nonrejecting (such as Eastman HT-00 used in this study on which dynamic membranes can be formed from sewage constituents) may find a place in treatment of sewage plant effluents where incomplete salt rejection can be tolerated and where substantially greater removal of turbidity and organic carbon is desired. Acknowledgment
The authors wish to acknowledge the support and encouragement of K. A. Kraus and the assistance of J. R. Love in carrying out these tests. literature Cited
Bregman, J. I., Environ. Sci. Technol., 4, 296 (1970). Hauck, A. R., Sourirajan, S., ibid., 3, 1269 (1969). Kraus, K. -I., Water Pollution Control Research Series ORI)-17030EOHOl/70, U.S. Dept. of the Interior, Federal Water Quality Administration, 1970. Merten, U., Bray, D. T., “Proc. 3rd I n t . Coni. hdvan. K a t e r Pollut. Re..,” Xunich, Germany, Vol. 3, pp 315-31, K a t e r Pollution Control Federation, TTashington, DC, September 1966.
Okey, R. W., Staverman, P. L., Proc. Symp. Xembr. Process. Ind., pp 127-56, Birmingham, -IL, X a y 19-20,1966. Rickert, D. A, Hunter, J. V., J. Water Pollut. Contr. Fed., 39, 1475 (1967). Savage, H. C., Bolton, X. E., Phillips, H . O., Kraus, K. .I., Johnson, J. S., Jr., V a t e r Sewage If-orks, 116 (3), 102-6 (1969). Sheppard, J. D., Thomas, D. G., Appl. Polym. Symp., Yo. 13, A. F. Turbak, Ed., 121-38 (197Oa). Sheppard, J. D., Thomas, D. G., Desalination, 8 , 1-12 (1970b). Sheppard, J . D., Thomas, D. G., AIChE J . , 17, 910-15 (1971). Thomas, D. G., ibid., 7,423-43 (1961). Thomas, D. G., ibid., 10,517-23 (1964). Thomas, D. G . , Griffith, IT. L., Keller, R. lI.,Desalination, 9, 33-50 (1971). RECEIVED for review September 22, 1970 ACCEPTED January 24, 1972 Research sponsored jointly by the K a t e r Quality Office, Eiivironmental Protection Agency, and the U.S. Xtomic Energy Commission under contract with the Union Carbide Corp.
Design of Cross-Flow Cooling Towers and Ammonia Stripping Towers Walter J. Wnek and Richard H. Snow1 I I T Research Institute, 10 TT’est 35th St., Chicago, I L 60616
A method of designing cross-flow cooling and ammonia stripping towers is presented which avoids the numerical analysis previously required for cooling tower design. Approximate analytical solutions are obtained for the simultaneous equations of conservation of energy and mass. The results agree with examples from the literature obtained b y a less general finite-difference method, and also with data from a pilot ammonia stripping tower. Equipment and operating cost correlations are presented, and also a method to optimize ammonia stripping tower design geometries. A sample calculation for a 1 -million g p d tower treating waste water from a typical municipal treatment plant shows that a tower 57 f t high with packing only 7 f t thick i s optimum to reduce ammonia concentration b y 40 : 1 . Reduction of 8 : 1 requires a 32-ft tower. Whether such towers can b e successfully operated, even if liquid distributors are added, has not been demonstrated; however, the results show that a short, wide tower will not perform as specified. The capital cost for the example tower i s $ 1 million, while the total unit cost i s 4 cents/l000 gal. Such high costs make the use of stripping towers unlikely for waste water treatment, although they cannot be ruled out because other feasible methods have so f a r not been found. The design methods are applicable to other cross-flow stripping systems, such as odor control.
C
rosa-flow towers for cooling and stripping operations are becoming increasingly important because of today’s more demaiiding situations. For example, it has been suggested that cross-flow towers may be more economical than countercurrent. towers for stripping ammonia from liquid waste n-here the concentration is on the order of mg, 1. or ppin. This is so became cros-flow towers allow use of a larger volume of air a t a lower fan power consumption. For t,his reason, the To whom correspondence ,\hould be addressed.
ammonia stripping tower built a t Lake Tahoe is of cross-flow design (Slechta and Culp, 1967). hlt,hough cross-flow towers offer a number of advantages over countercurrent towers, their use has been hampered b y a lack of adequate design procedures. Previous design procedures involve a numerical analysis using finite differences with a digital computer. Schechter and Kang (1959), extending the work of Zivi and Brand (1956), used this approach and presented set’s of design curves for cooling towers. Unfortunately, it is difficult to use their results for design purposes Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
343
The material gained b y the gas is equal t o
hcG G, - dxdydz dX where G, = the molar gas flow rate, lb mol/hr ft2,and CG = mol fraction of ammonia in gas. The mass lost by the liquid is equal to GAS
ie *COG
where L, = molar liquid rate, lb-mol/hr ft2, and C L fraction of ammonia in liquid. The mass interchange is equal to
KGa(C* - CG)dXdydZ Figure 1 . Schematic cross-flow tower
because interpolation between the curves and graphs is difficult to perform; graphical integration is required; and the liquid temperature profile across the width of the tower must be assumed, which necessitates a trial-and-error solution. Recently Thibodeaux (1969) and Roesler and Smith (1971) also used this approach for cross-flow stripping towers, but unfortunately, the same limitations hold as given before. This paper will present a more practical design method for two applications : cooling towers, and ammonia stripping towers with simultaneous cooling. Statement of the Problem
The main variables to be determined by the design procedure are the height, width, and length of the tower to achieve a given stripping or cooling specification. For a cooling tower, the design method is developed b y performing a n energy balance on a differential element of volume in the tower, from which a partial differential equation is obtained. The associated boundary conditions are that the liquid enters a t a given temperature and is required to leave a t a specified temperature, and the air enters at a given temperature. B y solving this boundary value problem, the necessary dimensions of the tower are determined. Similarly, for a stripping tower, the design method is developed by performing a material balance. However, there is the added complexity that Henry’s constant in the equilibrium relationship is temperature dependent, and there is a temperature profile in the tom-er due to the air-water contact. This means that the material and energy balances are coupled. The method of solution is to solve first for the temperature profile in the tower in the manner described above for cooling towers. Then one substitutes this profile into the expression for Henry’s constant and solves the resulting partial differential equation obtained from the material balance, with the boundary conditions that the liquid enters at a given concentration and is required to leave a t a specified concentration, and the air enters a t a given concentration. By solving this problem, the necessary dimensions of the tower are found. Basic Design of lower
The governing differential equations for the concentration and temperature profiles in the tower with the associated boundary conditions will be obtained. Assuming that steady state has been reached, the flow rates are constant, diffusional effects are negligible, and the mass transfer coefficient is constant, a material balance is performed on the differential element of volume in Figure 1. 344 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
=
mol
(3)
where C* = mol fraction of ammonia in gas in equilibrium with bulk liquid, and KGa = overall gas phase mass transfer coefficient, lb-mol/ft*-hr. Using Henry’s Law,
C* this becomes
=
mCL
(4)
KGa(mCL - cG)dXdydZ
(5)
where rn = Henry’s constant, dimensionless. Since the quantities expressed by Equations 1, 2, and 5 are all equal, the governing set of equations becomes
T o express the fact that the liquid and gas enter a t a given concentration, the boundary conditions become
CL = COL a t z C G = COG
=
=
0 at x
for all x
0 =
O
for all z
(7) (8)
Temperature Effect
If Henry’s constant were indeed a constant and did not vary with position in the tower, Equation 6 could be solved by analogy with methods for cross-flow heat exchangers. However, from the paper of Farrell et al. (1968), Henry’s constant for the ammonia-air-water system is a strong function of temperature. It may be written as m
=
0.10 exp (0.0284 t )
(9)
where t = temperature, O F for 32’F < t < 140’F. A solution is given below that accounts for the variation of Henry’s constant with position in the tower. An approximate solution assuming a constant wet-bulb temperature is also shown to be valid for ammonia stripping. The exact solution may be important for other systems where cooling and mass transfer occur a t more nearly equal rates. Zivi and Brand (1956) derived the governing set of equations to evaluate temperatures in a cross-flow cooling tower from a n energy balance
where L and G = liquid and gas mass flow rates, respectively, lb/hr ft2; c p ~= liquid specific heat, Btu/lb OF; t L = liquid temperature, O F ; and i = enthalpy of the air, Btu/lb. The rate equation is
G
ai
-
bz
=
Ka(i* - i)
where Ka = enthalpy transfer coefficient, lb/hr ft3; and i * = enthalpy of the air in equilibrium with bulk liquid temperature, given as (Fuller, 1956) i* = exp (1.77
+ 0.025 tL)
(12)
for 40°F 5 t~ 5 130°F. T o express the fact that the liquid enters at a given temperature and the gas a t a given enthalpy, the boundary conditions become
tL
tLe a t z
=
=
0
WATER INkET
WATER INLET1
ELIMINATORS
(13)
i = i , at s = O
(14)
These equations are based on the assumption t h a t steady state prevails; flow rates, heat capacity, and humid heat are independent of position; diffusional processes are negligible as compared to convective ones; the overall transfer coefficient is constant; and the Lewis relationship is valid. Solutions of Equations 10-14 will yield the temperature profile of the liquid in the tower. Substitution of this profile into Equation 9 then gives Henry’s constant as a function of position in the tower, which is now substituted intoEquation 6. Solution of Equation 6 will yield the composition of the liquid in the tower. The tower may be sized for a given separation by solving this equation. The solution is derived in the Appendix and is
OUTLET
COLLECTING BASIN
Figure 2. Cross-flow tower (The Marley Co., Inc.)
WATER
I
A
l
(15) where
CoL = the entering liquid concentration
CL,
=
the exit mean liquid concentration KGa
t o = __
G,
Figure3
20
Figure 3. Diagram of split flow packing
Y = (1, - 1)/Ie
I, = &/Io* I O *= exp (1.77 0.025 fLe) i, = entering enthalpy of the air € = l/I, q = aL,/KGaH H = 0.10 exp (0.0284 fLe) a = P I e [ l - exp (-Xo)/XoI = 0.028 KnIo*/cpLL Xo = Kaxo/G KGuH
+
L,
Approximate Design Equation
Z
This equation may be used to calculate either the height or width of the tower once the other has been set. For the type of cross-flow tower shown in Figures 2 and 3, the length of the tower yo is given b y yo = LT/2 SOL
(16)
The temperature profile in the tower was investigated in detail by Sno.iv and Wnek (1969). The important conclusion was that the heat transfer occurs more rapidly than the mass transfer because of the relatively large volume of air required for ammonia stripping. The liquid very quickly attains the wet-bulb temperature of the entering air, and the temperature varies only in a short section near the top of the tower. Using the wet-bulb temperature as the design temperature throughout, Equation 15 can be simplified by setting F([) = 1. As indicated in the Appendix, it becomes z =
where LT = liquid throughput, lb/hr. As derived in the Appendix, the mean temperature of the exiting liquid is tL,mean (2) = tLe
- 40 In [ y exp
+ €1
(17)
where y, a , and E are defined as above. Solving Equation 17 for z, the design equation for a cooling tower is obtained as
where ~ T W = B
tWB
=
0.10 exp (0.0284 twB)
(20)
wet-bulb temperature of entering air, O F .
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
345
For cooling towers, a term owing to droplets must be added (Kelly and Swenson, 1956), and a term for high-resistance mist eliminat ors.
dh
fV
pH Requirement of Entering Stream
Figure 4. Dimensions of splash bars for use in Equation 22
A fraction of dissolved ammonia must be in molecular form so that it can be stripped out. Snow and R n e k (1969) show that it is sufficient that the pH in the entering stream be pH
The results given by Equation 19 are in good agreement rvith those of the more exact Equation 15, Transfer Coefficients
Splash bar packing (Figure 4) is the type that has been used for cross-flow cooling towers aiid for the ammonia stripping tower a t Lake Tahoe. -1s discussed by Snow and Wnek (1969), the transfer coefficient for splash packing consists of a contribution from the liquid film on the bars and from the falling drops. Calculations based on correlations for transfer from liquid films and drops show that for the large (1 to 2 ft) vertical spacing of the spash bars used in cooling towers the contribution of the film action is negligible; on the other hand, for the close vertical spacing required for ammonia stripping ( 2 to 4 in.), the contribution of the drops is negligible. Therefore, the mass transfer coefficient can be found from correlations for the film coefficient, for example that of Johnstone aiid Singh ( 1 9 3 ) as given by
KGu = 1.5 V ~ G ' * U
(21)
where y o = a constant which varies according to the dimensions of the bars from 0.00525 for small bars with close spacing to 0.00249 for large bars with far spacing. a = 24 ( d h
+ dc)/shs,
(221
where dh, d,, sh, and s, are defined in Figure 4. The 1.5 factor is due to converting from the SOz system to the NH3 system by using the Schmidt number. Johnstone and Singh (1937) showed that analogy theory could be used for obtaining the enthalpy transfer coefficients, and the result as given by Snow and K n e k (1969) is
Ka
=
24.6 KGa
(23)
This equation is valid for ammonia stripping towers, but not for cooling towers, because of the above assumptions. Pressure Drop
A pressure-drop correlation for flow across slat packings has not been published. Until such data are available, the pressure-drop correlation given by Kelly and Swenson (1956) may be used for cross-flow towers by rotating his deck configurations through 90". The effect of the vertical spacing can be estimated by using a linear relationship (Berman, 1961). This gives the pressure drop (in. of 1320) across the tower as 0.0675
AP
=
12 B
(T) ( y )(:
G2
(24)
where pG = density of the gas, 1bjfb3; stref = spacing (perpendicular to the gas f l o ~ )for which the correlat,ion was determined; and B = empirical factor characteristic of the packing on the order of 4 x 10-9 as given b y Kelly and Swenson (1956). Owing to the dense packing used in ammoiiia stripping, the contribution of the falling drops t,o the pressure drop has been neglect,ed as verified by S n o and ~ K n e k (1969). 346
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
=
-log
[(")
Kb
COLcL - CL, ,meanmean
1
(25)
where K , = ion product for mater a t 25OC); K O = equilibrium constant for the T \ " 3 - & 0 system (1.8 X a t 2 5 O C ) ; CL,mean = effluent ammonia concentration averaged a c r o s the width of the liquid stream, mg;l.; and CoL = the influent ammonia concentration, mg/ 1. This equation is derived by basic chemical equilibrium methods. For CoL = 40 ppm and CL,m e a n = 1 ppm, the required pH is calculated to be 10.85. At Lake Tahoe, it was experimentally found that a p H of 10.8 gave the most successful operation of the tower. Verification of Model
I n Table I there is presented summer plant data on the cross-flow stripping ton-er in operation a t South Tahoe treatment plant (Smith and Roesler, 1969). The entering and exit ammonia concentrations of the liquid, the throughput, gas and liquid flow rates, and dimensions of the tower are given. The model was checked by using these data as input and predicting by the model the corresponding height for four operating conditions. The results are presented in Table I, mhere it is seen that the predicted heights are in agreement with the actual heights. The discrepancies are probably within the evperimental error since the outlet ammonia concentrations are accurate to only one ppm. Example Problem
It is desired to design the most economical cross-flow tower that can strip ammonia from 10 million gal. of waste water per day entering a t 70°F. The initial concentration of 40 mg/l. is to be lowered to 1 mg/l. S H 3 . The packing chosen is of Types C and D in Kelly and Swenson's (1956) paper, B = 4 X lop9, y o = 0.003, dh = 1.5 in., and d , = 0.375 in. The ambient wet-bulb temperature is 60°F. Equipment is to be amortized a t 5% over 15 years; structures a t 5% and 20 years. Take 50y0 as the efficiency of the fans and pumps. Electricity costs $0.008;kK-hr. The toJver is to be operated 96% of the time. Solution
Correlations (Jackson, 1951; 31cKelvey and Brooke, 1959; Peters and Timmerhaus, 1968; Sawistowski and Smith, 1963) for the capital cost of the various equipment such as the packing, fan, pumps, and basin and the operating costs such as electrical power, labor, and maintenance were used to evaluate the total annual cost. The solution is illustrated by the computer output shown in Tables 11-V. The procedure is to change one variable a t a time while holding the others constant until the total cost IS minimized for that variable, and repeating for the nest variables. This method may be justified on the grounds that the total cost is somewhat insensitive over certain ranges of the operating variables. The variables to be optimized are the gas and liquid flow rates, vertical and horizontal spacings of the spash bars, and the width of the tower. The initial values used were gas rate 2400 lblhr ft2, liquid rate 600 lb hr ft2,vertical spacing 1.5 in., horizontal spacing 2 in., and width 10 ft.
Table I. Comparison of Height Predicted by M o d e l with South Tahoe Data (Smith and Roesler, 1969) NHa concn, mg/l. Effl
lnfl
4
16.4 19.6 20.8 18.2 Width is 14 ft, length
4
Flow, million gpd
Ib/hr f t 2
G, Ib/hr ft2
Tahoe height,a f t
Predicted height, ft
630 812 698 1162
2480 2280 2346 2053
24 24 24 24
24.13 24.93 28.23 29.53
1.6 2.1 1.8 3.0
3.6 6.9 4.7 8.7 is 32 ft.
Table II. Solution for Example Problem Trial No.
Tower width, f t
Vertical spacing, sl, in.
Liquid rate, L
2
1.5
600
10 ...
1
2 3 4 5 6 7 8 9 10 11 12 13 14
2
120 29.5 100 31 . O Schechter and Kang, 1959.
3.0
Ib/hr f t 2
S
1000
Width, f t
1500 1200
100 200
15 6
1500 1200 Vouyoucalos, 1968.
The temperature profile of the liquid will be found first. Define the dimensionless variables
Io*
600
6.86 4.68 5.55 4.86 5.20 4.71 4.10 4.33 4 68 4.58 4.10 4.26 4.51 3.99
Ib/hr ft3
Appendix. Solution of Governing System of Equations for a Cross-Flow Tower
= = =
500 500 1000 600
2400 1000 500 1000 1500 1500 1000
Ib/hr ft2
'The optimized values for the height and width of the tower arc 57.44 ft and 7 ft, respectively. It is common practice in cooling tower design to use heightto-width ratios less than 2.5 to avoid maldistribution of both the air and water flows. If one arbitrarily increases the widt'h of the packing, not only will cost increase, but the desired removal will be unattainable with a reasonable height. This happens because the water flowing down the interior packing does not, contact fresh air; in a sense, it bypasses the stripping action. From Equation 19, it may be shown that as width is increased, height must also be increased, and that the limiting ratio is z / z o 2 29 LIO.18 mTWbG. I t may be practical to operate high, narrow towers if liquid redistributors are added. If practical operating problems remain, then we must conclude that cross-flow towers are not suitable for ammonia stripping. Cross-floiy towers are being proposed for other systems such as odor control, and our design method should be useful to evaluate these other applications, which may require less extreme size ratios.
Z
Total unit cost, 000 gal.
$11
Table 111. Comparison of Equation 35 and Numerical Solutions 1, G, KO, Height
Btu/lb
F
Gas rate, G Ib/hr ft2
3.0 6.0 3.0
1 5 1
5 4 15 7
1 e,
fLe, O
Horizontal spacing, SA, in.
0.025 KaIo*z;cpLL
Kax,.'G esp (1.77
+ 0.025 tLe)
given by computer, ft
Height given b y Equation 15, f t
30. 6b
31 4 6 45
Table IV. Effluent Concentration vs. Height of an Ammonia Stripping Towera
Influent concentration, 40 mg 1 Effluent concn, mg/l.
Removal,
%
Height, f t
1 97 5 57 4 2 95 0 46 5 3 92 5 40 3 4 90 0 35 8 5 87 5 32 3 a Operating conditions are the same as given in Table V and the example problem.
V I I,
= = =
0.025
(tL
-
tLe)
i/IO* ie/Io*
so that Equations 10-14 become
bV -aZ
=
exp ( V ) - I
ai
= -
b
x
V ( X , O )= 0 , I ( 0 , Z ) = I ,
(27)
From the papers of Schechter and Kang (1959) and Vouyoucalos (1968), the temperature change of the liquid across the width of the torTer is small in comparison to that across its height. This suggests seeking a solution of the form Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 3, 1972
347
Table V. Optimum Design for Example Problem for
whose solution is
Cross Flowa Dimensions of tower, ft Width 7.0 Height 57.44 Length 413.09 Volume 332,249.61 Capital costs (1969) Fan $ 666,195.38 Pump 20,105.82 Packing 186 ,890.40 Structure 134,462,49 Total capital cost $1,007,654.11 Debt service, cents/1000 gal. 2.67 Annual operating and maintenance cost, $/year Fan power 9,061.38 Pumping power 10,058.71 Labor and maintenance 26 , 834.66 Total operating cost/year 45,954.76 Operating cost, cents/1000 gal. 1.31 Total unit costs, cents/1000 gal. 3.99 Other data Exit temp, O F 60.00 Lb airilb water 13.67 Pressure drop, in. water 0.05 Max. entering pH required 10.87 Optimized values of the design variables are liquid flow rate, 600 lbihr ftz; gas flow rate, 1000 lb/hr ft2; vertical spacing of spash bars, 3 in.; horizontal spacing of splash bars, 2 in.; width of tower, 7 ft. Computer output for these values is presented in Table 11. Q
Solving for V ,
v,
=
(36) Substituting the original variables into Equation 36 and solving for the mean liquid temperature with respect to the width as a function of height yields
+
t ~ , ~ ~ =~ t~~ ~ (- z40) In [Y exp (az)
V(X,z)=
Y =
=
(28)
a = PIe[l
P
=
V,(Z)
dZ
=
-I
exp (V,)
(29)
dX
- exp (V,)
U
=
KGaHCL
fl'
=
KGaCG
H
=
0.10 exp (0.0284 f L e )
KGaH Lm
E=-
=
exp ( V m )
W
F(0
= -
ax
KGa G, [Y exp (-&
=
+
aLm 'P==
Equations 6-8 become
-I [ I , - exp (Vm)l
(32)
Integrating Equation 30 with respect to X from X = 0 to X = X O , where X O = (Ka/G)xo is the dimensionless width of the tower, yields
(33)
348
(38)
(39)
+ exp (-X)
Evaluating Equation 32 a t X in Equation 33 gives
e]-1.14
Z
Solution of Equation 31 yields
I
+
q = - 2
For a given 2, Equation 30 becomes
dI -_
CYL)
Substituting Equation 38 into Equation 6 and defining the variables
This approximation will be checked later, and if it proves t o be inadequate, a higher order approximation will be made. Equation 26 now becomes
dVm --
= 0.025 KaIo*/cpLL
0.10 exp (0.0284 tLe) [y exp (-
first term of the series. =
- exp (-Xo)I/Xo
T o determine if a first approximation is sufficient for sizing a tower, Equation 35 was checked against numerical solutions of Equations 10-14 by computer given in the literature (Schechter and Kang, 1959; Vouyoucalos, 1968). The results are given in Table I11 from which it can be concluded that higher order approximations are not necessary for design purposes. Substituting Equation 37 into Equation 9 gives Henry's constant as a function of position in the tower.
=
V ( X , Z ) = F,(Z)
(Ie - 1)/Ie
= 1/I,
E
X evaluated a t V = V,, n = an integer, and V averaged over X. -4s a first approximation, take the
where X ,
V,
C ( X - Xm)2pi(Z)
2=0
(37)
where
m n
el
=
uo
V ( 0 ,0
0
=
where U o = KGaHCoL. Proceeding as was done for the temperature profile, the mean value of I: with respect to the width of the tower is taken as a first approximation c(7,
Xo and substituting for Io
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
U ( q , 0) =
where
U,
E)
-
Crn(E)
= U averaged over 7 . Equation 39 then reduces to
For a given E, Equation 40 becomes
dW
- = F(€)U, dtl
approximation is sufficient, and that a higher order approximation is not necessary.
-w Nomenclature
whose solution is
W
= specific area of packing, ft2/ft3 = parameter for pressure-drop equation COG = entering mol fraction of ammonia in gas CoL = entering mol fraction of ammonia in liquid,
a =
F(C;)U,[1 - exp
(42)
(-7)l
Integrating Equation 39 with respect to
7
from
7 =
0 to
7 = qo, where 70 =
KGa xo GVl
(43)
~
is the dimensionless width of the tower and gives (44)
IVO is given b y Equation 42 evaluated a t solution is
7 = 70
whose
B
mg/l.
pressure drop, in. of H20 = horizontal thickness of a splash bar, f t d , = vertical thickness of spash bars, ft mTWB = Henry’s constant evaluated a t tTVB sh = horizontal spacing of splash bars, ft s, = vertical spacing of splash bars, ft s,ref = s, for which correlation was determined tLe = entering temperature of liquid, OF x = coordinate measured along width of tower, ft zo = width of tower, ft y = coordinate measured along length of tolver, ft yo = length of tower, ft z = coordinate measured along height of tower, ft AP dh
=
GREEKLETTER Y O = parameter for correlation of mass transfer literature Cited Berman, L. I]., “Evaporative Cooling of Circulation Water,” p 137, Pergamon Press, Kew York, S Y , 1961. Binnie, A. M.,Poole, E. G. C., Proc. Cambrfdge Phil. SOC.,33, 403 (1937). Farrell, J. B., Stern, G., Dean, R . B., “Sitrogen Removal from Wastewaters,” U.S. Dept. of Interior, FWPCA, Cincinnati Water Res. Lab.,Cincinnati, OH, January 1968. Fuller, A. L.,Petrol. Refiner., 35 (12), 211 (1956). Jackson, J., “Cooling Towers,!’ Butterworths, London, England, ,nz, IYJL.
+
The effect of taking the term [exp (-ppt;) ~ 1 O . outside l ~ the integral sign is negligible since for typical operating conditions it varies from 1.0 for E = 0 to around 1.02 for 4 = 30 or z = 40 ft. Thus, substituting Equation 47 into Equation 45 and rearranging 70 In
COL CLmean -
~
1 - exp (-70)
Johnstone, H. F., Singh, A . D., Ind. Eng. Chem., 29 (3), 286 (1937). Kelly, N. W.,Swenson, L. K., Chem. Eng. Progr., 52 (i),263 (1936).
M~K&ej-, K . K., Brooke, AI., “The Industrial Cooling Tower,” D 323, Eluevier, S e w York, NY, 1959. Peters, 11.S., Timmerhau., K. D., “Plant Design and Economics for Chemical Engineers,” p 640ff, McGraTy-Hill, New York, S Y . 1968. Roesler, J.,-Smith, R . , J . Sanit. Eng. Diu.) Amer. SOC.Civil Eng., SA3, 269-86 (1971). Sawistowski, H., Smith, W., “Mass Transfer Process Calculations,” Interscience, S e a York. S Y . 1063. Schechter, R. S., Kang, T. L., Ind. Eng. Chem., 51, 1373 (1959). Slechta, 4.F., Culp, G . L., J . TT-ater Pollut. Contr. Fed., 777, l l a y 1967. Smith, R., Roesler, J., T a f t Center, Cincinnati, Ohio. Drivate communication, 1969. Snow, R.H., PYnek, R. J., “rlmmonia Stripping Mathematical l l o d e l for LVastexater Treatment.” Final. I l..m o r t t o FKPC.4. I I T I U Rep. No.~C6152-6,September 1069. Thibodeaux, L. T., Chem. Eng., 76 (12), 165 (1969). Touroucalos, S., Brit. Chem. Enq., 13 (7), 1005 (1968). Wesiberg, N . , Gustafsson, B., T’attenhyg;en, 19, 2-10 (1963). Zivi, d. AI., Brand, B . B., Refrig. Eng., 64 (S), 31 (1956). I
Equation 48 may be used to calculate either the height or width of the tower once the other has been set. For one case Equation 45 may be checked. For heat eschangers f([) = 1, and for this case Binnie and Poole (1937) were able to obtain a n exact solution of Equation 39 in the form of a series. They gave an example problem in which water entered a t 40°F and another liquid a t 100”F, and the dimensions of the heat exchanger were 13.76 X 20 ft. They found the exit temperature of the liquid was 52.84OF while Equation 22 gave 54.15OF. This indicates that the first
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RECIXVED for review Sovember 6, 1970 ACCEPTED April 12, 19i2 P a r t of this research n-as performed pursuant to Contract S o . 14-12-463 with the Environmental Protection Agency.
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 3, 1972
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