Poppe’s model that a H a M a (i.e., the packing is completely wetted by the liquid), it turns out that Eq.(70) is
only an approximate version of Eq.(33) for the CDAWC model.
For the case of a bulk gas phase of supersaturated air, the lengthy algebraic simplification process shown in Appendix D.2 (Supplementary Data) leads to the following expressions for the energy balances [see Eqs. (D2.39) and (D2.54)]:
dT G dz
h G a (T L T G ) k Y a (Y W , I Y W , S ) [ C p , w (T L T G ) H vap , w (T G )] (g)
dY W , S (g) (L) G B C p , B YW , S C p , w (YW YW , S ) C p , w H vap , w (T G ) dT G
dT L dz
h G a (T L T G ) k Y a (YW , I YW , S ) H vap , w (T L ) (L)
(71)
(72)
L C p ,w
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39
A comparison of Eq.(71) with Eq.(50) for the CDAWC model indicates that due to the approximation made in Poppe’s model (Kloppers, 2003; Kloppers and Kröger, 2005a) that the specific heat capacities of water vapor, C p( g, w) , dry air, C p , B , and liquid water, C p( L, w) , are independent of temperature [see the text below Eq.(11) in Kloppers and Kröger’s paper], the product C p( g, w) (T L T G ) appears in the numerator at the right-hand side of Eq.(71) instead of the integral of C p( g, w) (T ) from T G to T L in Eq.(50), and also no value of temperature is specified for the evaluation of C p , B , C p( g, w) , and C p( L, w) in the denominator at the right-hand side of Eq.(71), in contrast to the T G value that is specified in Eq.(50). Due to these approximations and the assumption made in Poppe’s model that a H a M a (i.e., the packing is completely wetted by the liquid), it turns out that Eq.(71) is only an approximate version of Eq.(50) for the CDAWC model.
A comparison of Eq.(72) with Eq.(51) for the CDAWC model indicates that due to the approximation made in Poppe’s model (Kloppers, 2003; Kloppers and Kröger, 2005a) that the specific heat capacity of liquid water, C p( L, w) , is independent of temperature [see the text below Eq.(11) in Kloppers and Kröger’s paper], no value of temperature is specified for the evaluation of C p( L, w) in the denominator at the right-hand side of Eq.(72), in contrast to the T L value that is specified in Eq.(51). Due to this approximation and the assumption made in Poppe’s model that a H a M a (i.e., the packing is completely wetted by the liquid), it turns out that Eq.(72) is
only an approximate version of Eq.(51) for the CDAWC model.
It is noteworthy that in spite of the approximation made by Kloppers and Kröger (2005a) for the derivation of Poppe’s model that all the specific heat capacities are regarded as constant (see the text below Eq.(11) in that paper), for the numerical solution of the ODEs, those authors used temperature-dependent polynomial correlations [see Eqs.(A.2), (A.4), and (A.7) in Kloppers and Kröger’s paper) to evaluate the heat capacities at the arithmetic mean of the reference temperature T 0 0 °C and the temperature at which the enthalpies that are required by the model
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40 have to be calculated. Use of such an arithmetic mean temperature to compute the actual integral average of a heat capacity would make sense only if the heat capacity were correlated linearly with the temperature. 4.4 Klimanek and Bialecki’s model
In the model formulated by Klimanek and Bialecki (2009), the cooling tower packed height z was chosen as the integration variable of the four ODEs constituting that model. As for the CDAWC model, the equations describe the changes of the moist-air humidity mass ratio Y W , the liquid water mass flowrate (per unit area of cooling tower cross section) L, the moist-air dry-bulb temperature T G , and the liquid water temperature T L . Klimanek and Bialecki formulated their model for the two cases of unsaturated and supersaturated air as the bulk gas phase.
The text below Eq.(3) of the paper by Klimanek and Bialecki (2009) explains the notation used by those authors to write their model equations. They used superscripts attached to the symbols of the specific heat capacities to indicate whether a heat capacity is evaluated at the dry-bulb air temperature T G or at the liquid water temperature T L . A verbatim interpretation of that notation leads, in the case of unsaturated air as the bulk gas phase, to the form given in Appendix E1 (Supplementary Data) for Eqs.(E1.3) and (E1.4), and in the case of supersaturated air to the form given in Appendix E2 (Supplementary Data) for Eqs.(E2.3) and (E2.4), in which the heat capacities are evaluated at the specified values of temperature.
However, more details on the derivation of that model and the notation used for the specific heat capacites were given in a later paper by Klimanek (2013), in which, in the text below Eq.(6), it was stated that the specific heat capacities that appear in the equations of the model are actually average values obtained by integrating the temperature-dependent specific heat capacities from the reference temperature T 0 0 °C to the temperature indicated by the superscript attached to the symbol used for the heat capacities and dividing the calculated integral by the latter temperature, so that Eqs.(E1.5)(E1.7) may be written.
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41 As explained in detail in Appendix E (Supplementary Data), if such re-interpretation of the notation originally used in the paper by Klimanek and Bialecki (2009) is considered and Eqs.(E1.5)(E1.7) are applied only to the terms that are made by the product of the heat capacity and the temperature, it is possible to show, after an algebraic simplification process, that the ODEs for the energy balances in Klimanek and Bialecki’s model can be rewritten in the much simpler form obtained for the equations of the CDAWC model.
If Eqs.(E1.5)(E1.7) were also applied to every other heat-capacity term present in Klimanek and Bialecki’s model equations, then the expressions resulting from the simplification process would be in disagreement with the corresponding equations for the CDAWC model and, consequently, they would be incorrect. This is because in the CDAWC model equations, as follows from their detailed derivation, the heat capacities that turn out to be evaluated at specified temperature values are indeed point-value heat capacities and should not be confused with average heat capacities.
For the case of a bulk gas phase of unsaturated air, the algebraic simplification process shown in Appendix E.1 (Supplementary Data) starts from the more complicated form of Eqs.(E1.3) and (E1.4) and leads to the following much simpler form of the energy balances [see Eqs. (E1.14) and (E1.20)]: TL
dT G
h G a (T L T G ) k Y a (YW , I YW ) C p , w (T ) dT
dz
dT L dz
(g)
TG
G B [ C p , B (T G ) Y W C
(g) p ,w
(73)
(T G )]
h G a (T L T G ) k Y a (YW , I YW ) H vap , w (T L ) (L)
(74)
L C p , w (T L )
Since in Klimanek and Bialecki’s model the implicit assumption is made that a H a M a , then Eqs.(73) and (74) are strictly valid only if the packing is completely wetted by the liquid. In that case, it turns out that Eqs.(73) and (74) are in agreement with Eqs.(25) and (33), respectively, of the CDAWC model for the case of a bulk gas phase of unsaturated air.
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42 For the case of a bulk gas phase of supersaturated air, the algebraic simplification process shown in Appendix E.2 (Supplementary Data) starts from the more complicated form of Eqs.(E2.3) and (E2.4) and leads to the following much simpler form of the energy balances [see Eqs. (E2.19) and (E2.25)]: TL
dT G dz
h G a (T L T G ) k Y a (Y W , I Y W , S ) [ C p , w (T ) dT H vap , w (T G )]
(g)
TG
dY W , S (g) (L) G B C p , B (T G ) Y W , S C p , w (T G ) (Y W Y W , S ) C p , w (T G ) H vap , w (T G ) dT G
dT L
h G a (T L T G ) k Y a (YW , I YW , S ) H vap , w (T L )
(76)
(L)
dz
(75)
L C p , w (T L )
Since in Klimanek and Bialecki’s model the implicit assumption is made that a H a M a , then Eqs.(75) and (76) are strictly valid only if the packing is completely wetted by the liquid. In that case, it turns out that Eqs.(75) and (76) are in agreement with Eqs.(50) and (51) for the CDAWC model for the case of a bulk gas phase of supersaturated air. 4.5 Milosavljevic and Heikkilä’s model
In the model formulated by Milosavljevic and Heikkilä (2001), four balance equations were written and combined to yield a set of ODEs. The two material balances are given by Eqs. (8) and (11) if the approximate Lewis relation [Eq. (4) with a H a M ] is substituted into Eqs. (1) and (2) of Milosavljevic and Heikkilä's paper.
If the notation of the present work is used, then the two energy balances [see Eqs. (7) and (8) in Milosavljevic and Heikkilä's paper] are given by the following expressions:
h G a (T L T G ) ( N W a ) H vap , w (T L ) [ H vap , w (T 0 ) C p , w T G ] (g)
dT G dz
G B ( C p , B YW C
(g) p ,w
dL dz
(77)
)
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43
h G a (T L T G ) ( N W a ) H vap , w (T L ) C p , w T L (L)
dT L
dz
LC
dL dz
(L) p ,w
(78)
where the reference temperature T 0 in Eq. (77) was set equal to 0°C.
An equation similar to Eq. (32) can be written at the reference temperature T 0 , as follows: H vap , w (T 0 ) H w (T 0 ) H w (T 0 ) (g)
(L)
(79)
From Eqs. (32) and (79), and the defining expressions for H w( g ) and H w( L ) in terms of C p( g, w) and (L)
C p , w , respectively, it follows that
H vap , w (T L ) [ H vap , w (T 0 ) C p , w T G ] C p , w (T L T G ) C p , w T L (g)
(g)
(L)
(80)
Substitution of Eqs. (7), (10), (22), and (80) into Eq. (77) leads to the expression dT G
h G a (T L T G ) k Y a (Y W , I Y W ) [ C p , w (T L T G ) C p , w T L ] (g)
dz
(L)
G B C p ,G
(81)
A comparison of Eq. (81) with Eq. (25) for the CDAWC model indicates that besides involving the usual assumption that a H a M a (i.e., the packing is completely wetted by the liquid) and the approximate expression C p( g, w) (T L T G ) for the integral in the numerator at the right-hand side of Eq. (25), Eq. (81) has an additional extraneous term C p( L, w) T L being subtracted from the approximate form of the integral.
The overall negative sign at the right-hand side of Eq. (78) is wrong. Substitution of Eq. (10) into the revised form of Eq. (78), followed by use of Eq. (7), leads to the expression dT L dz
h G a (T L T G ) k Y a (YW , I YW ) [ H vap , w (T L ) C p , w T L ] (L)
(L)
(82)
L C p ,w
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44
A comparison of Eq. (82) with Eq. (33) for the CDAWC model indicates that besides involving the usual assumption that a H a M a (i.e., the packing is completely wetted by the liquid), Eq. (82) has an extraneous additional term C p( L, w) T L being added to the heat of vaporization term. Thus, it follows that due to the wrong overall negative sign at the right-hand side of Eq.(78) and the presence of extraneous terms in Eqs. (81) and (82), the two energy balances given by Eqs. (77) and (78) are incorrect. 4.6 Papaefthimiou et al.’s model
In the model formulated by Papaefthimiou et al. (2006), four balance equations were written and combined to yield a set of ODEs. The two material balances are given by Eq. (8) and the equation dL / dz G B dY W / dz resulting from the substitution of Eq. (10) into Eq.(6). If the notation of the present work is used, the energy balance for the gas phase is given by a simplified form of Eq. (25) in which the approximate expression C p( g, w) (T L T G ) for the integral in the numerator at the right-hand side is used, and the assumption that a H a M a (i.e., the packing is completely wetted by the liquid) is made.
In the notation of the present work, the energy balance for the liquid phase in Papaefthimiou et al.’s model is given by the following expression: dT L dz
GB LC
(L) p ,w
dT G dY W (L) (g) C p , G dz ( C p , w T L C p , w T G H vap , w ) dz
(83)
The overall negative sign at the right-hand side of Eq. (83) is wrong. Substitution of Eqs. (8) and (25) into the revised form of Eq. (83), followed by use of both the assumption that a H a M a and the approximate expression C p( g, w) (T L T G ) for the integral in the numerator at the right-hand side of Eq. (25), leads to the expression
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45 dT L
h G a (T L T G ) k Y a (YW , I YW ) [ H vap , w ( C p , w C p , w ) T L ] (g)
(L)
(L)
dz
(84)
L C p ,w
A comparison of Eq. (84) with Eq. (33) for the CDAWC model indicates that Eq. (84) has an extraneous additional term ( C p( g, w) C p( L, w) ) T L being added to the heat of vaporization term. Thus, it follows that due to the wrong overall negative sign at the right-hand side of Eq.(83) and the presence of an extraneous term in Eq. (84), the liquid phase energy balance given by Eq. (83) is incorrect. 4.7 Elsarrag’s model
In the model formulated by Elsarrag (2006), the material balance in the gas phase is given by Eq. (8), and no material balance was written for the liquid phase. If the notation of the present work is used, the energy balance for the gas phase [see Eq. (12) in Elsarrag's paper] is given by the following expression: dT G dz
h G a (T I T G )
(85)
G B C p ,G
where T I is the temperature at the gas-liquid interface. Elsarrag (2006) cited the textbook by Treybal (1980) as the source of Eq. (85). However, a calculational procedure to obtain T I was not given by Elsarrag. If assumption (5), given in Section 2, was implicitly made by Elsarrag and T I was considered to be equal to T L , then Eq. (85) is incomplete, since the second term in the
numerator at the right-hand side of Eq. (25) for the CDAWC model is missing in Eq. (85).
The energy balance for the liquid phase [see Eq. (14) in Elsarrag's paper] is given by the following expression: dT L dz
GB LC
(L) p ,w
dT G dY W (g) C p , G dz ( C p , w T G H vap , w ) dz
(86)
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46 Substitution of Eqs. (8) and (85) into Eq. (86) followed by the consideration that T I T L yields
dT L
h G a (T L T G ) k Y a (YW , I YW ) ( H vap , w C p , w T G ) (g)
dz
(L)
(87)
L C p ,w
A comparison of Eq. (87) with Eq. (33) for the CDAWC model indicates that besides involving the usual assumption that a H a M a (i.e., the packing is completely wetted by the liquid), Eq. (87) has an extraneous additional term C p( g, w) T G being added to the heat of vaporization term. Thus, if T I was indeed considered to be equal to T L by Elsarrag (2009), it would follow that due to the absence of one term in Eq. (85) and the presence of an extraneous term in Eq. (87), the energy balances given by Eqs. (85) and (86) are incorrect.
5. Simulation of the CDAWC model The continuous differential air-water contactor (CDAWC) model, presented in detail in Section 2, consists of four coupled first-order ordinary differential equations (ODEs): Eqs. (8) or (35) for the moist-air humidity mass ratio Y W , Eqs. (11) or (36) for the liquid water mass flowrate (per unit area of cooling tower cross section) L, Eqs. (25) or (50) for the moist-air dry bulb temperature T G , and Eq. (33) or (51) for the liquid water temperature T L . This set of equations must be integrated along the vertical distance z from the bottom of the cooling tower (i.e., at z 0 ) to the top (i.e., at z Z T , where Z T is the tower packed height) with the help of all the
subsidiary expressions and correlations given in Appendix A (Supplementary Data), in order for all thermophysical, psychrometric and transport properties involved in the model to be updated at each integration step.
However, since the liquid water and the air enter the cooling tower at opposite ends (see Fig. 2), simulation of the cooling tower is a two-point boundary value problem: boundary conditions T G ,1 and Y W ,1 (or T wb ,1 ) for atmospheric air are known at the bottom of the tower (starting integration
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47 point) and boundary conditions ( T L , 2 , L 2 ) for liquid water are known at the top of the tower (final integration point).
The shooting method (Hanna and Sandall, 1995) can be used to solve this problem, in the following general way. Values are assumed for the unknown conditions ( T L ,1 , L 1 ) of liquid water at z 0 , so that from the specified values T G ,1 , Y W ,1 (or T wb ,1 ), T L ,1 and L 1 for the four dependent variables, the numerical integration of the set of ODEs can be started at z 0 and ) carried on up to z Z T . The calculated values for the inlet water temperature T L(,calc and mass 2
flowrate L (2calc ) are then compared to the known values T L , 2 and L 2 , respectively, and the ) ( calc ) T L , 2 and L 2 corresponding discrepancies T L(,calc L 2 between the calculated and the known 2
values are used to make corrections to the assumed values of T L ,1 and L 1 in an iterative way (using the secant method), until the two discrepancies are reduced to values sufficiently close to zero. The final values of T L ,1 and L 1 together with the calculated values of T G , 2 and Y W , 2 (or T wb , 2 )
are the output data from the simulation.
5.1 Validation of the computer program In this work, the set of ODEs was numerically integrated by means of the fourth-order RungeKutta method, using the technique of step doubling to estimate the local error of each integrated variable (Press et al., 1989a). The CDAWC model and its solution algorithm were translated into a computer program coded in the Fortran 95 programming language. For the validation of our computer program, we considered some numerical examples previously presented in the literature. In the paper by Klimanek and Bialecki (2009), two examples were given, one for the case of a bulk gas phase of unsaturated air and the other for the case of supersaturated air, the latter of the examples corresponding to a case previously studied by Kloppers (2003), which was used for validation purposes by Klimanek and Bialecki. In the paper by Klimanek (2013), four examples were given, two for the case of unsaturated air and the other two for the case of supersaturated air, one of the latter corresponding again to a case previously studied by Kloppers (2003). Validation results for the CDAWC model are shown here only for the two examples
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48 given by Klimanek and Bialecki (2009) and for Case 1 of the paper by Klimanek (2013). The required input data for the three examples are listed in Table 2 and a comparison of the corresponding output data is shown in Tables 35 and Figs. 311.
Figs. 38 and Tables 3 and 4 show an excellent agreement between our simulation results for the temperature, humidity mass ratio, and water mass flowrate profiles along the cooling-tower packed height and those results previously reported by Klimanek and Bialecki (2009) and Klimanek (2013) for two cases of unsaturated air. The agreement of these results is in accordance with the mathematical equivalence (shown in Appendix E.1) between the CDAWC model and Klimanek and Bialecki’s model for the case of a bulk gas phase of unsaturated air.
Figs. 911 show a very good agreement between our simulation results for the liquid water temperature T L , moist-air humidity mass ratio Y W , and water mass flowrate and the results reported by Klimanek and Bialecki (2009) for the case of supersaturated air previously reported by Kloppers (2003). The CDAWC model predicts the onset of supersaturation at a packed height of 1.946 m, which is close to 2 m, in agreement with what had been stated by Klimanek and Bialecki. The agreement of these results is in accordance with the mathematical equivalence (shown in Appendix E.2) between the CDAWC model and Klimanek and Bialecki’s model for the case of a bulk gas phase of supersaturated air. In order to obtain a very accurate numerical solution (with a maximum local integration error of about 4 10 5 ) of the CDAWC model equations beyond the onset of supersaturation, Eqs.(35), (36), (50), and (51) had to be integrated using a step size of 1 10 4 m with the standard fourth-order Runge-Kutta method. Considering that a step size of 1 10 2 m leads to a maximum local integration error of about 6 10 13 for the solution of the CDAWC model before the onset of supersaturation, it follows that the value of the step size that is required right after the onset of supersaturation turns out to be rather small. From the numerical standpoint, this means that the system of ODEs becomes rigid (this is due to the very slow changes of T G in the supersaturation regime, as discussed below).
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49 With regard to the dry-bulb air temperature T G and the humidity mass ratio of saturated air Y W , S , the agreement between our simulation results and those reported by Klimanek and Bialecki (2009) is excellent up to the onset of supersaturation. However, a careful consideration of the simulation profiles shown in Figs. 9 and 10 for T G and Y W , S beyond the onset of supersaturation, indicates that a fundamental difference exists between the results obtained in the present work from the CDAWC model and the results reported by Klimanek and Bialecki. The two profiles obtained from the CDAWC model become almost flat beyond the onset of supersaturation whereas the profiles reported by Klimanek and Bialecki exhibit a steadily increasing trend.
To get an explanation for the flattening of the T G profile (and consequently, of the Y W , S profile), a careful examination of the numerical values taken on by all the terms involved in Eq. (50) is in order. It happens that the dry-bulb temperature gradient dT G / dz increases from 2.5 °C m1 at the bottom of the packing to 4.8 °C m1 before the onset of supersaturation, it abruptly decreases to 0.04 °C m1 at the onset of supersaturation and increases very slowly to 0.07 °C m1 at the top of the packing. The abrupt decrease in the dry-bulb temperature gradient at the onset of supersaturation is due to the very large contribution of the heat of vaporization in the term H vap , w (T G ) dY W , S / dT G
to the denominator at the right-hand side of Eq. (50). This term largely
exceeds (by a factor of about 700) the heat capacity of the supersaturated air so that the entire denominator could be regarded as a magnified heat capacity that makes the dry-bulb temperature gradient dT G / dz for the supersaturated air to decrease by a factor of about 120 at the onset of supersaturation. Figs. 9 and 10 show that the defining conditions Y W Y W , S and T wb T G for supersaturated air are always fulfilled for the results obtained from the CDAWC model, where T wb is the wet-bulb temperature calculated from Eqs. (A.9) and (A.10). In spite of the fact that TG
remains almost constant beyond the onset of supersaturation, the calculated wet-bulb
temperature T wb steadily increases beyond that point because the humidity mass ratio Y W , which accounts for the total mass of water (vapor and liquid) contained in the supersaturated air, also increases beyond that point.
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50 In industrial refrigeration, the term supersaturated air refers to an equilibrium mixture of saturated air and suspended liquid water droplets (Mago and Sherif, 2005). Even though the flattening of the T G and Y W , S profiles beyond the onset of supersaturation is a result obtained directly from the simulation of the CDAWC model, which is a combined heat and mass transfer model, it happens that such result resembles the behavior of supersaturated air at thermodynamic equilibrium. In thermodynamics, the Gibbs phase rule (Smith et al., 1996) indicates that the number F of intensive variables required to specify the intensive state of a system is given by the expression F C 2 r , where C is the number of components, is the number of coexisting phases, and r is the number of additional constraints. Since supersaturated air is an equilibrium mixture of saturated air (water vapor + air) and mist (liquid water drops), then C 2 (i.e., two components: air and water), 2 (i.e, two phases: gas and liquid), and r 2 (i.e, two constraints: the mole fraction of air in the liquid phase is equal to zero and the value of the total pressure P is known); therefore, F 0 , which means that at fixed pressure P, no other intensive variables (such as temperature T or any mole fraction in the gas phase) have to be specified in order to know the intensive state of supersaturated air. As a consequence, as long as the two phases (gas and liquid) are present in the system, the dry-bulb temperature T G of supersaturated air would be expected to remain constant at the value achieved at the onset of supersaturation. Since the supersaturated air in the cooling tower actually exchanges heat and mass with the liquid water being cooled, it is possible for the total humidity mass ratio Y W and the amount of mist YW YW , S
in the supersaturated air to change (due to the evaporation of the liquid water being
cooled and the condensation of the excess water vapor in the air), because those amounts are associated to the extensive state of the system and they are therefore not related to the phase rule, which involves only the intensive state.
5.2 Simulation of the model
As mentioned in Section 1.1, the classical early papers (London et al., 1940; Hutchison and Spivey, 1942; Lichtenstein, 1943; Simpson and Sherwood, 1946) reported experimental measurements of the performance of mechanical-draft counterflow wet-cooling towers. Full reports with a sufficiently large set of experimental data were given only by London et al. (1940)
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51 and Simpson and Sherwood (1946). For the simulation of the CDAWC model, we chose the set of 45 experimental runs reported by Simpson and Sherwood (1946) for an induced-draft counterflow wet-cooling tower (packed with 854 redwood slats) designated as Tower R-2. Simpson and Sherwood (1946) reported the following measured quantities (see Tables 2 and 5 of Simpson and Sherwood's paper): packed height ( Z T ) and cross section dimensions of the cooling tower, inlet ( T L , 2 ) and outlet ( T L ,1 ) water temperatures, inlet ( T G ,1 ) and outlet ( T G , 2 ) air dry-bulb temperatures, inlet ( T wb ,1 ) and outlet ( T wb , 2 ) air wet-bulb temperatures, and inlet water ( L 2 ) and dry air ( G B ) mass flowrates per unit area of tower cross section. The volumetric mass transfer coefficient k Y a M , calculated by means of the Merkel equation, Eq. (1), was also reported by Simpson and Sherwood (1946).
We carried out runs of our computer program to perform the following four simulation tasks:
Simulation Task 1. For each of the 45 experimental runs reported in Table 5 of Simpson and Sherwood's paper, calculate the packed height Z T( calc ) required to match the measured outlet water temperature T L ,1 by simulation of the CDAWC model with the value of the volumetric mass transfer coefficient k Y a M reported by Simpson and Sherwood.
Simulation Task 2. For each of the 45 experimental runs reported in Table 5 of Simpson and Sherwood's paper, calculate the volumetric mass transfer coefficient ( k Y a M ) ( calc ) required to match the measured outlet water temperature T L ,1 by simulation of the CDAWC model with the experimental value of the packed height Z T reported in Table 2 of Simpson and Sherwood's paper.
Simulation Task 3. Correlate the entire set of 45 values of k Y a M obtained from Simulation Task 2 in terms of cooling-tower inlet quantities and calculate (by simulation of the CDAWC model) the outlet temperature conditions ( T L ,1 , T G , 2 , T wb , 2 ) for the 45 runs using the values of kY aM
predicted by the best fitting correlation found.
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Simulation Task 4. Correlate only a subset of the entire set of values of k Y a M obtained from Simulation Task 2 in terms of cooling-tower inlet quantities and predict (by simulation of the CDAWC model) the outlet temperature conditions ( T L ,1 , T G , 2 , T wb , 2 ) for the remaining runs using the values of k Y a M predicted by the best fitting correlation found.
The usual assumptions that the cooling tower is completely wetted by the liquid (so that a H a M a ) and that the volumetric mass transfer coefficient k Y a M is constant along the
tower packed height were made for the implementation of the computer program. With a step size z 0 .01 m, implementation of the step doubling technique for the integration of the set of ODEs by means of the fourth-order Runge-Kutta method (Press et al., 1989a) yielded a maximum local integration error of the order of 10 10 . The four simulation tasks and the corresponding results are described in detail in the following sections.
5.2.1 Simulation Task 1
Since Simpson and Sherwood (1946) calculated the volumetric mass transfer coefficient k Y a M from their measurements by using the Merkel equation, Eq. (1), it is interesting to find out whether the values of the packed height that are required to match the measured outlet water temperatures by simulation of the CDAWC model with the values of k Y a M reported in Table 5 of Simpson and Sherwood's paper, agree or not with the measured value (1.05 m) of the packed height of cooling tower R-2 (see Table 2 of Simpson and Sherwood's paper).
To perform this task, simulation runs were carried out for the entire set of 45 experimental runs reported in Table 5 of Simpson and Sherwood's paper, in the following way. For the simulation of each experimental run, the value of k Y a M was set to the corresponding value reported by Simpson and Sherwood. A value was guessed for the cooling-tower packed height Z T and the shooting method was used to make iterative corrections (using the secant method) to the assumed values of the unknown conditions ( T L ,1 , L 1 ) of liquid water at z 0 until the measured
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53 conditions ( T L , 2 , L 2 ) (reported in Table 5 of Simpson and Sherwood's paper) were obtained by ) the end of the integration process (i.e., at z Z T ). If the converged value T L(,calc did not agree 1
with the corresponding measured value T L ,1 (reported in Table 5 of Simpson and Sherwood's paper), another value for the packed height Z T was tried and the simulation run was re-started. ) When an accurate match between T L(,calc and T L ,1 was reached, the last tried value of Z T was 1
taken as the required packed height Z T( calc ) corresponding to the value of k Y a M reported by Simpson and Sherwood.
The resulting values of Z T( calc ) are reported in Table F1 (in Appendix F, Supplementary Data) together with the corresponding input and output simulation data. It turns out that the use of mass transfer coefficients based on Merkel’s equation leads to an overestimation of the packed height necessary to match the measured outlet water temperature, the overestimation being on the average of 8.2% and reaching a maximum value of 18.1%. Such an overestimation confirms the approximate nature of Merkel’s model and points to using the simulated CDAWC model for the accurate determination of the volumetric mass transfer coefficient k Y a M , in order to match the measured outlet water temperature in the cooling tower R-2 with the known packed height of 1.05 m.
5.2.2 Simulation Task 2
Since the values of the mass transfer coefficient calculated by means of the Merkel equation and reported by Simpson and Sherwood (1946) lead (see Section 5.1) to an overestimation of the packed heights required to match the measured outlet water temperatures by simulation of the CDAWC model, then the simulated model was used to determine the values of the volumetric mass transfer coefficient k Y a M that are capable of matching those temperatures for the known value of 1.05 m for the cooling-tower packed height.
To perform this task, simulation runs were carried out for the entire set of 45 experimental runs reported in Table 5 of Simpson and Sherwood's paper, in the following way. For the simulation
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54 of each experimental run, the packed height Z T was set to the known value of 1.05 m, and the condition T L ,1 (at z 0 ) was set to the corresponding value reported in Table 5 of Simpson and Sherwood's paper. The shooting method was used to make iterative corrections (using the secant method) to the assumed values of k Y a M and the unknown condition L 1 (at z 0 ), until the measured conditions T L , 2 and L 2 (reported in Table 5 of Simpson and Sherwood's paper), respectively, were obtained by the end of the integration process (i.e, at z Z T ).
The resulting values of k Y a M are reported in Table F2 (in Appendix F, Supplementary Data) together with the corresponding input and output simulation data. From the calculated values of L 1 and the known measured values of L 2 , it follows that the water loss by evaporation is on the
average of 1.6% and reaches a maximum value of 2.1%.
Table F3 (in Appendix F, Supplementary Data) lists the percentage absolute relative deviations (ARD) for the calculated outlet air dry-bulb ( T G , 2 ) and wet-bulb ( T wb , 2 ) temperatures, and the volumetric mass transfer coefficient k Y a M with respect to the values reported in Table 5 of Simpson and Sherwood's paper. The average ARD values for T G , 2 and T wb , 2 are 1.6% and 0.9%, respectively. The calculated values of k Y a M are on the average 8.2% (on a maximum of 18.2%) larger than those obtained from the Merkel equation by Simpson and Sherwood. Due to the results obtained in Section 5.1 and the approximate nature of Merkel’s equation (see Section 3), the mass transfer coefficient values resulting from simulation of the CDAWC model can be considered to be more accurate and reliable than those based on Merkel’s equation.
From the consideration that use of the values of k Y a M calculated by means of the Merkel equation by Simpson and Sherwood led to an overestimation of the packed height Z T in Simulation Task 1, and due to the existence of an inverse proportionality relationship between Z T and k Y a M through Eq. (1), it follows that an unprevented use of the larger values of k Y a M
obtained from Simulation Task 2 (which are more reliable and could be regarded as “true” values) to predict the corresponding values of Z T by means of the simple Merkel equation, Eq.
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55 (1), would lead to an underestimation of Z T . , a result that is in agreement with what Sutherland (1983) had already found from a comparison of his performance-analysis method with Merkel’s model: an underestimation in the range from 5 to 15% in the values of Z T when combining “true” values of the volumetric mass transfer coefficient with the Merkel equation.
The temperature profiles (along the cooling-tower packed height) obtained from Simulation Task 2 can be classified into three morphological groups according to the relative position of the curves for the liquid water temperature T L and the air dry-bulb temperature T G . Group 1 includes those profiles in which the two curves T L vs. z and T G vs. z have an intersection point in the lower half of the packed section (i.e., at a distance z * 0 . 5 Z T ). Group 2 includes those profiles in which the two curves T L vs. z and T G vs. z have an intersection point in the upper half of the packed section (i.e., at a distance z * 0 . 5 Z T ). Group 3 includes those profiles in which the curve of T L vs. z is always above the curve of T G vs. z, so that these curves do not intersect. The classification of the profiles for the 45 simulation runs is given in Table F4 (in Appendix F, Supplementary Data). The profiles of 35 runs (78%) belong to Group 1 whereas the profiles of 8 runs (18%) belong to Group 2 and only 2 runs (4%) belong to Group 3. For the crossingtemperature profiles in groups 1 and 2, T L T G for z z * and T G T L for z z * . For Group 1, the crossing of temperatures occurs closer to the bottom of the packed section and for Group 2, the crossing occurs closer to the top of the packed section. For T L T G , sensible heat is transferred from the liquid water to the air so that the liquid water gets cooled not only by the transfer of energy due to the vaporization of liquid water at the interface into the bulk gas phase but also by the transfer of sensible heat from the bulk of the liquid to the bulk of the air. In contrast, for T G T L , the liquid water gets cooled by the transfer of energy due to the vaporization of liquid water at the interface into the bulk gas phase because that transfer of energy overcomes the unfavorable sensible heat transfer taking place from the bulk of the air to the bulk of the liquid. Also, as expected, for all the profiles in the three groups, the curve of T L vs. z is always above the curve of T wb vs. z, and the curve of T wb vs. z is always below the curve of T G vs. z, indicating that for all 45 profiles, a bulk gas phase of unsaturated air is always
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56 present. That this is indeed the case was confirmed from a calculation of the relative humidity ( H R , 2 ) of the outlet air. As follows from the results reported in the last column of Table F4, the condition H R , 2
