N A T U R A L D R A F T COOLING T O W E R Maximum Liquid Loading I. A. F U R Z E R University of Sydney, Sydney, Australia
A new important limiting condition has been found for the natural draft cooling tower.
There i s a maximum liquid loading which can be placed on the tower and still meet the cooling specification. The tower was simulated on a computer and a correlation was obtained for the rapid calculation of the maximum liquid loading. This correlation i s the basis for a very simple design method for finding the height and diameter of the tower.
paper (Furzer, 1 9 6 8 ) , an approximate solution of the equations for the natural draft cooling tower was presented in terms of dimensionless numbers. The tower number contained the resistance of the tower, the height of the tower, and the liquid loading, whereas the specification number was defined from four temperatures-the inlet and outlet water temperatures and the wet and dry bulb temperatures of the entering air. N A PREVIOUS
I
T = S [ L + i ]
-3
For constant values of b and S the maximum residual tower number occurs when
From Equation 9 d(TR) _ _ dNTU
NL2
T =
S
=
(z) AP
2P
3 + (NTU)2 '[NTU + ~
;I-'}
(10)
The required solution is
~
%P2
and
S 1 +bNTU (1 +;;TU
NTU where
-
-1
3
b
Atwa3
@A)
max
or Increasing the number of transfer units, N T U , for a constant specification number, S,results in larger tower numbers, T , as given by Equation 1 . But increasing N T U results in an increased packing resistance which contributes to an increase in the total tower resistance, N. Since both T and N increase with NTU, this indicates that the liquid loading, L, may pass through a maximum as given by Equation 2. Maximum liquid loading
NTU,,,,, can be substituted from Equation 12 into 13 and the right-hand side will contain only b and S terms or
Let the total tower resistance be given by
N
=
NE f NpZ
(3)
where NR is the residual tower resistance and Np is the resistance per foot of packing. Also 2 = H T U X NTU
*.
N = NR
+ N,HTU
X NTU
(4)
(5)
Defining a tower number based on the residual tower resistance by
TR(max)
= Sf(b)
(14)
wheref(b) is obtained from the substitution. Some typical values for b of 1 5 / 7 0 give, by Equation 12, an NTU(,,,, of 7 . 6 6 , outside the range of Merkel's approximation which is included in the solution of the equations for the natural draft cooling tower. So the maximum value of the residual tower number as given by Equation 14 will be in error, but it does indicate that a simple relationship may exist among TR(max), s, and b. Now
T = TR[1
then where
b=-
+ bNTU]
N,HTU NR
(7)
(8)
From Equations 1 and 7 TR =
So for constant NR and H a maximum value of the residual tower number corresponds to a maximum liquid loading on the tower-Le. , L(max)
(9)
=
S1'2f2(b, NR, H )
(15A)
A maximum liquid loading condition may occur to maintain a given specification number. Obviously this will be an imporVOL. 7
NO. 4
OCTOBER 1 9 6 8
561
tant limiting condition and needs to be examined in more detail. One limitation, the introduction of the approximate solution of the equations for the natural draft cooling tower, can be removed by returning to the original equations for the tower. Applying Bernoulli's equation to the tower gives
ApHg = N1/2pv2
Iterative Procedure
or
T
or
=
The four temperatures TI,, Tow, TDB, and T,, were selected and T was calculated from Equation 16A for various values of L/G. The number of transfer units for each value of L/G was obtained by evaluating the following integral by Simpson's rule-Le., N T U (S).
(G)
Ap L 2 P
containing N T U is in the denominator of Equation 17, T R will pass through a maximum. This would suggest that a trial and error method of searching for this maximum would be suitable; however, a highly convergent iterative procedure was used as outlined below.
From Equations 6,7, and 16A,
Now for a constant value of b it can be shown qualitatively that TR will pass through a maximum as N T U is altered as below. The slope of the operating line, ( L / G ) , and Ap are limited from zero to a maximum value when the air leaves saturated at the inlet water temperature. So the product Ap (L/G)2 will have a value between zero and a maximum value. However, increasing (L/G)results in an increasing number of transfer units, and an infinite number will be required if the air left saturated at the inlet water temperature. Since the term
Figure 1 contains the results of a calculation of T I , of 88' F., TO, of 72' F., TDB, of 54' F., and TvB of 50' F. for various values of L/G. But Equation 7 is a linear relationship between T and N T U and can be drawn on Figure l for values of TR and b. For b = 15/70, H = 300 feet, g = 32 X (3600)2 ft./hr.2, p = 0.075 lb./cu. foot and N R = 70, we have,
T = 10-7
I
2
4
Figure 1.
I
0
2
Figure 2. 562
[ + ;: 1
- NTU(S)
]
(g2) +
log [NR f iVp H T U NTU(S)]
I
I
6
NTU
~2
6
IO
(s)
Effect of NTU(S) on T
I
8
I
I
10
Various liquid loadings
I&EC PROCESS DESIGN A N D DEVELOPMENT
(1 8)
Figure 2 contains a plot of Equation 18 for liquid loadings, L, of 1000, 1500, and 1700 lb./hr. sq. foot. For a liquid loading of 1500 Ib/hr. sq. foot. The intersection of the line and the curve gives two operating conditions for the tower, so either 3.0 or 7.0 transfer units would meet the cooling specification. I n the case of the liquid loading of 1700 lb./hr. sq. foot, no intersection occurs, so the tower could not meet the cooling specification. Obviously some intermediate flow rate is the maximum liquid flow rate that can be applied to the tower and still maintain the cooling specification. This flow rate corresponds to the line tangent to the curve Taking logarithms of Equation 7 gives log T = log
0
di
II>.i*--i IOUT
NTU =
do Figure 3.
75+15 NTU(S)
Various cooling ranges
(19)
I
011
Figure 4.
S
I!O
I
Maximum liquid loadings NE = 70 N,HTU = 15
Figure 5.
Tower number (residual) and specification number
+
A plot of Equation 19, log T and log [ N E Np HTU NTU(S)], has a slope of 1. The other relationship between T and NTU(S) obtained by alteriilg L/G can be plotted on logarithmic paper (Figure 3). A point on this curve where the slope is 1 corresponds to a point on Equation 19. Since there is only one point where the slope is 1, this corresponds to the maximum value of log (Lz/Hgp2)from which L(,,,) can be
obtained. The iterative procedure starts a t the maximum value of L / G corresponding to an infinite value of NTU, then L/G decreases until the slope of the curve measured by finite differences is 1. Figure 3 is the log-log plot for cooling ranges of 8', 12', 16', 20°, and 24'F. Lines with a slope of 1 are drawn on the figure and for 16'F., L(max)is 1630 lb./hr. sq. foot. VOL. 7
NO. 4
OCTOBER 1 9 6 8
563
IC
Figure 6.
I-"
Effect of packing
I
N,HTU 0 10 A 20 30
IC
c Figure 7.
Effect of residual resistance I
NR 0
40 100
Table 1. N,HTU 10 15 20
30 15 15
Effect of N R and b on T R (,,,&a NR b 70 0.143 70 0.214 70 0.286 70 0,429 0.375 40 100 0.150
S for constant values of 6-i.e., 15/70. T R ( m a x )against S and is correlated by
Figure 5 is a plot of
TR(max) = 0.84 So."' (21) The effect of both N R and b on TRcmax)was investigated over the ranges given in Table I. Figure 6 shows the effect of N,HTU for an N R of 70 and lines with a slope of 0.808 have been drawn through the points. All lines on Figures 5,6, and 7 can be correlated by
Correlation
A computer program was written which allowed L,,,,) to be calculated for a wide range of dry bulb temperatures, depression of the wet bulb temperature, and water cooling ranges. Now Equation 15A indicates a relationship between L(,,,, and S for constant values of b, N E , and H-Le., 15/70, 70, 300. Figure 4 is a plot of L(-) against S and is correlated by =
3.03 X l o a
(g,
70, 300)
Also Equation 14 indicates a relationship between TR(max) and 564
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
Discussion
The approximate solution for T R ( m a x ) given by Equations 12 and 13 can be compared with the correlation of the computed results given by Equation 22. For a value of b of 15/70, Equation 12 gives an NTU(,,,) of 7.66 and Equation 13 gives a T R ( m a x ) of 1.50 for an S of 1, which compares with 0.82 from the correlation given by Equation 22. So the approximate solution overestimates TR(max) for an S of 1. The maximum liquid loadings as given on Figure 4 are close
to the operating liquid loadings of existing natural draft cooling towers. This would indicate that economic considerations result in operation of the tower close to the maximum liquid loading. This is very similar to the case of a gas stripping unit, where economic considerations result in the operating liquidgas ratio close to the maximum liquid-gas ratio. If a tower was designed to operate at a fraction, f, of the maximum liquid loading, then
L (operating) = fL(,,,)
g
G
(23)
Equation 23 is then a very simple design equation, since f, b, and S are known and the operating residual tower number can be obtained. For a circular tower of diameter D with a total liquid load of W , from Equation 6, (24) The determination of either H or D from this equation will depend on the ratio of D to H which was used to obtain factor f.
=
fraction
= acceleration due to gravity
= gas flow rate
H = height of tower HTU = height of a transfer unit
- enthalpy i L = liquid flow rate N = resistance N'TU = number of transfer units s = specification number - water temperature t T = tower number or air temperature depth of packing P = air density A = change a = Merkel's cooling factor
z
Tdoperating) = f T R ( m n x ) TR(operating) = 0.38 fzb-1/2So.808
f
=
SUBSCRIPTS DB IW max
ow
P
R W
WB
= = = = = = = =
dry bulb inlet water maximum outlet water packing residual water wet bulb
Acknowledgment
The author thanks the Mathematics Laboratory, University of Surrey, for time on the Sirius computer. Nomenclature = defined by Equation 8 b D = diameter
Literature Cited
Furzer, I. A., IND. ENG. CHEWPROCESS DESIGNDEVELOP. 7, 555 (1968). RECEIVED for review December 21, 1567 ACCEPTED June 10, 1968
MODIFIED JET PUMPING OF SOLID SPHERES H E R B E R T SUSSKIND, R O B E R T ODETTE,'
AND WALTER
BECKER
Brookhavan National Laboratory, Upton, N . Y . 17973 The removal of individual layers of balls from the surface of an ordered packed bed without disturbing the remaining packing was investigated. The modified jet pumping procedure could be used in the countercurrent fuel movement in the ordered bed fast reactor, as well as for the selective unpacking of beds of spheres in other industrial applications. Liquid jets entering two opposite sides of a square cross-section column through rectangular nozzles above the surface of the packed bed produced drag forces on the balls at a critical flow rote that were strong enough to entrain them and carry them out with the outgoing liquid. The distance of bed penetration increased with this critical flow rate. Such variables as liquid flow, ball density and diameter, nozzle velocity and inclination, and column size were investigated, and a mechanism 0.755 In X. for this model was postulated. The data could be correlated by the expression R = 2.63
+
TUDIES
were carried out at Brookhaven National Labora-
S tory to develop a sodium-cooled, fast breeder reactor concept in which packed beds of c '/b-inch-diameter spherical fuel particles were used in 12 X 12 X 48 inch steel containers (Susskind et al., 1 9 6 6 ~ ) . These studies culminated in a unique system in which perfectly ordered beds of spheres packed in a rhombohedral array were obtained consistently, even though the balls were dropped randomly into the containers. This method of packing provides the basis for the ordered bed fast reactor concept (Epel et al., 1966; Susskind et al., 1965). More generally, it applies to the packing of beds in any industrial application in which precisely known, high densities of 1 Present address, Massachusetts Institute of Technology, Cambridge, M a s . 02139.
solids and large heat and mass transfer surfaces are desirable. Countercurrent fuel movement (Epel et al., 1966) involves periodic rearrangement of the fuel balls, accomplished by mechanically or hydraulically handling them, either directly inside the reactor vessel or in an external hot cell. The purpose of the work described was to investigate the selective removal of individual layers of balls by hydraulic means from the top of a packed bed, ideally one layer a t a time, without disturbing the remaining packing. This was accomplished by the interaction between these balls and liquid jets entering the column just above them, representing a modification of the usual jet pump, which uses a high-velocity liquid to entrain a low-velocity or stagnant fluid. In this instance, solid spheres packed in the beds are effectively entrained. VOL. 7
NO. 4
OCTOBER 1 9 6 8
565
