Radial air distribution functions in water cooling towers

Chnicd EngineeringScience,1975,Vol. 30,pp. 349-351. PergamonPress. Printedin Great Britain

Radialairdistributionfunctionsinwatercoolingtowers (First receioed 8 March 1974;accepted 5 August 1974) tower. The second is for a round cooling tower, where the problem will be developed in cylindrical coordinates. Due to symmetry in the rectangular case we need only consider one half of the column. The air enters with a horizontal velocity IA, which decreases to zero at the plane of symmetry. Let the flow be isothermal and incompressible. In the open space under the packing we can neglect frictional terms. The vertical velocity u of the air through the packing depends on the pressure drop characteristics of the packing. If the half-width of the column is w/2, then Bernoulli’s equation for the horizontal gas flow gives,

INTRODUCTION

distribution of liquid in a packed column has been the subject of a number of recent papers[lJ] which account for various initial conditions (types of distributors) and boundary conditions at the wall. This has lead to a great improvement in our understanding of the liquid distributing mechanisms, and an ability to calculate radial liquid distribution functions for most of the common packings. To apply this work to the design of packed columns requires an equally good knowledge of the radial gas distribution functions. There appears to be an absence of theoretical work on this gas distribution function. Several experimental programs [4-6] have shown a rise in gas velocity near the wall due to irregularities in the void content. While the following analysis is general and can be applied to packed column design it is useful to make the analysis on a water cooling tower where, due to the large dimensions the effects are considerable and play a major part in the design. Figure 1 is a diagThe

P. = Pw,*t f p I& at the air inlet position P. =P, ttpu,‘atpointx

(1) (2)

A mass balance over a packing element of width dx gives, du u ---_=O dx H where H is the height of the air entry position. The pressure drop characteristic of the packing can be expressed as a pressure drop equivalent to N velocity heads. P, -P, = Nfpi

(4)

Where P, is the pressure above the packing. Separating IAfrom Eqs. (l-4) leads to I

I I I

I IO

&

! 1 ’

;-

-‘Pack ing Initial conditions u0= 0 The constraint, P, - P 5 0 must be satisfied as a reverse gas flow is physically impossible. The maximum value of the horizontal velocity given by u,, leads to the dimensionless terms U and X which reduce Eq. (5) as below:

.

Fig. 1.Modelof awater coolingtower. rammatic arrangement of a cooling tower, with air entering at ground level. The air leaves the packing with a vertical velocity u, which will vary along the dimension x. The problem is to theoretically predict the variation of u with x by an analysis of the air flow in the column.

The solution is given by TBJ?.ORY

The theory will be developed for two classes of problems. The first is for a square or rectangular cross-section to the cooling 349

U=SinXforXGf.

(6)

350

Shorter Communications

Definingsimilar terms for u,, and V leads to a solution for V

u_=

J[3%$];

v=e

V=CosXforXSi Equation (7) shows that the vertical velocity is distributed as a cosine function; with the maximum velocity at the plane of symmetry where X = 0. CYLINDRICAL COORDINATES

For a round cooling tower with air entering around the complete perimeter, we can write down Bernoulli’sequation and complete a mass balance for air about an annular element of width dr, which after a similar manipulation as for the rectangular case yields. dU U ;iirtj$l-U2)‘“=0;

R radius

v,=o

Fig. 3. Vertical gas velocity V and the dimensionless radius R, round tower.

where R is a dimensionless radius given by, R=’

Hfi whereas an exact solution for U was obtained in the rectangular case, it is necessary to obtain a numerical solution for Eq. (8). A CDC 6600computer was used for the integration and the results, showing the variation of U with R are given in Fig. 2. The dimensionless vertical velocity V is given by dU

U

V=$E

A numerical solution of Eq. (9) was also obtained and shown on Fig. 3.

Thus the maximum value of the width of the cooling tower W to the height of the air opening must be kept less than rfifor all the packing to usefully contribute to the heat and mass transfer processes. In round cooling towers the vertical air velocity is a maximumat the centre and a minimum at the diameter of the tower. The shape of the radial distribution function initially is similar to the cosine distribution but it never becomes zero. The value of u at any radius r can be obtained from the dimensionless plot on Fig. 3 and this can be used to obtain the ratio at any radius in the column, which is necessary for a detailed calculation of the heat and mass transfer processes. It is possible to treat the resistance to air flow in the tower as being the sum of separate resistances due to the packing, the air entrance, the spray eliminators, and an exit air loss. The air entrance resistance can be calculated from this theory and is given by, Ne, =

0.6-

W/W21 _N [Sin’(W/2H
(11)

For small values of W/H < 10, this approximates to 2

In cylindrical coordinates, the entrance resistance is given by,

0

NeN=O*25 $ ‘-N 0 2

I

R

3

(13)

4

radius

Fig. 2. Horizontal gas velocity P and the dimensionless radius R, round tower. CONCLUSION

vertical distribution of air through a packed column with a square or rectangular cross section with air entry in a symmetrical manner from two opposite sides, is described by a cosine function. The maximum velocity occurs at the centre of the tower; while the minimum velocity occurs at the air entry position. The minimum velocity can be reduced to zero when X = q/2; and this is equivalent to, The

0

I

I

I

100

200

300

4cm

Fig. 4. Resistance of gas entry position for a round tower of dia D, and with a height H of the gas entry position.

351

Shorter Communications where values of R and U must be obtained from Fig. 2, or N,, can be calculated during the integration of Eq. (8) A large range of N was covered from 20 to 60 and all points lie between the two curves shown on Fig. 4. for small values of D/H < 10, D2 N,, = 0.042 7i 0

vertical gas velocity and its dimensionless form width of a square or rectangular column horizontal distance and its dimensionless form gas density REFEmNckx

[l] Onda K., Gem. Engng Sci. 28 1677 1973. [2] Dutkai E. and Ruckenstein E., Gem. EngngSci. 197025 483. [3] Dutkai E. and Ruckenstein E., Chem. EngngSci. 196823 1365. [4] Schwartz C. E. and Smith J. M., Ind. Engng Chem. 195345 1209. [5] Benenati R. F. and Brosilow C. R., A. I. Ch. E. JI. 19628 359. [6] Price J. Australian Atomic Energy Comm. AAEC/E178 June 1%7.

NOTATION

H height of the gas entry position N resistance of the packing in velocity heads

NW resistance due to au entrance effect P pressure P‘7 atmospheric pressure p* pressure in the tower above the packing r,R radius and a dimensionless radius u,u horizontal gas velocity and its dimensionless form

Chemicof Engineering Science, 1975, Vol. 30, pp. 351-354.

v,V W x,X p

Pcrgmon

Press.

University of Sydney, Sydney NS. W. 2006 Australia.

I. A. FURZER

Printed in Great Britain

Bubble size distribution and interfacialareas of electrolyte solutions in bubble columns (Received 1August 1974;accepted 13September 1974) INTRODUCITON

In an earlier paper [ l] the oxygen transfer in tall bubble columns was studied. This study was carried out with water and aqueous solutions of NaCl and Na,SO., in two bubble columns of 723cm (BC I) and 440cm (BC II) height. In BC I a cross of nozzles with holes of 1 mm dia. was used as gas sparger, while in BC II the gas distributor is a glass sintered porous plate with a mean pore dia. of 150pm. The overall liquid phase mass transfer coefficients k,a measured in BC II are larger bv a factor of about 3 than the k,a-values obtained in BC i. This difference was assumed to be due to the ditferent gas distribution systems. In both columns it was found that the mass transfer rates increased only slightly-about 50 per cent-for the aqueous solutions of NaCl(O*l7N) and Na*SO.,(0.225N) compared with tap water. Whereas for the case of a 0.7 N solution of Na,SO, no increase in kLa was observed at all. With regard to the findings of other authorsl2-41 who studied the effect of electrolytes on gas-liquid dispersions the results reported in [l] are somewhat contradictory. The addition of electrolytes impedes bubble coalescence and owing to the smaller bubble diameter it could be expected that the interfacial area increases by a factor of 2 to 3 [4] for the concentrations used in [l]. On the other hand it was observed that the mass transfer coefficient k, decreases with decreasing bubble size[5-81. Therefore; it was assumed in [l] that the small increase of k,a is to be attributed to an increase of the interfacial area and a simultaneous &crease of k,. BUBBLE

SIZE DISTRIBUTION

To clarify this point the interfacial area was determined independently by taking photographs in approximately half the height of the bubble columns. The enlarged photographs of the gas-liquid dispersions were analyzed with a particle size analyzer

(Zeiss TGA) which classifies the bubbles in 48 sorts of diierent diameter. A number of 1000 bubbles were usually scaled. No change of the bubble size distribution was observed when more bubbles were counted. Bubble size distributions obtained at diierent gas velocities for sodium sulfate solution 0.7 N in BC I are shown in Fig. 1. Here at the ordinate AN$/AN,., is given, where AN, is the bubble number of diameter class i which is divided by AN,, the highest number of bubbles found in a class. The distributions are not symmetric but skew to right. At low gas velocities two subdistributions can be distinguished. This is also noted from Fig. 2 where the bubble size distributions for the air-water system in BC II are shown on probability paper. This plot really indicates that two bubble collectives with different mean diameters exist. In addition, the probability function is plotted against log d, in Fig. 2. Again no straight line is obtained. Because of the second bubble collective the slope of the distribution curve is increased at larger diameters. At higher gas velocities (~0 > 3 cmlsec) spherical cap bubbles turn up which have diameters larger than 5 cm. These large bubbles cannot be determined by photographic means. The amount and the size of these large bubbles is strongly dependent on the gas velocity and the height [9-IO]. The experimental bubble size distributions were split up in two normally distributed collectives. An optimization method proposed by Hasselblad[lf] was applied. This method determines the relative proportions, the means and variances of the subdistributions. To obtainan approximation for the iteration scheme, the experimental distribution was reflected round its first maximum, which yields the first approximated subdistribution. The estimated parameters of the second subdistribution were obtained by sub&acting the first subdistribution from the experimental distribution[lO]. Optimized subdistributions computed from this method are presented in Fig. 1.