NATURAL DRAFT COOLING TOWER A n Approximate Solution I . a. FURZER University of S y d n e y , S y d n e y , Australia
The theory of the natural draft cooling tower has been extended b y introducing an approximate relationship between the change in air density and enthalpy in passing through the tower. Combining this relation with Bernoulli's theorem, an over-all enthalpy balance, and Merkel's approximation o f the integrated heat and mass transfer rate equations leads to a simple tower equation containing the dimensionless tower and specification numbers.
HE power, oil, and chemical industries require the dissipaT t i o n of large quantities of heat and this is often satisfied by one or more cooling towers. The design of the cooling system requires a selection of the type of tower and the choice is limited to either the mechanical draft or natural draft types. The principal dimensions of the tower can be calculated by well established methods for the mechanical draft type. The natural draft cooling toivers suffer from the lack of an adequate simple design method and this has resulted in the use of empirical design methods. The difficulty with the design for the natural draft type is the absence of any freedom in selecting the air flow through the to\ver, whereas in the mechanical draft type the situation is flexible, in that a suitable air flow can be obtained from a careful selection of fan and motor sizes.
and enthalpy occurs if the air left saturated a t the inlet water temperature (100" F.). This approximation, while having no theoretical basis, is of importance in simplifying the equations for the cooling tower. An over-all enthalpy balance for the tower gives L A t m = GAi
(3)
Eliminating Ap and Ai from E'quations 1, 2, and 3 we have (4)
Equation 4 still contains the air flow rate, G, passing through the tower, but Ap has been eliminated. Heat and Mass Transfer Rates
Air Flow
Wood and Betts (1950) have applied Bernoulli's equation to the natural draft type by considering two planes on the same datum level, one inside and the other outside the tower extending to infinity. If the head losses in the system can be lumped into N velocity heads, we have ApgH = IV '/zpv2
A heat and mass balance taken over an element of packing, assuming uniform water and air flows, countercurrent flow conditions, and a Lewis number of 1, leads to the well known enthalpy transfer method which is the basis for the design of the mechanical draft cooling towers.
(1)
The problem in calculating u from this relationship for given values of g , H , and N is the determination of Ap, the difference in air densities inside and outside the tower. The density of the air entering the tower can be obtained from the wet and dry bulb temperatures as given in the appendix. The density of the air inside the tower could be evaluated from the wet and dry bulb temperatures if these were known. The difference in air density is very small, so the density must be known with high accuracy. Figure 1 is a plot of the air density and enthalpy, with the dry bulb temperature and the depression of the wet bulb temperature as parameters. A typical cooling tower would have air entering at T D B of 60' F. and TmBof 52' F. with the inlet water temperature in the region of 100'F. Line A on Figure 1 connects point ( TDB600F., TI,, 52' F.) with the saturation curve at 100' F. Air leaving the tower would be between 80' and 95" F. and saturated with water vapor, and would be represented by a point on the saturation curve, close to line A and the basis of the following approximation
2 (2) =
ma.
where subscript max refers to the maximum change in air density and enthalpy. The maximum difference in air density
I
I
I
I
ENTHALPY IO ao BTU/LB so
Figure 1 ,
+O
I
I
SAT. P I TEMP. d ' tw
m
Air density-enthalpy
VOL. 7
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1968
555
2 = H T U X NTU
G
HTU =
where
-
Ka
The integral must be accurately evaluated using numerical methods, since the equilibrium conditions are nonlinear. However, an approximate value can be obtained by assuming that the mean enthalpy driving force occurs at the mean water temperature, or mid point temperature.
of the entering air. Apmsr is obtained from the maximum density difference which occurs if the air is left saturated at the inlet water temperature Tlw. The value of Airnx is obtained from the maximum enthalpy difference which occurs if the air is left saturated at the inlet water temperature. This enthalpy at T I , and the inlet enthalpy are obtained from the appendix. The value of Atw in the cooling range is the difference between the inlet and outlet water temperatures. Merkel's cooling factor, a,can be found from its defining equation where iMP* is the saturated enthalpy at the midpoint water temperature and is obtained from the appendix. l o w e r Equation
Ai NTU = iMp* - iMp
r 1 Since T = S
This can be rearranged using Equation 3 to give
-1
G
;j
n L
AtW
or
+
NTU = __ HTU
- IN
iMp*
LNiU
it is possible to calculate a value of T , if Tlw, Tow, TDB, and TwBare specified so that S can be evaluated. The number of enthalpy transfer units, NTU, is obtained from
Merkel's cooling factor, a,is defined as f f =
4 7 - a
~
NTU
(5)
where H T U is a characteristic of the packing. To design a tower to meet a given specification for a given type of packing, values of T are calculated for different values of packing depth from
Eliminating L/G from Equations 4 and 5,
This equation is the basic relationship for the natural draft cooling tower, and contains several dimensionless groups which have been collected for the simple design and testing of the tower. Let T be the tower number, defined by
and let S be the specification number, defined as
s=-
~
Atwaa
P Aimas
If the depth of packing is given, only one value of the tower number will meet the cooling specification. Since T contains N , H , and D , any combination of the variables will be suitable, and the selected values will depend on an economic consideration of the cost factors involved in the construction and operation of the tower. If the tower is constructed, it must be tested and it is very unusual to find atmospheric conditions identical with those included in the specification. If the above theory is valid, for an existing tower with a fixed water loading and constant N , the tower number should be constant. Also since the value of H T U is substantially constant, the theory predicts that S should be constant under all conditions. If the water loading is different from the specified value, T will also be a variable. Equation 6 can be rearranged to give
.. a =
["Iii3
2Hgp
[L NTU -k
21 [
A pL2 __ Atw
]
(8A)
Aimes
l o w e r Number
The tower number contains the total resistance, N , of the tower, the height of the tower, H,and the diameter of the tower,
The first two terms on the right-hand side will be effectively constant for a given tower, so the equation is essentially a relationship between Merkel's cooling factor, a,and the last term in the equation. This term includes the following variables: water loading, Tlw, Tow,TDB, and TwB. A plot of a against L2 112 should be linear and the slope will have a value Aimas
D.
corresponding to
..
Specification Number
The temperature specification on a cooling tower is given by the need to cool water from the inlet water temperature T I , to the outlet water temperature Tow,with air entering with a dry and wet bulb temperature of TDB and Tw,. I t is possible to calculate S from Tlw, Tow, TDB, and TWB. The density of the inlet air, p, is obtained from the appendix, using TDB and TWB 556
I & E C PROCESS D E S I G N A N D DEVELOPMENT
[x]
'[
[L]'!' Wgp NTU +
;]
Experimental Results
Rish (1961) has reported tests on a natural draft cooling tower 238 feet high, 180 feet in diameter, and containing 7 feet 1 1 / 2 inches of corrugated asbestos-cement sheets. His Table 2
contains theory.
TIW,
Values of
CY
Tow,
and
and
TDB,
[
TWB,
L2
Atw]
~~
1'3
which is sufficient to test the
were calculated and plotted Since
Ai,,, in Figure 2. A regression analysis gives a
-
2.349 = 1.861 X lo-'
{[. ___
iMp* - iI N
a =
AtW
-
(9) 1.337 X loa\
AiUISX
The correlation coefficient of 0.94 is significantly different from zero. The standard error of a! about the regression line is 0.0643 and the confidence limits a t 1 2 6 have been included on Figure 2. The regression line does not pass through the origin so we need to test the hypothesis that there is no significant difference between the slope of the regression line and a line from the mean point of the data through the origin. The slope of the latter line is 1.757 and the standard error in the regression coefficient is 0.0755. Using Student's t test, we have
t =
The difficulty is accentuated by the rapid changes in atmospheric conditions and the large response time of the cooling water circuit.
1.861 - 1.757 0.0755
where atw is the error in calculating the cooling range, Atw. Now Atw is the difference between TTWand Tow, each of which can be measured to 0.1 ' F., so btw must be of the order of ~ due * to the error 0.2' F. The error in iMp* given as b i ~ is in calculating the midpoint water temperature,
hiMP*=
hiMp * btMP
~
btMP
where biMp*/btMp is the slope of equilibrium curve at the midpoint temperature. From the appendix, biMp*/btMp is of the order of 1.1 and if atMpis of the order of O.l'F., then
aiMp*= 0.11
(10 )
The error in i I N , the enthalpy of the entering air, is due to errors in measuring the dry and wet bulb temperatures.
t = 1.38 with 19 degrees of freedom
Inspection o f t tables shows there is no significant difference at the 5y0probability levels, so the theory provides a valid explanation of the experimental data.
bi 1
Values of - and atDB
Source of Residual Variation
Merkel's cooling factor, a, is calculated from measurements of TIT, Tow, T D B , and T w B , all of which include an experimental error which arises from the difficulty in obtaining a good sample of the inlet and outlet water streams and the inlet air.
bi 1 can
be obtained from the appendix,
3tWB
since they are the enthalpy gradients with respect to the dry and wet bulb temperatures
II O '
Figure 2.
Regression analysis VOL. 7
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557
The errors in estimating tDB ( b t D B ) and twa (dt,,) order of 0.2' F.
bils
=
0.07 (0.2)
Substituting values of bt,, At, = 10' F. and 01 = 2,
a01 = -1 [-2 10
are of the
+ 0.70 (0.2) = 0.15
(11)
ainrp*,and bilN in Equation 9 for (0.2)
estimate the total N T U is 1.80 and from Equation 12 we have N as 82. The large change in N from 61 to 82 when the number of transfer units changes from 1.50 to 1.80 indicates that the number of transfer units must be known accurately before N can be evaluated. The resistance of the packing is given from Equation 13A as 3.8 X 7.125 velocity heads-Le., 27-and Rish has given the other resistances due to the spray, eliminator, 5.2 10.5 13.5 outlet, inlet, and columns to be (3.8 2.3) velocity heads-Le., 35-so N is 62 velocity heads. This compares favorably with a tower resistance of 61 velocity heads when the end effects were neglected.
+ +
+ 0.11 - 0.151
The maximum error in bo is then 0.066. For a Merkel's cooling factor (Y of 3, ba is 0.086. These estimates for the error in 01 must be considered conservative, as water temperatures have been considered to have an error of 0.1 O F. and air temperatures of 0.2' F. The standard error in a about the regression line is 0.0643 and limits of two standard deviations amount to 0.128, which is somewhat higher than values of 0.066 and 0.086 from the error analysis. Evaluation of Tower Resistance
The good agreement between the standard error of the experimental results and that obtained by the error analysis indicates linearity between the variables on Figure 2. The slope of this line, given by Equation 8B, provides a method for obtaining I7
+
+
Effect of liquid-Gas Ratio on Tower Resistance
I n the analysis of the performance of the cooling tower both N and the height of a transfer unit have been considered as constants. Both vary with the operating conditions, since they are functions of the liquid-gas ratio, L/G. The effect is small for sheet packings but can be included in the analysis to lead to a more realistic treatment of the problem. The total tower resistance is composed of a number of fixed items as listed by Rish and a variable resistance due to the effect of L/G on the packing resistance. Data on the resistance of sheet packing (Rish, 1961) can be correlated by
IV.
The slope of the line on Figure 2 is 1.861 X theory 1.861
10-3 =
["I1:"
'[
2HgP
NTU
and from the
+
i]
where Np, is the packing resistance per foot of packing at a liquid-gas ratio of 1. For a sheet packing N p , = 3.8, so
with H = 238 feet g = 32 x (3600)* feet/hour* p = 0.076 lb./cu. foot
N p = 3.8 The total tower resistance is given by
N Any attempt to separate N will require a value for N T U and there will be various values of N depending on the method used to calculate N T U . The heat and mass transfer characteristics of the packing are commonly correlated for cooling tower packings in the form
9=
[g]-n
L
where X and n are experimentally determined constants. the definition of H T U we have HTU = HTUl where
[g]
From
Sheet-type packing as used in this tower has a value of X of 0.21 (Lowe and Christie, 1961) and for an L/G of 1, =
where NR is the residual resistance of the tower after removing the contribution due to the packing.
Equation 14 can be simplified by restricting the range of L/G from 0.8 to 3 and the range of N p , Z / N R from l / 4 to 1. A suitable correlation is
For the heat and mass transfer characteristics of the packing we 1 1 have from Equation 5 the term - I which can be simNTU plified for a range of N T U from 1 to 2 by the correlation
+
1.167
Substituting in Equation 12 gives
N = 61 Some-heat and mass transfer will occur above and below the packing, equivalent to an additional number of transfer units. Since no measurements have been reported on these end effects for a large diameter tower, we can only use an estimate of 0.3 N T U . A detailed analysis of the end effects requires countercurrent flow conditions to apply above the packing and a complex flow pattern in the spray under the packing. Using this 558
N R f NpZ
n -1
1 HTUl= X
[Al+ i]
=
l&EC PROCESS D E S I G N A N D DEVELOPMENT
the variable L/G can be eliminated from Equations 4, 5, 13, 15, and 16 giving 1
c " '
1
n-1 +-=-
Table 1.
Total l o w e r Resistance
No End Effects Constant .V, HTU Variable N , HTU
End Effects
61
82
63
85
The exponent on the last term can be evaluated, since n = 0.69 for sheet packing (Lowe and Christie, 1961), and has a value of 0.30
resistances. This would indicate that if the summation method is used for the total tower resistance, the end effects should be neglected. Equation 8A becomes for a liquid-gas ratio of 1
and Equation 6
or
The regression equation of a against [L2/Ap/Ai,,,Atw]1/3 passes through the mean point (1.337 X lo3, 2.349), SO where
L2
=
2.39
x
109 and S is defined by Equation 6A
Atw Aim,,
A plot of a against [L2/Ap/Ai,,,Atm]o.3 will have a mean point close to L2/Ap/Aim,,AtW = 2.39 X lo9 and a = 2.349. Substituting these values in Equation 18 and using the constants, 2 = 7.125 feet H T U l = 4.76 feet N p l = 3.8 velocity heads per foot
we have N E the residual resistance, as NE = 36
The total resistance at aliquid-gas ratio of 1 (NI)is
N1 = 36
+ 27 = 63
This value of N1has been obtained without any allowance for heat and mass transfer end effects. The number of transfer units at a liquid-gas ratio of 1 (NTU1) in the above example is obtained from Equation 16.
which corresponds to N T U l = 1.50 If the end effects amount to an additional 0.30 transfer unit so that NTUl = 1.80, we have
Substituting in Equation 18
N~
=
58
N1 = 58
+ 27 = 85
l o w e r Equation
The constant and variable N and H T U cases for a liquid-gas ratio of 1 are compared in Table I. The analysis based on the variable N and H T U is sounder than the previous case when N and H T U were considered as constants, but the result for N is almost identical if the liquidgas ratio is taken as 1 for the constant N and H T U case. However, including end effects results in a higher tower resistance than expected by the simple method of summing the separate
TI =
or
16N1W nHgp2D4 ~
(23)
Design of l o w e r
I n designing a tower to meet a given specification number,
A’, and total water flow W , the tower number can be evaluated for various depths, Z , of a packing with a given heat and mass transfer characteristic, HTU1, from Equation 21. The product HD4 can be obtained from the tower number as given by Equation 23, when the total tower resistance, N1 isc alculated for a liquid-gas ratio of 1. The determination of either N or D from the product HD4 depends on an economic evaluation of the tower. The final design would require a full optimization of the tower by considering the cost of the packing, the empty tower and the liquid pumping. l o w e r Testing
While the tower equation is suitable for the design of a tower, it is unsatisfactory for the testing of a tower, since both N1 and HTU1 must be considered as variables which can be selected by the designer. Equation 19 is in a more suitable form, since the completed tower has a given liquid loading, L, and the specification number contains a , Ap/Aimax,and Atw. Since a and [Lz/Ap/Ai,,,Atw]l’a are specified, they can be plotted, and a line drawn from the point to the origin, since these quantities are directly proportional. Confidence limits should be drawn about the line at *26, where u is an estimate of the standard error in a and is of the order of 0.07. Measured values of a and [L2/Ap/Ai,,xAtm]1’3 obtained during a test run should be plotted on the figure. Points falling above the +2u confidence limit should be considered unsatisfactory, points within the range =k2u should be acceptable, and points below the -2u confidence limit are well above the expected performance. Conclusions
The theory of the natural draft cooling tower has been extended from the contribution made by Wood and Betts by introducing an approximate relationship between the change in air density and enthalpy. Combining this relationship with Bernoulli’s theorem, an over-all enthalpy balance, and MerVOL. 7
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559
kel’s approximation of the integrated heat and mass transfer rate equation leads to a simple tower equation given by
where T and S are the dimensionless numbers known as the tower number and specification number, respectively. The theory was tested by considering the experimental results of Rish on a tower 238 feet high, and 180 feet in diameter, containing 7 feet l 1 / 2 inches of sheet packing of known pressure drop, and heat and mass transfer characteristics. A statistical analysis of the results confirms the direct relationship between Merkel’s cooling factor and [L2/Ap/AimSxA t w ] 1 / 3and from the slope of the line values of tower resistance can be obtained, in good agreement with the summation method used by Rish if the liquid-gas ratio is 1, and the end effects are neglected. This method allows for various diameters and heights of towers to be obtained to meet a cooling specification. The tower can be tested at conditions different from the specified conditions by testing whether the measured value of Merkel’s cooling factor is below or within the confidence limits of the line joining points CY and [L2/Ap/Ai,,,Atw]1~3 at the specified condition to the origin. Appendix.
Physical Properties of AircWater Vapor Mixtures
Density. The density of air-water vapor mixtures can be calculated from the ideal gas laws as p =
I
- [lo00MW,,,
RT
- p(MW,i, - MWE,,)]
where R is the gas constant, given by
R
=
740.1 millibars cu. feet/’ R. lb. mole
and T i s the temperature in degrees Rankine. The molecular weight of air has been calculated from the International Civil Aviation Organisation (I.C.A.O.) standard atmosphere as 28.95. The molecular weight of water is taken as 18, and the total pressure is 1000 millibars. The water vapor pressure is obtained by measurement of the wet and dry bulb temperatures and using the relationship for an Assmann hygrometer
p
= pwB*
-
0.444(t,B
- tWB)
The saturated vapor pressure at the wet bulb temperature bWB*)has been abstracted from the IHVE hygrometric tables 30’ to 136’ F. which derive from the data of Scheel (1910). A computer program was written to perform the calculations which are tabulated and is available from the author. The table is useful for the rapid determination of air densities from wet and dry bulb temperatures without obtaining the humidity as required for entry into other tables. Enthalpy
The enthalpy of air-water vapor mixtures can be calculated by assuming ideal behavior and summing the partial enthalpies. The specific enthalpy of water vapor has been obtained from Keenan and Keyes (1937) and correlated as
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I&EC PROCESS D E S I G N A N D DEVELOPMENT
iw = 1075.8
+ 0.432 (t - 32)
using a reference temperature of 32’ F. The enthalpy of the air-water vapor mixture at a total pressure of 1000 millibars is
[1075.8
+ 0.432(t - 32)]
The water vapor pressure p, was obtained from the wet and dry bulb temperatures using the previous relationship for the wet bulb thermometer and the saturated vapor pressures from the same source. A computer program was written to perform the calculations which are tabulated and are available from the author. Nomenclature
D
= diameter of tower
g G H HTU
= = = = = = = = = =
2
Ka L n N NTU
=
%’
t T
= = =
U
=
W Z
= =
p
=
A
= =
ff
acceleration due to gravity gas flow rate height of tower height of transfer unit enthalpy volumetric heat and mass transfer coefficient liquid flow rate exponent on heat transfer correlator total resistance of tower number of transfer units water vapor pressure specification number water temperature tower number, or air temperature air velocity total water flow rate height of packing air density change Merkel’s cooling factor
SUBSCRIPTS = liquid-gas ratio of 1 DB = dry bulb IN = inlet = inlet water IW O U T = outlet OW = outlet water max = maximum M P = midpoint = water W WB = wet bulb P = packing R = residual
I
Literature Cited
Keenan, J. H., Keyes, F. G., “Thermodynamic Properties of Steam,” Wiley, New York, 1937. Lowe, H. J., Christie, D. G., “International Developments in Heat Transfer,” International Heat Transfer Conference, Colorado, Part V, p. 993, 1961. Rish, R. F., “International Developments in Heat Transfer,” International Heat Transfer Conference, Colorado, Part V, p. 951, 1961. Scheel, K., Heuse, W., Ann. Physik31,715 (1910). Wood, B., Betts, P., Proc. Inst. Mech. Engrs. 163,54 (1950). RECEIVED for review December 21, 1967 ACCEPTED June 10, 1968
